DOCUMENT RESUME
SE 058 666
ED 411 138
AUTHOR
Booker, George, Ed.; Cobb, Paul, Ed.; de Mendicuti, Teresa
N
Ed
Proceedings of the Annual Conference of the International
Group for the Psychology of Mathematics Education with the
North American Chapter 12th PME-NA Conference (14th, Mexico,.
July 15-20, 1990), Volume 2.
International Group for the Psychology of Mathematics
Education.
1990-07-00
364p.; For volumes 1 and 3, see SE 058 665 and SE 058 667.
Proceedings (021)
Collected Works
French, Spanish, English
MF01/PC15 Plus Postage.
Educational Research; Elementary Secondary Education;
Foreign Countries; Higher Education; *Mathematics Education
,
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INSTITUTION
PUB DATE
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DESCRIPTORS
ABSTRACT
This proceedings of the annual conference of the
International Group for the Psychology of Mathematics Education (PME)
includes the following research papers: "Children's Connections among
Representations of Mathematical Ideas" (A. Alston & C.A. Maher); "Algebraic
Syntax Errors: A Study with Secondary School Children" (A. Avila, F. Garcia,
& T. Rojano); "The Development of Conceptual Structure as a Problem Solving
Activity" (V. Cifarelli); "From Arithmetic to Algebra: Negotiating a Jump in
the Learning Process" (A. Cortes, N. Kavafian, & G. Vergnaud); "Continuous
Analysis of One Year of Science Students' Work in Linear Algebra, in First
Year French University" .(J.L. Dorier); "Avoidance and Acknowledgement of
Negative Numbers in the Context of Linear Equations" (A. Gallardo);
"Introducing Algebra: A Functional Approach in a Computer Environment" (M.
Garancon, C. Kieran, & A. Boileau); "LOGO, to Teach the Concept of Function"
(D. Guin & I.G. Retamal); "The Concept of Function: Continuity Image versus
Discontinuity Image: Computer Experience" (F. Hitt); "Acquisition of
Algebraic Grammar" (D. Kirshner); "Embedded Figures and Structures of
Algebraic Expressions" (L. Linchevski & S. Vinner); "A Framework for
Understanding What Algebraic Thinking Is" (R.L. Lins); "Developing Knowledge
of Functions through Manipulation of a Physical Device" (L. de Lemos Meira);.
"Students' Interpretations of Linear Equations and Their Graphs" (J.
Moschkovich); "An Experience to Improve Pupil's Performance in Inverse
Problems" (A. Pesci); "Algebra Word Problems: A Numerical Approach for Its
Resolution: A Teaching Experiment in the Classroom" (G. Rubio); "Children's
Writing about the Idea of Variable in the Context of a Formula" (H. Sakonidis
& J. Bliss); "Observations on the 'Reversal Error' in Algebra Tasks" (F.
Seeger); "Generalization Process in Elementary Algebra: Interpretation and
Symbolization" (S.U. Legovich); "Effects of Teaching Methods on Mathematical
Abilities of Students in Secondary Education Compared by Means of a Transfer
Test" (J. Meijer); "On Long Term Development of Some General Skills in
Problem Solving: A Longitudinal Comparative Study" (P. Boero); "Cognitive
Dissonance versus Success as the Basis for Meaningful Mathematical Learning"
(N.F. Ellerton & M.A. Clements); "Time and Hypothetical Reasoning in Problem
Solving" (P.L. Ferrari); "The Interplay between Student Behaviors and the
Mathematical Structure of Problem Situations: Issues and Examples" (R.
Herschkowitz & A. Arcavi); "Paradigm of Open-Approach Method in the
+++++ ED411138 Has Multi-page SFR--- Level =l +++++
Mathematics Classroom Activities: Focus on Mathematical Problem Solving" (N.
Nohda); "Reflexions sur le Role du Maitre dans les Situations Didactiques a
Partir du Cas de l'Enseignement a des Eleves en Difficulte" (M.J.P. Glorian);
"Diagnosis and Response in Teaching Transformation Geometry" (A. Bell & D.
Birks); "Children's Recognition of Right Angled Triangles in Unlearned
Positions" (M. Cooper & K. Krainer); "The Role of Microworlds in the
Construction of Conceptual Entities" (L.D. Edwards); "The Cognitive Challenge
Involved in Escher's Potato Stamps Microworld" (R. Hadass); "Study of the
Degree of Acquisition of the Van Hiele Levels by Secondary School Students"
(A. Jaime & A. Gutierez); "Spatial Concepts in the Kalahari" (H. Lea);
"Integrating LOGO in the Regular Maths Curriculum: A Developmental Risk or
Opportunity?" (T. Lemerise); "Young Children Solving Spatial Problems" (H.
Mansfield & J. Scott); "The Role of Format in Students' Achievement of Proof"
(w.G. Martin); "L'influence des Aspects Figuratifs dans le Raisonnment des
Eleves en Geometrie" (A. Mesquita); "Children's Understanding of Congruence
According to the Van Hiele Model of Thinking" (L. Nasser); "Prospective
Primary Teachers' Conceptions of Area" (C. Tierney, C. Boyd, & G. Davis);
"Probability Concepts and Generative Learning Theory" (0. Bjorkqvist); "Some
Considerations on the Learning of Probability" (A.M.O. Salazar); "Gambling
and Ethnomathematics in Australia" (R. Peard); "Mathematization Project in
Class as a Collective Higher Order Learning Process" (H.G. Steiner). Also
includes a listing of author addresses. (MKR)
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INFORMATION CENTER (ERIC)
Fourteenth
PME Conference
With the North American Chapter
Twelfth PME-NA Conference
(July 15-20)
U.S. DEPARTISENT OF EDUCATION
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Mexico 1990
VOLUME
2
II
International Group For
the Psychology
of Mathematics Education
1
IIIIIIIIIIIIII,,
I,"' HI wii
PROCEEDINGS
Fourteenth
PME Conference
With the North American Chapter
Twelfth PME-NA Conference
(July 15-20)
Mexico 1990
VOLUME 1I
3
Published by the Program Committee
of the 14th PME Conference, Mexico.
All rights reserved.
Sponsored by:
Consejo Nacional de Ciencia y Tecnologia (CONACYT)
Gobierno del Estado de Morelos
IBM de Mexico, S.A.
Seccion de Matematica Educativa del CINVESTAV
Editors:
George Booker
Paul Cobb
Teresa N. de Mendicuti
4
Printed in Mexico.
TABLE OF CONTENTS
VOLUME.I
Preface
iii
Reviewers and Reviewer's Affiliation
History and Aims of the PME Group
vii
A Note on the Review Process
ix
A note on the Grouping of Research Report
Working Groups. Discussion Groups
xii
PLENARY SYMPOSIUM
The responsibilities of PME research community
Alan Bishop
PS:1
PLENARY ADDRESSES
The knowledge of cats: Epistemological foundations of mathematics education.
Robert B Davis
PME algebra research. A working perspective.
Eugenio Filloy
PI.1
PI I . 1
ADVANCED MATHEMATICAL THINKING
Some misconceptions in calculus. Anecdotes or the tip
an iceberg?
Miriam Amit, Shlomo Vinner
of
3
Difficultes cognitives et didactiques dans la construction
de relations entre cadre algebrique et cadre graphique.
Michele Artigue
Unbalance
and
recovery.
Categories
related
to
11
the
appropriation of a basis of meaning pertaining to the
domain of physical thinking.
Ricardo Cantoral
19
On difficulties with diagrams: Theoretical issues.
Tommy Dreyfus, Theodore Eisenberg
27
The two faces of the inverse function. Prospective teachers' use of "undoing".
Ruhama Even
37
5
Intuitive processes, mental image, and analytical and
graphic representations of the stationary state. (A case
study).
45
Rosa Maria Farfan, Fernando Hitt
The role of conceptual entities in learning
concepts at the undergraduate level.
Guershon Harel, James Kaput
mathematical
53
Mathematical concept formation in the individual.
Lena Lindenskov
61
Pupils' interpretations of the limit concept: A comparison
study between Greeks and English.
Joanna Mamona -Downs
69
Infinity in mathematics
cognitive psychology.
Rafael Nunez Errazuriz
as
a
scientific
subject
for
77
Organizations deductives et demonstration.
Luis Radford
85
The teaching experiment "Heuristic Mathematics Education"
Anno Van Streun
93
The understanding of limit: Three perspectives.
Steven R. Williams
101
AFFECT, BELIEFS AND NETACOGNITION
Self control in analyzing problem solving strategies.
Gerhard Becker
Influences of teacher cognitive/conceptual
problem-solving instruction.
Barbara J. Dougherty
levels
111
on
Can teachers evaluate problem solving ability?
Frederick 0. Flener, Julia Reedy
119
127
Teacher conceptions about problem solving and problem
solving instruction.
Douglas A.Grouws, Thomas A.Good, Barbara J.Dougherty
135
Math teachers and gender differences in math achievement,
math participation and attitudes towards math.
Hans Kuyper, H.P.C. Van Der Werf
143
Teaching students to be reflective: A study of two grade
seven classes.
Frank F. Lester, Diana Lambdin Kroll
151
Students' affective responses to non-routine mathematical
problems: An empirical study.
Douglas B. McLeod, Cathleen Craviotto, Michele Ortega
159
Accommodating curriculum change in mathematics: Teachers'
dilemmas.
Rita Nolder
167
Teachers'
characteristics and attitudes as mediating
variables in computer-based mathematics learning.
Richard Noss, Celia Hoyles, Rosamund Sutherland
175
Teachers' perceived roles of the computer in mathematics
education.
Joao Ponte
183
Mathematics process as mathematics content: A course for
teachers.
Deborah Schifter
191
Psychological/philosophical
aspects
of
activity: Does theory influence practice?
Rosalinde Scott-Hodgetts, Stephen Lerman
mathematical
199
A web of beliefs: Learning to teach in an environment
with conflicting messages.
Robert G. Underhill
207
Posters
Students' performances in
clinical interviews about
fractions
Lucia Arruda de Albuquerque
Tinoco, Univ. Fed. Do Rio de
Janeiro, Brasil.
215
A
cognitive approach to
instruction for prospective teachers
Nadine Bezuk, Judith Sowder,
Larry Sowder, San Diego State
University USA
216
The role of imagery in
mathematical reasoning
Dawn Brown, Grayson Wheatley
Florida State University USA
217
Matematica y realidad
propuesta de una didactica integradora de la
matemetica en ejercicios
de computaci6n
Yolanda Campos, Eloisa Beristain, Cesar Perez, Evangelina
Romero, Direc. Gral. de Educ.
Normal y Actualizacion del Magisterio. Mexico.
218
Piagetian and Van Miele
Theories: The search for
functoriality
Livia P. Denis, Ph. D., State
University of New York at
Albany SUNY-A. USA.
219
Tendencies of learning
thinking styles and
effect of mathematics
learning
Dr. Hamdy A. El-Faramawy Dept.
of Psychology-Menoufia University, Egypt.
Social constructivism as
philosophy of mathematics
Radical constructivism
rehabilitated?
Paul Ernest
University of Exeter
England
Students' preference for
numbers of variables in
translating algebraic
sentences involving two
quantities
Aparna B. Ganguli
General College, University
USA.
of Minnesota,
Perceived difficulty of
probability/statistics
concepts
William E. Geeslin
University of New Hampshire
Introducing teachers to
misconceptions in secondary school mathematics
Anna 0. Graeber
University of Maryland
USA.
224
The conception of inner
form: Nature and role in
learning mathematics
Keito Ito
Graduate Student, University
of Tsukuba, Japan.
225
Using metaphors for reflection in teacher education
Elizabeth H.Jakubowski
Florida State University
USA.
226
Mathematical features of
dyslexia/specific learning difficulty
Lynn S. Joffe
Joffe Consultancy Services
England
227
Images of an achievable
technological future
James J. Kaput
Southeastern Massachusetts
University
228
220
221
222
USA.
Mathematical lessons via
problem solving for
prospective elementary
teachers
Joanna Masingila
Vania Santos
Indiana University, USA
Exploraciones sobre el
razonamiento en Maternaticas
Eduardo Mancera Martinez
UNAM, UPN, Mexico
Mathematical concepts as
Learning about
tools:
angles in LOGO programming
Luciano L. Meira
University of California
Berkeley, USA.
223
229
230
231
Conflicts in computer
programing: Do empirical
contradictions affect
problem solving?
Lucino L. Meira
University of California
Berkeley, USA.
The shift of explanations
for the validity of conjecture: From explanations relied on actual action to
generic example
Mikio Miyazaki, Graduate
School of Education,
University of Tsukuba,
Ibaraki-Ken 305, Japan
Calculus as a bridge between intuitions and reality
Ricardo Nemirovsky
TERC. Cambridge MA,
USA.
234
Use of Van Miele phases
between levels in primary
school geometry
Norma C. Presmeg
University of Durban-Westville
South Africa
235
The relationship between
environmental and cognitive
factors and performance in
mathematics of indian pupils in the junior secondary phase
Anirud Rambaran
Tinus Van Rooy
University of South Africa
Cognitive strategies and
social classes: A comparative study of working
and middle class english
children
Antonio Roazzi
Universidade Federal de
Pernambuco, Brasil
Juegos matematicos
Ludwing J. Salazar Guerrero
Cornelio Yafiez Marquez
Instituto Politecnico Nacional
Walter Cross Buchanan,
.232
233
236
237
Mexico.
Prospective mathematics
teacher's conception of
fuction: The representational side of the coin
Dina Tirosh. School of Educ.
Tel Aviv University
Israel
Development of some aspects of mathematical
thinking in an Analytic
Geometry Course
Maria Trigueros
ITAM
Mexico
Computer graphics for
the acquisition of function concepts
Elfriede Wenzelburger C.
Maestria en Educacion en
Matematicas, UNAM
LIST OF AUTHORS
238
239
240
241
VOLUME II
ALGEBRAIC THINKING AND FUNCTIONS
Children's connections among representations of mathematical ideas.
Alice Alston, Carolyn A. Maher
3
Algebraic syntax errors: A study with secondary school
children.
Alfonso Avila, Francisco Garcia, Teresa Rojano
The development of conceptual
structure as
11
a
problem
solving activity.
Victor Cifarelli
19
Negotiating a jump in the
algebra:
learning process.
Anibal Cortes, Nelly Kavafian, Gerard Vergnaud
27
Continuos analysis of one year of science students' work,
in linear algebra, in first year of French University.
Jean Luc Dorier
35
Avoidance and acknowledgement of negative numbers in the
contex of linear equations.
Aurora Gallardo
43
From arithmetic to
Introducing algebra: A functional approach in a computer
environment.
Maurice Garancon. Carolyn Kieran, Andre Boileau
51
Logo, to teach the concept of function.
Dominique Guin, Ismenia Guzman Retamal
59
The concept of function: Continuity image versus discontinuity image (Computer experience).
Fernando Hitt
67
Acquisition of algebraic grammar.
David Kirshner
75
Embedded figures and structures of algebraic expressions
Liora Linchevski, Shlomo Vinner
85
A framework for understanding what algebraic thinking
Romulo L. Lins
93
is
Developing knowledge of functions through manipulation of
a physical device.
Luciano de Lemos Meira
Students'
interpretations of linear equations and their
10
101
graphs.
Judith Moschkovich
109
An experience to improve pupil's performance in inverse
problems.
Angela Pesci
117
Algebra word problems: A numerical approach for
resolution (A teaching experiment in the classroom).
Guillermo Rubio
its
Children's writing about the
the
idea
of
variable
in
125
context of a formula.
Haralambos Sakonidis, Joan Bliss
133
Observations on the "reversal error" in algebra tasks.
Falk Seeger
141
Generalization process in elementary algebra: Interpretation and symbolization.
Sonia Ursini Legovich
149
ASSESSMENT PROCEDURES
Effects of teaching methods on mathematical abilities cf
students in secondary education compared by means of a
transfertest
Joost Meijer
159
DIDACTICAL ANALYSIS
On long term development of some general skills in problem
solving: A longitudinal comparative study
Paolo Boero
Cognitive dissonance versus
success as
meaningful mathematical learning.
Nerida F. Ellerton, McKenzie A. Clements
the
basis
169
for
177
Time and hypothetical reasoning in problem solving.
Pier Luigi Ferrari
185
interplay
between
student
behaviors
and
the
mathematical structure of problem situations. Issues and
The
examples.
Rina Herschkowitz, Abraham Arcavi
Paradigm
of
open-approach
method
193
in
the
mathematics
classroom activities. Focus on mathematical problem solving.
Nobuhiko Nohda
201
le role du maitre dans les situations
didactiques a
partir du cas de l'enseignement A des
eleves en difficulte.
Marie Jeanne Perrin Glorian
209
Reflexions sur
GEOMETRY AND SPATIAL DEVELOPMENT
Diagnosis and response in teaching transformation geometry.
Alan Bell, Derrick Birks
219
Children's recognition of right
unlearned positions.
Martin Cooper, Konrad Krainer
227
angled
triangles
in
The role of microworlds in the construction of conceptual
entities.
Laurie D. Edwards
235
The cognitive challenge involved in Escher's potato stamps
microworld.
Rina Hadass
243
Study of the degree of acquisition of the Van Hiele levels
by secondary school students.
Adela Jaime, Angel Gutierez
251
Spatial concepts in the kalahari.
Hilda Lea
259
Integrating logo in the regular maths.
developmental risk or opportunity?
Tamara Lemerise
curriculum.
A
267
Young children solving spatial problems.
Helen Mansfield, Joy Scott
275
The role of format in students' achievement of proof.
W. Gary Martin
283
L'influence des aspects figuratifs dans le raisonnement
des eleves en Geometrie.
Ana Mesquita
291
Children's understanding of congruence according to the
Van Hiele model of thinking.
Lilian Nasser
MEASUREMENT
12
297
Prospective primary teachers' conceptions of area.
Cornelia Tierney, Christina Boyd, Gary Davis
307
PROBABILITY
Probability concepts and generative learning theory.
Ole BOrkqvist
319
Some considerations on the learning of probability.
Ana Maria Ojeda Salazar
327
Gambling and ethnomathematics in Australia.
Peard, Robert
335
Mathematization project in class as a collective higher
order learning process.
Hans George Steiner
343
LIST OF AUTHORS.
VOLUME III
RATIONAL NUMBER
The construct theory of rational numbers: Toward a semantic
analysis.
Merlyn Behr, Guershon Harel
Reflections on dealing: An analysis of one child's interpretations.
Gary Davis
About intuitional
school.
Joaquin Gimenez
knowledge
of
density
in
3
11
elementary
19
Understanding the multiplicative structure.Concepts at the
undergraduate level.
Guershon Harel, Merlyn Behr
27
A contextual approach to the teaching and learning of
mathematics: Outlining a teaching strategy that makes use
of pupil's real world experiences and strategies, and the
results of the first teaching experiment of project.
Tapio Olavi Keranto
35
On children's mathematics informal method.
Fou-Lai Lin, Lesley R. Booth
43
13
A case study of the role of unitizing operations with
natural numbers in the conceptualization of fractions.
Adalira Ludlow
51
Constructing fractions in computer microworlds.
John Olive, Lelie P. Steffe
59
Proportional
reasoning:
From
shopping
laboratories, and hopefully, schools.
Analucia Dias Schliemann, Magalhaes, V. P
kitchens,
to
67
The fraction
concept
in comprehensive school at grade
levels 3 - 6 in Finland.
Tuula Strang
SOCIAL INTERACTIONS,
75
COMMUNICATION AND LANGUAGE
Critical decisions in the generalization process:
A
methodology
for
researching
pupil
collaboration
in
computer and non computer environments.
Lulu Healy, Celia Boyles, Rosamund Jane Sutherland
83
"Scaffolding"
a
crutch
or
a
support
sense-making in learning mathematics.
Barbara Jaworski
91
The
role
of
mathematical
knowledge
understanding of geographical concepts.
Rochelle G. Kaplan
for
in
pupils'
children's
99
Speaking mathamatically in bilingual classrooms. An exploratory study of teacher discourse.
Lena Licon Khisty, Douglas B.
McLeod, Kathryn Bertilson..
105
The emergence of mathematical argumentation in the small
group interaction of second graders.
Gotz Krummheuer. Erna Yackel
113
Potential mathematics learning opportunities in grade
three class discussion.
Jane Jane Lo, Grayson, H. Wheatley and Adele C. Smith
121
Certain metonymic aspects of mathematical discourse.
David John Pimm
129
Inverse relations: The case of the quantity of matter.
Ruth Stavv, Tikva Rager
137
The development of mathematical discussion.
Terry Wood
147.
14-
STATISTICAL REASONING
Estrategias y argumentos en el estudio descriptivo de la
asociacion usando microordenadores.
Juan Diaz Godino, C. Batanero, A. Estepa Castro
157
Computerized tools and the process modeling.
Chris Hancock, James Kaput
165
WHOLE NUMBER AND INTEGERS
Examples of incorrect use of analogy in word problems.
Luciana Bazzini
175
Procedural
Children's pre-concept of multiplication:
understanding.
Candice Beattys, Nicolas Herscovics,Nicole Nantais
183
The kindergartners' knowledge of numerals.
Bergeron
Jacques C.
191
An analysis of the value and limitations of mathematical
representations used by teachers and young children.
Gilliam Marie Boulton- Lewis, Halford, G.
199
S
A study on the development of second-graders' ability in
solving two-step problems.
Feiyu Cao
207
Understanding the division algorithm from new perspectives
David William Carraher
215
Negative numbers without the minus sign.
Terezinha Nunes Carraher
223
Learning difficulties behind the notion of absolute value.
Chiarugi, I.,Fracassina, G., Fulvia Furinghetti
231
Le role de la representation
problemes additifs.
Regina F.
Datum
dans
resolution
la
239
Using a computerized tool to promote students'
sense" and problem solving strategies.
Nira
Hativa,
Victoria
Bill,
Sara
Machmandrow
des
"number
Hershkovitz,
Children's understanding of compensation,
subtraction in part/whole relationships.
Kathryn C. Irwin
Ivi
249
addition
and
257
Factors affecting children's strategies and success
in
estimation.
Candia Morgan
265
Assessment in primary mathematics: the effects of item
readability.
Judith Anne Mousley
273
Social sense-making in mathematics; children's ideas of
negative numbers.
Svapna Kukhopadhyay, Lauren B. Resnick, Schauble,L
281
Children's pre-concept of multiplication:. Logico physical
abstraction.
Nicole Nantais, Nicolas Herscovics
289
Building
on
young
children's
informal
arithmetical
knowledge.
Alwyn No Oliver, Murray, A., Piet Human
297
From protoquantities to number sense.
Lauren B.Resnick, Sharon Lesgold, & Bill V
305
Prospective elementary teachers' knowledge of division.
Martin A. Simon
313
Relative and absolute error in computational estimation.
Judith and Larry Sovder, Markovits, Z
321
A child generated multiplying scheme.
Leslie Steffe
329
Salient aspects of experience with concrete manipulatives.
Patrick Thompson, Alba G. Thompson
337
LIST OF AUTHORS.
16
Algebraic Thinking and Functions
17
Children's Connections Among Representations of Mathematical Ideas
Alice S. Alston and Carolyn A. Maher
Rutgers University
Analysis of the written problem protocols and videotape
segments of 11 children for seven problem-tasks containing
common structural elements over a 6 day period is made in
order to gain insight into the development of the
representations built by the children and the connections
made between and among the representations. A more
detailed analysis is given of the mathematical behavior of
two children. The observations provide descriptions of the
process by which children construct representations of
ideas in cooperative group problem-solving settings and in
individual written assessments. This information lends
insight to the results of a Larger study in which highly
significant gains Ln understanding were made by
participants in these activities as compared with a control
group.
Much attention has been directed recently to the need to study the processes by which
learners build-up systems of representation of mathematical ideas and relate them to other
systems (Davis, 1984; Kaput, 1987). One approach to assessing understanding of a
mathematical concept is recognition of that idea embedded within qualitatively different
representational systems (Lesh, Post, and Behr, 1987). The building-up of meaningful
experience may cane about by being aware of the structure of the activity and reflecting on
it (Steffe & Cobb, 1983). Another dimension of the building-up of representations is task
involvement by the learner (Cobb, Yackel, & Wood, 1989). Lave (1988) urges us to
consider, in studying the transfer of knowledge among representations, the learner's social
interaction or other factors that motivate problem solving. Wood and Yackel (1990)
demonstrate in their work the importance of peer group dialogue so that learners have an
opportunity to make sense of each other's interpretations and serve mutually supportive
roles. Brown, Collins, and Duguid (1989) also direct us to consider learning that arises out
of shared activity by other learners in a context in which representations of ideas are
constructed and discussed together. Our view is that learning mathematics is facilitated in
an environment that provides for social interactions in small group problem solving tasks
that enable learners to build-up representations, over time, of the structure of the idea(s).
(Maher, 1987; Maher, Alston, & O'Brien, 1986).
BEST COPY AVAILABLE
3
18
Background
As part of a larger investigation to measure the mathematical behavior of 12 and 13
year old 7th grade children, a teaching experiment was conducted in which 84 children
from two schools (one, a public school in a blue-collar community and the other, an
independent school in an affluent suburban community) were given a series of 7 problemtasks in a natural classroom setting over a five day period of time. The purpose of the
study was to observe and analyze children's mathematical thinking as they were engaged in
tasks dealing with the properties of closure, identity, inverse and commutativity and to
assess their ability to make connections among various representations of each of the
concepts. The population was stratified into three groups, the first two from a public K-8
school, and the second from an independent middle school: high ability prealgebra students
from the public school; students enrolled in regular seventh-grade math (students in this
heterogeneous group ranged from remedial to average in ability) from the public school;
and (non-honors) prealgebra students from the independent school. Children from each
school population were randomly assigned to two comparable groups for five class periods,
one, experimental and the other, control. The children in the experimental group were
given three nonnumerical problem-tasks each based on a different concrete embodiment of
the properties for the purpose of constructing solutions to the problems posed without any
teacher intervention, while those in the control group were similarly engaged but were
given problems that dealt with different content. In a final class session, each child was
given three final written assessments. Each of these was a numeric problem task that was
structurally isomorphic to one of the concrete tasks.
A logistic regression analysis was used to examine the relationship between problem
solving success and the factor of experimental versus comparison group for each of six
mathematical assessment categories across the three written postassessments. The
categories were defined as (a) Closure, (b) Identity, (c) Inverse, (d) Order, referring to
commutativity, (e) Transfer, that is generalization to other mathematical ideas, and (f)
Total, referring to total success on the preceding five sections for the particular written
assessment. The analysis of the Transfer category gave the probability of success for those
children who had participated in the experiment to be 97% for the high ability experimental
students; 79% for the average and heterogeneous experimental groups; and 32% for the
comparison groups for all three of the postassessment tasks.
19
4
This report provides a description of the mathematical behavior of a group of 11
experimental students. Because of limitations of space, a more detailed analysis is given of
only two of the children to provide insight into how they developed the concepts in their
small groups and made connections to the ideas across the task activities.
In particular, the study sought to investigate, for representations built by all eleven
children, the following three questions:
(1) Did the children make meaningful connections from the ideas considered in
each of the seven problem tasks to other mathematical ideas?
(2) Did the children make connections from one task to another?
(3) What references, if any, do children make to mathematical ideas that
are not specific to their task activities?
For the two more detailed analyses of children's mathematical behavior, two
additional questions were also addressed:
(4) What references, if any, do children make to structural similarities and differences
among representations?
(5) What connections, if any, do children make among the three concrete
representations? Among the three numeric representations?
Methods
Each child was given a written preassessment task (WPA). During the five following
sessions the children met in experimental or comparison classes. The members of each
experimental class were partitioned into groups of two or three children to work together to
construct solutions of the three nonnumerical problem tasks. The structure of Task One
(T1) was a Klein group using two small wooden figures, a boy and a girl. The elements of
the set were the four possible 180 degree turns of the two figures taken together and the
operation was one turn followed by a second. Task Two (T2) had a lattice structure and the
elements of the set were cards, each of which had cut out a different polygonal shape. The
operation was placing one card on top of a second to form a resulting polygonal shape.
The third task (T3) had a cyclic group structure based on index cards,called Road Cards,
each of which had a different set of four straight lines from beginning points on the left side
of the card to end points on the right. The result of the operation, in which one card was
followed by a second, was the single card with with the beginning points of the first card
20
and the end points of the second. Within each experimental class, two groups were
randomly chosen to be videotaped during all five sessions. On the final day to the teaching
experiment the children returned to their regular class groupings and each child was given
the three final written assessments (FWA I, FWA2, and FWA3). In each of the seven
problem-tasks, the children were asked to construct a table of results for the set of elements
and the given operation and then to answer a series of questions concerning closure,
identity, inverse, and commutativity for that particular operational system. The concluding
question of each task was whether the problem called to mind any other problems or ideas
about mathematics, and, if so, to describe them.
The seven boys and four girls considered in this investigation were four of the
cooperative problem-solving groups. The data included the children's solutions to each of
the four written assessments, each child's written solution to the nonnumerical problem
tasks, transcripts of the videotapes, and observer notes of the group problem-solving
sessions. (Note 1)
Results
Table 1 presents a summary of the data for the eleven children, In each task, evidence
of the presence of a meaningful connection is indicated by a Y, its absence by N, and when
the presence of a connection was in doubt, by U (e.g., student appeared to go along with
group consensus).
Categories of connections were also indicated. A reference to one of the concrete tasks
was coded as c; reference to a written numeric assessment was coded as a; references to
mathematical ideas that were not specific to the task activities were coded according to
their context (e.g., arithmetic operations (o); numbers (n); fractions (f); and geometric ideas
(g). Whenever children made specific reference to a particular property, the connection
was coded as p.
The table indicates that all children made some connections during the duration
of the study. Two of the eleven children made connections in the preassessment, one
to the property of zero and the other to the arithmetic operation of addition. In the
final assessment, all but one made connections to a variety of representations. An
analysis of two children's mathematical behavior, Ed (Cl) and Joe (C7), follows the
21
6
Table to illustrate the nature of the connections made among the various tasks and to
other mathematical ideas.
TABLE 1: CHILDREN'S CONNECTIONS AMONG REPRESENTATIONS
Problem Task
CHILD
Cl
WPA T1 T2 T3
FWAI FWA2 FWA3
Connections
p,a,n,f,o,g,c
a,f,c,p,o
a,f,c,o,p
Y
Y
Y
Y
Y
Y
Y
C2
N
Y
Y
U
Y
Y
Y
C3
N
U
U
U
Y
Y
Y
Y
Y
N
Y
Y
Y
Y
Y
Y
N
a,g,o
g,a,o
C5
NU
NY
C6
N
Y
Y
Y
Y
N
Y
a,g,c
C7
N
Y
Y
Y
Y
Y
Y
C8
N
Y
U
U
Y
Y
Y
p,o,c
o,p,n,c
C9
C10
C11
Y
Y
U
Y
Y
Y
Y
N
Y
Y
Y
N
N
Y
N
Y
U
Y
Y
Y
Y
C4
o,p,c,a
n.p,c,a
p,c,a
Note: Y = presence, N = absence, U = uncertain of connections.
p = properties (i.e. commutativity, identity), a = assessment problem tasks, c=
concrete tasks, o = arithmetic operations, n = numbers, f = fractions, g = geometric
ideas
Case 1: Ed (C1) worked on the classroom cooperative group tasks with two girls,
Trish (C2) and Natasha (C3), all children from the independent school. He was one
of the two children who indicated in WPA that the problem reminded him of another
mathematical idea. His written response stated that the problem reminded him of
problems about zero because zero and any other number is the same result. In
his written work for T1, he wrote: the problem task reminded him of WPA,
commenting aloud that the commands were like numbers. He also wrote that the
problem was like fractions -- like knowing how to cancel out the common
factor. While working with his group to figure out inverse elements for the set, he
announced to the others: It's like fractions; like 1/2 times 2/1; they cancel.
7
22
His reference to numbers was continued in T2. Ed wrote that this problem reminded
him of addition, subtraction, and multiplication with charts. He then referred
again to WPA: I did a chart like this with numbers that followed sort of the
same pattern. In discussing this problem with the group, Ed stated: It's the
commutative property; like 3 + 1= 1 + 3. It's like adding. While solving T3, Ed
pointed out to his group: This problem is like the other ones; this card is the
special card (identity element) like Nobody Turns (the identity element in Ti).
Ed wrote in FWA1 that the task reminded him of all of the problems that we have
done because all have a procedure to get the result and the results all follow a
pattern. In FWA2 he wrote that the problem reminded him of problems about 1
because (in the task) 1 and any of the numbers turns out to be 1 and in
multiplication 1 times any number equals 1. Finally in FWA3, Cl wrote: We did
another problem almost exactly like this a few days ago.
Case 2: Joe (C7) worked with Dave (C8), both students in the heterogeneous
public school group. Joe wrote in WPA that nothing about the problem reminded
him of any other mathematical ideas or problems. In T1, however, he wrote: It
reminds me of addition and multiplication, and in discussion with his partner he
stated: This is like the commutative property. OGT and OBT or OBT and OCT
(two elements of the set). Either way, they equal BT (a third element). It's like
please, my dear Aunt Sally; it's like addition and subtraction; no, I don't think
subtraction works -- only addition and multiplication. In T2 Joe wrote that the
problem reminded him of the commutative property and the property of 1. He
had pointed out to Dave earlier: It's (referring to Card D, the identity) like 1 and E
(another element of the set) is like 4. And D on E is E just like 1 times 4 gives you
4 because if you put D on anything nothing happens. It's just like Nobody
Turns (the identity element in T1). In his response for T3, Joe wrote: It reminds
me of the other two problems and problems about numbers because the cards
are like numbers. In discussing with Dave he said: Card A (the identity element)
is like 1 for multiplication. The property of 1 - You know! - 1 times anything is
the other number. In FWA1 Joe wrote: It reminds me of the problems that we
just did. Then in FWA2, Joe continued: The problems about the commutative
property. Finally, in FWA3 Joe concluded: They were about Roads and stuff.
23
8
Conclusions
The analysis indicated that each of the eleven children made at least one
connection in at least three of the 6 tasks (not including WPA) with the mode being
five. All children made at least one connection between tasks. Also, all children
made references to at least two different kinds of other mathematical ideas. Both
Ed and Joe made comparisons between the concrete tasks and operations with
numbers. Both indicated recognition of commutativity, comparing the concrete
elements with number representations. Both recognized the property of the identity
in the concrete tasks and each referred to the identity with numbers and in the other
concrete tasks. Both boys indicated in the final written assessments that the
numeric problems reminded them of the concrete tasks that they had done because
of properties such as commutativity and identity. The detailed analysis of the
mathematical thinking of Ed and Joe reported here is representative of the cases
developed for the other children.
A limitation in the study is that data were often obtained from video taped
episodes in which some children were more verbal than others. A design that
includes follow-up interviews could provide insight into the nature of the
uncertainty category as well as an opportunity to probe for meanings that are
unclear or inconsistent in written statements. For example, Ed's consistent
recognition of the identity among the seven tasks would lead one to expect that
what he meant to have written in his final assessment was that in multiplication, one
times any number equals that number rather than what he actually wrote (1 times
any number equals 1).
The detailed description of the representations articulated by the children and
the connections among them supports the statistical analysis of the larger study in
which the experimental group scored significantly better than the control. It is
important to understand how children build-up their mathematical ideas so that
appropriate task activities can be provided in classrooms to facilitate learning.
Note 1. For a detailed analysis of the children's problem-solving behavior in the group
activities as well as a description of the nonnumerical tasks (See Alston & Maher, 1988;
Alston, 1989).
9
References
Alston, A. (1990). Unpublished doctoral dissertation. Rutgers University.
Alston, A. (1989). Children's representations of arithmetic properties in small group problemsolving activities. In C. Maher, G. Goldin & R. Davis (Eds.), Proceedings of the Eleventh
Annual Meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education, (pp. 235-242). New Brunswick, New Jersey: Rutgers
University Press.
Alston, A. & Maher, C.A. (1988). The construction of arithmetic structures by a group of
three children across three tasks. In A. Borbas (Ed.), Proceedings of the Twelfth International
Conference for the Psychology of Mathematics Education. (pp. 117-124). Vezprem,
Hungary: OOK Printing House.
Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning.
Educational Researcher. 17 (1), 32-42.
Cobb, P., Yackel, E., & Wood, T. (1988). Curriculum and teacher development:
psychological and anthropological perspectives. In E. Fennema, T.P Carpenter, & J. Lamon
(Eds.0, Integrating Research on Teaching and Learning Mathematics.
Madison, WI:
Wisconsin Center for Education Research, University of Wisconsin.
Davis, R.B. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics
Education. Norwood, NJ: Ablex Publishing Company.
Kaput, J.J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of
Representation in the Teaching and Learning of Mathematics. (pp. 19-26). Hillsdale, NJ:
L.E. Erlbaum, Associates.
Lave, J. (1988). Cognition in Practice. Cambridge, England: Cambridge University Press.
Lesh, R., Post, P. & Behr, M. (1987). Representation systems and mathematics. In C.
Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics. (pp.
33-40). Hillsdale, NJ: L.E. Erlbaum, Associates.
Maher; C.A. (1987). The teacher as designer, implementer, and evaluator of children's
mathematical learning environments. The Journal of Mathematical Behavior. 6(3), 295-303.
Maher, C.A., Alston, A., & O'Brien, J.J. (1986). Examining the heuristic processes of mine to
twelve-year old children in small group problem-solving sessions. Proceedings of the Tenth
International Conference of the Psychology of Mathematics Education. (pp. 369-374).
London, England: University of London Institute of Education.
Steffe, L.P. & Cobb, P. (1984). Children's construction of multiplicative and divisional
concepts. Focus on Learning Problems in Mathematics, 6, (1 &2), 11-29.
Wood, T. & Yackel, E. (1990). The development of collaborative dialogue in small group
interaction. In L.P. Steffe & T. Wood (Eds.), Transforming Early Childhood Mathematics
Education: an International Perspective, (pp. 244-252). Hillsdale, NJ: L.E. Erlbaum,
Associates.
10
ALGEBRAIC SYNTAX ERRORS: A STUDY WITH
SECONDARY SCHOOL CHILDREN
Alfoneo Avila,
(4)Francisco Clarcici 'l" and
(0)
(**)
Teresa Rojano
Unidad
do
Apoyo
Didactico-Secretari a
y Bionostar Social del Retado do Mexico.
Centro
do
Investigation
Nacional
do
Formation
Matemeticas, "MEXICO.
y
do
y
do
**)
Educctcion,
.Cultura.
Estudioe
Avanzados
Programa
Actualization
do
Prof scores
do
ABSTRACT.- This paper reports on a study carried out whtth 221
Secondary School children of the State of Mexico.
Paper and
pencil
tests were administered in order
to detect
the
presence and frequency of albebraic syntax errors previously
reported in other studies CBooth C 1 .7, Hato C 8 .7, Kieran
5,6 2,
Kilchemann
C
7 2,
Collis f 2 2, Trujillo f 9 2,
Filloy/Rojano f 3 2).
Some of
these results were confirmed
for
the
mexican
data,
particularly
those
concerning
interpretation
and
manipulation
of
algebraic
symbols,
symbolization of generalizations, equation solving and word
problem soloing.
Other kinds of errors appeared which can be
interpreted as teaching effects.
Introduccion.1
1
y
De acuerdo a las investigaciones de L. Booth
D. KUckemann C 7 1, los niflos entre 11 y 16 aSos de
edad pueden tener distintos niveles de interpretaciOn de los
simbolos literales, cuando estos aparecen en expresiones algebraicas Cpor ejemplo,
la letra comp objeto, comp incognita
especifica, coma ntimero generalizado o come variable).
Estas
interpretaciones con frecuencia conducen a tipos especificos
de errores en el desempeflo do tareas algebraicas.
Por otro
lado.
el trabajo de M. Matz C 8 1
proporciona elementos
tedwicos que sugieren la presencia de procesos mentales tales
comp la extropolaciOn y la generalizaciOn,
capaces de generar
en el Algebra, tanto las respuestas correctas come las incorrectas. Tal es el caso de la bien conocida tendencia a aplicar
linealmente todo tipo de operadores
Cpor ejemplo, en
2
2
Ca + 1:0 2 se obtiene
a + b D,
al carecer de criterios de
discriminacion entre un dominio de extrapolacion vAlido y uno
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11
26
de los items del examen de Algebra del estudio Strategies and
1
1), para cubrir la
Errors in Secondary Mathematics CBooth
Tambidna se
section correspondiente a aritmetica generalizada.
para abarcar
los
temas
de
incluyeron series de items
resolution de
simplification de expresiones algebraicas,
ecuaciones y resoluciOn de problemas verbales, para lo cual se
Matz [83
adoptaron preguntas de los trabajos de Kieran [51,
y se elaboraron items
y Filloy/Rojano [ 3
Trujillo
9
[
1
[
exprofeso para la parte de sistemas de ecuaciones lineales.
Se completaron 23 preguntas C42 items) para la version
definitiva del cuestionario, algunos de los cuales se inciuyen
a fin de ilustrar los temas considerados.
a continuation
,
1.
,0111iie significa mn? Subraya todas las respuestas que cress
son correctas:
a)
m
b)
m X
c)
m +
m
n
a> z5 + 26
m) 25 X 26
f) Si tienes otra respuesta, por favor escribela
2.
6.
LCOmo escribirias 3 aumentado a
5v ?
Reduce, cuando sea posible, las siguientes expresiones
a) a + a + 3b + 5a =
b) 4 + 3y
=
c) 2a + 5b +
d) 5y - 2t
eD Ca
3a
=
=
b) + b =
LS4 puede obtener de la ecuacion
9.
Escribe Si
1) 2x
6 =4
o
1
la ecuacion 2 ?
No
2) 2x - 6 + 6 = 4 + 6
1)
3a + 5 + 4a = 19'
2) 12a = 19
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12
27
no-vAlido.
En otro orden de ideas, °studios teericos y empiricos han
revelado que el transit° de la aritmetica al Algebra, requiere
que
cambios
profundos en el nivel conceptual t.engan lugar
CFreudenthal [43, Filloy/Rojano [27, Kieran [6] 3
Ya que, de
no lograrse tales cambios, el anclaje en la manera aritmetica
de pensar
genera cierto tipo de operaciones aberrantes on
Algebra Ccomo
por
incognita,
16 x = 2 x + 5 ->
-> x = 2
en
ejemplo,
la
operation defectuosa de
C3 + 53
23 ->
16
la
8 =
o la redistribution del error, al afirmar que x + 37
= 150 tiene la misma soluciOn que x. + 37 - 10 . 150 + 103.
Uno de los
propOsitos del trabajo que aqui so expone es
el de verificar, hasty que punto, los errores algebraicos mAs
frecuentes, reportados en los estudios mencionados anteriormente, estAn presentes en la poblaciOn estudiantil de las
escuelas secundarias de una parte del
sistema educative
mexicano y confronter los resultados obtenidos en este contexto y nivel oscolar con los resultados de otras investigaciones
de la misma naturaleza.
Otro de los propOsitos es Ilevar a
cabo un.analisis de los tipos de error detectados, en terminos
de a) la interpretacien de los simbolos
y operaciones
algebraicas, per parte de los estudiantes; b) los procesos
mentales que pueden generar las respuestas erreneas; c) los
efectos que la enseFanza puede llegar a tenor en la generacion
y/o rectification de los errores tipicos.
Ya que los trabajos segalados con anterioridad se complementan unos a otros, en cuanto a dar explicaciones plausibles
de la presencia y uniformidad de los errores algebraicos.
algunos
aspectos de dichas investigaciones se tomaron on
cuenta para
conformar un marco teerico para el anAlisis de
los datos recabados en el estudio aqui resegado.
METODOLOGIA Y PASOS DE LA INVESTIGACIOM
ElaboraciOn y AplicaciOn del Cuestionario
Para la elaboration del cuestionario, se adoptaron algunos
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13
28
14.
LCuando es verdadera la siguiente expresion?
L +N+N=P+ N
18.
Resuelva las siguientes ecuaciones
aD 3x
1:06x-3
20.
Algunas veces
Nunca
Siempre
4 = 8
2x
=
+
1
Plantea la ecuacion que conduce a la solucion
del siguiente
problema; no se requiere que des la so-
1 uci 6n.
a)
El doble de u n ntimero disminuldo en 12
Simplifica, cuando sea posible,
22.
es igual a
LCual es ese ntimero?
26
las siguientes expre-
si ones:
2x + 3y
2x
preliminar del cuestionario a
Para
una administracion
una poblaciOn pequeRa de estudiantes, se hicieron ajustes de
algunos items en relaciOn al programa de estudios vigente y al
en los libros de text° usuales en la
lenguaje
utilizado
Una version definitiva del examen fue aplicada a 221
regi6n.
niflos del segundo grado de la ense?anza secundaria, provenientes de cuatro escuelas de zonas urbanas y sub- urbanas en el
Se verific6 que, en Ese momento, los ni.Eos
Estado de Mexico.
ya hubieran estudiado los temas del cuestionario, incluyendo
la
resolucion de sistemas simples de ecuaciones lineales.
Analisis de los resultados.
Se Ilevaron a cabo dos tipos de analisis de los resuluno, cuantitativo, el cual permiti6 clasificar los
en
items del cuestionario en cuatro niveles de dificultad:
se incluyen los items de menor dificultad,
el primer nivel
44%; en el sequndo
con un porcentaje de error entre lg' y
tados,
29
14
niv21, los items con porcentaje de error entre 48 y 64%,en el
tercero, los de porcentaje de error entre 67
y
84%; y el
cuarto nivel, el de mayor dificultad, los de porcentaje de
error entro 86 y 100 %..
Los items que resultaron ''mas faciles" para esta poblacion
corresponden a la resolution de enunciados
verbales tipo
abbaco U'encuentra un nfamere tal que su doble sea..."3 que
corresponden a ecuaciones lineales simples con ura ocurrencia
de la
incognita;
resolution de ecuaciones de "un solo pasoC6 x = 4 -> x = 4/63
y
simbolizacien de operaciones con
nOmeros y letras C4 sumado a 3n).
En las
franjas
de
-dificultad media",
se
encuentran items de substituci6n
numerics de la "variable" en expresiones simples; reduction de
expresiones, agrupando terminos semejantes; verificaciOn de
equivalencia de ecuaciones; equivalencia de expresiones con
notation literal; resolucion de ecuaciones entre un nilmero y
un binomio,
traduccion
a ecuaciones de enunciados verbales
simples; interpretaciOn de expresiones como 5n; expresiOn de
perimetros de poligonos con use do letras.
Finalmente, la
franja de mayor dificultad, la conforman 19 de los 42 items
considerados para el analisis cuantitativo e incluye tareas do
reduction de expresiones como a+ a+ 3b+ 5a y 4+ 3y;
simbolizacien de
-perimetros generalizados-; simplification
de expresiones rationales como
2x
6
2 x+ 3 y
x.
x + y
6
2 x
x
2
resolucion do sistemas
de ecuaciones como
2 x + y = 1
x
6
2 y = 8
x = 1
y = x + 3
;
operaciones entre binomios.
AdemAs del cuantitativo,
se neve a cabo un analisis
cualitativo de los tipos de error cometidos en cada item.
Del analisis
de las respuestas erreneas mss frecuentes
encontradas en
este estudio se desprenden, basicamente, dos
hechos:
a) La aparicion reiterada, en la mayoria de los items, de
15
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30
respuestas que manifiestan dificultades reportadas en
Trujillo,
Kieran,
Matz,
de Booth,
los estudios
A continuation se muestran algunas de
Filloy/Rojano.
dificultades, exhibiendo ejemplos, para cada
tales
una de ellas.
Concatenation de simbolos: mn = m + m
Sn = 5 + n
J + p = JP
- InterpretaciOn de las letras:
+ Letra no usada
A = 5 x 2
+ Asignacion de valores a las letras segOn el alfabeto:
mn = 25 x 26
+
Asignacion de valores espocificos diferentes
a
letras diferentes:
Nuncat+M+N es iguala L+ P+ N
- AplicaciOn de la regla -sumar nameros y anotar las
letras-:
3 + 5y = By
4 + 3y = 7y
2 + 5a = 7a
4 + 3n = 7n
Ausencia de significado para los parentesis:
5C 2a + bD = 10a + b
- Ambiguedad notational:
4a = 43, si a = 3
Incapacidad de.generalizacion:Este poligono tiene n lados, cada lado mide 2.
P = 20
P = 2
31_
16
10
Empl eo de metodos' primi.tivos en la so) uci On de
ecuaciones y problemas, como ensayo y error y hechos numericos.
Traduccian algebraica deficiente de enunciados:
+ Expresan 3 aumentando a Sy como 35, 35Y,
..5y
+ Expresan m+ 5 multiplicado por 3 como m + 5
+ Expresan el "triple" de un nOmero disminuido
18 es iqual a eso mismo numero come. 3x
18 =
b) La manifestacien repetida del empleo exagerado de
presiones en forma de potencies:
x
en
19.
ex-
a + a + 3b + Sa = 5a3 + 3b
2 +
2a + Sb + 3a = 5a
r+b
h +h+h+h+t= h4
x + x + 5 + 5 + 6 = x2 +- 52 + 6
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2.=
21 °
lo cual puede ser atribuido a la enseanza reciente de la
notacion exponencial.
CONCLUSIONES
a los propOsitos del estudio, puede decirse
ya
do la presencia de dificultades
reportadas en la literatura de investAgacion, nos remite a las
explicaciones teoricas de distintos autores, tales come, la
existencia de niveles do interpretaciein simbolica en algebra;
el
anclaje on la aritmetica cuando se abordan tareas de
resolucion de problemas y ecuaciones; la presencia de procesos
mentales de extrapolacien, que producen entre otras cosas, la
hipercieneralizacion de la linealidad entre operadoros; y la
necesidad de una semantica de la producciOn de simbolos compuestos Cletras y numeros) on las tareas de traduccien al
algebra de problemas verbales. Explicaciones que, en una
primera aproximacion a los datos recoloctados, pueden ser
otro lado,
no cancola
Esto, por
aceptadas como plausibles.
la posibilidad de pJantearse el atacar ostas dificultados, on
el ni vel do
la ensenarza, ya quo la poblaciOn ostudi ada la
En relation
que
la
confirmacion
17
conforman niFlos que se inician en el estudio del Algebra
para quienes puede tenor sontido crear acercamientos de
enseflanza que contemplen los aportes que las investigaciones
recientes nos han
proporcionado.
Bibliografia
Algebra:
(1] BOOTH, L.P.
NFER-NELSON, 1984.
Children's
strategies and errors.
A Study of Concrete and Formal
(2] COLLIS, K.F. C1975cD.
A Piagetian Viewpoint.
Operations in School Mathematics:
Melbourne: Australian Council for Educational Research.
&
E3l FILLOY, E.
Algebraic
From
ROJANO, T.
an
Arithmetical to
Thought CA clinical Study with 12-13
an
year
VI Annual Meeting of the International Group, for
the Psychology of Mathematics Education, North American
oldsD.
Chapter. Madison, Wisconsin, USA., 1984, pp.
51 -66.
(4) FREUDENTHAL, H. Didactical phenomenology of mathematical
D.
Reidel Pub lishing Company,
structures. Netherlands.
1983.
E6l KIERAN
of
ALGEBRA:
The
learning
Quebec, ERIC Reports, 1982.
C.
,
EXPERIMENT.
a
TEACHING
Concepts associated with the equality simbol,
(63 KIERAN, C.
Educational Studies in Mathematics, 12, pp. 317-326.
In
C1981D
Algebra.
177KOCKEMANN, D.E.
Children's Understanding of Mathematics:
Murray, pp.
102-119.
E8l
MATZ,
K.
CEd.D
London:
C1980D Towards a
M.
algebraic competence.
3C1D, pp. 93-166.
E9]
Hart,
11-16.
TRUJILLO,
M.
algebraico en
computational
theory
of
Journal of Mathematics Behaviour,
T .esis
de Maestria: Uso .del lenguaj,e
de problemas de aplicacion.
la resolution
Centro de Invesstigacian y de Estudios Av anzados
I.P.N., Mexico, 1987.
33
18
del.
THE DEVELOPMENT OF CONCEPTUAL STRUCTURE AS A PROBLEM SOLVING ACTIVITY
Victor Cifarelli
University of California at San Diego
This study examines the development of conceptual structures in problem solving
situations.
Nine college freshmen were interviewed as they solved a set of similar
algebra word problems. All interviews were videotaped and written transcripts of the
solvers' verbal responses were prepared.
Analysis of the solvers' solution activity
yielded four increasingly abstract levels of structural knowledge.
INTRODUCTION
The notion of conceptual structure underlies many goals for instruction in mathematical problem
solving.
Mathematics educators who have as their goal the development of "intellectual autonomy" (Kamii,
1985) in the problem solving actions of their students view conceptual structures in terms of their
interpretive qualities, as a means by which solvers can organize their problem solving experiences "with a
view to making predictions about experiences to come" (von Glasersfeld, 1987) (e.g., making conjectures
about one's potential solution activity in new situations).
Despite universal agreement about the importance of solvers developing such structural knowledge,
current work in situated cognition suggests the need to reexamine the traditional view of conceptual
structures as "decontextualized formal concepts" which are transferred across learning situations (Brown,
Collins, and Duguid, 1989).
The idea that learning and cognition are situated suggests that learners build
up their conceptual knowledge in the context of ongoing activity.
As a result, concepts continually evolve
with each occasion of use, "because new situations, negotiations, and activities inevitably recast it in a
new, more densely textured form ".
According to Lave (1988), a solver's articulation of structure in a
problem solving situation generates learning oppportunities in which exploration of "the plausibility of
both procedure and resolution in relation to previously recognized resolution shapes" can lead to a
restructure of one's prior solution activity.
This paper will argue that solvers construct such conceptual
organizations while performing mathematical problem solving activity and that further development of the
structure is a process of reconstruction resulting from subsequent problem solving activity.
OBJECTIVES
The purpose of the study was to acquire an understanding of the processes of constructing conceptual
knowledge during mathematical problem solving.
The study focused on the internal activity of the learner
with particular emphasis on the ways that learners elaborate, reorganize, and reconceptualize their solution
activity while engaged in mathematical problem solving.
Solvers face problematic situations in their mathematical activity when they can't see any way to
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34
achieve their goals (Pask, 1985).
When problem solving is related to one's goals as such, a variety of
situations qualify as genuine problem solving situations.
For example, solvers might face a problematic
situation when they attempt to make sense of or understand statements that describe a specific algebra word
problem.
Alternatively, the solver's problem might be to understand why a particular solution method led to
unanticipated success or why two different solution methods led to the same result.
These situations arise
in the course of goal directed activity and can serve as learning opportunities for solvers (Pask, 1985;
Successful resolution of such situations can be viewed as the construction of
Cobb, Yackel, & Wood, 1989).
conceptual understanding in the context of ongoing activity (Vergnaud, 1984; Lave, 1988) with the result
being that the solvers build structure for their current solution activity (or restructure their prior
solution activity).
Hence, the goals of the study were to provide clarification for these ideas by
observing solvers as they experience and resolve a range of problem solving situations and to characterize
their subsequent growth in structural knowledge.
METHODOLOGY AND DATA SOURCE
Subjects came from calculus classes at the University of California at San Diego.
participated in the study.
(see Table 1).
Nine subjects
Subjects were interviewed as they solved a set of similar algebra word problems
The interviews were videotaped for subsequent analysis.
In addition to the video protocols,
transcripts of the subjects' verbal responses as well as their paper- and pencil activity were used in the
analysis.
Table 1: SET OF LEARNING TASKS
TASK 1: Solve the Two Lakes Problem
The surface of Clear Lake is 35 feet above the surface of Blue Lake. Clear Lake is twice as deep as Blue
Lake. The bottom of Clear Lake is 12 feet above the bottom of Blue Lake. How deep are the two lakes?
TASK 2: Solve a Similar Problem Which Contains Superfluous Information
The northern edge of the city of Brownsburg is 200 miles north of the northern edge of Greenville. The
distance between the southern edges is 218 miles. Greenville is three times as long, north to south as
Brownsburg. A line drawn due north through the city center of Greenville falls 10 miles east of the city
center of Brownsburg. How many miles in length is each city, north to south?
TASK 3: Solve a Similar Problem Which Contains Insufficient Information
An oil storage drum is mounted on a stand. A water storage drum is mounted on a stand that is 8 feet taller
than the oil drum stand. The water level is 15 feet above the oil level. What is the depth of the oil in the
drum? Of the water?
TASK 4: Solve a Similar Problem In Which the Question is Omitted
An office building and an adjacent hotel each have a mirrored glass facade on the upper portions. The hotel
is 50 feet shorter than the office building. The bottom of the glass facade on the hotel extends 15 feet
below the bottom of the facade on the office building. The height of the facade on the office building is
twice that on the hotel.
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20
TASK 5: Solve a Similar Problem Which Contains Inconsistent Information
A mountain climber wishes to know the heights of Mt. Washburn and Mt. McCoy. The information he has is that
the top of Mt. Washburn is 2000 feet above the top of Mt. McCoy, and that the base of Mt. Washburn is 180
feet below the base of Mt. McCoy. Mt. McCoy is twice as high as Mt. Washburn. What is the height of each
mountain?
TASK 6: Solve a Similar Problem Which Contains the Same Implicit Information
A freight train and a passenger train are stopped on adjacent tracks. The engine of the freight is 100 yards
ahead of the engine of the passenger train. The end of the caboose of the freight train is 30 yards ahead of
the end of the caboose of the passenger train. The freight train is twice as long as the passenger train.
How long are the trains?
TASK 7: Solve a Similar Problem that is a Generalization
In constructing a tower of fixed height a contractor determines that he can use a 35 foot high base, 7 steel
tower segments and no aerial platform. Alternatively, he can construct the tower by using no base, 9 steel
tower segments and a 15 foot high aerial platform. What is the height of the tower he will construct?
TASK 8: Solve a Similar Simpler Problem
Green Lake and Fish Lake have surfaces at the same level. Green Lake is 3 times as deep as Fish Lake. The
bottom of Green Lake is 40 feet below the bottom of Fish Lake. How deep are the two lakes?
TASK 9: Make Up a Problem Which has a Similar Solution Method
The nonstandard format of the tasks provided opportunities to observe solvers as they faced problematic
situations.
For example, even though solvers might construct a solution to Task 1, they could conceivably
face problems while solving later tasks despite recognizing that similar solution methods are involved
(e.g., solvers could face a problematic situation while solving Task 3 if they try to do exactly the same
thing as they did in solving the earlier tasks).
Hence, such situations provide opportunities for solvers
to develop greater understanding about their solution activity.
In addition, the similarity among the tasks
allowed opportunities to observe how the solvers' newly constructed conceptual knowledge influenced
subsequent solution activity in similar situations (i.e., development of control of solution activity).
Using the written and video protocols, the analysis proceeded from detailed observation of the ways the
solvers resolved situations they found to be genuinely problematic while solving the tasks.
The solvers
were inferred to have experienced such situations when their initial anticipations of what to do in solving
a particular task proved incorrect when solution activity was carried out and novel activity was required.
In this way, the analysis focused on qualitative aspects of the solvers' solution activity (i.e., changes in
their anticipations and reflections) which indicated that constructive activity had occurred.
Based on the
results of the qualitative analysis, detailed written case studies of the solvers' performance were prepared.
RESULTS
Analysis of the solvers' solution activity indicated a gradual building up of their structural knowledge
they solved the tasks.
Procedures constructed while solving the earlier tasks were elaborated upon as
solvers solved later tasks.
This constructive activity was characterized in terms of distinct levels of
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as
solution activity.
Four increasingly abstract levels of solution activity were inferred from the solvers'
The levels are summarized in Table 2.
performance.
Table 2:
EXAMPLES
DEFINING ATTRIBUTES
LEVEL OF ACTIVITY
Structural Abstraction
Re-Presentation
Levels of Solution Activity
Solver can "run through"
Solver can draw inferences
potential solution activity in
from results of potential
thought and operate on its
activity without the need to
results
carry out solution activity
Solver can "run through" prior
Solver can anticipate
solution activity in thought
potential difficulties prior
to carrying out solution
activity
Recognition
Solver encounters new
Solver recognizes diagrammatic
situation and identifies
analysis activity as appropriate
activity from previous tasks as
for solving Tasks 2-9
relevant for solving current task
Instrumental
Solver demonstrates fragmented,
Solver uses mechanical coding activity
unreflective solution activity
as part of a translation strategy
The following paragraphs include episodes from the case study of solver KB and serve to illustrate
examples of the different levels of structural knowledge demonstrated by the solvers.
The solver's performance during the interview can be summarized in the following way.
struggled to construct a solution to Task 1.
The solver
She initially pursued a strategy where she coded all
information contained within the problem statements.
When this approach did not lead to a solution, she
pursued an alternate solution method incorporating a gecnetric approach (i.e., diagrams of the lakes were
constructed and relevant lengths from the diagrams were translated to a vertical axis which served as a
reference aid in constructing relationships).
This solution activity led to a correct solution and resulted
in the construction of an initial recognitionary structure.
Solution activity performed while solving Tasks
2-9 enabled the solver to elaborate and refine the initial structure, achieving higher levels of abstraction
and control with each successive task.
The following paragraphs describe this development.
The solver's initial attempt to solve Task 1 could be described as an unreflective, instrumental
approach (i.e., she did not appear to reflect on or think about the nature of potential solution activity
Prior to carrying it out).
She initially interpreted the task as a routine algebra word problem and
proceeded to code all information without attempting to develop a deeper understanding of the situation.
S:
That strikes me as an algebra problem with 2 variables.
So the first thing I should
do is assign variables to everything that is important.
She constructed a diagram and proceeded to generate all possible algebraic relationships.
22
Symbols
representing variables were manipulated in a mechanical fashion as the solver tried to code and relate
everything in the problem without reflecting to the extent necessary to consider whether such assignments
were relevant in finding a correct solution to the problem.
This activity resulted in the generation of
algebraic equations which she later found to be inappropriate.
S:
I have 4 unknowns and 3 equations.
And that's not good enough for me to solve an
algebra problem.
The solver realized she faced a genuine problem and proceeded to pursue an alternate method of solution.
She abandoned her unreflective approach (where symbols were manipulated mechanically without regard to
possible relationships) in favor of a more relational approach (where reflection upon entities signified by
the symbols led to the construction of a viable solution method).
This reflective approach was indicated by
the solver's conscious intention to use the drawing as an interpretive tool that would aid her
conceptualization and elaboration of potential relationships.
S:
I am going to look for a geometrical relationship for my drawing which I am going to
redraw because this is not accurate.
S:
This is the bottom, this is the surface of Blue Lake and this is the bottom of Blue
Lake.
This distance is 12 and this distance is 35.
twice that whole distance.
S:
And this whole distance is
(LONG PERIOD OF REFLECTION HERE)
Okay, if I label this whole distance X ... I can say ... that 12 plus X plus 35,
which is the height of Clear Lake, is going to equal twice X.
And that's the
relation in one variable I can solve.
S:
And the relation I was missing here is the fact that I'm looking at differences in
height, not absolute height.
This constructive activity culminated with the generation of an appropriate algebraic equation for the
problem, albeit an incorrect one (i.e., she made an error in her diagram).
This algebraic relationahtp
expressed a viable cohesive solution method rather than isolated relationships that corresponded to
fragments of the problem statement.
Upon discovery of an error in her diagram, the solver reconceptualized
the problem and generated a new algebraic equation which led to a correct solution.
S:
The bottom of Lake, ... and this lake is 12 feet above
the bottom of that lake.
So I didn't draw it that way.
S:
That means that my geometrical solution is probably off.
S:
So, the distance between these two is still 35.
S:
Yeah, but X doesn't mean the same anymore.
S:
So, 35 plus X equals 24 plus 2X.
S:
So Clear Lake is equal to 35 plus X which is 46.
11 which is ... 23.
I drew it 12 feet below.
The distance between these two is 12.
So 35 minus 24 equals ... X.
And Blue Lake is equal to 12 plus
That's the solution!
The solver's solution activity for Task 1 involved the construction of novel relationships which
expressed an initial conceptual structure.
This activity was novel in the sense that it involved meaning
making activity in genuinely problematic situations.
structured her solution activity.
The result of this novel activity was that the solver
Given this initial implicit initial structure, solution activity
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performed while solving Tasks 2-9 gave rise to opportunities for the solver to elaborate and
reconceptualize the relationships she constructed while solving Task 1.
To say that the solver constructed a conceptual structure for her solution activity while solving Task
1 is evident from her initial anticipations as she solves Task 2.
At this point her structure was primitive
in the sense that while she could recognize the appropriateness of using similar solution activity, she
could not anticipate a potential problem suggested by the additional information contained within the
problem statements.
S:
The first thing that strikes me is that this problem is alot like the previous one.
S:
And ... I think it would serve me well to start off in this one by just drawing a
picture.
The gradual discovery of the superfluous information puzzled the solver, suggesting that her initial
anticipation was based on a recognition of the relevance of activity similar to that which she had just
completed (i.e., at best she could only recognize diagrammatic analysis of the type performed in Task 1 as
appropriate to the new situation and could not anticipate potential difficulties).
She paused to reflect on
the situation.
I:
What are you thinking?
S:
I'm thinking that this line drawn due north doesn't seem to have anything to do with
the problem.
While the situation appeared to constitute a minor problem for her, she was not able to state with certainty
that the added information was indeed irrelevant.
She eventually chose to ignore the information ("So I'll
just look at the other relationships first") and constructed a solution.
Solution activity performed in Task 3 indicated that additional constructive activity had occurred and
that the solver had reorganized her structure.
construct a diagram.
After reading the problem statements, she proceeded to
The solver initially anticipated that she would use the same procedures that she had
used while solving earlier tasks.
However, she anticipated a potentially problematic situation soon after
constructing a diagram yet prior to carrying out the solution method.
S:
And here's the water level, here's the oil level.
S:
And the water level is 15 feet above the oil level.
S:
So solve it ... (ANTICIPATION) ... the same way. ...(ANTICIPATION)
Impossible:
The suddeness with which she was able to anticipate potential difficulty suggests that she had attained a
level of reflective activity not demonstrated while solving prior tasks (more precisely, she had "run
through" the potential solution activity in thought and could "see" the results as being problematic).
Further, this reflective activity served as a driving motivation for subsequent solution activity.
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It strikes me suddenly that there might not be enough information to solve this
S:
problem.
So I better check that. (LONG PERIOD OF REFLECTION HERE)
.
S:
I suspect I'm going to need to know the heights of one of these things.
S:
But I could be wrong so ... I'm going to go over here all the way through.
The solver spent much time and energy pursuing the elusive information.
She finally concluded that the
problem, as stated, could not be solved.
Tasks 2 and 3 presented opportunities for the solver to reflect on, elaborate, and generalize the
procedures she developed while solving Task 1.
In each case, the solver gave initial meaning to the task
she faced by assimilating the new situation to a conceptual structure that functioned at the level of
recognition (i.e., she recognized that the activity she performed in solving Task 1 was relevant for solving
Tasks 2 and 3).
In resolving problematic situations while solving Task 3, the solver was inferred to have
reorganized the structure at a higher level of abstraction (i.e., at the level of Re-Presentation).
The
solver appeared to further develop her structure as indicated by her solution activity in subsequent tasks.
The solver demonstrated this more abstract structure while solving Tasks 4 and 9.
Task 4 required the solver to construct a problem she could solve.
In constructing a problem to
solve, the solver reflected on potential solution activity in a powerful way which was not evident in
earlier tasks.
S:
The things they could ask for are things like ... (ANTICIPATION) ... the height of
one of the buildings but ... (ANTICIPATION) ... there's not enough information to
get that.
S:
...
(ANTICIPATION) ...
The only thing we have information about is ... (ANTICIPATION) ... Ah, the relative
heights of the two facades.
S:
So, if I were ... if somebody wanted me to solve any. problem, that's probably what
they're asking for.
This episode illustrates the solver's developing flexibility and control of her solution activity.
development continues throughout the remainder of the interview.
This
The solver's solution activity in Task 9
indicates that she had reorganized her stricture (i.e., at the level of Structural Abstraction) to the
extent that she could reflect on her potential solution activity and anticipate its results without the need
to carry out the activity.
The task required the solver to construct a novel situation which had a similar
solution method to the prior tasks.
S:
Okay, ... (ANTICIPATION) ... I'm thinking of something with different heights.
S:
Oh, ... (ANTICIPATION) ... bookshelves in a bookcase.
S:
No, ... (ANTICIPATION) ... that's no good. ... How about hot air balloons!
The solver ran through potential solution activity for the particular situation she proposed (i.e.,
bookshelves) and anticipated its results (i.e., that it would not work for "bookshelves" but that she could
solve it for "hot air balloons").
So, her structure allowed her to run through potential solution activity
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40
in thought, produce its results, and draw inferences from the results.
She routinely constructed
appropriate algebraic relationships and completed the task.
CONCLUSIONS
The study was exploratory and future work needs to focus on the following areas.
First, the
characterization of conceptual structures as actively constructed by solvers suggests the importance of self
Problematic situations were not given to solvers.
generated solution activity.
Rather, they were self
generated in the sense that they arose as solvers tried to achieve their goals and purposes.
In addition,
the solvers' ability to transform initial conceptual structures into more abstract forms was made possible
by the solvers' ability to generate new material to reflect on when they faced such situations.
results of the study suggest a relationship between cognitive and metacognitive activity.
Second, the
The cognitive act
of expressing their structure in new situations and the ways that they resolved problematic situations that
they faced along the way had a powerful influence on the solvers' subsequent solution activity performed
while solving later tasks.
More precisely, they were able to anticipate what it was they were to do and the
result of doing it before they carried out the activity.
In metacognitive terms it can be said that
planning and monitoring activity (i.e., anticipations about potential activity) developed as a result of the
solvers performing specific cognitive acts (i.e., the expressing of their structural knowledge in new
situations and the resolution of problematic situations in which they found themselves).
The crucial point
here is that their developing ability to monitor and plan their solution activity was made possible by their
cognitive advances.
This calls into question the notion that metacognitive skills can be treated as a
separate level of cognitive functioning (Brown, 1988).
REFERENCES
Brown, A. L. (1988). Preschool children can learn to transfer: Learning to learn and learning from example,
Cognitive Psychology, 20, 493-523.
Brown, J. S., Collins, A., Duguid, P. (1989). Situated cognition and the culture of learning, Educational
Researcher, 18-1, 32-42.
Cobb, P. , Yackel, E.
solving.
, & Wood, T. (1989).
Young children's emotional acts while doing mathematical problem
In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspective.
New York, NY: Springer-Verlag.
Kamii, C. (1985). Young children reinvent arithmetic: Implications of Planet's theory. New York: Teachers
College Press.
Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. New York, NY:
Cambridge University Press.
Pask, G. (1985). Problematic situations, Cybernetic, 1, 79-87..
Vergnaud, G. (1984, April). Didactics as a content oriented approach to research on the learning of physics,
mathematics and natural language. Paper presented at the annual meeting of the American Educational
Research Association, New Orleans.
von, Glasersfeld, E. (1987).
Learning as a constructive activity.
In C. Janvier (Ed.), Problems of
representation in the teaching and learning of mathematics. (pp. 3-17).
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Hillsdale, NJ: Lawrence Erlbaum
FROM ARITHMETIC TO ALGEBRA: NEGOTIATING A JUMP
IN THE LEARNING PROCESS
Anibal CORTES, Gerard VERGNAUD, Nelly KAVAFIAN
How can teacher and students negociate the move from
arithmetic to algebra during the very first phase of introduction
to algebra. The first problems that can be put into equation and
solved by algebraic means can also be solved by arithmetic.
Therefore some scaffolding and tutoring must be offered to
students for them to accept to deal with equations; unknowns, and
the transformation of equations.
INTRODUCTION
The learning of algebra constitutes a significant
epistemological jump for secondary school pupils. By this we mean
that the pupil has to shift suddenly from one state of
mathematical knowlege to another by rapidly assililating new
notions and procedures: (unknown, variable, equation, function,
graphic representation...) which build on previously acquired
knowlege but which require entirely new types of thinking.
Passing from elementary arthmetic to algebra, pupils will have to
substitute for the iterative treatment of problems stated in
natural language, the manipulation of algebraic expressions
according to explicit rules
a procedure which gives rise to a
(
succession of equations).
How to start teaching algebra? with which types of
problems? The answer is not immediate. In the course of a
previous experiment, we set pupils simple problems which put in
the form of equations; led to equations of the type a+x=b, ax=b
and ax+b=c. These problems are, in fact, easily solved through
arithmetic. Therefore, putting problems into equation form and
the algebraic treatment of equations is initially a response to
the teacher's request. Pupils learn,
for sure, but the
introductory process is slow and rests entirely on the pupils
acceptance of the didactical contract.
Algebra takes on a much clearer meaning in the solution
of problems which are insoluble or difficult to solve trough
arithmetic. Problems with two unknowns are generally good
examples by may be it would be setting too high a hurdle to start
the study of algebra with this type of problem.
Problems with one unknown which in the equation form
require an equation where the unknown appears on both sides (of
the type ax+b=cx+d) gives rise to serious difficulties for
beginners (we shall treat them only in the second didactical
sequence). Consequently we have chosen an intermediate approach
by starting the study of algebra with a problem with one unknown
giving rise to an equation of the type ax+b=c followed quickly by
problems with two unknowns. The present paper es concerned only
with the very first phase of introduction to algebra. We propose
to make a detailed analysis of the observed processes in 7thgrade (28 pupils) and 8th-grade (30 pupils).
Which are the conceptual difficulties encountered when first
working with algebra?
The concept of the equation: the fist step in solving a
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42
problem algebraically is to express it as an equation. This
consists in making explicit the mathematical relationship between
the unknown and given data in order to find a value for the
unknown. This tool-like characteristic of the equation may be
visible to the pupil when the equation is put in the form x....
it
is not visible when x
is incorporated in the analytical
expression of the relation ax+b=c.
Most pupils are not familiar with the concept of the
equation. For them an equation is an abbreviated way of writing
the terms of the problem: a summary. The purpose of the equation
largely escapes them. Arithmetic formulations are generally used
as a mnemonic device for arithmetic calculations. This cannot
easily be applied to algebra since it es necessary to work with
an unknown.
The concpt of the unknown: The concept of the unknown
is closely related to the concept of the equation. These two
concepts are constructed in parallel. One gives meaning to the
other and vice-versa. A broad definition of the unknown would be:
"what is not known in the terms of the problem". But one tacitly
calls "unknown" something that was to be calculated by jumping
over the problem of intermediate unknowns. In algebra the unknown
is symbolised by a character which represents an unknown number
(in the solutions of problems one should rather speak of an
unknown magnitude). One can see that characters written by pupils
can sometimes symbolise an object or a unit rather than a number
or a magnitude.
The meaning of the "=" sign. The "=" sign may have
several meanings:
a) It introduces a result. The "=" key of a pocket
calculator carries this meaning (it serves the purpose of
introducing the result by making it appear). Similarly; in the
most common usage of formulas, in V= L.l.h, for example, the "="
sign introduces the way to calulate V. For many pupils the "="
sign exclusively carries this meaning, which can sometimes lead
to writing incorrect equalities. For example, in 70-25 = 45+47 =
92-52 =.40, the number following the "=" sign is the result of
the algebraic sum expressed on the left.
b) Equivalence. In algebraic equations the "=" sign has
the following meaning: what is on the right of the "=" sign is
equivalent to what is on the left for an appropriatly selected
value of the unknown. This meaning takes shape at the same time
as the concept of equation and unknown.
c) Identity: For example,
in the transformation of
literal expressions.
d) Specification or definition. For example, in
f(x)= 2x + 52, the "=" sign introduces the analytical expression
of this function.
The homogeneity of the equation: in the expression of a
problem (of physics for example) the homogeneity of the written
terms of the equation is controlled. Now,
it is not at all
obvious to secondary school pupils that the terms of an equation
must be homogeneous; addition of values of the same kind (same
units, same meaning). We chose to ask our pupils to write the
units of the data from the very first session of study; this
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43
constitutes a first approach to the control of homogeneity. Later
these pupils will be confronted with problems in which control of
the units is not sufficient: for example, one does not add prices
and profits, neither weights and prices.
Numbers, the treatment of numbers: An average 7th-grade
pupil is supposed to be capable of operating within the D+ set;
he hardly knows fractions and directed numbers. In the 8th grade,
pupils are supposed to have become acquainted with fractions and
directed numbers. Nevertheless they also have many problems. Now,
the processes of putting into equation form and solving equations
algebraically call for a thorough grasp of operations with
numbers, especially with directed numbers. For example, it might
be necessary to multiply or divide by a negative number.
Algebraic calculation
the "detour" behaviour. One of
the most important aspects of the jump between arithmetic and
algebra is the acceptance of the "detour" behaviour: the pupil
must accept not to attempt immediatly to calculate the unknown or
intemediate unknowns (to put the problem into equation), accept
to forget the meaning of values and relationships represented by
algebraic expressions (succession of intermediate equations)
accept to rely on operations on written symbols which may not
have an arithmetical meaning, and nonetheless trust that the
solution thus found is both interpretable and correct:
conservation of the equality and the solution throughout the
algebraic calculation.
FIRST SITUATION
Putting into equation the first problem and solving it
algebraically require the use of concepts and procedures which
are barely understood or entirely misenderstood by our pupils
(equation, unknown, succession of equivalent equations,
conservation of the equality...). The gap between the problem to
be solved and the pupil's knowlege creates a paradox which can
only be resolved through the teacher's tutorial activity.
Tutorial activity: here the tutorial activity offered
to pupils consists in braking down into stages the process. This
way of providing guideposts for the task is designed on the one
hand to help define the steps in the process and on the other
hand to discourage the search for an arithmetic solution. At each
stage the pupil will be faced with a particular difficulty while
the observers will have the opportunity of establishing a
discussion with him. During the experience each class is
distributed into groups of four pupils (five groups in 7th-grade
and four en 8th-grade); a larger group is under the
responsability of the principal teacher (8 pupils in 7th-grade
and 14 pupils in 8th-grade).
The terms of the first problem are the following: On a
pair of scales in equilibrium, we have identical marbles and
weights labelled thus:
nrn000
1
soos
f vi
I
,13..
50,_
29
44
200 Qt0
a
SO4
a--
Item a) Write the equation which you think represents
this perfectly balanced pair of scales. The unknown in the
problem which we are going to calculate is the mass of a marble.
Most pupils launch themselves into an arithmetic
solution. The observer-teacher then ask the pupils in both
classes to write the equation. Most pupils then produce (and
think of) the equation as a summary of the terms of the problem.
To the question "what is the use of an equation?" most pupils
reply "it translates a text", "it simplifies like a shorthand"...
Nonotheless, a few pupils recognise the equation's properties as
a tool: "the unknown can be put on one side and it is possible to
calculate it more quickly". The results obtained are the
following:
Algebraic
6x + 500g + 50g =
6x + 550g =
=
6x + 500 + 50
6x + 550
=
equations
1Kg + 200g + 200g + 50 g
1450 g
1000 + 200 + 200 + 50
1450
8th
7th
7
6
9
2
2
1
'TOTAL
19
Neither 8th-grade nor 7th-grade pupils have ever solved
problems through algebra. However, 8th-grade pupils are more
capable of writing algebraic equations. The degree of elaboration
of the written mathematical expression is greater in the 8thgrade as well: 10 pupil write a reduced form of the equation.
We cannot be sure that the written characters (x,y,z,a)
have a correct meaning in all cases. Indeed the pupils can use
such symbols due to acquired training without for all that having
a correct representation of the unknown (the mass of a marble).
We have noticed that the pupils who write "x" have a tendency to
read their equations as
an equivalence between values
(equivalence of masses in our case). On the other hand, pupils
who use a symbol which is "closer" to the object referred in the
terms of the problem ("marbles, "m" or a drawing) seem to
interpret their equation as a simple juxtaposition of objects of
different kinds. For example:
"Juxtaposition" of objects
6m + 500g+50g=1Kg+200g+200g+50g
6 marbles+500g+50g=1Kg+200g+200g+50g
000 000 +500g+50g=1Kg+200g+200g+50g
These symbolisations of
8th
7th
3
2
2
5
the unknown can prove
operational in an arithmetical treatment (which is performed
closer to natural language) but can produce a shift of meaning in
an algebraic treatment. For example, the pupil whose algebraic
"one marble = 0,15 Kg" wants to signify
calculation results in:
that "the mass of a marble is equal to 0,15 Kg". This pupil has
navigated between the object (marbles) and the property of the
object that we wish to calculate (the mass of a marble).
8th
Equivalence of masses
6 masses of a marble +500g+50g=1Kg+200g+200g+50g
1
550g (6x) = 1Kg450g
6x+500g+50g is equal to 1Kg+200g+200g+50g
1
6y+(M=500g) +(M=50g) = (M=1Kg) + (M=...
.. 1,450Kg. On the other we have 550g and 6x
30
45
7th
3
2
1
The first equation is an equivalence of masses and very
probably the others are too; but algebraic notation is missing.
The second equation serves as the base for an arithmetical
calculation; in the third, the pupils resort to natural language
to express the equivalence. The new meaning of the "=" sign is
unknown to them. The fourth and fifth expressions are close to
natural language.
1
2
1
The "x"
7th
8th
Absence of the coefficient of the unknown
x +500g+50g = 1Kg+200g+200g+50g
1Kg450g
x 550g
500g+50g+1Kg+200g+200g+50g
x =
o + 550g = 1450g
1
1
1
in the first equation represents the mass of
six marbles: the pupil is therefore using an intermediate
unknown. The second expression resembles a "reduced" drawing and
serves as the base for an arithmetical calculation. The third
line is particularly interesting because the expression resembles
a formula: on one side theme is the unknown that we wish to
calculate; on the other are all the terms of the problem; in the
middle is the "=" sign which introduces a result. In order to
solve the problem,
this pupil has to write a mathematical
expression which he is not familiar with; he prefers to write the
mathematical expression he knows while neglecting the meaning of
the equality.
Four 7th-grade pupils calculate the value of the
unknown and then write a numerical equality: 900+550 = 1450.
These pupils have been able neither to make use of the adult's
tutoring nor to calculate the unknown by algebraic means. This
demonstrates the relevance of our thesis: it is necessery to
discourqge solutions by arithmetical means. Finally, a 8th -grade
pupil writes a false equality because he does'nt state the units
of the date: 6x + 500 +50 = 1 + 200 + 200 + 50
In their intuitive approach to the concept of equation
as a shortened of the problem, most pupil state units. We have
tried to include writing units in the didactical contract: with
the aim of intoducing control of the homogeneity of the equation.
Item b) Express all the terms of the equation in Kg.
The observers point out to the pupils that the equation
has to be reduced in order that it can be used to calculate the
unknown and that the reduction of the equation requires that all
its terms be expressed in the same units. We chose to calculate
the unknown in Kg in order to emphasize the constraint imposed by
homogeneity by means of a conversion of units which requires a
certain degree of elaboration. Several pupils make mistakes in
converting which we will not mention! The observers also point
out that the equation represents an equivalence of masses. The
pupils then write the following equations:
Algebraic equations
6x+0,5Kg+0,05Kg = 1Kg+0,2Kg+0,2Kg+0,05Kg
6x+0,55Kg = 1,450Kg
6x+0,5+0,05 = 1+ 0,2+0,2+0,05
6x + 0,55 = 1,450
6m+ 0,5Kg+ 0,05Kg = 1Kg+0,2Kg+0,2Kg+0,05Kg
TOTAL
31
BEST COPY AVAILABLE
46
8th
7th
13
5
8
7
2
2
1
4
23
19
Compared to the previous item there
is
a certain
elaboration of the mathematical expression, expecially in the
7th-grade: a larger number of pupils write algebraic equations,
in particular reduced equations.
The meaning of the unknown (mass of marble)
in
discussed within the groups of pupils. We notice that those who
were using the "m" symbol do not change it. On the other hand
three 7th-grade pupils doo not use the word "marble" anymore;
they
now
write:
6
masses
of
one
marble
+
0,5Kg+0,05Kg=1Kg+0,2Kg+... (five 7th-grade pupils at all).
Tutorial activity is more effective in small groups of
four pupils
than
in larger ones which are under the
responsability of the principal teacher. The following expression
belong to pupils from larger groups.
8th
7th
6 marbles+0,5Kg+0,05Kg=1Kg+0,2Kg+0,2Kg+0,05Kg
2
2
000 000+0,5Kg+0,05Kg = 1Kg+0,2Kg+0,2Kg+0,05Kg
2
6x = 0,9
x=0,15
2
0,5Kg+0,05Kg = 1Kg+0,2Kg+0,2Kg+0,05Kg
1
6y+(500g=0,05Kg)+(50g=0,005Kg) = 1Kg+(200g=0,2Kg...
1
;
Some pupils retain symbols which are "close to the
object in spite of the teacher's remarks about the meaning of the
unknown. Two pupils who had previously solved the problem
arithmetically in grams write teir calculations in Kg (third
line). One pupil connot manage to write an equivalence with an
unknown (he had previously written x=500g+50g+1Kg+200g+200g...).
One 7th-grade pupil introduces the conversion of units into his
expression; he remains close to natural language.
The equation with units
which could be called a
physical equation since it expresses a relation between
(
magnitudes) raises the problem of the treatment of units in
algebraic resolutions. It es necessary to be able to proceed to
equations without units.
Item c):Write the equation of the problem expressing
each term in Kg without stating units. Let "z" (for example) be
the unknown. The unknown number.... stands for.... expressed in...
Proposing the letter "z" to denote the unknown gives
rise to a debate about the relevance of the symbols used; nonalgebraic expressions ("marbles", "mass of a marble" and drawing)
are replaced by "z". The effect of the terms in which this item
is stated goes beyond what is desirable since many pupils who had
symbolised the unknown by a letter also change.
The expression "unknown number" draws the attention of
the pupils to the fact that the unknown is a number. The mayority
of the pupils complete the blanks in the sentence by: The unknown
number z stand for the mass of a marble expressed in Kg.
Algebraic equation
6z + 0,5 + 0,05 = 1 + 0,2 + 0,2+ 0,05
6z + 0,55 = 1,450
6z + 0,55 = 1,450
8th
11
14
TOTAL
32
47
7th
and
25
9
16
27
The passage to an equivalent reduced equation is not
obvious; several pupils link the two equations with an"=" sign at
the end of the first. There is therefore a shift of meaning: the
two distinct equations become a succession of transformations; a
kind of algebraic sum. The conservation of the equality (the
passage to another equivalent equation) gives rise to serious
difficulties:
7th
8th
6x+0,55 = 1+0,2+0,2+0,05 = 6x+0,55 = 1,45
6z+500+50 = 900+500+50 = 1,450
6z+0,5+0,05 = 1+0,2+0,2+0,05 = 0,55+1,45 = 1,85
0,5+0,05 = 1,0,2+0,2+0,05 = 0,55 = 1,45
x = 0,55/6
6x+0,5+0,05 = 6x+0,5+0,05 = 0,55
0,9+0,55 = 1,450
1
1
1
1
1
;
1
Both equations are written on the same line (first line). A
7th grade pupil (second line) writes the left hand side again
replacing the unknown with its value in grams, and comes to a
"result" in Kg (the right hand side). One pupil (third line)
groups together all the reduced numerical terms after his
equation and comes to a number: a shift towards an algebraic sum.
One pupil (4th line) is not capable of reducing his equation in
the presence of the unknown. One pupil (following line) is also
unable to operate on the numbes in the presence of the unknown;
he detaches the numerical part and puts forward for "x" the value
which solves for the equation 6x = 0,55. One pupil writes a
reduced equality without unknown (last line).
Algebraic solution of the equation. Item d): By substracting
0,55 from each side of the equation a new equation is obtained.
Which one?
The required notation (6z + 0,55
0,55 = 1,45
0,55
;
6z =
0,9) leaves a trace of the algebraic working, it allows the
pupils more easily to check their work;
it permits the
construction of a script-algorithm which is used to provide
guidance in the very beginning, when resolution strategies are
lacking. This notation does not appear naturally it has to be
required; it has to be constructed.
After the definition of the word "side" and "term" the
observers justify the algebraic operation (mathematically or by
refering to the scales) which the pupils can check by arithmetic.
The work on the equation, the passage to an equivalent equation
and the required notation raise problems (the observers
occasionally have to write it). Some pupils use this notation,
others write the resulting equation directly:
6z+0,55-0,55 = 1,45-0,55
6z = 0,9
6z = 1,45-0,55
6z
;
=
0,9
8th
7th
15
8
9
6
6z
;
=
0,9
1
Besides the notation, some pupils have problems writing two
distinct equations:
6z+0,55-0,55 = 1,45-0,55 = 0,9
6z +0,55 -0,55 = 1,45-0,55 =
;
;
6z = 0,9
6z = 0,9
6z = 1,45 - 0,55 = 0,9
33
48
3
-
1
4
2
1
Many pupils propose to treat each side of the equation
separately, some of them write it as
6z+0,55-0,55=6z
;
1,45-0,55 = 0,9
:
;
6z = 0,9
4
2
Two pupils substract 0,55 from the coefficient of the
unknown:
5,45 x = 0,9
5,45 Z + 0 = 1,395
1
1
This type of error (related to a misunderstanding of the
order in which operations must be carried and to the weakness of
the concept of the unknown) occurs frequently in the beginning
and desappears rapidly afterwards.
item e) Divide each side of the equation by
6.
What new
equation is obtained?
The proposed notation (6z/6 = 0,9/6
z = 0,15)
and its
justification are quickly accepted because they shed lighjt on
the step to the final equation: z = 0,15. The equation 6z = 0,9
is easily solved through arithmetic; pupil may therefore rely on
it as a mean of checking their work. Most of the pupils write
;
required notation others do not.
8th
6z/6 = 0,9/6
z = 0,15
z = 0,9/6
z = 0,15
z = 0,15
6z/6 = z
0,90/6 = 0,15
;
17
;
2
;
;
7th
21
11
6
30
28
z = 0,15
1
TOTAL
Later, in the course of the experiment, the majority of the
pupils adopt this notation.
CONCLUSION: We have dealt with the detailed analysis of the first
hour of the sequence rather than treating the whole of the 15
hours superficially. This first problem shows the beginning of a
conceptual construction which stretches over several years of
learning.
It seems important to point out the part played by the
organisation of the mathematical concepts in a conceptual field
(here, we have treated a small part of the whole). The study of
the conceptual field together with the epistemological approach
and the data concerning the pupils'difficulties allowed us to
analyse the mathematical contents as proposed to the students and
viewed by them; it also helped us to understand their behavior
and to build didactical sequences while keeping control.
A good collection of papers can be found in "The ideas of
1988
National Council of teachers of
mathematics USA, (especially the paper of Carolyn Kieran and the
paper of Zalman Usiskyn). We have also used the ideas of Y.
algebra, K-12
;
;
Chevallard and E. Filloy.
49
34
CONTINUOUS ANALYSIS OF ONE YEAR
OF SCIENCE STUDENTS' WORK, IN LINEAR ALGEBRA,
IN FIRST YEAR OF FRENCH UNIVERSITY
DORIER JEAN-LUC
Equipe de Didactique des Math6matiques, GRENOBLE (France).
Abstract
Linear algebra is one of the newest fields students discover in their first year at
university. Its abstract nature is often a problem for them. We wanted to know if notions in
basic logic are prerequired to succeed in linear algebra, and if yes what kind of previous
abilities are needed. We wanted as well to have a better appreciation of what teaching linear
algebra consists of and what kind of effects it produces on students. In the following article
we describe the methodology employed to analyse results of standard first year science
students in a pretest about basic logic and algebra notions, and in all the tests given to them in
linear algebra during a year. Then we try to answer the questions raised above with help of
this analysis and its statistical results. Finally we will propose a new organisation of teaching
linear algebra according to our hypotheses.
1- Introduction
The research presented in this paper is based on the analysis of results of the tests given
all the year through, in the field of linear algebra, to students in their first year at a French
university.
Our main goals were :
To better determine what teaching linear algebra consisted of, especially through the
analysis of tasks proposed to students, within the questions given in the tests.
- For a standard section of first year science students with a fairly standard teaching of
linear algebra, we wanted to determine the methods, procedures and mistakes of students in
relation to the tasks proposed to them and relatively to their individual previous abilities in
basic logic and algebra notions.
- Being then able to draw a diagnosis on the different effects of this teaching, we may
propose some hypotheses for its possible change.
2- The methodology and the hypothesis
We analysed copies of eight different tests.
We first took eighty-four copies from a pretest on basic notions in logic and algebra.
This had been given to students in their first weeks at university, before any specific teaching
in these fields. The evaluation of this test gave us the individual level of acquisition to what
we thought may be prerequired for linear algebra.
For the analysis itself, we used a methodology introduced by A. Robert and F.
Boschet, in their work on the acquisition of real analysis notions in first year at a science
university ( [1] et [2] ).
35
Among the questions on the test, we sorted out four main types : quantification (QA),
implication and equivalence (EQ), numerical algebra (AN) and algebraic structure (AS). For
the first two types we distinguished three different levels, in the tasks induced by the
questions. The first one is a purely formal setting, but seen from the outside, since it is asked
to say whether a proposition expressed in formalised language is true or false (QA1 and
EQ1). The second one is formal as well, but the task is this time internal, since it is asked to
give the negation of a formalised proposition.Only the QA-type of questions appeared at this
level (QA2). The last level corresponds to an interplay between the formal setting and another
setting, in the meaning introduced by R. Douady [3]. The questions, this time, consist in
translating a proposition from a formalised language into an every-day or a graphic
formulation, or vice-versa (QA3 and EQ3).
So we obtained seven different types of questions, which we can associate to seven
different bodies or "blocks" of knowledge.
The hypothesis we made and which is induced by Piaget's work, is, in outline, that the
acquisition of new knowledge is usually made possible by the destabilisation of old
knowledge followed by its reorganisation through complex cognitive mecanisms of
destabilisation/reequilibration. The necessary destabilisation is not usually part of the explicit
teaching, and the process described above is then of course unconscious. Yet, R. Douady [3]
showed that (at least for primary school pupils), didactical situations, in which a notion,
meant to be taught, may be seen in at least two different settings, in which the pupils have
different levels of ability, is suitable to start this dialectical process in good conditions.
After A. Robert's and F. Boschet's work ([1] and [2]), we think that former
knowledge, efficient in different settings, may be a better guarantee for the acquisition of a
new notion. More precisely we may raise such questions as : will a student who is very good
at formal logic (EQ1, QA1 and QA2) but not very good at dealing with the interaction
between formal and natural languages, learn linear algebra less well than a student , who is
globally of the same level in logic, but having more homogeneous abilities ? For every
prerequired block of knowledge, is there a minimal threshold of acquisition beyond which
the probability of success is much higher ?
To be able to answer these questions, we have defined for every block, three different
states : full (2), half-full (1), empty (0), according to a mark given to the questions related to
it. We have also considered the parameter B, giving the number of empty blocks, which
mesure the number of gaps in previous abilities
We then obtained nine different variables (the global mark of the test, the seven blocks,
and the number of empty blocks), which evaluate the level of acquisition of basic logic and
algebra notions, for each student. A statistical study of the results of the tested population,
led us to build a new block Q, with the QAi, to summarise the level of different abilities in
questions dealing with quantification.
We finally kept only the following nine variables, which we show to be the most
significant ones : the global mark N, the number of empty blocks B and seven variables
51
36
(being 1 or 0) for the blocks : EQ1 (2) (full), EQ3 (2), EQ3(0) (empty), Q(2), Q(0), AN(2),
AS(2). This seemed to be, with minimal loss, the best way to keep information compact
enough and suitable to our further purpose.
Nevertheless in addition to the specific methodology developed here, some restrictions
about the test itself, are to be considered, to give the real value of this evaluation, which is of
course only a partial way of considering the contents as well as the level of acquisition of
preriquired notions in basic logic and algebra. Indeed, if the questions about logic, in the
test, seem to be suitable, although necessarily incomplete, the ones about algebra appeared to
be less satisfaying : numerical questions were a bit too imprecise to give a good evaluation
and the ones about structure were too "cultural" to give a real idea of the level of acquisition
(for instance : asking someone to give an example of a group is not enough to evaluate his
knowledge about groups).
The seven other tests were : four "ordinary" two-weekly tests, the mid-term and the
final exam, and a special mid-course true/false-test. Except for the last one all these tests
included questions on real analysis subjects.
For each of these tests, we made an a-priori analysis, which includes a explanation of
the tasks induced by the questions and the different procedures that could possibly be
developed by students. We then gave the statistical results, with marks given to every
question and codes to identify special procedures, which we gathered in a table, whose
arrays represent the students. We also obtained a global mark for each test. We divided every
sample into three categories, according to these marks, we managed to balance the
distribution numerically.
We analysed, in this order, thirty-nine papers from the first ordinary test (T1),
seventy-four from the mid-term exam (El), forty-six from T2, fifty-eight from the
true/false-test (TF.), fifty-eight from T3, fifty from T4 and seventy-three from the final exam
(E2). Apart from the mid-term and final exams, none of these tests were compulsory, besides
we had to photocopy the papers in the short time while the correctors had them ; those two
material reasons explain the difference in numbers of papers analysed for each test. In the
end, we got unfortunately only nineteen complete sets of papers of the eight tests.Each paper
analysed corresponds to a student whose pretest we have analysed anyway, so that the
students analysed at each test form a sample of the main population analysed for the pretest.
For each test, we made a short analysis of the new data obtained for the pretest with the
new sample. We compared the mean-value of marks, their standard deviation, the
percentages of students having EQ1(2), EQ3(2), EQ3(0), Q(2), Q(0), AN(2), AS(2), B=0,
B.1; 1352, with the equivalent data for the whole population. In each case, we noticed only
little variations, which always have rather obvious explanations. The samples imposed on us
under material circumstances seem then to be representative enough of the whole population,
to give a certain validation to our general conclusions.
For every test we finally gave a crossed table, giving for each of the three different
groups of students defined by the mark of the test, the mean value and standard deviation
37
52
from the pretest, and the distribution of the same ten variables as above. We gave a table with
percentage on the line and one with percentage on the column, which gave an easily read
representation of the correlations between each test and the pretest.
Finally, we analysed more precisely the results of all the tests (including the pretest) for
the nineteen students, whose eight papers we had . We made several factorial analyses
(Analyse en Composantes Principales) of some of the different characters definied on the
sample, although the small number of students did not allow us to make a real statistical
analysis. Nevertheless, we got quite a lot of information on every students, which would not
have been possible with too many students. More over, we took the results as they appeared
in a real teaching situation, with all its complexity. This kind of analysis, for linear algebra
had not been made before, as far as we know, in France. So we claim that our work, was a
necessary step before carrying out a statistical analysis over many more students. To be able
to look at the correlations between the different components of the knowledge in linear
algebra over a large statistical population (a few hundred), we have to be more familiar with
the contents of the teaching, the different tasks and procedures involved in linear algebra, and
we must be able to draw some hypotheses that will help us to build tests according to them.
We hope that the kind of analysis, we propose, meets these aims.
3 - The results
a - Global analysis of the contents of the teaching
In most French universities, first year students in science classes follow a
two-hour-a-week course over one semester, which represents more or less a fourth of their
annual teaching in mathematics. The course usually starts with the axiomatic definition of a
vector space, and finishes with the results about diagonalisation of matrices. This is of course
an average estimation. In fact linear algebra having completely disappeared from secondary
teaching, even for geometry, a new tendacy consists, in first year at university, of teaching a
bit less abstact linear algebra and a bit more linear algebra for geometry.
The abstract part of this teaching is usually feared by students, because of its esoteric
nature and by teachers, because of the bare obviousness of most reasonings, which leaves
them without arguments faced with their students' incomprehension.
On another hand, a historical study (cf also [4]), has confirmed us in the idea that linear
algebra is a simplifying and unifying concept. For this reason, it is usually very difficult, if
not impossible (?), to find "the suitable problem" to introduce a notion related to it, as we
would like to do according to G. Brousseau's "Theorie des situations" [5] or R. Douady's
"Dialectique outil/objet" [3]. There is no problem, except a few, far too complicated for
students, for which linear algebra is an absolute necessity. Besides, even if linear notions
give a more elaborated or a more general answer to a problem, it is often too subtile for
students to realise, because they already have many difficulties in using concepts, which they
are not familiar with, to be able to have a critical look at their work.
This nature, quite specific to linear algebra, leads to a dichotomous attitude in teaching,
which is reflected in two different kinds of problems.
53
38
The problems of the first kind present applications of linear notions to questions about
polynomials, functions or series... They include interplay between different settings, and
change of point of view. Most of them are both real problems and good illustrations of the
simplification and generalisation given by solutions using linear algebra, but only to someone
who has first no difficulty in using linear notions and who is secondly quite familiar with the
subject involved. For instance most of the problems of interpolation with polynomials have
very elegant and generalisable solutions with use of the theory of vector spaces, but one
needs to have quite a lot of calculations to do, to see the simplification given. Besides, in
those problems one usually needs to obtain a lot of results, before being able to reach the first
questions really concerning linear algebra. So if such problems are given to students, one
may have to deal with the following two difficulties :
1) The use of linear algebra will be only an effect of the didactical contract, as it is not
absolutely necessary to solve the problem and the students cannot appreciate the
simplification it provides. Students will follow the process of resolution induced by the
questions even if they see a solution not using linear algebra.
2) The first questions necessary to approach the linear questions may need so many
abilities in different fields that only a few students will manage to answer the questions
dealing with the notions of linear algebra. The evaluation of the result of such problems is
then more on these questions than on linear algebra.
In the second kind of problems, linear concepts are used in a formal setting without
interplay with any other setting. Those might be either very formal and difficult questions
about subtile notions, such as supplementary spaces..., or on the contrary mechanical use of
algorithms such as the search of eigenvalues and eigenvectors of a matrix... In the first case
they give useful results, although very hard to obtain, in the second case they are only
training for calculation and easily evaluated contents for tests!... These problems do not use
"real" vector space, but very general ones, mostly 1Rn.
In our analysis, tests T1, El and T2, are of the first kind.
El is a typical example. The goal of the problem was to obtain Gregory's formula,
which gives a polynomial in terms of the values of the P(n+1) - P(n) (n=0 to deg(P)). There
is a very attractive solution, through the study of the operator D : P --> Q s.t.
Q(X)=P(X+1)-P(X). Yet it is quite long and difficult, it is then really useful only for
theoritical reasons or if you need to calculate quite a lot of polynomials. In fact, most of the
students didn't succeed in proving all the steps leading to the formula, mostly through lack of
technical ability in algebraic calculations. But when they were asked in the last question, to
find the polynomials of degree less than three, whose values in 0, 1, 2 and 3 were given,
although Gregory's formula had been given, they used a direct method and solved a system
of four linear equations with the four coefficients of the polynomial as unknown!
In T1, the question was to find the polynomials of degree less than four, whose values
in 0 and 1, as well as the ones of the derivated polynomial were given. The solution induced
by the test, was to first find the polynomials, whose all four known values are 0, and then to
39
54
deduce the general solution by addition of any solution, for instance the one of the third
degree. Of course the first question is obvious, for such polynomials can be divided both by
X2 and (X-1)2, but most of the students did not realise that, and again solved a system of
four equations with five unknowns! As they have to solve another system to find the solution
to the third degree, they ended with more calculations, plus a theoritical proof, than if they
had directly solved the system of four linear equations given by the conditions.
In T2, the questions preparing the linear solution were so technical (they used
polynomials with two variables), that hardly no students had a chance to answer any question
about linear algebra.
It is clear that there is a real difficulty here.We think that such problems should be
introduced by explicit metamathematical approach and that the "technical" points they raise in
the field of algebraic calculation or logical reasonning should not be under-estimated.
TF, T3, T4 and E2 are of the second kind.T3, T4 and E2 are mostly applications of
numerical algorithms about the search for eigenvalues and eigenvectors, diagonalisation or
reduction to a triangle form of matrices ... But in T3 and E2, we find also some very
theoritical questions. The true-false test is typically about abstract notions, although nearly all
of them refer to R3. It would be too long here to develop all the results to this quite specific
test, it shows in outlines that formal questions about basic notions of linear algebra such as
linear independance, generating subsets, supplementary etc... bring to light some sharp
misunderstandings from students.
Generally one of the most obvious difficulties for students in all tasks about linear
algebra is to be able to keep control of what they are doing. This goes from the confusion
between variables and parameters in the resolution of linear systems and leads to one of the
best illustrations of it
:
in E2, students were asked to find an orthogonal basis of
eigenvectors, after they found three eigenvectors of two different eigenvalues, they proved,
in all details that they were independent, without shortening the proof to the independance of
the only two of same eigenvalue, and then that they were orthogonal.
b - The main statistical results
The factorial analysis on the seven series of marks (all but the pretest's) for the nineteen
students reveals two sets of tests : Tl, E2 and T4 on one side and T2, TF and T3 on the other
side, E2 being just between those two groups.
This is a different distribution to above. These two groups separate the calculating tasks
from the more conceptual tasks. Indeed in the first group of tests there were quite a lot of
resolutions of linear systems, asked explicitly (Tl and E2) or appearing as the suitable ways
to solve questions (determinations of polynomials, eigenvalues or eigenvectors...). T4
consists mainly in the use of algorithms for calculations with matrices. T1 and E2 use also
algebraic calculation notions for polynomial or integral. On the other hand T2 and T3 and
mostly TF deal with more conceptual problems. The final exam seems to be quite a
well-balanced compromise of the two.
40
55
This separation is given by the second factorial axis of the analysis, the first one
separates the globally successful students from the ones who failed. The distribution of
students on the first factorial plan is quite harmoniously spread out, which seems to induce
that both numerical and conceptual abilities are useful, but independent, to succeed in linear
algebra. For instance, it shows that students can globally succeed in linear algebra, without
having a good conceptual basis. For instance they can find the triangle form of a matrix
without having a good knowledge of the concept of supplementary subspace, although it is a
basic notion for the theory of matrices' reduction. This points out a contradiction in teaching
linear algebra. The choice made in most French universities' curricula to teach linear algebra
from the definition of a vector space to the diagonalisation of matrices all in one year, induces
a restriction in the teaching of basic concepts to the benefits of more easily taught and
evaluated notions such as reduction of matrices. This is of couse the effect of the difficulties
and the failure encountered in the teaching of abstract notions.
c - Correlations with the pretest
The correlations with the pretest are globally quite strong. The main correlation appears
with the number of empty blocks. This confirms our hypothesis about the existence of a
minimal threshold in the acquisition of previous abilities beyond which chances of success in
linear algebra are greater. The AN and AS blocks are not very correlated, and among the
blocks related to logic, Q is the most correlated of all. These results seem to show that a
certain level of previous abilities in basic logic, mainly abilities in the use of quantification, is
required to reach a minimal success in linear algebra.
But some results of the more detailed correlations are a bit surprising.
For instance in the true/false test, there were the two following propositions given for
any linear map f : IR3--> IR3 :
- If (U,V) are two independent vectors of R3, then (f(U), f(V)) are also independent.
If (f(U),f(V)) are independent, then (U,V) are also independent.
Many students got mixed up in the use of the definition of independent vectors so that
they say exactly the contrary of what was true. It first seemed to be a difficulty related to
logical notions about quantification and implication. But it appeared that it is only slightly
correlated with the results to the pretest. Other similar phenomena may be noticed in our
analysis. This leads us to think that logical difficulties specific to linear algebra might exist,
and cannot be solved by any former teaching in logic.
The last step in our analysis was to reorganise our data in terms of several tasks as well
as procedures in the different fields of linear algebra. We defined 23 variables and made
factorial analyses on several groups of them. This gave us answers or enlargement to local
hypotheses and the results, which could not be easily summarized, and would take too long
to be developed here. We'll try to fill in this gap during the oral presentation.
4 - Conclusions - Outlook
Presented in so few pages, this work may seem very disorganised and partial. It
compiled quite a large amount of data and had to deal with a field, which was nearly
41
56
unexplored by didacticians, so if the conclusions it drew are incomplete, it is nearly by
necessity.
We have now to answer our last goal. If notions in basic logic seem to be prerequired
for linear algebra, it seems that prerequistes extend to more general abilities in different areas
of algebraic calculations such as polynomials, integral or differential calculus etc... which are
not necessarily part of standard mathematical teaching for first year science students, and may
have specific aspects in linear questions. As some logical problems seem to be specific to
linear algebra as well, we propose the following reorganisation of the teaching of linear
algebra.
The first step would be to teach only basic notions but over a longer period and quite
separated in time from the calculations with matrices, which could be only a further part of
the teaching, not necessarily in the same year.
This first part should be illustrated through the solving of several problems dealing with
varied vector spaces. In those problems, there should be an explicit metamathematical
approach, in which the student should have an active participation (like comparing two or
more different ways to obtain the same solution with or without linear algebra). It should
include as well the explicit teaching of logic and algebraic calculation notions useful to solve
the problems.
We think that this could help to reduce the difficulties raised by abstraction as logic
would be part of the explicit teaching. Finally it seems to be a more satisfaying approach
from the epistemological point of view, as basic concepts would really appear as unifying
and simplifying notions used in various fields in which the students will be given sufficient
abilities.
The content of this paper is developed in our doctoral thesis, that should be defended
and published by the end of the year. This thesis will also include a historical presentation of
the emergence of linear algebra basic concepts and some elements for a new teaching
approach.
References and bibliography :
[1] F. BOSCHET and A. ROBERT : " Acquisition des premiers concepts d'analyse sur
R dans une section ordinaire de premiere annee de DEUG." Cahier de didactique des
mathematiques n°7. IREM de Paris VII.
[2] A. ROBERT : " Rapport enseignement /apprentissage (debut de l'analyse sur R).
Fascicule 1 : analyse dune section de DEUG A lere armee (les connaissances anterieures et
l'apprentissage).Cahier de didactique des mathematiques n°181. IREM de Paris VII.
[3] R. DOUADY "Jeu de cadres et dialectique outil-objet". These d'etat. Universite de
Paris VII-1984. Or in Recherche en Didactique de Mathematiques, Vol. 7-2, pp 5-31, 1986.
[4] J. ROBINET : "Esquisse dune genese des notions d'algebre lineaire enseignees en
DEUG" . Cahier de didactique des mathematiques n°29. IREM de Paris VII. Mai 1986.
[5] G. BROUSSEAU : "Fondements et methodes de la didactique des mathematiques".
Recherches en Didactique des Mathematiques, Vol. 7-2, pp 33-115, 1986.
42
57
AVOIDANCE AND ACKNOWLEDGMENT OF NEGATIVE
NUMBERS IN THE CONTEXT OF LINEAR EQUATIONS.
AURORA GALLARDO and TERESA ROJANO
Centro de Investigation y de Estudios
Avanzados del IPN
Mexico
Abstract.- This paper reports on the difficulties with negative
numbers displayed by 12-13 year olds in a clinical studyonlinear
equation solving. As a methodological counterpart, we describe
acknowledgment and avoidance manifestations concerning negative
two
cultures: the chinese and the greek one. In the
numbers in
conclusions, a first hypothesis is outlined with regards to possible
causes of avoidance and conditions underwhich acknowledgment arises
in individuals.
Introduction.- In the last few decades, outstanding efforts have
been made in the research field trying to elucidate the problem
of misconceptions and operative deficiencies associated to negative
numbers.
Such efforts have developed in different directions,
towards: the teaching field [e.g. Bell, A. 1,2], the psychology
[e.g. Resnik, L. et al 13], the history [e.g. Glaesser, G. 9].
In the present work we appeal to history in order to find out
explanative elements of the observations made in the clinical
study "Operating on the Unknown" concerning negative numbers.
Thus, once this history analysis is carried out, the methodological
cycle: epistemological level-clinical observation- history level
will be completed.
This methodology characterizes our research
since its very beginning, seeing that the clinical stage was
preceded by a history investigation about the pre-symbolic algebra
methods (XIII-XV centuries) for equation solving [3]. This article
comprises three sections:
1.- Difficulties with negative numbers in the study "Operating
on the Unknown". 2.- Negative numbers in two antique civilizations.
3.- Conclusions.
1.- Difficulties
with negative numbers in
the study
"Operating on the
unknown".
This clinical study was carried out in the period 1981-1986 and
analyses transition phenomena from arithmetic to algebraic thought
[4,5,6]. It consists of 22 videotaped interviews with 12-13 year
olds, who face for the first time simple linear equations with
occurrences of the unknown on both sides. A previous classification
of children in three pre-algebraic proficiency levels was made
(upper, medium and low level). Among the results reported in other
papers [6,7,8] we will refer to those related to different difficulty
areas in the learning of algebra, in particular, we will focus on
the specific area of negative numbers in which, manifestations of
avoidance are present, for instance when a negative solution is
not conceived in an solving equation process. Such an avoidance
appeared in different ways in the interviewees; for example:
1) Interprating the symbol 'X' as a positive number, cancelling
in this manner the possibility of solving equations of the type
X+1568=392 (children of all levels). 2) Changing the equation's
structure: when the written equation (in the item above-mentioned)
is read as "I have to find out a positive number such that added
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58
to 1568 sums 392", some children tended to replace the operation
sign '+' by another one. 3) Assigning different numerical values
to different occurrences of 'X' in the same equation. For example,
with the aim to achieve to a numerical identity in 4x+6=2x some
children anticipated that the 'X' on the left hand side would be
smaller than the 'X' on the right (children of the medium level).
Besides the avoidance manifestations, cases of avoidance
acknowledgment of negative numbers were detected, for example:
a girl (of the upper level) was taught to solve equations of the
type Ax + B = Cx + D (where A, B, C and D were particular natural
numbers) by means of translating the equation's elements into a
geometric situation, where figures with equivalent area were
involved:
equivalent to
A
Once the geometric model was understood by the student, she spontaneously extended it to other mode of equations, including
those with negative constant terms (Ax + B = Cx - D). She interpreted
the "negative term" as an action of "removing a piece of the
figure with an area equivalent to D". This interpretation corresponds
to considering those terms as subtrahends.
Thus, the geometric translation of 9x + 33 = 5x - 17 Was:
9
The pupil carried out the following actions in the model:
331
4
which led to the reduced equation (with one occurrence of 'X'):
4x + 33 + 17 = 0. Nevertheless, at this point of the solving process, the student showed an avoidance syntome, she kept quiet for
a few minutes because of the presence of a negative solution.
This, as it can be seen in this case, although the elements of
the equation are provided with geometric meanings and the negative
terms as well as its transposition are interpreted as removing,
adding and composing actions, this does not result in a total
acknowledgment of negative numbers. There exists an essential
difference between interpreting a negative number as subtrahend
(a
b) and conceiving it as an isolated number (possibly, as a
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44
,negative solution of an equation).
2.- Negative Numbers in two Antique Civilizations
We start with the fiu zhang suanshu (Nine chapter of the
mathematical art), one of the earliest mathematical text in
China [11]. The various mathematical concepts and techniques
embodied in the nine chapters of the text were, in fact, the
culmination of knowledge and practical experiences of chinese
mathematicians prior to the beginning and the early year of the
Christial, era.* Let us examine the eighth chapter entitled fang
cheng.
Just like the other chapters in the text, the present
version of the fang cheng chapter contains a number of problems
together with their respective solutions. Firstly, we find in
the chapter the use of negative numbers, showing that the ancient
Chinese had a clear concept of them and were able to apply it in
mathematical considerations as we would do nowadays. Secondly,
the fang cheng chapter shows the formulations and solution of
simultaneous linear equations of up to five unknowns.
Thirdly,
the fang cheng chapter introduces the methods of solving equations
by tabulating the coefficients of the unknowns and the absolute
terms in the form of a matrix on the counting board, thereby
facilitating the elimination of the unknowns, one by one.
In explaining the content of the chapter, we shall be using modern
notation. However, it must be emphasized that ancient civilization
had no ready made sets of notations. Conceptualizations were in
a verbalized form, though the Chinese took a forward step when
they used rod numerals to convert concepts onto the counting
board.
An overview of the fang cheng chapter. There are only two methods in
this chapter. The first, called fang cheng or calculation
by
tabulation, is on solving a set of equations. The second method
called the positive
negative rules (sheng fu shu), comprises
of rules for the subtraction and addition of positive and negative
numbers.
Fang Cheng The method of calculation by tabulation. Lui Hui defines
the term fang cheng as the arrangement of a series of things in
columns for the purpose of mutual verification. The number of
columns to beset up is determined by the number of things involved.
Each column has two sections; the top section consists of the
quantities a..
(i j = 1,2,...,n) of the various things while the
ij
bottom one shows the absolute terms b.(i = 1,2,...,n). Such an
arrangement on the counting board can be shown as follows:
*Neither the author non the date of the composition of this text is known to us.
commentary remained the principal source of the study for the fiu zhang.
BEST COPY AVAILABLE
95
60
Liu Hui's
Left
Top
a
Bottom
Right
0
a21
all
Thing 1
Thing 2
ant
a 22
a 12
ann
a 2n
a.
Thing n
b,
b2
b1
Absolute terms
The whole process of operation is done on the counting board
using the rod numerals to represent the various quantities**
The unique place
value feature of this method of computation
renders the use of symbols unnecessary. In each column of things
of the counting board, the space between alm and b., has the
implicit function of an equal sign.
The former matrix arrangement is transformed in such a way that
all the numbers in the upper side of the main diagonal are equal
to zero (only columns are operated on). This transformed matrix
corresponds to a diagonalized set of equations, from which all
the unknowns are successively determined.
One can see that it is essentially the usual method in present
day algebra.
Zheng fu shu
The positive
negative rules. Since the process of the
fang cheng solution is the succesive elimination of numbers
through mutual subtraction of columns, there could be cases when
a number to be subtracted from in one column is smaller than the
corresponding one in the other column. The opposite result
obtained has to be indicated and certain rules have also to be
established in order to continue the eliminating process. This
gives rise to the creation of names: the term fu to indicate the
resulting opposite amount to the term zheng for the normal
difference. The concept of zheng and fu seems to have evolved
from such ideas as "gain" and "loss" as clearly shown in Problem
8 which reads: "By selling 2 cows and 5 goats to buy 13 pigs,
there is a surplus of 1000 cash. The money obtained from selling
3 cows and 3 pigs is just enough to buy 9 goats. By selling 6
goats and 8 pigs to buy 5 cows, there is a deficit of 600 cash.
What is the price of a cow, a goat and a pig? "The text considers
** There are two types of numerals as shown below:
A
1
11
111
lilt
11111 T IT TT Tlir
EE
The
B
A
_L J_ 1 1_
type numerals is for representing units, hundreds, ten thousands, etc., while the
type is for tens, thousands, etc.
61
46
the selling price zheng because of the money received and the
buying price fu because of the money spent. The surplus amount
is considered zheng and the deficit fu. Liu Hui points out that
these terms are merely names to.indicate the nature of numbers.
For the purpose of computation, numbers described bythese terms
have to be transcribed into a concrete form. He tells us that
there are two ways of doing this with rod numerals. If different
colored rods are used, then red ones represent zheng and black
ones represent fu. Alternatively, if the rods are of one color
only, the fu numeral is indicated by an extra rod placed diagonally across its last non-zero digit. He explains: when anumber
is said to be negative, it does not necessarily mean that there
is a deficit. Similary, a positive number does not necessarily
imply that there is a gain. Therefore, even though there are red
and black numerals in each column, a change in their color
resulting from the operations will not jeopardis the calculation. Liu Hui's exposition on'negative numbers shows that he
conceptualizes them as a class of numbers in the mathematical
sense that is familiar to us today. The concept of positive and
negative, which initially evolved from opposing entities such as
"gain and loss", "add and minus" and "sell and buy"
is
now
,
detached from linguistic associations. Its development has resulted
in negative numbers being regarded as one group of numbers with
properties which are connected with the other group of "normal"
or positive numbers. These properties are defined by these posi
tive-negative rules [12] which may be represented in modern symbols as follows:
Suppose
substraction
A > B > 0, then for
t A- (t B)=±(A-B),
t A- (-7- 13)=t(A+B),
o - ( t A ) = +A
for addition
t A+(tB) =t (A+B)
t A+ (-TB) =t (A-B)
0+ (tA)=t A
Problem 8 involves selling and buying which equate to the concept
of positive and negative respectively. The corresponding set of
equations in tabulated form becomes:
5,
6
3
-
2.
5
9
8
3
600
0'
-
13
1000
As it can be seen, the fin zhang has provided substantial evidence
that, by the first century, the Chinese not only accepted the
validity of negative numbers but understood their relationships
47
62
with positive ones and were able to formulate rules and to compute
with them. Outside China, the recognitions of negative numbers
as a separate class of numbers came much later.
first
The
mention of these numbers in an occidental work is in the Arithmetica of Diophantus [10], where the equation 4x + 20 = 4 is
spoken of as absurd, since it would give x = - 4. On the other
hand the greeks knew the geometric equivalent of
(a
b) 2and
of (a + b)(b - a); and hence, without recognizing negative
numbers, they, knew the results of the operations (-b)(-b) and
(+b).(-b). In fact, we could assert that with the greek culture,
a history of avoidance of negative numbers is initiated and, it
was not until the 15th century that these numbers were gradually
accepted in their own right.
Conclusions.- The eight chapter of the fiu zhang gives the earliestgeneral method of solving a system of linear equations. By
tabulating numbers in an array, the Chinese invented a notation
and raised this branch of algebra from a rhetoric form to a
notational one.
When the fan cheng method was applied to the
various problems, it was inevitable that this led to the concept
of
a class of numbers different from the class of numbers that
was known. Thus, the negative numbers emerged from this computational language,
freed from the concrete meanings that they
used to have in the context of specific word problems. On the
other hand,
in the greek culture, the numeric domain of the
solutions of an algebraic equation was restricted to positive
numbers (probably due to their geometric interpretation of the
elements of the equation). This led to avoidance manifestations.
Nevertheless it can be said that the greeks had a partial
acknowledgment of these numbers, since the geometric language
admitted the subtraction of areas.
The different conceptions
extracted from these two cultures provide elements to build up
hypothesis worthy of being proved at the level of individuals,
concerning the possible causes of avoidance and the conditions
which may further a full acknowledgment of integer number.
Considering the clinical evidence as well as the history findings, up to the moment , we can conclude that the kind of
language conferred to the elements of the equation determines
the acceptance of a negative solution. Further studies at both,
the clinical and the history levels, will provide new elements
of analysis to the problem initially stated.
References
[1]
Bell, A.
Looking at Children and Directed Numbers.
Mathematics Teaching. No. 100 sept 1982.
[2]
Bell, A.
[3]
Filloy, E. and Rojano, T.
La Aparicion del Lenguaje Arit
metico- Algebraico. L'Educazione Matematica, anno V
(3). Cagliari, Italia, 1984, pp. 278
306.
Directed Numbers and the Bottom Up Curriculum.
Mathematics Teaching. No. 102 March 1983.
63
48
[4]
Filloy, E. and Rojano, T. From an Arithmeticl to an
Algebraic Thought (a clinical study with 12 - 13 year
olds). Proceeding of the Sixth Annual Conference for the
Psychology of Mathematicas Education-North American
Chapter, Madison, Wisconsin, 1984, pp 51 - 56.
[5]
Filloy, E. and Rojano, T. Operating the Unknown and
Models of Teaching (A clinical study among children 12
-13 with high proficiency in pre-algebra).
Proceedings of the Seventh Conference for the Psychology
of Mathematics Education
North American Chapter.
Columbus, Ohio, 1985, pp 75 - 79.
[6]
Filloy, E. and Rojano, T. Solving Equations: The
Transition from Arithmetic to Algebra.
For the Learning
of Mathematics 9, 2 (june 1989) FLM Publishing.
[7]
Gallardo, A. and Rojano T. Common Difficulties in the
Learning of Algebra among children displaying low and
medium Pre-Algebraic proficiency levels.
Eleventh
Annual Meeting for the Psychology of Mathematics Education
Quebec University, Montreal Canada, 1987 (1) pp 301
307
[8]
Gallardo, A. and Rojano T. Difficulty Areas in the
Acquisition of the Arithmetic and Algebraic Language.
Recherches en Didactique des Mathematiques. Vol. 9
No. 2. pp. 155 - 188, 1988.
[9]
Glaesser, G. Epistemologie des nombres relatifs.
Recherches en Didactique des Mathematique, Vol 2, No.
1981 pp 303
346.
3
[10] Heath, Thomas L.
Diophantus of Alejandria: a study in
the history of greek algebra.
Dover Books 1964.
[11] Lam Lay
Young & Ang Tian.
Se. 1987. The earliest
negative numbers: How they emerged from a solution of
simultaneous linear equations. Archives Internationales
d'Histoire des Sciences 37, 222 - 262.
[12] Li Yan and Dih and Shiran. Chinese Mathematics.
A Concise History. Clarendon Press, Oxford, 1987.
[13] Peled, I., Mukhopadhyay, S. and Resnik, L. Formal and
Informal Sources of Mental Models for Negative Numbers.
Thirteenth Annual Conference for the Psychology of
Mathematics Education, Paris, 1989 pp 106 - 110.
49
64
INTRODUCING ALGEBRA: A FUNCTIONAL APPROACH IN A COMPUTER
ENVIRONMENTI
Maurice Garangon, Carolyn Kieran and Andre Boileau
Departement de mathematiques et d'informatique
Universite du Quebec a Montreal
In the first phase of a long-term project, we have been studying a functional approach to
introducing algebra in a computer environment. The 13- and 14-year-old students have been
learning to represent algebraic word problems in the form of computable algorithms, which
serve as intermediate representations in the process of developing standard algebraic
representations, and which also permit guess-and-test numerical strategies. In a study of trial-
and-error numerical strategies in a computer environment, we found that the students: 1) do
not refer to the context of the problem to help them in their numerical search, 2) operate on
the implicit hypothesis that the function is increasing, and 3) rely on partial pattern-matching of
the digits. A second study that investigated the algorithms used by students to represent
algebra problems showed that a functional approach, based on separating the situation from
the question, was extremely accessible to all students; it also helped to avoid some of the
difficulties that are traditionally exerienced by students when translating problems of the type
ax ± b = cx ± d into equations.
Theoretical Framework
As the first phase of a long-term project, we have been studying for the past few years a functional
approach to introducing algebra in a computer environment. At PME-XI we described part of the
theoretical framework supporting this research (Boileau, Kieran, Garangon, 1987); however, since
1987, we have not presented any update at international PME conferences. We have therefore
decided to present at this time a summary of our work over the past three years.
The students have been learning to represent word problems as computable algorithms, a form of
representation which we believe constitutes an intermediate step in the development of standard,
algebraic representations. A characteristic of this approach is that the development of the algorithms is
based, at first, on the students' operational knowledge of arithmetic and that the resulting sequence of
instructions is also operational, in that it can be executed in the computer environment.
To support this approach, we may cite several recent studies which have shown that, for a given
concept, operational representations are more accessible to novice students than are structural
representations.
1This research was supported by the Quebec Ministry of Education, FCAR Grant #89EQ4159,
and by the Social Sciences and Humanities Research Council of Canada, Grant #410-88-0798.
We thank Anne Luckow for translating from French to English.
Une version frangaise de cet article est aussi disponible aupres des auteurs.
51
In a theoretical paper, Sfard (1989) presents an analysis of different mathematical concepts (number,
function, and others) to show that abstract notions may be conceptualized in two fundamentally
different ways: structurally (as objects) and operationally (as processes). For example, the notion of
natural number may be conceptualized as a counting process, or at the other extreme as the cardinality
of a class of equivalent sets. Likewise, a function may be seen operationally as a calculation process or
structurally as a subset of the Cartesian product of two sets. Sfard also shows that from an historical
perspective, operational conceptions generally preceded structural ones, and suggests that there may
be a parallel development at the psychological level. In fact, in a study (Sfard, 1987) of sixty 16- to 18year-old students who had a good knowledge of the concept of function, and in particular its structural
definition, she found that the dominant conception remained operational rather than structural. In
another part of the same study, 96 students aged 14 to 16 years were asked to translate four word
problems into equations, and also to give verbal algorithmic descriptions to calculate the solutions to
the same problems. Results showed that the students were much more successful in the verbal
description task.
In the same vein, the work of Clement, Lochhead, and Soloway (1980) shows that for certain word
problems, students find ft easier to arrive at a correct algorithmic representation in a computer language
(BASIC in this case) than to formulate an algebraic equation. The authors attribute the difficulties
encountered with algebraic equations to the absence of a procedural interpretation of these
equations.
Representing word problems by computer programs presents other attractive features. It enables the
students to use trial-and-error and successive approximation techniques, both of which are linked to
the arithmetic experience of beginning algebra students and which also favor a functional view of
algebra. Such techniques have been recommended for the early teaching of algebra (Fey, 1989).
The Computer Environment
1. Writing the algorithms
Our approach to introducing algebra uses CARAPACE, a computer environment specially created to
meet our research objectives. CARAPACE is a tool to aid in the solving of algebra problems, allowing
the writing and computing of algorithms.
To be computable in CARAPACE, an algorithm must be specified in terms of independent variables
(input variables) and the functional relations among variables. These functional relations must be
ordered such that a variable needed to evaluate another variable must have been previously identified
as an input variable or evaluated by a preceding relation. For writing these algorithms, CARAPACE
provides a screen divided into three parts. In the first part, entitled "Ask for the following values," the
input variables are entered, so that upon execution CARAPACE will ask for trial values of these
variables with which it will execute the algorithm. The second part of the screen, "Carry out the
following calculations," is for entering the functional relations, ordered from top to bottom, one per line.
The third part is called "Display the following results," and indicates the names of the variables which will
be displayed at the end of the execution.
It should be noted that the only restrictions on naming a variable are that it not begin with a number, nor
contain any arithmetic operating symbols.
66
For example, "The price of the 3rd house" is perfectly
52
admissible. This allows for the creation of algorithms that retain a significant portion of the semantics of
the original problem.
The writing of the functional relations is done without using the equal sign, in order to avoid confusion
with its different meanings (indicating an equivalence or the result of a calculation). Instead we use the
word gives (accessible with one touch of a key), to indicate that this is a calculation which gives a
result. The functional relation, then, is a calculation involving variables, constants (the givens of the
problem), and operators (+, x, 4-, exponent, parentheses), followed by the word gives and the name
of a variable. An example is presented in Figure 1.
The oroblem
The situation: The town of
Verval has twice the number
of inhabitants as Beaubourg,
and Beaubourg has 87 654
inhabitants fewer than
Montclerc.
The auestion: If the total
population of these three
towns is 567 890, what is
the population of each town?
Programming
Editing
Executing
Program al
fAsK for the following values:)
Beaubourg
cerny out the followina calculations:)
Beaubourg
2 gives Verval
Beaubourg
Verve! gives
Beaubourg
87 654 gives flontclerc
Population of two towns gives Total population
tiontolerc
Population of two towns
IDisolau the followino results:1
Total population
Figure 1. A problem (left) and the corresponding algorithm entered by the user
Once the algorithm has been written, the syntax and algorithmic structure may be verified by
CARAPACE, and if the algoritm is not "executable", CARAPACE gives a message indicating the
nature of the error.
CARAPACE offers six levels of use which impose restrictions on the accepted level of generality of the
algebraic expression. The first level allows only one operation per line (in addition to gives). The
students are therefore required to name (using semantically-laden variable names) the intermediate
results of every calculation. Difficulties associated with the order of operations are therefore temporarily
delayed. The first level is the only one that we have actually used in our experimentation. Therefore, a
description of the other levels will be very brief:
- The second level allows more than one operation per line on the left hand side, as long as the
expression is completely parenthesized.
- The third level allows partial use of parentheses. For example, a + b + c is admissible, whereas at the
second level, we would have to write a + (b + c) or (a + b) + c. On the other hand, a + b - c is not
accepted and would have to be written (a + b) c or a + (b - c).
- At the fourth level, the traditional order of operations is observed.
At the fifth level, implicit multiplication before parentheses is accepted.
- Finally, at the sixth level, implicit multiplication is accepted throughout, on condition that one-letter
variables are used. This is a necessary restriction in order to distinguish between the product ab and
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66
the variable name ab.
These levels have been enumerated according to an increasing order of generality such that an
algorithm which works at one level will also work at any of the higher levels, with the exception of Level
six which requires single-letter variables.
2. Executing the algorithms
Once an algorithm has been written in CARAPACE, its execution may be presented in two different
modes, one a detailed calculation and the other a table of values. The execution always begins with a
request for values for the input variables. The detailed calculation mode simultaneously presents a
step-by-step rewriting of the algorithm and an evaluation of the variables and expressions of the
functional relations, taking into account the order of operations and parentheses, if necessary. The
second mode of execution presents a table of values with the variable names as column headings. As
soon as an input variable is entered, a new line is added to the table. No calculations are shown.
At any time in the execution, the user may switch from one mode to the other, and the input and output
values (as well as all intermediate values that may have been named in the 'Display the following
results" zone) of the last 15 executions are always available for review in the table of values. (See
Figure 2.)
Programming
Editing
Programming
INPUTS
Beaubourg
[Ask for the following veluesi
Beaubourg
10 000
125
120
122
120
121
120
120
120
120
120
120
'Carly out the following calculations:I
Beaubourg
10 000
2 giyag Verve!
Beaubourg
Venial elves Population of two towns
30 000
20 000
10 000
Beaubourg
10 000
2
glIsi 20 000
87 654
Editing
gives
87 654 glues
000
000
000
240
000
010
100
090
020
030
040
Figure 2. The execution of the algorithm of Figure 1. On the left
587
567
575
568
571
567
568
654
654
654
614
654
694
054
568 014
567 734
567 774
567814
a partial view of the
detailed calculation mode in which the user has entered the value of 10 000 for
"Beaubourg". At a signal from the user, the value of "Montclerc" will be calculated, before
going to the next line. On the right : the tabular mode, in which a new value for "Beaubourg"
is awaited. The goal is to obtain 567 890 for "Total population" (see the problem of Figure 1).
Pedagogical approach used in this environment
1. Separating the question from the problem
Consider a word problem which may be symbolized as F(X) =Y. On one hand, we have a situation
involving X and Y and the relations between them symbolized by F, and on the other hand we have a
question. If the question is to find X given Y, we have an algebraic problem which involves (if possible)
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68
inserting F. If, however, the question is to find Y given X, we have an arithmetic problem which simply
requires the evaluation of F using X.
It is this simple remark that forms the basis of our approach to initiating students to the representation of
word problems by algorithms and to their solution by successive approximation. The main idea is to
separate the question from the problem situation, to model the situation by posing questions which
generate arithmetic problems, and to represent these solutions in a progressively more generalized
way.
Consider this problem as an example: "Corinne works part-time selling magazine subscriptions. She
earns 20$ per week, plus a bonus of 4$ for each subscription sold." The question is: "How many
subscriptions must she sell in order to earn 124$ in a week?" During the first session, the situation is
presented without the question, and the "interviewer" asks questions like "How much will she earn if
she sells 3 subscriptions?,... 5 subscriptions?,... 8 subscriptions?,... etc." Note that in order to answer
these arithmetic questions, the student must possess an operational understanding of the implied
functional relations. The next step is to have the student formalize these functional relations. He/She is
asked first to verbalize and then write his calculations, line by line, as follows (for 8 subscriptions):
8 X 4 aives 32
32 + 20 aives 52
(Note that from the start we encourage the use of "gives" to demarcate the result of a calculation.)
After a few calculations of this type, with different numbers of subscriptions, we ask the student to
create a table of all the trial values and the corresponding values calculated. The goal of this exercise is
to encourage the student to consider names for the table headings by engaging him/her in a
discussion aimed at recognizing and naming variables; these variable names in turn serve as input and
output variables in CARAPACE. Typical variable names in our example might be "number of
subscriptions," "salary," and "salary with bonus."
Once this exercise is done, the students are asked to write a series of generalized instructions (with no
given value for the number of subscriptions), using the table headings as names, to arrive at something
like this:
number of subscriptions X 4 alveg salary
salary + 20 alveg salary with bonus
We now have an algorithm which may be entered into CARAPACE and executed, once we have
identified "number of subscriptions" as an input variable.
2. The numerical search for solutions
Once the algorithm has been written in CARAPACE, we can now return to the original problem and ask
the student the planned question: "How many subscriptions must she sell in order to earn 124$ in a
week?" As it stands, the algorithm allows the calculation of "salary with bonus," given the "number of
subscriptions." The question may now be reformulated as follows: For what value of "number of
subscriptions" will "salary with bonus" have the value of 124? The student then tries different values for
"number of subscriptions," with CARAPACE calculating the corresponding "salary with bonus." The
resulting tabular display shows the student how different values of "salary with bonus" are functions of
"number of subscriptions." Usually the target value of 124 is not achieved on the frst try; the student
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must reevaluate his choice before making another guess. The students' strategies become apparent
as they gradually refine their guesses to ultimately arrive at their goal. This is the solving process that
we call numerical trial-and-error search.
Study of Numerical Searches Used in CARAPACE (Kieran and al., 1988)
During the 1987-88 school year, we worked with two 13-year-old students of slightly above average
mathematical ability. Our goal was to document both their numerical search strategies and the influence
of a known problem context on these strategies. We used two methods. In the first, the student began
with a word problem and created his own algorithm in CARAPACE before proceeding to a solution
search. In the second method, the problem and corresponding function were hidden, and the student
attempted to arrive at a target output with a series of trial input values. For both methods, the students
used the tabular display mode.
The results:
Effect of Context: When the students began to solve their problems on paper, their input values
reflected their knowledge of the context of the problems (with respect to both external semantics, like
the price of an object, and internal semantics, like choosing an even number when there was a division
by two and the result had to be a whole number). However, once the work was at the computer; we
noticed no difference in the numerical search strategies, whether the context of the problem was
known or hidden.
Search strategies: We recorded two main strategies which we call increasing and decreasing. An
increasing strategy is: If the result is too small (or respectively, too large) with respect to the target
value, then increase (or respectively, decrease) the trial value. This strategy was associated with such
substrategies as: bisection, comparison of variation, asymmetry, digit-by-digit, additivity, and partial
additivity. Certain expected strategies were not employed, notably proportionality, interpolation and
reliance upon the given relations of the word problems.
A decreasing strategy is: If the result is too small (or respectively, too large) with respect to the goal,
then decrease (or respectively, increase) the trial value. The students generally had difficulty using this
strategy, and even in the case of decreasing functions, frequently returned to their preferred
increasing strategy.
Our subjects were reluctant to choose a truly random value in their first trial. They tended to choose a
number that was the solution to a previous problem, or they would calculate a value from the givens of
the problem, knowing full well that their guess was most likely incorrect.
Study of Algorithmic Representation Processes in a Functional Approach (Kieran
and al., 1989)
In this study during the school year 1988-89, our subjects were 12 seventh-grade students of average
mathematical ability. We divided the research into three phases, according to the algebraic structure of
the word problems. In the first phase, the students were introduced to CARAPACE using the
functional approach described earlier in this paper. The problems were structured in the form a x ± b =
c, and the problem situation was presented without the question. The major finding which emerged
from this phase was the facility with which the students were able to develop ordered algorithms and
transpose them to CARAPACE. On the other hand, as soon as the students were given the question
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which completed the algebra problem, most of them attempted to use inverse operations to solve the
problem directly. The algorithm which they had just developed did not seem to provide any useful
purpose for them.
In the second phase, the students were presented with the complete word problem (both situation
and question). Problem types included: a x + x = c, b - (d x + e.a x)
c, x+ax+bx = c, x + a x + (x +
b) = c. At the beginning, we noticed that many students persisted in using inverse operations (but this
time unsuccessfully) to attempt to find the solution directly from the text of the word problem,
bypassing completely any intermediate written representation. After several sessions, however, they
began to realize that their success with inverse operations was diminishing as the problems became
more difficult. They then decided to use CARAPACE to represent the problem situations and to help
them in their numerical search.
For the third phase, complete problems were presented, and most were of the type a x ± b = cx ± d.
Note that In order to solve these problems using CARAPACE, where equality does not exist, the
student is required to produce two functional representations of the forms a x ± b and c x ± d, and give
trial values for x, with the goal of obtaining the same result for both functions. Representing these
problems in algorithmic form presented no difficulties for the students, and with these problems, no
one attempted to use inverse operations.
Our results go in a different direction from those of Filloy and Rojano (1984), who have proposed the
existence of a "didactic cut" between problems of the type ax + b = c and the type ax + b = cx + d; they
have suggested that the first type can be solved arithmetically (with inverse operations), whereas the
second type require an algebraic representation involving direct (or forward) operations. Our students
represented both kinds of problems with equal facility using the algoritmic approach of CARAPACE.
Research to come
In her analysis of the passage from operational conceptions to structural conceptions of a mathematical
notion, Sfard (1989) identifies three phases which she calls interiorization, condensation and
reification. Our goal In 1989-90 is to study the phases of interiorization and condensation for the
concepts of variable and algebraic expression. By working at the higher levels of CARAPACE with
word problems of increased algebraic complexity, the subjects of our study will gradually approach the
formal algebraic notation of equations. In particular, we will be observing the roles of a) parentheses,
b) the ordering of operations, c) the shortening of variable names to arrive finally at single-letter
variables, and d) implicit multiplication, in making the transition from procedural representations to more
standard algebraic representations. We will also be looking at the nature of those situations which
provoke the use of inverse operations directly from the word problem statement, as well as those
situations in which the use of inverse operations would be useful.
In 1990-91, we plan to study a representational form which traditionally has presented difficulties to
students: Cartesian graphing of algebraic relations. To provide continuity with our current projects, we
will be adding a graphing module to CARAPACE which will give an algorithmic and dynamic character to
graphic representation. In this new environment, the student will be able to draw a point or series of
points on the graph, with the possibility of specifying and following step-by-step the calculations and
geometric constructions carried out, at the student's own chosen level of detail. In this way, we will be
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able to study the effects of this student-controlled construction mechanism on the qualitative and
quantitative interpretation which the students bring to the graphs they produce. The data will be
analyzed in terms of operational and structural conceptions of graphs (Sfard, 1989), in addition to the
nature of the links made by students among graphical, procedural, and algebraic representations.
In 1991-92, we will examine the problem of algebraic manipulations. Often these manipulations are
carried out mechanically, without any consideration of the numerical models on which they are based.
We plan to Create a computer environment which will allow the students not only to do algebraic
manipulations but also to interpret them numerically. From this, we hope to acquire a better
understanding of the interaction between meaningful manipulations of expressions/equations and
students' conceptions of these mathematical objects. We also hope to be able to document the kinds
of learning engaged in by beginning algebra students when using our modified form of symbolic
manipulator.
References
Boileau, A. Kieran, C., & Garangon, M. (1987). La pensee algoritmique dans ('initiation a
l'algebre. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.) Proceedings of the
Eleventh International Conference of the Psvcholoav of Mathematics Education (Vol. I, pp.
183-189). Montreal, Quebec: Universite de Montreal.
Clement, J., Lochhead, J., & Soloway, E. (1980). Positive effects of computer orogramming
on student understandina of algebra (Technical Report). Amherst: University of
Massachusetts, Department of Physics and Astronomy, Cognitive Development Project.
Fey, J. T. (1989). School algebra for the year 2000. In S. Wagner & C. Kieran (Eds.),
Research issues in the learnina and teachina of algebra (pp. 199-213). Reston, VA:
National Council of Teachers of Mathematics.
Filloy, E., & Rojano, T. (1984). From an arithmetical to an algebraic thought. In J. M. Moser
(Ed.) Proceedinas of the Sixth Annual Meetina of PME-NA (pp 51-56). Madison:
University of Wisconsin.
Kieran, C., Garangon, M., Boileau, A., & Pelletier, M. (1988). Numerical approaches to
algebra problem solving in a computer environment. In M. Behr, C. Lacampagne, & M. M.
Wheeler (Eds.) Proceedinas of the Tenth Annual Meetina of PME-NA (pp. 141-149).
Dekalb, IL: Northern Illinois University.
Kieran, C., Boileau, A., Garangon, M. (1989). Processes of mathematization in algebra
problem solving within a computer environment: a functional approach. In C. A. Maher, G.
A. Goldin, & R. B. Davis (Eds.) proceedings of the Eleventh Annual Meetina of PME-NA
(pp. 26-34). New Brunswick, NJ : Rutgers University.
Sfard, A. (1987). Two conceptions of mathematical notions: Operational and structural. In J.
C. Bergeron, N. Herscovics, & C. Kieran (Eds.) Proceedings of the Eleventh International
Conference of the Psycholoav of Mathematics Education (Vol. III, pp. 162-169).
Montreal, Quebec: University de Montreal.
Sfard , A. (1989). On the dual nature of mathematical conceptions: theoretical reflections on processes
And objects as different sides of the same coin. (Manuscript submitted for publication)
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Lan TO TEACH THE CONCEPT GIF FUNCTEDN
.
Dominique GUIN et Ismenia GUZMAN - RETAMAL
Institut de Recherche de Mathemafigue Avancee,
10 rue du General Zimmer ,
67084 Strasbourg Cedex , France
Summary : Our aim is to elaborate teaching situations with a constructivist approach underlying
the various aspects of the function notion on one hand , on the other hand the relationships between
them.We assume that a computer environnement can give an efficient contribution to our goal in two
different ways: programming activities in an applicative language LOGO , and using microcomputer as a
tool We present some teaching situations and first issues on a comparison of productions with a
traditional teaching in an other classroom : there are significant differences between the two groups
according to the concerning aspects of the function concept . After these first results , we have altered our
didactical situations for a new experiment : we discuss similarities and difficulties about it . The research
concerns twenty.14 -15 years old students , having a LOGO experience prior to the study
OBJECTIVE
.
:
Objectives of french curriculum on functions are:
- Recognition of a function in various situations ( graphical, algebraic , common life ).
- Manipulating functions .
- Applications to equations of lines .
Our objectives are to contribute answering the questions raised in ( Tall D.( 1987 ) I in the
restricted domain of teaching the function notion :
- In what ways can multiple links representations be integrated into the curriculum for
learning, teaching problem solving ?
- In what ways can computer environments be designed an used to provide intelligent support to
the learning process ?
- In what way are programming and the use of prepared software ( computer as a tool)
complementary , and what constitutes an optimum combination of the two in terms of understanding and
efficiency ( time on task) ? "
To find an answer for these questions , it is necessary to elaborate and experiment teaching
situations underlying :
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63
- the various aspects of the notion of function
,
- the relationships between them .
A computer environnement can give an efficient contribution in two different ways :
- programming activities in an applicative language LOGO ,
- and using microcomputer as a tool .
LOGO language has been chosen for its good adaptability to programming activity for
problem solving because of its procedures , and to work on functions because it is an applicative
language. Functional programming languages such as Logo are very close to the language of
mathematics [ Klotz S. (1986) ] .
THEORETICAL
FRAMEWORK :
Vinner already introduced a distinction between two aspects of the function notion : concept-
definition and concept-image [ Vinner S. ( 1983 )
.
A function is not just a :
a table of values
a graphical representation
a formula
a correspondence
it is all of them at once . It seems necessary to define different registers that I. Guzman
[Guzman I. ( 1989)
displays in this way :
(
Conceptual
Algebraic
)
Programming
(Language)
Table
Graphic
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a) registers of processing on the same plane : the algebraic (formulas) ) , the graphic, table (of
values ) and programming activity LOGO .
b) registers of conceptualization ( relationship , correspondence ) and language which
allows communication between registers .
We have noticed that usually , in France , the teacher gives the function definition (relationship
or correspondence between two sets ) . Then , the exercises are essentially calculus on algebraic
expressions and graphic representations of functions . Links between different registers are rarely
explicited ( for example , the link graphic <--> algebraic ) and the use of these registers to analyze
empirical situations is left out .
Our cognitivist and didactical hypothesis
1) Constructivist hypothesis
:
:
mathematical knowledge is constructed by problem
solving .
2) The use of a programming activity in mathematical teaching requires a real
alphabetisation in Computer Science [ Rogalski J. ( 1985 )
3) An appropriate use
.
of computer as a tool can underlie links between different
registers.
Several studies were carried out in this way :
- Link algebraic register <--> graphic register, [ Dreyfus T. , Eisenberg T. (1987)],
[Goldenberg E. ( 1987 )] , [ Zehavi N. , et alii (1987)1.
-
Link algebraic register <--> LOGO programming register , [ Leron U., Zazkis R.
(1986)].
METHOD
The LOGO project ( 87)
:
In french curiculum , it is the first teaching on the function notion .We had to build didactical
situations favouring interactions between different
registers optimizing the use of a
microcomputer environment :
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1) Introduction to the function notion as a procedure linked to a table of values This activity
deals with the table and LOGO programming registers dealing with the conceptual aspect
(correspondence).
- introduction of vocabulary on fonctions : image , domain of definition , etc...
large scope of examples ( empirical situations ).
-
F : length of a word
X
Cat
Chicken
Dog
Elephant
Stork
Marmot
F
F(X)
2) Experiments with the function notion :
- with various examples according to the curriculum of such students ( constant , linear and
affine functions ) .
- by investigations on hidden functions [ Leron U., Zazkis R. ( 1986)) : students have to
find out the LOGO procedure-function with an experiment on different values of the variable.This activity
deals with table , algebraic , LOGO programming registers , language and empirical
situations .
3) Introduction to graphic representation of a function as an execution of a procedure
representing the set of points with coordinates ( x , f ( x) ) . This activity deals with table , algebraic ,
graphic and LOGO programming registers , language and empirical situations . This process
points out the role of parameters a and b in the expression f(x)=ax+b through a systematic
variation of these parameters .
Experiment environment
:
This experiment took place in a classroom ( 20 students, 14-15 years old ) with the participation
of the mathematic teacher. The fifteen sessions occured within 4 weeks . There was 7 micro-computers
in the classroom . Functional programming requires a good level of programming abilities . It is the
reason for having choosen students with a LOGO experience of more than one year :
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I - Basic teaching of LOGO programming turned to structured programming [ Dupuis C.,
Egret MA., Guin D. ( 1988) ( 6 months ) .
2 - Teaching on recursion [ Dupuis C. , Guin D . ( 1988) - Gain D. ( 1986 ) ) , ( 6 months).
First results
:
A test was submitted to a reference classroom [ Guzman I. ( 1989) , leading to the following
observations :
1) first erouo z The answers which are better in the reference classroom are essentially related
with conceptual register and its link with language register :
- In the recognition task of a function from empirical situations , the vocabulary on
functions (as domain of definition , image ...) is rarely used in the experiment classroom : It points out a
weakness between conceptual register and empirical situations
.
- In the recognition task of a function from a graphic representation , the students of
the reference classroom used a graphic criterion : the function notion was introduced in this classroom
from conceptual and graphic registers . In the experiment classroom, the link graphic register <-->
conceptual register was not enough emphazised
During the mathematics lesson , the teacher asks for examples of functions and receives the
followinganswers from his students :
- Alain proposes weight as function of size for all the students of the school .
- Bernadette proposes weight as function of age for a child between 0 and 5 years old .
- Claude proposes size as function of age for an adult between 0 and 40 years old .
In your opinion , are these examples good or bad and tell why?
****************************************************** ********* *******************
- In the experiment classroom , the function notion is seen as a way to calculate one object
from the other , if there is not such a possibility , there is no function : its points out an unbalanced
practice between LOGO programming and conceptual registers .
Common language was not used for recognition of a function : students do not handle
key-words related to function concept , for example : dependence , although they have feeling of it .
A specific work on conceptual register and links with language and graphic registers in
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7?
empirical situations seems to be lacking . For this it is important in analysis of tasks to locate the
words linked to conceptual aspects ( conceptual register ) necessary to build models on empirical
situations .
211gcordzroulu The answers which are better in the experiment classroom are essentially
related to production tasks . We can conjecture that these results come from an active practice on
empirical situations , functional programming , table registers and links between them
with are rarely tackled in traditional teaching : for example , there are no attempts from students in the
reference classroom to find a correspondence between graphic situation and algebraic expression ( link
graphic register <--> algebraic register ).The functional vocabulary as " linear or constant function
"
etc...
is more used in experiment classroom . The answers which are better in the experiment
classroom are also related to application tasks using properties requiring that a relationship or a
correspondence be set up , but also in producing or recognizing algebraic expressions
The graphic project ( 88)
.
:
According to the first results , we are aiming to alter our didactical situations :
1) Simultaneous introduction in graphical register ( as in traditional process ) and table
register of function notion
.
2) Introduction in graphical register of functional vocabulary
3) Recognition activities which point out the correspondence aspect of a function in table
and graphical registers which allowed students to find out by themselves the graphical criterion
.
4) Investigations on hidden functions with microcomputer : during this second experiment ,
students had too weak a level in LOGO for programming
activities
,
they only used
microcomputers as a tool .
5) Investigations on graphical representations with microcomputer .
A comparison between the graphic
(87):
project ( 88 ) and the logo project
Between these two experiments there are differences and similarities. The differences which
are external to the structure of the project are relative to the training of the students : class 88 was
weaker and had not enough abilities in LOGO programming. Therefore, the programming register has
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only been implemented as a tool, especially in the work sheets
hidden function " and " graphic
representation ": Programming activities requires a real alphabetisation in LOGO.
The two projects have the same aim as far as teaching is concerned : to bring into play all the
registers involved in the concept of function. External factors were the cause of a modification in
methodology. In order to present the concept of function , the logo 87 project has highlighbed the
programming register and has presented a function as a procedure. The table register has also been
simulancously brought in , with a work on functional vocabulary . On the other hand , the graphic 88
project presents the concept of function in the graphic register pointing out intuitively the
correspondence aspect by developing a graphic criterion used to identify a function. Therefore , the
first work sheet about "Functions" is different in the two projects. The "Hidden Functions" work sheet
and the "Graphic Representation" work sheet are almost the same one . The fourth work sheet which is
entitled "Affine Functions and Lines Equations" is different because the lines equations in 1988 were
taught before the function chapter .
Discussion
_
Results arc only relevant on the qualitative level because the populations in the two
experiments were very different .There are still a few variables that we did not grasp such as training of
students for example . From a qualitative view point the following facts were pointed out :
- A slight improvement at the conceptual level in the experimental class 88 in relation
to the experimental Class 87.
- The link between the graphic and algebraic registers did not improve during the first
experiment ( in 1987 ). The behaviour of students in the experimental and control class was the same one
. In others words, their behaviour was the classical behaviour of students at this level. We do not have a
didactical explanation for this fact. During the second experiment ( 88 ) , we have discovered that
students have great difficulties in linking the algebraic and graphic registers . They stumbled in
interpreting the algebraic expression of a function and translating it using tables and graphs.
Nevertheless , the inverse operation of interpreting graphs and graphic representations was more
successful! that in the traditional process . It is obvious that the graphic strategy used by the
students of class 88 has not been transferred to the others registers.
To conclude :
- In the first experiment ( hight level ) , there were very good results for manipulations
and very poor results at the conceptual level .
65
'7 9
- In the second experiment ( low level ) , there were an improvement at the conceptual level
and very poor results for manipulations .
This leads us to ask the following question
:
Is there an independence between the
understanding of the conceptual aspect of correspondence of a function and the manipulation of
this correspondence in the others registers and their links ? To answer this question , we have to find
good conditions of comparison .
References
:
On the deep structure of functions ". Proceedings
DREYFUS T. , EISENBERG T. ( 1987) :
of the PME-X1 Conference pp 190 -196 . Montreal .
DUPUIS C. , GUIN D . ( 1988 ) : " Ddcouverte de la rdcursivite en LOGO dans une classe ",
Acres du premier colloaue franco-allemand de Didactique des Mathdmatiques et de l'Inforrnatique
Ed La Pensee Sauvage , F-38002 .
DUPUIS C. , EGRET M. - A. ,GUIN D . ( 1988 ) : " Pour une approche multi-criteres
dactivites de programmation en LOGO " Annales de Didactique et des Sciences Cognitives ,
Vol 1 , pp. 111 - 131. Ed IREM de Strasbourg .
GOLDENBERG E. ( 1987) : "Believing is seeing". Proceedings of the PME-XI . Conference
pp 197 -203. Montreal .
GUIN D. ( 1986 ) : " Recursion with LOGO in a classroom ". proceedings of the 2MI
Jntemational Conference for LOGO and Mathematics Education pp 1- 8 London .
GUZMAN I. ( 1989) : "Registres mis en jeu par la notion de fonction
et des
Annales de Didactique
Sciences Cognitives , Vol 2 pp. 230 -260. Ed IREM de Strasbourg .
KLOTZ S. (1986) : " When is a program like a function ?". Proceedings of the 2011 International
Conference for LOGO and Mathematics Education pp. 14- 23 .London .
S.LERON U., ZAZKIS R. ( 1986 ) :
Functions and variables , a case study of learning
mathematics through Logo ".Proceedings of the Zad International Conference for LOGO and
Mathematics Education pp. 186 - 192 .London .
ROGALSKI J. ( 1985) : " Alphabetisation informatique , problemes conceptuels et didactiques",
Bulletin de I'APMEP no 347 , pp. 61-74 .
TALL D.( 1987 ) : " Algebra in a computer environment ". Proceedings of the PME-XI
Conference pp. 262 - 271 .Montreal .
VINNER S. ( 1983) :
Concept definition , concept image and the notion of function". The
International Journal of Mathematical Education in Science and Technology . vol 14 , pp. 293 .
ZEHAVI N. , GONEN R. , OMER S. , TAIZI N. ( 1987) : " The effects of
microcomputersoftware on intuitive understanding of graphs of quantitative relationships ".
Proceedings of the PME-XI Conference pp 255 -261. Montreal .
8-0
66
THE CONCEPT OF FUNCTION :CONTINUITY IMAGE VERSUS DISCONTINUITY
IMAGE (Computer experience)
Fernando Hitt
Seccion de Matematica Educativa del CINVESTAV. PNFAPM, Mexico.
SUMMARY
With the increasing use of computers in the classroom and also
of
graphics packages,
a
of
concept
functions is
being
encouraged. This is the concept of function defined by a single
a
formula,
continuous
Here
function.
we
shall
consider
functions expressed by more than one formula in a
pencil
and
Logo"
with
"paper,
the aim of
developing in
secondary school pupils a broader notion of function than that
which is generally possessed by such pupils.
context,
INTRODUCTION
In previous studies of the concept of function, we find
that the problem of
obstacles to understanding has been
approached in a number of ways. One such approach is that of
Tall and Vinner [1981] and Vinner [1983] who interpret certain
obstacles in terms of a lack of interaction between what they
call "concept definition" and "concept image ".
Markovits. Eylon and Bruckheimer
[1096] report answers
given by pupils in relation to a graph of a discrete function
that
was
it
not
a
function,
since
points
-the
are
not
In the same study, the authors state: "Only one
student drew the graph of the following function correctly
connected ".
r.fnatural
numbers
J.
(natural
fCx) = 3
numbers }
.
Host answers
subconsciously replaced the natural by the real numbers ".
Another study, this time with teachers [Hitt. 1989], we can
see that the teachers showed a strong tendency to think in
terms of continuous functions (spontaneous behavior). Their
concept image is linked to an idea of function continuity
expressed by a single formula.
Our
activities
provide the pupil
related
with
to
this
work,
are
designed
to
problems and exercises whereby the
67
81
different variables mentioned above can, in a structured way,
play a positive role in developing a concept image.
MATHEMATICS TEXTS Cdefinition of function and images used)
As is well known, different authors introduce the concept
of function in different ways. Generally, we can classify them
into four main definition types CHitt Cidem]). The experience
of teachers and researchers has shown that some definitions are
not all equivalent in an educational context (Malik, 1980].
THE COMPUTER AND THE GRAPHICAL EXPRESSION OF FUNCTIONS
With the use ever-increasing use of the computer and the
production of new graphics software, a new concept image is
being generated which is different from the one being purveyed
in textbooks. We use the phrase "new concept image", because,
unlike textbooks, which show graphs of discontinuous functions,
the software that has been available up to now does not allow
graphical functions defined by more than one formula.
Let us consider another problem.
the functions: fCxD=
( 3 ifx<0
If we take, for example,
and gCx)=4x+8, if xdN. The two
10 ifx>0'
.
functions are continuous in their definition domain. However,
when the graphs of the functions are shown to the pupils, they
interpret each one in a different way. Some will think that the
functions are not continuous C"The pencil was taken off the
intuition
primary
Our
the
curve").
paper
when drawing
IFishbein. 1987) prevents us from seeing them as continuous.
What can we do to develop a secondary intuition, in Fhisbein's
sense?.
Another
computer
problem is
screens,
that
functions
because
such
as
of
the limitations
fCx) =C1 /x)
sin x
of
,for
xdOR-(0>, where the limit at 0 on the left is equal to the limit
at 0 on the right, appear to be continuous functions in R. In
some cases, for example in Tall (Supergraph, 1985), when the
computer is drawing the graph of the function and finds that it
not defined at one point, the computer makes a beep.
However, the problem with the image on the screen remains.
is
82
68
To summarize, an unsolved problem with currently available
software is that the concept Cdefinition and image) which they
are implicitly reinforcing can be described as
* Function defined by a single formula
m Function-continuity
* Continuous domain (connected)
These problems may be resolved in the near future. That is,
new versions of the graphics packages will be produced with the
capacity to overcome these kind of problems.Much work needs to
be done on reconciling textbooks with computer languages and
:
educational software.
DESIGNING ACTIVITIES WITH PAPER, PENCIL AND COMPUTER
Our
aim was
expressing
to concentrate specifically on graphically
functions,
both
continuous
and
discontinuous:
Logo graphic tasks" [Hitt, 1980]. We
also wished to place special emphasis on the domain and set
"Functions and Graphs.
image of functions.
It will be seen that the proposed activity is attempting to
build a bridge in the pupils, between the concept of function
and their mental image of it by using the computer. A further
objective was to provide the mathematics teacher with "paper,
pencil and computer" activities for use in the classroom. Thus,
we have attempted to link a language, Logo, and a concept,
namely, that of function.
The development of a secondary intuition in Fishbeins
sense [ibid] would have to be developed through mathematical
activities before making use of graphic software which would
introduce obstacles into the change from one level of intuition
to another.
It is our hypothesis that the proposed activities will help
to bring about this change in the level of intuition. In our
experiment,
we only try to prove that no knowledge of Logo
required by the pupil in order to work with it.
Two approaches were adopted in the experiment: a laboratory
BEST COPY AVAILABLE
69
approach, and a normal working situation in the classroom. The
sample consisted of five pupils. The interview with pupil
called 1 was undertaken entirely in the University laboratory
Ctwo and a half hour session). He was 14 year old (end of 3rd
year in secondary school). He had little previous experience
with Logo in the context of Turtle Geometry.
The remaining pupils, aged between 16 and 17, were given
activities to work on in a normal individual work session in
their computer laboratory C2 hours). These pupils were in the
6th year of secondary school. They already had some knowledge
of Basic but had not previously worked with Logo.
We now set up the activities that the pupils were required:
1. Write down the concept of function.
2.
Read the definition of function which we would use in
our context and our examples.
3.
To write again about the concept of function.4.
graphs of f1,..
Draw
.f7 and write down their set image.
E. To discover functions FUNONE, FUNTVO.....FUNSEVEN
8. To write down the function related to the graph showed.
7. To write once again about the concept of function.
ANALYSIS OF THE RESULTS
I. Definition of function Activities 1, 2, 3 and 7.
Pupil 1 wrote that the concept of function was a process
and that function can be represented by a graph. He also gave
an example,
3 =L. 6. This answer suggests that a formal
definition was lacking. The experiment had a strong effect on
the concept image of this pupil. In fact, at the end Cactivity
73 he wrote three pages and provided three graphs of continuous
functions, one graph of a discontinuous function, and two
examples of functions expressed by more than one formula.
Pupil 2 did not remember what the definition of function
was.
Pupil 3 The word -operation- was used by this pupil in the
three definitions that were written by him. In his final
definition, we can see that the functions expressed by more
70
than one formula had an influence on his original
idea,
as
evidenced by the use of the phrase "combination of operations".
Furthermore, he added to his definition the words "there
only one imago for each number".
Pupil 4.
is
This pupil provided a definition of function which
was associated with activities of differential calculus and
locating maxima and minima.
Pupil
5.
This
'operation'
and
definition
there
presentation.
pupil's
definition
'mathematical
was
momentary
was
process'.
break
in
In
terms
his
influenced
of
second
by our
At the end he returned to his original
idea,
adding the words 'function contain different operations'.
In the light of the foregoing paragraphs, pupil 1 can be
seen to have assimilated the ideas that were presented to him
much more fully than did the other pupils, owing to being at an
'intuitive stage'
That is. his knowledge of the concept of
function was intuitive rather than formal.
.
Surprisingly,
definition,
none
of
the
students,
in
their
first
included any of the graphs they had previously
acquired in the course of their studies.
II. Transfer from algebraic to graphical form. Activity 4.
Pupils were then asked to do some tasks with paper and
pencil, transferring functions written in algebraic form to the
graphical form, and giving an image set for each one.
The five pupils were new to compound functions, saying that
they were used to having functions in the form y =
suggested that they analyze the function in parts.
They
answered that this would be sufficient to know what to do.
The main problems arose with functions f5 and fes. Generally
pupils coped with f1, f2, f3, f4 and f7 without difficulty. In
the case of pupils 2 and 4, the concept image is linked to
function continuity, with the result that the curves they drew
were not functions.
Pupil 3 shows considerable confusion.
BEST COPY AVAILABLE
71
85
drawing a correct graph of f4 but then changing this to a
vertical straight line. This same pupil drew an incorrect graph
for f5 (drawing two straight lines which did not represent any
function), and then corrected himself, giving another wrong
answer. It is possible that he failed to read f7 correctly and
inverted the two parts of the function. Pupil 5 followed the
TRANSFER FROM
ALOEBRIC TO
GRAPHIC FORM
f
f
5
d
(x)..
5
4
correct
answer
2 if m'So
-2 if x>0
(x).-
PUPIL No.
3
2
1
correct
x -2
if
x.2
if x>0
x:50
IP
same procedure as the previous pupil with f5. When this pupil
drew the graph of f6, he must have thought that f(x) =x -2 ,for
x50, would have to pass through the point C-2,03 instead of
(0,-2). And for f(x) =x +2, x>0, he thought that the line should
pass through C2,03. It is important to note that this pupil
realized that at x=0 there was a "change in the function-. It
can in fact be seen that he left a gap in the graph.
III. Transferring an algebraic form (inside the computer) to an
algebraic and graphical form. Activity 5.
At the end of the activity 4, the pupils were asked to work
with functions rumohnt, FUNTWO".., FUNSEVEN (paper, pencil and
computer).
GUESS MY FUNCTION
AND MAKE A GRAPH
f
s
tx)..
(x).
f
45
7
PUPIL
1
1
if
if
x5o correct
x>0 answer
x-1
x+1
if
if
x.So
-1
if
if
correct correct
answer
x >O
correct correct
answer answer
x9
correct correct
86
No
3
2
72
4
--jf
471
5
correct
answer
/
/
Pupil 3 made several attempts to find a graph for f6, finally
producing a one which was not a graph of a function. Pupil 4
tried out several relative whole numbers CZ3,and in defining
functions fs, f6 and f7 he did not take account of the interval
C0,13. Here it is worth mentioning that this pupil wrote down
the functions wrongly, committing syntactic errors.
It is
possible that these errors had been committed in the past, but
had been corrected when they came up in the course of
activities, but in coming up against a new situation (quite
complicated in some cases] the error appeared again in a new
context.
f 5C
=
x
1
= 1
x >
f 8C
1
= x +
=
1
;
f 7C x.) ={
x < 0 = x
x >
x
1
=
0 = x-1
A further difficulty with function f7 is that pupils 3, 4,
5 interpreted the graph
for
x53 as if it had to be
positioned beneath the x axis. It may be that when pupil 5 drew
the graphs of functions composed of more than one expression,
was expecting that the two graphs would be of the same type.
and
IV.
Transferring functions from the graphical
form to the
algebraic form. Activity 6.
Pupil 1 was the only one who did this part completely
(committing one error), the others did not have enough time to
write down the answers to the functions f 11 , f 12 , f
13 and f 14 .
CONCLUSIONS
Functions defined by a single formula did not seem to cause
any great problem in a connected domain (except as far as the
subconcept image set was concerned). The results show that it
is possible to work with functions expressed by more than one
formula and that pupils could become better at handling these
if they were given more practice with them.
A knowledge deficiency will resurface as errors with the
passage of time. The definition which we provided did not,
except very briefly, replace the definition that had been
acquired by the pupils in earlier years. The results show that
BEST COPY AVAILABLE
73
in only one case were new elements incorporated and only in one
case did the definition help a pupil to recall parts of his
definition that he had forgotten.
we
If
compare
the
results
of
the
activity
in
which
were transferred from the algebraic form to
graphical form Cpapor and pencil). with the results of
functions
the
the
"guess my function" activity Cpaper. pencil and computer), it
is seen that learning has taken place in the course of these
activities. The drawing of straight lines not representing the
graph of any function virtually ceased. Some of the corrections
which the pupils made and the graphs of discontinuous functions
that they produced lead us to believe that some progress in the
mental image of these pupils was made.
The emphasis that we gave to the subconcepts domain and
image set on the coordinate axes Cwhen graphs were shown) dici
not appear to have any positive effect on the behavior of the
pupils.
REFERENCES
Fishbein,E. [1987) Intuition in Science and Mathematics. An
Educational Approach. D. Reidel Publishing C., Dordrecht.
Hitt F. C1989]. Obstacles related to the concept of function.
British Society for Research into Learning Mathematics.
Malik
M.
(1980]
Historical and pedagogical aspects of
the
definition of function. International Journal of Mathematics
Education in Science and Technology. Vol.11, p.489 -492.
Markovits Z.,Eylon B.,Bruckheimer M. (1986) Functions today and
yesterday. For the Learning of Mathematics 6, 2. June.
[1985] Supergraph (BBC). Glentop Publishers. London.
Tall D..Vinner S. [1981] Concept image and concept definition
to
limits and
in mathematics with particular reference
continuity. Educational Studies in Mathematics 12, p.151-169.
concept image and the
Vinner S.[1983) Concept definition,
notion of function. International Journal of Mathematics
Education in Science and Technology, Vol.14, p.293-305.
Tall D.
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AVAIIIMILEH
74
ACQUISITION OF ALGEBRAIC GRAMMAR
David Kirshner
Louisiana State University
A new rule of algebra is proposed and evidence for its psychological reality, presented. A
model for acquisition of this rule is explored.
Usually it is presumed that the rules of the algebr'a game are explicitly available; having first
germinated in a mathematical mind, next been transplanted to textbook and teachers manual, and
finally been harvested in the classroom for distribution to the populace. For instance the focus of
Carry, Lewis and Bernard (1980) (following Bundy, 1975) on strategic decisions of selecting and
sequencing rules suggests that the character of the rules themselves is relatively unproblematic.
For Wagner, Rachlin and Jensen (1984) the available rules can be captured by "rote memorization
of formulas and algorithms" (p. 7). Others, (e.g. Matz, 1980) postulate intermediate processes
between the available rules and the rules actually used in solving problems. She proposes that
extrapolation techniques may be required to bridge the gap between the base rules
of the
curriculum and problem contexts for which no available rule exactly fits. For instance she
describes how the new situation ax + ay + az might be handled by deriving the needed rule
A.(B + C +...+W)=(A.B)+(A-C)+ ...+(A.W)
from
the
given
rule
A '(B + C) = (A -B )+ (A 'C) (p. 104). In all of these instances, however, the rules underlying
successful algebraic performance are introspectively obvious.
Elsewhere (Kirshner, 1987a; 1987b; 1989) I have proposed rules of algebra that are not
introspectively obvious, and argued for a reassessment of traditional assumptions about the nature
of algebraic knowledge. The present paper also proposes a new rule of algebra and offers support
for its psychological reality; however, the focus here is on the possible processes of acquisition of
the rule, and on such characteristics of the human cognitive system as can be inferred from the
acquisition processes.
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75
89
THE GENERALIZED DISTRIBUTIVE LAW
The Generalized Distributive Law (GDL) presented here as a psychological theory was first
introduced by Schwartzman (1977) (in a slightly less rigorous form), but as a pedagogical
technique. It is based upon a simple hierarchy of operation levels which groups together inverse
operations:
Level 1 operations are addition and subtraction
Level 2 operations are multiplication and division
Level 3 operations are exponentiation and radical
(If " " is an operation, then " I I" represents its level.)
Using this convention, the GDL can be simply stated:
(a & b) c = (a
c), whenever 1"I=I&I+1
c)& (b
Note that this generalized rule subsumes eight other rules usually presumed to be discrete entries
in the rule system of algebra:
Level 3 over Level 2
Level 2 over Level 1
(a
b)c = ac
a+b
c
[
be
a
b
c
c
ab a
b
c
c
b.
(ab)`
(a + b)c = ac + bc
=
a'
b
srci
1
b
The claim is that the GDL is not just an interesting formalism, potentially useful as a pedagogical
rule, but rather an integral part of the knowledge that is acquired in the development of algebraic
skill.
Support for the psychological reality of the GDL involves analysis of the oft reported errors
of the form (a ± b )` = a'
± b`
and
b=
`,/b- (Budden, 1972, p. 8; Schwartzman,
1977, p. 595; Laursen, 1978, p. 194; Davis & McKnight, 1979, p. 37 and p. 98; Matz, 1980, pp.
ilhe rules involving the radical operation appears in surface form to be left-distributive; however, Kirshner (1987, p. 93)
argues that the deep representation of the radical operation is reversed from its surface form.
90
76
98-99; and Smith, 1981, p. 310). These errors can be seen to satisfy the Overgeneralized
Distributive Law,
(a & b)* c = (a * c) & (b * c), whenever I* I > I & I
,
proposed here as a developmental precursor to the GDL. The present analysis is that the
Overgeneralized Distributive Law represents a phase of covert 'experimentation' with contextual
constraints on the application of distributivity. In this account, the GDL results front honing down
distributivity to its maximally permissible context of application: l* I = I & I + 1.
Matz (1980) also has attempted to account for the (a ± b)` = a' ± 6' and
4ct ±b =
errors as overgeneralization of distributivity rules; but without postulating an
-4,72 ±
introspectively unobvious rule like the GDL. In her analysis, some normally useful processes for
extrapolating from base rules to new situations has gone awry.
There are a number of deficiencies with Matz's explanation of these errors that are avoided
in the present account. Firstly the extrapolation techniques that are presumed to have gone awry in
the overgeneralization errors are not explicated in her theory. She gives illustrations, but does not
detail the actual mechanisms at work. As a consequence of this lack of specificity, Matz's theory
can be used to describe errors that do occur, but provides no theoretical basis to predict which
error should occur. In contrast, the present theory predicts exactly the observed errors.
What is more, the present theory can extend in its prediction to a range of data that Matz's
theory cannot explain even after-the-fact: Matz (1980, pp. 98-99) notes the occurrence of other
linearity errors including a (bc) = ab -ac , a"" = a'" a" , a"*" = a" + a", and
b+c
=
b
+
c
These errors can be described as fulfilling yet more elementary versions of the Overgeneralized
Distributive Law in which right or left distributivity holds, or operations distribute over
themselves. For lack of a more precise analysis, call this Open Context Distributivity. The present
theory, therefore, predicts that the first set of errors should prove more tenacious than this latter
class,
since
open context
as
the
student
progresses
from
l*I>l& I ) I* l=l& I + 1,
wider
contexts
to
narrower
contexts,
the latter class of errors falls away before
the former. This prediction, I believe, corresponds with the facts1; facts that Matz's framework
'Unfortunately, longtitudinal records of student behavior are completely absent, and even systematic crosssectional data
are scarce; nevertheless, the anecdotal evidence is strong.
BEST COPY AVAILABLE
77
91
cannot account for.
The greater specificity, predictive rather than descriptive adequacy, and greater range of
applicability of the present account would seem to make it a far stronger explanation of the
(a ±b)` =
± b' and
Nta ±b =
±
46 errors than the account of Matz (1980). But the
existence of introspectively unobvious rules like the GDL raises a host of new questions, among
them questions about rule acquisition, to which we now turn.
ACQUISITION
A variety of approaches to the question of rule acquisition are possible ranging from a
Chomskyan innatist model in which the processes of induction are presumed to lie far beneath
conscious cognition, to Anderson's (1983) ACT* (ACT STAR) theory of learning in which it is
presumed that "all incoming knowledge is encoded declaratively; specifically, the information is
encoded as a set of facts in a semantic network" (Neves and Anderson 1981, p. 60). For the
purposes of coming to terms with the relatively radical notion of introspectively unobvious rules
of algebra, it seems prudent to select the framework that is most compatible with usual
assumptions about mathematical knowledge; namely the ACT* theory.
ACT* has been applied extensively to teaming from direct instruction in such domains as
geometry proof (Anderson, 1983b), computer programming (Anderson & Reiser, 1985), and word
processing (Singly & Anderson, 1985). The approach taken in the theory is to trace processes of
proceduralization and composition whereby new knowledge which enters the system in
declarative form --for instance as text book rules or teacher instructions-- is compiled into
automatically executed procedures as skill is developed.
Generally speaking, knowledge compilation results in the evolution of less abstract rules as
more general declarative structures gradually becomes adapted to the specific conditions of the
task environment. But ACT* does invoke inductive tuning for the creation of more abstract rules.
Inductive tuning involves the complementary processes of generalization and discrimination. In
generalization, conditions on the applicability of a rule are relaxed, resulting in a new version that
applies to a broader range of contexts than the original rule. Discrimination tightens up rules that
92
78
have been overgeneralized. Anderson (1986) illustrates the tuning mechanism with an example
from language acquisition:
Suppose a child has compiled the following two productions from experience with verb
forms:
IF the goal is to generate the present tense of KICK
THEN say KICK + S
IF the goal is to generate the present tense of HUG
THEN say HUG + S
The generalization mechanism would try to extract a more general rule that would cover
these cases and others:
IF the goal is to generate the present tense of X
THEN say X + S
where Xis a variable.
Discrimination deals with the fact that such rules may be overly general and need to
be restricted. For instance, this example rule generates the same form, whether the subject
of the sentence is singular and plural. Thus, it will generate errors. By considering different
features in the
successful and unsuccessful situations and using the appropriate
discrimination mechanisms, the child would generate the following two productions:
IF the goal is to generate the present tense of X
and the subject of the sentence is singular
THEN say X + S
IF the goal is to generate the present tense of X
and the subject of the sentence is plural
THEN say X
These learning mechanisms have proven to be quite powerful, acquiring, for instance,
nontrivial subsets of natural language (J. R. Anderson, 1983). (p. 205)
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93
These processes of generalization and discrimination can be applied to model acquisition of
the GDL. Assume the existence of rules of the following form which capture students applicative
knowledge of the (a + b)c = ac + bc , (ab )e =aebe, and other. "distributive" rules of
the
curriculum:
IF the goal is to generate an expression that has addition as its dominant operation
and the current expression has multiplication as its dominant operation
and the next-most-dominant operation is the goal-dominant operation (addition)
and the dominant operation is to the right of the next-most-dominant operation
THEN create a new expression with the goal-dominant operation (addition) as dominant
and the previously dominant operation (multiplication) as next-most-dominant
(and assign the subexpressions appropriately)'
IF the goal is to generate an expression that has multiplication as its dominant operation,
and the current expression has exponentiation as its dominant operation
and the next-most-dominant operation is the goal-dominant operation (multiplication)
and the dominant operation is to the right of the next-most-dominant operation
THEN create a new expression with the goal-dominant operation (multiplication) as dominant
and the previously-dominant-operation (exponentiation) as next-most-dominant
(and assign subexpressions appropriately)
Generalization across operations would produce the new production:
IF the goal is to generate an expression that has & as its dominant operation
and the current expression has * as its dominant operation
and the next-most-dominant operation is the goal-dominant operation (&)
and the dominant operation is to the right of the next-most-dominant operation
THEN create a new expression with the goal-dominant operation (multiplication) as dominant
1111e dominant (or !exit precedent) operation of an expression is the last one to be performed if variables are assigned
values and the expression evaluated. The next most dominant operation is second -to -last to be performed in
evaluating the expression. The goal
dominant
operation is the dominant operation in the goal expression.
80
94
and the previously-dominant-operation (exponentiation) as next-most-dominant
(and assign subexpressions appropriately)
With such a generalized but undiscriminated rule in place, the student, faced with an
expression like (a + b )2, and having as a goal to generate an expression that has "+" as its
dominant operation, but not yet having mastered the appropriate rule for achieving this goal, will
call upon the (over)generalized rule to derive a2 + b2. Eventually processes of discrimination
would constrain
such overgeneralization,
achieving
the
appropriately constrained rule
corresponding to the GDL:
IF the goal is to generate an expression that has & as its dominant operation
and the current expression has
as its dominant operation
and the next-most-dominant operation is the goal-dominant operation (&)
and the dominant operation is to the right of the next-most-dominant operation
and I* I = l&I+1
THEN create a new expression with the goal-dominant operation (&) as dominant
and the previously-dominant-operation () as next-most-dominant
(and assign subexpressions appropriately)
Such derivations go some way toward explicating the development of rule structures;
however they are incomplete. A complete theory also must account for the presence of the
original rules, in their given form. In Anderson's (1986) linguistic example (above), the goal
structure "to generate the present tense of HUG" (p. 205) implies that the category of present
tense previously has been abstracted from the child's linguistic experience. Thus a full account of
the present-tense rule would have to explicate this process of abstraction. (It seems plausible that
tense differentiation could be motivated in terms of pragmatic communication needs of the child;
but such an account must be given for the above explanation to be complete.)
In the GDL derivation it can be observed that the operations in the given and derived
expressions of the curricular rules are linked (recall the reference to "goal-dominant operation" in
both the condition and action statements). If this were not so, and the rules were presented as
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95
IF the goal is to generate an expression that has & as its dominant operation
and the current expression has as its dominant operation
and the next-most-dominant operation is &
and the dominant operation is to the right of the next-most-dominant operation
THEN create a new expression with & as dominant
and
as next-most-dominant
(and assign the subexpressions appropriately)
with no mechanism to link together occurrences of & or', the structure would be too complex for
generalization to occur.
Can the richer representation required for generalization be justified in the terms of
traditional cognitive (computer science-inspired) theory? Perhaps not! In standard applications
(e.g. Bundy, 1975; Carry, Lewis & Bernard, 1980), computational production rules are
condition/action pairs with no necessary rational association between the condition and the action.
It is perhaps a peculiarly human form of representation that results in the perception of
(a +b)c = ac +bc
not as a rule for writing a new expression from an existing one, but as a rule for re-forming a
single expression. In this sense, distributivity may be an emergent property of formally fixed
expressions, rather as movement in a motion picture (movie) emerges from individually fixed
stills.
This analysis is suggestive, not conclusive. It appears that the GDL could not easily be
induced by a computer programmed with unembellished condition/action productions, but that it
might be induced by a cognitive system endowed with richer representations. Furthermore, the
human cognitive system would seem to be predisposed to such rich representation, either as an
artifact of visual/perceptual functioning, or perhaps as a result of vast natural-language experience
with syntactic forms identical in structure to algebraic rules:
e.g. (Dogs and cats) are animals) -4 (Dogs are animals) and (cats are animals)
BEST COPY AVAILABLE
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1. ,REFERENCES
Anderson, J. R. (1986). Knowledge compilation: The general learning mechanism. in R.S.
Michalski, J.G. Carbonell, & T.M. Mitchell (Eds.). Los Altos, Ca: Morgan Kaufmann:
Machine learning: An artificial intelligence approach, V olll, 289-310.
Anderson, J. R. (1983a). The architecture of cognition. Cambridge, MA: Harvard University
Press.
Anderson, J., R. (1983b). Acquisition of proof skill in geometry. In Michalski, R., S., Carbonell,
J., G. & Mitchell, T., M. (Eds.) Machine learning: An artificial intelligence approach. Palo
Alto, CA: Tioga Publishing Company, 191-219.
Anderson, J. R., & Reiser, B. J. (1985). The lisp tutor. Byte, 10, 159-175.
Budden, F. (1972). Why structure? Mathematics in Schools, 1(3), 8-9.
Bundy, A. (1975). Analysing mathematical proofs. DAI Report No. 2, Department of Artificial
Intelligence, University of Edinburgh.
Carry, L. R., Lewis, C. & Bernard, J. E. (1980).
Psychology of equation solving: An information processing study Tech Rep. SED 78-
22293). Austin: The University of Texas at Austin, Department of Curriculum &
Instruction.
Davis, R. B. (1979). Error analysis in high school mathematics conceived as informationprocessing pathology. Paper presented at the Annual Meeting of the American Educational
Research Association, San Francisco. (ERIC Document Reproduction Service No. ED 171
551).
Davis, R. B., & McKnight, C. C. (1979). The comeptualimtinn of mathematics learning as a
foundation of improved measurement (Development report number 4. Final report).
Washington, D. C.: University of Illinois, Urbana/Champaign. (ERIC Document
Reproduction Service No. ED 180 786).
Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Afathematics
Education, 20(3), 274-287.
Kirshner, D. (1987a). The grammar of symbolic elementary algebra. Unpublished doctoral
dissertation. Vancouver: University of British Columbia.
Kirshner, D. (1987b). The myth about binary representation in algebra. Proceedings of the 1987
Annual Meeting of the International Group for the Psychology of Mathematics Education.
Lamer], K. W. (1978). Errors in first-year Algebra. Mathematics Teacher, 71 (3, 194-195.
Matz, M. (1980). Towards a computational theory of algebraic competence. The Journal of
Mathematical Behavior, 3.93 -166.
Neves, D. M. & Anderson, J. R. (1981). Knowledge compilation: Mechanisms for the
automatization of cognitive skill. In 1. R. Anderson (Ed.) Cognitive skills and their
acquisition. Hillsdale, NJ: Lawrence Erlbaum Associates, 57-84.
Schwartzman, S. (1977). Helping students understand the distributive property. The Mathematics
Teacher, 70(7), 594-595.
Singley, M. K., & Anderson, J. R. (1985). The uatisfel of text-editing moil. Journal of ManMachine Studies, 22, 403-423.
Smith, B. D. (1981). Misguided mathematical maxim-makers. Two-Year College Mathematics
Journal.
Wagner, S., Rachlin, S. L. & Jensen, R. J. (19841.
Algebra learning project: Final report (Contract No. 400-81-0028). Washington, DC:
National Institute of Education.
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EMBEDDED FIGURES AND STRUCTURES OF ALGEBRIC EXPRESSIONS
Liora Linchevski and Shlomo Vinner, Hebrew University Jerusalem
In this paper we try to clarify some relationships between success
in the well known embedded figure test and success in what we call a
hidden algebraic structure test. We claim that both tests require
certain visual-analytical abilities. The visual-analytical ability
required for the hidden algebraic structure test is probably a major
component of the ability to handle high school mathematics and
therefore there is a high correlation between success in the hidden
algebraic structure test and a common mathematics test. Analysing our
present research data one can hypothesize that the ability to
identify hidden algebraic structures does not depend on age but does
depend on the immediate algebraic experience in the period prior to
the day on which the test was taken.
In many algebraic tasks it is crucial for the student to identify
certain structures in given algebraic expressions, structures that
sometimes cannot easily be seen on the surface. For instance, when a
student is asked to add 1/(a2
b') + 1/(a4
helpful if he or she realizes that a4
(a')2
b4) it is more than
b can be considered as
(132)2 and therefore can be written as (a2
he or she should solve: (x + 1)2
b2)(a2 + b2). If
7(x + 1) + 12 = 0 it will be much
easier to consider x + 1 as one "entity" and solve z2 - 7z + 12 = O.
It is only natural to hypothesize that this ability to identify
"hidden structures" in algebraic terms is one of the components of
success in school mathematics. Therefore, it is natural to expect a
certain correlation between this ability and success in common
mathematics achievement tests. On the other hand, if you want to
think of a general ability from which the above particular ability is
derived, it seems that this general ability should be the ability to
identify certain simple figures hidden or embedded in a complex
configuration. This ability is measured by the well known embedded
figure test (EFT).
Because of the common way to report about psychological research
(namely, without including the test items),
a mysterious predictive
power is associated with the EFT. The reason for this is that the EFT
correlates with too many "things", especially with intellectual
achievements (Witkin (1977), McNaught (1982), El-Famamaury (1988) and
85
many others). However, a simple analysis of the EFT items indicates
that it measures the ability to distinguish certain details from
their context. This is a general analytical ability and when implied
in particular situations, which require analytical ability, results
in high success. This was already implied by Witkin and Goodenough
(1981) who associated "field-independence" with being analytical.
We, here, do not wish to involve the entire theory of field
dependence with our research questions. It will be totally redundant.
What we said above will be enough to explain our findings.
Furthermore, we do not wish to elaborate on the dispute whether
field-independence (high success in EFT) is a cognitive style or
ability. We will call it ability being aware that Witkin and
researchers consider it as a cognitive style. Our first task was to
construct a mathematical test that will measure directly the
algebraic ability mentioned above, namely, the ability to identify
certain hidden algebraic structures in given algebraic terms. We
denote it by NAST. Kieren (1988) related to the above situation using
different terminology. She speaks about the surface structure and the
systematic structure of a given algebraic term. The surface structure
is, more or less, what you see on the surface. The systematic
structure is "all the equivalent forms of the expression according to
the properties of the given operations (p. 434). Thus, theoretically,
according to Kieren, the systematic structure of a given algebraic
expression is an infinite set of equivalent algebraic expressions.
However, from a practical point of view, we are not interested in the
set of all equivalent expressions. We are interested only in one or
two expressions which are relevant to our algebraic task. That is the
reason we prefer to speak about hidden algebraic structures and not
about the systematic structure. In addition to that, a hidden
algebraic structure in our approach can be a surface structure in
Kieren's approach. For instance, when considering 5 + 3(x + 2)(Kieren
(1988), p. 434) in a certain context we can claim that 5 + 3z is a
hidden structure of 5 + 3(x + 2) and preserving this structure while
carrying out a certain algebraic task can be very helpful. For Kieren
5 + 3Z is the surface structure.
Kieren, of course, formed her
terminology for theoretical purposes different than ours.
Our research questions were:
1.
Is there a statistically significant correlation between our MST
and EFT?
99
86
Is there a statistically significant correlation between our HAST
2.
and a common mathematical achievement test?
Is success in HAST correlated with age?
3.
The first two parts of HAST included items which are taught already
in grade 8. Hence, if the ability to identify hidden algebraic
structure is mainly innate and it is formed basically by the
introduction of the relevant algebraic topic, then elder students are
not expected to do better than younger students. In other words,
algebraic experience will not have a major role in the formation of
the above ability.
Method
1) Sample: Our sample included four grade levels (grade 8 - grade
11), each of them devided into two groups, high level and low level
students. The division was done by the school and as every division
it is not hundred percent reliable. Groups 1, 3, 5,
7 are the low
level students of grades 8, 9, 10, 11 respectively and groups 2, 4,
6, 8 are the high level students of grades 8, 9, 10, 11
respectively.
2) Questionnaire: Our hidden algebraic structure test (HAST) had
three parts. In the first part, after seeing one example, the student
was asked whether a given algebraic expression could be obtained from
a + b by substitution. If the student answered positively, he was
asked to state exactly what should be written instead of a and what
should be written instead of b in order to obtain the given algebraic
expression. Only positive answers with correct substitutions were
considered as correct answers. The algebraic expressions of part A
were:
1)
3x + 2y
2)
a2 -b2
3)
x2 + 3y + z
4)
a + b + c
5)
-b + a
6)
x2
7)
b + 5(c - d)
(the items will be denoted by Al
A7).
In part B the situation was quite similar with the only difference
that the expression in which the student was supposed to substitute
was: (a + b) (a
1)
(1 + x2)(1
3)
(-b + a) (-b
5)
(b + 462
b). The algebraic expressions of part B were:
x2)
2)
(6 + x)(x
a)
4)
(a + b + c) (a
4ac)(b - .1b2
6)
4ac)
1-00
87
b + c)
B5.
These items will be denoted by B1
The third part of the questionnaire was administered only to grades
10 and 11, since its items were related to the solution formula of
the quadratic equation, a topic which was studied only by the 10th
and the 11th grades in schools where the questionnaire was
administered. The third part started with the following introduction:
It is common to present a quadratic equation with the unknown x as:
ax' + bx + c = 0, where a, b, c are the coefficients that can be
specific numbers but also letters. It is known that the solutions of
the quadratic equation,if exist, are given by:
-b + 4b3
4ac
2a
x1,2In the following there are several equations. For which of them it is
possible to substitute in the above formula in order to find the
soultion. When it is possible, please make the appropriate
substitution. You are not asked to calculate the final solution.
The equations were:
5 = 0
1)
x2 + 2x + 1 = 0
2)
x2 + mx
3)
x2 + ax + cx = 0
4)
x2
6)
-x2 + 7x = 0
5)
9x2
3mx - 6x
m + 1 =
0
2x + 1
m2 = 0
C6. It is worthwhile to mention
These items will be denoted by Cl
that part B and C included some items that have not been reported
above. These were items with negative answers. Many weak students who
could not see the appropriate substitutions in the positive cases
claim also about these items that it is impossible to obtain them
from the given expression or it is impossible to solve the equation.
We considered these answers as "false positive" and excluded them
from our analysis. The items were: (1 + x)(1
y) and (-x + y)(x
y)
in part B and ax + b + 1 = 0 in part C.
The second test which was meant to be administered to our sample was
a simple group form of the embedded figure test. Because of
administrative difficulties only 120 students of the entire sample
(N = 322) wrote this test being a partial population of groups
3, 4, 5, 6 (the 9th and 10th graders). A representative item of the
EFT can be the following:
There is a simple figure (Fig. B).
Does this figure appear in
configuration A and where?
101
as
The student is supposed to draw his answer as in Fig C.
There was also a common achievement test that was administered to all
the 10th graders. It was a classification test which was designed to
distinguish between the low level and the high level students at the
end of the school year.
Three typical questions out of 10 in this test were:
I
A two digit number is given. Write 1 at its right side, then
write 3 at its left side and then add the two numbers which were
formed. The result is 598. What is the given number?
II
Find the area of a quilateral triangle if the diameter of the
circle inscribed in it is 8 cm.
III Find a, b if it is known that a, b are integers such that
(a + b)(a
b) = 41.
Results
Before bringing the data which relates directly to our 3 research
questions we would like to bring some information concerning the
HAST. In the following we will list the questions of each part of the
test according to their difficulty order. In parentheses we will note
the percentage of the correct answers. The number of respondents to
parts A and B is 322. The number of respondents to part C is 179.
A1(88.5), A5(71.5), A7(50.0), A2(47.0), A3(42.0), A4(35.5), A6(32.5)
B1(69.5), B3(52.5), B5(41.5), B2(17.0), B4(2.0)
C1(70.5), C2(55.5), C6(22.5), C4(22.5), C3(13.0), C5(9.5)
Note that only 1/3 of the respondents could see that x2 can be
obtained by substitution from a + b (for instance, a = x', b = 0).
The fact that the plus sign does not appear in the expression was
probably the cause that so many students did not see the hidden
structure a + b in it. The items in part B were harder and even much
harder than the items in part A. Only 17% could see that
(6 + x)
(x
0 is equal to (x + 6)(x
6) and therefore can be
obtained by substitution from (a + b)(a
b). Only 7 students out of
322 that answered the questionnaire could see that
(a + b + c) (a
b + c) is equal to ((a + c) + b) ((a + c)
therefore could be obtained from (a + b)(a
b) and
b) by substitution. It
is interesting that C6 and C4 had the same degree of difficulty in
our sample. The difficulty in each of them is related to the free
coefficient of the quadratic equation. In C6 the problem is to
identify 0 (which is not written) as the free coefficient. In C5 the
problem is to identify 1
m2, a complex expression, as the free
coefficient, which in the schema of the quadratic equation is denoted
89
102
by a single letter. The three above sequences are Gutman scales
(roughly speaking, almost everybody who answered correctly a given
item, had answered correctly all the items which are easier than this
item). The coefficients of reproducibility are 0.84 for the first
sequence and 0.94 for the second and the third sequences. Additional
information about part C is that only 45.5% thought that the solution
formula of the quadratic equation is inappropriate for solving ax + b
+ 1 = 0 (an item which we only mentioned in the previous section but
did not include in the above sequences). Taking into account the
information in the third sequence it can be claimed that at least 3/4
of the respondents cannot identify the structure of the quadratic
equation when its form is "much" different from the common
prototypical form (items C6, C4, C3 and C5). Only 70.5% of the
students know how to use the solution formula even in the simplest
case (C1). This is amazing because this topic is a central one in the
curriculum and the solution formula itself was given in the
questionnaire. In C2, where a very simple parametric form of the
quadratic equation appeared, the success level dropped to 55.5%.
The coefficients of correlation between HAST and EFT and between HAST
and the mathematics classification test are given in Table 1.
Table 1 Coefficients of correlation between the various tests
Mathematics classification test
HAST
EFT
N = 81
N = 120
r = 0.85
r = 0.75
p = 0.001
p = 0.001
Thus the first' two of our research questions are answered positively.
The answer to the third question is not so clear and in order to
relate in a non-superficial way we would like to present to the
reader some tables.
Table 2 Percentages of correct answers and means
to questions Al
A7 in groups 1
8
1(N
2(N
3(N
4(N
5(N
6(N
7(N
8(N
=
=
=
=
=
=
=
=
18)
31)
30)
64)
82)
59)
17)
21)
Al
A2
78
97
93
97
76
93
82
95
A3
A4
AS
A6
A7
mean
11
17
11
6
42
32
33
45
32
32
18
61
21
39
65
69
56
71
87
83
60
86
35
67
6
38
43
26.4
53.9
46.7
65.0
42.0
70.1
29.4
54.4
39
61
24
38
103
33
71
12
52
44
22
63
12
38
90
29
24
20
56
44
78
12
67
Table 3 Percentages of correct answers and means to questions B1 B5
Question
B1
B2
B3
B4
B5
mean
Group
28
74
53
86
59
90
41
76
1
2
3
4
5
6
7
8
0
11
29
45
50
77
39
7
22
12
27
0
19
73
18
52
0
0
0
36
7
3
3
0
48
34
2
81
0
6
10
66
7.8
36.8
24.0
47.2
28.8
54.6
13.0
44.6
Table 4 Percentages of correct answers and means
to questions Cl
C6 in groups 5
8
Question
Cl
C2
C3
C4
C5
C6
mean
7
70
86
18
8
71
5
6
49
78
6
57
11
12
6
29
13
5
39
0
29
15
0
19
20
25
12
33
28.0
42.5
7.0
39.7
Table 5 Means of correct answers to parts A and B of BAST
Group
1
2
3
4
5
6
7
8
Mean
50.3
18.6
46.8
37.3
57.3
36.5
63.7
22.6
Table 6 Means of correct answers to the entire HAST
Group
5
6
7
8
Mean
33.7
56.6
17.4
46.8
If we look only at the means of part A, the easiest part of BAST, we
discover that there is almost no difference between either good or weak
students of the 8th grade and the either good or weak students of the
11th grade. There is an improvement in grades 9 and 10 but there is
almost no difference between grades 9 and 10. Our guess is that this
improvement is due to the fact that in grades 9 and 10 a lot of
attention is given to the manipulation of algebraic expressions. Hence,
the experience with algebraic expressions contributes quite a lot to the
ability to identify hidden algebraic structures. On the other hand,
after the period of intensive manipulations on algebraic expressions is
over, the ability decreases and stabilizes around the level it was in
the 8th grade.
As to the means of part B, the picture is even more complicated. There
is an improvement from grade 8 to 9 and from grade 9 to 10. However,
there is a regression from grade 10 to grade 11. These results can be
explained in a similar way to the previous one. The items of part B
belong to a repertoire of exercises which appear very frequently in
grades 9 and 10.
These exercises usually disappear from the 11th grade
repertoire. Thus, the ability to discover hidden algebraic structures
91104
decreases and stabilizes somewhere between the level of the 8th grade
and the level of the 9th grade. The same picture is discovered if you
look at the means of part A and part B together (Table 5). The above
arguments also explain Tables 4 and 6. Namely, immediate experience with
algebraic expressions improves the ability to identify hidden algebraic
structures but the moment this experience stops, the ability decreases
and stabilizes quite close to the point of its function. Note that we
analysed our data as if it were developmental data whereas, what we
really did was comparing different groups of different ages. This is
quite common in educational research when comparing age levels and it is
based on some reasonable assumptions, however it should be noted.
In order to neutralize the effect of immediate algebraic experience on
the ability to identify hidden algebraic structures, perhaps a different
and more sophisticated research design is needed.
References
El-Faramawy, H. (1988). Some cognitive preference styles in studying
mathematics. Proceedings of the 12th International Conference for the
Psychology of Mathematics Education, Hungary, 1, 271
279.
Kieran, C. (1988). Learning the structure of Algebraic expressions and
equations. Proceedings of the 12th International Conference for the
Psychology of Mathematics Education, Hungary, 2, 433
440.
McNaught, C. (1982). Relationship between cognitive performances and
achievement in Chemestry. Journal of Research in Science Teaching,
10, 77
86.
Witkin, H. A. (1977). Educational implication of cognitive style. Review
of Educational Research, 47(1), 1
64.
Witkin, H. A., Goodenough, D. R. (1981). Cognitive Styles: Essence and
Origin. International Universitk Press, Inc. New-York.
105
92
II
I
II
.
is it
.1
I
I
I
I
Romulo C. Lins
Shell Center for Mathematical Education / CNPq- Brasil
On this paper a framework (the Numerical-Analogical framework) is
proposed in order to provide a reference for investigations (both theoretical and
experimental) on the nature of Algebraic Thinking. The framework is described and
its adequacy is demonstrated by examining: experimental evidence from students'
work (both new and previous findings), the historical development of algebra and
algebra as a subject-matter in Mathematics. A characterisation of Algebraic
Thinking on the basis of the Numerical- Analogical framework is provided. The
belief that Algebraic Thinking can only happen in the context of algebraic
symbolism is shown to be erroneous and misleading.
But neither of them was able to prove the theorem, and Waring
confessed that the demonstration seemed more difficult because no
notation can be devised to express a prime number. But in our opinion
truths of this kind should be drawn from notions rather than from
notations"
C.F. Gauss, on Wilson's theorem, in Disquisitiones
Arithmeticae
j. Introduction
Until now, a substantial amount of information has been gathered on the learning of
school algebra (eg, Collis,1982;KLichemann,1984; Wheeler & Lee,1987; Be11,1987), but
nevertheless, a clear characterisation for "Algebraic Thinking" is still missing (Kieran,1989;
Lee,1987).
As a whole, that research has been strongly focused on investigating Algebraic
Thinking as the mode of thinking that goes with "doing algebra" (either interpreting or
manipulating algebraic statements or using algebra to solve problems and explore situations),
rather than the mode of thinking that allows the development of algebra.. A consequence of this
"content-driven" approach is that the students' "infomml" solutions have been characterised more
in terms of misinterpretations and failure to "understand" and less in terms of what they are
actually doinR .
L. Booth suggested that the sources of those misunderstandings (or lack of
understanding, as it might be more adequate) are to be found in an incompatibility between the
"informal" methods used by the students and the methods of algebra rather than in developmental
obstacles (in the sense of Piaget) (see, for example, Booth, 1984). We strongly share this point
of view, and investigating the nature of those -informal"'solutions at the same time we
investigate the nature of "algebraic" solutions, has been the central objective of a set of studies
carried out by the author for the last two years, aiming at identifying possible source(s) for that
incompatibility.
A framework that helps us to understand the twofold nature of this question, is one that
enables us to handle the different meanings that can be attached to the elements involved in the
situation that is being dealt with by the students: numbers, operations and arithmetical and
algebraic symbolism (where they are involved), but also the imagery suggested or provided by
the situation or used as a support for reasoning (the context of "realistic" problems, diagrams,
etc). In speaking of "meaning" we are inevitably led to referentials, and this is what our
framework has to provide in the first place: a description of different fields of reference in
which different interpretations of those elements produce solutions of different nature.
This paper is a result of the work being carried out by the author as pan of his PhD studies, under the
supervision of Dr Alan W Bell, at Nottingham University.
106
A first important consequence of thinking in terms of distinct fields of reference within
which the elements of a situation are interpreted, is that our approach is not content-driven: the
same framework can be applied to the analysis of solutions of "realistic" and "purely numerical"
problems, problems set in algebraic language and "verbal" problems. Also of considerable
importance, such framework can be applied to the analysis of the algebra of the "ancients", and
this might shed some new light onto a possible paralellism between the historical development of
algebra and the aqcisition of algebraic thinking by individuals.
In the next four sections such a framework is sketched and support for its adequacy is
drawn from three sources: the historical development of algebra, algebra as a theoretical
discipline and empirical evidence from investigations on students' solutions. On the last section
we return to the framework and its characteristics are fully described.
2.THE NUMERICAL-ANALOGICAL FRAMEWORK
Our framework distinguishes between two basic fields of reference: the Numerical
field of reference and the Analogical field of reference.
To operate within the Numerical field of reference means that only the
"arithmetical" environment is relevant to the process of manipulating or exploring a situation. If it
is the case of solving a problem, the problem is solved through the manipulation of the numerical
relationships contained in or described or allowed by it, and this process is guided by the
arithmetical structure of those relationships and by the principles that are recognized as governing
the arithmetical environment.
To operate within the Analogical field of reference means that a situation is
manipulated or explored by manipulating features of the situation itself. Arithmetical operations
are used to evaluate parts, and the choice of operation to be used is made on the basis of a
qualitative analysis of the situation or problem that is being examined.
The framework we propose here has two fundamental characteristics:
(i) it rejects the idea of a "pre-algebraic" mode of thinking, something that when
extended or further developed leads to an "algebraic" mode of thinking; we use instead the idea of
a "non-algebraic" mode of thinking; the "meaninglessness" pointed out by students is interpreted
not in terms of the "meaninglessness" of algebra itself , but in terms of the shift of referential
that is necessary to operate within the Numerical field of reference.
(ii) the N-A framework is concerned with the process of solution, not with the
problems to be solved or the situations to be structured. As a result, the use of algebraic (literal)
notation does not characterise any of the two modes. Although solving a "purely" algebraic
problem using algebra (eg, formally solving an equation written in symbolic notation) is certainly
an activity that develops within a Numerical field of reference, the same "purely" algebraic
problem might be solved within an Analogical field of reference (for example, modelling it with a
scale balance). Also, the general description of the number of, say, dots on a geometrical pattern
"using letters", for example, is typically Analogical, because the choice of operations to be used
in the description depends only on the way in which the pattern is visually perceived, but a
"purely arithmetical" problem can be handled in a typically Numerical way (eg,
[157+157+157+157+1571
5 = 157 because there are five 157's, etc.).
3.From the historical development of Algebra
Westernly, the historical development of Algebra has been referred to as a succession
of three phases: rethorical, syncopated and symbolic (Joseph, 1988, is a brief but excelent
appraisal of Eurocentrism in Mathematics). The first phase is associated to pre-greek 'algebra',
the second with the work of Diophantus and the third with the work of Viete and Descartes. (eg,
Hogben, 1957). This description clearly corresponds to a development of algebra as a subject
matter, given our modern definition of Algebra as a form of "symbolic calculation", and this is
thoroughly expressed on the usual assertion that Viete was the first to produce "truly" algebra.
Jacob Klein's work (Klein,1968, originally published between 1934 and 1936)
radically departs from this line of analysis. It shows, based on a deep reading of Greek classical
texts and on a careful study of Viete's work and of the cultural and conceptual context
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surrounding him, that Viete's deeper achievement was not simply the development of a symbolic
notation (his, after all, was to some extent still "syncopated" and full of geometrical
suggestions...), but shifting algebra from "solving problems" _to "a method for solving
problems" . Viete himself comments on his work saying "TO LEAVE NO PROBLEM
UNSOLVED". The way in which Viete achieves his goal is by bringing the solution of the
problems entirely into the context of number and for this reason his work is about how to
proceed within a (general) numerical context. Klein's work, however, does not consider similar
developments outside the Diophantus-Viete axis.
The work of arabic mathematicians from al-Khwarizmi (c.800) onwards share the same
Numerical character of Viete's, and if in many instances careful attention is paid to the process of
'translating' the problems into a suitable Numerical form (Rashed,1984, p20), this does not
mean that "solving problems" was the 'raison d'etre' of their work . In fact, the arabic algebra
extends itself over "algebraic" powers, operating with polinomials, normal form of an equation,
polinomial equations of higher degree, and a number of topics in Number Theory, a body of
knowledge that makes Viete's "Introduction to the Analytical Art" look like a first book in school-
algebra . It has to be stressed however, that until at least the 12th century the
arabic algebra is totally "rethorical", and even the work of al-Qalasacli
-
15th
century - is still in a "syncopated" form (for example, the use of distinct symbols for x
and x2).(Cajori,1928, items 115,116,118,124)
The nature of the mode of thinking that generates such knowledge is partially explained
in the words of an arabic mathematician As Samaw'al (12th century) who said that algebra
was concerned with "...operating on the unknown using all the instruments of arithmetics, in the
same way in which the arithmetician operates on the known [values]" (Rashed, p27). This
comment is better understood in the context of the process of "arithmetisation" which algebra
underwent after the pioneer work of alKhwarizmi, a process that consisted in restricting the
methods of algebra to those of "arithmetics" (Rashed, p32, but also analysed in many other
places in the book. It is particularly interesting to consider the link that Rashed establishes (p25)
between al-Khwarizmi restricting himself to equations of the 1st and 2nd degrees and his
conception of proof [to a great extent geometrical]). The process of "arithmetisation" undergone
by algebra in this period corresponds, in the context of the epoch, to the process of "abstraction"
that algebra underwent during the 19th and 20th centuries: the substitution of a collection of
procedures for solving "classes" of problems (later: a collection of results about specific
systems, "arithmetical" and "non-arithmetical") by a method that allows us to attack problems in
any of those classes (later: an "abstract" system the results from which can be applied to all those
particular instances of systems). Algebra becomes an autonomous discipline (later: Abstract
Algebra becomes an autonomous discipline).
A less explicit but equally distinctive aspect of the arabic algebra, is the fact that
once a "contextualized" problem is represented in terms of arithmetical relationships, the process
of solution develops entirely within the Numerical field of reference. It is for this reason that
careful attention is given to the process of "translation": from that point on. the "context" would
pot provide a source of reference: if the arithmetical relationships do not accurately correspond to
the problem. the algebraic method could not detect the mistake and the Numerical process of
solution would result in a waste of time (to say the least). This "internalism" is made possible by
the development of algebra as a "theoretical" discipline (Rashed, p20) already clear in alKhwarizmi's use of normal forms of equations at the same time it makes possible further
developments in algebra. As Klein points out throughout Part II of his book, this kind of
"internalism" was not possible in Diophantus, especially because of his conception of number
(the conflict between the "pure" number and the "number of things" and the concept of eidos as
the only possible form of "general number").
Those two principles "arithmeticity" and "internalism" are also characteristic of
Viete's work, and to such an extent implicitly taken by him that they become almost transparent
by staying always in the background of the symbolic invention. However "hidden", these are
exactly the principles that support Viete's creation of a "symbolic calculus". (for those who
wishfully think that Vieta's algebra is totally context-free, let us remember that he had different
symbols for subtractions where one number was known to be greater than the other and
subtractions where this was not known)
What becomes evident with this picture in view, is that a content-driven approach to
understanding the Algebraic mode of thinking leads us to miss the point that the "symbolic
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108
calculus" of algebra was but a consequence of the development of a body of knowledge that
already embodied the calculus (hisab, for al-Khwarizmi) that is progressively made
"symbolic".
We think that it is totally adequate, then, to characterise Algebraic Thinking as the
mode of thinking that produced
from the arabic mathematicians on, to our
knowledge
the "theoretical" discipline we know as Algebra. As a consequence,
"arithmeticity" and "internalism" are features of thinking algebraically. As we said before,
"abstraction" would replace "arithmeticity" in a more general characterisation, but we will keep
the latter for two reasons:
(i) Our primary interest is in the development of an algebraic mode of thinking; schoolalgebra is an algebra of numbers, as Algebra was for a very long period of time ;
(ii) We think that by using "abstraction" one reinforces the idea of an absolute "lack-ofmeaning", which we deny as misleading.
4. From Aleehra as a subject-matter in Mathematics
A simple way of defining Abstract Algebra is to say it is "the study of algebraic
systems", an algebraic system being composed by a set, one or more algebraic operations defined
on it and a set of axioms which have to be satisfied by the operations. An algebraic operation on a
set A however, is a function from A" onto A, and this means that the set A is mentioned
separetely not because its elements are relevant in any sense, but because we want all the
operations to refer to the same set. This is, in a sense, the result of the evolution of the
"internalism" mentioned in the previous section: the operations are defined internally and they all
refer to same set of elements; no other reference is needed. Because we do not want to refer to
anything else "external" (particular), the elements are "abstract", and the only way to do any kind
of manipulation within this system is on the basis of the properties of the operations. This allows
us generality, as operations are "globally" defined. In a very similar way, if one is solving an
equation in a "purely numerical way", one has to do it on the basis of properties of the
arithmetical operations.
This characteristic of Algebra means that in Algebra operations become objects,
ie, they are a source of reference, they have properties. This is true both for "number
algebra" as it is for Abstract Algebra.
When dealing with school-algebra, it is usually useful to think in terms of operators
(eg, "1-2") instead of in terms of bynary operations (Kirshner, 1987), but this does not
essentially alter our point, because the operators are built from the arithmetical operations.
Moreover, as a consequence of Algebra being used as a method, ie, generally applicable, we are
left in fact with only four arithmetical operators (viz., +a, -a, xa, +a).
This analysis of Algebra as a subject-matter helps us to understand an aspect central to
much of the discussion about Algebraic Thinking: that of meaning.
When a problem or situation is modelled in terms of arithmetical relationships, the
objects that provide information on "what can be done to manipulate those expressions" are, as
we saw, the operations and their properties, this corresponding to an algebraic treatment. On the
other hand, when an Analogical model is used the numbers are associated, as "measures" (or
operators operating on "measures", eg, "3 buckets"), to some other object; if one is dealing with
a "purely numerical" problem, the numbers might be associated, for example, to parts and
wholes; those other objects and their "qualitative" structure are the elements which provide us
with information on "what can be done to solve the problem". One knows which operation to
perform and with which numbers because each operation corresponds to an evaluation and the
numbers are "attached" to the parts involved.
What is "lost" in a Numerical process of solution is exactly this Analogical reference on
"what to do with the quantities", and this is the meaning of "meaningless" that could be applied to
an algebraic solution. ("it is meaningless"
"I can't see how those elements tell me this is
what I should have done")
I
a
11
.1
I
1
1,
I.
(I) Harper (1987) analysed solutions to the problem "If you are given the sum and the
difference of any two numbers, show that you can always find out what the numbers are", and
109
96
identified three groups of answers that correspond to "rethorical" (totally verbal), "Diophantine"
(or "syncopated"; symbols only for the unknowns) and "Vietan" (or "symbolic"; symbols for the
unknowns and for the given [general] values) answers.
One has to notice however, that all three kinds of solutions are general, in the sense of
being generally applicable to any sum and difference given and they are thus
undistinguishable from that point of view. Moreover, Viete's answer to the problem
(p88) totally corresponds to the "rethorical" answer presented on p81, apart, of course, the use of
letters (and this is correct even to the extent that Viete's answer is 'I2D t /2B and not 1/2 ID-B1
). Whenever a correct "rethorical" answer is not accompanied by an explanation as to how the
result was obtained (as it is the case with the one presented on p81), one has to consider that the
process of "thinking out the problem" (p80) could correspond to anything, including Viete's
method.
The important point here is that although lacking symbolic generality, "rethorical" and
"Diophantine" solutions might eventually involve much of the same mode of thinking that a
"Vietan" solution does (we emphasised the "eventually" because an Analogical solution is also
possible on all three 'styles'
).
Harper's classification of solutions is certainly useful to describe differences in the use
of mathematical symbolism, but by itself it does not provide a framework that enables us to
distinguish different modes of thinking.
As a consequence we are again led to the necessity of a framework that takes into
consideration the ways in which solutions are produced, ie, which are the sources of reference
used, and this is exactly the focus of attention of the N-A framework.
(II) Lesley Booth's follow-up study of the CSMS survey (Booth, 1984) produced a
number of important findings. Although primarily concerned with situations that involve the use
of letters, Booth's conclusions point out to the necessity of understanding children's "informal"
methods if we are to understand the nature of the gap between non-algebraic and algebraic modes
of thinking.
Of particular interest to us is her characterisation of the "child methods"(p37): "(1)
intuitive, ie, based upon instinctive knowledge: not systematically reflected upon and not checked
for consistency within a general framework; (2) primitivt, ie, tied closely to early experiences in
mathematics; (3) context-bound ie, elicited by the features of the particular problem; (4)
indicative of little or no formal symbolized method; (5) worked almost entirely within the system
of whole numbers (and halves)".
If those "methods" are seen as based on a qualitative analysis of the situation presented
(an Analogical approach), the first four characteristics follow as a consequence: context-bound
because the solution depends on understanding a particular situation and the possibility of
manipulating its elements to perform evaluations; non-systematiized because of the obvious "one-
off" (or even "few-off') character of the solutions; intuitive because non-systematic, but also
probably because the knowledge required to perform the qualitative analysis is not seen as
mathematical knowledge; little or no symbolization both because the strategies actually used to
"think the problem out" comparing, decomposing and recomposing wholes, for example are
easily and accurately described by verbal statements, and because "thinking the problem out"
(using the strategies) is of a different nature than "working the problem out" (the actual
evaluations, the performance of the operations). Symbolic notation might be used to describe
but this does not contribute to the process of solution itself . [This is not the case with a
Numerical solution, because the operations are at one time the source of reference and the
instruments used to manipulate the information: a concise and homogeneous notation which is
intended to be manipulated is adequate and possible] Those "methods" are primitive because the
operations can retain their original role, that of being tools for evaluation. (the latter idea is
also conveyed, in a slightly different form, in the assertion that children see operations as
"something to be performed" [Booth,op. cit. , pp90-91]).
Three of Booth's research findings (pp85 and following) also provide evidence that an
Analogical approach is probably preferential to those students (the item numbers correspond to
the original text):
(1.c) " Some children are confused over the distinction between letters as representing
the value(s) or number(s) relating to a measure or object, and letters as representing the measure
or object itself. ...". From the point of view of the N-A framework, this could be interpreted as a
97
consequence of the students operating Analogically, ie, as the numbers are "numbers of things"
and as those "things" are the source of reference on what to do or on bow it works (more
specifically, the qualitative structure involving those "things"), it would be more natural to
represent primarily the "things" and not the numbers that correspond to them.
(3.b.i) "The context of the problem determines the order of operation" and (3.b.ii) "In
the absence of a specific context, operations are performed from left to right". Those two points
indicate the extent to which the operations are not constituted as objects and their use remain
subjected to other sources of reference.
(III) On a exploratory study carried out in Nottingham, England, in 1989 and reported
in Lins(1990), two groups of 3rd year secondary school students and a group of 4th year
primary school students were asked to solve a set of five "verbal" problems and to explain why
they did it that way. Both correct and incorrect solutions, together with the explanations, were
then analysed to determine whenever possible the source(s) of reference used by the students
to work the problems out.
Two of the problems used:
Carpenter: The stick on the top is 28cm longer than the one in the bottom; altogether they measure 160cm.
How long is each of them?
Buckets : From a tank filled with 210 liters of water I took 3 full buckets: Now I have only 156 liters left.
How many liters go into a bucket?
The analysis showed that in many cases the solutions were Analogical (eg, "to take 156
away from 210 to determine how much was taken by the 3 buckets" on BUCKETS), but it also
showed that in those cases where only the calculations were provided they corresponded in all
but one instance
to those that would be used with the simplest Analogical solution (for
example, when solving the Carpenter's problem, to begin with 160-28 but not with 160+28 and
never representing the difference as the result of a subtraction [as in xy=28] ). The overall
result of the analysis suggests that: (i) the use of an Analogical approach, as we define it, is
experimentally generally verifiable, and (ii) those students used mainly an Analogical approach.
The following fragments of an interview from another study (Laura , 10yrs5mths)
provide a clear example of the use of an Analogical approach: (the problem is "George and Sam
have £1.60 altogether, but Sam has 38p more than George does. How much does each of them
have?"; the emphasis on the transcription is ours)
int... how did you know that you had to take 38p away and not to add 38p?
Laura: If you added 38p... then... ahnn... if you added 38p then you wouldn't have, ahnn... you would have
more than £1.60 to start off with.., and it says you only have f 1.60.
Int: But if you take 38 away, then you have less than you had...
Laura: yeah... I think I was just trying to_get the 38p out of die wav for a bit ! and then...
Features of the situation act as constraints and source of reference in the process of
solving the problem.
(IV) In another study, we investigated the sources of reference used by six
postgraduate students in the University of Nottingham to validate given symbolic representations
as correctly describing a verbally given situation (a brief discussion is in Lins,1988). One of
them was the well known "students and professors" situation ("In a school tht are six students
for each professor,...", etc.). Two basic strategies were identified: (i) always tol refer back to the
verbalised situation, and (ii) to determine one correct symbolic representation and from it to
derive the correctness or incorrectness of the others on the basis of algebraic manipulation. One
of the students, who otherwise always referred back to the text and adopted as correct the
"wrong" representation 6S=P, when faced with the item 18P=3S simply divided both sides by
3 to obtain 6P=S and concluded it was not in agreement with the verbal description. Moreover,
she proved quite able to solve formally set equations and had no difficulty with the CSMS item
"Which is greater: 2n or n+2". The information gathered by this exploratory study suggests that
using an Analogical approach (in that above case modelling the situation by putting "blocks" into
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correspondence) is not necessarily the result of an inability to deal with "unclosed" or
"symbolic" expressions, but rather the result of structuring the situation using a referential that is
different from the referential that would produce a representation in terms of arithmetical
relationships.
(V) Friedlander et al.(1989) investigated, among other things, differences between
in the context of the problem
visual and numerical justifications, a distinction that corresponds
analysed, a "geometrical" problem to our N-A distinction.
7. Conclusion
The N-A framework was developed as part of our effort to provide a clear
characterisation of Algebraic Thinking. On its foundation is the assumption of two distinct fields
of reference (Numerical and Analogical).
Analogical
Nurnerical
Operating within the Numerical field of reference means that
only the arithmetical structure is relevant.
the objective of any manipulation is to derive new
arithmetical relationships that, because of its form, bring
with it new information about the initial relationships; In
doing so, one is guided exclusively by the operations
involved and their properties. Operations can have
properties because they are OBJECTS.
Operating within the Analogical field of reference means that
the relevant information is provided by the "qualitative"
structure (eg, bigger/smaller, decrease/increase, wholes/parts).
the objective of any manipulation is to make evaluations
possible; this is done through the manipulation of the
elements of the situation; comparing wholes and
decomposing wholes and rearranging the parts thus obtained
are typical Analogical strategies.
out.
because the guiding principles apply irrespective of the
particular arithmetical structure dealt with with the few
canonical exceptions that also apply to arithmetics, like
operating within the Numerical
division by zero, etc
field of reference has a strong character of
_
method; meaning belongs thus to the process as a whole.
(A METHOD TO SOLVE PROBLEMS)
Limits of the context arc taken as limits for the answer but
not for the process of solution
Operations are the
TOOLS with which the evaluations are carried
_
operating within the Analogical field of reference is
an activity bound by the specific "qualitative" structure, and
thus presents itself as a procedure' meaning belongs to
each step of the solution process. as the "qualitative" structure
changes with each new evaluation. (TO SOLVE A
PROBLEM)
Limits of the context are taken as limits for the process of
solution
Other important general features of the N-A framework are:
(1) The use of symbolic notation is not characteristic of operating within any of the two
fields of reference nevertheless, a symbolic notation that is intended to be manipulated is
possible and adequate when operating within a Numerical field of reference but not when
operating within an Analogical field of reference.
(2) The central distinction being made is between ways of interpreting the elements of
problems and situations and not between the consequences of different interpretations;
(3) It avoids the idea of "pre-algebraic" and "algebraic" modes of thinking that is
inherent to the content-driven arithmetical-algebraic distinction; this offers us a perspective of
analysis of the learning process different from that of developmental stages.
99
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AVAIL&
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From the point of view of the N-A framework, Algebraic Thinking is naturally defined
as the mode of thinking that enables one to operate within the Numerical field of reference.
Nevertheless, Algebraic Thinking applies to fields of reference other than the Numerical
(applied to sets it might lead for example to Boolean algebra); for this reason it is adequate to use
Numerical instead of Algebraic field of reference, once we are examining the development of
Algebraic Thinking in the context of school-algebra, which is certainly an algebra of numbers.
Also, algebra being the study of the properties of an algebraic system (as defined in
section 3) Algebraic Thinking is the mode of thinking that leads to the development of algebra
and the symbolic system that corresponds to the calculus embodied in the ideas of algebra is a
possible consequence of thinking algebraically, not a characteristic of it.
The N-A framework enables us to examine the development of an algebraic mode of
thinking in more depth, both because it links Algebraic Thinking to a field of reference (and ;hen
as a consequence to what is possible and necessary when thinking algebraically) and because
non-algebraic thinking is characterised in itself and not as "inability-to-think-algebraically". This
positive characterisation of a non-algebraic mode of thinking is essential if we are to understand
the "misconceptions ", "failures" and "rejections" related to the learning and use of algebra. Also,
the NA framework provides a non-circumstancial explanation for the inadequacy of "algebra as
a language", by exposing the impossibility of a "translation" producing by itself the required shift
of reference that takes one into the Numerical field of reference.
Because Numerical and Analogical fields of reference are distinct operating within one
of them cannot be reduced to operating within the other. This means that each of them provide
distinct approaches that are more or less adequate depending on the task in hand; non-algebraic
approaches are not weaker a priori (see, for example, Janvier, 1989 and Fischbein,1988) and the
fact that this conclusion follows from the way in which our definition for Algebraic Thinking is
built is certainly an indication of the its adequacy.
Bibliography
Bell,A ; Malone,J A ; Taylor, P C (1988): "Algebra
an exploratory leaching experiment"; Curtin
University, Australia
Booth, L (1984): "Algebra: children's strategics and errors"; NFER-Nclson
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Collis, K F (1975): "The development of formal reasoning"; University of Newcastle, N.S.W., Australia
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Kieran, C (1989): "A perspective on algebraic thinking"; PME, Paris
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Kuchemann, D (1984): "Algebra" in "Children's understanding of Mathematics", K Hart (c,d); NFERNelson
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100
DEVELOPING KNOWLEDGE OF FUNCTIONS
THROUGH MANIPULATION OF A PHYSICAL DEVICE
Luciano L. Meira
Mathematics Education
University of California, Berkeley
Abstract
In this paper, I discuss the usefulness of algebra instruction that provides
students with dynamic physical systems as models of algebraic notations,
and a curriculum that profits from their intuitions about mechanisms and
causality. I analyze one student's emerging understanding of linear
functions and algebra as he uses mathematical concepts, principles, and
symbols as modeling tools to explore a simple winch machine.
Introduction
By the end of middle school, children are typically introduced to new levels
of mathematical abstraction in the study of algebra and functions. Current
mathematics instruction at that grade level too often over-emphasizes symbol
manipulation in ways that obscures children's understanding of the objects,
both mathematical and concrete, that the symbols are about (Kaput, 1987;
Schoenfeld, in press; Greeno, 1988; Brown, Collins & Duguid, 1989). Physical
referents of mathematical abstractions are typically overlooked, under the claim
that symbol manipulation promotes robust "context-free knowledge." This study
examines learning of algebraic functions fostered by physical operations on a
mechanical winch. I will argue that mathematics instruction characterized by
manipulation of dynamic physical systems provides students with a sense of
mechanism and causal relation that facilitates learning (White & Frederiksen,
1989), and that helps to engage learners in meaningful mathematical activity.
The analysis describes one student's emerging understanding of linear
functions and algebra as he uses mathematical concepts, principles, and
symbols as modeling tools to explore a physical event. The activity discussed
in this paper involves the student's attempts to write equations that model the
functioning of a simple "winch machine."
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Background to the Analysis
The subject was a 12 year old 7th grade student, named CC. Nearly 5 hours
of interviews were conducted in which CC solved problems and manipulated a
winch mechanism. The device consists of two spools fixed in a common axle
(Figure 1). As the axle is turned, the spools drag small blocks, labelled A and
B, along a numbered track. The spools circumference and the blocks initial
positions at the track can be set to several values. Mathematically, the relation
between block position and number of spool turns map to the function y= mx+b
as "Position(final) = Spool
Turns + Position(initiao."
Figure 1- The winch mechanism.
Before the study, CC had done some simple work solving one or two-step
equations of one variable, but had not studied intensively either word problems
or modeling of the type described here. The following is an example of the
problems in the learning curriculum: "[The equations "embodied" in the winch
were A: y = 4x+8 and B: y = 6x+3) Would there ever be a point at which block B
is ahead of the other block? (If 'yes') After how many turns? (If 'no':) Why not?"
The curriculum did not include teaching interventions that explicitly dealt with
topics such as formal algebraic structures or strategies to record and/or
interpret experimental data. A micro-developmental analysis of the student's
work on the learning curriculum was conducted. Pre and post-tests were
employed to contrast the student's incoming and final knowledge states. All
sessions were video taped.
Protocol analysis focused on obtaining a microscopic trace of the understandings developed by the student. The following activities were considered:
(1) generating equations, graphs and tables; (2) handling the physical device;
(3) describing properties and relationships observed in the device. Two
questions guided the analysis of the reasoning processes underlying the
genesis and evolution of conceptual understanding:
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(1) What aspects of the situation did the subject represent? and
(2) How did these representations evolve?
The segment of protocol discussed below focuses on the student's use of
algebraic notations to record and interpret a physical event. The analysis
illustrates mathematics understanding as a gradual process that depends on
connecting pieces of physical, arithmetic and algebraic knowledge constructed in activity.
Analysis
The child first worked a winch word problem (which served as a pre-test)
and wrote a symbolic expression designed to capture the situation described.
He then worked on the physical device solving many practical problems and
managed, by means of the dialectic between his naive algebraic knowledge
and his perception of the device, to evolve an expression close to the correct
equation. I describe below the episodes that formed the basis of the subject's
initial grasp of algebraic notations and meanings in the physical situation.
In the pre-test, CC correctly solved winch word-problems that described a
scaffold used by window cleaners on a building. The problems were similar to
those used in the learning curriculum. His solutions were empirical, using
number sequences. The underlying equation in the described situation was y
3x+2. The subject was then asked to "write an equation to show the relationship between number of spool turns and height of the board in each scaffold."
The following summarizes his answer:
CC: "I know I just have to add 3 onto the next answer... (Writes n+3 =
n) because on my 6th spool turn I found that it was 20 meters
high, and so with that 20 meters I add another 3 meters and
that's my next answer."
In the following session the student worked with the actual winch machine,
which was set up to embody the equation y = 4x+8 correctly solving many
practical problems. For example, when given the question "How many turns
will it take for the block to be at the 72 mark? the subject mentally computed 72
minus 8 equals 64, estimated 16 as the number that multiplies 4 to yield 64
(writing down '4'16 = 64'), and gave 16 as the answer. Requested then to write
toi
116
an equation to show the relationship between "number of turns and where the
block is at", he wrote 4n = n:
CC: Tour times the number of turns you want to go/ should I put n in
there?... n, equals n. (It goes) 4 inches every turn, so 4 times the
number of spool turns equals the answer."
Note that the use of a multiplicative relationship is already an advance over
the answer for the word problem (n+3 = n). Note further that the use of the same
literal, n, with different index values in the situation did not seem to confuse CC,
as he managed to keep distinct the assigned meanings. The problem, however,
is to specify the assigned meaning to what he names "the answer." Given the
problem context at this point and other segments of the protocol, "the answer"
seems to indicate 'how far the block goes.' Yet, it remains to be known whether
the subject meant 'displacement' or 'position' of the block.
There also seemed to be important links between the equation 4n = n and
the arithmetic procedures used to solve practical problems with the winch.
Immediately following the segment above, CC provided the explanation
transcribed below:
CC: "(...so 4 times the number of spool turns equals the answer)
...just like down here I did 16 times 4, this will be/ the 4 is right
there (points to 4 in '4n = n'), 16 is the n (first from left to write),
equals 64 and that's the answer."
Having interpreted the reference as including the whole procedure (72-8 =
64; 4'16 = 64), I pointed out to CC that his equation did not include the
subtraction operation. He then proceeded to revise the equation:
CC: "Oh, yeah! ...N would be the place where it's starting, minus
(writes 4n = n-a)... wait, you have to do it first... in order to find
how far we want it to... I'm thinking if I put minus n, the n would
be how far it starts out at... you'd have to find out how far it
would go to be able to minus how far that is..."
Not satisfied with 4n = n-n, he then suggested the expression 4n = n+n:
CC: "I just found that if you added like... This (4n =n) gives you the
answer of how far it would go like this (points to 4'16 = 64), but
then we could add the place where it starts; put another n right
there (completes expression 4n = n to 4n = n+n) and that's the
number of inches times the number of turns you want to go,
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equals the number that its gonna go, and then... in order...
well... and then add the 8 (block initial position)... to it, and
that's your answer. Because like this (4'16=64) it went 64
inches but since it started at 8 it went 72 inches, because you
added the 8 to the 64."
LM: "You said 'that's your answer.' What the answer is? Is it the
number of turns? What is it?"
CC: " Where it would stop; where it would be."
At this point, CC seemed to realize that he had focused on distance
travelled (as in 4n = n or 4n = s-n) rather than final position (as in 4n = at).
Indeed, he then made the first spontaneous reference to final position:
CC: "(4n = n+n) is 4 times the number of turns, equals how far it will
go plus how far it started off, and that gives you where the block
will be at."
This reasoning appears to be robust and sensitive to the situation, though
misleading from a strictly formal stance. The subject is then asked to "check
with the apparatus whether the equation 4n=n+n works." He turns the spool 6
times, which makes the block arrive at the 32 mark. His reaction is transcribed
below:
CC: "Ok, so it's at 32, and then... so you did, 4 times my 6 equals
thirty... yeah, no... yeah, 4 times 6, no... equals 241_ and then...
you added the 8 and that's 32..."
LM: "What are you thinking?"
CC: "...I said 4 times 6 turns equals 24; but I want it to 32, in 6 turns...
but wait... this equation can work with this (device) but you have
to say you did the 6 turns and then added 8 on; so it's like
saying 4 times 6 and you added 8 to it... it doesn't seem right,
because it started at 81"
In this segment, we see a clash among the student's understanding of equa-
tions, of the arithmetic procedures that worked in practical problems, and his
model of the physical mechanism. The equation is then rewritten as 844n1= n.
This time, the last 'n' in the equation is labelled "the overall answer." Note the
match between the position of terms in the equation and the sequence of states
and events in the physical device: (1) block starts at position 8; (2) handle
linked to a 4 inch spool is turned n times; (3) block arrives at position n, the
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118
"overall answer." We observe that, save for the use of 'n' to represent both the
number of turns and where the block finally lands, nia final equatiortiaGg
Moreover, it was generated by a heavy reliance on the mechanism of the situation, which did not appear in the paper-and-pencil winch problem in the initial
test.
The change in CC's understanding of the modeling task can also be detected through a contrast between his initial and final assessment tests. Figure
2 shows CC's answers to the scaffold word problem discussed earlier. The
question read as follows: "Draw a graph and write an equation to show the
relationship between number of spool turns and height of the board in each
scaffold (A and B)."
initial Assessment
Target equations- A: y = x+9
Final Assessment
Target equations- A: y = 2x+6
B: v = 3x+2
B: v 3x+2
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Two advances are notable: (1) CC's graphs in the final test are far more
comprehensible and sophisticated than in the initial test; in particular, he has
evolved from bar to line graphs based on data points inferred from the de-
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106
scribed situation; (2) he is clearly able to apply the algebraic knowledge
developed during the study and analyzed above e.g., compare pa = r to
2+(N31 =
H.
Conclusion
In the segments of protocol presented, we observe three elements of the
subject's evolving understanding of the situation: (1) a mental model of the
physical mechanism, inferred from his overt simulations of the functioning of the
device or verbal accounts of its mechanism; (2) arithmetic knowledge, in the
form of calculations of unknowns in specific problems directly involving the apparatus; (3) algebraic knowledge, used to annotate quantitative (and physical)
relationships observed in the situation. The excerpts above present CC's
mathematical understanding as constituted of pieces of physical, arithmetic
and algebraic knowledge.
I interpret the subject's evolving algebraic knowledge as fostered by his
perception of the physical winch. The device provided the means by which CC
could manipulate quantities (as opposed to symbols) and test his intuitions
about patterns and algebraic structures. This case study lends support to White
& Frederiksen's (1989) claim that science learning proceeds from an
understanding of causal principles:
The evolution of knowledge can be captured as a progression of
increasingly sophisticated causal models that are qualitative
early on but that can later be mapped into quantitative models as
students' understanding progresses." (p. 94)
School mathematics too often over-emphasizes symbolic manipulation and
the symbol systems taught have only other symbols as referents. As an
alternative for algebra instruction at the middle-school level, I suggest the
value of dynamic physical systems as powerful aids in promoting students'
understanding of symbol systems and concepts.
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fieferenceg
Brown, J., Collins, A. & Duguid, P. (1989) Situated cognition and the culture of
learning. Educational Researcher, January-February: 32-42.
Greeno, J. (1988) The situated activities of learning and knowing mathematics.
Paper presented at meeting of the Group for Psychology of Mathematics
Education (PME- North America).
Kaput, J. (1987) PME XI algebra papers: A representational framework.
Proceedings of the International Conference for Psychology of Mathematics
Education (PME), Montreal, Canada.
Schoenfeld, A. (in press) On mathematics as sense-making: An informal attack
on the unfortunate divorce of formal and informal mathematics. In D. N.
Perkins et al (Eds.), Informal reasoning and education. Eribaum: Hillsdale,
NJ.
White, B. & Frederiksen, J. (1989) Causal models as intelligent learning
environment for science and engineering education. Aoplied Artificial
Jntelligence. Hemisphere: Washington, D.C.
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Students' interpretations of Linear Equations and Their Graphs
Judit Moschkovich, University of California at Berkeley, U.S.A.
This study examines data from two algebra classrooms identifying common student interpretations
of linear equations and their graphs. These interpretations are consistent with previous research in
this content area, and arise even after direct instruction and experience graphing lines. Peer
discussions of these alternative interpretations are also analyzed for evidence of resolutions of
alternative conjectures. Lastly, suggestions are made for iencouraging the transfer of authority fror
the teacher to peer discussions.
Introduction
Within a constructivist framework, the mistakes students make as they learn mathematics are a
crucial aspect of instruction. As Lampert (1986) puts it, students "need to be treated like sensemakers, rather than rememberers or forgetters" (P. 340), or mistake-makers. Students'
alternative interpretations should be taken into account in two ways. First, we need to identify
common alternative interpretations that students generate in different content areas. Second, we
must develop instructional methods to address specific interpretations. This paper explores how
students interpret linear equations and their graphs in different ways than experts do. Previous
research shows that student interpretations include: changing the y intercept moves lines
horizontally (Goldenberg, 1988), and using a three-slot schema which includes the x intercept for
equations of the form y=mx+b (Schoenfeld , Arcavi, and Smith, in press).
Within a Vygotskian framework that views knowledge as socially constructed, classroom
discussions of students' interpretations are crucial contexts for students to develop meaning for
the mathematics they are engaged in. Many researchers have proposed learning through social
interaction, and peer collaboration specifically, as an important element in constructing classroom
environments where students can make sense of mathematics (Resnick, 1989; Brown and
Pallincsar, in press). Combining aspects of the constructivist and Vygotskian frameworks, this
study examines how students in two classrooms generated alternative interpretations of equations
and lines, and how their discussions did or did not resolve conflicting viewpoints.
Linear equations and their graphs
After typical school instruction, students may or may not be able to perform standard procedures
such as graphing equations, solving for variables, or changing the forms of equations. However,
experts' knowledge of functions extends beyond procedural competence (Schoenfeld et al, in
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press; Moschkovich, 1989). For example, an expert's view of this domain includes treating curves
as conceptual objects which can be manipulated and seeing changes in the parameters of
equations as having corresponding changes in the curves. Both of these aspects of expertise are
crucial for using functions in advanced mathematics courses such as calculus. Guided exploration
of functions, their equations, and graphs with graphing software, as opposed to direct instruction
or pencil and paper tasks, has been proposed as a useful tool in the development of this
elaborated view of functions (Schoenfeld, in press).
Subjects and Methods
The site for classroom observation was an urban high school in California where a pilot version of
Math 9, a college preparatory course, is being tested. The students in the two classrooms
observed are 9th and 10th graders following a college track curriculum; that is, they are neither
remedial nor honors students.
The curriculum was designed to encourage exploration, discovery,
and discussions of alternative understandings. Thus, the classrooms are an excellent environment
for exploring students' active construction of mathematical knowledge through interaction with
peers or teachers. The curriculum unit observed lasted approximately five weeks. The chapter
included modeling of real world situations with equations and graphs, interpreting graphs, use of
graphing calculators and computer software, and student group work with some class discussions.
The lessons that will be discussed in detail involved graphing on a calculator (Day 13), producing
lines on the computer screen to match lines in a handout using Superplot (Day 14); playing Green
Globs, a game where students use straight lines to shoot random globs (Day 15); and graphing
on a calculator (Day 16). In each of the two classrooms I observed peer group work and teacherstudent interactions. I observed and audiotaped students working in groups during four lessons,
and videotaped four pairs of students working on a computer using Green Globs (Dugdale, 1982)
and Super Plot. The classroom notes, audiotapes, and videotapes were analyzed in terms of two
themes: students' alternative interpretations of linear equations and their graphs, and instances of
peer discussions of these interpretations.
Analysis
Following previous research on students' knowledge of linear functions (Schoenfeld, Arcavi, and
Smith, in preparation), my own pilot work, and recurring themes during these five lessons, I focus
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on four areas of common alternative interpretations: the role of the y intercept, separating x and y
intercepts, slope as location, and separating slope and y intercept.
1. The role of the y intercept is not obvious
In Classroom B, on Day 13 of the chapter, the teacher directed a whole class discussion on the
role of the slope and the y intercept in the equations and graphs of straight lines. The students
had graphed the equations y=2x+1, y=2x-1, and y=2x on their calculators and the teacher graphed
these three lines on the board. The teacher then posed the following question:
Teacher B: Can anyone tell me what the significance was of that number there? (pointing to the 1
in y=2x+1).Does anybody know where this plus one and this minus one came in to play on these
graphs? (silence) Can you see it on there?
Student: Yeah...(silence)
Teacher B: To make along story short, there are little blip marks on the x and the y axis, right?
Mt: Yeah
Chorus: yes
Teacher B: Which little blip mark did this graph go through?
M: Two, the second one.
Students: Two and two...
Teacher B: This one you drew, which blip mark did it go through?
Student: Negative two
Chorus: Negative two
Teacher B: Negative two?
Mt: Yes
M: And positive two.
Student: And negative one...
Teacher B: That one went through here didn't it (pointing to (0,1) on the graph)?
Mt: Yeah
Student: Through the middle
Chorus: It went through the middle.
The teacher proceeded to ask the students what y values were produced for different x values.
However, the students never resolved the question of what the role of the +1 or -1 in the
equations was in the graphs during this lesson. This could be explained by the fact that the
students had not yet had enough experience graphing lines, and thus were not yet ready to
discover the role of the y intercept. However, episodes from subsequent days show that even
with more graphing experience and direct instruction, the role of the y intercept remained
problematic.
In Classroom A, on Day 16, students worked on graphing lines with the same intercept or the
same slope on their calculators. They were asked to answer two questions in their groups: "What
does the number in front of the
do to the lines?", and "What does the number being added or
subtracted do to the lines?" The three previous lessons had involved graphing lines with the same
slopes or intercepts on the calculator (Day 13), reproducing on the computer computer screen
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lines given in live exercises (day 14), playing Green Globs (day 15), and a summary by the teacher
on how to rotate or translate lines (Day 15). It seems reasonable that by this lesson students
should have noticed what the y intercept does, and used this knowledge to either produce lines or
hit globs. However, the two students that were audiotaped were again unable to resolve the
second question.
2. The x intercept is important (when using the form y=mx+131
In Classroom B, on Day 15, the students played Green Globs. The teachers introduction to the
activity summarized how to translate lines up and own by changing the b in the equation, and
rotate lines by changing the m in the equation. The following is an excerpt from the video
transcript of two students who had played five games using mostly horizontal and vertical lines.
(Game 6: M and K have tried the equations: y=x-3, y=x-2, y=x=y-3, and x=y-3.)
Mt: Negative y...OK (he types in the equation x = -y -3 and then x = -y)
K: X minus, y equals x minus....y equals x minus 1,2,3,4 (counts along the x axis and keep s his
finger on (4,0)).
Mt: Four...ah yep. Y equals x...
K: Minus four
Mt: X minus...Sure? That won't be up here? (traces a line from the IV to the II quadrant)
K: No..(shakes his head)
Mt: (Types in the equation y= x-4)
In this episode K used the x intercept (4,0) to generate the equation y=x-4. Unfortunately, the line
y=x-4 did hit the globs they had selected. The x and the y intercept for the line they wanted to
produce were respectively (4,0) and (0,-4). Thus, K's use of the x intercept to generate a line was
not challenged by the result on the screen.
After class, on Day 16, I worked with two students (M and C) who had questions about the
previous lessons and their homework. For one of the problems they were working on they had
generated and graphed three lines on the board: y=x, y=x+3, y=x-2 on the blackboard. I asked the
two students if they could write an equation for the top line, y=x+3. One student looked and
pointed at the x intercept, saw it was (-2,0) and generated the equation y=-2x. When I suggested
using a table of values to check whether that was the right equation, the three of us showed that
y=-2x didn't work for the line in question (using a table for y=-2x gave a different line).
3.Slooe and location are related
Returning to Classroom A, Day 16, students D and C tried to determine the role of the m in their
equations.
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D: Ok, one question. The number before x . Hey, what does the number before x do?
C: See, this is y, right, and then this is x...
D: Yes
C: So y equals x is over here, is five. I don't know how to explain it! I know what it means.
D: Try it
C: See here is 1,2,3,4,5 so it's all the way there and you'll run all the way over there. No, that's not
how I would say it. Something like that, I don't know. See like here's the spaces. Here I think I got
it.
See here it is. Over here it's like y equal to negative three x, and it's over here, so this side is
negative and so this side is positive. So over here it's five, like that.
D: Oh, the number before gives it the side like the positive, it starts from the positive side, right?
C: Yeah, like that!
These two students appear to have figured out something about the sign of the slope. However,
they refer to lines as "starting from" somewhere. This is a reasonable result of their experience
with graphing calculators and software. On both screens lines are graphed starting from left to
right. This means lines with positive slope "start" in the Ill quadrant, and lines with negative slope
"start" in the II quadrant. There is nothing inherently wrong with this, if what they are talking
about is a student version of "lines with positive slope rise to the right, and lines with negative
slope rise to the left". However, in the subsequent whole class discussion, this is not the
interpretation of the slope that D presented, or that the rest of the class supported.
After graphing lines on their calculators and discussing the questions in their groups, one student
from each group (7 in all) went to the overhead projector to give their answer to the question
"What does the number multiplying the x (in y=mx+b) do?". Three different students insisted that
the line for y=5x looked like the line for y.x. They graphed it on the overhead projector by
counting from the origin to (-5,-5) and to (5,5) and connecting these two points. Answers to the
question included "it tells the calculator where to draw the line", "it tells where it starts from", "it
directs where the line should start". When prompted to explain why their lines were y=5x, several
students pointed to (-5,-5) as the starting point. None of the students generated any other ways
to prove or disprove this proposition. The teacher then suggested checking what y values were
produced for different x values when using the equation y=5x. Subsequently, two students used
this process to show that the lines they had graphed were in effect y=5x and y=2x.
4. Separating slope and y intercept
In the example presented above in section 2 (Day 16, M and C), not only did the student try to
use the x intercept in the equation, she also tried to place the x intercept in the m slot. Thus,
separating the role of the m and the b in the equation is also problematic. On Day 16, while two
students in Classroom A used Superplot, they also faced the issue of separating the role of the
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slope and the y intercept. On this day students were given a series of lines to reproduce on the
screen (Exercise 1: y=4, y=2, y--1, y=-5. Exercise 2: yzor, y=2x, y=3x, y=-3x, y=-2x, y=-x. Exercise
3: y=2x+3, y=x+3, y=.5x+3, y=-2x+3, y.-x+3, y=-2x+3.) Students S and E worked together on
exercises 1 through 3. They successfully produced four horizontal lines to match the ones in
Exercise , and the six lines for Exercise 2. However, they were stumped when they came to
Exercise 3. They asked the teacher for help in taking the line y=3x and 'putting it more up", that is
changing the y Intercept from (0,0) to (0,3). The teacher helped them to notice that all the lines
went through the point (0,3). After the teacher left, S and E tried the equations y=3.3x and
y=4.4x. The teacher returned and suggested they try ''plus or minus something'. Next, they tried
the equations y=0.3 +5x and y=3+5x. They matched one line in Exercise 3 with this last equation
and attempted to change the slope of this line by changing the b (y=3.5 +5x, y=3.8+5x, y=3.7+5x)
until they realized that this was not affecting the slope of their lines. They finally moved to trying
the equations y=3 +x, y=3+2x, and y=3+3x, as the way to translate lines up and down. Thus, for
these two students, slope and intercept were initially neither independent on the graph nor did
they show up in different places in the equation.
Peer Discussions
Some of the students I observed were engaged In 'finding the right answer (Lampert, in
preparation). They looked to see what other students were doing for inspiration, and asked other
students, the teacher, or the researcher for answers. In their case, conflicting interpretations did
not generate peer discussions. Instead, these students accepted conflicting interpretations or
results, and moved on to another activity. Other students were attempting to make sense of the
mathematics for themselves. That is, they looked for patterns, generated conjectures,
propositions, or questions, and searched for explanations.
For example, S and E above discussed every decision they made, attempted to justify their own
viewpoints when they disagreed, used the computer to check their conjectures, and then modified
their conjectures to be consistent with the computer feedback. In their case, at least at the level of
generating equations, the computer feedback played a crucial role in resolving conflicting
viewpoints. As far as providing explanations of why equations and lines behaved as they did,
however, the computer proved insufficient. For example, while D and Mi where playing Green
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Globs (Day 15), they asked each other to explain why lines with positive and negative slopes
looked the way they did several times. Neither student attempted to provide an explanation or use
the computer to explore this question further.
Summary
Expert interpretations of equations and their graphs are packed with meaning. Experts know that
the variables and parameters In an equation are relevant in different situations (Goldenberg, 1988)
and they know that the m and the b in the equation are the relevant parameters for comparing
equations of the form y=mx+b (Moschkovich, 1989). The classroom and peer discussions outlined
above show that even with experience graphing lines and some direct instruction, students
generated alternative interpretations. The most common ones were: the x intercept is Important
(i.e. it should show up somewhere in the equation); m and b are not independent (i.e. if you
change one in the equation,the other might change in the graph; if you want to translate a line,
change m or b; if you want to rotate a I ine change m or b); slope and location are related (i.e. the
line for y=5x starts from the point (-5,-5)). Students did not seem to parse equations of the form
p.mx+b as y="x +A, that is with m and b as the relevant parameters that rotate or translate lines
lines and make the equations different.
Peer discussion was a good context for generating conjectures, but not for choosing between
different alternatives. In terms of the examples presented above, the fact that b is the intercept
because (0,b) satisfies the equation y=mx+b and (0,b) lies on the yaxis, or the Cartesian
connection (Schoenfeld, Smith, and Arcavi,in press), could have resolved conflicting
interpretations of the role of the y intercept. Again, using a table of values generated by a
proposed equation could have resolved conflicting interpretations of the role of the slope. As
students are introduced to the definition of slope as directed line segments, another element of
the Cartesian connection between the graphical and algebraic representations, students could also
use this piece to resolve conflicts involving slope. These are the sorst of mathematical tools that
Lampert (in preparation) suggests "enable students to make arguments of a substantially different
sort than they would be able to make without them (p. 17)."
Beyond providing students with specific methods such as these, instruction also needs to provide
students with legitimate processes for exploring parameters and choosing between alternative
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conjectures. As Lampert (in preparation) proposes, doing mathematics and thinking mathematically
involves mathematical tools, activities, such as gathering information, organizing it strategically,
generating and testing hypotheses, and producing and evaluating solutions, and discourse
processes. While mathematical tools are a crucial element of doing mathematics, discourse
processes f roe evaluating conjectures through discussions are also essential.
The examples presented above are not meant as evidence of the poor performance of these
students or teachers. On the contrary, many of the students observed were engaged in "sensemaking' (Schoenfeld, in preparation). The teachers encouraged students to talk about their
interpretations and conjectures, and tried to address them in subsequent lessons. Students should
be expected to construct alternative interpretations, even if these Interpretations look
mathematically 'wrong". Moreover, instruction needs to include not only a discussion of alternative
interpretations but also tools, activities and discourse processes for choosing between alternative
conjectures. If students are to move from seeing mathematical knowledge as something that the
teacher possesses and magically transmits into students' heads to evaluating their own
conjectures, we need to consider In detail the activities and discourse processes students can
practice to become 'authorities' in the process of constructing mathematical knowledge for
themselves and with each other.
References
Brown, A. and Pallincsar, A. (in press). Guided cooperative learning and individual knowledge
acquisition In L. resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert
Glaser. Hillsdale, NJ: Elrbaum.
Dugdale, S., (1982, March). Green Globs: A microcomputer application for graphing equations
Mathematics Teacher, 75, 208-214.
Goldenberg, P. (1988). Mathematics, metaphors, and human factors: Mathematical, technical, and
pedagogical challenges in the educational use of graphical representations of functions. Journal
of Mathematical Behavior, 7, 135-173.
Lampert, M. (1986). Knowing, doing and teaching multiplication. Cognition and Instruction, 3 (4):
305-342.
Lampert, M. (in preparation). The teachers role in reinventing the meaning of mathematical
knowing in the classroom. To appear in American Educational Research Journal (?).
Moschkovich, J. (1989) Constructing a problem space through appropriation: A case study of
guided computer exploration of linear functions. Paper presented at the 1989 Annual Meeting
of the American Educational Research Association, San Francisco.
Resnick, L. (1989). Treating mathematics as an ill-structured discipline. In R. Charles and A. Silver
(Eds.), The teaching and assessing of mathematical problem solving, vol. 3. Hillsdale, NJ:
Elrbaum.
Schoenfeld, A. (in preparation). Learning to think mathematically: problem solving, metacognition,
and sense-making in mathematics. To appear in D. Grouws (Ed.), Handbook for Research on
Mathematics Teaching and Learning. New York: Macmillan.
Schoenfeld, A.H. (in press). GRAPHER: A case study in educational technology, research and
development. In A. diSessa, M. Gardner, J. Greeno, F. Reif, A. Schoenfeld, and E. Stage
(Eds.), Towards a scientific practice of science education. Hillsdale, NJ: Erlbaum.
Schoenfeld, A.H., Smith, J.P., and Arcavi, A.A. (in press). Leaming. To appear in R. Glaser (Ed.),
Advances in instructional psychology, vol. 4. Hillsdale, NJ: Erlbaum.
129
116
AN EXPERIENCE TO IMPROVE PUPILS' PERFORMANCE
IN INVERSE PROBLEMS (*)(**)
A. PESCI, Dept. of Mathematics, University of Pavia, Italy.
SUMMARY
The
work
experimental
problems
with
meaning of
to improve
solution
strategies
in
old pupils:
year
The
"inverse problems" and the steps necessary to solve
are described:
inverse
intends
inverse procedures with 11-12
them
The significance of the use of arrows planned to
problems
is
intended particularly
also described:
to
The present didactic
help weak students
face
plan
is
a problem
understand
through different representations of the situation itself:
1. INTRODUCTION
This
work
plans
is framed in a research,
to study how to improve,
strategies
Referring
in
started three years
in 11-14 year
old
problems with inverse procedures
to
the
concept of reversibility:
pupils,
(inverse
to literature on reversible thought,
importance
ago,
which
solution
problems).
Piaget gave a lot' of
the
reversibility
of
thought operations, which requires the mobility of mind in the forward
and reverse directions,
is placed,
in the mental development of
the
child, in the period of formal operations. The mastery of such ability
is considered essential for the exsperimental and logical-mathematical
thought (Piaget,
p. 334). From a mathematical point of view there are
many activities which require to reconstruct the direction of a mental
process and then to change the direction of the train of thought:
instance,
when
for
we deal with direct or inverse arithmetics operators,
with direct or inverse theorems or with a formula which is to be
from left to right or from right to left.
The psychological basis
read
in
these situations is considered the same (Kruteskii, p. 143).
Several
psychological
reconstruct,
(*)
in
a
studies
train of
proved
thought,
moreover that
two
the
directions,
skill
direct
Research supported by the C.N.R. and the M.P.L. (40%).
(**) The psychologist M.G. Grossi collaborated in this research.
117
130
to
and
reverse, is essential to master many situations, not only mathematical
and it is not easily reached by all pupils (Kruteskii, p. 288).
our work by inverse problem we mean a problem which requires us to
In
go backwards in a given succession of (arithmetical or not) operators,
when
the result of such a succession is known.
cases
In simple
the
succession consists of just one operator.
The solution of an inverse problem requires, mainly, the understanding
of the succession,
the
of
order,
given in more or less explicit way,
the awareness
need to go backwards and the skill to inverte,
the given path.
In what follows we will see,
in the
right
synthetic
in a
way, the work plan realized in the last year for 11-12 year old pupils
and what we are doing now for the same age-group.
2. THE DIDACTIC PROPOSAL
The didactic itinerary proposed to 11-12 year old pupils, described in
details in Pesci, deals, essentially, with the concept of arithmetical
operators (+k,
of
-k,
arithmetical
xk, :k) as binary relations, with the composition
operators
and
with the
inversion
of
composed
a
relation. All that with the essential use of language of arrows.
The main objectives linked to the didactic plan are the following:
- to
the concept and the visualization of a binary relation
use
inverse to face the usual inverse problems (in
its
and
arithmetic,
in
geometry, in proportionality problems,...);
to
stress
without
with
everytime
the structure
of
a
problematic
situation,
taking into account the nature of a numerical data (numbers
or without point,
misconceptions'
greater or less than 1,...),
so
to
avoid
influence on the choice of operations (Bell et al.;
Fischbein et al.; Mariotti et al.).
As
far
as
the use of the language of arrows
is
concerned,
important to mention that it has been thought at two different
it
is
levels
which we call "concret" and "abstract" respectively.
At the first level the arrow represents a situation where it is clear,
in a logical-temporal sense,
the starting point, the point of arrival
and the operator in use.
At this level, the dynamics of the situation, explicited on the paper,
118
131
handling
is
in a concret way and it simplifies the reasoning
back to the starting point (when is known,
to
go
obviously, that +k and -k,
xk and :k respectively, are one the inverse of the other).
this
At
"concret"
level the scheme with arrows is
"diagrammatic
a
model" which semplifies mental processes (Fischbein, pp. 165-166).
language
The
of
arrows is also important in order
improve
to
the
construction of mental images ("abstract" level).
With arrows a particular simple scheme may be constructed:
transferred
into
it can
mind as it is and it can be enriched by many
be
other
meanings, at more abstract and more formal levels.
The
psychologist widely recognize the power of images' code
partial
autonomy from the verbal one (Cornoldi,
importance
skill
91).
its
Hence
the
to develop and train abilities of visualization as a basic
young pupils (Lean-Clements;
in
p.
and
therefore,
Bishop) and
the
of schemes which for their simplicity can be
importance,
internalized
as mental images.
3. RESULTS FROM THE FIRST VERIFICATION
To
study the influence of the didactic proposal (mentioned in 2.) -on
the
strategies of solution of inverse problems,
three questionnaires
were given to 2 experimental classes (33 pupils in all) and to 1 class
of control (21 pupils).
Every
questionnaire has 8 problems,
inverse.
4 direct (as distractors) and 4
The text of the questionnaire, the way of administration and
othe details are in Pesci.
Here
year,
would complete with the final results obtained
I
emerged
from
the
comparison between the exit
the
in
of
the
last
first
questionnaire (before the didactic proposal) and the third one (at the
end of the scholastic year).
In the two following tables,
the
control
solutions in
in
order
classes,
to
group.
S is for experimental group and C is for
In table 1 there are the percentages of
correct
direct and inverse problems. But it is more significant,
not
take into account the
initial
situation
of
the
to look at percentage variations of the correct problems
the third questionnaire with respect to the first one (see table 2).
119
132
in
TABLE 1
DIR.
INV.
PR.
PR.
45%
S
I
TABLE 2
18%
DIR.
INV.
PR.
PR.
S
+38%
+129%
C
0%
+ 59%
I-Ill Q.
Q.
C
61%
26%
S
63%
42%
C
61%
42%
III Q.
I limit myself to underlining the positive influence (+129%)
Here
of
the didactic proposal on the solution of inverse problems.
Since
the
result is only indicative,
for the low number
tested
of
pupils, an analogous experiment is now in course, as descibed below.
4. THE PRESENT PLAN
the present scholastic year the work-plan for pupils of
In
the
same
age-group (11-12) has the two following aims:
- to reconfirm the positive influence of the didactical proposal in 10
experimental
classes
(about
200 "pupils)
through
the
same
questionnaires mentioned above;
place particular attention to pupils with difficulty of learning
- to
who have been identified, beyond the teacher's judgment, by a double
tests (see 4.1).
The objective of the activities
to
strengthen
working
their
ability to
planned for those pupils (see 4.2) is
represent
problematic
situations,
in such a way as to arrive at the necessary skills to use the
language of arrows.
The
hypothesis which is to be verified is whether the use of
as said above,
simplifies the reasoning and allows good
arrows,
performance,
also to weak pupils, in inverse procedures.
We think that the work with problematic pupils may make it possible to
characterise better the potentialities of image language versus verbal
language
and the complementarity of one language with respect to
the
other.
4.1 Test to identify weak pupils
The
main objective of the double test,
classes,
presented to the experimental
was to identify the least able pupils with reference to
133
120
the
basic skills which are required to face our didactic proposal centered
on inverse procedures and on the use of arrows.
The
first
part of the test has 9 items:
the first three
with
deal
logical-temporal sequences, 4 and 5 with the symbolisation and spatiotemporal
abilities,
the
last
three
astraction,
the
namely
regularities' identification and production.
The first 9 items are the following:
1) Every morning Luca,
before arriving at school,
does the following
actions. Put them in time order, numbering them from 1 to 5: he leaves
home,
he pays for the bun, he wakes up and gets dressed, he goes to a
bakery and buys a bun, he arrives at school and greets his friends.
Look
2)
following pictures and put
the
at
them
in
time
order,
numbering them from 1 to 6.(For sake of brevity pictures are omitted).
in the right order,
Write,
3)
5 actions you do when you
your
wash
hands.
4)
I have invented a secret code for phone numbers, here it is:
O
1
A
c
2
4
3
5
6
7
v
What phone number is this
II 8 8 II
A
B
9
44
4.
7
5) Invent your own secret code and write "we have won".
In a game there are
6)
obstacles
of different forms:
When
a
ball meets one
obstacles it
changes
as indicated
below
on
it
until
these
of
a
direction
and
goes
it
meets
another
a
obstacle.
:
At 4
t
Draw
enters
the
route of a
as
indicated
which
ball
by
the
arrow:
7) Look at the sequence of squares and draw the missing square:
S
121
134
8) Explain,
with your own words, how the following sequence has been
constructed:
5, 8, 11, 14, 17, 20
9) Look at the following picture:
Ka
L___J
With which of the following little square would you complete it?
a
b
d
ag
JP
EgE
The second part of the test has 8 problems, each of them requires only
one operation. They are the following:
Today
my
parents
yesterday
my
grand-parents gave me 15.000 lire.
1)
have given me 10.000
lire
money
pocket
and
wallet
If in my
I
already had 7.500 lire how much have I got now?
2) To make a cake you need 3 eggs. How many eggs do you need to make 7
cakes?
3)
To go on holiday I drove 355 Km in my car.
Coming back I
different way and I drove only 317 Km. How many kilometers did
4)
I
have to put 120 books on shewes of equal
which contains 15 books. How many shewes do
5)
I
dimensions,
came
I
a
save?
each
of
need?
I would like to buy some pens and I have 8400 lire.
If a pen costs
600 lire how many 'pens can I buy?
6) A cork weighs 3.2 gr. How much do 25 corks weigh?
7) For a ring you need 2.5 gr.
of gold. How many rings can you obtain
with 35 gr. of gold?
8) In Carla's wardrobe there are 8 skirts and 12 yumpers.
In how many
different ways could Carla dress herself?
Table 3 shows the percentages of correct solutions in the first and in
the second part of the test respectively. Tested pupils were 215.
135
TABLE 3
1
2
3
4
5
6
7
8
9
PART
98%
83%
89%
94%
77%
59%
86%
64%
37%
II PART
98%
97%
90%
80%
80%
88%
63%
44%
I
Given
0
right
item,
points
to every wrong or omited item and 1 point
every
to
the average score is 6.94 in the first part and 6.45
in
the second one. Pupils for whom reinforcement activity is planned (see
are those who obtained a score less than 4 at least in one part
4.2)
the
of
test and who have been also considered weak by
the
previous
the
explicit
judgment of the teacher.
4.2 Reinforcement Activity
It can be described, synthetically, in the following way:
a) activity
request
aiming
understand a given text,
to
represent
to
in different
ways
with
(figural,
verbal) the situation given in verbal way or,
symbolic
with the
viceversa,
request to construct a text around a situation given in
or
non-verbal
way;
b) activity aiming to strengthen logical-temporal abilities;
c) activity aiming to improve awareness of symbolisation moment.
every
In
class the most weak pupils (in the sense before
mentioned)
have been placed in the same group and an "average" pupil has been put
in their group,
not with a leader function but with the aim to favour
the work itself of the group.
Even the rest of the class, divided in groups, works with the activity
described in a),
since we consider essential,
in learning,
to
work
with translations from one mode of representation to another (Janvier,
pp.
27-32). In every class the groups of pupils are etherogeneous and
the proposed activities are sometimes differentiated.
REFERENCES
Bell
A.,
performance
descriptive
Greer B.,
on
multiplicative
theory,
Mangan
Grimison L.,
word
C.,
problems:
(1989),
Children's
elements
of
a
Journal for. Research in Mathematics Education,
Vol. 20, 434-449.
123
136
3
- Bishop
Space and Geometry,
(1983),
A.J.,
in Lesh R.
Landau
and
Acquisition of Mathematics Concepts and Processes, Academic
(eds.),
Press, New York, 175-203.
Libreria
Memoria e Immaginazione,
(1976),
C.,
- Cornoldi
Editrice
Universitaria Patron, Padova.
Deri M., Nello M.S., Marino M.S., (1985), The role of
- Fischbein E.,
division,
implicit models in solving problems in multiplication and
Journal for Research in Mathematics Education, Vol. 16, 3-17.
Fischbein
(1987),
E.,
Intuition
in Science and
Mathematics,
D.
Reidel Publishing Company, Dordrecht, Holland.
ed., (1987), Problems of representation in the teaching
- Janvier C.,
and learning of mathematics, Lawrence Eearlbaum Associates.
- Krutetskii V.A., (1976), The Psychology of Mathematical Abilities in
Schoolchildren, Chicago U.P., ed. by J. Kilpatrick and I. Wirszup.
- Lean G., Clements M.A., (1981), Spatial ability, visual imagery, and
mathematical performance,
Educational Studies in Mathematics,
Vol.
12, 267-299.
- Mameli M.,
Zan R.,
Organizzazione spazio-temporale 1,
(1987),
La
Scuola, Brescia.
- Mariotti
M.A.,
ragionamento
parts,
Sainati Nello M.,
proporzionale
Sciolis Marino
nei ragazzi di 13-14
M.,
anni,
della matematica e delle scienze
L'insegnamento
(1988),
11
and
II
I
integrate,
Vol. 11, n. 2, 105-136, n. 4, 313-339.
- Pesci A.,
(1989),
Inverse procedures: the influence
of a didactic
proposal on pupils' strategies, in Vergnaud G., Rogalski J., Artique
M.
(eds.),
International
Proceedings
Group
of
the
13th
Annual
for the Psychology of
Conference
Mathematics
of
the
Education,
Paris, Vol. 3, 111-118.
- Piaget
J.,
Inhelder B.,
(1971),
Dalla logica del fanciullo
alla
logica dell'adolescente, Giunti-Barbera, Firenze.
- Vergnaud G.,
International
(1988), Frameworks and Facts, Proceedings of the Sixth
Congress
on Mathematical Education,
Hirst (eds.), Budapest, 29-47.
137
124
Ann
and
Keith
ALGEBRA WORD PROBLEMS: A NUMERICAL APPROACH
FOR ITS RESOLUTION (A TEACHING EXPERIMENT IN THE CLASSROOM)
Guillermo Rubio
Universidod Aut6nomo de Mexico. CCH Sur
Mexico
This paper reports the findings of a teaching experiment carried out
in the classroom. It is a proposal for teaching how to solve algebra
word problems using numerical approaches. The research had two prime
aims. One of them was to investigate if the teaching approach made
students get better results than those obtained in a pre =test. The
other one was to analyze the changes produced in the student's thinking by the teaching scheme in the olgebrizotion of the word problems.
In general terms, the proposal was succesful. The students acquired
more flexibility to interpret and to translate word problems to equations, the latter not being, as usual, in a literal form. The following was another important observation of the study: if the students
do not keep track of the operations carried out and the meanings linked with those in a proper numerical interpretation procedure the olgebrizotion process of the word problems is not possible.
The purpose of this paper is to communicate some of the
findings of a study carried out with
a
of
group
28
Senior high School level students (15-17 years old), during
the
academic year
1988-1989.
The
investigation
was
focused
on
the modification of the student's algebra
knowledge acquired at the secondary school; in particular, on
the development of their ability to algebrize word problems
using a numerical approach to solve them. Underlying the
teaching proposal's
outline
is
the intention to intermix
the arithmetic background
with
algebraic elements in such
form that the teaching process
enables to go to-and-fro
between both interpretations. The aim of this procedure is
to visualize and to solve problems,
which
treated in a
conventional way would need working out a
great
of
deal
algebraic
sintaxis
and
the
semantics.
Within
teaching
experiment described in this paper, the last assumption of
the
proposal's outline was not fully verified. A further
investigation will be carried out for that purpose.
Background and Theoretical Framework of the Proposal
Cervantes and Rubio, in 1983, investigated
the
posibility to implement an algebra course taking
into
account the ideas discused by Piaget (1979) and Aleksandrov
et al (1956)
related with the
process
formation
of
the
scientific
knowledge
of Humanity.
Later,
within the
structuring
procedure of the present
proposal,
it
was
considered that, an adaptation of such ideas to the teaching
process is linked with the constructivist psycological current
related to the need to face the individual with problematical
situations that will enable him to
construct meaningful
knowledge (Rubio, 1987; 1988; 1989; 1990).
The principal aspect of the teaching proposal, which at
the same time is not
common,
is
the
need
to
make
a
numerical interpretation of the word problems. The latter must
lead
to a proper algebraic interpretation and its solution.
The assumption is
that this numerical interpretationwhich
12
38
involves an iterative procedure trial,interpretation,error-is
in an epistemological way to the first two stages
connected
related
to
are
by Piaget (1973). These stages
described
logicalthe
the assimilation process of the real world to
of
development
way
of thinking within the
mathematical
contemporaneous
physics. Piaget argues that those stages
preced the data translation to a system of equations.
The
numerical interpretation captures both stages. The first one,
from the real world is
the establishment of facts or data
as:
such
modeling
of
mathematics
not
independent
classification, relationships, correspondences, measurement,
etc. The second one, refered to the building of intuitive and
towards
a
core
guide
qualitative
schemes
constitute
the
algebraic
third
stage,
formalization.
The
interpretation underlies the numerical interpretation of the
problem.
Likewise, this teaching approach takes into account the
solve word
pre-algebraic trends to
spontaneous
problems (see Bell, 1977; Trujillo, 1987). Those trends could
be a consequence of the iterative use of numerical values (and
as
school,
the
primary
its operations),
made
since
geometrical
physical
and
of
abstract
representations
the
student's
world within
magnitudes
the real
of
those
to solve a word problem
environment. In an attempt
"thought concreteness". It is a
values
become
numerical
belief that such process of concreteness enables, in many
the operations'
adquisition
of
cases, the formation
and
meanings, which are established between the unknowns and the
can
problem's data. In this way, the numerical approach
the student to face and solve a
provide a means to enable
problem with a new conceptual framework. This structure is an
organization of the preceding student's conceptual system.
students'
The Teaching Proposal
The following
proposal:
phases
are
differenciated
in
the
teaching
1. The understanding of the problem.
2. Numerical approach of the solution (trial and error
process).
3. The interpretation of operations and relationships.
4. The obtention o the equation derived from the pattern
determined by the trial and error process.
5. Algebraic and/or numerical resolution of the equation.
Illustration using
two problems solved
in
the classroom
A teacher hands out 120 chocolate bars and 192
sweets between the students in a classroom. Each student
recieves three sweets more than chocolate bars. How many
Problem 1.
students are there
in the classroom?
9
126
4ur,11 bear. posed
as
Phases 1 and 2
Number of students
Phase 3 :
Nwnbelt of chocolate bars received by each student 120/20 = 6
NwnbeA of sweets received by each student
192/20 = 9.6
6
,
3
+
?
9.6
=
numbers of sweets
= {Numer s of}
swets
e
The methodology establishes a proCess to recover the operations
carried out, that is:
120/20
3
+
192/20 ?
=
After the analysis of similar trials, a letter
is
precise solution of the problem ( Phase 4)
120/ y +
3
= 192/y
where y = number of students.
posed as the
shows the different types of equations found
after the numerical approach .to the problem:
The folllowing
3 = 192/y
3) = 192
3 = (120 192)y
120/y
(120/y
120/y . 120/y
192/y - 3 = 120/y
Problem 2. CCH has twice as much students as the Colegio de
Bachilleres and the latter has 87654 students less than the UAM.
The total of students in the three institutions is 567890. How
students does each institution has? Observation.
problem does not have a whole number solution.
many
Phase 1.
The
Setting up the unknowns.
Number of students In CCH.
Number of students In Colegio de Bachilleres
Number of students In UAM
Phase 2.
Numerical values were chosen for one of the three
unknowns and afterwards the values of the other two were
computed. The three interpretations which emerged in the
classroom are described in the following paragraphs.
First case. A value for the number of students In CCH Is posed.
quantity
2400000
+
{Students}
of CCH
400000
Students
of
number
2
87654 =
+
{number
Total of
=
st udents
1
=
UAM
{
students
of
567890?
}
students of
1
{Bachilleres}
Bachl I leres
Total
400000
+
2
1 number
Total of
students
After several trials, the process lead to an equation- of the
following type:
x
where
4.
+
2
87654
=
567890
X = number of students of CCH
Second case:
A value was posed for the number of students of
Colegio de Bachilleres.
127
14 0
quposedy
90000
L 2(90000) +
+
90000
{StVents
{Students).
+
Beall I leres
of CCH
=
567890
=
number of
students
87654
{ Student a of
).
BMA
Total
A process several trials conduced to the following equation:
2(x) +
+
x + 87654 = 567890
x
where X = number of students of BachIlleres.
Third case. A group of students in the classroom posed a value
for the number of LIAIN's students.
P:2:11tY
4 120000
+
(120000 - 87654) +
{students}
(120000
f students of
Bachl I leres
of LIAM
x
where X =
of
CCH
The following equation was obtained from the
procedure:
+
87654)2
f students
(x - 87654) + (x - 87654)
number of students of CCH.
trial
=
=
=
567890
total
{number of
students
and error
567890
The Word Problems
In the class sessions, twenty two word problems were
solved using the
numerical approach. Sixteen of them were
linear problems with one, or more unknowns. The latter
type
could be reduced to equations with one unknown. The other six
comprised areas of rectangles, which generated a quadratic
equation. The selection of the problems and the
order
of
presentation took into account aspects such as: a) the number
of times the unknown appeared in the equation associated with
the
word problem,
b) the side of the equality in which the
unknown appeared in the equation: on both or only on one side,
c) in equations with more than one unknown, considerations
were made with respect
to
the difficulty of expressing an
unknown as a function of the others, in order to get a first
order equation, d)
equations in which the unknown is a
divisor, e) the use of parenthesis to group properly the
terms of an equation, f) the difficulties to determine the
equivalence relationship.
It is a belief that the preceding elements are related
to semantic
and/or
sintactic difficulties
caused
by the
word problem's
textual entities.
The
former .type
of
hindrances
are difficult to characterize however, it was
observed that students found less
obstacles
when
faced
with problems
linked more directly with their "realworld".
The words used in the problem were "meaningful" for the pupil.
Furthermore,
it
was
considered
thatalgebraic
semantic
difficulties are also prompted
by
the number of composite
unknowns which are necessary to be generated in a problem
(Trujillo (1987)) (e.g., if x = number of students, "120/x =
number of chocolate bars per student" is a composite unknown
141
128
resulting from the interrelationship of
two quantities with
linked
different meaning). Supposedly, this last aspect is
with the structure's complexity level of the operations
established beteween the unknown magnitude and the data;
aspect which is originated by the conditions given in the word
problem.
The area problems were thought to be sufficiently meaningful
for the pupils, though, in almost all the levels, they had
several geometrical difficuties. A lack of development of the
ability to "imagine" and represent in a
or
numerical
way
algebraic
the dimensions of a rectangle, which
are
being
increased
or diminished, was shown.
The classification tests. Three
paper
and
pencil
tests
were applied: a pre-test, a post-test and a
delayed
posttest.
The post-test was applied immediately at the
end
of
the
teaching sesions (12 weeks), during three week
periods (two of them of 50 minutes and the third one of 100
minutes).
The
delayed
post-test
was
applied
33
days
afterwards; during this period of time no academic activity
was carried out. The tests used were of the same type. Each of
those was composed by three axis or subthemes of algebra:
operativity (29 items), equations (22 items) and semanticresolution
of
problems
and
interpretation
of
word
formulations-(24
items).
The
experimental
group
was
classified with respect to each subtheme of the three tests.
Pupils were ordered according to their performance and items
to the order of difficulty. The classification was a means to
prepare tables to record the changes of the students' results
in the three tests. Likewise, to enable the
identification
of interrelationships between the three axis in each test,
a
further classification was carried out.
The data. The data comprises: the students' answered
tests (pre-test,
post-test
and
delayed
post-test),
the
classroom annotations of two pupils,
the
teacher's
daily
observational notes, the sequence of 22 written problems
worked
out
in
the teaching sessions during the term and
the aforesaid classification tables.
Development of the Proposal
The group was subdivided in subgroups of 4 pupils and a
problem was posed.
The dynamic
method of the teaching
proposal consists of encouraging pupils
to
express
the
unknown in a written form (during
the course of the
development of several problems
pupils
were
reticent. In
many cases, the prime difficulty encountered in solving word
problems was to have a clear and explicit idea of what had to
be found).
Pupils are then urged to pose random numerical
values for one of the unknowns in order to get them involved
in a verifying process of the
hypothetical value as a
solution. This procedure comprises a new reading of
the
problem
and
a
search
for
relationships between the
mentioned value
and
the
data,
using
"meanigful"
arithmetic operations linked with the conditions pointed out
in the problem's formulation and
the
analysis
of
the
element's unities which intervene in each operation (e.g.,
multiplying 120 chocolate bars by 24 students would have no
129
142
sense in the sweet's problem to indicate how many chocolate
bars would each student receive).
Once this first mental interpretation is carried out,
pupils are encouraged to perform operations
and
interpret
the elements of the operations and its results, as well as,
the potencial equivalence relationship which emerges from
the
comparative
process
derived
from
the
problem's
conditions (as shown in problem 1). In each trial, pupils are
asked
to
write horizontally each operation carried out-to
enable an insight of the actions involved. Furthermore, they
should observe the role of the hypothetical solution value in
the operation. After several trials, students are asked to
search
for
a
proper pattern, for all the trials performed
(by marking the numerical value posed as solution to the
problem).
Finally,
pupils
are asked to use a letter to
represent the "exact" solution of theproblem, by this process
an equation which models it the word problem is obtained. At
this point, two different alternatives are used.
For
the
first problems of the sequence only a numerical solution is
required; the procedure is to give different values for the
literal and to verify,
making
further estimations, the
obtention of
an
equality.
In
a
progressive manner, the
algebraic
resolution of the equation using
an Eulerian
type process (carrying out the same operations on both sides
of the equality) is taken over.
Results
Results related to the selection of the unknowns. An
attempt was made, in the class sessions, to aid students the
least.
The strategy lead pupils to choose unknowns in a
problem to start working out the method trial-interpretationerror. It was observed that students acquired more flexibility
to interpret a problem without restricting themselves to
literal
translation. In several cases, they started to pose
numerical values for
the first unknown which appeared
in
the problem's formulation. However, in some of the preceding
problems, the pupils
selected that unknown which comprised
less mathematical difficultiesin a numerical and sintactical
sensewhen a value was given to it. Understood the problem's
semantics,
the pupils
got
intensively involved
in
the
numerical resolution. They showed good competence to operate
positive decimal numbers; a vast number of students approached
the solution using various
decimal
cyphers even without a
calculator.
Results related to the search of an equivalence
relationship.
A qualitative analysis of the problems worked
out by the
students shows that, the numerical approach
becomes an aid to build up a comprehension of the equivalence
relationship between
two
sets of operations necessarily
while
compared
solving a
problem. However,
this was not
an homogeneous process; the whole population did not achieve
that goal. For example, in problem 7, 30 percent of the
pupils
continued comparing
the operations separately, that
is, without relating them to the equal sign (the latter was
being used twice
as
143
a
connective).
130
Nevertheles,
in
a
manner,
established an
progressive
more
students
equivalence relationship. It is a belief that the numerical
enabling
from
the
approach,
to
go
to-and-fro
numerical
interpretation
(which
a
gives
rise
to
visualization
of
the equalness between two numbers) to the
one,
algebraic
generates consciousness of the equivalence
between two expressions.
It
is easier to capture that
understanding by this procedure since it allows a comparison
between numerical values
of
two
algebraic expressions
(meaningful for the students) derived from the word problem,
which at
the
same
time
is,
actually
solved
in
a
numerical way.
Results related to the setting up and the resolution of
equation.
Arithmetic operations
are not
frequently
presented in a horizontal way. At the begining of the course,
that situation caused some resistance of the pupils.to write
or
rewrite the arithmetic operations relating the unknown
with
the data
Similarly, in the process to capture in a
written form
X11
the operations carried out to solve a
problem, in order
to
assign meaning to each operation and
its results, it was observed that some students left out some
of those operations (particularly, when those were mentally
done).
As
a
consequence,
pupils obtained equations which
did not represent the problem properly. In the first phase of
the
teaching
experiment,
an
algebraic resolution of the
equation derived
from
the problem
was
not required. The
the
purpose was to obtain an algebraic representation. The next
phase aimed at a gradual involvement of the students in a
Eulerian type process showed that the understanding of this
method required a reconsideration of the knowledge
acquired
in preceding courses, when these have emphazised a mechanical
use of transposition. From this observation
the
following
hypothesis emerges: those pupil who are not able to give off
the
transposition
method
face
serious
difficulties
to
understand the equation
as
an
equivalence
relationship
and
therefore
to establish and assign meaning to the
equations derived
from
the word problem.
Some quantitive results. One of the fundamental aims of
teaching proposal, mentioned at the begining of this
paper, concerns
the
semantic
aspect
of
word
problem
solving.
In
quantitive terms,
an improvement
of
the
student's
performance for the semantic axis items was
achivied. The comparison of
the results between the pre-test
and 'the delayed pos-test showed: a) an increase of the
percentage, between 40 to 80 percent
for
15 items, b) an
increase between 15 to 38 percent for 9
items
and c) no
the
improvement in only one item. The error percentage of the
population's mean decreased 36 percent. Even though, a great
impact on the other aspects-operativity
and
equations-was
not expected in this first trial, the comparison between the
results obtained for the items of these axis in the pre-test
and
the delayed post-test
showed
that
the
error
percentage of the population's mean decreased 18 percent.
Further studies related to the project. In the preceding
of the
axis
has
been
analysis searching for the
parragraphs,
a
first analyses
presented. However, a further
131
144
interrelatioships between different axis should be carried out
second
to complete this first stage of the investigation. A
planned, using this teaching proposal as a basis
trial is
in
order to further understand the building up process of
composite unknowns and the efficient use of the arithmetic
operations immersed in an algebraic setting.
References
Aleksondrov A.D., Kolmogorov A.N. y otros (1973). "Lo Motem6tico su contenido, metodos y significodos"..Porte 1. "VisiOn General de lo Motematicas.
Editorial Alianza Universidad. Primera edici6n.
Bell A.W. y Galvin W.P. (1977). "Aspects of Difficulties in the Solution
of Problems Involving the Formation of Equations". Shell Centre for Math"ematicol Education. University of Nottingham.
Cervantes S. y Rubio G. (1983) "Proposicien curricular pars los cursos de
y II del Colegio de Ciencias y Humanidades: bosodo en un onelisis epistemologico historic° critico de los contenidos progrometicos".
Tesis de Moestria. Secci6n de Motematica Educotiva del CIEA del IPN. Mexico.
Matemarticas
I
Pioget J. (19791. "Tratado de Logic° y conocimiento cientifico". Volumen
III: "Epistemologies de la matemetica". Editorial Paid6s, Buenos Aires, Primer° Edicion, 1979.
Rubio G. (1987). "Una Propuesto Metodolegico par° Resolver Problemas en
Algebra". Memorios del IX Congreso de lo Asociaci6n Nacional de Profesores
de Matematicas. Xalopo, Veracruz. Mexico.
Rubio G. (1988). "La Ense6onza del Algebra en el Bachilleroto o troves de
Problemas. Uno Propuesto Metodologica". Memorios de la Segundo Reuni6n Centroomericono.y del Coribe sabre Formacien de Profesores e Investigacion en
MatemOtica Educativo. Guatemala, Guatemala.
Rubio G. (1989). "Lo resoluci6n de problemas en Algebro bosado en ocercomientos numericos. tUno opci6n actual porn cerror el abismo existente entre
el desarrollo sintactico y semantic° en el Algebra del Bachillerato?". Memorios del Primer Simposio Internacional sobre Investigocion en Educoci6n
Motemotica. CIMAT. Guanajuato, Gto. Mexico.
Rubio G. (1989). "Un ocercomiento numerico a la resolucion de problemas en
Algebra.
Una tendencies actual en la ense6onzo del Algebra?". Memorios de to
Tercero Reuni6n Centroomericono y del Coribe sobre Formacien de Profesores e
Investigocion en Motematico Educative. San Jose. Costa Rico.
Rubio G. (1990). "Apuntes del Seminario de Lenguaje Algebroico". Edicion
en Cuadernos de investigocion del Programa Nacional de FormaciOn y Actualizocion de Profesores de Motematicas (Par aparecer).
Trujillo M. (1987). "Use del Lenguoje Algebraico en lo Resoluci6n de Problemos de Aplicocien". Tesis de Moestria. Secci6n de Matemertica Educative del
CIEA del IPN. Mexico. 1987.
145
132
CHILDREN'S WRITING ABOUT THE IDEA OF VARIABLE IN THE
CONTEXT OF A FORMULA
H. SAKONIDIS and JOAN BLISS
King's College London, University of London
ABSTRACT: This paper presents the analysis of the responses of 394 pupils
from 13 to 16 years old to three questions which consider aspects of the
algebraic idea of a variable via its role in a formula. The results show a
tendency of the pupils i) to focus on the operation rather than on the
variables of the formula ii) to give explanations either through utilitarian
considerations or by focusing on the operation or by restating the given
information and iii) to refer to the idea of a variable at different levels, possibly
dependent on pupils' cognitive level and the nature of the given task.
INTRODUCTION
The concept of a variable is one of the keystones of the discipline of mathematics
because its understanding is decisive for the comprehension and appreciation of a considerable
number of mathematical ideas. Much research has been concentrated on the idea of a variable
and how it is understood in the context of school mathematics. This research shows that
children have considerable difficulties not only with the idea itself but also with its
representation, or both (Kuchemann, 1981, Booth, 1984). Piaget's work also dealt with
variable in the scientific sense and pupils' methods of isolating and controlling variables which is seen as a formal level ability. He (1958) argued that children can only conceive of a
variable in late formal operational stage and that this becomes apparent when they start to
reflect upon reciprocal relationships between several variables.
Although the idea of a variable has been investigated, variable set within the algebraic
idea of a formula has been given little attention. However, the importance of this idea and its
particular characteristics are far from any doubt since the development both of the concept of
variable and the interdependence of variables take place in the context of a formula.
This piece of research considers aspects of the concept of a variable as it is taught in
the context of school algebra, examining ways in which pupils write about it through a
consideration of its role in a formula.
133
146
THE STUDY
The focus of this paper is on three questions about different aspects of the algebraic
idea of formula which have been taken from a larger questionnaire on children's ideas about
algebra. 394 pupils between the ages of 13 to 16 years old were involved in the study, taken
from four urban schools: one boys, two girls and one mixed. There were 155 3rd year pupils
( 90 boys and 65 girls), 153 4th years ( 73 boys and 80 girls) and 86 5th years ( 44 boys and
42 girls). All the subjects had at least one year of formal teaching of algebra. The schools were
banded for mathematics and a top and a middle group were taken from each school in the 3rd
and 4th years and a top group only in the 5th year.
In the following, an analysis of the responses in each of the three questions is given
both in terms of the type of mathematical focus of the response and the explanation given by
the pupil for the response. Examples of the pupils' written explanations are given for each
question. Finally, some discussion and conclusions are provided.
For each of the three
items, the analysis includes two components:
(i) Mathematical focus: (a) Focus on Variable: when the child focuses on the variable (s) or
the representation of the variable(s) of the problem; (b) Focus on Operation: when the child
refers to the operation relating the constituents of the given formula and (c) Dual Focus: when
the response focuses on both the variable(s) and the operation(s) involved.
(ii) Content focus: Categories specific to each question are described in the relevant part of the
presentation below.
We illustrate below examples of the three types of mathematical focus described in (i)
above. These are all taken from the three questions which are the focus of this study.
It is
suggested that reader refers to these when familiar with the question:
Focus on operation: "C is the biggest number because the result must be bigger if
p is being added to something else first" (q. 2); "It tells us that to find the time travelled
you have to divide the distance by the speed" (q. 3).
Focus on variable: "The least helpful answer is Tom's because letters can stand for
different things in formula" (question 1); "C is the largest number because we are
adding a positive number to an undefined number so the result must be larger than the
original" (q.2).
Focus on both operation and on variable: "The new formula' tell us that you
.147
134
have to know the values of d and s and then you divide them to find the time"
(question 3); "C has to be bigger because whatever value is given to p, you have to
add 2 to get C" (q.2).
QUESTION ONE
The concept of a formula becomes object of focus through its representation.
Therefore, the understanding of what constitutes an appropriate representation of a formula is
of interest. This question presents to the pupils a well known formula, that of the area of a
rectangle, in three different forms: 1) area = width x length 2) A = a x b and 3) a x b, and
children are asked to make a choice of the most helpful (part one) and the least helpful (part
two) formula, giving each time the reason for their choice.
(i) choice of formula
In part 1, formula 1 is the most frequent choice by the majority of 4th and 5th year
pupils, however for the 3rd years their choice splits almost evenly between formula 1 and
formula 2. In part 2, for the 4th year the choice of the least helpful is clearly formula 3 (67%)
whereas for the 3rd and 5th year the choice is divided between formula 2 and formula 3
(formula 2: 35% and 38% respectively and formula 3: 37% and 49% respectively).
(ii) The mathematical focus
The analysis of the data shows that none of the responses focus on operation. A
number of children refer to the variables of the formula in both parts, with a small increase in
the four year ( for the 3rd year approx.25% in both parts; for the 4th and 5th years about
35% of the responses in part 1 and about 40% in part 2).
(iii) The content of the explanations
The analysis of the data showed that the reasons for choice could classified in the
following 3 categories based on a consideration of whether or not the formula is: (a)
confusing or misleading, (b) sufficiently explicit and (c) efficient, that is, "it allows you to do
things".
In part 1 the responses in the efficiency category are a little more predominant than
those in the explicitness category and this difference between the two increases in the 5th year
(ratio of responses: 3rd year: 7 : 5; 4th year, 6: 5; 5th year 2:1). In part 2 the efficiency and
135 148
explicitness responses have very similar profiles for the 3rd and 5th year pupils, with a
frequency of response of around 40% for both. In the fourth year responses appealing to the
explicitness criterion are a little more predominant than those referring to efficiency in a 3:2
ratio.
Examples of the children's writing to illustrate these categories are given below:
Efficiency: "The most helpful answer is area = width x length because you can work
it out very easily"; "The most helpful answer is A =axb because it is better expressed
and easier to rearrange."
Explicitness: "The most helpful answer is area = width x length because it is very
detailed, all the information you need is there"; 'The most helpful answer is area =
width x length because people can understand what the three components of the
formula are immediately because they do not have to remember what any substituted
letter stands for."
Misleading: "The least helpful answer is A= a x b because the two a's can be
confusing."
Summarising for question one, in both parts, pupils, particularly the older ones, choose
formula 1 to be the most helpful and formula 3 to be the least helpful. The only type of
mathematical focus is on variable (maximum 40%) and the majority of the explanations are
based on criteria of "explicitness" and "efficiency" in both parts. In their explanations pupils
talk about variables as concrete objects where, for example, the variable for length "a" is a
"thing name".
QUESTION TWO
The focus of this question is on the role of the constituents of a formula the variables
- and their interrelationships. Pupils are given a problem where the relationship between two
variables C and p is expressed through the formula C = p + 2. They are presented with an
answer given by a child and which they are told "is wrong". The problem concerns which
variable in the formula represents the bigger number and the imaginary pupil replies "C
because it's on the left-hand side." Pupils are asked to imagine explaining the problem to the
pupil who is wrong.
149
136
(i) The mathematical focus
In all three years, the responses where the focus is on operation, are the most frequent,
with a frequency which is very similar across the years (approx. 45%). However about 30% of
the responses in all years focus in some way on variable.
(ii) The content of the explanations
Four possible explanations
which were in fact procedures for comparing the size of variables
- were identified: (a) The response compares C and p+2 but the expression p+2 is seen as
a whole and not in terms of its constituents related by an operation, (b) The response
compares C with p+2, based on the operation which relates p and 2, (c) The answer
compares C and p+2 by focusing on the relative size of the variables and (d) The response
compares
C and p+2 either using by substituting values or or by making generalised
statements about the nature of formulae.
In all three years, the most frequent type of explanation is that which compares the size
of the variable through the operation, with about half of the pupils giving this type of
response. The next most frequent type of explanation relies on the relative size of the variables
and is given by approximately 15% of the pupils in each year. Examples of the two most
frequent types of answers for (ii) are:
Explanations using operation for comparison:
"C is the biggest number
because if the reverse formula is used (C=p-2) C is bigger because you take off 2
to get C" ;"She is wrong because when you add P and 2 together then you find out
the answer to
C".
Explanations using relative size: "C represents the largest number because C is
2 more than P"; "P is always 2 less than C".
In summary, the majority of the responses focus on operation, with a frequency which
is similar across the years. However a focus on variable is observed in about a third of the
responses. The emphasis on operation is reiterated in the children's explanations because the
majority of pupils discuss the interdependence of the variables in terms of the operation which
relates them. In those explanations where variable is referred to it is seen as a 'varying
number' in relation to other 'varying numbers'
137
150
QUESTION THREE
In this third task, the focus shifts to the effects of manipulating the representation of a
formula in certain ways. In particular, the rearrangement of a formula is the subject of this
question. The formula given is a well known one, that of the relation between speed, time and
distance. The formula d=st was presented followed by the rearrangement: t=d/s and the
pupils were asked to explain "what the new formula tell us ".
(i) The mathematical focus
The "focus on operation" type of responses are by far the most frequent, with a
frequency which peaks a little in the 4th year (72%), and which is very similar in the 3rd and
5th years approximately 65%. A focus on variable is only about 15% in all years.
(ii) The content of the explanations
Inspection of pupils' responses gave rise to the construction of the following four
categories of types of explanation: (a) Static approach : When the child does not add anything
new to the given information about the formula, (b) Pragmatic or functional approach : When
the child sees the formula as having a functional purpose, (c) Inter-relational approach : When
the child considers that there is a relationship among the variables of the formula and (d)
Logico-mathematical : When the child sees the numerical solution as dependent on knowing
the values of the other variables.
The category of "static" responses are the most frequent in all years; their frequency is
similar in the 3rd and 5th years and decreases in the 4th year (57%, 46% and 55%
respectively). The"logico-mathematical" type of responses are the next most frequent their
incidence peaking in the 4th year, but staying approximately the same in the 3rd and 5th years
(10%, 28%, 13% respectively)
Examples of the pupils' written responses in these two categories are as follows:
Static: "The time taken can be found by dividing d by s"; "Time is equal to the
distance covered over the speed"; "It tells us that t for time is being made equal with
distance and speed divided."
Logico-mathematical: It tells us the time to travel a certain distance at s speed.
where s must he known in order to get t"; "It tells us that if you know what d and s
equal you then can find t".
151
138
Summarising, the algebraic focus of the answers is again on operation. Also in
children's explanations operation plays a role since in about half of them there is simply a
repetition of the information in the formula which is held together by the operation. There is a
small minority of pupils in all years who use explanations of an "if... then" type of reasoning.
When the pupils refer to variables in their written explanations they treat them again as "thing
objects" denoting either abstract but familiar entities, e.g. time , or more concrete ones such
as distance and speed.
DISCUSSION AND CONCLUSIONS
The main characteristic of the three tasks is that they all deal with the notion of variable
through its role in a formula. Despite this, the results above show a strong persistence on the
part of the pupils to avoid any concern with variables and as in the two last questions to
concentrate on operations. This fairly consistent absence of reference to variables could,
perhaps, be understood in terms of the way in which pupils interpret letter symbols in an
algebraic context.
Although the letter symbols used in all three questions can be seen as variables, the
nature of the given tasks may determine the approach to the notion of variable adopted by
pupils. We would suggest that when the task "takes away" the difficulty and the abstraction
of the notion of variable by providing the means to handle it either as a concrete or familiar
entity or as a "number object", pupils reject the abstract idea of variable, adopting a surrogate
object to deal with it. Thus in question 1 they use the "thing-objects" to approach the task
pragmatically and since all the given formulae have the same linear expression, they do not
have to worry about the relationships between the variables. The same could be the case for
question 3, but now the "thing objects" are linked by an operation which has to be considered.
However, when the task does not provide any means of avoiding the notion of variable at an
abstract level as in question 2, pupils seem to be forced to take a step towards using it . They
adopt a "variable-like" approach, considering the letter as "a varying number in relation to
other varying numbers" .
The above would suggest that in the attempt to overcome or avoid difficulties with the
139
152
abstract notion of variable, children are likely to rely on those elements of the context that will
allow them to go back to previous representations. Piaget argued that concrete thinking
remains essentially attached to empirical reality, whereas formal thought deals with verbal
statements substituted for objects. Mathematical variable is apparently more than a verbal
statement replacing an object; in fact, it is a symbolic statement replacing a value which
makes the notion more abstract. Therefore, it would seem reasonable for children to attempt
to find ways of embedding this idea in a reality which has more meaning. The results support
this general proposition of Piaget in that pupils are seen to be approaching this notion by
resorting either to pragmatic or to familiar aspects of the context of the problem in order to
cope with the abstractness of the idea of a variable.
Clearly it is difficult to compare the content of the explanations since the three
questions are different. However, it appears that pupils see the relationship between the
variables in a formula mainly via the operation which relates them and not in relation to one
another. Pupils only consider variables in relation to one another when the question provides
a framework for doing so. Question 2 is the only question which actually sets out such a
framework of reference but even then pupils do not see the overall nature of the relationship
between variables but rather how in their terms "one variable influences or operates on another
and how that, in turn, influences the next". Furthermore, pupils show little appreciation for
the fact that the mathematical manipulations of a formula give successive equivalent
mathematical statements rather they see a rearranged formula in isolation from the given
formula (question 3).
REFERENCES
Booth, L. (1984): Algebra: Children's strategies and errors, Windsor, NFER Nelson.
Kuchemann, D. (1981): Cognitive demand of secondary school mathematics items,
Educational Studies in mathematics, 12, 301-316.
Piaget, J. and Inhelder, B. (1958): The growth of logical thinking from childhood to
adolescence, London, Rout ledge and Kegan Paul.
153
140
OBSERVATIONS ON THE "REVERSAL ERROR"
IN ALGEBRA TASKS
Falk Seeger
Institut fur Didaktik der Mathematik
University of Bielefeld - West Germany
Summary: The aim of the present contribution is first to present new empirical
findings on the wellknown "reversal-error" in algebra tasks. Following this the
implications of these findings are discussed for the more general problem of tasks
in algebra and empirical work. The new empirical findings indicate that "semantic
confusion" might not give an adequate explanation of what is happening when
students adopt the reversal-error-strategy and that the concept of "variable" did not
play an important role. Most of the 549 students in our sample were able to come
to a correct solution with the "wrong" algebraic equation when asked to apply it to
a task that required arithmetical operations. The implications of these findings for
the role of tasks in educational as well as in research settings are discussed.
The empirical studies that will be reported in this paper started from the
wellknown observation of Clement & Kaput (1979) given in their "Letter to the
Editor". Their observation gave rise to a bunch of follow-up studies (e.g. Clement
1982; Clement, Lochhead & Monk 1981; Clement, Narode & Rosnick 1981;
Cooper 1986; Fisher 1988; Kaput & Sims-Knight 1983; Lochhead 1980; Rosnick
& Clement 1980; Wollman 1983). These studies generally confirmed the first
impression by Kaput and Clement that a large proportion of students couldn't give
a correct solution to the following task:
Write an equation using the variables S and P to represent the
following statement: "There are six times as many students as
there are professors at this university". Use S for the number
of students and P for the number of professors.
The solutions to the above task were showing that approximately 50% of the
answers were algebraically "wrong" having the form of "6S=P". This was called
the reversal error. That a similar proportion of reversal errors could be found not
only with students but also with faculty members made this observation still more
astonishing.
In the first part of the present contribution I would like to report some findings
from own empirical studies. In the second part I will add some observations and
theoretical speculations about the role of tasks used in empirical studies and tasks
in math education in general.
141
154
The empirical study
The goal of the empirical studies was to test the following idea: the "reversal"
strategy from the point of view of the subjects that follow it cannot be understood
as totally "wrong". Actually, the reversal strategy makes sense from an arithmetical point of view: if the equation is understood as establishing a relation
between the set of students and the set of professors.
The most interesting point now was, to what extent students adopting the reversal
strategy were able to carry out arithmetical operations with the "wrong" equation.
Our hypothesis was, that actually the "wrong" equation had nothing to do with
their ability to correctly perform on arithmetical tasks that were related to the
original problem. In order to test the above hypothesis we administered written
test questions to a sample of 549 students. The age of the students was between
13 and 24, 70% being 15 to 17 years old. The tests were completed in class, the
teachers were distributing them, collecting them and mailing them back to us. So
the test situation was very similar like a written examination, but the teacher was
told to explain the purpose of the test to the students.
The tasks
There were four different tasks that were imbedded into a common task context:
the relation of students to teachers as actually recorded in one of the "Lander" of
the Federal Republic of Germany and as projected in educational planning. In the
first task the "classical" question of Kaput & Clement was put, the second task
asked for an application of that equation, the third task required some arithmetical
operations, while the fourth task asked for the equation that expressed the studentteacher relation in the third task. So, the first two tasks should represent the case
of having an equation and applying it to a certain arithmetical context, in short:
"equation" (task 1) -> "arithmetical application" (task 2), while the third and the
fourth task should represent the inverse case, "arithmetical application" (task 3) ->
"equation" (task 4).
Task 1 "There are twenty times as many students as there are
teachers in Northrhine-Westfalia. Find an equation for this situation, where S is the number of students and T is the number
of teachers"
Task 2
"There are 1.400.000 students in Northrhine-Westfalia.
How many teachers are there then?' (use the equation from task
1)"
155
142
Task 3
"Educational planning in 1973 was assuming that 1985
there should be 17 students for each teacher. According to this,
how many teachers should there be in the following Lander in
1985? (Fill in the number of teachers below with the help of
your pocket calculator!)"
students
385.000
627.000
1.400.000
290.000
79.000
224.000_
Hessen
Niedersachsen
Nordrhein-Westfalen
Rheinland-Pfalz
Saarland
Schleswig-Holstein
teachers
Task 4
"Find an equation that expresses the relation of students
and teachers given in Task 3, where S is the number of students
and T the number of teachers."
Results
The results for Task 1 and 4 ("equation" context) showed a variety of different
forms of equations:
Task
equation
Task 4
1
%
equation
%
20T = S
T = 20S
T:S = 1:20
S:T = 1:20
S:T = 20:1
T = S:20
S = T:20
43.7
28.8
30.2
18.4
1.3
17T = S
T = 17S
T:S = 1:17
S:T = 1: 17
S:T = 17:1
T = S:17
S = T:17
other
no answer
5.7
3.6
other
no answer
12.2
10.6
.7
.2
1.1
14.8
1.3
2.4
23.5
1.3
The results showed a considerable decrease of the reversal error from 28.8% in
Task 1 to 18.4% in Task 4. A correct answer was given by 57.4% in Task 4
compared to 60.3% in Task 1. The equation T = S:20 was used by 14.8% in
Task 1 and T = S:17 by 23.5% in Task 4. For Task 2 it was not very suprising
that 89.6% of the students gave a correct answer. Having used the equation with
the reversal error was no obstacle for coming to a correct arithmetical result. For
143
156
Task 3 the task difficulty across the six subtasks was still lower with 91.4%
correct answers.
In the following table the differences between the use of equations aT = S, T =
aS, and T = S:a are shown in absolute frequencies:
Task 1
Task 4
aT = S
T = aS
241
166
158
T=S:a
other
no answer
101
91
39
20
14 9
74
59
N
549
549
Chi-Square = 70.48
DF = 4
..p..2;201
The adoption of one of the different equations listed above could be seen as
reflecting the use of different "strategies" by the students. The results indicate that
the use of a certain strategy depended to a considerable extent on task-context. In
our study "task-context" was playing a role in two dimensions: first, as an overall
"applied" context that was relevant for all four tasks leading to a relatively high
proportion of correct answers (as compared to related studies from other
authors), second as a subcontext resulting from the different requirements in task
1 -> task 2 (equation -> application), and task 3 -> task 4 (application ->
equation). In subcontext A (task 1 -> task 2) the equation aT = S was used by
43.7%, going back to 30.2% in subcontext B (task 3 -> task 4). The "reversal
error" also was reduced from 28.8% in subcontext A to 18.4% in subcontext B.
The reduction of the "reversal error" as well as the reduction of the algebraically
correct equation in subcontext B was largely due to the increased use of equation
T = S:a. These equation obviously is very close to the arithmetic procedure that
was required in the "applied" tasks, because it literally describes the order of
procedural steps to take: given a number of students and a multiplication factor to
calculate the number of teachers by dividing the number of students by the
multiplication factor.
Even the students that start with the "reversal error" equation are finding highly
creative - albeit mathematically incorrect ways to transform the original equation
into the form where the number of students is divided by the multiplication factor.
For reason of space only one example from the test is taken to illustrate the
157
149
fundamental clash between the (incorrect) algebraic concept and the procedural
concept:
7.0 1:20
L
Lox occ
L
-9 70
-Lo
= 70 coo
There was only one case out of 549 where a student was "correctly" filling in the
numbers in the "reversal error" equation reaching a total of 28 million teachers for
1.4 million students. Maybe some kind of wishful thinking was involved here.
Task and context
If we try to explain - as we did - the findings as influenced by the variation of
task-context, it must be said that "context" as an important theoretical construct
was in our study only dealt with on the level of the task itself. That is to say, that
the context or the situation of working on the tasks was not controlled in our
study. If already on such a restricted level, task context is an important factor, it
should be more important on a larger scale. This lead to a critical evaluation of the
design of our study and the tasks that were used. In the following I would like to
sketch some of the apparent shortcomings of our study relating them to important
issues of empirical research in math education. The focal point of interest here is
how tasks are employed in empirical research.
1. We have to ask ourselves how the design of our standard test restricted
the interpretability of the obtained results. It is quite clear that we can say only
very few things about the processes that underly or accompany the solution of the
tasks. Theses processes could only be dealt with in an indirect way, whereas a
clinical interview study or a transcript/protocol analysis could have told more
about that. However, what is gained with one method seems to be lost with the
other one: the insight into the distribution of certain solutions on a large scale
could not be gained when the reconstruction of solution processes via clinical
interview or transcript/protocol analysis is the aim of a study. These two
approaches basically differ in relation to time.
145
15
2. A closer look at the distribution of solution processes in the of the
reversal-error was instructive. It could show that the situatedness of thinking (cf.
Brown, Collins & Duguid 1989; Lave 1988; Suchman 1987) about algebra tasks
could not be understood as a "misconception". Context-boundedness in clinical
interview studies often was seen as a major obstacle to algebraic thinking (cf.
Booth 1984, p. 37). A central problem in clinical studies often seemed to be how
children make sense of the interview situation and consequently on the tasks that
were presented to them. One central problem of standard test situation is that it
nearly automatically is identified with an examination situation followed by the
positive implications on the motivational level this has for some students while it
has negative consequences for others.
3. The results obtained in this study confirm the view that a change in the
very notion of "task" is overdue. The tasks presented to children cannot be
understood as "objective" stimulus conditions being the same for each child. It
rather must be seen that children actually work on tasks that differ from the given
task and from the tasks other children work on. Even for the seemingly simple
case of the division algorithm Newman et al. (1989) could show that children
turn the same "objective" task into very different "personal" tasks. They
understand tasks as "strategic fictions" that arise in social interaction.
4. If our notion of the "task" should change this also entails the notion of
"error". There is important evidence from cognitive psychology (Norman 1987;
Norman & Draper 1986; Seifert & Hutchins 1989) and from the psychology of
work (Wehner & Mehl 1986; Wehner & Stadler 1988) that a strategy that is
designed for the avoidance of errrors might no be as effective as a strategy
designed to exploit the vital importance of errors (cf. Bromme, Seeger &
Steinbring 1990). Errors should be understood as productive and creative
achievements. Consequently, tasks and task systems should be designed "usercentered" or"user-friendly" instead of following the philosophy to minimize and
avoid errors. The idea is that errors should not lead to a system crash. Repair
strategies (Brown & van Lehn 1980) seem to be a suitable means in this context.
Concluding, remarks
To reconcile the standard test procedure and the methods of clinical interviews
and transcript/protocol analysis is an important issue for research in math
education (cf. Ginsburg 1981). Rather than seeing the different methods as
belonging to different research paradigms, it could be tried to project them onto
different levels of the process-structure of math education. Obviously, in addition
to that, new methods could be adopted that allow for the empirical research of the
"situatedness" of learning.
146
159
References
Booth, L. R. (1984). Algebra: Children's strategies and errors. Windsor: NFERWilson.
Bromme, R., Seeger, F. & Steinbring, H. (1990). Aufgaben, Fehler und Aufgabensysteme. In
R. Bromme, F. Seeger & H. Steinbring (Eds.), Aufgaben als Anforderungen an
Lehrer und Schiller - Empirische Untersuchungen, p. 1-30. KO ln: Aulis Verlag
Deubner & Co.
Brown, J.S., Collins, A. & Duguid, P. (1989). Situated cognition and the culture of learning.
Educational Researcher, January/February, 32-42.
Clement, H. & Kaput, J.J. (1979). Letter to the editor. The Journal of Children's
Mathematical Behavior, 2, 208.
Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common
misconception. Journal for Research in Mathematics Education, 13, 16-30.
Clement, J., Lochhead, J. & Monk, G. (1981). Translation difficulties in learning
mathematics. American Mathematical Monthly, 88, 286-290.
Clement, J., Narode, R. & Rosnick, P. (1981). Intuitive misconceptions in algebra as a source
of math anxiety. Focus on Learning Problems in Mathematics, 3, 36-45.
Cooper, M. (1986). The dependence of multiplicative reversal on equation format. Journal
of Mathematical Behavior, 5, 115-120.
Fisher, K.M. (1988). The students-and-professors problem revisited. Journal for Research
in Mathematics Education, 19, 260-262.
Ginsburg, H. (1981). The clinical interview in psychological research on mathematical
thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 4-11.
Kaput, J.J. & Sims-Knight, J.E. (1983). Errors in translations to algebraic equations: Roots
and implications. Focus on Learning Problems in Mathematics, 5, 63-78.
Lave, J (1988). Cognition in Practice. Mind, mathematics and culture in everyday
life. Cambridge: Cambridge University Press.
Lochhead, J. (1980). Faculty interpretations of simple algebraic statements: The professors
side of the equation. Journal of Mathematical Behavior, 3(1), 29-37.
Newman, D., Griffin, P. & Cole, M. (1989). The construction zone. Working for
cognitive change in school. Cambridge: Cambridge University Press.
Norman, D.A. (1987). The psychology of everyday things. New York: Basic Books.
Norman, D.A. & Draper, S.W. (1986) (Eds.). User centered system design: New
perspectives on human-computer interaction. Hillsdale, N.J.: Lawrence Erlbaum.
Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Mathematics
Teacher, 74, 418-420, 450.
Rosnick, P. & Clement, J. (1980). Learning without understanding: The effect of tutoring
strategies on algebra misconceptions. Journal of Mathematical Behavior, 3(1), 3-27.
147 160
Seifert, C.M. & Hutchins, E. L. (1989). Learning within a distributed system. The
Quarterly 74ewsletter of the Laboratory of Comparative Human Cognition,
11(4), 108 114.
Suchman, L.A. (1987). Plans and situated actions: The problem of human-machine
communication. Cambridge: Cambridge University Press.
Wehner, T. & Mehl, K. (1986). Handlungsfehlerforschung und die Analyse von kritischen
Ereignissen und industriellen Arbeitsunfallen - Ein Integrationsversuch. In: Amelang, M.
(Hrsg), Bericht fiber den 35. Kongress der Deutschen Gesellschaft fiir
Psychologie in Heidelberg 1986. Band 2, 581-593. Gottingen: Hogrefe.
Wehner, T. & Stadler, M. (1988). Fehler und Fehlhandlungen. In: Greif, S., Hol ling, H. &
Nicholsen, N. (Hrsg.), Europitisches Handbuch der Arbeits- und Organisationspsychologie. Miinchen: Psychologie-Verlags-Union.
Wollmann, W. (1983). Determining the sources of error in a translation from sentence to
equation. Journal for Research in Mathematics Education, 14, 169-181.
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148
GENERALIZATION PROCESSES IN ELEMENTARY ALGEBRA: INTERPRETATION
AND SYMBOLIZATION.
SONIA URSINI
SecciOn de Matematica Educativa
CINVESTAV-IPN
The results obtained through a written test concerning the
symbolization of situations Involving a generalization process
and the concept of generalized number, are reported. 65
children, 11 - 14 years old, starting with the study of algebra
were tested. One of the most interesting results was the
regularity of the answers obtained, together with the
instability of a particular individual's answers. This study is
a part of a wider project concerning the feasibility of
diminishing, in a computational environment, the difficulties
children have with the different characterization of variables.
Children have great difficulties and insecurities when
that involve literal symbols ([1], [3],
[5], [6), [9], [11), [12], [14)).
There are a lot of data
concerning the most common errors they commit [1] and very
interesting results on how they interpret literal symbols [6].
faced with expressions
The focus of this article is on the way children, starting the
study of elementary algebra, symbolize, on their own, situations
involving a generalization process and the concept of
generalized number. The main objective of the study was to find
out: 1) How children symbolize such situations and if they use
literal symbols for it; 2) If their answers present some kind of
classifiable regularity; 3) If the answers given by a particular
child are stable in a certain class.
Methodology.
To answer the questions mentioned above, a questionnaire
was designed where children were asked to: 1) Interpret literal
symbols representing unknown or generalized numbers, (14 items);
2) Symbolize situations that involved unknowns or generalized
numbers,
(16
items).
The questionnaire
was
partly based
on
[2],
[4),
[8],
[10)
It was applied to 65 students, aged 11 - 14, entering
the first year of Secondary school in Mexico City and other
mexican towns. None of them had had previous instruction in
and [11].
algebra. Overviews of results of questionnaire indicated however
that almost the totality had some notion about the use of
literal symbols and considered them as representing unknown
numbers.
149
162 -
For each item an analysis of the answers obtained was done,
and for each student the answers given to all the items were
analyzed. The results suggest a classification of the answers
obtained, that we will verify further with a wider population.
The main topics of the study, along with the rationale for
each topic and some examples of the items we used to approach
them, are showed in the table (see next page).
Results.
For each topic it has been possible to classify the answers
obtained.
Symbolization of simple verbal statements involving an unknown
or a generalized number. (7 items)
The answers given pointed out that there were children who:
1) Could not symbolize algebraically and gave a numerical answer
to these items; (21/65)
Answered writing a single letter; (4/65)
3) Were able to symbolize simple statements that implied writing
an equation where they have only to add to or multiply a literal
symbol by a number; (31/65)
and also symbolize statements that implied
4) Could do (3)
writing an open expression, eg. 8N(3+X), using letters, numbers
and brackets; (9/65).
Interpretation of the symbolization of a generalization.
2)
(5 items)
When asked to interpret a letter that represented a
generalized number in an expression, we found that there were
children
who:
1) Could not interpret the letter in any way; (19/65)
2) Interpreted it as 'letter evaluated' [6], assigning it an
arbitrary but specific value; (26/65)
3) Interpreted it as 'specific unknown'
[6], and without giving
the value they specified that it can have only one value; (2/65)
20 of the 65 subjects showed inconsistency in their answers; in
similar circumstances, they did not interpret the letter in the
same way. Some of them could not interpret it consistently and,
for some items, gave no answer (7/65); others interpreted it as
'letter evaluated' but also as an 'object' [6] (12/65); only 1
child interpreted it as 'letter evaluated' and gave also a- range
of variation interpreting the letter as 'generalized number'
[6]. The interpretation of 'letter as an object' appeared
163
150
TOPIC
RATIONALE
Generalization
processes and
symbolization
of a generalization
Interpretation of a
functional relations
solution of equations
with one or more
appearances of the
unknown
Interpretation and
symbolization of
some known geometrical
Mow do children
interpret the
symbolization of a
generalization?
How do children
symbolize a given
generalization?
How do they
generalize and
express it?
How do children
interpret a functional
relation? Can they
solve simple equations,
with one unknown; with
more than one unknown?
How do children interpret
letters that are indicating
figures' dimensions?
How do they symbolize an
area and a perimeter of a
known geometric shape?
In the following
expressions write
down all the values
you think X can
haves
(10)A racing track is
divided in 16 parts
of equal length. Each
part is X kilometers
long. Write a formula
to express the total
the racing
length
track.
iln the following
'expressions write down
'all the values you
Write
formula for
the area of the following
figures'
Symbolization of
simple verbal
statement involving
an unknown or a
generalized number
Interpretation of
the symbolization
of a generalization
How do children
symbolize verbal
statement that
implies the use of
unknowns and
generalized numbers?
Do they use letters?
Write a formula
which means.
An unknown number
multiplied by 13
is equal to 127.
concepts
I
Write a formula
which means,
8 multiplies the
sum of 3 and an
'unknown number.
X
2
2+ X
I think X can haves
I
13 + X . V
I
IX + 5 . X + X
'
5 . X
I
lx + 3 . 5
Writ, the meaning
of the following
expressions using
words,
2
p
X
7
ITEMS
2
j
(11)The following
shape is not
completely visible.
We do not know how
how many sides it
has; we will say it
has N sides. Each
side is 2 centimeters
long. Write a formula'
to calculate the
perimeter of the
shape.
(12)A road which was
X kilometers long
was extended by 25
kilometers. How long
is it now?
BEST COPY AVAILABLE
a
1
X
164
1
1
b
1
a
4
5
X
clearly when there were letters different from X and it was
possible to relate them with another subject, eg. geometry: 2*p
was thought of as 2 times the perimeter.
Generalization processes and symbolization of a generalization.
(7 items)
children generalize and symbolize a
generalization, two groups of items were given:
a) An already generalized situation with the primary
symbols given (items 10, 11, 12) was presented and children were
asked to conceive the general situation and create a new symbol
using the given one. There were children who could not symbolize
any item (20/65). The others could symbolize one or more items
correctly: 40 of the 65 subjects symbolized correctly the item
10; 19 the item 11; 30 the item 12. The items 10 and 12 were
apparently of the same degree of difficulty, but while the item
12 has only a verbal explanation, the item 10 has also a drawing
schema. It seems that the presence of the drawing helped many
children in their process of generalization and symbolization.
But the number of correct symbolizations diminished
substantially for the item 11, where a partially hidden figure
were shown. This caused confusion in many children, whose
answers to items 10 and 12 were correct. Answers such as L+L or
H+N , which were trying to give the total number of sides
(visible and not) of the figure, were given; some children
ignored the hidden part of the figure, multiplying the number of
visible sides by 2; others gave lis2/2 as the answer.
To
b)
see
how
A sequence of geometrical shapes with a sequence of
numbers in correspondence (eg. number of sides) (4 items), were
presented. Children were asked to find out and symbolize a
general rule, which will produce for any further figure of the
sequence its corresponding number. There were children who:
Could not answer any item; (5/65)
Could generalize only by drawing; (26/65)
3) Could generalize by drawing and by numbers; (8/65)
4) Could symbolize algebraically the simplest item; (8/65).
25 of the 65 children showed inconsistency in their answers,
giving one or another of the previously mentioned answers. Some
of them, besides symbolizing by drawing could also, for some
items, go on with the numerical sequence, but only when small
numbers were involved. (15/65)
1)
2)
In the answers to these items as well as to thOse referring
to the symbolization of simple verbal statements, it was clear
that when faced with a process that involved more than one
operation, the majority considered only one of them.
We may
compare this behavior to the partially executed procedures found
by Matz
however remarking that
this case we are
referring to the symbolization and not to the solution process.
Interpretation of a functional relation; solution of arithmetic
[9),
in
tone appearance of the unknown) and non-arithmetic (more than
one appearance of the unknown) equations. (7 items)
When asked to interpret a functional relation, there were
children
who:
1)
Gave no answer;
2)
Assigned a unique value to X and calculated the corresponding
value
for
Y;
(28/65)
(33/65)
3) Gave arbitrary and unrelated values to X and Y;
4) Gave a range of values for X; (1/65)
(2/65)
Some children tested had difficulties when solving an arithmetic
equation. No one could solve the non-arithmetic one; when trying
to do it, almost all of them assigned different values to the
different appearances of the unknown [3).
Interpretation and
concepts.
(2
symbolization
of
some Known geometrical
items)
In primary school, in Mexico, children are faced with the
use of literal symbols when dealing with general formulae for
the area and perimeter of geometric figures. To see the extent
of their understanding of these formulae and the meaning they
attached to the literal symbols presented in them, they were
faced with Known geometric figures the dimensions of which were
indicated:
1) With literal symbols;
2) Combining numbers . and letters
In both cases they were asked to symbolize the
perimeter.
area and the
When faced with figures where the dimensions were indicated
only by letters, there were children who:
Could not give any answer; (15/65)
2) Remembered the general formulae already learned and wrote
them using letters different from those indicated; (25/65)
1)
153
166
3) Assigned an arbitrary value to the letters and calculated the
area or perimeter, or assigned an arbitrary value directly to
the area or perimeter; (11/65)
4) Were able to symbolize using the given dimensions; (13/65)
Only 3 of the 65 children tested showed some inconsistency:
sometimes they wrote a general formula learned by heart and
sometimes considered the letters given.
When faced with shapes where the dimensions were indicated
by letters and numbers, there were children who:
1) Could not give any answer; (7/65)
2) Assigned an arbitrary value to the letter and calculated the
area or perimeter, or they assigned an arbitrary value directly
to the area or perimeter; (6/65)
Remembered the general formulae and ignored the indicated
3)
dimensions; (8/65)
4) Considered the letters as generalized numbers and were able
to manipulate them. They considered the indicated dimensions and
were able to adapt their previous Knowledge to the new
circumstances;
(3/65)
35 of the 65 children tested showed inconsistencies in their
answers. For similar questions they gave different answers of
Sometimes they assigned an arbitrary
the type listed above.
value to the letter and sometimes they tried to manipulate it
without giving a value and instead invented their own way of
doing it (5/65). Sometimes they assigned an arbitrary value to
the letter and sometimes they ignored the letter and considered
only the numbers (20/65). Sometimes they ignored the letters and
sometimes they manipulated them as generalized numbers (7/65).
It seems that many children when evoking the general formulae,
were unable to consider as a symbol, the letter that indicated
the dimension of the figure. It was clear that for the majority,
the letters that appear in the general formulae did not
represent a generalization but were considered as labels.
Conclusions.
The answers children gave confirm some results already
i.
found in other studies: they had difficulties with the use of
brackets [1]; there existed confusion between the signs of
addition ( +) and multiplication 0 [31 they assigned different
values to the same letter when it appeared several times in an
expression [31 they were unable to interpret the conventional
167
154
algebraic notation
some of the categories established by
Kuchemann (6) appeared when interpreting the literal symbols.
2.
All the children used literal symbols to express
themselves. The great majority interpreted them as specific
unknown when they were asked to interpret a given symbolization.
It was not always so when they were asked to symbolize; in this
case some of them even were able to state algebraically a
generalization (items 10 and 12). However, there was no clear
evidence that they could interpret the letter as generalized
number, nor that they were capable of interpreting the
expression written by themselves, as a general expression and
operate on it.
[5);
3. The detailed analysis of the answers obtained led to a
classification of these answers concerning capability in
manipulating situations that involve generalization processes
and their symbolization, and the use and interpretation of
letters as generalized numbers. In spite of the fact that all
the answers given by the children tested fitted into the given
classification, the answers of a particular child were not
stable in any one class. We therefore have a classification of
responSes not of children.
4. Because the sample tested was quite general, including
children from Mexico City and other Mexican towns, and because
we found: a) the presence, albeit in an inconsistent way, of the
literal symbols in children's answers for an unknown or for
symbolizing a generalization; b) the presence of some correct
answers although also incorrect ones to similar items, showing
inconsistencies in children's answers; we consider justified the
hypothesis that children entering the first year of secondary
school, in Mexico, belong to a
'zone of proximal development'
[13) concerning their capabfllty of dealing with the concepts of
unknown and generalized number and their symbolization. We will
investigate this hypothesis in a LOGO environment specially
designed for this purpose, that is we anticipate that in such an
environment in contrast to the common school algebra environment
(see (1)), children can come to learn generalized number by
provision of carefully structured and sequenced activities.
155
188
REFERENCES.
(1)BOOTH,L.R.
Errors.
'Algebra: Children's Strategies and
(1984).
A report of the strategies and errors in secondary
mathematics project', NFER-NELSON.
Understanding of Mathematics: 11-16'.
(2)The CSMS Mathematics Team, HART,K. et al.
(1981).
'Children's
A. and ROJAHO. T. (1989). 'Areas de Dificultades en
la Adquisicion del Lenguaje Aritmetico-Algebraico', to be
[3)GALLARDO.
published in Recherche en Didactique des Mathematiques.
[11)GEOFF GILES, DIME Pre-Algebra Project, 'Number Patterns 1
Simple Mappings', Department of Education, University of
Stirling.
'Pre-Algebraic Notions among 12 and 13
INICIERAN,C. (1981).
Year olds', Proceedings of the Fifth Conference of the
International Group of Psychology of Mathematics Education,
'The understanding of Generalized
[6)1CUCHEMANN,D.
(1980).
Arithmetic (Algebra) by Secondary School Children', PhD
Thesis, University of London.
'Algebraic Thinking in High
(7)LEE,L and WHEELER,D. (1987).
School Students: their Conceptions of Generalization and
Justification', Department of Mathematics, Concordia
University, Montreal.
(8)MASON,J. et al.(1985). 'Routes to/Roots of Algebra', Center
for Mathematics Education, The Open University,Great Britain.
(9)MATZ,M. (1982). 'Towards a Process Model for High School
Algebra Errors', in D. Sleeman and J.S.Brown (Eds.),
Intelligent Tutoring Systems, pp. 25-50. London, Academic
Press.
[10)NOSS,R. (1986).
'Constructing a Conceptual Framework for
Elementary Algebra Through LOGO Programming', in Educational
Studies In Mathematics 17, pp. 335-357.
(11)ROJANO,T. (1985).
'De la Aritmetica al Algebra. (Un estudio
clinico con niflos de 12 a 13 aflos de edad)', Tesis Doctoral,
SecciOn de Matematica Educativa, CINVESTAV-IPH, Mexico.
(i2)SLEEMAN,D. (1986). 'Introductory Algebra: A Case Study of
Student Misconceptions', in The Journal of Mathematical
Behavior 5, pp. 25-52.
'Mind in Society, the Development of
[I3WIGOTSKY,L. (1978).
Higher Psychological Processes', Cambridge, Mass.; Harvard
University Press.
[14)WAGNER,S. (1981). 'Investigating Learning Difficulties in
Algebra', Proceedings of the 3rd Annual Meeting of the North
America Chapter of the IG for PME, Minneapolis.
Grenoble.
.
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156
Assessment Procedures
170
EFFECTS OF TEACHING METHODS ON MATHEMATICAL ABILITIES
OF STUDENTS IN SECONDARY EDUCATION
COMPARED BY MEANS OF A TRANSFERTEST
Author boost Meijer
Center for Educational Research
University of Amsterdam
Grote Bickersstraat 72
1013 KS Amsterdam, The Netherlands
ABSTRACT
In the research project' to be described here, a transfertest for mathematics was constructed wherein optional help was provided by
means of hints, that were presented on the screen of a micro-computer. The first aim of the project was to investigate the claim,
that the scores on the items, where hints were consulted, would contribute to the predictive validity of the test. This implies that
such scores could be weighted in an objective way and that this procedure would render a more refined measurement of
mathematical transfer capacity, compared to a test where oil items have to be solved independently.
The second aim was to compare the effects of different teaching methods by means of this test. The results show that offering help
in testing situations seems to be a valid approach and adds to the amount of observed variance in transfer capacities between
students, thus indicating more refined measurement. Only marginal differences in effects of teaching methods were found, which
also seem to depend on student characteristics.
Introduction
At the 1985 PME conference we presented preliminary results of a project involving the measurement of mathematical learning
potential of students in secundary education in the Netherlands (see Meijer et. al., 1985). This project fmished in 1987. We will
report about its outcomes at this conference (the delay is due to various circumstances).
The central task which we took upon ourselves at the start of the project was the composition of a transfertest for mathematics,
wherein, contrary to ordinary mathematics tests, students could ask for guiding information ("hints") if they were unable to solve a
test-item independently. In this way we hoped to arrive at more refined measures of mathematical transfer capacity than
measurements resulting from the use of conventional tests, which contain only "all-or-nothing" items. Because transfer of
mathematical knowledge and skills is a very difficult task, such tests usually result in very crude distinctions between, the transfer
capacities of subjects.
The idea was originally derived from Vygotksy's theory (1964), which states that, since learning is'an interactive, social process,
accurate measurement of (mathematical) performance level can occur only if a subject has art-ms to help in any way. In other
words, Vygotsky assumed that measuring independent achievement is.insufficient for predicting future performance. Taking into
account performance levels of subjects given hints when needed will contribute to the predictive validity of a measuring
procedure, if certain methodological requirements, such as reliability and equality of "information impact" on each subject, are
satisfied. The difference between the level of performance one can achieve independently and the level of performance one can
achieve with help, be it from elders, peers, books, charts or even computers, was dubbed "the zone of proximal development" by
Vygotsky. It may be conceived of as "thinking structures in embryonic form" (Ginsburg, 1983).
In the inquest of Krutetskii (1976) concerning the structure of mathematical abilities of schoolchildren such a procedure was
applied. However, since help for subjects was available from experimenters in individual testing situations, the reliability of this
procedure for measuring learning potential is questionable. If rigid psychometrical demands are taken into account, it is essential
that every student requiring help receives the same information. Moreover, this information should have equal value for each
subject, i.e. all subjects should be pushed equally further to the solution of the item by every hint, independent of their initial
ability.
The results show that such strict psychometrical demands could hardly be met. Nonetheless the experiment proved succesful
because its outcomes highlight the possibility of educational use of tests with availability of hints and shed light upon the factors
influencing mathematical performance.
A secondary goal of the research project was to assess the differential effects of teaching methods for mathematics. Recently, so
called "realistic" methods for teaching mathematics have been propagated its the Netherlands (see ao. Treffers, 1987). These
methods are more loosely structured than conventional methods and emphasize "reinvention" of mathematical principles by
students in stead of extensive explanation by teachers.
Concluding, the goal of the research project to be described here was twofold:
1.
Development of a test wherein help can he obtained by students if they cannot solve the mathematical problems
independently;
2.
Comparison of the effect of teaching methods for mathematics on the mathematical ability of students.
Method
Attainment of the first goal of the project implied the development of a test for mathematical ability with a sufficient level of
difficulty, so that the effect of offering help could be assessed. Since transfer of mathematical knowledge and skills is one of the
most important goals of mathematics education and usually hard to measure, it was decided to construct a transfertest for
mathematics, containing hints. This leads to the opportunity to measure transfer in a more refuted way than conventional tests,
because the inability to solve transfer problems independently can be alleviated by offering help. Partial credit can be given for
correct answers given after consultance of hints.
The project vas partially sponsored by the Foundation for Educational Reeearch, The Mara,
The
Netherlands, grant no. 1128. It wan conducted at the Free University of Amsterdam, DO Boelelaan 1115, Amsterdam.
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159
171_
The question of the usefulness of offering help during a testing situation confronted us with the question of the predictive validity
of the test to be developed. Obviously, measuring mathematical performance when help is available only makes sense if such
measurements contribute to the validity of the measurement of independent mathematical performance level.
In order to be able to judge the usefulness of the testing-procedure, a criterion-test was constructed that was administered six
months alter the administration of the transfertest.
A nrv.Psary condition for transfer of mathematical knowledge is availability of preliminary knowledge. Therefore, it is important
to control for mastery of taught subject-matter. This was done by construction of a mastery test. Items in this test only called
upon knowledge of the subject-matter that was taught, i.e. knowledge structures did not have to be transformed to fmd the
solution. Most of the items in this test were derived from a test, constructed by the Dutch National Institute for the Development
of Educational Tests.
If one wants to compare the effects of teaching methods in a pre-posttest design it is important to control for initial mathematical
aptitude of subjects. No clear operationalization for this latter concept could be found, neither in the form of a definition (see
Meijer et, al., 1985) nor in the form of a test, specifically designed for this purpose.
In view of the lack of tests for initial mathematical aptitude it was decided to administer two tests for cognitive ability, that
correlate high with mathematical achievement. Two subtests of a test for measuring learning progress in secondary education,
constructed by Horn (1969) were used for this purpose. The subtests concerned were a) a test consisting of series of figures. b) a
test consisting of series of alphanumeric characters. In both tests the element that did not fit in the series had to be eliminated by
the subject. Aurin (1968) reports high correlations of both tests with mathematics achievement.
Many problems in the test for measuring mathematical transfer capacity required recognition of the underlying mathematical
structure in a situation that is described in common language. It was assumed that the capability to recognize this structure is
influenced by field-dependency (Witkin Cl. al.. 1977). It was established by several investigations that expertise correlates with the
ability to type domain-specific problems in terms of domain-specific principles in stead of superficial characteristics of the
problem-statement, which novices use to categorize a problem (Chi. Feltovich & Glaser, 1981). This is similar to the ability to
distinguish figure and background, a feature that the Embedded Figures Test pretends to measure. Therefore, the GroupEmbedded
Figures Test was administered in order to be able to correct for initial differences in this ability between students.
The usefulness of offering help during a testing situation can be tested by investigating the contribution to the predictive validity
of scores on items where hints were consulted, compared to a test wherein independent achievement is assessed only. That means
that if we add learning-potential scores to a regression equation containing the criteriontest scores as a criterion and a measure for
independent achievement as a predictor, the beta-weight for this added predictor should be statistically significantly greater than
nought.
Differences between the effect of teaching methods on the abilities of students should strictly be studied by imposing rigid
restrictions on an experiment, in order to prevent plausibility of alternative explanations. In educational research this is hardly
possible. In this research project, we compared the mathematical abilities of students in secondary education, learning mathematics
from different series of textbooks. The contents of each of three commercial series was studied carefully by experts on the
didactics of mathematics (see De Leeuw, Meijer, Groen & Perrenct, 1988). It was concluded that the fast series of textbooks
could be characterized as highly structured, but at the same time only teaching algorithms, "mechanistic', as De Lange (1987)
types it. The second series clearly aimed at insight, but on a very formal, abstract level ('structuralistic'). The final series was
most similar to the 'realistic approach', as mentioned in the introduction. In these methods context-problems are used to introduce
mathematical concepts as well as for application of taught concepts. The teachers role should be to build on the intuitive notions
of students and creating conditions that allow students to discover mathematical solution methods for different kinds of realistic
problems.
It was hypothesized that students using the last method would score highest on the transfertest because they should be most used
to using mathematical solution methods for unacqainted problems. Some of the items in the transfertest required generalization of
known mathematical principles, which was called "vertical transfer' by De Leeuw (1983). It was expected that students learning
from the structuralistic method would show highest performance on these items. Transfer-scores should be corrected for mastery of
subject-matter in testing these hypotheses, since the availibility of knowledge is a prerequisite for transfering it.
Finally, it was hypothesized that students characterized by a high need for prestructuring subject-matter would benefit most from
highly structured teaching-methods and would be disadvantaged by loosely structured teaching-methods. The 'realistic' teaching
method for mathematics can be typed as relatively unstructured, because much emphasis is put on proper construction and
inventions of students, i.e. students have to discover mathematical structures, which are only implicitly present in the contextproblems that serve as subject-matter, by themselves.
On the other hand students that are low in need of prestructuring may be disadvantaged by highly structured methods, because
they will be continuously disturbed by their lack of freedom to impose structure by their own effort. It was assumed that fielddependency is a good measure for need of prestructuring, since distinguishing figure from background depends oa structuring
activity. In other words, an interaction between teaching method and field-dependency in their effect on mathematical ability was
expected.
Procedure
I. Development of the transfertest
The transfertest for mathematics contained relatively difficult items. We started out with a test of eightteen items, each supplied
with six hints. All items concerned the subject-matter of functions, linear as well as quadratic. All hints were open indications, but
gradually increased in specificity, building on Selz' ideas (1935) about 'kleinst moegliche hilfe", i.e. we did not want to offer
solutions for the problems, but only structuring information. Hints could be made visible by tearing of pieces of paper on the
answer sheet. It appeared that most items were too difficult for our subjects, and that the effect of the hints differed greatly. Also,
the hints did not yield very many extra solutions. The advocated scoring method was to give partial credit for correct answers,
depending on the number of hints used. The main problem was however that there was no way to verify whether a hint had been
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160
understood by the subject. This implied that credit points would be subtracted unjustfully if subjects did not assimilate or already
were aware of the information that was contained in the hint.
A solution for this problem was found by changing the format of the hints. In stead of offering open ended indications, a
question was being asked with four answering possibilities. As in most multiple choice questions, only one of these answers was
correct, but only two were incorrect. The remaining answer was 1 don't know". This was added to avoid guessing, because
subjects automatically received extra explanation if they chose a wrong answer. In this way verification of assimilation of the
content of the hint was made possible. It was assumed that subjects that answered the hint-question correctly needed no further
information. In this case, only one credit-point was subtracted. If a subject gave a wrong answer or admitted he/she did not know
the right answer, extra explanation was given and two credit-points were subtracted.
Also, it was attempted to make the test-problems easier, mainly by deleting items with very low p-values and adding simpler
ones. Sometimes test-problems were simplified by incorporating the content of a hint in the problem-statement or by clarifying the
structure of the problem in advance.
This procedure was tried in individual testing situations, where experimenters recorded the sessions on audio-tapes. Subjects were
instructed to think aloud while trying to solve the test-items, during independent solving efforts as well as after having been
offered a hint by the experimenter. Hints were presented on slips of paper. Experimenters did not offer any extra explanation by
their own initiative.
After trying the new procedure on this small scale, is was repeated collectively with 210 students in grade four of secondary
education. The items in theirselves had clearly become easier, the percentage of test.problems that was solved independently rose
from 12 to 44. Although the percentage of items that was answered incorrectly notwithstanding consultation of hints declined from
56 to 24, it could not be shown that the effect of the hints had actually increased. The percentage of items that was solved after
using one or more hints remained 16.
The pilot-researches resulted in 15 items, that were deemed suitable for the final version of the test. During the pilot-experiments
it was also made clear that it took quite long to solve the test-problems. One the one hand, this was due to the difficulty of the
problems, one the other hand it obviously takes time to read and process a hint As a consequence, the total test was split up in
seven partially overlapping subtests of five items each.
The final version of the test-problems consisted of open-end questions, presented on paper sheets, that served as answer sheets at
the same time. Hints could be obtained on the screen of a micro-computer. The new procedure for presenting hints was retained,
i.e. depending on their answer on a hint-question subjects received extra explanation. If the answer was correct only one point
was subtracted from the score for the item, if extra explanation was offered after a wrong answer or choosing the alternative "I
don't know" one more point was subtracted. If an item was solved without help five points were obtained. Only two hints could
be consulted, so asking for both hints and answering them incorrectly, but giving the right end-solution resulted in a score of one
point. Wrong solutions to test-problems always yielded a zero score, regardless of hint usage. The final experiment was conducted
on eight schools, twelve classes were involved, totalling 325 students.
2. Administration of the reference tests for mathematical ability
The test for measuring actual mastery of subject matter was administered in the same period as the transfertest. Half of the
subjects first completed the test for actual mastery and then worked on the transfertest. The order was reversed for the other half
of the subjects. This was necessary because there were only sufficient micro-computers available to serve half a class of students.
The test consisted of 19 items. all except one were multiple-choice items. The only open ended item, that was scored by judges,
concerned the quality of a drawing of a graph of a quadratic function.
Both tests were administered in june 1986, when subjects were at the end of their third year in secondary education.
The criteriontest for mathematical ability was administered between decernber 1986 and january 1987, about half a year after the
first testing session. Because of the delay between the testing sessions, the predictive validity of the transfertest could be assessed
in a meaningful way.
The criteriontest consisted of 10 items. Two parallel versions were constructed in order to avoid possible fraude, because students
were arranged in adjoining seats. The test was carefully designed so that items represented subject-matter taught between the
period of administration of the first and second battery of tests. Solutions given by subjects were scored by judges based on
guidelines that were established in advance.
3. Administration of the reference tests for cognitive ability
The test for fieldindependency was administered at the same time as the test for mastery of mathematical subject-matter and the
transfertest. A period of four lessons hours (50 minutes each) was available for this first testing period on all schools. Two
lesson-hours were reserved for completion of the transfertest, while the other two were devoted to four tests in total, ie. the
Group Embedded Figures Test, the mathematical mastery test, a mathematics attitude test and a test measuring achievement
motivation and fear of failure. On the relationship of these latter tests with mathematical performance I will not divert here but a
report of this is in progress (Meijer, paper submitted to the Journal of Educational Research, Meijer, in preparation).
The Group Embedded Figures Test consists of several example-items and 13 test-items. In every item the problem is to identify a
simple geometrical figure in a rather complex drawing. Subjects must answer the problems by stressing relevant lines in the
drawing.
The other tests for measuring cognitive ability (series of figures and alphanurnerics, subtests of the PSB) were administered
simultaneously with the criteriontest for mathematical ability. One of the reasons to use these tests was that their administrationtime is very limited (respectively 8 and 5 minutes). Since there was only slightly more than one lesson-hour available for all tests,
it was important to save time.
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lei
Concluding, the following table shows which tests were administered when:
Table 1 Time table for test-administration
mathematics tests
june 1986
test for actual mastery
cognitive ability
embedded figures test
january 1987
criteriontest
transfenest
series of figures
series of alphanumerics
Results
The hypothesis that performance on items where help could be obtained contributes to the predictive validity of a test which also
accounts for independent performance was confirmed.
Scores on the transfertest were divided in the proportion of correct solutions arrived at independently and the proportion of correct
solutions found after consulting hints. Since only five items were administered, independent achievement was operationalized as
the amount of items solved correctly without help, divided by 5. Learning potential scores (operationalized by performance level
when given assistance) were calculated by dividing scores on items where hints were used by 5. Therefore, the maximum
learning-potential score to be obtained was 4 (i.e. 5 items times 4 = 20 divided by 5 = 4). Subjects solving all items without
consulting hints were excluded from the analysis. In this context, the problem with the impeccable performance of such eminent
students means per definition that their learning-potential is nil, which is not very plausible.
Initially 325 students took part in the experiment. Except four students, all of these completed the tests administered in june 1986
(actual mastery test, transfertest and embedded figures test plus some other tests that are not of interest here).
Because the second testing period took place after the selection of students between their third and fourth year in secondary
education, 86 subjects could not participate in the second testing period, due to non-promotion to fourth grade or dropout.
Complete data (on the actual mastery test, the transfer test and the criteriontest) were available for 224 subjects. Table 2 contains
the results of a regression analysis wherein independent achievement and learning-potential scores predict criteriontest performance:
Table 2 Prediction of criteriontest performance
criterion:
criteriontest for mathematical ability
R'
B
R
predictors:
transfertest
1. items solved without help
2. items solved with help
.289
.137
.108
.125
.329
.354
F
26.95
4.34
p
<.001
.038
Obviously, when using the proportion of items solved correctly without consulting hints as a predictor in the first place, the score
on items solved correctly after consulting hints turns out to contribute to the amount of explained variance in the criteriontest
scores significantly. This means that measuring performance when help can be obtained is not only a valid predictor for future
performance (as independent performance), but also shows sufficient discriminative validity. If the correlation between criteriontest
performance and independent achievement is partialled out, there still remains a portion of criteriontest score variance, that can be
explained by learning potential scores. However when using actual mastery test scores as a third predictor, the effects of learningpotential scores are no longer statistically significant:
Table 3 Prediction of criteriontest performance revisited
predictors:
transfertest
actual mastery
criterion:
criteriontest for mathematical ability
R
1. items solved without help
2. actual mastery test scores
3. items solved with help
.325
.383
392
B
.105
.147
.154
.227
.197
.091
25.81
10.49
11.85
p
<.001
.001
.175
We must conclude that independent transfer capability of mathematics knowledge and skills is the best predictor for future
mathematics performance, even after controlling for initial mastery of mathematics subject-matter. Measurement of transfer
capability in situations where help can be obtained ("learning potential") can contribute to the predictive validity of tests, albeit
marginally. Apparently partial credit can be given in a very objective fashion.
The results of the final experiment also show that the effectiveness of the hints had increased. The results of the second
experiment showed no improvement in the effectiveness of hints compared to the first experiment (both experiments resulted in 16
% items, solved correctly after consulting hints). In the final administration this percentage increased quite dramatically to 38.
This is probably partly due to the fact that the second pilot experiment took place in grade four, while the final experiment was
conducted in grade three, Because of this, the difficulty of the test problems increased comparatively. Items were relatively easy
for subjects in the second pilot experiment (illustrated by the fact that 44 % of the items was solved without help), while they
were quite difficult for subjects in the final experiment (only 23 % of the items was solved without help). The results show that a
scoring method where the amount of consulted hints is taken into account can be deviced in an objective way and that this
procedure renders a more refined measure for mathematical ability than measuring independent mathematical achievement only.
We will now turn to the comparison of the effect of teaching-methods. Three commercial teaching-methods (i.e. series of textbooks for mathematics) were compared (see Method).
Method A can be characterized by a highly structured approach, leaning heavily on algorithmization, stronly emphasizing practice
in recognizing known problem-types and using the appropriate solving-algorithm.
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1 62
Method B is typified by a relatively high level of abstraction, aiming at insight. Although algorithmization is also important in
this method, theory is explained extensively before application. Practice in solving many similar problems consecutively is
stressed.
Method C relies heavily on the "reinvention" principle (Freudenthal, 1978), i.e. the capacity of students to construct mathematical
principles and rules by themselves. Context problems are used for developing mathematical solving methods as well as for
applying such methods. The role of students is very important teachers should appreciate solving attempts of students and
instruction should be based on the constructions put forward by students.
The mean scores on the three measures of mathematical ability of students using these three methods are displayed in table 4.
Table 0 Mean scores on mathematics tests per teaching method
Mathematics tests
Teaching methods
Transfer
Mastery
A
B
C
Criterion
10.4
10.5
11.5
9.4
9.7
10.8
12.2
14.2
11.9
Although students using method C score highest on both tests administered in grade three, their mean score on the criteriontest is
lowest.
Because it was hypothesized that method C would score highest on transfer, after controlling for initial ability (measured by actual
mastery) a covariance-analysis was performed wherein transfertest.scores were the dependent variable, teaching methods were
conceived of as quasi-experimental treatments and actual mastery of subject-matter was the covariate. In order to make sure that
effects of teaching-methods would not be confused with the effect of school-environment or the grouping of students in classes,
groups of students in the same class within teaching methods were used as a nested factor. In this way the effect of teaching
method can be distinguished from the effect of the grouping of students. Table 5 outlines the results.
Table 5 Effects of teaching methods
corrected for initial mathematical ability
df
SS
Within cells variance
Mastery test
Teaching methods
Classes
9948
2043
3.4
240
308
1
MS
32.3
2043
2
9
1.7
F
63.3
.05
.83
26.6
p
<.0001
.948
.593
In can be seen that practically all variance in mathematical transfer capability can be explained by initial mathematical knowledge
of subjects; teaching methods and grouping of students in classes hardly matter. The fact that classes hardly make any difference
also points out that the effect of teachers on the development of mathematical ability of students should not be overestimated,
since all 12 classes had different teachers.
One of the assumptions of covariance analysis is that dependent variables regress with equal slopes on covariates in every cell of
the design matrix. This implies that the regression of mathematical on cognitive ability should be equal for all students in all three
teaching methods, in order to fmd out if one of these methods results in a higher mean mathematics achievement of students,
after controlling for cognitive ability.
It was found that several measures for mathematical ability show quite different strengths of the relationship with measures for
cognitive ability in method C, compared to methods A and B. Contrary to our expectations it appears that mathematical learning
results for students using the most loosely structured teaching method hardly correlate with their cognitive ability.
This means that the results of the covariance analysis indicating that there is no difference between the effect of the three methods
should be interpreted with great care. In order to shed more light on the relationship between mathematical and cognitive ability,
and on the differential effects of the teaching methods, an analysis-method proposed by ltireskog and SOrbom (1986) was applied.
A computer program developed by the authors mentioned first (L1SREL, short for Analysis of Linear Structural Equations) was
used to estimate: 1) the regression of mathematical on cognitive ability for each method, 2) the difference in means of
mathematical ability for each method, corrected for mean cognitive ability of students using every method.
On the basis of the covariance matrix and the vector of means of the mathematical ability measures (mastery, transfer and
criterion) and the cognitive ability measures (the Group Embedded Figures Test and the Series of Figures) for each method, the
linear structural model. depicted on page 6, was analyzed.
Eta, and eta, are latent (unobserved) variables, respectively representing mathematical and cognitive ability. Lambda, lambda and
lambdan are the regressions of the three tests used for mastery, transfer and criterion performance on 'true" mathematical ability;
lambda. and lambda. are the regressions of the scores on the Group Embedded Figures Test and the Series of Figures on "true"
cognitive ability.
The variable ksi, is a dummy variable. The parameters gamma, and gamma, represent the difference between means on the latent
variables for the students using the three teaching methods. Since no absolute value for the mean of a latent variable can be
estimated, the mean of one method for these variables is set to nil, so that only deviations from this standard have to be
estimated. Beta. represents the regression of latent mathematical ability on latent cognitive ability.
For estimating the differential effects of the teaching methods on mathematical ability, corrected for cognitive ability, the
parameter gamma, is of interest. If this parameter differs significantly for teaching methods, it may be concluded that students
educated with these methods differ in mathematical ability. The restriction is however that other parameters (the observed means
on measurements of mathematical and cognitive ability and the regression of observed on latent variables) do not differ. Only
differences in the means of the latent variables (unobserved mathematical and cognitive ability) are of interest.
189
En COPY AVAILABLE
1 75
Actual Mastery
Transferten
Criteriontesa
Field-dependency
Sena of Figures
52
Figure 1 A linear structural model for estimating differential effects of learning methods
These restrictions do not meet the data. Because of the suspection that the regression of mathematical on cognitive ability was
different for teaching method C and the fact that students using this method also scored lowest on the criteriontest, betas and
lambda were estimated separately for method C. Results are summarized in table 6:
Table 6 Estimated parameters per teaching method for the LISREL model, depicted in figure 1
(standard errors between brackets)
Teaching method:
A
B
Parameter:
lambda
lambda
lambda
lambda.
lambda.
beta
gamma,
gamma,
1
(.000)
2.334 (.349)
1.836 (.289)
(.000)
358 (.144)
.689 (.198)
.576 (.425)
-.395 (.576)
1
1
(.000)
2.334 (.349)
1.836 (.289)
I
(.000)
.558 (.144)
.689 (.198)
-.583 (.463)
.305 (.562)
I
(.000)
(.349)
(.332)
1
(.000)
.558 (.144)
.126 (.127)
.000 (.000)
.000 (.000)
2.334
.632
(chi' = 48.98, df = 24, p = .002, Goodness of Fit = .885)
Although the model shows only very moderate correspondence to the data. it is obvious that the regression of mathematical on
cognitive ability is not significant for students using method C. while teaching methods A en B show significant influence of
cognitive on mathematical ability. Also the regression of criteriontest performance on latent mathematical ability is much smaller
for method C than for methods A and B.
There appears to be no difference between mathematical or cognitive ability between the three methods, although it seems that
students educated with method B show slightly superior performance. At the same time, they seem to have lower mean cognitive
ability.
However, these differences are statistically insignificant and open to doubt because of the differing values of lambda, and beta.
for method C, compared to methods A and B.
Since the regression of criteriontest performance on latent mathematical ability and the regression of latent mathematical ability on
latent cognitive ability appear to be similar for methods Ay and B, an adequate comparison of the effects of these methods on
mathematical ability by means of this method of analysis may be made. The results of this analysis are summarized in table 7:
76
184
Table 7 Estimated parameters per teaching method for the LISREL model, depicted In figure 1
(standard errors between brackets)
Teaching method:
A
B
Parameter:
lambda
lambda.
lambda,,
lambda*,
lambda.
beta,'
gamma,
gamma,
1
(.000)
2.153 (.322)
1.763 (.269)
1
(.000)
.546 (.145)
.711 (.204)
1.224 (.442)
-.698 (.565)
(.000)
(.322)
(.269)
1
(.000)
.546 (.145)
.711 (.204)
.000 (.000)
.000 (.000)
1
2.153
1.763
Method A was used as the standard in stead of method C here. i.e. gamma, and gamma, were set to nil for this method. All
other parameters were set equal for both teaching methods. The model fits the data remarkably well (chi' = 10.53, df = 15. p =
.785).
Obviously for both methods mathematical achievement is quite strongly influenced by cognitive ability, as measured by the score
on the Group Embedded Figures Test and Series of Figures. Again, it seems that students using method B start out with a slight
handicap in cognitive ability, but this difference is not statistically significant. In spite of their initial disadvantage these students
perform significantly better on all mathematical tests taken than students educated with method A.
Discussion
The results of the experiment described in this paper show that there is definite perspective in constructing transfertests for
mathematics wherein the twee can obtain help if needed. This is of great importance in the light of the fact that transfer of
domain - specific knowledge and skills usually occurs on a very small scale. It is often only observed in the case of great similarity
between problems.
The lack of evidence for transfer may very well be due to the poor discriminative power of transfertests, wherein items have to
be solved independently.
It was established in this research-project that obtaining help during testing-situations can be scored in an objective way.
Therefore, basic methodological objections against such procedures hardly seem valid. Equal help was available for all subjects,
offered by means of a computer, without the intervention of an experimenter.
Although questions concerning equal impact of information contained in every hint for each subject still remain unresolved, we
think that the development of this type of test could be of major importance for education.
Since there are so many confounding variables involved (for example: social economic background of students, quality of teachers
and school-climate), comparison of the effects of teaching methods is a very difficult matter. Dubin & Taveggia (1968) even
contend that no differences can be found in the effects of teaching students individually, in small groups or in lecturing groups.
involving rather large amounts of students. Therefore, chances of finding differences in mathematical achievement between
students educated by different series of textbooks seem remote. The influence of school environment, teachers, student population,
must be a greater potential source of influence compared to what school mathematics book is being used.
In this study no support could be found for the hypothesis that a loosely structured teaching method (method C) renders relatively
bad learning results for students low in cognitive ability. On the contrary, cognitive ability appeared hardly relevant for students
educated by such a method. This may imply that loosely structured teaching methods are indeed beneficial for all students.
Furthermore a significant advantage was revealed for students using a "structuralistic" teaching method (e.g. method B), compared
to students educated with a 'mechanistic" teaching method (method A).
Therefore, we may conclude that structuralistic teaching methods should be preferred over mechanistic teaching methods. However.
the effect of structuralistic teaching methods depends on cognitive abilities of students equally strong as the effect of mechanistic
methods. In contrast, the relationship of the effect of realistic teaching methods with cognitive ability of students appears hardly
important. In order to ensure optimal results for all students, the application of such teaching methods seems very promising.
At present we are conducting a preliminary investigation into the effects of realistic teaching methods for mathematics in the
Netherlands in different types of secondary education. We hope to be able to present results of this study at future conferences.
REFERENCES
Aurin, K. (1966), Bildung in new Sicht Ermittlung and Erschliessung von Begabungen im ldndlichen Rawn, Villingen, Neckar
Verlag.
Chi, M.T.H.. Feltovich, P.I. and Glaser, R. (1981), CategorizatiOn and Representation of Physics Problems by Experts and
Novices, Cognitive Science, 5, 121-152.
Dubin, R. and Taveggia, T.C. (1968), The teaching-learning paradox; a comparative analysis of college teaching methods, Eugene,
Oregon. University of Oregon.
Freudenthal. H. (1978). Weeding and Sowing; Preface to a Science of Mathematics Education, Dordrecht/Boston, Reidel Publishing
Co.
Ginsburg. H. (1983), Development of Mathematical Thinking, New York, Academic Press.
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Horn, W. (1969), Prdfsystem fur Schuh and Bildungsberarung (landanweisung), Gottingen. Verlag fUr Psychologie, Hogrefe.
Joreskog, K.G. and Sorbom, S. (1986), Analysis of Linear Structural Relationships by the Method of Marimum Likelihood, Uppsala,
University of Uppsala.
Krutetskii, VA. (1976), The Psychology of Mathematical Abilities in School Children. Chicago, University of Chicago.
Lange, 1. de (1987), Mathematics, Insight and Meaning, Utrecht, State University of Utrecht.
Leeuw, L. de (1983), Teaching Problem Solving; An ATI-study of the Effects of Teaching Algorithmic and Heuristic Solution
Methods, Instructional Science, 12, 1-48.
Leeuw, L. de, Meijer, 1.. Groen, W.E. and Perrenet, .I.C. (1988). Construction and Validation of a Transfertest for Mathematics
Education using Items with Optional Help: Amsterdam, Free University of Amsterdam (in Dutch).
Meijer, 1. et. al. (7985), A Transfertest for Mathematics, containing Items with Cumulative Hints, Proceedings of the Ninth
International Conference for the Psychology of Mathematics Education. 393-412.
Selz, 0. (1935), Versuche zur Hebung des Intelligenznivos, Zeitsclvift far Psychologie, 134, 236-301
Treffers, A. (1987), Three Dimensions, Dordrecht, Reidel.
Vygotsky, L.S. (1964), Denken and Sprechen, Berlin. Akademie Verlag.
Witkin, HA., Moore. CA., Goodenough, DR. and Cox, P.W. (1977), Field-dependent and Field-independent Cognitive Styles and
their Educational Implications. Review of Educational Research, 77, 7-64.
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166
Didactical Analysis
179
ON LONG TERM DEVELOPMENT OF SOME GENERAL SKILLS IN
PROBLEM SOLVING: A LONGITUDINAL COMPARATIVE STUDY
Paolo Boero, Dipartimento di Matematica, University di Genova
In this report I relate upon a research performed on written texts of 65 grade II
to the grade IV primary school pupils, concerning the development of their
hypothetical reasoning skills in problem solving. Some comparisons are made
on the nature of the problem situations in which these skills appear for the first
time: mathematical or non-mathematical problems; situations contextualized or
not in some "experience field" of our project.
1.Introduction
In the researches concerning applied mathematical problem solving performed during the next
fifteen years,a growing importance is attributed to the "content" of the problem situations,to the
"context" in which they take place and to the children's involvement in the "context " , in order
to interpret both the conceptual and the procedural acquisition of the pupils, and the difficulties
they meet in some circumstances (Carpenter & Moser,1983 ; De Corte & Verschaffe1,1987;
Nesher,1980 ; Lesh,1981 ; Lesh,1985 ) . Carraher's (1988) results concerning the acquisition
("in the street situations") of important problem solving skills of "strategical" type for problems
concerning "money" give further insights in the same direction.
In (Boero,1988) I examined the pupils' "sensitivity" to the "content " and to the "context "of
some problem situations taking place in the Genoa group's project for the primary school.In
(Boero,1989) I proposed
a theoretical framework for the "context" problem , in relation to the
experiences of the curricular projects developed by my group and current researches. In that
conference, I advanced the hypothesis that the choice of suitable "fields of experience" might
influence the development of general problem solving skills concerning the "representation"
processes,hypothetical reasoning and metacognition.
Ferrari (1989) considered the hypothesis (derived from the classroom observations of many
teachers working in our group) that the child "who uses properly hypothetical reasoning in
mathematical problem solving is already able to use it properly in other settings".
In this paper I will provide more precise elements to support the hypothesis that children's
involvement in suitable "contexts", in classroom activities, may influence the development of
some general problem solving skills.These elements might also contribute to clarify the
relations existing between the development of general problem solving skills in mathematical
and in non-mathematical problem situations.The research I relate upon concerns some of the
skills involved in generating and managing hypotheses during the construction of strategies in
problem solving . I wish to compare the moments at which these skills appear in written form
during non-mathematical activities,and in contextualized or noncontexualized mathematical
problems, for children followed from the age of 7 to the age of 10.
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180
2.The educational context
As described in more details in (Boero,1988 and Boero,1989 ) ,the educational context to which
I will refer is a curricular project concerning mathematics and the other main subjects taught in
the Italian primary school.This project consists in a systematic work in suitable "experience
fileds"(Boero,1989) concerning the natural and social reality (for instance: "productions in the
classroom", "history of the family", "economical exchanges", "the sun shadows", and so on).
Most of the mathematical problem situations proposed to the children are inserted in these
"experience fields " .However,the work in each "experience field" also concerns many
non-mathematical activities (for instance,performing experiments and writing reports about
them; studying historical and geographical topics about the experience fields; and so on).These
activities naturally produce many non-mathematical problem situations.
Particular importance is given in the project to the verbalization processes and to the activities
aimed at developing verbal competencies.
Each year,the controlled experimentation of the project concerns about 40 classes of each grade
from the age of 6 to the age of 10.For each grade 4 classes are "observation classes", and these
classes are followed from the first to the fifth grade, usually by the same teachers (according to
the Italian tradition) who collect all the pupils' written texts with detailed information about the
conditions in which they were produced.For the observation classes we have at our disposal,
amongst the others materials, individual written texts concerning: non-mathematical activities
performed in the "experience fields"; mathematical problem solving contextualized in the
"experience fields"; decontextualized problem solving. For each child belonging to an
"observation class" we collect, during five years, about 1000 written texts (about 1 each day)
About 40% of these materials concerns (partly or completely) mathematical activities.
3.Posing the research problem
The structure of the project and the materials we collect in the "observation classes" allow us to
perform some systematic analyses concerning the development of general problem solving
skills and the influence of the "experience fields" on it.However,it is necessary to distinguish
carefully the analyses which may really give a reliable insight to the questions considered For
instance,the materials at our disposal allow us to establish when and in which context some
skills appeared in written form,for the first time,in order to study the transfer to other contexts
and situations, and so on.Then we must focus on the skills whose development we wish to
analyze.With regard to the hypotheses quoted in the introduction, we must choose skills
involved both in non-mathematical and mathematical problem solving activities (and, in this
case, in contextualized and noncontextualized situations).For a first longitudinal studyd
consider in this paper some skills concerning "generatina_and managing hypotheses in problem
solving " in order to ascertain the influence of the context.of the content of the problem situation
and of the teacher's request on their appearance and transfer These skills may be described in
more detail by considering the following two kinds of performances:
TRIALS-type performances : the pupil
makes (heuristic) trials and,after analyzing their
170
outcome, plans the following activities, in order to reach the solution ( "./ make the assumtion
that...; than I see that...; and so 1 must take...")
BONDS-type performances : the pupil discovers the bonds inherent to the problem situation
and builds a strategy leading to the solution according to these bonds ("I must take into account
that...; and then if ....else...")
These kinds of performances are relevant in many problem solving situations (Ferrari,1989),and
their importance is growing in the computer age.As we will see in the examples,frequently both
performances need to be taken into account in the same problem solving process and in many
cases they are interwoven.
The requests activating the production of the written texts, which reveal the skills we are
considering, may be of three kinds:
(VD) Verbalizing the reasoning "During" the performance(" write down what you are thinking ")
(VA) Verbalizing the reasoning immediately "After" the solution has been attained ("relate about
your resolution of this problem")
(VG) Verbalizing a "General" method of resolution (" explain to a friend of yours how he may
solve this problem")
It is necessary to point out the fact that under one of these requests many children combine
elements referring to the others.For instance in the texts produced under a (VG)-request they
frequently combine (VG) -pieces and (VA)-pieces (especially if children have already solved the
problem in a particular case),In the texts produced under a (VD)- request they frequently insert
(VA)-pieces (because many children prefer to get a partial solution and then to relate upon the
method utilized to get it).We may observe also that in the same problem situation the nature of
the request may pull towards TRIALS-type or BONDS-type performances: in many problem
situations a (VG)- request pulls towards BONDS-type performances which are less important
under a (VD)- request.
As typical problem solving situations demanding the skills we are considering,we may quote
(from our project):
(EXAMPLE 1) To set a wood table in an horizontal position over an "irregular" ground,
utilizing a spirit level.. It is a problem situation which only partially refers to mathematics ( it
will be classified later as a "mixed situation"); in our project this problem situation is
contextualized in the "experience field" of the "sun shadows"; TRIALS-type and BONDS-type
performances are required, the latter especially under (VG)-requests.
(EXAMPLE 2) To relate upon an experience of production in the classroom, demanding to
perform some "controls" and choices consequent to the outcomes of these controls. It is a
problem situation without (or with poor) mathematical content, contextualized in the experience
field of the "class productions", and relevant especially for BONDS-type performances
(particularly under a (VG)-request: "write down the recipe to prepare the cake...")
(EXAMPLE 3) To divide an expense (like 107000 liras) amongst some children (for instance,
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182
34 pupils) : before a written calculation technique has been taught, this problem situation (of
mathematical content) demands TRIALS-type performances (Boero, Ferrari & Ferrero,1989);
the problem may be posed as a completely noncontextualized one,or as a question naturally
contextualized in the classroom activities.
Referring to these examples and to the conceptual tools which are utilized during problem
solving processes, we may distinguish (but see also §6 ) three kinds of problems:
"non-mathematical problems", like in EXAMPLE 2; "mixed problems", like in EXAMPLE 1;
and "mathematical problems", like in EXAMPLE 3
.
4.Available materials and utilized materials
I have considered the individual written texts collected in four "observation classes" (73 pupils)
followed by the same teachers during the school years 1983/84 to 1988/89 . I have restricted my
analyses to grade 11,111, and IV (where the activities aiming to develop hypothetical reasoning are
more frequent), and to written texts referring to mathematical and non mathematical problem
situations in which TRIALS-type and/or BONDS-type performances are demanded. With these
restrictions, I have taken into account 5510 texts referring to 128 problem situations (many of
them common to 3 or 4 classes) which may be classified as follows: about 75% contextualized in
some "experience field"(and 25% non-contextualized); about 32% of non mathematical type;
about 20% of mixed type; about 48% (23% contextualized, 25% non-contextualized) of
prevailing mathematical type. All the non-contextualized problem situations are of mathematical
type.This distribution is almost uniform for every grade and for every class considered,and
almost uniform during each year.This "uniformity" may
be explained by the fact that the
classes adopt the same project and that the project maintains, at every grade, an equilibrium
amongst the different kinds of activities we are considering.
On every written text considered the teacher noted the conditions in which it was written. The
texts were produced during the first or the third phase of the usual work of our classes on a
given problem situation:
phase I:individual work,with individual writing of a text (under requests of VD,or VA,or VGtype) ; we will refer to these texts as "Autonomous Without Discussion" (briefly, AWD-) texts
- phase II: discussion of the strategies proposed by some pupils (who illustrate them to the other
pupils); if there are no valid strategies,the teacher gives some suggestions and points out the
inadequacy of the proposed strategies
- phase III: individual work, with individual writing of a text.The pupils having already
produced a good AWD-text are asked some further question (not considered for the aims of this
analysis); the others pupils produce "Autonomous After Discussion" (briefly, AAD-) texts, or
need the individualized support of the teacher in order to produce " Supported texts"
- phase IV: comparison,correction and enregistration on the individual "copybook" of the whole
work performed in the classroom on the problem situation considered.Incidentally [ observe that
the analysis of these "copybooks" may be very useful to reconstruct, after some years, the
conditions in which the texts at our disposal were written and to complete any information
1S3
172
eventually lacking or ambiguous on the written texts collected.
Examining with the teachers (M.G.Bondesan,A.Carlucci,E.Ferrero, G.Pontiglione, A.Rondini)
the records of the 73 pupils ,we excluded 8 pupils for the following reasons:1 was frequently
absent at grades II and III and followed the activities of another class during five months in
grade IV; 3 had a record containing many scarce and/or confused "traces" (expecially in grades
II and 110;4 revealed a complete mastery of hypothetical reasoning in mathematical and non
mathematical situations already at the beginning of grade II.
Considering the other 65 pupils,we had 4975 written texts at our disposal; by eliminating 1312
texts (unintelligible; or too scarce; or containing strategies not aimed at solving the posed
problems- the greatest majority of these texts were "Autonomous Without Discussion" texts), I
considered the remaining 3663 texts.It must be pointed out that these texts may contain "good"
strategies, or also partly wrong resolutions The reason for this choice is the fact that a partly
wrong resolution may contain a good approach to hypothetical reasoning .
TABLE 1 gives some information about the distribution of the texts which have been taken into
consideration for the following analyses. The percentage is evaluated on the data of each line.
TABLE 1
distribution of the 3663 written texts utilized (65 pupils)
CONTEXTUALIZED PROBLEMS
NON MATH.PROB. MIXED PROB. MATH.PROB
(20%)
(23%)
(32%)
ON CONTEXT.PROBL
(MATH.PROB.)
( 25%)
kWD-texts
463 (35.5%)
298 (22.8%)
283 (21.7%)
261 (20.0%)
k AD-texts
452 (36.0%)
265 (21.1%)
261 (20.8%)
278 (22.1%)
upported
texts
326 (29.6%)
210 (19.1%)
268 (24.3%)
298 (27.0%)
It may be observed that the first line contains only 1305 texts, with a distribution which leaves to
mathematical problems (48% of the total) only 41.7% of the texts; in fact, mathematical
problems (especially the non contextualized ones ,25% of the total) give a larger contribution to
"scarce" or
"completely wrong" strategies). It must be pointed out,however,that
non-contextualized mathematical problems are generally not more "difficult" (as far as the
mathematical concepts and procedures involved are concerned) than the contextualized problems
proposed at the same time.Then we find here a first element which indirectly supports the
hypothesis that the "content" and the "context" of a problem situation influence the quality
(autonomy..) of the performances related to hypothetical reasoning.
5.Some analyses performed and their results
We tried to get information pertinent to the problem posed in § 3 from the 3663 selected texts.
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184
A first analysis was performed on the first occurrence of an hypothetical reasoning
BONDS-type performances) in the "Autonomous Without Discussion"
(AWD) -texts fa in the "Autonomous After Discussion" (AAD)-texts produced by the
pupils.The data obtained is shown in TABLE 2 (2 pupils out of 65 had never produced an
(TRIALS-type or
hypothetical reasoning in problem solving activities up to grade IV):
TABLE 2 : first occurrence of hypothetical reasoning
NON-CONTEXT.PROB.
CONTEXTUALIZED PROBLEMS
(MATH.PROB.)
NON MATH.PROB. MIXED PROB. MATH.PROB.
AWD-texts
7
4
2
0
AAD-texts
23
13
10
4
Some remarks:
- the 13 pupils of the first line are all considered by their teachers to be "good problem solvers"
in any kind of problems
- amongst the 50 pupils of the second line,we have considered the 19 pupils who needed more
than 3 individualized interventions (registered in "supported texts") before producing an
AAD-text revealing the presence of hypothetical reasoning; they are considered to be "poor
problem solvers" by their teachers.They produced their first hypothetical reasoning in a
non-mathematical contextualized problem in
12 cases, and in a non-contextualized
(mathematical) problem in 2 cases.
Another analysis was performed on the 59 pupils (out of 65) who produced an AWD-text
revealing the presence of hypothetical reasoning before the end of grade IV
TABLE 3 : first occurence of hypothetical reasoning in AWD- texts
NON-CONTEXT.PROB.
CONTEXTUALIZED PROBLEMS
NON MATH.PROB. MIXED PROB. MATH.PROB.
AWD-texts
28
16
13
(MATH.PROB.)
2
We have also performed an analysis about the time delay separating the first hypothetical
reasoning performance from the transfer to other kind of situations in which it occurred; we got
the following results:
transfer from "non- mathematical" to "mathematical" problems in AWD-texts: 26 cases ; mean
value of the delay, 3.4 months (standard deviation: 1,4 months);transfer not realized in 2 cases
- transfer from "contextualized" to "non- contextualized" problems in AWD-texts: 45 cases; mean
value of the delay: 7.1 months (standard deviation:3.1 months);transfer not realized in 12 cases
Other analyses performed concern:
the age at which pupils,on average, reveal for the first time an hypothetical reasoning in an
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174
AWD-text (63 cases,adding the 4 cases already revealing the mastery of hypothetical reasoning
at the age of 7,evaluated with the age of 7) : we have found an age of 8 years and 5 months,
with a standard deviation of 8 months
- the kind of requests under which pupils produce their first written hypothetical reasoning : for
the reasons considered in § 3, it is not easy to answer this question; considering only 28 pupils
producing a text perfectly coherent with the kind of request, we see that in 24 cases the request
"forcing" the hypothetical reasoning is a (VD)- or an (VA)-request .This result may be explained
in two manners: by the fact that in our classes the (VG)-requests are more frequent at the IV
grade than at the II grade; and/or by the fact that the (VG)-requests are more" difficult" than the
others (pupils need to take into account also "cases" not experienced while solving the problem
in a particular situation).
6.Discussion
The results seem to prove that:
i) the classroom work contextualized in the "esperience fields" of our project anticipates (in
comparison with non-contextualized problem situations) the development of the skills of
hypothetical reasoning considered in this paper.This result agrees (for those problems
demanding TRIALS-type or BONDS-type performances) with the general results quoted by
Lesh (1985) concerning the success of good problem solvers "experts" in a given domain "who
tend to use powerful content-related processes", and the failures of pupils "who do not have
relevant ideas in a particular domain".
ii) the classroom work in non-mathematical,well contextualized problem situations favours
some anticipation in the development of hypothetical reasoning in comparison with problem
situations strongly referring to mathematical contents
Possible limitations to these results may depend on:
the arbitrary classification of problems,both
concerning the distinction between
"mathematical", "mixed" and "non-mathematical" situations, and the distinction between
"contextualized" and "non-contextualized" situations (for instance,a problem like EXAMPLE 3
in § 3 may be proposed as a "contextualized" problem in a class working on "class
productions",or as a "non-contextualized" problem one month later...but children may easily
refer the problem to their past experience...).This limitation appears to be intrinsic to the
research.
- the small number of pupils involved (but this limitation might be overcome by extending the
analysis to the other "observation classes" of our group); I observe however that all the results of
the analysis performed on this group of 65 pupils agree with i) and ii)
- the "didactical contract " (Brousseau,1984) taking place in the non- contextualized situations ;
often they are "evaluation tests",and then many pupils consider them in any case as "evaluation
tests" (also if they are proposed without this aim); this might reduce the "acceptance of risk"
which favours the TRIALS-type performances . However we observe that many pupils do not
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186
transfer the skills already extensively revealed in contextualized situations of lower level of
difficulty to non- contextualized situations
- evaluation of the solving strategies: generally the evaluation is not difficult for the good
problem-solvers ; on the contrary, it may be arbitrary in many cases concerning the low level of
the classes (confused verbal "traces", interference of suggestions coming from the schoolmates
- interpretation of the verbal productions of pupils: generally the syntactic analysis of the verbal
"traces" of the solving strategies is not sufficient to ascertain the presence of an hypothetical
reasoning (which frequently is expressed without an hypothetical period); then it is necessary to
make a semantic type analysis,and this implies (in some cases) rather arbitrary choices.
Keeping these limitations in mind, the internal coherence of the results of the analysis
performed, in agreement with the "impressions" of many teachers of our group who have
observed the same phenomena in their classes , and the external coherence with the results of
other researches on problem solving seem, however, to enhance the validity of the conclusions
(i) and (ii).The next step of the research will be to deepen the analysis of the factors which allow
the "experience fields" to act on the development of the skills concerned in this paper : my
present opinion is that the "motivating" factor is not the most important one, and that ,on the
contrary, to discover the most relevant factors it is necessary to analyze the mental processes
which bring the pupil to a total mastery of the "experience fields" .In any case, the impact of this
kind of research on mathematical education is not negligible, due to the fact that TRIALS-type
and BONDS-type performances are of great importance for problem solving with the computers;
and that most of the problem situations proposed to pupils in our primary school are of
non-contextualized, mathematical type.
References
Boero,P.(1988)- Acquisition of meanings and evolution of strategies in problem solving ...
Proceedings P.M.E.-X11, 1 (pp.177-184).Veszprem: OOK
Boero,P.(1989)-Mathematical literacy for all:experiences and problems. Proceedings PME-X111,
1 (pp.62-86). Paris.
Boero,P.,Ferrari,P.L.,& Ferrero,E.(1989)- Division problems: meanings and procedures in the
transition to a written algorithm. For the Learning of Mathematics, 9 ,17-25
Brousseau,G.(1984)- The crucial role of the didactical contract..., in Steiner et al.,Theory of
mathematics education, occasional paper 54,(pp.110-119).Bielefeld, I.D.M.,
Carpenter,T.P.,& Moser,J. (1983)-The acquisition of addition and subtraction concepts.In
R.Lesh & M.Landau (Eds.). Acquisition of mathematical concepts and processes
(pp.7-44).New York: Academic Press.
Carraher,T.(1988)- Street mathematics and school mathematics. Proceedings P.M.E.-X11,1
(pp.1-23).Veszprem: OOK
De Corte,E.,& Verschaffel,L. (1987)- The effect of semantic structure on first graders'
strategies ... Journal for Research in Mathematics Education. 18 , 363-381
Ferrari,P.L.(1989)- Hypothetical reasoning in the resolution of applied mathematical problems at
the age of 8-10. Proceedings P.M.E.-X111. 1 (pp.260-267).Paris.
Lesh,R.(1981)-Applied mathematical problem solving.Educational Studies in Math.12,
235-264
Lesh,R.(1985)- Conceptual analyses of mathematical ideas and problem solving processes.
Proceedings P.M.E.-1X. 11 (pp.73-96).Utrecht : OW & OC
Nesher,P.(1980)- The stereotyped nature of school word problems. For the Learning of
Mathematics 1, 41-48
187,
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COGNITIVE DISSONANCE VERSUS SUCCESS AS THE BASIS FOR
MEANINGFUL MATHEMATICAL LEARNING
Nerida F. El lerton and M.A. (Ken) Clements
Deakin University, Australia
Abstract
Cognitive dissonance theory has often been advocated as a guide to
mathematics teachers iruerested in creating stimulating learning environments
for their students. This paper contrasts cognitive dissonance theory with the
success-based theory of Karmiloff-Smith, and argues that the latter theory is
more compatible with natural learning environments in the mathematics
classroom. A case study which involved 28 pupils and 2 teachers in a
primary school is outlined. The natural learning environment created by the
teachers enabled significant worthwhile mathematical learning to occur learning that contrasted with that arising out of more traditional mathematics
classroom environments.
Learning as a Result of Cognitive Conflict Versus Learning Based on Success
Cognitive Conflict Theory
There is a considerable body of data supporting the idea that children best learn mathematics
by being exposed to their misconceptions before actively resolving their inner cognitive conflict
(see, for example, Bell 1986; Bell and Bassford, 1989). Typically, Piaget's equilibration
principle, with its twin notions of assimilation and accommodation, is called upon to provide a
theoretical basis for this conflict-resolution theory. Concerning the role of the teacher, Piaget
(1975) himself wrote:
The teacher as organiser becomes indispensable in onder to create the situations, and
construct the initial devices which present useful problems to the child . . . he [the
teacher] is needed to provide counter-examples, that compel reflection and
reconsideration of over -hasty solutions. (p. 16)
Vygotsky's (1986) notion of a zone of proximal development is also invoked to support the
theory (see, for example, Brown and Campione, 1984, pp. 145-146). The zone of proximal
development is said to refer to the distance between the level of performance the child can reach
unaided and the level of participation the child can accomplish when guided by someone else
who is more knowledgable in that domain. For a particular child in a certain domain, this zone
may be quite small; that is to say, the child is not yet ready to participate at a more mature level
than his/her unaided perfomance would indicate. For another child in the same domain,
however, the zone of proximal development can be quite dramatically large, indicating that,
with teacher assistance, and sometimes minimal assistance at that, the child can participate much
more fully and maturely in the activity than one might have supposed. In traditional terms,
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these are notions of "readiness," and Vygotsky's theory is invoked to justify the teacher
manipulating a learning environment so that the child will experience cognitive dissonance and,
as a result, move rapidly within his/her zone of proximal development.
Learning Mathematics Naturally
Michael Cole and his colleagues (Laboratory of Comparative Human Cognition, 1983) have
appealed to Vygotskian theory to develop a theory of learning which differs from cognitive
conflict theory. Focusing on Vygotsky's notion of expert scaffolding, and in keeping with his
long-standing in "natural learning," Cole points out that children of many cultures are initiated
into adult work-activities gradually, and without explicit instruction. The adults simply get on
with doing their work, and the children participate, first as spectators, then as novices
(responsible for very little of the actual work), and then increasingly as maturing participants.
They become capable of performing more complex aspects of the work that they have seen
modelled by adults many times before (Brown & Campione, 1984, p. 146). In this situation,
the main agenda is the natural one of getting the task done, and the idea of helping children learn
is less important than the work activity itself; there is no suggestion of a teacher-learner
relationship in the situation, yet the natural learning environment is powerful, perhaps even
more powerful than a cognitive dissonance teacher-guided environment.
Mathematics educators have long taken the idea of a natural learning environment for
mathematics seriously (see Clements & Del Campo, 1987, pp. 4-39), and there is evidence to
support the osmosis theory of mathematics learning (Ellerton, 1988). It needs to be recognised,
however, that the natural learning thesis differs sharply from cognitive dissonance theory. As
Pe led and Resnick (1987) pointed out:
The natural-environmental approach suggests that understanding of a concept
emerges from dealing with real world situations; therefore the exemplifications
should be the situations themselves, rather than a representation of the abstract
mathematical entities. The structural approach, on the other hand, treats the
abstract mathematical entities and their mathematical senses as the reference of the
exemplification. Real world situations, according to the structural approach, should
be introduced instructionally only after the formal system has been established.
(p. 185)
Pe led and Resnick (1987) went on to argue that the natural approach is not necessarily superior,
from a learning perspective, to the more structured approach. They outlined an investigation in
which they defined numbers and operations first, and only later introduced real world situations
(p. 189).
Success rather than cognitive dissonance as the basis of learning. The cognitive
developmental psychologist, Annette Karmiloff-Smith, is another who has questioned some
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aspects of the cognitive dissonance theory of learning. Karmiloff-Smith (1984, p. 40) has
argued that neither failure nor economy are the major motivations of developmental change. In
her view, cognitive change, in an individual, stems from a constant motivation for control, both
over the external environment and over one's internal representations - in this respect she is in
line with cognitive dissonance theory. However, she focuses on the gradual process of gaining
such control, demonstrating how children spontaneously go beyond initial success, achieved
through their adaption to environmental feedback, to working subsequently on their internal
representations as a form of problem solving in its own right. Success, and not failure, she
argues, is the essential prerequisite of fundamental developmental change; in fact, failure is
rarely the prerequisite of representational change. According to Karmiloff -Smith (1984):
Failure generates behavioral change during which the system evaluates, and
narrows the distance between, the child's goal and the child's present output. By
contrast, representational changes are the result of representational
reorganizations, the prerequisite for which is not failure, but procedural success.
(p. 40).
Thus, Karmiloff-Smith (1984, p. 40) claims, "once children have obtained a robust initial
success, they go beyond it and try to understand why certain procedures are successful,
unpacking what is implicit in them, and unifying separate instances of success into a single
framework."
Cognitive Conflict Versus Success-Based Mathematics Classrooms
The previous discussion outlining the differences between learning theories based on
cognitive dissonance and "success" theories has important implications for mathematics
classrooms.
Cognitive dissonance classrooms. In these classrooms, it is the role of the teacher to arrange
for learners to experience cognitive conflict situations at just the right time. It follows that
teachers need to be constantly assessing what experiences need to be provided for different
learners, and trying to ensure that the appropriate conditions for these experiences are present in
the classroom.
A likely consequence of the cognitive dissonance approach is that teachers will make most
of the decisions concerning what, how, and when individual children should learn. In such
circumstances, it could hardly be claimed that the children "own" what they are learning; rather,
the teacher and the textbook writer are likely to be seen by the students as the controllers of their
mathematical destinies. The quest to understand becomes an individual pusuit, and is therefore
more likely to be a competitve, rather than a collaborative act.
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Success-based classrooms. Here it is likely that teachers will be more accepting of what
students do. Consequently, adult-imperfect responses will be accepted as appropriate provided
they are consistent with perceived cognitive growth. The teacher's focus will be more on what
children can do, and less on what they cannot do. The classroom environment is likely to be
closer to a so-called "natural" learning environment. In a success-based classroom, the teacher
will not feel the need to assess constantly what the child is ready to learn, or what conflict
situation needs to be imposed to ensure maximum learning. Therefore, learning from other
children, both through the process of osmosis and through discussion, is more likely to take
place. The "ownership" of knowledge is not seen as resting with the teacher or the textbook,
and the quest to understand can be seen as deriving from a combination of individual and group
activities.
Strathbogie Primary School: An Example of a Success-Based Mathematics Classroom
Videotapes were made of 28 children, aged 5 to 12, involved in mathematical activities with
two teachers in two composite grades (covering the whole range of early childhood and primary
schooling) in a small rural school in Strathbogie in north-eastern Victoria. The teachers (one
male, one female), each with eight ye.ars' teaching experience, were totally committed to a
success-based approach. Our observations of the children doing mathematics in their
classrooms convinced us that, not only were the teachers respected by the children, but the
children realised that they themselves were responsible for their own learning of mathematics.
The children themselves decided what aspect of mathematics they would investigate on any
particular day, whether they would work individually or in a group, whether they needed
equipment, what books they wouli use, and how they would record their findings (no student
questioned the need to record). The students also decided whether they wanted to consult with
their teacher about what they were doing.
Importantly, both teachers believed that their role was to be seen doing mathematics that was
relevant to them (i.e. the teachers) personally (see Waters & Montgomery, 1989, for a more
detailed statement of how the teachers saw their role). They worked quietly on their own
mathematical problems (on a particular day, one teacher worked on the costing for a school
camp that was imminent, and the other on the meaning of the Richter scale for earthquakes).
The children did not expect the teachers to move around the classroom asking them questions.
There was a definite impression that they, and not the teachers, owned the mathematics they
were doing.
We interviewed some of the children on videotape, and in every case their responses were
delightfully fresh, creative, and unihibited. They were unashamed of learning from others either
by questioning or by observation; there was no sense that it might be "cheating" to watch how
someone else approached a problem.
We do not wish to give the impression that initially, at least, some of our observations did
not cause us concern. We wanted to correct the child who consistently reversed digits (though
she read them correctly); we wanted to intervene and create cognitive conflict (in fact, we
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attempted to do so on a few occasions); and we continually asked ourselves "how well do these
children know their basic mathematical facts?"
Yet, on reflection, we are certain that, if we had attempted to find out how well the children
their basics by administering a standard, norm-referenced, pencil-and-paper test of basic
mathematical skills, we would have been raping a system which had given children confidence
to explore, to ask, to cooperate, to feel comfortable, and to learn mathematics. Probably, such a
norm-referenced test would have found the class below the mean. However, we believe that the
international mathematics education research community needs to question the validity of such
an assessment for classroom situations such as the one we have described. Validity, of course,
is related to objectives, and we believe that a willingness to explore, ask, and cooperate in
mathematics, and a spontaneous enjoyment reflect higher order objectives than do skill-based
objectives.
The Teachers' Perceptions of their Task
In a jointly authored article about the Strathbogie mathematics program, the two teachers
commented that they had drawn together what they knew about learners and what they knew
about mathematics in developing strategies aimed at assisting the children in their care to
become better mathematicians (Waters & Montgomery, 1989, p. 81). In describing the
program, they explained that some children work in pairs or in groups, but most work
individually; there is constant discussion, often between an older and a younger child. The
children ask each other for assistance regularly, and usually "some children are working in the
classroom, some in the corridor/kitchen area, some are in the office, and some are outside" (p.
82). After stating that the children generated their own mathematical tasks, the teachers gave
examples of activities that took place. These included:
* Drawing shapes with a ruler and measuring the corners with a protractor (Tristan, aged 6)
*
Measuring the dimensions of a football (Jason, aged 7)
* Trying to cram a matchbox with the maximum number of different items (Roslyn, aged 11)
* Using a bead-frame to record sets of counting (1, 2, 3, ..., 10) while timing a minute using
a stopwatch (Ben aged 6, and Shannon aged 7)
* Writing a description of what is understood of a short division algorithm (Loretta, aged 10)
* Measuring the distance from one set of goalposts to the other [in the school grounds]
(Maren, aged 6)
* Making a scale model of the monkey bars using wire (Robyn aged 9, and Terry aged 10)
* Recording subtraction equations that give a negative number display on a calculator (Ryan,
aged 7)
Waters and Montgomery (1989, p. 81) added that at 10.30am the children pack their equipment
away and write individual descriptions of their work. The teachers also do this.
BEST COPY AVAILABLE
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Among a series of individual "snapshots" provided by Waters and Montgomery (1989, p.
84) was the case of Bernadette, aged 11, who wanted to investigate how much burning time
one box of matches would provide. She worked at the sink with a box of matches, holding one
match at the very end with a pair of tweezers. For a while she experimented, lighting some
matches and seeing them burn. She then asked Roslyn (aged 11) to help her for a few minutes.
Bernadette had Roslyn used a stopwatch to time the burning of one match. There was much
discussion over when to start and stop timing. "From the time the flame starts till the time the
flame dies." They ran five trials, and Bernadette recorded each time.
Bernadette took her workbook back to the classroom. She had five various burning times
between 31 and 33 seconds. She used these two measurments as minimum and maximum
burning times, and disregarded the other three times. She then multiplied both numbers by 50
(there are 50 matches in a box) using a standard algorithm. To convert her answers from
seconds of burning time to minutes, she divided both numbers by 60. To do this she used a
calculator. Bernadette found that a box of matches has a burning time of between 25.83 and
27.50 minutes.
After this, Bernadette wrote about the mathematics she had done. "Today I burnt a match
right to the bottom with a pair of tweezers, and timed it to find out how long it would take to
burn a whole box of matches, one after the other. "33 x 50 = 1650, 31 x 50 = 1550."
At the conclusion of their article on the Strathbogie mathematics program, the two teachers
said they believe that to be mathematics teachers, they must be practising mathematicians. That
is why they themselves always do their own mathematics alongside their students. By doing
this, they model both the scope of the mathematics course, and what it is to be a healthy learner
of mathematics - self-motivated, self-directed and self-regulated. They said that they attempt to
create an atmosphere that is "risk-free": learners' attempts are valued. When their pupils talk to
them about mathematics, they are particularly interested in whether the pupils:
- see themselves as mathematicians;
want to take responsibility for their own learning and to make sense of what they
are learning;
- use mathematics frequently without inhibition
believe that making mathematical sense of the world, and learning more
mathematics isn't hard work, but is engaging and exciting;
- are willing to seek help from the Strathbogie Primary School community of
mathematicians (pupils and teachers) which responds to their challenges,
frustrations and successes (p. 85 )
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A Concluding Comment
Recently we claimed that "despite the universal rhetoric about school mathematics being
integrally linked to scientific, technological and economic development . . . the main lesson
learned by most school leavers after year of being forced to study mathematics is that they can't
do it" (Ellerton & Clements, 1989, p. vii). It is possible that all around the world there needs to
be a reconceptualisation of what school mathematics should be about (the "why"), what
mathematics should be studied in schools (the "what"), how it should be presented and how it
should be assessed (the "how"). Our observations of mathematics being done in a small rural
school, some of which we have captured on video (Ellerton & Clements, 1990), suggest that a
success-based theory of mathematics learning, linked with "natural" classrooms, might offer
children far more, so far as their future mathematical growth is concerned than do traditional
cognitive-conflict, teacher-textbook-owned approaches to school mathematics.
Perhaps there needs to be a whole new approach to mathematics curricular design (see
Steffe, 1989). We are concerned that mathematics teachers and educators around the world,
bolstered by the high status accorded to the subject they teach, have burried their heads in the
sand and therefore remain oblivious to the irrelevance of an adult-defined, adult-monitored,
adult-assessed, middle-class, largely male-inspired school mathematics agenda.
References
Bell, A. (1986). Outcomes of the diagnostic teaching project. In Proceedings of the Tenth
International Conference of the Psychology of Mathematics Education. (pp. 331-335).
London: International Group for the Psychology of Mathematics Education.
Bell, A., & Bassford, D. (1989). A conflict and investigation teaching method and an
individualised learning scheme - A comparative experiment on the teaching of fractions. In
G. Vergnaud, J. Rogalski, & M. Artique (Eds.) Acres de la 13e Conference Internationale:
Psychology of Mathematics Education. (Vol. 1, pp.125-132). Paris: International Group
for the Psychology of Mathematics Education.
Brown, A.L., & Campione, J.C. (1984). Three faces of transfer: Implications for early
competence, individual differences, and instruction. In M.E. Lamb, A.L., Brown and B
Rogoff, B. (EdS). Advances in developmental psychology (Vol. 3. pp. 143-192).
Hillsdale, N.J.: Lawrence Erlbaum.
Clements, M.A., & Del Campo, G. (1987). A manual for the professional development of
teachers of beginning mathematicians. Melbourne: Association of Independent Schools of
Victoria, and the Catholic Education Office of Victoria.
Elleron, N.F. (1988). Logo, learning and osmosis. Paper presented at the Eleventh Annual
Conference of the Mathematics Education Research Group of Australasia, Geelong, 10-13
July, 1988.
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Ellerton, N.F., & Clements, M.A. (Eds.) (1989). School mathematics: The challenge to
change. Geelong: Deakin University.
Ellerton, N.F., & Clements, M.A. (1990). A glimpse at some future mathematicians:
Mathematics at Strathbogie Primary School (Videotape). Geelong: Deakin University.
Karmiloff- Smith, A. (1984). Children's problem solving. In M.E. Lamb, A.L., Brown and B
Rogoff, B. (Eds). Advances in developmental psychology (Vol. 3. pp. 39-90). Hillsdale,
N.J.: Lawrence Erlbaum.
Pe led, I., & Resnick, L.B. (1987). Building semantic computer models for teaching number
systems and word problems. In J.C. Bergeron, N. Herscovics, & C. Kieran (eds),
Proceedings of the Eleventh International Conference for the Psychology of Mathematics
Education (Vol. 2, pp. 184-190). Montreal: International Group for the Psychology of
Mathematics Education.
Piaget J. (1975). To understand is to invent: The future of education. New York: Viking.
Steffe, L.P. (1989). Principles of mathematics curricular design: A constructivist perspective.
In J. Malone, H. Burkhardt, & C. Keitel (Eds.), The mathematics curriculum: Towards
the Year 2000 (pp. 453-465). Perth: Curtin University of Technology.
Vygotsky, L.S. (1986). Thought and language. Cambridge (Mass.): MIT Press.
Waters, M., & Montgomery, P. (1989). Simulating the early learning environment in a maths
classroom. In B. Doig (Ed.), Every one counts (pp. 76-85). Melbourne: Mathematical
Association of Victoria.
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TIME AND HYPOTHETICAL REASONING IN PROBLEM-SOLVING
Pier Luigi Ferrari, Dipartimento di Matematica, University di Genova, Italia
Hypothetical reasoning plays an important function in problem-solving and, in particular, in
resolution processes of complex problems. Many clues suggest that hypothetical reasoning
cannot be analysed without taking into account the role of time. For example, I have
observed that the 'if P then Q' construction has, for children, a meaning which is contiguous
to the meaning of constructions such as 'when P, Q' and, in problem-solving, the ordering
of the steps of the solving procedure is related both to time and logical consequence.
I try to explain the role of time in hypothetical reasoning as regards both the interplay
between the logical and the chronological structure of events and the situations which allow
a more frequent production of hypothetical reasoning. This may be related to some findings
regarding the role of space-time representations in problem-solving.
1.Introduction
In (Ferrari, 1989] I discussed the role of hypothetical reasoning in problem-solving at the
age of 8-10. It seems to be crucial as regards relatively complex problems, which need the
construction of a strategy with 2 or more steps. Examining more closely this issue I have
observed a lot of phenomena which emphasize the importance of the variable 'time' in problem
situations and the role of children's mental time when constructing a procedure. In particular,
many clues suggest that the management of hypothetical reasoning is strictly related to the
management of mental time by the child.
At this regard, out of the phenomena I have noticed, I report:
al) Children very often use connectives related to time (as 'when', 'till when' and so on) in
order to denote hypothetical reasoning.
a2) In most of arithmetical word problems with all necessary data explicitly given in the text
of the problem, children, when asked to record 'a posteriori' their procedure and reasoning,
perform it with a wide use of other connectives related to time (as 'then', 'so', 'after' and so
on), but without explicit hypothetical constructions.
a3) The situations in which the explicit production of hypothetical reasoning seem more
frequent are those which allow the children to reflect about their own (or other children's)
reasoning and to work in conditions of cognitive detachment. Situations like those allow
children to use time freely as a basis for simulation (leaving from real time).
These observations are included in a wider frame of phenomena regarding the role of time in
problem-solving. I am not going to examine them closely in this report but I shall refer to them
on many occasions:
b ) more or less 'expanded' management of problem-solving strategies (in particular,
division strategies); the child, after understanding an algorithm, uses it without reconstructing
in his mind the 'expanded' procedure which has generated that algorithm (e.g. the Greenwood
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algorithm for division, which in our curriculum is introduced as a natural outlet for more and
more organized and effective trial-and-error strategies see [Boero, Ferrari, Ferrero, 1989]).
In this way there is a sort of time-contraction which seems to affect even children's
reasonings as they are expressed.
b2) Specific difficulties children usually get into in problems involving time as a physical
variable and correlations between problem-solving skills and space-and-time-managing skills.
The main aim of this report is, in short, to explain the role of time in hypothetical reasoning
as regards both the interplay between the chronological and the logical structure of events and
the situations which allow a more frequent production of hypothetical reasoning (in
a 'spontaneous' and explicit way). As far as this last aspect is concerned, I shall refer to some
findings regarding question b1 related to the effects that the contraction process seems to
generate on children's behaviour, on the management of problem-solving procedures and even
on the kind of reasoning recorded.
2.The context
In this section I provide some information essential to understand the paper. A further
information on the educational frame in which the research is included can be found in
[Boero,1988], [Boero,1989], [Boero - Ferrari Ferrero, 1989], [Ferrari, 1989].
In our project, 'experience fields' are strongly stressed; in particular problem-solving is dealt
with mainly in 'experience fields% children are often asked to build, in a context providing
meaningful constraints, strategies in order to calculate arithmetical operations (as division) and
the management of trial-and-error strategies is strongly enhanced.
- A. wide space is assured for activities such as verbalizing, reflecting on the meaning of
connectives (if...then..., while, whereas, when, till when, ...), analysing and describing
complex machines and procedures.
The didactical context provides, anyway, many occasions of cognitive detachment, as, for
example, when comparing different strategies or distinguishing between "how the machine
works" and "how we can use the machine",...
3. Methodology of the research
We have a lot of materials from 'observation classes' (from which we gather, from grade 1
to grade 5, all texts individually produced by each child) and 'assessment tests' (administered at
the half and at the end of the school-year) from alla the classes. As regards connectives, most of
the protocols are 'spontaneous' productions by children. By 'spontaneous' we intend to refer to
texts produced freely by the child, without any direct intervention by the teacher or his
schoolfellows, but in a learning context which is planned to guide him towards a wide usage of
complex syntactical constructions.
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When examining children's 'spontaneous' productions it is necessary (as will be shown) to
deal with the problem of the relationship between children's thought and the text which
represents it.
Furthermore, when selecting and analysing children's spontaneous productions, it is
necessary to take into account the time when verbalization has been performed; particularly,
when the verbalization is performed while a problem is being solved, a wider presence of
hypothetical reasoning and other syntactically complex forms has been observed; if it happens
after a resolution procedure has been found, I have observed a great amount of sequentially-
structured texts, with a wide usage of connectives as 'so', 'then', 'hence', ..., which are
referred to both time-ordering and logical consequence. The ways in which the verbalizations
are usually elicited fulfils the conditions stated by Ericsson & Simon [1980] not to affect child's
mental process.
As regards the kind of materials, I have selected and analysed:
- normal working protocols, referred to situations in which the child is at ease but may be
influenced by the teacher, on the ground of verbalization, since in our curriculum in some
occasions the teacher 'lends the words' to the child in order to support him in expressing his
thoughts.
- assessment protocols, which are produced in somewhat unnatural situations but are useful,
related to the usual working conditions in class, to analyse children's behaviour without direct
and immediate influences.
In this study I shall refer to these materials:
m1) materials related to arithmetical word problems, with verbalization performed by
children while (possibly with a tape-recorder) or after solving a problem.
m2) written descriptions of everyday-life processes (e.g. how to prepare a coffee) and of the
working of a machine (e.g. a slot-machine);
m3) reports of discussions performed in class about the strategies each child has built in
order to solve a problem (not necessarily a standard problem);
m4) non-numerical word problems administered as assessment-tests at the end of primary
school;
m5) non-mathematical texts (e.g.: "Describe everyday-life in the Middle-Ages and tell if you
would like living in the Middle-Ages").
The study will refer to materials selected among those produced in 2 classes of grade 4 and 2
classes of grade 5; for any grade considered there is 1 class from the suburbs of a big town and
1 class from a little town in the neighbourhood of another big town. Nevertheless, part of the
findings I am going to present are supported by a greater amount of data, as it will be specified
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everytime. For example, the findings related to materials m1 are supported by data from about
20 classes.
4. Some data from the analysis of the protocols
In various situations children use connectives as 'when', 'till when' and so on to express
hypothetical reasoning:
a) When describing the working of complex machines (e.g. a slot-machine ) children uses
either 'if... then...' or 'when' to express conditional controls; this seems not to be random,
because to test each coin all children who insert a conditional control at that step use the form
'if... then...' ( "if the coin is 'good', the machine ..."), whereas to test the total amount of the
coin already inserted some 80% of the children who insert a conditional control use 'when'
("When the amount of the coin inserted is 400 lire, the machine ..."). Such a different usage
might be accounted in relation to the different meanings the two controls assume for children.
But it is not possible to claim certainly, without further evidence, that the usage of different
linguistic forms is a signal of different ways of thinking, even if many of the materials I have
examined seem to exclude that the usage of either of the forms should be random. The problem
of the relationship between children's thought and linguistic forms they adopt to report them
will not be examined closely in this report.
b) When describing other processes (e.g. the preparation of a cup of coffee) about 40% of
the children uses almost once constructions such as "when P, Q" or "P till when Q" (e.g.,
"when the coffee-pot is ready, put it on the fire" or "put water into the coffee-pot till when it is
full"). About 40% uses almost once 'if P then Q' (e.g. "if the water is not yet boiling, wait a
bit"). About 15% of the children (all good problem-solvers) use both constructions. These
children, when take into account the final amount of coffee use 'if...then...' ("if the coffee is
not enough, 1 must put more water, if it is too much 1 must put less water"), whereas 'when' is
mainly used related to more 'intrinsic' steps of the process, which are more difficult to master
from the outside ("when water boils, / must put the fire out"). About 35% of the children
(generally, poor problem-solvers) do not use any of the constructions I have mentioned, not
making explicit any conditional control but introducing constraints in other ways (e.g. "you
rnuct put enough of water into the coffee-pot, in order to prepare the ri2ht amount of coffee").
c) The same children of example 1 and 2 have been invited to describe the aspect of MiddleAges everyday-life most striking for them, and to tell if they should like living in the MiddleAges. About 50% of the children uses properly the 'if P then Q' construction almost once, and
about 60% uses properly a conditional form with 'when; the first group is contained in the
second. Among children who use either form I have observed that constructions with 'when'
are mainly used to speak of normal or unavoidable facts ("when there was a war, many
peasants would be killed"), and those with 'if...then...' mainly to speak of facts more
dependent on free choice of people or related to everyday-life ( "if a slave did not work, they
punished him").
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d) When discussing in class children's strategies and reasonings, about 80% of them uses
almost once a construction with 'if...then...'. About 45% uses also a construction with 'when'.
A very interesting example is the following. Reporting a discussion (performed in class) on the
criteria proposed by the pupils to know whether a pin picked into a wooden board is vertical,
Simone (a 4th grader, good problem-solver) writes: "It is not true that daze pin is vertical, then
the 'shadow' is long; it is quite the contrary, when the shadow is long, the pin is not vertical."
Here, if the verbal forms adopted by the child reflect his thoughts, it is likely that the first
construction ( "j.[ the pin is vertical... then the 'shadow...") is referred to a proposition to be
falsified and the second ("when the shadow is...") to a procedure to be performed ("everytime
I find a long shadow, I can argue...").
A general remark which can be done correlating the analysis of the protocols with a general
information on the children who have produced them is that the children who never use any of
the constructions mentioned are generally poor problem-solvers.
Related to the linguistic constructions used in the protocols, I have not found significant
differences between 4th and 5th graders.
When solving problems with numbers, explicit forms of hypothetical reasoning can hardly
be found if children already know an algorithm they can apply effectively to compute
arithmetical operations; in conditions like these, in almost all the problems examined, no child
uses explicitly a construction with 'if...then...' or 'when; only a 10% of children uses in more
than one problem constructions such as 'P since Q', which could be related to hypothetical
reasoning. In most cases, the text is organized in an inferential, not hypothetical way: true
statements are inferred from true statements and almost never explicit hypotheses are stated.
The style is mainly procedural: children mainly connect and organize their actions ("then I
do..., and so I find..., after that 1 compute...") and hardly connect properties of objects with
other properties of objects (e.g. number facts). Nevertheless, it is likely that even an inferential
organization of the text may hidden forms of hypothetical thinking.
5. Space, time and hypothetical reasoning in problem-solving
I have found that, in the resolution of problems with numbers, children use widely
hypothetical reasoning when they are forced to invent a strategy to compute an arithmetical
operation (in particular, division). Nevertheless, (as already remarked at the end of section 4)
these forms are no longer used when children can apply more contracted algorithms, which
need not rebuilding every time the complete reasoning, though they can understand the meaning
of what they are doing.
In my report at PME XIII I provided some examples which may contribute to a better
explanation of these phenomena. In example 3 [Ferrari, 1989, p. 262], a child designes a
resolution strategy for a division problem (to represent on a wall of the classroom a given
period of time), organizing, with a wide resort to hypothetical forms, a trial-and-error strategy
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in which the child's spatial representation of the situation play an important function, as well as
the representation of the procedure in a sort of mental time the child can manage quite easily.
Protocols like this suggest that making hypotheses is strictly related to the mastery of mental
time which allows the child to simulate in mind the possible developments of the situation.
Another example is the following: at the end of primary school, children have been asked to
design a general procedure to order alphabetically an arbitrary set of surnames. In the texts
produced by the children I have found a wide presence of reasonings grounded on space and
time, where the spatial representation of words (e.g. written from the left to the right) and the
flexible management of time related to making different hypotheses (in a sort of mental
experiment) play an important function.
The main findings of the analysis of the protocols of the primary school pupils who
experiment our curricular project at this regard are the following:
= all good problem-solvers can design resolution procedures in which space and time
dimensions and their interplay seem to play a major role, and manage very well this interplay in
a sort of 'game of hypotheses';
= the ability of designing resolution procedures with hypothetical reasonings strongly
grounded on space and time is strongly correlated to the ability of dealing, in an effective way,
with problems in which time appears as explicit variable in the text.
With regard to this last finding, Boero, Ferrari and Ferrero [1989] have discussed some
examples of phenomena of this kind; for example, with similar numerical values, it is much
harder for children to state "how many times this has become as big as before" than to state
"how many times this is as big as that", and the strategies adopted are clearly different.
Analogous phenomena have been observed also in the classes of comprehensive school
which experiment the project we have designed for this kind of school.
6. Discussion
From the data and observations reported in sections 4 and 5 the following conclusions can
be drawn:
= Explicit forms of hypothetical reasoning in problem-solving are not produced in a
spontaneous and uniform way, but mainly in particular contexts which can induce 'cognitive
detachment'. In these contexts the child must be able to manage consciously the procedure and
to take into account different alternatives.
= In problems with numbers, the child seems to meet with difficulties in describing objective
relations among the elements of the problem and prefers to describe the organization of his
actions ("I do..., then I compute..., so I find..."). In other words, the relations and properties
among the elements of the problem are implicit in his resolution procedure. Procedures are
customly represented as chronologically ordered sequences of operations, and then it is quite
natural for children to ground their reasoning (and even the 'logical' - or arithmetical or
geometrical ...- relations they may have found) on time. Then a statement such as "I compute P
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and then I find Q" represents both a relation of logical consequence (computing P implies that Q
can be found) and a chronological ordering (before I compute P and afterwards I find Q").
These meanings seem to be joined in the child's mind.
= Also in the descriptions of procedures or of the working of machines there is some
connection among hypothetical reasoning and time. Children who use hypothetical reasoning
clearly seem to prefer 'if...then...' in the situations in which both alternatives are noteworthy
and the control is related to something external to the machine or the procedure. The fact that
slot-machine could refuse a coin (or that the coffee could be not enough, and so on) is a real
possibility for children. The control on the amount of money they have inserted into a slot
machine is not regarded as meaningful before they have inserted all the coins requested: to
insert the exact amount of money, if they have already it in their hands, is only a problem of
time for them. In the same way, a child knows that if he put a coffee-pot on the fire, it must
boil, sooner or later; it is only a question of time.
= The analysis of the texts on the 'Middle Ages' corroborates these results. Also in this
context, among the children who use both constructions, forms with 'if...then...' are preferred
to describe events which may or may not happen (as the premature death of a slave) or
depending upon people's will (as the flight of a slave), whereas forms with 'when...' are
preferred to describe events children regard as ineluctable or more strictly related to time (as
wars, hunting, seasons).
= Two different forms of reasoning, which are referred to the presence of alternative
hypotheses, could be seemingly singled out when analyzing the children's protocols: in the first
form (strongly guided by the context), though in general there are two or more alternatives, the
context allows to single out which is true, and then the others are not even taken into account.
On the linguistic ground, this form of reasoning is verbally represented without explicit
hypothetical constructions. The second form of reasoning can be found in situations in which
both the alternatives must be taken into account. It is clear that the task (e.g. comparation of
different strategies, simulations, ...) or the particular situation (e.g. a complex situation, with
the necessary data not all explicitly given) prevent the 'contraction' of this kind of reasoning
(i.e. the 'evaluation'of the alternatives and the elimination of those which do not happen).
Nevertheless, it seems clear that the important role of time (related to the reconstruction and
development of alternatives) and of the context prevent any identification between the mastery
of hypothetical constructions in verbal language and the formal management of the proposition
'P implies Q' based on classical propositional logic.
= The strong dependence on time and meanings of the management of hypothetical
constructions can contribute to explain some difficulties widely reported in literature about the
learning of conditional sentences (regarded as sentences defined by means of truth-tables) [e.g.
O'Brien et al., 1971; Johnson-Laird, 1975; Markovits, 1986].
In particular, the findings of Markovits on the effect of pictures in some tests on implication
can be regarded in this way, because pictures as far as they neglect time and point out the
statical aspects, may hidden more than verbal language the chronological dimension.
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REFERENCES
Boero, P. - Acquisition of meanings and evolution of strategies in problem solving from
the age of 7 to the age of 11 in a curricular environment, Proceedings PME XII, 177-184,
Vezprem: OOK (1988).
Boero, P. - Mathematical literacy for all: experiences and problems, Proceedings PME XIII vol.1, 62-76, Paris (1989).
Boero, P., Ferrari, P.L., Ferrero, E. Division Problems
:
Meanings and
Procedures in the Transition to a Written Algorithm, For the Learning of Mathematics, 9-3
(1989).
Caron, J. - la comprehension d'un connecteur polysemique: la conjonction "si", Bulletin
de Psycologie, tome XXXII N°341.
Ericsson, K.A., Simon H.A. - Verbal reports as data, Psycological Review vol.87
(May 1980), 215-251.
Ferrari, P.L. Hypothetical reasoning in the resolution of applied mathematical problems
at the age of 8-10, Proceedings PME - XIII vol.', 260-267, Paris (1989).
French, L.A. Acquiring and using words to express logical relationship, from
S.A.Kuczaj & M.D.Barrett (Eds.), The development of word meaning, Springer-Verlag, 1985,
303-337.
Johnson - Laird, P.N. - Models of reasoning, in Reasoning: representation and process
in children and adults, R.J.Falmagne and N.J.Hillsdale eds., Lawrence Erlbaum ass., 1975.
Laborde, C. Longue naturelle et ecriture symbolique, These- d'Etat, University de
Grenoble, 1982.
Lesh, R. - Applied Mathematical Problem Solving, Educational studies in mathematics,
vol.12 (1981), 235-264.
Lesh, R. Conceptual Analysis of Mathematical Ideas and Problem Solving Processes,
Proceedings P.M.E. , 1985, 235-264.
Markovits, H. - The curious effect of using drawings in conditional reasoning problems,
Educational studies in mathematics, vol. 17 (1986) 81-87.
O'Brien, T.C.; Shapiro, B.J.; Reali, N.C. - Logical thinking
language and context
Educational studies in mathematics, 4 (1971) 201-219.
Rumain, B., Connell, J., Braine, D.S. Conversational comprehension processes
are responsible for ..., Developmental Psycology, vol.19 (1983), 4, 471-481.
Vygotsky, L.S., Mind in society: the development of higher psychological processes,
Harvard University Press, 1978.
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THE INTERPLAY BETWEEN STUDENT BEHAVIORS AND
THE MATHEMATICAL STRUCTURE OF PROBLEM
SITUATIONS - ISSUES AND EXAMPLES
Abraham Arcavi
Department of Science Teaching
Weizmann Institute of Science
76100 Rehovot, Israel
JRina Hershkowitz
In Friedlander et al (1989) we analyzed the mathematical behavior
of seventh graders in generalization and justification processes.
The analysis of the data presented there and additional data led
us to focus our attention on the interplay between aspects of the
mathematical structure of the problem situations we designed and
the spectrum of observed student behaviors in these problem
situations. Our aim is to tackle this issue by analyzing some
epistemological aspects of problem situations in the first part of
this paper. In the second part, we analyze the "traces" of the
mathematical structure of the problems on student behavior.
The epistemological aspect
There are several epistemological aspects in the light of which problem
situations can be examined. We would like to concentrate on the
relationships between single examples of the problem situation domain (and
actions one can perform on these examples), and processes of
generalization and justification. In order to sharpen our description we will
proceed to make a distinction between two "extreme cases".
Type 1 problems.
The general process of justification is based on actions and processes which
are analogous to the processes and actions carried out on one single
example.
Suppose one asks the following question:
"Solve 1/2 1/3= , 1/3 1/4= , 1/4 - 1/5=
. Do you observe any pattern?
Can you generalize? Can you justify your generalization?"
The process of formal justification of the general pattern [by means of
algebraic manipulations of 1/n 1/n+1=1/n(n+1)] is completely analogous to
the arithmetic process by which one solves any single example, as follows.
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2O4
Single example
Formal justification
1
1
7
8
_2_ = 1
56 56
1 _1_ - n+ 1
n n+1 n(n+ 1 )
56
n(n+ 1 )
n(n+1 )
Even when the generalization and justification is not expressed by means of
mathematical symbols, it will still be no more than a reflection of one
particular arithmetical process. In other words, the justification can be
"carried on the shoulders" of the single example, if one just looks at the
single example with "general spectacles".
Type 2 problems:
The processes of generalization and justification in the problem situation are
completely different from the actions on a single example of the problem
situation.
Suppose one asks the following question:
"What can you say about the numbers resulting from the differences between
the third power of a whole number and the number itself [n3 - n]?" (This
problem is also used in the study by Fischbein and Kedem, 1982). By trying
different numbers, one may notice that all the differences are multiples of 6.
But, in order to provide a universal justification of this generalization (or, in
other words, to prove this conjecture formally) one needs some extra steps,
in this case: i) the appropriate algebraic manipulations, and ii) their
interpretation.
In other words, to produce:
i) the factorization n3 - n = n(n-1)(n+1),
and
ii) its interpretation: the factors are always three consecutive numbers; at
least one of the three consecutive numbers will always be even (divisible by
two), and one of them will be always divisible by 3, therefore the product will
always be divisible by 6.
Here, the numerical examples, no matter how many of them one may
produce, are not "transparent": they will not let the general mechanism be
seen or appreciated. Again, as in the Type 1 problem described above, the
algebra provides us with the appropriate "general" spectacles for the general
justification, but this time it does more than that. It enables us to see and
express general relationships between numbers by laying down the
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structure of these relationships. Such a structure is invisible through
numerical examples.
Most of the typical problems in an Euclidean Geometry course are of this
type. Current educational software (e.g. The Geometric Supposer, the
CABRI) are excellent tools designed to support for conjecture-making of
general properties, an aspect much overlooked in traditional courses. These
computerized tools indeed sustain conjecture-making by providing easy and
"cheap" ways to experiment by measuring, constructing quickly and
efficiently several examples, and manipulating those constructions. But in
order to produce a ,general justification of the conjecture, one usually needs,
also here, extra steps, which in this case consist of a deductive reasoning
chain, probably some auxiliary construction of general validity, and perhaps
a good dose of insight to put things together. These extra steps seldom arise
during the empirical phase of conjecture-making.
Consider for example the sum of the internal angles of a triangle. One can
easily measure different types of triangles and conjecture quite quickly that
the sum is 1804 (or about 180 °). However, in order to prove this conjecture
one has to take extra steps: an auxiliary construction (a line parallel to one of
the sides of the triangle through the third vertex, and a translation of the three
angles) not present in the conjecture making process.
Our above description deliberately utilized "extreme cases" for the purpose
of clarification. Obviously each problem has its own peculiarities, but
elements from the above distinction can be identified as intertwined in the
problem's "fabric". Consider the following.
A- The same problem situation can be "attacked" in different ways.
For example, one may notice that the general justification is, by virtue of its
generality, obviously reducible to any of the single examples. In the case of
the sum of the internal angles of a triangle, it is certainly possible to make the
auxiliary construction needed for the proof while playing with a specific
triangle. As a matter of fact, the general proof is usually accompanied by the
drawing of "any" triangle (which can always be regarded as one specific
example) and applying the "general" construction to it. However, in the case
of n3-n, it is quite unlikely to "discover" the structure of say 73-7 as
7(7-1)(7+1) and thus to have a glimmer of the general justification.
Here we need to notice one central difference between generalization and
justification processes in school algebra and school geometry. General
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processes in geometry rely on seeing one single example as representative
of a whole class, whereas algebra resorts to a symbolic language. This
language enables handling "general" patterns and also laying down the
structure of relationships, which are invisible from single examples.
B- There can be a more or less "smooth" transition from working on single
examples to the general justification of the generalization.
Consider the following problem situation: "Find the sum of the internal angles
of a polygon". One can proceed to work with, say, pentagons first, and start
measuring the angles, and arriving at the conclusion in the same way we
described for the triangles. If one generates the extra steps of dividing the
pentagon into three triangles (whose sum of angles is already known) by
means of the diagonals from one vertex, the general justification for all the
pentagons arises: one has to multiply 180Q by the number of triangles. And if
one wishes to generalize further for the case of all polygons, an additional
extra step is required, whose outcome will relate the number of sides (or
vertices) of the polygon to the number of triangles (or diagonals) created.
We suggest that, in spite of the complexity raised by our distinction, the
elements observed in the "extreme cases" are useful in shedding additional
light on understanding student behaviors.
Student behaviors
The situations we describe in this section are borrowed from existing studies.
We expect to bring to our oral presentation additional data from the study we
are currently running.
I
The example as "judicial evidence".
A typical student behavior, while making general arguments and trying to
justify them, consists of placing a single instance as "judicial evidence": like,
in court, clear-cut evidence is necessary and sufficient to convince a jury
about the certainty of an event'. In other words, a convincing justification is
regarded as the presentation of a fact (an example) which confirms the
general claim at stake. K., one of the seventh graders in our study, expressed
This can be reinforced by the language. For example, the Hebrew word for "proof" is used
both in mathematics and in law (as evidence). Seventh graders, who have rarely met the
mathematical connotation of proof, may associate its meaning with that of "legal evidence", a
well known word frequently heard in everyday life.
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it very clearly: "I think that to prove something means to show some
examples".
In such cases, the students confer to the single example, as a "prover of a
general claim, the same status conferred by mathematical standards to a
counterexample as a "disprover" of a general claim.
In a more elaborated version of the example as "judicial evidence", we found
students who check evidence from different domains of examples (small
versus large numbers, different types of triangles, etc.) in order to make the
"evidence" more convincing for themselves and/or others.
In any case, when students are requested to justify a general claim (given by
others or even produced by themselves), they return to an example, or to
domains of examples, which by their mere existence provide the justification
requested.
This use of the example as "judicial evidence" is a justification tool in the
hands of many students regardless of the epistemological type of the
problem situation on which they work.
II. Beyond " judicial evidence" - the role of examples in Type 1 problems.
Once a generalization of a Type 1 problem (like the subtraction of two
consecutive unit fractions) is achieved, and the student is requested to justify,
(s)he does not resort to the example as a confirmation, but as a prototype
from which a general property or mechanism can be abstracted. The
following quotation is from a pre-algebra student (a seventh grader),
attempting to justify verbally the general statement 1/n -1/(n+1)=1/n(n+1),
by "riding on the shoulders" of the single example 1/8 -1/9=1/72 .
"...each time we have, let's say, 72 divided by 8 we get 9, and when we
divide by 9 we get 8... one number less the other is always 1... that's clear.
The phrases "each time", "let's say, 72 divided by 8...", and "one number less
the other" are indications that, for lack of any other appropriate tool, the
particular values of the example are not invoked as such, but as tokens or
potential placeholders involved in a general mechanism. For some students,
this use of the example is an intermediate step towards an example-free
verbal formulation of the justification.
In sum, the very nature of Type 1 problems seems to encourage the use of
examples as generators of justifications.
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III. Beyond judicial evidence" - the role of examples in Type 2 problems.
The following vivid anecdote will illustrate this point. During an in-service
teacher training course in our department, teachers were presented with the
curious equality 62x39=26x93. Generating more examples of two-digit
multiplication convinced them very quickly that reversing the digits of the
factors does not always lead to the same result. Puzzled by this rarity they
were invited to investigate for which two-digit numbers the phenomenon will
take place. Among those who did realize that they need to resort to algebra
was one teacher who seemed the brightest of the group and who had no
trouble producing the algebraic manipulations and reaching the right
conclusion (the equality will take place when the product of the tens equals
the product of the units of the two-digit numbers). However, when we noticed
that she was checking for additional numerical examples after she
completed the proof, we were puzzled. When asked, she told us that she was
not checking in order to see whether she obtained the right result with the
numbers, because the general validity of the algebraic tool was obvious to
her. However, in light of the algebraic justification she had produced, she
needed to Ilk what happens when one translates the mechanism of the
algebraic general justification "to actual numbers", namely how do they
combine and how do they behave as compared to letters.
It seems that mathematically-able people, in their search for meaningfulness
would often use examples in this way, in order to get a feeling of those extra
steps that they themselves (or somebody else) were able to generate.
We conjecture that this use of examples would be rarely encountered in
Type 1 problems, in which the general proof is based on a repetition of the
mechanism of a single example.
IV. The struggle for the extra steps.
In Friedlander et al (1989), we described a pair of pre-alaebra students
working with the following problem. They were given a calendar sheet of a
given month, for example:
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1
2
3
4
5
8
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
The students were requested to observe general patterns and to justify them.
One of the students conjectured that, in any 2x2 cell arrangement, the
difference between the two products of the numbers located in the diagonals
is 7. In the process of trying to produce a general justification, they were able
to express the products of the diagonals using letters [namely ax(a+7+1)
and (a+1)x(a+7)]. At this stage they substitute different numbers for a and
confirmed empirically their conjecture.
One of the students was quite happy with the algebraic "machine" they had
created, and since this "machine" generated the expected numerical result,
he though that the task was completed. He considered the creation of the
algebraic "machine" as the justification sought. However, the other student
expressed his dissatisfaction by saying that he still wanted to "show it [the
justification of the general pattern] with letters". Since this is a Type 2
problem, and the extra steps needed required algebraic manipulations, he
could not make progress. The lack of knowledge of algebraic manipulations
needed for producing such justification, namely to prove that ax(a+7+1) and
(a+1)x(a+7) differ by 7, did not prevent him from feeling its necessity.
Epilogue
We suggest that aspects of the epistemological nature of a problem may turn
out to be crucial variables in understanding and possibly predicting student
behavior. If further studies confirm this view, the findings can have important
instructional implications regarding the kinds and timing of the problem
situations students should encounter in order to foster the need for general
justifications and proofs.
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Referencea
Fischbein, E. and Kedem, I. (1982) Proof and certitude in the development of
mathematical thinking. Proceedings of the VI PME Conference, Antwerp, pp.
128-131.
Friedlander, A. Hershkowitz, R. and Arcavi, A. (1989) Incipient "algebraic"
thinking in pre-algebra students. Proceedings of the XIII PME Conference,
Paris, vol. 1, pp. 283-290.
Laboratoire Structures Discretes et Didactique (1988) Cabri Geometric?.
Grenoble, France: Institut IMAG.
Schwartz, J.L., Yerushalmy, M. & Education Development Center (1985) The
Geometric Supposer [computer program]. Pleasantville, NY: Sunburst
Communications.
21i
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Paradigm of 'Open-Approach' Method in the Mathematics Classroom Activities
-Focus on Mathematical Problem-Solving---
Nobuhiko NOIIDA,
Jerry P.
Becker,
University of Tsukuba, Japan
SIU at Carbondale, USA
Sunnary
Our study on analyzing students' strategies and difficulties in problem
solving is considered indispensable to improve teaching and learning in
mathematics classroom activities.
It seems that these strategies and difficulties
are influenced greatly by some social and cultural factors, such as languages,
symbols and representations etc.. This study is planned in order to make exact
the effects of teaching and learning of teacher and students who engage in
problem-solving by means of the 'Open-Approach' method, particularly with
reference to share [mathematical ideas of problem and use of mathematical patterns
involved in problem solving. We have to become more aware of the information
processes which consist of the communications and interactions between the
teacher's explanations and pupil's approach to problem-solving.
The sixth-grade class (Kale;
Female; 22,
18,
Totale; 40) we used in this
study was composed of pupils in amnia elementary school near Tsukuba City.
Ms.
K.
Mashiko is an excellent teacher, who had come to University of Tsukuba for
studying mathematical problem-solving for about three months, three years ago.
The lesson was held on January 26,
I.
1987.
Mathematics Classroom Activities
Several difficulties concerning problem solving are,
in our opinion, due to
the narrow and isolated conceptions of the basic didactical. category on Imoblemsolving'. We,
therefore, attempt to reveal the global and relational character of
problem solving', which is to call attention to the necessity of dealing with a
broad spectrum of activities related to Japanese culture and society.
These new demands can be found in Christiansen and Walter (1986), which
necessitate changes in the teacher's role and moves:
1.
changes in the distribution of emplmsis on the different types of activity,
2.
changes in the types of teacher's moves and in the sequencing of these in
the teaching pavers,
3.
changes in the ways in which the teacher serves as a mediator of
mathematimal meaning.
The process of problem-solving becomes evident when teaching is seen as a
process of interaction between the teacher and learner-and among the learners-in
which the teacher attempts to provide learners with access to mathematical.
thinking in accordance with given problems. This teaching/learning process (like
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all processes between learners) is influenced by a number of social and
developmental aspects and factors which can be included in problem solving. The
carmunication between teacher and learner is thus nest only conditioned by formal
decisions about goals, content and teaching methods, but it is also strongly
dependent on even more informal aspects in initiative stages of problemrsolving,
such as the teacher's words and explanations to the problem-solver, and the
students' motivation to solve the problem and to he concerned with it.
We will cite an example of the problem-solving activities between teacher and
learners (Fig.1). Some of the roles of the teacher at different stages of the
teaching/learning process arc: instructor to teach mathematical knowledges and
skills (Top-Down);'teacher to help students in problem- solving (Bottom-Up): and
decision maker to judge whether teaching goes ahead or not. The teacher' s
explication of such roles is integrated with his specific actions and serves in
establishing his background and context for the interactions between his
students' actual and inner activities in connection with their subjective words.
Teachers' Experiences
Helle-Teacciing
Teacher's Instruction
tural
Social-
Background
Background
Problem-Solving
Top-Dawn
Activities
Bottom-lip
T
Students' Learning
I
Hetp-leau?ning
Students' Experiencm
Fig. 1 Problem-Solving Activities
The above sentences illustrate the essential and relational character of
communications between teacher and learners. Accordingly. coammication through
Ineblem-solving' as an organizing principle in Japanese mathematics learning
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calls for meta-learning under the teacher' s support. This communication is
considered mathematics classroom teaching as controlling the organization and
dynamics of the classroom activities for the purloses of sharing and developing
nothemat I cal thinking.
What is the open-approach?
2.
The aim of open-approach instruction is to foster both the creative
activities of the students and the mathematical thinking in problem solving
simultaneously. In other words, both the activities of the students and the
nothmatical thinking Riga be carried out to the fullest extent. Then, it is
necessary for each student to have the individual freedom to progress in
problem solving according to their ()um abilities and interests. Finally, it
allows them to cultivate mathematical intelligence. Class activities with
mathematical ideas are assumed, and at the same time students with higher)
abilities take (Art in a variety of mathematical activities, and also strulents
with lower abilities can still enjoy unthernti cal activities according to their
own abilities.
By doing so, it enables the students to perform the mathenotical problem
solving. It also offers than the oHoetunity to investigate with strategies in
the manner they feel confident, and allows the possibility of greater
elaboration within nothearntical problem solving. As a result, it is possible to
have a richer development in mathematical, thi.nki»g, and at the same time,
foster the creative activities of each student:. This is the idea of the 'opon..
approach', which is defined as an instruction in which the activities of
interned-On between nothmoties and students are open to varied problem solving
appLoaches.
Next, it is necessary to make clear that the meaning of the activities of
interaction between nothenot Ica] ideas and students' behaviors are open in
problem solving. This has been explained from three aspects:
(I) Students' activities are developed by the open-alio/weir
A problem using the open-approach involves nothenotical ideas
Open-approach
should be in harmony with intersection activities
(3)
(2)
between (1) and (2)
Chruneterizatiors of the `Oiren-Approach' problem and method
We hope to become more aware of the informtion processes which consist in
the Open-Approach of relationships between the problem and method. We use here
'Open Approach' problan as like non-routine problems: problem situations, process
problems and open search problems (Christiansen & Walter, 1986). in actual
practice, each teacher will have to take his or her own classroom ()rotations and
tencbtog objectives into consideration. The, method we use in 'Open-Approach'
depends on the problems which consist of problem situations, process probleos and
3.
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21,
open-ended problem, and procedures of these problems inrludiog classroom
conditions and teaching objectives (No hda,
1903,
I986).
We use here the problem: [trebling situations, process problem and open-ended
problems. We define a problem as follows: A problem occurs when pupils are
confronted with a task which is usually given by the teacher and there is 1/0
pr icribed way of solving the problem. It is generally not a problem when it can
be immediately solved by the students. Problco situations, process problems and
open-ended problems are defined as follows:
Figure 2. Open-Approach Problem and Itching!
On going lesson
*
Regining of
LeS5.011
Original Problem(s)
R Solving of Problem
Solut
Solution(s)
Solution(s)
Problem Sii:uatiotrs
Process* Prelrlerns
C.
Fuld of lesson
New Problem(s)
--- New Problem(%)
New Problem(s)
Genereti ye Pt ebl ems;
Open-approach teaching differs from ordinary problem solving teaching.
lime we use the problem as open-approach pi-11AM; mentioned above. Treatments
of these problems will depend on the teacher's intentions for his "her
objectives:
A. What kind of problem does the teacher want the strulents to formulate fulmi
given problon situations ? (Relation with Problem Situations)
teacher want the students to solve
R. Row many ways of thinking does the
the problem given ? (Relat ion'witli Process Problems)
C 1Lat kind of advanced problem does the teacher want the students to site
flog the oriel-tat preblon ? (Rola,. ion with Generative Problems)
1. Actual problem-solving activity in sixth-grade classroom
In every day life, pupils are confronted with many problem-solving as
problem situations where they can take a variety of solution. The method for
solving the problem of daily life, seem to Include some regular rule or
procedures.
To foster their mathanatical thinking, mathematics teacher should emphasize
problem-solving, in which pupils would discover better way of drinking through
discussions of their various solutions of the ploblem.
Here is used the process problem. The sixth -grade class (t ble: 18. Fenele: 22,
Totale; 10) we used in this study was composed of pupils in a coral elementary
school near Tsularba City. Hs. K. Hashilio is an excellent teacher, who had orie to
204
no for studying mathematical. problem-solving for about three months, three years
ago. The lesson was held on January 26, 1987.
(1) Teach& s plan of problem solving
(a) Original problem in Japanese textbook
Squares are made by using swill bars as shown
in figure 3. When the number of squares is 8,
Fig. 3
how army small bars are need ?
(b) Change Problem to Process Problem
Squares are constructed by using small bars
as shown in figure 4. Men the number of
squares is 10, bow many small bats are used ?
Find the lots of ways of counting of the
Fig.
snail bars. as possible.
(2) Actual lesson of problem-solving: First class, sixth-grade
4
(a) She started as follows: (5 minutes)
Each pupil was given a picture of 'small bars' and the teacher asked the
',evils flow many squares are there in this figure?" and she put the real snail
bars on the blackboard. She explained some notions to them; "Arrange the sonll
bars to shape two squares like Figure 4 and count the number of stroll bars one by
one" as follows:
(b) Give hints to help pupils of lower abilities urwlerstand the problem.
(5 minutes)
T: When the number of squares is 2, how runny siert bars are used ?
P: 7 bars
T: When the timber of squares is 4, how nwmy small tors are used ?
P: 12 bars
T: O.K., you come up with various ways of solving the problem.
After she explained the problem pupils worked on the problem individually.
(c) Let's find various ways of counting on the sheet given.
Answer: 27 bars
(ii) 7x3 +3 x2
(1) One to one counting
I
(15 minutes)
"'
I431 nn
3
7
I
I
3
7
I
I
7
I
I
1---n1
(iv) 4x6 +3
(iii) 4 x10 (Wrong)
I_I
III1H
1_1
I
FolTri
I
1:12_
BEST COPY AVAILABLE
205
216
(v)
5x3 + 6x2
*(vii) 7+ 5x4
*(vi) 2+ 5x5
il-4"Fer"fel fel-1-171
1Tirfel
(d) Discuss pupils' hays of counting. Which one do you think is the hest way ?
And Why ? What happens when the number of squares increases ?(I5 minutes)
T: Which is the easiest way to count when we have 20 squares ?
P: *(vi) 2+ 5x5 or (vi) *(vii) 7+ 5x4
(e) Formula expressed i.n words: (5 minutes)
+ [5 Imslx [Sets of
[Two squares
with 7 bars]
= [Total number]
increase number]
7
I-
71 5x (5
1)
= 27
Ans. 27 Inns
(f) Give open-ended problems to ptipils: (Iloinewor.k)
(i) When the number of equilateral. triangles is 8,
how irony small hats arc used?
Ans.
(ii)
2x8+
=1
7
When the number of squares is 7,
how twiny small bars are used?
Ans.
(iii)
3 X 7 4.
1
22
When the number of squares is 15,
how irony snod.1, bans are rased?
Ans.
PI obl ern
(i)
7x4
10=38
variations of fomula
Ptrle(18)
2 x8 t
3 +2 x7
3 x8 --7
1
3
206
-91
(3)
Totale(40)
8
(8)
10
(10)
9
(9)
23
(23)
0
others
3 x7 +1
Fame 1 e (22 )
1
(1)
5
(1)
6
(I)
9
(9)
12
(12)
4 +3 x6
1 x8 +7 x2
(iii)
8
(8)
22
(22)
2
(2)
2
(2)
3
(1)
4
(I)
(3)
4
(4)
7
(7)
(8)
5
(5)
13
(13)
1
(1)
0
I
(I)
6
(0)
13
(0)
19
(0)
14
(14)
others
1
(0)
7 x5 +3
3
10 +7 x4
8
5 x4 +3 x6
others
Note ( ): Number of corrects
(3) Disseussion on sixth grade problan-solving
The variations of strategies used in this problem-solving were as follows:
No pupil used one by one though they often used strategy in the lesson. Ilost
pupils used the equations adjusted to the structure of the problem. This was
dependent on the following cannunication between the teacher and the pupils: The
teacher advised the pupils who could find the suitable configuration of the
problem. We are impressed that pupils have a real appreciation of sharing the
inthematiesil structures through communications among pupils themselves under
the teacher's orientation and understand the mathematical formula through the
results of pupils' homework. Dr. Jerry P. Becker gives the torment as follows: I
have found that challenging problems bringing students together in thiking
about: the situation. searching to understand the problem and then trying to
solve it. Sometimes 1 almost seise that a "spirit of community" ensues with
students reflecting and building on each other's ideas this is a heathy state
of affairs.
5. Instructional inplications
The instructional implications from our study of the elementary school
level in the context of problem-solving in the rrnthematies classroom consist of
the following:
a.
In the study of pupils' strategies and difficulties in problem-solving,
we should concentrate on both the structure of the problem and the mode of the
pupils' acts of problem-solving. We suggest here that the pupils need to
initially act by themselves to solve the problem and then through convonication
establish matharntical structures in the modification of their initial, acts: for
example, pupils rake the equation as like 74- 5x4 flan 7 x 3 + 3x2.
b. Some excellent pupils can solve the problem by finding the nnthenotical
structures underlying in the problem. The teacher has to support these pupils to
promote their more advanced solution after they use the teacher's primitive
method. They are willing to independently find the advanced solutions. The
excellent cournunication is the most important for the teacher. Thus, they become
the good problem solvers for the future.
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c. Many normal pupils cannot solve the problem at hand. In these situations,
the teacher has to advise them to be ready for their ['manlier problems which
they have solved in the past. After they feel an appreciation for carrying out
the problem-solving individually, they are able to solve the problem in the
near future. This is the effect of the classroom activity supporting them by
the teacher.
References
Balacheff,
(1988), 'Off-print of English impels', Univers' te Joseph Fourier
N.
Becker, J., P.
(1986),
111 theariti cs?' ,
iqhy Are Students Unable to Solve Non--Routine Problens in
Proceedings of the U. S. -Japn Seminar on 1kr them t eel
Problem Solving.
Becker, J & Silver, E., (1988) 'A Cross-Cultural Study of Japanese and .Ameoican
Students' Problem-Solving Behaviors', Revised Description.
Christiansen, B. &Walter, G. (1986), Task and Activity' , Christensen, B et at (Eds. )
"Perspectives on nathenatics Education".
Kumagai,
D.
Reidel Publishing Co.
(1987), 'A study on 'Sharing Process"in the Mathematics Classrooai
K.
Annual Report of Graduate Studies in Education, Vol 11.
Lindquist,
M.
M.
(1989), 'It' s Time to Change', NUM "New Directions for Perot-into
ry School 1tathanatics",
1989 Yeaetxx)k.
(1985), 'Implication of Cognitive Psychology for Instruction in
1titheentical Problem Solving', Silver, F. A. (Ed. ), 'Teaching and
llayer,
R.
E.
Learning Ilathemati cal. Problem- Solving: 111,1 ti pl e Research Perspect i yes",
Lawrence &Miura Associates.
Nohda,
(1983), "A Study of 'Open-Approach'
N.
Strntegy, in School Katheantics
Teaching", Touyoulain Publishing Co., Doctoral Dissertation. (in Jeirmese)
This abstract is ' The Heart of "Open-Approach" in ItItheorities Teaching'
The Proceedings of IGTE-JSME Regional. Conference on Mathematical
Education
Nohda,
1903.
(1986), "Open Mind of Children in API thmetie Teaching", Noubunshoin
N.
Publishing Co., (in Japanese).
Polya, G. (1962), "Mathematical Discovery-On Understanding, learning and
Teaching Problem Solving" , Vol I and 2. John Wiley & Sons.
Silver,
E.
A., (1979), 'Student Perceptions of Relatedness awing liathenatical
Verbal Problems' ,
Silver,
F.
N. C. T. M. ,
.1. R. a F. Vol. 10, No. 3, .
A. , (1987), 'Problem Solving, Ti ps for Teachers' ,
N. C. T. 11. ,
A. T. ,
Vol. 34, No. 5, .
Shimoda, S , (1977), "Open-Eded Approach in Arithmetic and ttithematics--A New Plan for Improvement of lessons ", Mizoorni-Sholxm (in .Japanese)
(Jeremy 25, 1990)
21:9
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REFLEXIONS SUR LE ROLE DU MAITRE DANS LES SITUATIONS
DIDACTIQUES A PARTIR DU CAS DE L'ENSEIGNEMENT A DES ELEVES
EN DIFFICULTE
Marie-Jeanne PERRIN-GLORIAN
IREM Universit6 PARIS 7
We worked in mathematics with classes mainly made up of pupils (9 - 13 years old)
meeting with difficulty at school in many subjects, most of them coming from lower classes.
Basing ourselves on results coming from cognitive psychology, sociology, social
psychology..., we ay to interpret difficulties in mathematics of these pupils in the theoritical
frame of didactics of mathematics as it has extented in France. In particular, we draw the
constrain= bearing on negotiation of the "didactic contract", and coming from teachers as well
as from pupils. This leads us to reconsider the part of the master in the theory of didactic
situations. We study more specially processes of "devolution" and "institutionnalisation" of
knowledge. This study induces us to identify a kind of situations, called here "recall situations"
that seems to us particularly important for pupils who are in trouble at school. The matter is, for
pupils, to account orally for a situation of action which has taken place in a previous session,
when action is no more possible
Introduction et cadre theorique de reference.
Nous nous placons dans le cadre theorique de la didactique des mathematiques tel qu'il
s'est developpe en France et a déjà ete evoque dans les rencontres de PME (R. Douady, PME 9
et 11, C. Laborde PME 13). Nous utilisons plus particulierement la theorie des situations
didactiques elaboree par G. Brousseau (1987) et les notions de jeu de cadres et dialectique
outil-objet introduites par R. Douady (1987). Cette theorie s'est developpee en adoptant un point
de vue epistemologique qui donne une grande importance a la resolution de problemes, aussi
bien dans la construction du savoir que comme critere du savoir. Cette position n'est pas
toujours conforme aux conceptions sur l'enseignement et l'apprentissage des professeurs qui
utilisent les resultats des recherches. Les Cleves ne sont pas non plus toujours prets a engager
leur responsabilite dans une resolution de probleme. Ces distorsions s'observent
particulierement lorsqu'il s'agit d'enseigner les mathematiques a des eaves en grande difficulte
scolaire, issus pour la plupart de milieu social defavorise. Pour les Ctudier, nous empruntons
avec A. Robert et J. Robinet (1989) le cadre theorique de la representation sociale (Abric,
1987).
Dans ce texte, nous donnerons d'abord les grandes lignes de l'interpretation que nous
faisons, en nous servant de ce cadre theorique, de l'echec d'eleves en grande difficulte a l'Ccole.
En relation avec cette interpretation, nous analyserons ensuite des aspects qui nous paraissent
importants clans le role du maitre.
Interpretation des difficultes des eleves.
Absence de creation de representations mentales et de projet implicite de reinvestissement.
Nous avons constate qu'il y avait souvent, chez les enfants en difficulte, un divorce net
entre les situations d'action qui devaient servir a donner du sens aux notions enseignees et
l'institutionnalisation qui est faite ensuite par le maitre : au tours de l'action, dans les premieres
situations qui permettent d'aborder une notion nouvelle, on ne voit pas beaucoup de differences
entre Cleves. En revanche la difference s'accentue trios vite des qu'il s'agit de reutiliser les
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220
connaissances nouvelles clans d'autres situations. Le savoir institutionnalise par le maitre, meme
dans le cas oir il est memorise, semble coupe des situations d'action qui lui ont donne naissance
et ne peut etre utilise pour resoudre de nouveaux problemes.
Une des principales explications que nous avancons est que les eleves qui ne rencontrent
pas ce genre de difficulte ont un projet, meme implicite, de de.contextualisation des le moment ou
ils travaillent sur la situation d'action. Us savent qu'il y aura peut-etre lieu de reutiliser
l'experience acquise. Ils se creent des representations mentales non seulement pour resoudre le
probleme pose actuellement mais pour pouvoir en rappeler et riutiliser des elements dans
d'autres occasions. Ceci leur permet de reinvestir partiellement une connaissance, meme si elle
n'est pas encore totalement identifiee. Pour d'autres enfants, ce "transfert" ne se fait pas parce
qu'ils ne font que resoudre le probleme pose, dans les termes oir il est pose, sans avoir de projet
de connaissance. Il n'y a pas creation de representations mentales qui ont déjà valeur
symbolique et sur lesquelles on pourra travailler ensuite. II n'y a pas non plus de mises en
relation, "d'accrochage" 3 l'ancien pour le renforcer ou le remettre en question. Tout ceci
empeche la capitalisation et la memorisation des connaissances. Ainsi, chaque experience est
nouvelle, ou plus exactement, seul le contexte est feconnu : "on a plie des bandes de papier, on a
&coupe des rectangles"...
Manque de fiabilite des connaissances anciennes.
L'absence de connaissances anterieures solides auxquelles se referer contribue d ce manque
d'organisation et d'integration des savoirs nouveaux : pour certains enfants, rien n'est sill-, tout
peut toujours etre remis en question, puisqu'ils ont l'habitude de se tramper.
Absence d'identification de l'enjeu des situations didactiques.
Une autre cause nous parait etre la non reconnaissance du veritable enjeu des situations
proposees en classe et l'absence d'identification de l'objet du travail propose par l'enseignant :
par exemple, si celui-ci propose des decoupages de rectangles pour travailler sur les fractions
alors que, pour l'eleve, it s'agit d'apprendre 3 partager les rectangles, il n'y a pas de lien entre
cette activite et le pliage de bandes de papier. Ainsi, les fractions utilisees dans les deux
contextes n'ont pas de rapport entre elles, l'eleve n'a donc pas de souci de coherence.
Cela a des consequences au niveau didactique, par exemple l'usure rapide des situations :
les eleves qui identifient la situation a son contexte se lassent avant qu'on puisse avoir une
identification et une clecontextualisation locale des savoirs en jeu suffisantes pour permettre leur
reinvestissement ulterieur. Nous allons voir que cette usure participe 3 l'enclenchement d'un
cercle vicieux renforce ensuite par les choix des maitres.
Simplification des situations et enclenchement d'un cercle vicieux.
La difficulte de reinvestissement des eaves est particulierement grande dans le cas de
situations complexes oil it y a a identifier un probleme connu a l'interieur dune situation oir
interviennent d'autres eldments. Cela renforce l'idee qu'on facilite l'apprentissage en simplifiant
le probleme, en mettant des paliers intermediaires. Cela entrailie aussi, chez les enseignants
comme chez les eleves, le desir de recourir le plus possible a l'apprentissage de procedures de
traitement stereotypees, plus securisantes. En effet, les eleves en difficulte quetent l'approbation
221
210
du maitre a chaque pas des qu'ils sortent de la routine. Bs reclament des algorithmes. Par
ailleurs, du cote des maitres, on fait moins confiance aux eleves, on a tendance a les aider
davantage et on pense leur dormer ainsi des moyens de reussir au moins quelque chose.
II est vrai que les algorithmes eux-memes sont souvent insuffisamment memorises par ces
&eves. Ceci entrain une charge en memoire insupportable lors de la resolution de problemes,
leur fait perdre le fil de la resolution et encourage donc l'enseignant a dormer plus de place
encore A l'apprentissage des lecons et des algorithmes.
En outre, avec les eaves en difficulte, les professeurs ont tendance a se concentrer sur le
cadre numerique en negligeant des activites geometriques ou graphiques qui pourraient dormer
d'autres references.) Comme un changement de point de vue est toujours difficile, Hs pensent
generalement que, pour ces eleves, it faut faire le moins de melanges possible. Cela contribue
accroitre le deficit de connaissances solides dans des cadres differents et empeche le
fonctionnement de jeux de cadres, ce qui diminue encore les occasions d'apprendre h mettre en
relation differents savour.
On assiste ainsi l'enclenchement d'un processus "boule de neige" : les eleves ne se
representent pas les actions, ne pert oivent pas les enjeux > les eleves ne memorisent pas >
le professeur se concentre sur l'apprentissage des resultats du cours et de savoir-faire
algorithmises > les situations proposees aux eleves se resument a la repetition de problemes
de contrede stereotypes > les eleves ne se representent pas, ne mettent pas en relation >
et l'apprentissage se resume au renforcement d'algorithmes dont les situations d'utilisation ne
sont jamais maitrisees.
Autres aspects
Nous ne pouvons divelopper ici d'autres aspects importants dans l'enclenchement du
cercle vicieux dont nous avons parle :
Les problemes de langage, expression et lecture, sont aussi bien sOr d l'origine de
difficultes en mathematiques, de trois facons au moins : au niveau de la prise d'information, au
niveau de la conceptualisation, an niveau des productions.
Une autre difficulte tient a la capacite d'interpretation du niveau de discours du maitre.
Dans le deroulement de l'enseignement, en effet, le maitre utilise plusieurs niveaux de discours
qui sont souvent assez imbriques et que l'eleve doit reussir a decoder avec leur signification
dans la situation. II doit etre capable de reperer ces changements de niveau et de tirer profit du
discours non mathematique - que nous appelons ici un peu rapidement, metamathematique mais
qui recouvre de nombreux registres que nous n'avons pas la place de distinguer ici - pour
s'approprier plus facilement le discours mathematique du maitre et des autres eleves.
Les situations du quotidien avec lesquelles les eleves ont une certaine familiarite, utilisent
souvent des modes de raisonnement non conformes a ceux qu'on attend clans un cours de
mathematiques. II peut ainsi s'installer un veritable malentendu et une communication absurde
entre le professeur et certains eleves. Ceci ne veut pas dire que l'experience des enfants dans la
vie quotidienne ne peut pas 8tre utilisee, mais faut alors batir, comme le font par exemple
certains chercheurs italiens autour de P. Boero (1989), des situations qui s'appuient sur la rialite
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familiere et permettent de la depasser en posant aux enfants de veritables problemes theoriques.
Rapport de l'eleve d recole, d son metier d'eleve. Plusieurs des explications que nous venons
d'avancer sons a relier au plan plus general des attentes et des representations sur recole, a la
presence ou a l'absence de projet general, et a ce que Y. Chevallard (1988) appelle le rapport a
recole, au métier d'eleve... Par exemple, un clove peut trouver illegitime qu'on lui propose un
probleme dont on ne lui a pas enseigne la reponse et refuser cette responsabilite.
Rapport de l'enseignant d son metier d'enseignant.
L'idee que l'enseignant se fait de son métier et de revaluation des eleves peut parfois
l'amener a vider l'enseignement de son contenu, en particulier clans le cas ou it s'adresse a des
eleves en difficulte. 11 peut etre ainsi conduit a "surinstitutionnaliser" des resultats ou des
methodes rencontres dans des resolutions de problemes et a remplacer le veritable enjeu de
renseignement par des intermediaires introduits pour faciliter racces a une connaissance.
Les representations des enseignants sur les capacites des eleves se conjuguent avec celles qua
renseignant sur la bonne maniere d'apprendre, ce qu'est une formation mathematique, le
contenu vise. L'enseignant choisit en fonction de ces representations qui lui font estimer le coat
par rapport a la rentabilite attendue, les methodes qui lui paraissent convenir compte tenu du
contenu et du public.
Representation de soi de l'eleve. Leur situation d'echec a recole contribue donner aux eaves
en grande difficulte une image d'eux-memes devalorisie. Cette image et la representation gulls
se font de leur place par rapport aux autres &eves de la classe ont des repercussions sur toute
leur vie scolaire, y compris racceptation de certaines formes de travail (en groupes, notamment)
Le role du maitre.
Le role du mitre dans la thiorie des situations.
Dans la theorie des situations didactiques telle que la developpe G. Brousseau (1987) et
que nous schematisons ici, le role du maitre dans les situations didactiques se situe
essentiellement a trois niveaux : choix d'un probleme et dune situation a-didactique et
determination des variables didactiques de facon a mettre en jeu la connaissance visie,
devolution de cette situation a l'eleve et institutionnalisation des connaissances. La premiere
phase nest pas forcement entierement a la charge de l'enseignant qui peut avoir recours a des
trays= d'ingenierie didactique. Nous nous interesserons dans la suite aux deux autres phases.
Nous laissons volontairement de cote revaluation qui intervient aussi de facon importante dans
rinstitutionnalisation des connaissances et clans les representations que les eleves se font des
concepts mathematiques et des mathematiques en general.
La &volution.
Pour que reeve construise un savoir, il est necessaire, d'apres G. Brousseau (1987), qu'il
produise ses connaissances, les false fonctionner ou les modifie comme reponses aux exigences
du milieu et non au &sir du maitre. Pour cela, it faut que releve accepte que la resolution du
probleme soit de sa responsabilite, qu'il accepte de prendre en charge ce que Brousseau appelle
une situation "a-didactique'', c'est-a-dire une situation depouillee de ses intentions didactiques.
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L'eleve dolt faire sienne la question posee et chercher a la resoudre sous sa propre
responsabilit6, sans essayer de deviner les intentions du maitre ni chercher a lui faire plaisir. La
devolution est alors un processus necessaire parce que l'acces de l'eleve a la situation
a-didactique ne va pas de soi car elle est au depart tres imbriquee a la situation didactique qui la
contient : "La situation a-didactique finale de tiference, celle qui caracterise le savoir (...) est une
sorte d'idial vers lequel it s'agit de converger" (p.50)
La question que nous nous posons est donc la suivante : qu'est-ce qui permet a l'eleve de
converger vers la situation a-didactique, qu'est-ce qui fait qu'il met un savoir mathematique en
jeu en tentant de resoudre le probleme pose par le maitre ? G. Brousseau donne lui-meme par
avance une premiere reponse a cette question : "L'eleve salt bien que le probleme a ete choisi
pour lui faire acquerir une connaissance nouvelle mais it dolt savoir aussi que cette connaissance
est entierement justifide par la logique interne de la situation et qu'il peut la construire sans faire
appel a des raisons didactiques" (p. 49). La devolution est essentiellement ce qui le lui fait savoir
et it est vrai que c'est une condition pour que l'eleve fonctionne de fagon scientifique et non en
reponse a des indices extCrieurs. Mais it nous semble que l'affirmation de la premiere pantie de in
phrase "l'eleve salt bien" n'est pas Cvidente. Suivant leur origin culturelle ou leur experience
scolaire anterieure, certains Cleves savent bien en effet qu'il y a toujours un objectif
d'apprentissage dans ce qu'on leur propose et on a l'habitude dans l'enseignement de faire
comme si cette evidence Ctait partagee. Or nos observations sur les Cleves en difficultC nous
laissent penser qu'elle ne Pest pas. La question didactique qui se pose alors est de savoir de quel
projet faut-il faire devolution a reeve avec le probleme (ou avant pour permettre ensuite la
devolution du probleme a l'eleve), ou, en d'autres termes, comment faire devolution a l'eleve de
la prise en charge de son propre apprentissage ? Cette question se pose, pour le maitre, dans la
nCgociation du contrat didactique a plusieurs niveaux : au niveau general de l'enseignement des
mathCmatiques clans la classe considCrCe, au niveau de l'ensemble du processus d'apprentissage
d'un concept donne et au niveau de chacune des situations composant ce processus.
L'institurionnalisarion.
Ceci nous amen a considCrer l'institutionnalisation comme un processus qui se dCroule
tout au long de l'enseignement, un moteur de l'avancement du contrat didactique et du temps
didactique et non comme une phase en fin de processus on le maitre fait son cours.
L'institutionnalisation des connaissances commence pour nous des le tout debut de la devolution
puisqu'il faut déjà que le maitre donne a fileve, s'il ne l'a pas, le projet d'acquerir ces
connaissances. Evidemment, nous trouvons IA un des paradoxes du contrat didactique que
Brousseau a mis en evidence : le maitre ne peut pas parler de la connaissance nouvelle puisque
c'est justement l'enjeu de l'apprentissage, it peut au plus dire qu'on va apprendre quelque chose
de nouveau et Cclairer les Cleves sur les connaissances anciennes a mobiliser pour "accrocher"
cette connaissance nouvelle. En fait le maitre tend a l'institutionnalisation tout au long du
processus : s'il veut que l'institutionnalisation puisse se faire pour les Cleves dans de tonnes
conditions, avec du sens, it ne peut aller droit au but mais l'a toujours present a l'esprit pour
mCnager des le depart et tout au long du processus d'enseignement les conditions qui vont lui
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224
permettre de negocier le contrat didactique dans ce sens.
Avec des eleves en difficulte, les contraintes qui pesent sur l'institutionnalisation sont
particulierement visibles et on se sent comme un funambule sur son fil : si is la suite de la
resolution d'un probleme, aucune decontextualisation nest amorcee par le maitre, les eleves ne
retiennent rien et ne peuvent parler que du contexte du probleme et non de son enjeu, s'il y a
decontextualisation par le maitre, on assiste a un derapage formel qui amen les eleves 3 prendre
les &Mures mathematiques sans les crediter du sens qu'elles pouvaient avoir dans le probleme
traite. L'equilibre est difficile a trouver. Nous en deduisons que, pour certains eleves au moins,
l'institutionnalisation ne peut se faire que de facon tits progressive avec de nombreux cycles
contextualisation - decontextualisation.
Ceci nous amen 3 distinguer des etapes dans ]'institutionalisation
- institutionnalisations locales dans divers contextes, au sens de R. Douady (1987)
- reinvestissement d'un contexte dans un autre
- cours construit par le professeur au sens traditionnel, donnant un statist d'objet mathematique a
certaines des notions rencontries.
Ces etapes ne correspondent pas entierement it un ordre chronologique, le reinvestissement
se placant tout au long, avec des ciegres de decontextualisation differents : des que les eleves ont
rencontai une premiere situation sur la notion, ils peuvent reinvestir des pratiques en
reconnaissant une analogie entre deux situations, jusqu'apres le cours oir ils pourront peut -titre
reinvestir le savoir en tant qu'objet mathematique.
Les situations de "rappel ".
Un des temps forts dans le processus de depersonnalisation et decontextualisation des
savoirs construits en classe se situe au cours des bilans qui suivent une phase de recherche des
eleves. Des chercheurs ont analyse le role du maitre dans ces bilans et distingue a cote des
moments d'institutionnalisation, des moments oil le maitre cherche a homogeneiser la classe et
oit s'effectue une premiere depersonnalisation des procedures mises au point par les eleves clans
la phase de recherche. Nous avons, pour notre part, identifie un autre type de situations qui
nous semblent jouer un role important dans ce processus de depersonnalisation et
decontextualisation, a deux moments au moins : d'une part elks vont permettre d'adapter
finstitutionnalisation locale aux conceptions actuelles des eleves ; d'autre part, avant le cours
proprement dit, elks vont permettre d'accrocher les notions qu'on va exposer aux problemes qui
ont permis de leur donner du sens.
Nous les appelons pour le moment, et faute d'avoir trouve un meilleur terme, des situations
"de rappel". 11 faut tout de suite dissiper un malentendu possible : it ne s'agit pas de revision ni
de rappel par le maitre de ce qui a ete fait, it s'agit plutot pour les eleves de se rappeler une ou
plusieurs situations déjà traitees dans des seances precedentes sur un m8me theme, avec un peu
de recul donc, de faire un retour par la pensee et la parole sur ces seances. En essayant de dire
collectivement ce qui s'est passe, quel probleme on a traite, les eleves sont amends a repenser le
probleme, les procedures de traitement envisagees dans la classe. Les eleves qui ne se sont pas
construit de representation mentale au cours de l'action trouvent la une nouvelle occasion et une
22
214
raison de le faire puisqu'ils vont devoir parler de ce qui s'est passé et le decrire sans pouvoir
agir a nouveau. II se peut que pour certains eleves l'action soit a nouveau necessaire mais elle est
alors placee clans une nouvelle perspective : it faut agir non seulement pour trouver une solution
mais aussi pour pouvoir en parler.
Dune part it se produit alors une depersonnalisation des solutions dans la mesure oir elles
sont reprises et exposees par d'autres eleves que ceux qui les ont trouvees, d'autre part it se
produit une pre-decontextualisation : en reprenant a froid ce qui s'est passe, on elague les. details
pour identifier ce qui est important. A cette occasion, le sens cache, le role pour rapprentissage
de l'un ou l'autre des problemes poses peut se reveler a certains eleves. De plus, s'il y a une
suite de problemes sur un theme, chacun des problemes traites est integre dans un processus, II
est interiorise avec un sens nouveau. Au cotes d'une telle situation, les formulations evoluent,
on peut avoir des retours sur des &bats de validation qui ont deja eu lieu ou rencontrer la
necessite de nouveaux. On n'est pas a proprement parler dans une situation de formulation oe ii
s'agit de produire un nouveau langage, ni dans une situation de validation, mais on retravaille
les formulations et les arguments deja produits. En meme temps, par le retour reflexif sur
l'action que ces situations supposent, elles favorisent la construction de representations naentales
par les eleves.
Darts ce type de situation, le role du maitre est tres important. Le choix de donner la parole
a un eleve plutot qu'a un autre donne a la situation une signification toute differente : s'il veut
que la fonction d'homogeneisation et de depersonnalisation soit remplie, it va donner la parole
aux eleves qui n'ont pas trouve de solution ou qui n'ont pas abouti pour verifier qu'ils suivent et
reprennent a leur compte les methodes utilisees, s'il veut avancer dans la decontextualisation et
la formulation, ii va davantage donner la parole aux "leaders", quitte a faire reprendre les
nouvelles formulations du probleme par l'ensemble de la classe dans le courant de la seance ou
ulterieurement. On voit ainsi une evolution par rapport a la phase de bilan oir ce sont plut8t les
"leaders" qui exposent les methodes de resolution qu'ils ont trouvees, les "suiveurs" se
contentant d'ecouter ou d'intervenir sur des points de detail qui sont dans le domaine de
l'ancien. Ses marges de manceuvre se situent aussi dans le choix des questions, dans ce qu'il
reprend ou non des interventions des eleves, dans ses commentaires.
II peut agir sur ces marges pour ancrer "le nouveau" dans les connaissances anciennes et
dans ce que les eleves ont reellement fait ou faire avancer la connaissance en s'ecartant un peu
du probleme reellement traite, en proposant un debut de generalisation ou de reinvestissement
dans un contexte legerement different.
Le role du maitre est essentiel dans le processus d'institutionnalisation, quel que soit le
style d'enseignement. II doit notamment choisir ce qui est a retenir dans chaque séance et decider
en mEme temps quel "ancien" remobiliser, que reprendre dans les activites des eleves, jusqu'oe
aller dans la decontextualisation.
Ces decisions vont dependre de ce que les eleves ont reellement fait et de revaluation qu'en
fait le professeur : est-ce que ce qu'il considere comme ancien est reellement acquis par
suffisamment d'eleves, est-ce que ('appropriation des methodes de resolution est suffisamment
215
226
generalisee dans la classe... Il s'agit la dune evaluation globale, intuitive des eleves, qui a des
liens avec revaluation officielle realisee par ailleurs mais qui ne s'y reduit pas (cf Perrenoud
1986). Cette lecture par le professeur du travail des eleves va faire intervenir les representations
gull a sur le savoir vise ainsi que sur la maniere d'apprendre,,sur son role clans l'apprentissage
des eleves. De leur cote, les eleves sont inegalement prets a suivre le maitre dans une
decontextualisation de ce qui a ere vraiment traite. II revient encore au maitre de laisser ou non la
possibilite de refaire ce chetnin a d'autres moments pour ceux qui n'etaient pas encore prets.
Conclusion
En analysant les clifficultes d'eleves de 9 a 13 ans en echec scolaire et le fonctionnement
didactique, nous avons rencontre des phenomenes qui peuvent provoquer, a notre avis, le
renforcement de rechec de ces eleves. Une question didactique importance est celle du choix de
situations de complexite optimale pour ces eleves. Une autre, que nous avons commence a
etudier id, est le role du maitre clans le processus d'institutionnalisation des connaissances. A
l'issue de ce travail, il nous parait important d'approfondir cette etude, en particulier dans les
situations que nous appelons "de rappel" car elks nous paraissent un lieu possible pour briser le
cerele vicieux qui maintient ces ayes en echec.
References
Abric J. C. (1987) Cooperation, competition et representations sociales Ed. Del Val, Suisse
Bautier-Castaing E. et Robert A. (1988) Reflexions sur le role des representations metacognitives dans l'apprentissage des mathematiques.Revue Frangaise de pedagogie n°84 INRP Paris
Boero P. (1989) Mathematical literacy for all experiences and problems Acres de PME 13
Brousseau G. (1987) Fondements et methodes de la didactique des mathematiques. Recherches
en didactique des mathematiques Vol. 72
Chevallard (1988) Notes sur la question de !'echec scolaire IREM de Marseille
Douady R. (1985) The interplays between different settings. Tool-object dialectic PME 9
Douady R. (1987) L'ingenierie didactique, un instrument privilegie pour une prise en compte de
la complexite de la classe Acres de PME 11
Douady R. (1987) Jeux de cadres et dialectique outil - objet Recherches en didactique des
mathematiques Vol. 7.2
Laborde (1989) Hardiesse et raison des recherches francaises en didactique des mathematiques.
(english version available) Acres de la rencontre PME 13, Paris
Perrenoud P. (1984) La fabrication de !'excellence scolaire Droz, Geneve
Perrin-Glorian M.J. et al. (1989) Une experience d'enseignement des mathematiques a des
eleves de 6eme en difficultes Cahier DIDIREM n °5 IREM de Paris 7
Robert A. et Robinet J. (1989) Representations des enseignants de mathematiques sur les
mathematiques et leur enseignement Cahiers DIDIREM n° 1 et 4 IREM de Paris 7
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216
Geometry and Spatial Development
228
DIAGNOSIS AND RESPONSE IN TEACHING TRANSFORMATION
GEOMETRY
Alan Bell and Derrick Birks
Shell Centre for Mathematical Education, University of Nottingham, UK
In a comparative teaching experiment involving two classes taught reflection
geometry by the same teacher, a conflict and discussion method showed superior
initial learning and good retention over two months, while a widely used scheme
of individual guided-discovery booklets shows very poor retention. The
development through piloting of the conflict method involved discarding easier
tasks and retaining those harder ones which provoked discussion. These led to
an analytic awareness of the essential properties of reflection as distinct form a
global perception of 'balance'.
Among the topics treated in previous . researches on pupils'
developing mathematical concepts and on the design of diagnostic modes
of teaching which make use of this knowledge, geometry has been
somewhat under represented. The main part of this paper will help to
redress this balance. It will bring into play a different set of didactic
variables and a somewhat different way of designing tasks from, for
example, the experiment reported last year on the teaching of fractions.
(Bell and Bassford, 1989). In that study, the learning tasks consisted mainly
of challenges embodying the fundamental notions concerning fractions finding many ways of displaying half a square, finding all the fractions
equivalent to a given one, finding how to compare 3/4 and 4/5, then any two
fractions, finding how to add fractions - the essence of the challenge being
to decide the meaning of the question and to choose suitable
representations in which to seek an answer. The main didactic variable,
which was manipulated to provoke generalisation, was the size of the
integers comprising the fraction.
The present study concerns the
transformation of reflection; a quite different set of didactic varibles arises,
and different ways of generating the field of examples.
The range of problems considered, and most of the misconceptions
observed are illustrated by Figure 1, part of the last worksheet of the
teaching sequence. The aim was the construction of plane reflections in a
line and the identification of the lines of reflective symmetry in plane
figures. Previous research by Kidder (1976), Schultz (1978) and Kiichemann
(1981) had identified as relevant variables
219
229
1.
2.
3.
4.
the direction of the mirror (horizontal, vertical, 450, other)
the complexity of the figure being reflected
the presence of a grid
the size of the figures and distance from the mirror.
The first three of these were incorporated by Kiichemann in a
structured sequence of questions. He identified levels of response as global,
semi-analytic, analytic and analytic-synthetic. In global responses the object
is considered and reflected as a whole with no reference to particular parts,
angles or distances; in semi-analytic responses, a part of the object, usually
an end point, is reflected first and the rest drawn from it matching the
original in shape and size. In fully analytic responses, the object is reduced
to a set of key points, each reflected individually. These are connected and
the result accepted even though sometimes the image looks wrong. In
analytic-synthetic responses, the analytic and global responses are
co-ordinated so that the final image is accurate and also looks correct.
The present study consisted of interviews and a pilot teaching
experiment with a mixed ability secondary school class aged 11 to 12 years,
followed by a comparative teaching study with two other parallel classes of
the same age. We shall report here the main misconceptions found in the
interviews, the design of the materials for the teaching experiment, noting
particularly modifications made following the pilot work, and finally give
the results of the comparative teaching.
Pupils' Concepts
The first group of misconceptions comprised beliefs that horizontal
objects must have horizontal images and vertical objects vertical images or
that horizontal objects have vertical images and vice versa. These can be
seen in questions 2, 4, 5 of Figure 1.
Approximately 40% of the sample made errors corresponding to one
or more of these misconceptions during the pilot testing. The next
misconception consisted of associating reflecting with various pairs of
opposites such as forwards and backwards, towards and away, left and right,
upwards and downwards. For example, one pupil producing a response
somewhat like that in number 2 of Figure 1 said,
"this one is on the left and points up so that one must be on the right
and point down".
220
230
The following worksheet was given to Edward Green for Homework. Mark the work,
correcting all the mistakes. In your book, explain where Edward is going wrong'
1
4
-yr
4.....
A
a .*.
1
i
298
6
C
0
A....:
Figure 1
Figure 2
Such verbal descriptions of the relation between object and image
which might be derived initially from some correct observation, are thus
transferred to other situations to which they do not apply. In a similar
way, the term 'straight across' was in some circumstances interpreted by
some of the sample in a way dependent on the nature of the object and/or
the presence of the grid and/or the slope of the mirror line. Thus the term
might be used in item4 of Figure 1, and in cases like that shown in Figure 2,
where axes 2 and 4 are taken as lines of symmetry.
The somewhat unexpected misconception that there could be more
than one possible correct image was displayed by some 14% of the pupils
interviewed; all of these were pupils who saw reflection as a mirror image,
rather than as a folding and they often justified their conclusion by
showing how the mirror could be moved, still standing on the same line,
to produce a movement of the image. The same connection with physical
mirrors rather than folding gave rise to another misconception, that the
image might simply be similar to the original object and not necessarily
221
231
congruent to it (Figure 1, No. 6). Over-generalisations of verbal statements
were also used to justify erroneous placings of lines of symmetry on, for
example, the letter N or as a diagonal of a rectangle; the shape was held to
have a line of symmetry if it could be split into two equ'al parts even if they
were inverted or displaced. Pupils said, for example, "it is the same on both
sides". Another difficulty arose when a fairly complex figure might have
symmetry if certain details were ignored. It would seem important that this
type of example should be given, and pupils encouraged to give alternative
statements about its symmetry according to whether or not various details
are considered (figure 4).
Teaching Experiment
The experimental teaching occupied 10 one hour lessons for each of
the two groups. In the diagnostic method the pattern of each lesson was
that pupils, in groups of about 4, discussed the problems on a given
worksheet and arrived at agreed conclusions.
Following this, there was a
class discussion in which the conclusions from each group were
contributed and defended and conflicts among the various interpretations
were resolved. Extract from the worksheets for lessons 1, 3, 5 and 10 are
shown in Figures 3, 5 and 1.
4
0
Figure 3
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232
111
30 pupils were asked to find the point that
was straight across from A. Ten different
suggestions were given and are shown above.
Circle the point that you think is correct or
suggest another point if you do not agree
with any of their answers
Figure 4
Figure 5
Some modifications were made to the sheets and to the mode of
conduct of the lesson following the pilot work. These were as follows:
1.
Certain easier tasks were omitted (marked 0 in Figure 3). These
were answered correctly by most pupils, which left some of them
with the impression that they were doing reasonably well and did
not need to change their strategy even if they in fact possessed serious
misconceptions causing errors on the harder questions.
2.
The worksheet following Figure 5, which asked the pupils to write an
explanation of why their choice in Figure 5 was correct. Some groups
responded with rather weak explanations. In the modified lesson,
the teacher intervened by playing 'devil's advocate' and so
provoking them to produce more cogent arguments.
3.
More time was taken at the beginning of the teaching to discuss the
positive aspects of making errors, the importance of explanation and
of listening skills, and the need for mutual respect of other's
opinions.
223
,233
B4
B2 Copy these on spotty paper.
Draw each reflection with the help of mirror.
(b)
(a)
You need triangular spotty paper (or these.
Check each one with mirror.
(I)
(c)
/
(b)
...
3
Ile does not have reflection
symmetry when he stands
This lizard has reflection
symmetry when he stands
like this.
like this.
Check with your mirror.
Figure 6
The alternative teaching method was based on two booklets on
Reflections (from the SMP 11-16 course). These were in use in the school as
part of an individual learning scheme during the first two years. The
pedagogical method embodied in these is that of examples with
explanation, followed by questions for individual practice. The first booklet
concentrates on reflection as mirror images, the second booklet on folding.
The questions become increasingly complex, but the learners are not asked
to devise their own methods and none of the situations demands a high
level of thought or enquiry. This contrasted with the diagnostic method, in
which the aim was to lead pupils, through the discussion of difficult
questions, to recognise and to state explicitly and carefully the general
properties of reflection. The booklets were well received and enjoyed. In
this group, the teacher was fully occupied in managing the issue of the
booklets and administering the review and check tests, and in answering
pupils' individual questions relating to the material. An example of the
material is given in Figure 6.
Results
A 23 item test containing a mixture of items of the types illustrated
here from both types of teaching was given to both groups before and
immediately after the teaching and again 10 weeks later.
224
3'4
The graphs in Figures 8 and 9 show the performance of each pupil in
each of the two groups. The superiority of the experimental method for
retention, and correspondingly the long term inadequacy of the booklets
teaching, is very evident. For a full report, see Birks (1987).
At'
fat
rstr
Scores of pupils in booklets group
Scores of each pupil in diagnostic
group
Figure 7
Figure 8
225
235
Implications
Teaching of the type represented by the booklets method in this
experiment is currently very common. It can best be characterised as
'guided discovery'. The initial explanation shows the pupils how to
approach the questions, and as these are worked through different aspects
of the embodied principles are called into play. Two elements commonly
missing are (a) feedback and (b) awareness. Errors made are not generally
discovered until sometime later when responses to the whole set of
exercises are checked by the teacher or by reference to answers and at that
stage, a score of 60 or 70% correct is regarded as satisfactory. Thus
misconceptions brought into play by the remaining questions remain
untreated, indeed, they are reinforced through the act of use. These
materials also ignore the importance of making the correct principles, and
the way in which they are manifested in various contexts, explicity through
discussion. Other research has shown that what is actually learnt from
studying a given piece of material is strongly influenced by the learner's
orientation towards it, and this depends on the learner's expectation of the
use to which this learning is to be put; for example, whether a factual
recall test will be given or a test requiring comprehension of the material,
or its application to fresh situations (Mayer and Greeno, 1972). In many
current classroom environments, the expectation of future testing is
minimal and, in some cases non-existent, and the pupil's orientation is
towards the completion of assignments and the attainment of grades based
on successful work. The distinction between doing and learning is often
not made by pupils, nor sometimes by teachers, successful performance
being what is rewarded rather than the acquisition of new knowledge or
skills not possessed before, or the eradication of erroneous conceptions.
These considerations suggest that metacognition, in the shape of pupils'
awareness of their learning processes, is an important field for study and
development at the present time. We intend during the next two years to
make a study of pupils' learning concepts in a number of typical and
innovative mathematics classroom environments, to develop approaches
aimed at improving pupils' self awareness of learning, and to study the
effects of the implementation of these.
236
226
CHILDREN'S RECOGNITION OF RIGHT-ANGLED TRIANGLES
IN UNLEARNED POSITIONS
Martin Cooper
University of New South Wales, Australia
Konrad Krainer
Universitat Klagenfurt, IFF, Austria
Are children better able to identify right triangles in orientations
in which they have been learned or does the horizontal-vertical have
an over-riding effect? Austrian primary school children were taught
to recognize right triangles in particular orientations. On testing,
they were able to identify them better when the shorter sides were
horizontal and vertical, even when the triangles had been learned in
other orientations.
In classrooms and geometry textbooks, right triangles are
frequently presented standing on one of the non-hypotenuse
sides
a "standard" orientation.
Indeed, school children
often experience difficulty in identifying figures such as
squares when they are presented in non-standard orientations.
Cooper and Shepard (1973) found that when university
students tended to take longer to say whether non-symmetric
numerals and upper-case letters were "normal" or laterally
inverted as the angle of inclination of the characters to their
usual upright became greater.
This suggested that people
identify a tilted test character by mentally rotating an
internal representation of the character into congruence with a
long-term memory representation (a schema) of the character.
Cooper (1975) found similar results with unfamiliar random
angular forms. Herschkowitz et al (1987) have shown that
children are able to recognize right triangles best when they
are presented in "the upright position as usually drawn", have
less success when the triangle is rotated through about 45°,
with success decreasing "drastically" when the right-angle is
at the top".
Eley (1982) found that children were able to
identify letter-like symbols with more accuracy when the
symbols were presented in orientations closer to the trained
orientation.
It may well be that when children are taught the definition
of a particular figure by means of illustrations presented in a
standard orientation they, too, form a mental image, or schema,
of the figure in this standard orientation. When later faced
BEST COPY AVAILABLE
237
with the same figure in a non-standard orientation, the child
may then mentally rotate an internal representation of the
figure into congruence with the already internalized
standard-orientation representation.
Herschkowitz has suggested that the horizontal and vertical
sides of the page on which a triangle is drawn may act as a
"surrounding field" from which many people have difficulty in
The effect is observed also in the
isolating the triangle.
case of isosceles triangles, the accuracy of recognition being
greatest when the "base" is horizontal and at the bottom of the
figure.
Both Hart (1981) and Grenier (1985), mention this
effect in relation to plane reflections when a
vertical-horizontal grid is used as the background.
The purpose of the research presented in this paper was to
investigate the variation in accuracy of recognition of a right
triangle, learned in certain standard and non-standard
positions, when later viewed in a set variety of orientations.
METHOD
Twenty-four 35 mm slides were prepared for use in both the
Each slide depicted a
learning phase and the testing phase.
clear circular disk set against a black background, the disk
Half the triangles
containing a triangle drawn in black.
(Set R) were identical right triangles; the other half (Set N)
were matching isosceles triangles having the same area and the
same shortest-side length as the right triangles. The
triangles in each set were oriented so
030
y
000
4)1Adl.
ISO
300
010
330 degrees (clockwise) to the
120
49.0k150
k0
.0
i%0
that their shortest sides were
respectively inclined at 0, 30, 60, 90,
120, 150, 180, 210, 240, 270, 300 and
For ease of
reference, the slides were encoded as
left-hand horizontal.
Rxxx or Nxxx, where the first character
denotes the set and the 'xxx' stands
for the angle of orientation.
The
R-set orientations are shown opposite
Fifty-five 7-8 year-old children from three primary
schools in Klagenfurt (Austria) took part in the study.
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228
All
were familiar with triangles and all had been taught to recogChildren were interviewed individually.
nize right-angles.
In the initial, screening phase, the researcher questioned
the child to ensure that he or she could recognize a
right-angle. The child was then shown drawings of a right
triangle and an isosceles triangle (both in standard
orientation) and asked what the figures were called. All
The researcher then
subjects said that they were triangles.
pointed to the right triangle and said that it was a special
triangle because one of its corners was a right-angle (pointing
It was demonstrated that none of the corners of the
Again referring to the right
other triangle was a right-angle.
triangle, the researcher told the child that a triangle having
to it).
a right-angle in one corner was called a "right triangle".
After the child had repeated the term, he or she was asked
whether the isosceles triangle was a "right triangle" or not.
All children gave the correct answer, many describing the
isosceles triangle, later, as a "false" triangle.
After passing the screening phase, each child was randomly
assigned to one of four methods in the training phase:
A
B
C
M
orientation group
trained orientations
Method
000,
030,
060,
180,
090,
120,
150,
240,
180,
210,
240,
300,
270
300
330
330,
[twice each]
[twice each]
[twice each]
150,1090, 060, 030
"mod 000"
"mod 030"
" mod 060"
(mixture)
In each method, 16 slides (eight right triangles in the
"trained orientations" mixed with the eight corresponding
isosceles triangles) were projected on to the white wall in
front of the child.
For each of the first eight slides
("assisted"), the researcher told the child whether or not the
triangle was right-angled, either pointing out the right-angle
or demonstrating that none of the angles was a right-angle, as
appropriate.
After a short pause, the second sequence of eight
slides ("unassisted") was screened, the child being asked to
say whether or not the triangle was right-angled and, if so, to
If the child gave an incorrect
response, the preceding slide was re-screened and the sequence
indicate the right-angle.
continued from that point.
If any child gave more than three
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239
incorrect responses, it was considered that sufficient learning
had not taken place, and his or her results were discarded.
After a short pause, each child who successfully completed
the training phase entered the testing phase and was shown the
entire series of twenty-four slides in the following order:
R000,
N090, R180, N270, N060,
R300, N030,
R150, R270, N120, R030, N300,
R120, N180,
N210, R240, R330, N150, R210, N000, R090, N330,
N240, R060
For each slide, the child was asked to say whether or not the
image represented a right-angled triangle and, if so, to
indicate the right-angle.
In this phase, neither assistance
nor reinforcement was provided.
RESULTS
Training phase
The number of children entering each of the training phase
methods and the number and percentage successfully completing
it, were as follows:
method
A
number entering
number (%) completing
C
8
10
10 (100%)
M
17
15
9 (60%)
10 (59%)
all
13
8 (62%)
55
37 (67%)
About two-thirds of the children successfully completed the
training phase. There was a large difference between those who
experienced Method A (100% completing) and those who
experienced Methods B, C or M (about 60% completing).
Testing phase
The numbers and percentages giving correct responses to the
"R-set" slides triangles in the respective methods were:
orientation
000 030 060 090 120 150 180 210 240 270 300 330
Method
10
100
A
Method
N
%
N
B
Method
C
Method
3
8
5
4
8
5
5
9
4
5
30
80
50
40
80
50
50
90
40
50
7
7
7
7
4
6
8
6
5
6
7
8
78
78
78
78
44
67
89
56
67
78
89
6
8
6
8
60
80
60
80
N
10
6
7
7
7
8
8
67
4
%
100
60
70
70
70
80
80
40
5
N
8
%
100
total N
35
M
4
40
8
7
7
4
7
7
5
7
88
6
75
6
63 100
75
88
50
88
88
63
88
22
29
22
24
31
19
23
30
22
28
240
25
230
INSPECTION OF DATA AND DISCUSSION
Individual methods
Method A
trained orientations: mod 000 (000, 090, 180, 270)
The average success rate in the trained orientations was always
80% or greater. These orientations represent local maxima of
performance, or "peak performances".
For untrained
orientations, this group identified between 30% and 50% of the
right-angled triangles.
Method B
trained orientations: mod 030 (030, 120, 210, 300)
Average performance peaked at none of the trained orientations
(although that for 030 was part of a plateau).
Method C
trained orientations: mod 060 (060, 150, 240, 330)
With the exception of orientations 000 (100%) and 210 (40%),
all correct-identification percentages lay between 60% and 80%.
None of the trained orientations shows a peak, although 060 and
150 belong to plateaux.
The untrained mod 030 orientations
have the worst results.
Method M
trained orientations:
030, 060, 090, 150, 180, 240, 300, 330
On average, the horizontal mod 000 (000 and 180) and trained
mod 060 (060, 150, 240 and 330) orientations show good results
(about 90%).
No easily discernable pattern is evident.
It is not surprising that when 7-8 year-old children are
taught to recognize right-angled triangles in the standard mod
000 orientations, it is subsequently very easy for them to
identify such figures in such orientations (100% of the
research sample successfully completed the training). On the
other hand, it is much more difficult for 7-8 year-olds to
identify right-angled triangles in non-standard orientations,
in spite of training in these positions under the same temporal
conditions (only 60% completed the training). Thus, the
training items were generally "easier" for children who
experienced Method A than for those in Methods B, C or M.
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241-
Combinations of methods
Below, we present the numbers and percentages of correct
responses for "non-A" method groups and for all groups:
orientation
000 030 060 090 120 150 180 210 240 270 300 330
25
93
18
22
21
%
67
81
78. 63
20
74
23
85
Methods f N
A+B+C+M
%
35
95
22
59
25
68
29
78
24
65
Methods if N
B+C+M
1
17
22
59
14
52
18
67
31
19
84
51
23
62
21
78
30
81
18
67
23
85
22
59
28
76
A graph of percentage of correct responses against angle of
orientation is given below for Method A and for Methods B, C
and M combined.
A glance at the table and graph given above suggests that,
even for non-A methods, performance tends to be biased in
favour of mod 000 orientations.
For the Method A group,
performance was very good at the trained orientations but
relatively poor at all other orientations. The possibility of
mental rotation between the mod 000 orientations cannot be
ruled out.
For all other method groups combined, subjects
tended to perform better when the right triangles were
presented in the "mod 000" orientations (apart from a small
perturbation at orientation 090), even though these
orientations were not trained ones. It is clear that the mod
000 orientations are more easily identified than any other
group of orientations irregardless of which orientations were
used in the training. The results for each of the mod-group
orientations are summarized below:
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232
method
mod 000
00,03,06,09
A
B
C
M
B+C+M
A+B+C+M
orientations
mod 030
mod 060
01,04,07,10 02,05,08,11
(A)
(B)
(C)
87.5%
77.8%
82.5%
90.6%
83.6%
84.6%
45.0%
66.7%
57.5%
62.6%
62.2%
57.9%
42.5%
72.2%
72.5%
87.5%
77.4%
68.7%
all
combined
(A+B+C)
58.3%
77.2%
70.8%
80.0%
77.4%
70.4%
The last line in the above table suggests that, without regard
to the method of training, the orientations may be classified
into three groups: those which are most accurately, moderately
accurately, and least accurately identified as right triangles.
The most accurately recognized group contains the triangles
whose shortest and longest sides are horizontal and vertical.
Of these, the triangles whose shortest sides are horizontal
fare better than those whose shortest sides are vertical.
In
each of these subgroups, the triangle with the third vertex at
the top is recognized with greater accuracy than that with the
third vertex at the bottom.
The "000" triangle and the "090"
triangle - respectively the most and least easily recognized
figures in this group - differ in both these characteristics;
the "000" triangle has the shortest side horizontal and the
response
rate
orientation
35
000
short
top
31
180
horizontal
bottom
30
270
top
29
090
short
side
vertical
28
25
330
060
24
23
150
240
33
horizontal
or vertical
side(s)
si de
no Ode
horizontal
or vertical
location
of top or
bottom vertex
bottom
top
top
bottom
bottom
hypotenuse
top
horizontal
bottom
22
030
hypotenuse
to
19
210
vertical
bottom
acute angle at the top, whereas the "090" triangle has the
22
22
120
300
shortest side vertical and the acute angle at the bottom.
The second most accurately recognized group consists of the
triangles with no side either horizontal or vertical. Within
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this group, triangles with the acute angle uppermost fare
better than those with the acute angle at the lowest position.
The least accurately recognized group consists of
Within
triangles having a horizontal or vertical hypotenuse.
this group, one triangle is recognized less accurately than the
others (which are all identified equally accurately). This
triangle (orientation 210) has a vertical hypotenuse and the
Indeed, this triangle is
acute angle is the lowest position.
the least accurately recognized triangle of the complete set.
In conclusion, the research shows that only a few 7-8 year
old children can recognize right triangles in non-standard
orientations. Even when they have been trained in these
orientations, they can generally recognize right triangles
better in the untrained, standard orientations than in the
It appears that children's natural body-axis
trained ones.
direction (Piaget et al, 1972) over-rides any training of the
"Upward-pointing" triangles tend to be easier to
sort given.
again, children's
recognize than "downward-pointing" ones;
natural standing orientation is upward from the base.
REFERENCES
Cooper, L.A. (1975): Mental rotation of random two-dimensional shapes,
Cognitive Psychology, 7, 20-43
Cooper, L.A. and Shepard, R.N. (1973): Chronometric studies of the rotation
of mental images, in M.G. Chase (Ed.), Visual Information Processing,
Academic Press, New York, 75-175
Corballis, M.C., Zbrodoff, N.J., Shetzer, L.I. and Butler, P.B. (1978):
Decisions about identity and orientation of rotated figures, Memory and
Cognition, 6, 98-107
Eley, M.G. (1982): Identifying rotated letter-like symbols, Memory and
Cognition, 10, 25-32
Grenier, D. (1985): Middle school pupils' conceptions about reflections
according to a task of construction, Proc. Ninth Internat. Conference for
the Psychology of Mathematics Education, Noordwijkerhout, 183-188
Hart, K.M. (1981): Reflections and rotations; chapter 10 of Children's
Understanding of Mathematics: 11-16, London, Murray, 137-157
Hershkowitz, R., Bruckheimer, M. and Vinner, S. (1987): Activities with
teachers based on cognitive research, in Lindquist, M.M. and Shulte, A.P.
(Eds): Learning and Teaching Geometry K-12 (1987 NCTM Yearbook), NCTM,
Reston, 222-235
Piaget, J. et al (1974),: Die naturliche Geometrie des Kindes, Klett,
Stuttgart
The authors thank Erika Kohlmaier, Principal of Volksschule 7,
Klagenfurt, for her support and participation in the research.
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THE ROLE OF MICROWORLDS IN THE CONSTRUCTION OF
CONCEPTUAL ENTITIES
Laurie D. Edwards
University of Washington
This paper discusses the learning of a small group of middle-school children in a
mathematical domain which was new to them. Their learning is described in terms of
the construction of new "conceptual entities" which corresponded in important ways
to the mathematical entities introduced during instruction. These mathematical
entities, specifically, certain transformations of the plane (translation, rotation,
reflection and dilation), were presented to the students in the context of an interactive
computer micro world. By using linked visual and symbolic representations of the
transformations in the microworld, the students were able to build their own partial
understandings of these entities, and then go on to use them in problems-solving
activities of various kinds. In addition, the microworld provided the feedback
necessary for the students to "debug" or refine these conceptual entities so that they
became increasingly close to the correct mathematical versions of the transformations.
Introduction
In a traditional textbook-based curriculum, students are often introduced to a new
mathematical topic or domain by means of definitions and teacher-centered demonstrations. This
kind of introduction is often followed by extensive practice with the new concept or procedure, and
may culminate in actual applications or use of the concept in problem-solving. As an example,
transformation (or motion) geometry is introduced in two recent textbooks as follows:
Definition of Transformation
A transformation is a one-to-one mapping whose domain and range are the set of
all points in the plane (Bumby & Klutch, 1982, p. 440).
The motion of an elevator is called a translation. You may think of a translation as
a motion along a straight line without any turning (DeVault, Frehmeyer, Greenberg &
Bezuska, 1978, p.290).
In the research reported here, children were introduced to transformation geometry in a very
different way. Instead of presenting definitions and asking the students to apply these definitions,
the researcher began by physically modeling simple motions of the plane in a very concrete manner
(specifically, using cut-outs and transparencies on an overhead projector). She then elicited the
students' own descriptions of the motions, and introduced a simple vocabulary for naming the
transformations which was clearly related to what the students had seen and described. The
remainder of the work with transformations took place as the students interacted with a computerbased "microworld" for transformation geometry. This microworld, called TGEO, linked the
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newly-introduced vocabulary (symbolic representation) to a dynamically-changing graphic display of
the transformations (visual representation).
This paper will present the results of research carried out over a 6-week period with twelve
middle-school children who used the TGEO microworld, and will discuss the role played by the
microworld in the children's construction of new knowledge about transformations. Preliminary to
presenting the details of the study and the results of the research, I will present a brief clarification of
the terms " microworld" and "conceptual entity."
A microworld can be thought of as an embodiment of some abstract or idealized domain in a
concrete or semi-concrete form which is accessible to new learners. Papert describes the microworld
of the Logo graphics turtle as defining "a self-contained world in which certain questions are relevant
and others are not" (Papert, 1980, p. 117). Microworlds typically present multiple, linked
representations of the objects and operations in the domain. For example, in the Newtonian
microworld known as the "dynaturtle" (diSessa, 1982), the motion of a graphical turtle on the
computer screen is linked to "kicks" input from the keyboard. The turtle reacts to the kicks as if it
existed in an idealized, friction-free Newtonian universe. Thus, the laws of Newtonian dynamics
are embodied in the dynaturtle microworld; however, rather than being spelled out to the students as
explicti laws, they are left implicit, waiting to be discovered. In creating an instructional context for
using microworlds, activities must be designed which can help students to encounter the regularities
in the domain, and to construct their own understanding of these regularities. It is the thesis of this
paper that an important component in understanding a new domain lies in the construction and
refinement of conceptual entities corresponding to the idealized mathematical or scientific entities of
interest.
The term conceptual entities is introduced by Greeno (1983) in a discussion of problemsolving in mathematical and scientific domains. Greeno talks about the "ontology" of a domain, by
which he means "the entities that are available for representing problem situations" (ibid., p 277.).
Such mental "objects" are contrasted with attributes, relations, and operations which make use of the
objects. As Greeno uses the term, "conceptual entities" refers to "cognitive objects that the system
can reason about in a relatively direct way, and that are included continuously in the representation"
of a problem or situation (loc. cit.). In the context of the research described here, the "system" is
the learner, it is in the learner's mind that new conceptual entities are constructed. It is proposed here
that when learners encounter a new domain, a significant part of their learning involves building new
conceptual entities for the domain, distinguishing these entities from similar existing mental objects,
and refining their understanding of the characteristics of the conceptual entities. The thesis of the
research reported here is that a well-designed computer microworld can provide the conditions under
which students can construct and "debug" conceptual entities in a new domain.
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246
Objectives of the research
The study was both an exercise in the principled design and evaluation of a new computer
microworld for mathematics and a detailed qualitative investigation into children's learning in an
intellectual domain which was new to them. The objectives of the research were to:
(1) design and implement a microworld for transformation geometry which would be
effective in supporting students' learning in the domain; and
(2) to investigate the nature of children's learning as they interacted with the microworld.
The aim was both to build a detailed qualitative model of what the students learned, and also to
propose conceptual mechanisms which could at least partially account for the learning that occurred.
The students' learning was assessed both via quantitative measures (performance on paper
and pencil worksheets and on a final exam) and by gathering extensive qualitative data (videotape
and computer records of the students' interactions with the microworld, with the investigator, and
with each other). This combination of quantitative and qualitative measures was intended to provide
a sufficiently rich empirical base to at least begin to answer the following questions:
(1) Was the microworld and its associated curriculum of activities effective in helping the
children to construct an initial understanding of the domain?
(2) How did the microworld support this learning? What were the characteristics of the
microworld itself and the children's use of it which contributed to the students' learning in this new
mathematical domain?
Methodology
The study was carried out with twelve middle-school students, ages 11 to 14, from a private
school in Oakland, California. There were nine boys and three girls in the group; of the group, one
boy was Asian-American, one African-American, and the remaining children, Caucasian. The
students worked in pairs after school in the computer lab one hour a week for a period of six weeks.
Thus, their total exposure to the microworld was limited to about seven hours (including an initial
introduction to the microworld in a whole-class setting).
The microworld itself is illustrated in Figure 1. Three euclidean (distance-preserving)
transformations were instantiated in the microworld, called SLIDE, ROTATE/PIVOT and
REFLECT/FLIP, as well as change of scale transformations (SCALE/SIZE).
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[1,
Rotate 25 25 45
Slide 10 -20
Pivot 45
;
Reflect 20 0 45
Flip
Scale -20 -20 3
Figure 1: Transformations
The students were introduced to the transformations concretely, as described in the
introduction. They were then immediately placed in a problem-solving situation, in which they
were asked to use the transformations to play a game, called the Match game, on the computer. The
purpose of the game is to apply a sequence of transformations in order to superimpose two
congruent shapes on the screen. To succeed at the game, the students needed to understand each of
the transformations, and in order to get the best score (by using the smallest number of moves) they
also needed to compare the transformations with each other so as to find the most efficient sequence
of moves.
Thus, the initial activity of the curriculum involved the students in problem-solving iAkgi the
transformations (as contrasted with a more traditional approach of learning definitions and then
practicing procedures with paper and pencil exercises). Later activities during the study asked the
students to investigate inverses and combinations of the transformations and to use the vocabulary
of transformation geometry to describe the symmetries of geometric shapes. The overall goal of the
curriculum was to present the students with a range of increasingly-challenging contexts in which to
use the transformations. This emphasis on using a concept as the initial step in learning is
consistent with the Using-Discriminating-Generalizing-Synthesizing model proposed by Hoy les and
Noss (1987). Not only was it hoped that these active, problem-solving contexts would be
motivating for the children, but it was hypothesized that their understanding of the new
mathematical entities would be richer and more flexible if they were constructed by the students
themselves while solving problems.
24S,
238
Results
The results of the study, in brief, indicated that the students were successful in using the
microworld and the curriculum to build an initial and generally correct understanding of the euclidean
transformations, and in applying this new understanding to problems in the domain. In the written
final exam, which consisted of 12 tasks identical to those used in a British study (Hart, 1981), and
12 additional tasks, the students performed above the average for Hart's population on 10 of the 12
items. Thus, this group of students, who had a total of about seven hours of experience with the
microworld, performed at a level comparable to the students in Hart's study, who were taught the
topics of transformation geometry, as one part of their mathematics curriculum, over a period over
several years.
In addition to this quantitative measure of the students' learning in the microworld, detailed
protocol analysis was used to create a "learning paths chart" tracing the development of the students'
understanding of the transformations. The students progressively discriminated more of the
properties of each of the transformations as they worked through the curriculum, and they also
showed development in their general and specific problem-solving strategies. The portion of the
learning paths chart dealing with specific knowledge of the euclidean transformations is shown in
Figure 2.
Reflect Knowledge
Rotate Knowledge
Slide Knowledge
Slide as way to
change location of
Ruling out reflection
when sense doesn't
Rotate/Pivot as way to
change heading of shape
shape
In specific, can use
slide to superimpose
change
In specific, heading of
target shape can be read
off from screen Info.
starting vertices
Necessity of reflection
to change sense
Disambiguate
positive/negative
directions for slide
Can use 0 appropriately
for horizontal or
Disambiguate relative change
in heading from absolute
heading (used in Reflect)
Required heading for
reflection can be
calculated (JJ only)
Odd and even number
vertical slide
of reflects (Dan only)
Use of screen Info. to
accurately determine
slide amount
Rotation as composition of
slide and pivot (Rotate Bug)
I
Rotation as
whole-plane motion
Figure 2: Learning Paths Chart
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249
An example of a refinement of a conceptual entity is found in this learning paths chart, in the
progression of the students' understanding of rotation is noted. An early misconception or
alternative conceptualization of rotation (the "Rotate Bug") was found in 3 out of the 4 pilot subjects
and 2 out of the the 12 main study subjects. In this conceptual bug, students believed that the
rotation command was actually a combination of a translation and a pivot in place, rather than a
turning of the whole plane around a single fixed center point. This bug was found much less often
during the main study, when the transformations were introduced by using rotating sheets of acetate,
rather than directly on the computer (as happened with the pilot group). In both cases, however, it is
important to note that the students discovered and corrected this conceptual bug for themselves,
when they found that their expectations about how ROTATE worked were not met in the
microworld. In other words, they were able to use the visual feedback from the microworld in a
process of conceptual "debugging" during which they refined their emerging conceptual entity for
the rotation operation.
This process of constructing and refining conceptual entities is central to what makes the
microworld effective as a learning environment. I will cite only one additional example of this
process before turning to a discussion of some general characteristics of conceptual entities and
microworlds.
In addition to constructing conceptual entities corresponding to each transformation, the
students were asked to investigate new mathematical entities, including inverses, compositions and
symmetries. The development of the students' understanding of inverse is another example of the
construction of a conceptual entity, this time at a rather more mathematically-abstract level.
In the context of transformations of the plane, the inverse is the operation which "undoes"
the previous mapping or motion. Thus, for example, the inverse of SLIDE 50 30 is SLIDE -50
30, the inverse of a rotation would be a rotation in the opposite direction, and the inverse of any
reflection would be the same reflection again. The term "inverse" was not introduced to the students
until the second session, when they were asked to find the inverses for the various transformations,
and to generalize by writing them as "formulas" (for example, the inverse of SLIDE A B would be
written SLIDE -A -B). Even though inverse was not introduced explicitly in the early sessions, the
students did use inverses while playing the Match game. If they missed the target shape, a common
strategy was to invert the previous move, and re-enter a closer guess. In this case, the students were
implicitly using the concept of inverse, but they showed no signs of being aware of inverse as a
separate, identifiable conceptual entity. They did not have their own name for "undoing" operations,
and they were unaware of any general characteristics of such operations. In other words, for them,
the concept of inverse lacked both indexicality (having a name or way to refer to it) and internal
structure.
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290
When the Finding Inverses worksheet was given to the students, the opportunity was present
for them to construct "inverse" as a conceptual entity. That is, once the students had worked at
finding inverses explicitly, the idea of an "inverse" could be reasoned about directly, both in special
cases and in its generalized form. The students were also able to determine the characteristics of
inverses for each of the different euclidean transformations. They used the microworld to enter and
test their candidates for inverses, and were successful at completing the worksheet and finding
general versions for each of SLIDE, PIVOT, ROTATE, FLIP and REFLECT. In later activities,
they were also able to use inverses in specific problem-solving situations. And finally, during a
transfer task involving two new transformations, SIZE and SCALE, the children were asked to find
the inverses for each operation. They understood this task immediately, and were able to carry it out
easily (even though the inverses for size and scale were multiplicative rather than additive, as
previous inverses had been).
Without a longer-term follow-up, it is impossible to assess the robustness of the students'
construction of inverse. However, the children's work with this concept in the microworld showed
a nice developmental sequence, from an implicit and informal use of inverses in playing the Match
game, to an explicit focus on the term and its meaning for euclidean transformations, culminating in
transfer of the idea to the context of a new set of transformations, SIZE and SCALE. Thus, in
addition to constructing the individual transformations as a conceptual entities, each with its own
name, internal structure and place in the children's reasoning processes, the students were able to
construct and use an entity corresponding to an important and more mathematically-general concept,
that of inverse.
Discussion
In conclusion, I have proposed that it is through the construction of conceptual entities that
learners made sense of their new experiences in the TGEO microworld. Greeno has stated that
conceptual entities are objects about which people can reason directly, and which exist continuously
in the representation of a problem situation. I would also propose that important characteristics of
conceptual entities include indexicality, a representable internal structure, and a place in an emerging
reasoning system. Conceptual debugging, in the context of the right set of curricular activities, is the
process whereby students construct and refine conceptual entities. These entities, if they are
productive, will be:
useful in the immediate problem-solving or game-playing context;
increasingly connected to other entities in the domain;
ideally, more in line with standard mathematical entities; and
the roots for the construction of new conceptual entities at the next level of abstraction.
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The process of conceptual debugging is supported by a microworld environment because of
a number of characteristics incorporated into its design. These include multiple, linked
representations of the mathematical entities, which are presented in a way which is accessible to the
new learner at his/her current state of understanding. The use of visual feedback is particularly
effective in assisting students to "see" the differences between their emerging models of the
transformations and the correct versions embodied in the microworld. Problem-solving activities
centered around a computer microworld create a context where new entities are needed in order to
succeed. And finally, a microworld which presents a symbolic system for representing mathematical
operations and entities also provides students with a vocabulary, a way to attach names to the new
conceptual objects which emerge as they interact with the microworld.
Paths for future research in this area will investigate commonalties among microworlds in
different domains. Are the characteristics listed above essential for supporting learning in
microworlds? What are the strengths and limitations of these interactive learning environments? In
particular, an important issue concerns the difference between the kind of exploratory, inductive
reasoning which is easy to do in a microworld, and an approach to discovery in mathematics which
is more rigorous, analytic, and sensitive to the requirements of deductive proof.
References
Bumby, D. & Klutch, R. (1982). Mathematics: A topical approach, Course 1. Columbus, Ohio:
Merrill.
De Vault, M., Frehmeyer, H. Greenberg, H. & Bezuska, S. (1978). SRA Mathematics. Level 8.
Chicago: SRA.
diSessa, A. (1982). Unlearning Aristotelian physics: a study of knowledge-based learning.
Cognitive Science 6. 37-75.
Greeno, J. (1983). Conceptual entities. In Gentner, D. & Stevens, A., Mental Models. Hillsdale,
NJ: Lawrence Erlbaum Associates.
Hart, K. (1981). Children's understanding of mathematics: 11-16. London: CSMS, University of
London.
Papert, S. (1980). Mindstorms: children, computers, and powerful ideas. New York: Basic
Books.
252
242
THE COGNITIVE CHALLENGE INVOLVED IN
ESCHER'S POTATO STAMPS MICROWORLD
Rina Hadass, Oranim, University of Haifa, Israel;
Nitsa Movshovitz-Hadar, Technion, Haifa, Israel;
Yehoshafat Give'on, Beit Berl, Kfar Saba, Israel.
Abstract
This paper presents the preparatory stage of a
study of the challenges embedded in computerized
Escher-game environment. The instrument and the
background are described, and results of a small
scale pilot study of children operating in that
environment are presented.
Introduction
It is widely accepted that there are both practical and
theoretical reasons for taking an interest in geometrical
transformations. Usiskin (1974) has given a number of reasons
for adopting a transformation approach to high school
geometry. He claims that transformation approach is
especially well suited for slower students. Kiichemann (1981),
who carried, within CSMS, a study of children's understanding
of transformation geometry claimed: "The fact that the
transformations can be defined in terms of actions (folding
and turning), and their results represented in a very direct
manner by drawings, means that the topic is ideally suited to
a practical and investigative approach... in ways that are
meaningful to most children." (Ibid p. 157).
The ultimate goal of our present work is to study slower
students' intellectual functioning in an investigative
environment of geometrical transformations. This is, within
the framework of a curriculum designed for this population
(Hadass and Movshovitz-Hadar, 1989; Movshovitz-Hadar, 1989).
243
253
The Environment
In developing the investigative environment, we were
inspired by a game with engraved potato stamps invented in
1942 by the mathematician-artist M.C. Escher (Ernst, 1976).
Escher's son, George, described a simplified version of this
game, with which his father used to entertain him in winter
evenings (Coxeter et als, 1986). We developed a computerized
version of that game.
In our software, the potato stamps are replaced by
squares. Each "stamp" has a different "engraved" pattern, and
the various stamps are of such design, that when they are
placed side by side, the lines connect to each other, and
thus form beautiful "carpets". Twenty different stamps are
stored in the computer memory (see appendix). Each can be
called to the screen by a numerical code. The user can take a
look at all the twenty stamps at one time, or see each of
them separately.
The software enables a variety of activities:
(a) Changing the basic stamps by transformations of rotations
and reflections. E.g. by entering "4R" we get stamp
number 4 rotated by 90° to the right:
4
254
4R
4RR
244
4RRR
(b) Generating a carpet from a selected 4-tuple of the
stamps,
or their rotations and reflections, according to
the user's choice. E.g.
1
8
7
13
(c) Detecting the basic stamps in a carpet, created randomly
by the computer. E.g. Which 4-tuple of stamps generates
the following carpet?
245
255
Operating the software can stimulate an investigation of
many mathematical issues. For instance:
Which basic stamps won't change under rotations or
reflections and why?
How many different stamps can be created from a given
basic stamp by rotations (of 900, 1800 and 2700)
and by reflections?
What is the influence of symmetry on the number of
distinct stamps obtained by rotations or reflections?
How many different carpets can be created from four basic
stamps by changing their order?
How does the symmetry of the basic stamps affect the
number of distinct carpets? The symmetry of the carpet
obtained?
In addition, the software has the potential to introduce to
students some basic notions of computer literacy.
Preliminary Findings
We confine this report to findings obtained during the
first exposure of nine children aged 11-17 to the software.
Each child was given about 30 minutes to operate the
software, in the presence of one researcher. All nine
sessions were recorded. The reactions were then analyzed
according to children's expressions, which can be attributed
to intellectual functions. We bring here a few representative
examples:
(1) Questioning the flexibility of the environment:
(17 year old), after looking the first time at the twenty
basic stamps followed by a demonstration of a carpet created
from 4 of them, by the researcher, asked:
Is it possible to create new stamps?
D.
Other children asked:
Can I use more than four basic stamps to create a carpet?
Is it possible to enlarge a given basic stamp?
G.
(12 year old) commented also:
It would be interesting if each stamp had a different
sound.
It would be nice if one could create additional stamps on
the given ones or inside them.
256
246
A.
-
(13 year old) suggested:
It's worthwhile adding a colour to each stamp or to the
whole carpet.
(2) Investigational behavior:
(13 year old) tried a carpet out of the basic stamps
6,7.6,7, and said:
If we take 6,6,7,7, it will give another carpet, I guess.
K.
She then confirmed it on the screen. Then she tested the
differences between 4,4,7,7 and 4,7,4,7. She tried additional
carpets and said:
I am looking for non-symmetrical shapes. On the other
hand, I think that with symmetrical stamps maybe the
carpet will look nicer. That's why I keep trying.
(3) Concrete Observations:
After demonstration of a reflection of stamps 7 and 12,
T (11
Q:
A:
Q:
A:
Q:
A:
Q:
A:
year old) was- asked:
What happens to stamp 15 by reflection?
You won't be able to notice any change, because the stamp
is the same from all sides.
And if I rotate stamp 15 by 90 0?
You won't be able to notice anything either, because the
stamp is the same from all sides.
In which other 8tamps you won't notice a difference in
rotation (by 90.)?
In stamps 2 and 20.
What about stamp 1?
In reflection you won't see a difference, but in rotation
(by 90
you will see a difference, as it's long here
and short there (pointing at the right places in the
stamp, respectively).
)
(4) Generalization:
(12 year old) said:
Symmetric stamps remain the same under rotation and
reflection.
O.
A.
(13 year old) said:
I know why each stamp can be connected to another one:
it's because each side of the square is divided into
three equal parts, and the connecting lines start out
from two fixed points.
(5) Attitudes:
One girl, who seemed to be very practically oriented, said at
the end:
It's fun, but it lacks a defined aim.
After trying to discover the stamps from which a carpet was
created, G. (12 year old) said:
It's interesting, but difficult because the stamps
interlock. If I spent one day playing with it, I would
know whatever there is in it.
A.
(13 year old) said:
You can play with it as much as you like. There are 1001
possibilities.
247
257
K.
(13 year old) said:
You can make greeting cards out of it.
You should write which combinations you prefer, so that
you will be able to return to them, whenever you wish.
D.
(17 year old):
It's beautiful for textile designing.
Summary
Our preliminary observations indicate that Escherinspired computer environment, we created, provides fertile
ground for a variety of intellectual activities. Moreover, it
can
make the topic of transformation geometry enjoyable,
thus following Lesh (1976), giving a response to critics who
charge that laboratory activities tend to "make fun topics
important rather than making important topics fun".
The learning experience in this micro-world is different
from the routine learning in school, and its advantages and
limitations should be checked for the use of populations of
students having different qualifications. It would be
especially interesting to check whether low achievers, having
a history of failure in school, can profit from this
software. This study is in progress now.
References
Coxeter, H.S.M., Emmer, M., Penrose, R., and Teuber, M.L.
(Eds.), (1986), M.C. Escher: Art and Science, NorthHolland, Elsevier Science Publishing Comp. Inc., N.Y.,
PP. 9-11.
Ernst, B.
(1976), The Magic Mirror of M.C. Escher,
Ballantine, New York, p. 40.
Hadass, R. and Movshovitz-Hadar, N. (1989), "Low, ,Mathematics
Achievers' Test Anxiety" in Vergnaud G. et al's. (Eds.),
Proceedings of the 13th International Conference of PME,
Paris, Vol. 2, pp. 39-44.
258
248
Kilchemann,
D.
Hart, K.M.
(1981), "Reflections and Rotations" in
(Ed.), 'Children's Understanding of
Mathematics: 11-16, John Murray Publishers, London,
pp. 137-157.
Lesh, R.
(1976), "Transformation Geometry in Elementary
School: Some Research Issues" in Martin, L. (Ed.), Space
and Geometry, Eric/Smeac, Ohio, pp. 185-243.
Movshovitz-Hadar, N. (1989), "Mass-Mathics" in Vinner S. at
als. (Eds.), Proceedings of the 2nd International
Jerusalem Convention on Education, p. A-51.
Usiskin, Z.P. (1974), "The Case for Transformations in School
Geometry", Texas Mathematics Teacher.
Appendix
The 20 Basic Stamps
L
hr
16
%4
ri
17
J
r
18
19
20
11
12
13
14
15
6
7
8
9
10
1
2
4
249
259
A STUDY OF THE DEGREE OF ACQUISITION OF THE VAN HIELE LEVELS IN
SECONDARY SCHOOL STUDENTS *
Ade la JAIME and AucteLELIERBEZ.
Depto. de Didactica de la Matematica. Universidad de Valencia (Spain)
Abstract
In this report we describe a method for assessing the degree of
acquisition of every Van Hie le level of thinking by the students Our core
assumption is that for an accurate assessment of the students' level of thinking
it is necessary to observe their way of using every thinking level.
We have administered a test on plane Geometry (namely on polygons) to a
group of Secondary School students The test has been analyzed according to the
mentioned method and we discuss the results This way of working allows us to
recognize different interesting students' behaviors
The knowledge of the students' level of reasoning plays an important role
in most research carried out on the Van Hiele model, as it provides a way for
checking the theoretical hypothesis of researchers. It seems, therefore,
Important to define a method of evaluation which gives a good idea about the
students' thinking level.
In this work, we have considered the Van Hiele levels to 4, and we have
excluded level 5. You can find a detailed description of the Van Hiele levels in
several of the main references; see Gutierrez, Jaime (1989) for a complete
compilation of this references.
1
The existing literature about the Van Hiele model shows that, up to now,
the methods of evaluating a student's thinking have resulted in the allocation of
one level to the student. This has some problems, as there are students whose
answers reflect the presence of various levels. In this paper we present a way
of evaluation of the Van Hiele levels which considers such situation.
* This paper is part of a research project funded by the "Concurso Nacional de
Proyectos de InvestigaciOn Educativa" (1989) of the Spanish Ministry of
Education and Science.
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251
There are two main points to consider:
1) The student's answers to several activities often reveal different Van
Hie le levels of reasoning. This probably means that the acquisition of the levels
Is not absolutely linear (as stated in the theoretical descriptions of the Van
Hie le model) but that the student is making progress within more than one level.
We do not reject the hypothesis of the hierarchical structure of the Van Hie le
level, but we propose to consider it in a wider meaning.
2) The acquisition of a thinking level by a student does not happen
suddenly, but progressively. This progress can be recognized by the way how the
student uses the thinking types specific to the level, from an initial period of
lacking of awareness of the abilities of the level (no acquisition of the level of
reasoning) to a complete mastery of the corresponding way of thinking
(complete acquisition of the level), with several intermediate behavior patterns
easily recognizable in the student's answers to problems. Of course, this comes
in support of the hypothesis of continuity of the Van Hie le levels.
This progress towards the acquisition of the level is considered in our
method of evaluation by means of the determination of a "degree of
acquisition" of this level by the student. Therefore, the evaluation of a
student's reasoning results in four values which reflect the student's degree of
use of each Van Hie le level of reasoning. If we quantify the process of
acquisition of a level of thinking, by representing it as a graduate segment from
0% to 100%, figure
shows the various periods of the progress through the
1
segment divisions (acquisition of one level) that we have identified. The
specific values of the partition are subjective and can be modified according to
the researcher's point of view.
Low
No
Intermediate
High
Complete
acquisit. acquisition acquisition acquisition acquisit.
0%
15%
40%
60%
85%
100%
Figure I
In order to determine a student's degree of acquisition of the Van Hie le
levels, first we have to determine the Van Hie le level of each student's
answer (levels
to 4). But the completeness of the answers and their
1
252
mathematical accuracy should be taken also into account. We concrete these
aspects of the answers by assigning each one of them to one of eight types of
answer. To determine to which type an answer belongs, it is necessary to
consider it from the point of view of the Van Hie le level it reflects; that is, the
answer could be correct according to the requirements of a level, but incorrect
according to the requirements of a higher level. Any answer to an open-ended
item may be assigned to one of the following types:
Type 0: No reply or answers which cannot be codified.
Type 1: Answers which indicate that the learner has not attained a given level
but which give no information about any lower level
Type 2: Wrong and insufficiently worked out answers which contain incorrect
and very reduced reasoning explanations but give some indication of a
given level of reasoning
Type 3: Correct but insufficiently worked out answers which contain very few
explanations or very incomplete results but give some indication of a
given level of reasoning
Type 4: Correct or incorrect answers which clearly reflect characteristic
features of two consecutive levels of reasoning.
Type 5: Incorrect answers which clearly reflect a level of reasoning.
Type 6: Correct answers which clearly reflect a level of reasoning, but which
are incomplete or insufficiently justified.
Type 7: Correct, complete, and sufficiently justified answers which clearly
reflect a level of reasoning.
Conseguently, these types of answer may reflect the various periods of
acquisition of the Van Hiele levels of thinking represented in figure 1: Types 0
indicate no acquisition; types 2 and 3 indicate low acquisition of the
level; type 4 indicates an intermediate acquisition; types 5 and 6 indicate a
and
I
Nab acquisition; and type 7 Indicates a complete acquisition.
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2832
253
is the level
Thus, we assign a vector OM to each answer, where
reflected by the answer and t is the type of answer (the component I is empty
when t = 0). By weighting the types of answer t in terms of the percentage of
1
acquisition of the reflected level of reasoning (from 0% to 100%) and by
considering the vectors (1,t) of all the questions which could have been
answered at each level, we obtain the student's degree of acquisition of the Van
Hie le levels.
Application of the method of evaluation to a specific test
As a part of an ongoing research project aiming the evaluation of the Van
Hie le levels of thinking of students in Primary and Secondary Schools, we have
administered a test on polygons to a group of secondary school students, and we
have determined their Van Hie le levels with the method that we have just
described.
Sample: The test was administered to 19 secondary school students (aged
15-16) in a Spanish Professional Training School.
The test: It was a paper and pencil test which consisted of nine open-ended
items. Each sheet contained one item and they had a lot of blank space;
furthermore the statement of the items encouraged the students to explain
their answers. The questions dealt with several plane geometry topics:
Triangles, quadrilaterals and polygons in general. Each student had a ruler and a
protractor.
Items 1 and 2 are intended to identify specific sorts of figures: Regular,
irregular, concave, and convex polygons in item 1; square, rhombus, and
rectangle in item 2. The students were presented several figures and they had
to identify them; they were also asked several questions aimed to know their
ways of identification. Item 2 had also several questions about classification
of figures, like: "Write whether there are quadrilaterals being rhombi but not
squares. Justify your answer".
Item 3 began with formal definitions for "square" and "rectangle" that had
to be used to answer to questions of identification and classification similar to
those in item 2.
263
254
items 4 and 5 were based on the definition of a polygon (not a known
polygon) called ANLA. Item 4 was similar to item 1, and item 5 consisted of
questions about classification of ANLAs and other kinds of polygons.
In item 6 students were given a list of properties, and they had to select
all those which were true in an obtuse triangle; they had also to select two
minimal sets of conditions which enabled to define an obtuse triangle.
Items 7 and 8 were based on the sum of the angles of a triangle. In item 7
students were asked to prove that property for an acute triangle; students were
provided with several hints so as to help them to write the proof. In item 8
students were given a complete proof for acute triangles and they were asked
to prove the property for right and obtuse triangles.
In item 9 students were asked to prove that the diagonals of a rectangle
have the same length and that the diagonals of a rhombus bisect and are
perpendicular.
We do not evaluate the statement of the items (as done by Usiskin (1982)
and Mayberry (1983)) but the students' answers (as done by Burger, Shaughnessy
(1986) and Fuys et al. (1988)); then each item was assigned to a range of levels
where it could be answered by the students (table 1). Fro'rn our prior knowledge
of the students, we suspected that most of them would have level 2 or perhaps
levels. This
assignation was first made by the researchers, and later it was improved by
pilot testing.
level 3; therefore most items were intended to cover these
Item
Levels
Item
Levels
Item
I
1,2
4
2,3
7
2
1, 2, 3
5
8
3
2, 3
6
2, 3
2, 3
Levels
2, 3, 4
2, 3
9
2, 3, 4
Table 1. Range of levels where the items can be answered.
The administration of the test: It took place in two sessions as a part of
the class of mathematics. The students were allowed to take as long as they
needed to answers the questions (time ranged from 25 to 45 minutes per
session).
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255
264
The evaluation of the test: We have assessed the students' level of
reasoning by applying the twofold method described above (Van Hie le level and
type of answers). First each researcher has assigned levels and types
separately, and later we have compared our assignations, looking for a
consensus when they were different.
To obtain the degree of acquisition of a given Van Hie le level, we weighed
the student's answers to all the items which could have been answered in that
level, according to the range of levels shown in table 1; that is, for level 1 we
have considered items 1 and 2, for level 2 all the items, for level 3 items from
2 to 9, and for level 4 items 7 and 9.
Analysis of the results: Here we are not trying to generalize the results
from the point of view of the general reasoning of the students, as the sample
is restricted to one group of one High School. Our aim is rather to show how our
proposal of considering the degree of acquisition of each Van Hie le level results
in a more detailed information about the development of the students' reasoning
than the classical assignation of one level of reasoning to the students.
The different patterns of acquisition of the Van Hie le levels obtained from
the sample are shown in table 2, which depicts the acquisition of each level
according to figure 1. Figure 2 shows graphs of the degrees of acquisition of the
levels by students of groups A to F.
Level 1
Complete
Level 2
Intermed.
Low
D
Complete
Complete
Intermed.
E
Intermed.
F
Low
A
B
C
Level 3
Low
Level 4
4' of stud.
3
9
2
Low
2
1
2
Table 2. Number of students and their acquisition of each Van Hie le level.
285
256
100
100
100
85
85
85
0 60
EI) 60
60
40
40
40
15
15
15
0
0
2
3
0
4
2
Levels
3
4
A
B
100
100
85
85
g, 60
60
RI' 60
40
40
40
15
15
15
0
0
4
4
C
85
3
3
Levels
100
2
2
1
Levels
0
1
2
Levels
3
4
2
1
Levels
D
3
4
Levels
E
F
Figure 2. Patterns of the students' degrees of acquisition of the levels.
The classical assignation of a single Van Hiele level of reasoning would
probably have resulted in the assignation of level to the students in groups A
1
to C and no level (or level 0) to the students in groups D to F; or, perhaps, it
would have resulted in the assignation of level 2 to the students in group A,
level
to the students in groups B to E, and no level to the students in group F.
Anyway, it is clear that such kind of assignation implies an oversimplified view
of the students' thinking abilities.
1
On the contrary, table 2 and figure 2 provide an evidence of important
differences among the students: Some of them (group F) need strong instruction
directed to the attainment of level 1, whereas other students (groups D and E)
only need a reinforcement to complete the acquisition of level I; the rest of the
students (groups A, B, and C) has completely acquired level
and they need
instruction for attaining level 2, some from the very beginning (groups B and C)
and some from a more advanced point (group A).
1
257
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266-
Another interesting point that can be observed in table 2 is, referring to
groups A and D, that a significant number of students begins the acquisition of
the abilities of a thinking level before they have completely acquired the
previous one. This Is a situation certainly caused by the curriculum of
mathematics and the way students have been taught, which has to be identified
and investigated.
In short, this kind of assessment of the Van Hie le levels provides an
accurate and detailed picture of the current students' thinking abilities.
References
Burger, W.F.; Shaughnessy, J.M. (1986): Characterizing the Van Hie le levels of
development in Geometry, Journal for Research in Mathematics Education
vol. 17, pp. 31-48.
Fuys, D.; Geddes, D.; Tisch ler, R. (1988): The Van Hie le model of thinking In
Geometry among adolescents (Journal for Research in Mathematics
Education Monograph no. 3). (N.C.T.M.: USA).
Gutierrez, A.; Jaime, A. (1987): Estudio de las caracteristicas de los niveles de
Van Hie le [Study of the characteristics of the Van Hie le levels),
in
Proceedings of the 11th International Conference of the P.M E vol. 3, pp.
131-137.
Gutierrez, A.; Jaime, A. (1989): Bibliografia sobre el modelo de razonamiento
geometrico de Van Hiele, Ensenanza de las Ciencias vol. 7, pp. 89-95
Mayberry, J. (1983): The Van Hiele levels of Geometric thought in undergraduate
pre-service teachers, Journal for Research in Mathematics Education vol.
14, pp. 58-69.
Usiskin, Z. (1982): Van Hiele levels and achievement in secondary school
Geometry. (ERIC: USA).
26.7
258
Spatial Concepts in the Kalahari
by Hilda Lea
University of Botswana.
Abstract.
Hunters and herdsmen in the Kalahari, who have never been
to school and who have lived in very remote areas all their
lives, were interviewed on two occasions to ascertain how far
their spatial concepts have developed. When asked how they
recognised animal footprints, and how they found their way in
the desert, they were seen to have a very good visual memory,
and to be aware of the minutest detail in recognising shapes.
When given a visual thinking test, they performed with a high
degree of skill on items related to their environment.
Introduction.
Before Independence in Botswana, people's lives were largely untouched
by the technological world, and many of those in remote areas of the
Kalahari have been living all their lives in the same way and in the same
environment as generations before them. It was therefore felt to be
useful to investigate how far their spatial concepts have developed, as a
result of the interaction with their particular environment.
Hunting in Botswana is carefully controlled, though Batswana can hunt
non protected animals for food in areas where they live, during specific
periods..
Seventy per cent of the national herd grazes in the Kalahari. Cattle do not
normally stay near villages but are kept at cattle posts which are usually
a long way off. Herdsmen spend their time in the sandvelt looking after
cattle belonging to other people.
University students interviewed hunters and herdsmen known to them,
at cattle posts in the Kalahari.
Spatial ability is a complex set of interlocking skills. Good visual memory
requires an ability to retain, recall and manipulate information
concerning shapes and spatial relationships. Visualisation depends on the
degree to which the perception, retention and recognition of the
configuration is seen as an organised whole. Orientation is an ability to
manipulate a shape, to transform it mentally by moving or enlarging it or
seeing it from a different point of view.
Skills include aspects of distance, direction, perception, movement, and
relationship of part to whole and objects to each other.
259
268
The Commonwealth Secretariat (1970) commissioned a review of
research in different countries relating to difficulties students face in
pictorial perception, in various cultural settings. Literature was reviewed
on the subject and case studies discussed regarding the acquisition of
particular skills. Eskimos were shown to have a high level of spatial
ability (Berry 1966, 1974). In Papua New Guinea students from rural
backgrounds were shown to have a highly developed visual memory
(Bishop 1977). South Pacific studies on the navigation skills required
when travelling by canoe among the islands, showed a highly developed
sense of direction. Gladwin (1964) analysed the navigation skills of
Trukese adults, which showed a concrete level of thinking. Navigation is
by the stars, wind direction and wave patterns, and on a dark night by
the sound of the waves and the feel of the boat. The Trukese knows
where he is in relation to every island though cannot give a verbal
account.
Lewis (1972) identified mental mapping in the orientation behaviour of
Aborigines in Australia in finding the way. They seemed to have a
dynamic mental map which was constantly updated in terms of time,
distance and bearing, and realigned at each change of direction. The
Aborigine also seemed to have Treamings* related in some way to paths
which aiss crossed the land, which ancestors had followed. They had
great acuity of perception of natural signs and an ability to interpret
them, and almost total recall of every topological feature of any country
they had ever crossed.
ResearCh shows that each society develops its own way of understanding
and adapting to the environment. Different groups do not necessarily
follow the same development path, since their particular goals and
requirements are different. Visual memory is seen to be highly
developed in many pre technological societies.
In Botswana, work has already been done on informal mathematics (Lea
1990), looking at mathematical activities in traditional daily life, and in
the way of life of the Bushmen. This study looks at spatial abilities in the
Kalahari to identify spatial skills acquired in the daily life of hunters and
herdsmen, and to see how far these skills can be transferred to more
structured situations.
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260
fjcperiment 1.
Method.
During the Christmas vacation 1988, students carried out interviews with
hunters and herdsmen in the Kalahari outside the game reserves, to
ascertain a) how animal footprints are recognised b) how people find
their way in the desert and c) if prints can be recognised when on paper,
42 subjects were interviewed of whom 26 were Bushmen.
Results.
a) Though people live in an area where there are large herds of wild
animals, it is still quite difficult to find them and to track them.
The hunters explained with words, sand drawings and hand gestures
how they know the difference between footprints. They said they
sized them with their eyes and looked for distinguishing features
such as presence of claw marks, distance between front and back
prints, distance between toes and paw, distance between the two
parts of a hoof, the depth and overall structure of the footprint.
Prints of hyenas and jackals are similar though differ in size. Their
claws mark the ground but jackals' claws dig a little deeper.
Leopards and lions have similar marks and do not show their claws.
The leopard leaves tiny fur marks because its claws are hidden in
their sheaths and covered with fur, and a lion's footprints are
preceeded by a mark of fur as it tends to drag its paw. The general
shape of the leopard's footprint is more circular than the lion's.
loved animals have similar prints as all have sharp pointed hooves
except wildebeest and buffalo. Zebra prints are like a donkey's only
larger. Buffalo and cows make similar marks but the hooves of the
buffalo have an opening in the middle and make deeper marks. The
four footprints of one animal never show the same mark.
They commented 'Point to any spoor on the ground whether old or
new, and the answer will be certain'.
b) The Kalahari is a vast and desolate area, and those unaccustomed to
the desert would find it apparently featureless. This is not so for the
people who live there. Important features to be noted are particular
trees, particular vegetation and vegetation under trees, and if
travelling in unknown territory, these must be remembered in the
correct order. The sun, shadow and direction of a breeze can help
also.
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270
Those interviewed said they would not get lost nor would they lose
the track of animals they were trailing. Some said that if really
lost, they would go to sleep and in the morning when the mind
was refreshed they would know the way. Some said that a good
method of finding the way back by donkey was to ride it without
controlling it, and it would retrace the path followed earlier. Some
said they would wait till night till the donkeys cry. Others said
that if they were really lost when walking, they would look for a
very straight tree with few branches, climb up and go to sleep, and
in the morning the tree would tell them which way to go. Another
said that when lost, shout "Beee.ee.ee.' and if anyone hears he will
come. Of those interviewed all had been to unfamiliar places and
no one had ever been lost. One said 'It may be easy to get lost in
a city or village, but not out there, not in the wilderness'.
c) Pictures of footprints were shown in three forms -- on sand coloured
paper, solid black prints on white paper, and black outline on white
paper. In each case there was no problem and all were identified.
There was some argument over tiny detail for example that one
print should be more pointed than another, or more curved at the
edges.
Discussion.
In any society, abilities best developed are those necessary for a way of
life. Whilst most of those interviewed have little use for the printed
word or picture, they were nevertheless able to recognise footprints
irrespective of the context.
Mental mapping seems to be used in finding the way. Instead of
referring to a map on paper at a particular time, positions and
orientations are carried in the head, and these are realigned after every
change of direction.
In Piagetian terms it would seem that thinking is at the concrete
operations level, having no need to move to abstract thinking. In the
concrete - iconic - symbolic mode of intellectual development, thinking is
at the iconic stage because there is no need to move to the symbolic
mode. The level reached is determined by the need of culture and
environment.
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271
Experiment 2.
Method.
During the Christmas vacation in 1989, students gave the "X test on visual
thinking' to 70 herdsmen in the Kalahari, from 8 different regions. The
test contained 38 items requiring recognition and manipulation of shapes.
Items 1 12 were chosen as having some relationship to the immediate
environment, comparing lengths, tracing paths, following mazes,
unravelling knots, and identifying right and left hands: Items 13 - 16
required the identification of animals from composite pictures from
'Signs of the wild'. Items 21 38 were more formal and were taken
from Dale Seymour Set B visual thinking cards. Concepts tested were
congruence, direction, geometric shapes, magnitude, part whole
relationships, patterns, similarity, and rotation and position. All items
gained 5 marks.
Discussion.
Performance gave an overall of 56%. Questions related to the
environment, Nos 3 - 18, 22 and 25 averaged 64%. Other questions which
were more structured, many not having been encountered before,
averaged 51%. This suggests that the subjects had a very good visual
memory, and had acquired skills which were transferred to new
situations.
As topological shapes do not have rules regarding length, number of sides
and size of angles, they can probably be identified more easily, such as 32
simple figures, 29 key, and 25 irregular piece. 36 jig saw piece was more
difficult as it had to be mentally rotated.
Knots questions were easy because a common activity is setting snares.
21 chain question would seem to be related, but this was not a high score.
22 arrows was done well though 35 arrows was not as many of the
subjects considered the arrows or the spots, but not both together.
7 and8 hands were easily identified though 9 and 10 to identify the odd
one out was more difficult.
3 and 4 were following pathways by eye when ten paths intertwined. 23
faces would seem to have some similarity to footprint identification, but
scores were not high. Perhaps the instructions were confusing.
20 embedded figures was well done.
18 orientation had quite a good score.
38 did not have a very high score, because unless turns were exactly
through ninety degrees the cumulative effect of small errors gave a
wrong direction.
BEST COPY AVAILABLE
263
272
Item
.1,
12
82%
11,-.'--
MI
)/..-4
Item
21
30%
#
74%
e&:43
From 8 find 2 the same.
Will the knot pull tight.
22
(-.
34%
From 12 find 3 same &
direction.
From 15 find 2 exactly alike.
32
23
78 %
324:
444
From 12 list pairs same
From 4 find piece like A
shape; size.
29
25
72%
56%
From 6 choose 2 alike.
From 5 find piece like A.
20
33
From 7 same shape
different size.
ila els
24
MI,
? If
irg
I
86%
fts
0
36%
From 6 find 2 alike.
A
18
60%
36
34%
32%
Face N turn L L R...
End up.
(_(:
4/-
From 6 choose correct piece.
saz
From 6 choose piece
to complete.
273
......6
Identify slopes of A & B.
38
38%
V
0
30
From 8 find 2 alike.
28
pzi )
.Ii
From 5 embedded figure
like Y.
From 12 find 2 alike.
26%
fi
78% U
26%
''
,
0
From 5 choose piece to
complete.
264
27 completing a lino pattern gained a higher score than 28 completing a
zebra type pattern.
34 matching triangles gained a high score, but 33 to find triangles of the
same shape and different size, was difficult.
13 - 17 animal pictures were enjoyed though not done particularly well.
Two of these were composite pictures of animals in a group, where it was
necessary to identify them in a holistic way.
Shapes with two variables were easily compared, though there was
greater difficulty with three variables. With more than this, some sort of
classification is needed to compare in a systematic way. Those
interviewing said this was a problem with 31 where there were different
arrangements of curved outline, circle, octagon and crossed lines. It was
surprising that 24 had a high score as there were twelve, pieces to choose
from. These were coloured and this seemed to have made a big
difference.
Items made use of spatial or visual imagery, and required the perception
and retention of visual forms, and / or the mental manipulation of
shapes, as well as a skill in making logical comparisons.
Summary.
This paper attempts to show a relationship between cultural and
ecological characteristics of a particular group of people, and the
perceptual skills developed by that society. As in other cultures, certain
perceptual skills must be developed for survival. The similarity to visual
thinking of the Aborigines, Eskimos and others supports the idea that the
development of perceptual skills is embedded in the individual's total
environmental and cultural context.
It is clear that those who live in remote areas have highly developed
spatial skills necessary for their way of life. In trying to measure the
nature of these abilities, it would appear that they have excellent visual
memory in identifying footprints, and the context in which these are
presented is not important. They have a good sense of position and
direction in their environment, and have a mental map which they can
update quite easily. Performance was good on the visual thinking test,
and it would be interesting to compare results on the same test given to
secondary school pupils.
An abstract analytical way of thinking may be considered to be better in
a technological society but not in a non technological society where visual
thinking is very necessary. People tend to assume that those who have
been to school are more intelligent, but this is not necessarily so. The
type of intelligence differs and so does the experience.
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274
Acknowledgements.
Thanks are expressed to the University of Botswana for funding this
project. Thanks are also given to Dale Seymour Publications for kindly
allowing their Visual Thinking cards to be used, and to Mr C. Walker for
giving permission for picture from 'Signs of the Wild' to be reproduced.
It would not have been possible to carry out this research without the
help of students. Thanks are due to Mr Luckson Mabona, Mosiemamang,
Namane, Petso, Kaisara, Sethapelo, Tinye and Tseleng who interviewed
people known to them.
References.
Berry J. Temne & Eskimo Perceptual Skills International Journal of
Psychology Vol 1 No 3 207-229 1966.
Berry J. Ecological & Cultural Factors in Spatial Perceptual Development
Canadian Journal of Behavioural Sciences Vol 3 No 4 1971.
Bruner J. Beyond the Information Given Allen & Unwin 1971.
Lea H. Informal Mathematics in Botswana Proceedings International
Commission on the Study and Improvement of Mathematics Teaching
1990.
Lewis D. Route Finding by Desert Aborigines in Australia The Journal of
Navigation Vol 29 No 1 21-39 1976.
Mac Arthur R. Some Ability Patterns - Central Eskimos & Nsenga Africans
International Journal of Psychology Vol 8 No 4 239-247 1973.
Seymour D. Visual Thinking Cards Set B Dale Seymour Publications 1983.
Walker C. Signs of the Wild Struik Publishing Company 1987.
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266
INTEGRATING LOGO IN THE REGULAR MATHEMATICS'
CURRICULUM. A DEVELOPMENTAL RISK OR OPPORTUNITY?
Tamara Lemerise, Ph.D.
Departement de Psychologie
Universith du Quebec a Montreal
Summary
Logo is at a turning point th its history Certain tyucial chokes must be made
in order to assure the development if not the survival of Logo in the current
educational system. On the 0/119 hates Logo still needs to penetrate more
deeply into the educational milieu. maftiplvkg its agents and its
contributiOns On the other hand certain fflAaSZifeS must be taken to aSS111-e
that the fundamental link between Logo and mathematics he maintained
ThiS rep.rt examines the extent to Whie:h the current trend favoring the
integration of Logo into the school curriculum responds to these .11e&S.
Theoritkal considerations ba..grld on experimental data are presented
The adolescent period of Logo
It has now been nearly fifteen years that Logo has been known and used in
primary schools for the purpose of creating an educational context that
favors the development of mathematical thinking. Although it would be an
exaggeration to say that Logo is currently undergoing an "adolescent crisis",
it would not be farfetched to suggest, given the fundamental questions that
one encounters nowadays, that Logo is very much in search of its identity.
Among the many questions that have arisen, three in particular stand out.
(1) What is going to be the impact of Logo on the educational system?
Will it be seen as the trigger that set off a revolution or as a factor of
evolution in the educational system at large and especially in the field of
mathematics?
(2) What is the connection between Logo and school mathematics? Is
Logo a_new_way. of _doing_mathematis:s to the exclusion of the old? a
mathematical alternative that is complementary to the traditional one? or a.
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lifebuoy_ that has come to the rescue of a sinking traditionnal mathematics
curriculum?
(3) Is the growing variety of ways of using Logo, which are actually
observed in the schools, a sign of development and enrichment for Logo? a
mere survival measure? or the premonition sign of an imminent death?
The impact of Logo: revolution or evolution?
It is by now obvious that the initial goal of revolutionizing the educational
milieu will not be achieved in a straightforward manner. It is more likely
that any revolution that actually took place will be consolidated through the
slow process of evolution: through successive waves of pedagogical changes
that, step by step, embrace an ever increasing pool of active agents and
significant educational topics. What data are available that support the
hypothesis that Logo is a factor of evolution in the educational system?
On the one hand, there is clear evidence that Logo already went and is still
going through important developments. Logo-Writer and Lego-Logo, to
name but two recent innovations, are arms newly developed by Logo to help
it reach out in the educational environment (Weir 1987). So in the last
decade, not only have we witnessed the continuous technical sophistication
of the early Logo but, more importantly, we have also seen Logo linked to
different kinds of abilities that were not initially easely accessible to it:
writing, mechanical and physical abilities with both theoretical and practical
applications (diSessa, 1982, Weir, 1987, Weir, in press.). With this trend
Logo tends to be more and more multidisciplinary slowly infiltrating its way
into new fields after its initial start in programming and mathematics.
Parallel to this phenomenon of outward expansion, another kind of expansion
can be observed, a more inward and subtle phenomenon, namely the
development of cleaner and clearer connections with the mathematical
universe. Such well-known authors as Hoy les, Noss (1987) Hi llel, Kieran
&
Gurtner (1989) and Gurtner (in press) have recently underscored the
pernicious possibility of children doing Logo without ever really getting in
touch with mathematical entities or mathematizing the solution. There is a
current today that favors a tighter and surer link to mathematical thinking.
It is no longer enough to make loose associations between Logo and problem
solving abilities or turtle geometry; special care must now be taken to assure
direct and solid connections with authentic 122.-71VitvziatAul types of solutions.
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In its quest for rapprochement with the mathematical domain, Logo finds
itself more and more in contact with elements that are already covered by
the traditionnal mathematics' curriculum . A review of the literature points
to an increase in the nature and in the frequency of the interaction. The
extent to which these new liaisons with traditionnal mathematics can be
beneficial to Logo or could constitute dangerous liaisons -- is an important
question that need to be clarified in a general discussion of Logo's role in the
evolution of the educational process.
The relation between Logo and the traditionnal mathematics'
curriculum: a risk or an opportunity?
Simply stated, the risk of making tighter and tighter links between Logo and
traditional mathematics is that Logo could be gobbled up by the traditionnal
approach. This is often called the recuperation phenomenon. The old system
annihilates the innovative approach by slowly adapting it to its own. Logo
would then be treated, for example, as one exercice among others, an
element of the curriculum mechanically "covered" by the teacher, which is
what often happens to other mathematical topics. More tragically, the Logo
spirit and philosophy could be muzzled for many years to come. Were Logo
to be so ensnared, all hope would be lost for Logo as an active agent of
change in the learning and teaching of mathematical thinking.
On the other hand, the opportunity that arises from forging tighter links to
the traditional curriculum has to do with some Logo's contemporary needs.
How, for example, could the insertion of Logo into the mathematics
curriculum foster a real mathematical spirit and context when doing Logo.
How could it favor the evolutionary role of Logo in contemporary education.
Let us examine the opportunity and how the risks might be minimized.
The link to the nmtliem3tics' curriculum. Away o f mattrematiIng litro
As stated above, Logo does at times encounter difficulties in bringing
children to think mathematically. Gurtner (in press) uses the metaphor of a
tunnel to express how characteristics of Logo situations sometimes make
"students miss nice view-points on mathematic and geometry"; and he asks
that windows be opened in the Logo tunnels in order for children to have a
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perspective on related realities while working on specific Logo tasks. At the
same time, Gurtner notes the need for bridges that permit students to go
back and forth actively between Logo actions and basic mathematical
principles, laws or notions. In order to avoid progressive isolation, Logo
needs to be consolidated and enriched by significant links to the field of
mathematics. As Cote (1989) emphasized when writing about his new
microworld of two turtles'
"les deux tortues" -- many interesting
connections can be made to concepts already in the primary and secondary
mathematics' curriculum. Hoy les & Noss (1987a) have already start to work
in this direction. Thus, from a general point of view, links with traditional
mathematical content could be benefical to Logo.
In a way, Hoy les & Noss (1987a), de-dramatise the necessity for Logo to link
up with mathematical concepts. The whole of mathematics' teaching seems
to suffer from a similar but stronger malaise : "the separation of any sort of
meaningful activity and the separation of pupil's conceptions from their
formalisation'. A first response to such a malaise resides in a general
awareness of the need for links, concrete and abstract, for whatever
problem-situation is being worked on. Many authors ( Cote & Kayler 1987;
COW, 1989; Gurtner, 1988; Hillel et al 1989; Hoyles & Noss, 1987a) have
Proposed the creation of mathematical microworlds as an interesting solution
to this particular problem for Logo and to the more general problem of
mathematics' education. The microworld notion can, of course, present
subtle difference of definition from one author to the next, but what is most
important is the view of working on a given topic from different points of
view and with different kinds of tools (computer, paper and pencil ruler,
compass -, etc.). If that were done for all pertinent mathematical concepts
(number, measure, area, variable, operation, function, etc.) the future of Logo
and the future of mathematics would be in better hands!
In sum, the
confrontation of Logo with the mathematics' curriculum could be benificial to
both, but especially to Logo given its chances of influencing the whole of
mathematics' teaching.
The link to m3thematks' curriculum.: A way to support the evolutlOnary
role of Lego
Historically, Logo has now reached the point where progress in the evolution
of the learning and teaching of mathematics is, for the most part, in the
hands of the teachers. In the beginning, Logo was actively supported by a
nucleus of keyed up teachers and by a lot of researchers; then, after a short
period of adaptation that in many cases brought along better infrastructural
school support (more equipment, direct support in class, better information
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and training), a larger group of teachers became active in Logo. Today, with
Logo having more direct links to a content that is known and judged
important by teachers, a larger group could become positively involved with
the Logo approach. This is the successive wave phenomenon mentionned
above. Such a phenomenon is not particular to Logo and has often been
observed in the past with other kinds of innovation. In the end what
matters is that the spirit and crucial philosophy underlying the innovation be
not lost in the successive phases of implantation and adaptation.
Authors who favor more direct links with the mathematics' curriculum ha ,*
described the necessary conditions for not losing contact with the Logo
philosophy (C8t4, 1989; Hoy les, Noss 1987a; Hoy les, 1985; Gurtner, in press).
What appears essential in implementing Logo in schools is not the form of
presentation but the spirit in which it is presented, and the maintenance of
specific pedagogical goals in whatever modality is chosen.
The growth of variety in employing Logo what counts?
What evidence do we have that what counts is the nature of the goals
pursued rather than the external means of presentation? An apparently
"good" way of presenting something does not guarantee the respect of
important goals: it is not because Logo is offered in an open non-directive
environment that such developmental goals as the acquisition of autonomy,
mathematical knowledge and thinking skills are neccessarily attained. Nor
is it because Logo is offered in a relatively structured environment that such
goals are not attained. As such ecologists as Bronfenbrenner (1979) and
Garbarino (1982) have said : it all depends!
There is a wide variety of contexts in which Logo is offered today. A
supervisor can choose basic Logo or opt for an expanded version such
asLego-Logo. The context can be open, that is, centered on childrens*
projects, structured in such a way that the situations are chosen in advance,
or semi-structured, which alternates the two. It is possible to focus on
aspects of visual art, programming and/or mathematics. Promoters of a
mathematical framework can either choose to link it to mathematics
curricula or find another way of assuring the process of mathematization
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.
Although it was once seen as heresy to do Logo in way different from what
Papert (1979, 1980) first proposed, many now see it as a favor to Logo to
vary its type of implementation. Of course it all depends on how it is done.
Hoy les (1985a, 1985b) and Hoy les & Noss (1987a) have clearly described
the needed conditions for an adequate integration of Logo in the school
mathematics' curriculum. Research by Lemerise (in press), and Hoyles &
Noss (1987b) has showed that a structured approach can facilitate the
realization of many Logo goals.. Cote (1989) and Weir (1987) even talk
about "structured exploration" and "structured discovery" as a way of
reaching some of Logo's goals; the pedagogical agent makes certain
predetermined choices in order to favor exploration or discovery in a
particular domain or situation. Some data now exist (Lemerise, in
preparation) on how 4th, 5th and 6th graders behave in a specific
microworld (COte's (1989) "two turtles") that is tightly related to their school
mathematics' curriculum. In a class' context where work on computer
alternates with paper and pencil's work children construct, explore, compare
and generate laws. Sometimes, of course, the way certain tasks are presented
can trigger reactions of dependance, or guessing, but such problems arise in
all contexts. What matters is that they are flushed out and dealt with
intelligently.
Variety, in conclusion, is more often a strengh than a weakness, rigidity
more deadly than flexibility. A revolution tends to be totalitarian, evolution
more democratic.
References
Bronfenbrenner, U. 1979. The eailtyr of human developement eAperimeats
hatvre and eittSer/S: Cambridge, Havard University Press.
COte, B. 1989. Les deux tortues: logiciel de construction et d'exploration en
mathematiques pour le primaire et le secondaire. Unpublished paper.
Section Didactique, University du Quebec a Montreal.
Calk, B., & Kayler, H. 1987. Geometric constructions with turtle and on paper
by 5th and 6th grade children. Proceedings of the third Conference for
Logo and Mathematics Education, Dept. of Mathematics, Concordia .
University, Montreal.
DiSessa A..1982. Unlearning Aristotelian physics: a study of knowledgebased learning. Cognitive Science, 6, 37-75.
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281
Garbarino, J. 1982. ?."2.1.Liteirt9/2,.vadfamilks th tb.
twviratimont.c Aldine
Publishing Co., New York.
Gurtner, J.L. 1988. Equa logo. In C. Hoy les (Ed.) Proceedings of the working
group on. Logo. Budapest, ICME-6.
Gurtner, J.L. (in press). Between Logo and mathematics: a road of tunnels
and bridges. In C. Hoy les, & R. Noss (Eds.). Lturning mathem,.?tias
and Logi? Cambridge, MA, MIT Press.
Hillel, J., Kieran, C., Gurtrier, J.L. 1989. Solving structured geometric tasks on
the computer: the role of feedback in generating strategies.
Educational Studies in Mathematics. 20, 1-39.
Hoyles, C. 1985(a). Developing a context for Logo in School. Mathematics. The
lo2irnal of mathematical behavior. 4, 237-256.
Hoyles, C. (1985(b). What is the point of group discussion in mathematics?
Educational studies in mathematics. 16, 205-214.
Hoyles, C., & Noss, R. 1987(a). Synthesizing mathematical conceptions and
their formalization through the construction of a Logo-based school
mathematics curriculum. International Journal of Mathematics
Education. in Science. and Technology: 18, 4, 581-595.
Hoyles,C., & Noss, R., 1987(b). Children working in a structured Logo
environment: from doing to understanding. Recherches en didactique
cles.Mathernatiques, 8, 12, 131 -174.
Lemerise, T. (in press). On intra- and inter-individual differences in
children's learning styles. In C. Hoyles, & R. Noss (Eds.). Lawn/4
matimmaties a,,,,d1:15-3 Cambridge, MA, MIT Press.
Lemerise, T. (in preparation). La fusee tortue. Analyse des comportements
d'enfants de 4ieme, 5ieme et bierne armee. Rapport dexperimentation.
Papert, S., Watt, D., diSessa, A. & Weir, S. (1979). Final report of the Brokline
Logo project. Logo Memos 53 and 54, MIT, Cambridge, Mass.
Papert, S. 1980. ilithlist.iytas:.
ay./yr:veers
;writ,/ .tiltv$ Basic
Books, New York.
Weir, S. 1937. CiatiFat.thg ihrteiS: a Liv a:t-s1.4k9c)A New-York, Harper &
Row.
Weir, S. (in press). Lego-Logo: a vehicle for learning. in C. Hoyles, & R. Noss
(Eds.). L-Part2(171 Mg17101128tk
Liyo. Cambridge, MA, MIT Press.
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Young Children Solving Spatial Problems
Helen Mansfield and Joy Scott
Curtin University of Technology
Western Australia
In our research with young children, we have begun to explore young children's actions when they
are engaged in spatial problems. Our interest has not been so much in whether children are
successful in solving the problems we pose to them, nor in tracing the success rates of children of
different ages. Rather, we have been interested in exploring the procedures that children who are
successful are able to use, and in trying to identify differences between the procedures employed
by children who are successful in solving the problems and those who are unsuccessful.
Our research falls within the general framework of a constructivist view of learning. One aspect
of this framework is the belief that the form taken by new knowledge constructed by learners is
dependent on the form of the knowledge they already possess. Young children who already possess
appropriate procedures and use them successfully on simple spatial problems seem likely to use
those procedures more readily and successfully in subsequent more difficult problem-solving
situations than children who do not already possess such procedures.
One of our basic assumptions is that children construct their own mathematical realities, which
may differ significantly from the reality of the adult researcher. While there may be an
inherently logical structure to a problem as it is perceived by adults, the structure the child
imposes on the task may be different.
Lester (1983) has suggested that three main questions constitute the core of all mathematical
problem-solving research:
(1) what the individual does, correctly, incorrectly, efficiently and inefficiently;
(2) what the individual should do; and
(3) how individual problem-solving can be improved.
The goal of the research reported here was to improve our understanding of how young children
solve simple tangram-like problems. We were, therefore, interested in exploring the first of
Lester's questions within the context of some simple spatial tasks.
In the tasks that we used, the children were requested to use two or three cardboard shapes to
cover completely a region drawn on card.
Clearly, these tangram-like puzzles require simple
shape and size recognition and discrimination abilities. In particular, as well as recognizing the
overall shape configuration of a region, children must be able to judge angles as equal and line
segments as equal.
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X83
.e children may also draw on some planning skills to enable the tasks to be carried out
successfully. Initially, there may be several ways that the first piece can be placed on the target
egion. Usually, only one of these placements leaves a region that is the correct shape for the
remaining pieces to cover it. Other initial placements cannot lead to a correct solution. The child
then must recognize that the first piece prevents a solution and must be willing to remove it and
place it again in a different position. The recognition that the first placement is unfruitful
therefore requires further shape and size recognition and discrimination as well as willingness to
remove and re-position a piece that seemed correct initially.
When children place a piece on the region, they may pick up the piece to be placed and by chance
position it appropriately. If this does not occur, the child may be able to position the piece
appropriately by removing it and trying again, or by rotating, reflecting, or translating the piece
until it does fit satisfactorily. We speculated that children who are able to rotate or reflect the
pieces before placing them on the region, or after they have been placed on the region, are more
successful at solving the tangram-like problems that we presented to them. We also hypothesized
that children who are able to recognize an incorrect placement and who are willing to remove a
piece and try to find alternative placements are also more likely to be successful in solving the
problems than those who do not display these behaviours.
The research reported here is exploratory. Our purpose was to observe children in a clinical
situation as they attempted to solve a variety of tangram-like tasks, with a view to documenting
the children's actions.
We sought to identify the sequence of actions the children used, and to
identify those actions that were efficient or successful. We also attempted to identify planning
skills employed by the children.
The samples
We worked with three groups of young children.
1.
Pre-school children. This sample consisted of 4 boys and 3 girls. These children were
interviewed on one occasion. Their ages ranged from 45 months to 56 months, with a mean age of
52 months.
2.
Pre-primary children. This sample consisted of 6 boys and 2 girls. These children were
interviewed on two occasions. At the time of the first interview, they ranged in age from 55
months to 63 months, with a mean age of 59 months. At the time of the second interview, the
mean age was 66 months.
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276
3. Year one children. This sample consisted of 6 boys and 4 girls. These children were
interviewed on one occasion. Their ages ranged from 70 months to 85 months, with a mean age of
75.5 months.
Procedure
The two questions whose results are described here were questions three and four in a five
question sequence. In the first two questions, we explored the children's knowledge of the names
of common geometric shapes, presented as regions or as boundaries. In the fifth question, we
explored whether the children were able to construct common geometric shapes from a variety of
sticks of different lengths. Only the results of questions three and four are discussed here.
For questions three and four, the child was first presented with a set of geometric shapes made
from pieces of card. The set consisted of congruent right isosceles triangles, congruent squares,
congruent rectangles, and congruent equilateral triangles. Each square could be covered exactly
by two of the right isosceles triangles, and exactly by two of the rectangles. The child was invited
to handle the pieces and to sort them according to shape. Most children in fact did this without our
asking them.
For question three, the child was then presented with four shapes drawn on card. Each of these
shapes could be covered by two of the cardboard pieces which the child had already handled. There
were diVisions drawn on the shapes which represented the boundary between the constituent
pieces and were intended to provide a clue to the placement of the required pieces. The child was
asked to find the required piece from the set already sorted, or if the child did not know the name
of the shape, the child was given the two pieces that were required. The following question was
then asked: Can you put the two shapes on top of this shape (shape on card indicated by gesture)
so that they cover it exactly?
A similar procedure was followed for question four. In this question, nine shapes to be covered
were presented to the child on cards. The shapes were shown without divisions drawn on them to
indicate the boundary between the pieces to be placed on them. Each shape could be covered by two
or three of the cardboard pieces.
Results
Reported here are the children's responses to questions three and four. Table 1 shows the number
of children who were unable to complete the various problems presented in questions three and
four.
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285,
Table 1
Numbers of Children who were Unable to Solve Problems in Questions 3 and 4
Question 3
1
Sample 1
Sample 2(1)
Sample 2(2)
Sample 3
23
1
Question 4
4
1
3
2
6
2
2 3 4 5
3 2
67 89
52 43
1
2
1
2
1
1
1
1
1
2
The figures given in Table 1 suggest that the shapes that caused the most difficulty were shape 4
in question three, and shapes 1, 6, 7, 8, and 9 in question 4.
Shape 4 in question three consisted of a right isosceles triangle, presented with its longest side
horizontal and at the base of the figure. The internal marking showed the boundary between the
two smaller right isosceles triangles with which it had to be covered. Shape 1 in question four
also consisted of a right isosceles triangle, with its longest side horizontal but at the top of the
figure. Some of the youngest children's actions in attempting to solve these two problems are
discussed in some detail below.
In question four, shape 6 was a parallelogram that could be covered by two right isosceles
triangles, shape 7 a rhombus that could be covered by two equilateral triangles, shape 8 required
a square and a right isosceles triangle, and shape 9 required a square and two right isosceles
triangles.
The results summarized in Table 1 also show that the youngest sample (sample 1) had the
greatest difficulty in completing the problems presented to them. While the table shows that most
children in the other two samples were able to solve the problems, observation of the children as
they attempted the problems showed that the older children did not necessarily find the problems
easy to solve. Indeed, some of the problems proved to be quite difficult, but the children in
samples 2 and 3 were very persistent in trying to reach a solution, and were also more prepared
than the children in the youngest sample to remove and re-position a piece whose initial
placement prevented completion of the problem.
A simple tally of success or failure at completing the problems also masks the quite different
approaches the children used. The responses of four of the children from sample 1 as they
attempted to solve the two isosceles triangle problems illustrate four quite different sets of
actions.
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278
Laura, who at 45 months was the youngest child interviewed, was successful at solving both the
isosceles triangle problems, that is problem 4 in question three, in which the internal
boundaries of the constituent triangles were shown, and problem 1 in question four, in which no
boundaries were shown.
In solving the first of these problems, Laura rotated one of the given
right isosceles triangle pieces in the air until it matched one of the regions drawn on the target
shape. She held the cardboard piece by its right angle, and appeared to match the other two angles
of the piece to its region. In the second of these two problems, she was not told which two pieces
to use, but picked up one of the correct right isosceles triangle pieces immediately. Again, she
rotated the piece in the air until she was satisfied that it matched half of the target region before
she placed it in position. She then easily placed the second piece.
Murray was unsuccessful at solving either of these problems.
In the first problem, he placed a
right isosceles piece so that its right angle matched the right angle of the target region.
In doing
this, he appeared to ignore the boundary drawn as a clue on the region. He then placed another
triangle at the base of the target region, matching their two base angles. He was then left with an
uncovered region for which there was not a matching shape. He chose a rectangle, and placed it so
that it covered all of the remaining region but overlapped the boundary. It seemed that for
Murray, the important objective was to cover all the region, without necessarily matching the
shape of the target region. Apparently, the internal boundaries were no help to him. He also
seemed to operate by matching congruent angles. In the second of these problems, he chose the
same three pieces and placed them in the same way.
Jesse was successful in solving the first of these two problems but unsuccessful in the second. In
the first problem, he initially tried an equilateral triangle, but after recognizing that the angles
did not match, he chose a right isosceles triangle and was able to place it without difficulty. The
second triangle piece was then easily positioned. In the second problem, he placed a triangle piece
so that its base angle matched one of the base angles of the target region. He then considered the
trapezoidal region that remained, and declared that it would not work, since there was not a shape
like that available. He then re-positioned the first triangle piece so that its right angle matched
that of the target region. Again, faced with a trapezoidal region remaining, he was unable to find a
piece to match and made no further attempt at a solution. Jesse was able to recognize when a
solution was not possible by considering the remaining region and attempting a match with the
available pieces. He also attempted to re-position an initial placement, in contrast to Murray,
who was unable to recognize that an initial placement did not enable a further correct placement
to be made.
Teneka was also successful on the first of these problems but unsuccessful on the second. Teneka's
approach seemed to be more arbitrary than that of the other children discussed here. In the first
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problem, she picked up the correct triangle in an orientation that by chance matched the target
region. She seemed to recognize immediately that this piece was correctly placed. She then
picked up and discarded a succession of triangle pieces until one that she picked up was in the
correct orientation for the remaining region and she was able to complete the problem. She did
not attempt to rotate the triangles, either in the air before placing them or after they had been
positioned on the target region. In the second problem, Teneka made an initial placement of a
triangle so that its base angle matched the base angle of the target region. She then tried a
succession of triangle pieces, discarding those that did not fit in the way she had picked them up
and selecting another. In this problem, she appeared to be matching the lengths of the second
piece selected to the first piece she had placed on the target region. She appeared not to recognize
that her first move could not lead to a correct solution, and was unable to re-position a piece that
appeared initially to be correct.
The responses of these four young children show that they each used different sets of actions in an
attempt to solve the two problems discussed here. One action that was used by Laura, who solved
both problems, was rotation of the pieces. We ranked the children in all three samples according
to how many of the problems presented in questions three and four they were able to solve. The
children in the second sample, who were interviewed on two occasions, were entered twice in this
ranking. Of the children who were ranked in the top ten according to the number of problems they
solved, eight used rotation of the pieces in their solution attempts. Of the ten children who were
ranked in the lowest ten according to the number of problems they solved, only two used rotations
in their solution attempts.
The four children from sample one whose problem solving approaches we have described above
used different procedures and had varying rates of success on these problems. From an adult
perspective, Teneka's principal strategy of picking up and discarding pieces seemingly at random
until they matched the target region was inefficient, and seems unlikely to enable her to solve the
more difficult spatial problems with which she will be faced in her schooling. Yet Teneka was
eventually successful at solving all the problems in question three and six of the nine problems in
question four. There is no very compelling reason from Teneka's point of view to find a more
efficient strategy. When we return to interview Teneka in a few months time, we will be
interested to find whether she has retained this strategy or whether she has been able to or needed
to find another strategy.
One question in which we were interested was whether young children continue to use the same
procedures for similar problems over an extended period of time, that is whether the procedures
they have developed well before starting school are persistent. It was to explore this question
that we interviewed the children in sample two twice, seven months apart.
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Our study of the actions of these children revealed some interesting features of their problem-
solving strategies. At the time of the second interview, they were even more persistent than they
had been in the first interview, and were generally willing and able to re-position a piece that
would not allow a solution. As Table 1 shows, they were also more successful at solving ,the
problems. We were also struck by the similarities that most children showed in their actions in
the two different interviews.
In her first interview, Jessica was unsuccessful in the first triangle problem and successful in
the second. Throughout her attempts at both problems, Jessica had both pieces she was trying to
place on the puzzle in her hands. She would rotate and place one piece, then try to place the
second. When that was unsuccessful, she would remove the first piece and rotate it to a new
position. She appeared to be matching the lengths of the sides of the two pieces she was using,
while also trying to cover the target region. At one stage in the second problem, she placed the
first piece so that its right angle matched that of the target region and then placed the other
triangle so that they formed a square.
In her second interview, Jessica was successful at solving both problems. Her actions were very
similar to those she had used in her first interview. She had both pieces she was trying to place
in her hands throughout, and placed first one and then the other, rotating them to try to place
them. Again, she appeared to be matching the lengths of the two pieces, and she even constructed a
square in the same way she had in the first interview. Perhaps the most obvious difference in her
two interviews was that in the second, she did at one stage turn one piece over, which she had not
done in the first interview.
While the similarities in the two interviews with Jessica were particularly striking, we noticed
marked similarities in the actions of the other children in this sample. For example, Will more
than any other child turned the pieces over several times in both interviews. He also placed the
same pieces in the same inappropriate positions in the two interviews. However, by the time of
the second interview, he also rotated the shapes in order to place them successfully.
Conclusion
Our interviews with these young children showed that they had already developed some procedures
for solving the problems we presented to them. Naturally, some of the children were more
successful than others, and the youngest children were the least successful. The children who
were successful showed an ability to recognize when a shape would not lead to a solution and a
willingness to reposition pieces. Generally, the children were quite persistent in their efforts
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to solve the problems, with the children in the older samples being more persistent than those in
the youngest.
Even though we interviewed only a small number of children, we found many quite different
procedures used by the children.
We also found that the actions of individual children who were
interviewed twice were remarkably similar in the two interviews. The changes that were noticed
were additions to the repertoire of actions that the children employed.
Our observations suggested to us that children who rotated the pieces were more successful at
solving the problems than those who did not. Very few of the children actually turned the pieces
over. It seems likely that children who try to rotate or turn pieces have developed actions that
are particularly useful in these spatial problems. What we do not know is whether most children
learn these actions for themselves or whether and how these actions can be taught.
Some children in all three samples used actions that from an adult perspective were inefficient.
If we believe that children construct the procedures that they will eventually employ in solving
spatial problems by trying out actions for themselves, then some children may retain their
inefficient initial procedures over a long term, and may not construct the more efficient
procedures that their peers are able to employ.
In the next phase of our work, we will be returning to interview the young children we
interviewed here. We will be looking to see whether the procedures they display as they become
older retain resemblance to the ones they have already used. We also want to work with older
children to see whether the great variety of procedures we observed with these children are also
observed with older children and whether older children are able to employ different procedures.
We will also be exploring the procedures used by young children in a variety of other spatial
problems, particularly three-dimensional problems. This work is directed towards gaining a
greater understanding of how young children solve spatial problems and observing the genesis of
the correct, incorrect, efficient, and inefficient strategies that older children use.
Reference
Lester, F. K. (1983). Problem solving: Is it a problem? In M. M. Lindquist (Ed.), Selected
issues in mathematics education. Berkeley, Calif.: McCutchan.
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282
THE ROLE OF FORMAT IN STUDENTS'
ACHIEVEMENT OF PROOF
W. Gary Martin
University of Hawaii at Manoa
The two-column proof format is widely used in high school geometry in the USA.
While many have suggested that alternative formats be used, little investigation of the
impact of using other formats on students' ability to write proofs has been undertaken. The preferences of a class of 30 high school geometry students for twocolumn, flow, and paragraph formats were investigated, as was the relationship
between the format used and success in writing proofs. This study suggests that,
when given a choice, most students develop a marked preference for a particular
format of proof, while others prefer to use a mix of formats. More students
preferred two-column proofs than the other formats. The format of proof used was
not found to be related to achievement in writing proofs.
Developing students' ability to write and understand proofs has been one of the important
objectives of high school geometry in the USA throughout the past century. However, for many
students this objective is not being satisfied. Senk (1985) found that less than a third of high
school students in proof-oriented geometry courses in the USA have "mastered" the ability to
write proofs. Moreover, evidence exists that students frequently do not see a proof as a series of
logical connections that guarantee the truth of a conclusion, given a set of hypotheses. For example, Martin and Harel (1989) found that many students base their judgment of the validity of a
mathematical argument on whether it appears to be a proof, rather than on an analysis of its cor-
rectness. Fischbein and Kedem (1982) found that even students who accept a proof as being
correct may not believe that this guarantees the universal truth of the statement.
One explanation for this lack of understanding, at least for students in the USA, may be the
manner in which proof writing is presented. In the USA, proof has traditionally been presented
to secondary geometry students using a rigid two-column format, in which the left column con-
tains inferred statements and the\ right column contains reasons, generally definitions or theorems, to support each statement. Farrell (1987) describes the "awkward complexity" of the two-
column proof.
[(p
q) A p]
The typical high school geometry proof relies on modus ponens;
q. In the two-column format, the particular antecedent(s) (found in the preced-
ing statements of the proof) are first presented, then a particular consequence (in a statement of
the proof), and finally the general implication on which the inference is based (in the reason for
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291
A
LABC is isosceles.
GIVEN:
AM bisects LBAC
/1\
AM 1 BM
PROVE:
BMC
CBM bisects LABC)
i
Def. of bisector
( AABC isosceles )
(
Def. of isosceles
BC a AB
)
Reflexive
S.A.S. princip e
( AAMB 6,AMC )
C. P.
A linear pair is supplementary
T. C.
( LAMB a LAMC)
Algebra
(LAMB & LAMC
are right angles
Def. of perpendicular
BM1 AC
)
Figure L. A sample flow proof, based on a student response.
the statement). This ordering does not follow the logical flow of modus ponens. Furthermore,
the two-column format does not emphasize the logical connections between statements as the
antecedent(s) for an inference are not specified but rather are buried in the preceding steps; this
may prove confusing to students beginning the study of proof in geometry (MacMurray, 1978).
The use of two-column proofs has also been criticized on curricular grounds. The
Conference Board of the Mathematical Sciences (1982) advocated "playing down" two-column
proofs. The Curriculum and Evaluation Standards for School Mathematics, developed by the
National Council of Teachers of Mathematics (NCTM, 1989), suggests that use of the two-col-
umn format should be greatly decreased. In particular, the document suggests that verbal paragraph proofs be emphasized and that the use of a particular format not be enforced.
Several alternative formats to the two-column proof have been suggested, in addition to
paragraph proofs. For example, Retzer (1984) advocated adding numbers corresponding to the
statements on which an inference relies to the "reasons" column of a two-column proof.
MacMurray (1978) suggested the use of a "flow proof," in which logical connections are diagrammatically represented; an example produced by a student of this study is shown in Fig. 1.
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Despite advocacies of increased attention to proof formats other than the two-column format, little attention has been given to the implications particular formats may have for students. If
given a choice, will students prefer using paragraph or flow formats over the two-column format? Will they consistently use the same format? Why do they choose to use one format over
another? Is use of a particular format associated with better achievement of proof? This study
addresses these and other questions with respect to a class of students who were enrolled in a
high school geometry course.
Subjects
A class of thirty tenth-grade students enrolled in a high school geometry class at the
University of Hawaii Laboratory School (UHS) formed the subjects for this study. By state
mandate, UHS is maintained for the purpose of curriculum research and development. In order
to create curricula that are reflective of the needs of the students of the state, the student body is
representative of all students attending public high schools in the state, based on intellectual abil-
ity, ethnicity, and socioeconomic level. The class participating in this study consisted of the top
thirty of fifty-eight tenth-graders with respect to performance in previous mathematics classes at
UHS and scores on standardized tests. Thus, students in the study were representative of average and above-average students of the state of Hawaii.
The class was conducted as a pilot study of an on-going curriculum research and development project in high school geometry. The emphases of this project include developing problem-
solving processes (Rachlin, 1987), developing important concepts of geometry in accordance
with the van Hie le levels (van Hiele-Geldof & van Hide, 1984), and developing concepts of
proof. Proofs were initially introduced the third week of the course by having students write
informal paragraphs justifying why certain properties should be true for a given figure; this is
consistent with the suggestions of the Curriculum and Evaluation Standards (NCTM, 1989, p.
144). Proof writing was not explicitly addressed until the fifth week of the course, when the
given versus to-be-proven parts of a statement were discussed, along with general strategies for
developing proofs. In the seventh week of the course, flow proofs were introduced. While twocolumn proofs were never formally presented in class and were never modeled by the classroom
teacher, students were exposed to the two-columri format in a textbook provided to them for ref-
erence. Students were initially required to use either the flow or paragraph format (or both
285
formats) on several examples. They were later allowed to use the format of their choice on both
homework and tests.
Method
Three data sources were used in answering the questions of the study. Firit, four written
assessments of students' proof writing were made at intervals throughout the school year; a
follow-up assessment was made at the beginning of the second semester of the following year.
The proofs were rated from two perspectivesthe proof format used and the "correctness" of the
proof. Correctness was rated on a 0-4 scale adapted from Senk (1985), as follows: 4a proof
reflecting all necessary aspects of the proof, with only minor omissions or errors; 3a proof
which is generally right, and has only one serious omission or error; 2a proof which has a
sequence of correct inferences but which is based on a faulty premise or fails to support the final
conclusion; 1a proof which includes one correct inference; 0a proof with no correct inferences. Each response was independently rated by the investigator and a research assistant.
The second data source consisted of the complete written work of several students covering
the entire school year. The work of five of these students was analyzed to provide a more detailed view of the impact of proof format on students' proof-writing. One proof was chosen for
analysis from each week of the course in which proof was considered; the proofs were again ana-
lyzed by proof format and by correctness. In addition, the formats used in all homework problems throughout the school year were tabulated.
Finally, students were given a questionnaire concerning their preferred proof format in the
third month of the course and again in the fifth month of the following year. In this questionnaire
they were asked to identify their favorite proof format and why they liked or did not like each of
the formats.
Results
Responses from the five assessments of proof-writing were categorized by the format of
proof used and by correctness; see Table 1. In the first assessment, taken early in their experiences with proof, most of the students still used paragraph proofs since this was the initial format
introduced. By the second assessment, a distinct shift had taken place; the majority of the students were now using two-column proofs. This preference for two-column proofs was generally
consistent throughout the remainder of the assessments, with relatively few students using
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paragraphs and almost no students using flow proofs. In the follow-up assessment the following
school year, only five students used paragraphs and only one student used a flow proof. Note
that missing scores are due to student absences from class.
Table 1. Achievement of Proof by Proof Format
Proof format
Flow Proof
Total
Testing date
Two-column
Paragraph
Mean
n
Mean
n
4
3.50
19
2.53
6
3.00
2
3.00
5
2.20
18
3.28
2.00
8
2.50
18
2.22
2
2.00
8
2.50
20
2.75
2.78
1
2.00
5
3.00
21
2.76
2.69
11
2.73
43
2.65
83
2.77
n
Mean
n
Year 1, Month 3
29
2.76
Year 1, Month 5
25
3.04
Year 1, Month 7
28
2.29
2
Year 1, Month 9
30
2.63
Year 2, Month 5
27
Total Responses
139
Mean
To provide a view of how individual students' usage of proof formats changed over time,
the consistency of the formats they used in these assessments was also analyzed, as seen in Table
2. The minimum possible agreement is 40%, in which case the student would have used one
format once and the other two formats twice. More than two-thirds of the class had a consistency
of use of 80% or above, with a third being completely consistent in their use of a format. Thus,
a picture of relatively consistent use of a particular proof format for a given student emerges, with
most of the students using two-column proofs.
Table 2. Frequencies of Consistency of Use of Proof Formats
Format
Agreement
Total
60%
8
80%
10
100%
12
Total
30
Flow Proof
Paragraph
Two-column
0
7
0
1
9
1
4
7
2
5
23
1
The correctness of the students' proofs does not appear to be related to the proof format
used. In each of the assessments given, the mean scores for the formats appear to be very close
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to each other, as do the total mean scores for the formats. A similar result was found when com-
paring students' proof scores to the proof format they predominantly used; see Table 3. A
Kruskal-Wallis analysis yielded H = 0.68, with p > 0.71, for this table.
Table 3. Relationship of Achievement of Proof to Preferred Proof Format
n
Mean
Proof Format
Flow proof
Paragraph
Two-column
2
2.51
5
2.71
23
2.71
The complete written work of five students was considered to obtain a more detailed view
of the role of format in students' ability to write proofs. One proof from each week of the course
was analyzed by format used and by correctness; see Table 4. Furthermore, the formats for all
attempted proofs were tabulated. All the students initially used paragraph proofs, as this was the
first format introduced. Several distinct patterns of use of and success with the various formats
can be identified. Note that inferences relating format used to success must be viewed with caution since variables such as maturation may confound the inferred relationship.
Table 4. Use of Proof Formats by Individual Students
Format
Flow
Subject
Paragraph
Two-column
n
Mean
n
Mean
n
Mean
M.
1
4.00
20
3.70
2
4.00
S.
0
12
2.67
8
3.38
K.
0
15
2.40
8
2.88
L.
2
3.00
13
2.54
3
2.67
B.
1
3.00
10
2.50
12
3.83
Three of the students showed a clear, long-standing preference for using a particular format. M. continued using paragraphs throughout the course, with infrequent deviations. M. was
very successful in writing proofs; no differences in his use of the formats can be inferred. B.
began using two-column proofs during the twelfth week of the course, after which he rarely used
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any other format. B. tended to be more successful with two-column proofs than with paragraphs. K. used paragraph proofs almost exclusively through the eighteenth week of the course,
when he switched to primarily using two-column proofs, along with an occasional paragraph
proof. K. tended to be somewhat more successful with two-column proofs. None of these three
students used flow proofs more than four times in all of their written work.
The other two students used a more mixed set of formats. S. began using two-column
proofs the twelfth week of the course and used them heavily for around a month; she also wrote a
number of flow proofs during the twelfth and thirteenth weeks. By the sixteenth week she was
again primarily using paragraph proofs, with an occasional two column proof. She also sometimes used a mix of the two formats, a two-column proof in which reasons were presented as
short paragraphs. She was somewhat more successful writing two-column than paragraph
proofs. L. began using two-column and flow proofs during the thirteenth week of the course.
Like S., L. continued to use some paragraph proofs. She rejected the use of flow proofs by the
fourteenth week and never used another after that time. By the nineteenth week, she was again
primarily using paragraph proofs. Unlike S., L. eventually completely abandoned the two-column format, never using it after the twentieth week. L. was equally successful in using the two
formats. Both of these students experimented with flow proofs early in the course, but rarely
used them later in the course.
The students were given a questionnaire the fourth month of the course and again the fol-
lowing year, asking them to identify the proof format they prefer and why. Their self-reports
closely matched (p<0.001) the formats they used in their workx2=21.86 (df=4) and x2=35.96
(df=4), respectively.
Explanations for their preferences fell into the following categories.
Preferences for two-column proof focused on organization ("Easier to organize my thoughts"),
readability ("It's easy to read because it is like a list"), and understandability ("I can understand
why what makes what"). For paragraph proofs, reasons focused on a flow of consciousness ("I
just write it as I think it out") and a preference for writing ("I sort of like to write things out").
Several students expressed the opinion that flow proofs were especially good for short proofs;
one student mentioned its adaptability ("Can adjust arrows, less erasing"), while others pointed
to its quickness ("Much faster when you're short of time").
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Discussion and Conclusions
The data of this study suggest that, when given a choice, most students develop a preference for a particular format of proof, while others continue to use a mix of formats. Contrary to
what might be expected, most students preferred to use two-column proofs. This is unexpected,
both based on an analysis of the two-column proof from a student point of view and on the lack
of emphasis on two-column proofs in their geometry course. Further, students produced plau-
sible reasons for preferring the two-column format, including organization, readability, and
understandability. The format of proof used was not related to achievement in proof-writing.
Many people have advocated a major deemphasis in the use of two-column proofs. This
study seems to imply that such a change cannot be justified on the basis of student preferences or
on the basis of achievement of proof-writing, although other reasons for the deemphasis (such as
curricular considerations) may still be valid. In any case, given the strong preferences that students may develop for a particular format (such as.M., who wrote "I love the paragraph, I like to
thoroughly explain myself..."), the advice of the NCTM Curriculum and Evaluation Standards
(1989) to not enforce a particular format for writing proofs seems particularly appropriate.
Bibliography
Conference Board of the Mathematical Sciences (CBMS). (1982). The mathematical sciences
curriculum K-12: What is still fundamental and what is not. Washington, DC: CBMS.
Farrell, M. A. (1987). Geometry for secondary school teachers. In M. M. Lindquist (Ed.),
Learning and Teaching Geometry: K-12 (Yearbook of the National Council of Teachers of
Mathematics, pp. 236-250). Reston, VA: National Council of Teachers of Mathematics.
Fischbein, E. & Kedem, I. (1982). Proof and certitude in the development of mathematical
thinking. In A. Vermandel (Ed.), Proceedings of the Sixth International Conference for the
Psychology of Mathematics Education (pp. 128-131). Antwerp: International Group for the
Psychology of Mathematics Education.
MacMurray, R. (1978). Flow proofs in geometry. Mathematics Teacher, 7/, 592-595.
Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary school teachers.
Journal for Research in Mathematics Education, 12, 53-62.
National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: NCTM.
Rachlin, S. (1987). Using research to design a problem-solving approach for teaching algebra.
In Proceedings of Fourth Southeast Asian Conference on Mathematics Education (pp. 156-
161). Singapore: Singapore Institute of Education.
Retzer, K. A. (1984). Proofs with visible inference schemes. School Science and Mathematics,
84, 367-376.
Senk, S. L. (1985). Flow well do students write geometry proofs? Mathematics Teacber, 78,
448-456.
van Hiele- Geldof, D. & van Hiele, P. M. (1984). English translation of selected writings of
Dina van Hiele-Gehlof and Pierre M. van Hide. Washington, DC: NSF.
298
290
L'INFLUENCE DES ASPECTS FIGURATIFS DANS LE RAISONNEMENT
DES ELEVES EN GEOMETRIE
A. L. Mesquita*
DEFCL
Lisbonne - Portugal
La geometrie est un domaine des mathematiques dans lequel on fait en permanence appel a trois
registres, celui du registre figuratif, lie au systeme perceptif visuel, avec des lois d'organisation
propres a ce sisteme, celui du langage nature], avec ses possibilites de description et d'explicitation
du statut des &tomes et celui du langage symbolique, avec ses possibilites propres d'ecriture et de
recours a des formules. Parmi ces registres, ce sont surtout les deux registres de langage qui ont ete
etudids dans les recherches en didactique des mathematiques (C. Laborde, 1982). Les recherches
qui ont pour but analyser les difficult& soulevees par les problemes en geometrie euclidienne se
sont principalement focalisees sur les differences entre le mode de raisonnement des Cleves et les
exigences propres au raisonnement mathCmathique. Tel est le cas de A. Bell (1976), de M. Stein
(1986) et de N. Balacheff (1988), par exemple.
On a beaucoup moins prete attention au role _loud par la figure dans les problemes en gdometrie.
Examinons brievement ce role des figures. Dun point de vue mathCmatique, la question des figures
semble etre clair : un mathematicien sait ce qu'une figure peut lui apporter. Frenkel (1973) exprimait
bien cet apport : "les figures permettent de mobiliser simultandment les multiples relations que la
parole ou l'ecriture - qui se dCroule linCairement dans le temps
ne peuvent enoncer que
successivement".
Les mathematiciens tout en reconnaissant ce role, reconnaissent aussi que les figures sont
dispensables. "On s'en passe fort bien" (J. DieudonnC, 1964). L'absence des figures, prone dans
l'Cpoque des mathematiques modernes est un "manque" toutefois bien suppled par les
mathCmaticiens. Bien entendu, les mathCmaticiens utilisent habilement ces trois registres et les
aruculent convenablement.
Pour les Cleves, contrairement aux mathCmaticiens, le role des figures peut etre ambigu 1. En effet,
a) ou bien les &yes ne parviennent pas a voir sur la figure ce qui peut amener a une solution, b) ou
alors les figures attirent leur attention sur des pistes qui n'ont rien a voir avec le probleme (et dans ce
cas, la figure est un obstacle au dCveloppement du raisonnement) ; c) ou alors les figures
291
299
remplissent un role heuristique pour le probleme en question, et risquent de suggerer les demarches
de raisonnement qui peuvent etre selon les situations, correctes ou inconipletes.
D'un autre cote, les informations issues de ces trois registres ne sont pas necessairement les
memes. Le passage d'un registre a l'autre peut ne pas se faire directement. 11 peut exiger alors une
ou plusieurs transformations intermediaires : c'est le phenomene de la non-congruence qui
provoque un coin cognitif et qui constitue un obstacle pour les &yes, comme l'a montre R. Duval
(1988a). D'un autre cote, due a la pregnance relative de l'information figurative, celle-ci domine
naturellement l'information issue des autres registres (A. L. Mesquita, 1989a).
Nous presentons ici des criteres d'un modele d'analyse de figures qui vise a expliquer le pouvoir
heuristique dune figure dans un probleme et qui cherche en particulier a determiner des facteurs qui
pour une situation mathematique dorm& font que les elements dune solution soient plus ou moins
visibles sur une figure ; par d'autres mots, qui cherchent a &gager les conditions de visibilite et de
reorganisation dune figure. Ces conditions de visibilite sont tits variables, comment le suggerent
les recherches de J.-W. Pellegrino et R. Kail (1982). Ces auteurs ont mis en valeur le coat des
operations elementaires (tels que la rotation et le &placement) requises dans la recomposition dune
figure : les temps de reaction dans des Caches de reconnaissance sont variables, pouvant atteindre dix
secondes dans le cas les plus complexes (ceux ou la rotation et le &placement sont mis
simultanement en jeu).
D'un point de vue de la geometric et des figures, la distinction suivante, déjà signalee par
Merleau-Ponty (1945), a la suite des gestaltistes, a une importance fondamentale pour les
traitements exigeant une reorganisation de la figure : "une ligne objective isolee et la meme ligne
prise dans une figure cesse d'être, pour la perception, la meme. Elle nest identifiable dans ces deux
fonctions que pour une perception analytique qui nest pas naturelle'(ibid., p.18). Cette distinction
est a la base de notre modele d'analyse de figures.
Pour notre analyse, nous prenons aussi en consideration des criteres lies a l'articulation entre les
registres impliques par les traitements. En particulier, nous mentionnons ici des criteres lies a
('articulation entre le registre figuratif et le registre de langue naturelle. R. Duval (1988b) a montre
l'importance de ce type de congruence, semantique, entre ces deux registres, en mettant en evidence
la difference de resultats obtenus dans une meme Oche (presentee en deux versions, l'une
congruente, l'autre non) : la tiche non-congruente obtient un taux de reussite moins eleve.
Cette articulation entre le registre figuratif et le registre de langue naturelle est a la base de deux
criteres importants2.
Un premier critere est le role de la figure. 11 nest pas le meme dans tous les problemes de
geometrie. Ce role peut etre soit descriptif, quand it se reduit a une apprehension synoptique des
292
300
proprietes en presence, soit heuristique, si la figure agit comme un declencheur de demarches.
Le role de la figure est en general associe a ce que R. Duval (1988b) appelle une apprehension
operatoire de la figure, c'est-a-dire, a une forme d'apprehension centree sur les modifications
possibles de la figure et a sa reorganisation en des sous-figures autres que la figure de depart.
L'apprehension operatoire permet de mettre en evidence l'existence de figures fondamentales
suggerant des traitements. Par exemple, dans le probleme suivant, le role de la figure est
heuristique:
Dans la figure suivante, AI est Ia diagonals du rectangle ASIE.
Comparer les airs des dcux rectangles hachures OURS et LUNE .
a
A
(cocker la case correspondanle a la reponse)
OURS a l'aire
la plus grandc
Los dcux aims
sons egales
LUNE a rake
Ia plus gran&
A
E
La reconnaissance de deux reconfigurations, l'incluante et la complementaire, sont des facteurs
indispensables pour une justification complete de l'egalite des aires (A. L. Mesquita, 1989a). La
non-reconnaissance de ces deux reconfigurations est alors un obstacle heuristique.
Si la figure privilegie une certaine forme d'apprehension, cela peut mettre d'cmblee sur des
&marches de resolution, correctes ou non. Ainsi, nous avons vu (ibid.) que ce que nous avons
appele d'identification analytique de la figure -celle qui se centre sur les parties elementairessuggere naturellement l'operation de reunion (fig.l). Une identification globale -celle qui se fonde
sur le partage de la figure- oriente plutot vers le passage au complementaire et les traitements
soustractifs (fig.2).
(fig.1)
(fig2)
Notons que l'utilisation de la reunion ne permet pas d'obtenir directement l'egalite des aires des
rectangles hachures. Un raisonnement par l'absurde ou par contraposition est alors necessaire. Ces
293
30t-
formes de raisonnement ont ete utilises par deux binomes de 10-11 ans :
A: Je sais! ga
c'est la meme aire que ga ...donc ils
sont pareils3 les 2 rectangles (...) disons que ga,
c'est 2, ga c'est 1, des deux cotes, done ca fait (...)
3, it faut qu'en tout, ca fasse 4, disons...(...) it est
oblige que les 2... fassent 1 cm ... sinon un est
plus grand que I'autre... et vu que c'est bien
refs. 1 et 6, 3 et 4
refs. 2 et 5 ; 3 et 4
ref. 1 et 6
ref. 2 et 5
ref. R
divise en 2... c'est forcement pareil des 2 cotes...
Un autre critere que nous avons considers, le statut de la figure, est lie au type de traitement admis.
La figure peut avoir un statut d'objet , si les relations geometriques utilisees pour sa construction
peuvent etre reutilisees. Nous disons que la figure a un statut d'illustration quand on ne peut en
extraire directement aucune relation geometrique. Meme si certaines relations d'incidence et
d'alignement, par exemple, semble respectees.
Ces distinctions ne sont pas automatiquement pet-cues par les eleves. D'ob une non-congruence
possible entre le statut de la figure tel qu it est engendre par la ache, et 1' interpretation de la figure
tale qu'elle est percue par chaque eleve.
Par exemple, dans le problertie suivant, le statut d'illustration de la figure n'est pas pet-v.1 par les
eleves qui utilisent la mesure et la proportionnalitei, procedures incorrectes pour une figure avec un
tel statut :
Paul regarde, d'en bas, unecathtdrale. II fait le croquis ci.dcssous, ou it dessine cc qu it a obscrvd.
II note aussi les
indications suivantes, prises aux archives de la cathedrale:
I est un triangle Equilateral;
2 est un rectangle;
- 3 et 4 sons des carres:
- la figure form& par 3, 4 ct 5 est tin carrel;
- la longueur de AC est dc 12m.
D'apres ccs indications. que pcut-on dire des longucurs suivantcs?
Kocher la use conespoodanle n la reponse)
C'cst 12 m
LF
Cc West pas 12 m
On ne pcut pas savoir
Cc West pas 12 m
On nc pcut pas savoir
CATIIE2
On nc pcut pas savoir
CATIIE3
CATHEI
El
Ccst 12 m
FG
CD
Ccst 12 m
Cc West pas 12 in
O
t=1
294
302
Les resultats des reponses des eleves a cette question le montrent bien :
Tableau I : Les faux de reussite lechec a
CATHE1
CATHE2
CATHE3
Reussite
64
50
37
Echec
31b
42C
57d
5
8
7
Non-reponse
Note:
a
b
c
d
en pourcentage
20% des reponses concement LF*I2
24% des reponses concement FG-* i2
48% des reponses concement CD*12
Le statut d'illustration est d'ailleurs la difficulte majeure de ce probleme. Une fois &passe cet
obstacle, les substitutions necessaires pour sa resolution sont facilement faites. Notons que la
figure a ici un role descriptif : neanmoins, l'apprehension des proprietes exige une correcte
interpretation du statut d'illustration de la figure.
En guise de conclusion
Ces criteres nous donnent une base objective pour ('identification de certains obstacles lies aux
problemes. Ils apparaissent comme des criteres efficaces pour une analyse de taches et en
particulier de leurs difficultes. Les aspects figuratifs ont, en effet, une influence dans le
raisonnement et les criteres riper& contribuent a reclaircir.
La distinction entre les types d'apprehension semble etre un moyen indispensable pour effectuer
une analyse utile des taches geometriques. Cette distinction ainsi que les concepts utilises
constituent des premiers elements dune theorie cognitive de la resolution des problemes de
geometrie. Ils se presentent comme des outils dont la finalite est double. D'un cote, ils permettent
d'etablir une gradation des difficult& de resolution de problemes de geometric en fonction du statut
et du role des figures et des criteres de congruence. Dans ce sens, une hierarchic de difficult& peut
etre etablie. Dun autre cote, en choisissant convenablement ces criteres, on peut, par sa variation,
s'attendre a que certaines questions revelent des differences individuelles dans les reponses des
eleves5.
NOTES
I
II nest pas alors etonnant de constatcr que dans l'enseignement des mathematiques la place des figures a change
avec le temps. Ce changement reflecte le role variable et ambigu que les figures peuvent avoir pour les eleves.
295
303
2
Nous developpons en A. L. Mesquita (1989b) le modele d'analyse mentionne. Id, nous nous occupons
specialement de quelques criteres de ce modele, et de leur influence sur le raisonnement.
Ont la meme aire. A voter aussi que nous utilisons'ici le codage introduit dans les figures preadentes.
4
II s'agit de reponses d'eleves de 14 ans.
5
L'analyse de ces differences individuelles donne lieu a une typologie des comportements des eleves en geometric,
que nous decrivons en A. L. Mesquita (1989b).
REFERENCES BIBLIOGRAPHIQUES
BALACHEFF N. (1988) line Etude des processus de preuve en mathematique chez des &eves de
College, These de Doctoral d'Etat, Universite Joseph Fourier, Grenoble.
BELL A.W. (1976) A study of pupils' proof-explanations in mathematical situations. Educational
Studies in Mathematics, 7, 23-30.
DIEUDONNE J. (1964) Algebre Line-sire et Geornetrie, Paris : Hermann.
DUVAL R. (1988a) Ecarts semantiques et coherence mathematique, Annales de Didactique et de Sciences
Cognitives, 1,7 - 25.
DUVAL R. (1988b) Pour une approche cognitive des problemes de geometrie en termes de congruence, Annales de
Didactique et de Sciences Cognitives, 1 , 57 - 74.
FRENKEL J. (1973) Geometric pour l'eleve-professeur, Paris : Hermann.
LABORDE C. (1982) Langue naturelle et Ccriture symbolique : Deux codes en interaction dons
renseignement mathimatique, These de Doctoral d'Etat, Universite Scientifique et Medicate,
Grenoble.
MERLEAU-PONTY M. (1945) Phenotnenologie de la perception, Paris : Gallimard.
MESQUITA A. L. (1989a) Sur une situation d'eveil a la deduction en geometric, Educational Studies in Mathematics.
20 , 55 - 77.
MESQUITA
A. L. (1989b) L'influence des aspects figuratifs dans ['argumentation des eleves en
geometric
:
elements pour une typologie, These de Doctorat, Universite Louis Pasteur,
Strasbourg.
STEIN M. (1986) Beweisen , Bad Salzdetfurih : Franzbecker.
PELLEGRINO J.-W. et
KAIL Jr. R. (1982) Process analyses of spatial aptitude, in R.J.Stcrnberg
(ed.), Advances in the psychology of human intelligence (Vol.1), pp.311-365, Londres
Lawrence Erlbaum Associates.
Boursiere de FundacZo Calouste Gulbenkian
Departamento de Educacab da Faculdade de Ciencias de Lisboa
R. Ernesto Vasconcelos, Cl- piso 3
P- 1600 LISBOA
304
296
:
CHILDREN'S UNDERSTANDING OF CONGRUENCE ACCORDING
TO THE VAN HIELE MODEL OF THINKING
LILIAN NASSER
UNIVERSIDADE FEDERAL DO RIO DE JANEIRO - BRAZIL
KING'S COLLEGE
UNIVERSITY OF LONDON. - ENGLAND
In this work, descriptors for the van Hiele levels
of
thinking in the concept of "congruence of -shapes" are
suggested. Four activities were designed and used
in
clinical interviews with English and Brazilian students
aged 13 to 16 years. In Brazil congruence is taught for
malty, in a Euclidean approach, while English students
learn congruence informally, through transformations.
The analysis of the interviews give evidence that,
de
spite the approach used, the responses fit the
leveT
descriptors suggested, and that the levels are hierarchical.
The van Hiele model of thinking in Geometry establi
shed by Pierre van Hiele and Dina van Hiele-Geldof in the late
50's (van Hiele, 1959) has been investigated according to seve
ral points of view in the last decade.
Some research studies
addressed the relation between the levels achieved by a student
in different tipics of Geometry (Mayberry, 1983; Gutierrez and
Jaime, 1987; Nasser, 1989).
Usiskin (1982) and Senk (1985) in
the relation between the van Hiele level achieved
by a student at the beginning of the year with
his
(her)
achievement on Geometry tasks
during the year,
while
Fuys,
vestigated
Geddes and Tischler (1988)
studied the effects of instruction
modules on students' van Hiele levels.
Burger and Shaughnessy
(1986) provided a characterization of the van Hiele levels of
development in Geometry based on responses to clinical inter
view tasks concerning triangles and quadrilaterals, and sugges
ted that the same kind of investigation
should be carried out
concerning other geometric concepts.
This work is part of research to investigate if the
learning and understanding of "congruence" by Brazilian secon
dary school students can be improved when the instruction is
based on the van Hiele theory.
To obtain a picture of how the
concept of "congruence"
is acquired, this work was developed
297
305
with the purposes of:
(a) Suggest descriptors for van Hiele levels in "congruence";
(b) Develop activities fitting the descriptors in (a) to be used
in clinical interviews;
(c) Through the analysis of the interviews, check if the
leyel
descriptors suggested in (a) are acceptable.
The topic of congruence is taught in different
in England and in Brazil. In England,
ways
through a transformation
approach, congruence appears informally, as "the same shape and
size". There is no attempt to prove the congruence of triangles,
but transformations that preserve length can be used to justify
the congruence of shapes. On the other hand, in Brazil, through
a traditional approach to Euclidean Geometry,
taught using deductive reasoning.
congruence
is
Students are asked to write
proofs based on the cases of congruence of triangles (SSS, SAS,
ASA) to justify other properties of shapes.
Analysing the descriptors of the van Hiele
levels
given for traditional Geometry by van Hiele (1959),
Hoffer
(1983), Burger and Shaughnessy (1986) and Fuys, Geddes
and
Tischler (1988), the following descriptors can be suggested for
the van Hiele levels in congruence:
Basic level - Recognition of congruent shapes
only based
on
appearance. Orientation is considered as a relevant
attri
bute for congruence. Corresponding elements of
congruent
shapes are not yet perceived in isolation.
Level 1 - Recognition of congruent shapes
relying on measure
ments and/or fitting on top of each other.
Orientation is
seen as irrelevant.
Properties of congruent
analysed (necessary conditions).
Level 2
Establishment and understanding
tions for the congruence of triangles.
fy formally the congruence of shapes.
Level 3 - Ability to reason logically, in
congruence of triangles.
of
shapes
are
sufficient condi
No attempt to justi
order to justify the
Informal proofs can be
attempted
(using the cases of congruence or transformations).
Level 4 - The importance
understood.
of rigour
Ability to
write
298
36
in
a
demonstrations
formal proof using
is
the
cases of congruence or triangles or transformations.
Four activities were
designed fitting the level des
The activities are described in detail bellow
together with the expected responses to them.
criptors above.
Activity 1 - Recognition of congruent shapes:
The five cards on fig. 1
were shown to the student,
who was asked if the pair of shapes in each card was "congruent"
(or "the same shape and size") or not, and to explain.
DD
N]
\D
Fig. 1: Cards used in Activity 1
Students were offered measurement instruments
or, if
the possibility was mentioned, they could use tracing paper or
fold the card to check if the shapes matched. Expected respon
se: Basic level-recognition relying only on appearance, orienta
tion considered to be relevant; Level 1 - recognition relying
on measurements or transformations.
Activity 2 - Sorting congruent triangles:
The material for the second activity consisted of ten
cutouts of triangles numbered, and with different colours
in
each face (fig. 2).
Fig. 2: Triangles to be sorted (Reduced size)
299
The student was asked to sort the triangles in groups
of congruent triangles and to
mention
common
properties
exhibited by congruent triangles. Expected response: Basic level
- sorting based only on appearance; Level 1 - the
superimposing the shapes is used to justify the
strategy
sorting;
of
state
ment of necessary conditions for the congruence of triangles.
Activity 3 - Establishment of sufficient conditions:
In this activity, the student was asked whether it was
possible to draw triangles with different shapes
having
the
features shown in each card (fig. 3).
Sides measuring 3 cm
A 60° angle
and
5 cm
a side measuring
7 cm
'5 cm
Sides measuring 4 cm and 7 cm
forming
a
50°
Fig. 3: Cards
angle
Used in Activity 3
If the student could not answer promptly,
encouraged to try drawing one or more triangles
(s)he
fitting
was
the
features on each card and, then, answer the question. Expected
response: Level 2 - at the end of the activity, sufficient
con
ditions for the congruence of triangles (cases
of congruence)
could be established by the student, when required by the inter
viewer.
Activity 4 - Logical explanation:
This task was designed in order to evaluate students'
logical reasoning and ability to justify the congruence of
two
triangles in a more complex figure. The task is shown in fig. 4.
Explain why AABC is congruent to
Given: BC = CE
LSCDE.
A
E
AC = CD
_
Fig. 4: Task requiring a logical explanation
308
300
Expected response: Level 4 - rigorous proof
but
for
the
logical
congruence of the triangles; Level 3 - informal,
explanation for the congruence. Acceptable answers at this
tracing paper)
that
el could vary from showing (with
triangle was the image of the other after a rotation,
or
lev
one
to
having
observe that angles AeB and OD were opposite
the same measurement, and using activity 3or the cases ofcongruen
angles,
ce, conclude that the triangles were congruent.
Sample: The sample for the interviews was composed of
varying
attainments
15 English and 10 Brazilian students of
aged 13 to 16 years.
Results of the interviews: Only two students have con
sidered orientation as a relevant attribute for congruence (one
could
correctly
English and one Brazilian). All the others
solve the first activity. One English student used folding to
verify the congruence of the shapes in cards one and five, while
activity
all Brazilian children used measurement. The sorting
was easily solved by all students. The strategy of superimposi
tion was used by all the English children, and by three Brazi
Tian ones. The other seven measured the sides of the triangles.
When asked about the necessary conditions for the congruence of
sides.
triangles, all students mentioned the same lengths of
Some necessary but not sufficient conditions were mentioned:ten
of
English and six Brazilian students mentioned the equality
the angles, and four English students said that
the
triangles
the
stu
had the same area. Activity 3 was more demanding for
dents without a formal Geometry course. Only seven of the Eng
lish children could come to a conclusion about the sufficient
conditions for the congruence of triangles. For the seven Bra
congru
zilian students which had already studied the cases
of
ence, the answer should be easy, but three of these
could
not
without a
remember anything. On the other hand, two students
had
Geometry course could reason based on the triangles they
drawn and conclude the cases of congruence. Only three students
in the English sample could give an acceptable explanation for
at
students
the congruence of triangles in activity 4. Some
tempted to explain using tracing paper, but failed to conclude
the task. From the Brazilian sample, five students with a Geom
two
but only
course tryed to answer activity 4,
etry
301
BEST COPY HAMA LE
or.
309
succeeded. The others had some ideas of how the proof should be
written, but could not organize them clearly.
Comments:
The responses obtained seem to fit the lev
el descriptors, despite the approach experienced by the student.
Also, the levels appear to be hierarchical, since, in general,
students performing at a certain level were sucessful in tasks
demanding a lower level performance.
The only exception were
the two students who considered shapes with different orientati
ons as non congruent, but could solve activity 2 at a level 1
performance. English and Brazilian students used different stra
tegies to solve the tasks due to the approach to Geometry they
had.
The low familiarity of the Brazilian students with manipu
lations and concrete materials was shown by the strategies they
used to solve the tasks. Although used to manipulations,
the
English students very seldom used transformations when solving
the activities.
References
BURGER, W. and SHAUGHNESSY, J.M. (1986): Characterizing the van
Hiele Levels of Development in Geometry. JRME, vol. 17, n2
pp. 31-48.
1
FUYS, D., GEDDES. D. and TISHLER, R. (Eds.) (1984):
The
van
Hiele Model of Thinking in Geometry among Adolescents. JRME,
Monograph n2 3. Reston, VA: NCTM.
GUTIERREZ, A. and JAIME, A. (1987): Estudio de las Caracteristi
cas de los Niveles de van Hiele. Proceedings of PME-XI, vol.
3. pp. 131-7. Montreal.
HOFFER, A. (1983): Van Hiele based Research. In: Lesh, R.
Landau, M. (Eds): Acquisition of Mathematics Concepts
Processes. New York: Academic Press.
and
and
MAYBERRY, J. (1983): The van Hiele Levels of Geometry
Thought
in Undergraduate Preservice Teachers. JRME, vol. 14,
n2 1,
pp. 58-69.
NASSER, L. (1989): Are the van Hiele Levels Applicable to Trans
formation Geometry? Proceedings of PME-13, vol. 3, pp. 25-32,
Paris.
3 10
302
SENK, S. (1985): How Well do Students Write
Geometry
Mathematics Teacher, vol 78, n2 6, pp. 448-456.
Proofs?
USISKIN, Z. (1982): Van Hiele Levels and Achievement in Seconda
ry School Geometry. Columbus, OH: ERIC.
VAN HIELE, P.M. (1959): La Pensee de l'Enfant et la
Geometrie.
Bulletin de l'Association de Professeurs de Mathematiques de
l'Enseignment Public, 38e anne, n2 198.
303
311
Measurement
312
PROSPECTIVE PRIMARY TEACHERS' CONCEPTIONS OF AREA
Cornelia TIERNEY, Leslie College, Cambridge MA, and Phillip Institute of Technology, Melbourne
Christina BOYD, Philip Institute of Technology, Melbourne
Gary DAVIS, La Trobe University, Melbourne
Area misconceptions of a population of prospective primary students were examined. Relations with
other studies of area misconceptions are drawn, and use is made of the notion of cognitive
"signpost" to explain what it is that the students do in their work on area. The emphasis in the
student tasks was on comparison of regions lry cut and past methods prior to judgements about
numerical values of area.
Area seems to be one of those concepts that is so intuitive and deeply embedded in everyday life that
attempts to formulate carefully what is meant by it are seen as pedantry by a majority of elementary
student teachers. Student teachers expect that what they learned, or imagine they were taught, in high
school is an adequate base for their teaching of area concepts to children. However, as soon as one
begins to probe their understanding of area one discovers that there are significant numbers of them
who have no mental image of area at all, depend on memorized formulas, and incorrectly use linear
measures for computations of area.
Deborah Ball (1988) writes about her concern that mathematics educators themselves take for
granted that prospective teachers are well enough prepared in school mathematics content from material
they dealt with when they last studied these topics in school. In practice these future teachers reveal
some basic misconceptions that would make it difficult for them to to teach correct applications of
formulas, let alone concepts. Ball uses as an example the prospective teachers' lack of knowledge of the
relationship between area and perimeter. By asking students to respond to a hypothetical pupil's
conjecture, she found that 80 percent of primary teacher trainees, and an even greater proportion of high
school mathematics teacher trainees, believed that area always grows with perimeter and they were
satisfied that a single example "proves" such a conjecture.
We have recently looked at the views of area and perimeter held by two populations of prospective
primary teachers, those from a university where the entrance standards are reasonably high as well as
those from a teachers' college where the entrance standards are somewhat lower. Responses of our
university students to a question similar to that of Ball's was one of the things that drew our attention to
the inadequacy of their conceptions of area and perimeter. Those of us who teach at the teachers'
college had noticed many different occasions when our students demonstrated misconceptions about
area in their college mathematics work or in designing curriculum for primary students. Some examples
are:
Students frequently generalized the formula for finding the area of a rectangle to plane figures other
than rectangles.
Many students think area is "Length by Width". When we asked our college students what they would
teach a ten year old child about area, 80% of them drew a rectangle and wrote "L X W" or "L by W"
near it. Some of these students placed arrows around a rectangle in a way which denoted perimeter
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rather than area. The 20% of the students who did not mention Length x Width mentioned no formula:
they were the only students to draw or name figures other than rectangles and they defined area as the
space inside a figure.
Many students who used the Length x Width formula for calculation did not move from linear units to
square units. They either presented their answer in terms of the formula, without a numerical result, as
in "the area of the board is 1.2 metres by 2.4 metres", or they labelled answers with linear units.
Students often generalized changes in linear dimensions to changes in area. In responding to
questions about the effect of halving or doubling the lengths of the sides of a square, most students said
that the area was also halved or doubled.
It was common for students to use whatever numbers were available (often lengths of sides when an
altitude was appropriate) to get answers. When numbers were not available students would count
something such as the nails or squares around the perimeter of a shape.
As we discussed our observations from the two populations of students, we realised that them was
evidence to indicate that a very high proportion of elementary student teachers do not have an
understanding of area which would support their teaching of it even with the aid of a reasonable
textbook. In one class from the college, all of the five students who chose area as a topic to teach to
children directly taught incorrect interpretations of area. For example a student-teacher demonstrated, to
a grade six class, Length x Width as the way of finding area and then asked children to find areas of
some rectangles and some non-rectangular parallelograms for which she provided measures of sides but
not altitudes.
We believed that the situation is as Ball describes. However, the task of mathematics educators is
not only to extend the concepts developed by students in their pm-college schooling but to undo much
of what students mistakenly believe to be true. If our students had any informal primary school
experiences which allowed them to develop concepts of area and perimeter, these have largely been
replaced by strongly held beliefs based on the Length x Width area formula for rectangles.
SIGNPOSTS
In talking about these misconceptions we began to call them "signposts... We chose the term
signposts because it refers to the things on which students focus attention when they feel lost.
This notion of signpost is an analogy. It stems from our own experience that in travelling around a
large and only partially familiar city some individuals are comfortable in heading in the general direction
of where they want to be, discerning where they are in relation to their starting point by certain
signposts. These signposts are most commonly landmarks, but may also be less obvious things such as
the quality of housing in, the style of dress in, or general affluence of, a region. These signposts assist
an individual in feeling.comfortable that they are heading towards their goal and, just as important, that
they can get back to where they started. For other individuals a detailed map is a necessary aid, and the
route needs to follow a prescription worked out from the map. Similar signposts occur in all intellectual
and physical activities. The question is one of orientation and a feeling of balance for an individual: that
they can relate where they am to where they began and where they want to be, and that they do not feel
unbalanced or out of control.
In our view it is this feeling of disorientation that students try to overcome in unfamiliar
mathematical settings by reverting to familiar signposts, as irrelevant as these signposts might appear to
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an observer. The question of how close a student is to a solution of a mathematical problem is not
something that can be judged objectively by an observer. The very notion of distance from a goal is a
subjective notion, different for different people, and depends on how far someone feels that they have
moved towards the goal and away from familiar signposts. Signposts for the teacher may have no use
to the student. For example, to respond to a question about the area of a square formed by doubling
the sides of a square of area nine, one student dutifully drew two squares, labelled the sides of one 3
and 3, and the other 6 and 6, but concluded, apparently without considering the drawings, that the new
square would have an area twice that of the smaller square, or 18.
When students try to calculate the area of an irregular quadrilateral, for example, by using the length
by width formula, what they are doing we postulate is returning to a familiar scene around which they
feel comfortable and do not feel disoriented. It is for the student not at all a question of whether the
formula is inappropriate, but rather a question of how, emotionally, they could move into uncharted
territory without new signposts to help them orient themselves. This view of the way students approach
problems has considerable implications for curriculum and the role of a teacher.
The signposts which the students most frequently in their area work were Length x Width, measures
of sides of figures, and counting. We found we could not separate these misconceptions and deal with
them individually. They were, and perhaps always are, interconnected into a unified construction of
what area might mean to these students. For example, for many students:
area is a measured by the formula Length x Width
to apply the Length x Width formula , one needs to know the measures of the length and width (or of
two adjacent sides) and as these linear measures are in centimetres or other linear units, so the answer is
in the same units
when measures are not given, one should count something (nails or pins, or squares around edges of
figures, for example) in order to get numbers.
These signposts all represent actions which can be taken immediately - namely multiplying,
measuring, counting - and they all result in a numerical answer. However their salient point is that they
are actions with which the students feel comfortable: they are cognitive signposts that prevent them from
feeling lost.
THIS STUDY: PRE-NUMERIC AREA COMPARISONS
We were able to look more closely at the occurrences of the students' use of these signposts in a
mathematics content course at the teachers' college. We spent approximately six hours on tasks related
to perimeter and area of closed figures. After each session students wrote journals about what they had
done and what they had learned and at the end of the work, they did a written 'test' paper. The
observations discussed in this section are of the students' responses made in class and in their journals
to the exercises described above and to the test questions.
The lessons were not planned from the beginning but in the way we usually plan our teaching, by
continually responding to our observations of students. Instead of attacking each of these overused
signposts separately, we tried to move students to a broader conception of area which would include
representations built from experience as well as those learned as social conventions from previous
teaching.
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A mature quantitative thinker's view of the area of a planar region is as a number. This number is
obtained by discovering how many standard unit regions usually squares exactly cover the region
whose area is sought. Of course this might require infinitely many ever smaller copies of the unit, and
so the number that gives the area measure might be quite a sophisticated number - such as 7t. A notion
that is historically anterior, and seems to be also cognitively anterior, to this relatively sophisticated
numerical notion of area measure is that of area comparison of two planar regions: one region has
smaller area than another if it can be moved by rigid motions (or more generally by other area
preserving transformations) so as to lie inside the other region. More generally one allows certain "cuts"
of the first region before rearrangement: these cuts are in practice usually straight line cuts, but could, in
principle, be something more general.
For many students the situation is often quite different. They want to find area as a number, but
often any number at all will do. For example, one of our students told us that she was counting the nails
around the sides of a geoboard so that she would have a number. Another told us that area had never
existed for her because we have no tool to measure it. Of course calculating area is a number task: it is,
as we have indicated above, the task of finding how many standard unit regions, or small copies of a
unit region, it takes to cover the region whose area is sought. A major difficulty for students is that they
confuse what it is that they are counting in order to find area, and they confuse what it is they count
when they measure length. They also appear to have no mental perception that there is a region to be
covered. Without this they seem to count or compute whatever they see. Thus, our first goal in this
study was to encourage students to pay attention to the region whose area was sought, as well as to a
measure of that region. We thought it plausible that if students learned to work with area directly by
covering a region with unit squares or partitioning into regions of known area, they would not be so
dependent on formulas which involved linear measures. We were really asking them to address the
question: "Which of these regions has larger area?" without calculating the area of either region as a
number.
We believe that in order to make sense of formulas, our students need to construct a mental image of
the area as a region which they could focus on and talk about before attaching a number. We wanted
them to change shapes and compare areas to see when the amount of area stayed invariant, when it
halved, and which relationships of area to linear lengths stayed invariant. We wanted them to have the
kind of experience generally provided for primary school children of finding areas directly without
formulas by partitioning regions into measurable rectangles or triangles or covering them with unit
squares and, when helpful, by moving portions of shapes.
We first posed a problem which we believed would require the students to attend to area without
attending to the length of the sides. The problem was this: "Without using a ruler, put some cardboard
shapes in order by area." The shapes which became the focus of attention were a square and a
parallelogram which had the same area (and same base and height) but different perimeters. We
expected that students would compare areas directly by placing one shape on top of another and by
mentally "cutting and pasting" to test if the shapes were congruent.
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JDENT RESPONSES TO SOME COMPARISON TASKS
Instead of placing shapes on top of another, about half the students held the cardboard shapes edge
to edge to compare perimeters. They were attempting to measure shapes not with unit squares as we
suggest but with linear measure of the edges . They seemed to believe that if a shape has a larger
perimeter it also has a larger area.
In trying to order the cardboard shapes by area, students who held shapes edge to edge noticed in a
square and a parallelogram, with the same area, that the lengths of two sides of the parallelogram were a
bit longer than the sides of the square. Other students showed the parallelogram and square to have the
same area by placing them on top of one another and showing the congruence of the pieces which
extended beyond the overlap. After much discussion, a number of students were convinced that a
shape can be the same in area, and at the same time different in area, to another shape depending how
they placed one shape on top of the other. While they agreed that two shapes appeared to have the same
area, they thought that if one of the shapes were rotated ninety degrees, the two shapes would no longer
have the same area. A student supported her theory that longer sides meant larger area pointing out that
"one sticks out longer". Thus these students maintained their belief that if the side lengths were longer
the area would be larger, while at the same time agreeing that if two shapes are held in a certain way one
can see that they have the same area.
This activity exposed confusions between area and perimeter which we had not expected. These
confusions were riot just a matter of word usage. The students had not just reversed the two labels. I n
fact, they had not distinguished for themselves two separate entities which would require two separate
labels. Some students, like Robyn, commented on this: "This was a successful learning experience for
me because before the class I could not distinguish area from perimeter." Others like Kylie didn't seem
to have a concept of area at all: although she writes "area" we suspect that she is equating the word
"area" with the boundary of the shapes. Kylie said: "It helped me understand how different areas can
really equal the same area."
The disequilibrium created by trying to assimilate new notions of area is described by one student
Cloe in her journal:
"We were given a number of shapes and told to organize them according to their area. It was
difficult because we weren't allowed to calculate the area because of a few factors [we did not
allow students to use rulers ] and sometimes when you thought you have them correctly
ordered, you placed two shapes on one another and came up with a different answer. I also
learnt that the perimeter calculations sometimes don't account for anything....I was still unclear
and confused how to work out the area if you can't always calculate length of the object by the
width. I walked out of the class knowing that Length x Width is not always true but was
unsure what to do to come up with the answer to fmd the area of an object if the situation arose
again."
The issue of sorting out differences in perimeter from differences in area came up again when
students were asked to compare the areas of a set of different shapes with a unit square. The set of
shapes included a parallelogram A which increased the length of sides and maintained area as in the
previous example, and another parallelogram B which maintained perimeter instead. Students most
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often concluded that both parallelograms had the same area as the unit square since the first comparison
activity "convinced" them of this.
To demonstrate the difference between A and B we put a "straw" square with pins in the corners
onto an overhead projector and demonstrated what happened when the square was gradually collapsed
to the smallest possible area. When asked at which point the area got smaller they agreed with one of
the most confident students that it was half way down, but found it difficult to decide at which point the
area became smaller in tilting the square a little. Students could see that the area was smaller at the
extreme when there was an obvious visual difference.
As we read the students comments about their understanding, we noticed a pull between what they
observed and what they expected or thought logical. Errors occurred when they were guided by only
their observations or only what they thought logical. Effie, for example, doubted her perceptions when
she measured and found that her ruler and school diary had the same perimeter, but the diary appeared
to her to have a larger area: "I also learnt that shapes can be deceiving. The perimeter may be the same
but the area of the ruler is more smaller or appears to be ? For example because of its boundaries."
Later, when she had made a triangle, a trapezium, and a parallelogram from three identical rectangles of
paper and measured all of the perimeters, Effie was convinced by her logic that the figures must have
the same perimeter as well as the same area. She did not believe that she or her classmates had measured
the perimeter accurately:
"It was good because we were involved in making these shapes and also finding out that all had the
same perimeter [we think she means areal because they were originally the same rectangle. The
perimeter was also the same apart from the errors in cutting the paper and having a slightly bigger or
smaller piece of paper."
Stella, in responding to the same activity, tried to combine her sense that the shapes must be the
same because they were made with the same size piece of paper, with her sense that some of them were
bigger
"The point of the activity was to make us decide whether the area of all the shapes were the
same. The areas were the same, but some larger shapes, e.g. larger trapeziums, gave larger
areas....The trapezium can be 32 square centimetres because the area is still there, its just
structured in a different way."
In the final written test, nearly all of the students described area as the minority of students had at the
beginning--as the space inside a bounded figure. Although the students now expressed agreement with
the instructors in viewing area as a region, some still used Length x Width as the way to compute area
for all polygons. The majority however attempted to integrate their new notion of comparing areas of
regions with area as a fomiula. Some created more complicated routines than multiplying length by
width which however still did not work , for example they multiplied lengths of all the sides of a figure.
Thus, although most students did not integrate notions of comparisons of areas of regions with area
formulas, or correct their confusions about relationships between area and perimeter, they no longer
accepted Length x Width as the all-encompassing solution to finding area. They had apparently given up
some signposts without constructing new ones which would give them more success in finding area.
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DISCUSSION
It seems that three kinds of data are entering into students' responses to these area questions:
social knowledge in the form of a formula their teacher taught them: Length x Width
their visual perceptions
their own supposed logical arguments.
The cut and paste argument makes sense to many students in that they see that if a piece is only
moved rigidly then its area has not changed . This is a part of so called conservation of area as defined
by Piaget and coworkers. Invariance of area under the group of rigid motions is a key basic feature of
area. Indeed it was just this feature of area that Piaget took as the basis for his studies of conservation.
What is not so obvious however, is that there is in fact no external reality of the concept of area to
which we can refer, independent of its invariance under a group of motions, that tells us that it should
be so invariant. In other words if people come to conserve area, as some seem to do, and if we take this
conservation of area to mean its invariance under rigid motions, then they must have constructed this for
themselves as part of their building of the concept of area. A curious question arises here however. If
students accept "conservation of area" under rigid motions yet still imagine that shapes with the same
perimeter have the same area, then is it really area, in the numerical sense, that they are conserving, or is
it just a notion of equality of figures that are equivalent under rigid motions ?
When we saw that students tried to use perimeter to rank shapes by area, we posed questions to
direct their attention to the lack of constant relationship between area and perimeter. Students seem to
confuse area and perimeter and refer to "size" and "big" for both of them. It is entirely possible that the
only perceptual cue they have is the perimeter. We believe that many of them cannot focus on the
region whose area they have to compare or find, and they have learned to associate perimeter with a
formula for area. An often expressed point of view is that the area of a region is the "amount of stuff in
that region". This idea leads naturally for many people to a conviction that if the length of the boundary
of a region is kept fixed as the shape of the boundary is changed then the area stays constant because:
"the amount of stuff hasn't changed". This indicates a failure to understand area as the number of
standard unit shapes needed to cover a region.
Hart (1984) deals with children's understanding of area in her chapter on measurement. She has a
discussion of some nice experiments with 12
14 year old children that tease out apparent confusion
between area and perimeter. These results must however be considered in light of the results in the
doctoral thesis of Izzard (1979). He reports clear indications that perceptual boundary cues of shapes
used in area and perimeter questions have a strong influence on a child's view of which feature is
salient. Many of the secondary school children's misconceptions of area detailed by Hart are paralleled
by our observations of student teachers' misconceptions of area.
This makes us suspect that we are seeing in these young adults a resurfacing of misconceptions that
have lain dormant and unchallenged for a number of years. The recourse to numbers and formulas by
these students is critical in their understanding, or rather their misunderstanding. It is critical in their
ability to adapt their understanding of concepts taken for granted. New situations for learning require
that students at this level synthesize intuition, experience, and logical deduction.The misconceptions
have appeared, perhaps for the first time, in high school use of area formulas. This raises the issue of
mathematical judgements depending on social authority, especially the authority of numbers and
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formulas, intuition and experience, and logical deduction. These students may not have been asked to
consider area in primary school when it might have been acceptable for them to depend on their
perceptions. Perhaps, on the other hand, they feel that only logical reasoning is acceptable, and if they
can't develop their own then they must accept someone else's, in the form of what teachers tell them.
They no longer test their assumptions with perception. Rather, they perceive what they decide to
believe.
Hirstein, Lamb and Osborne (1978) discuss the results of interviews with 106 grade 3, 4, 5 and 6
children, designed to study how children incorporate numbers into their judgement of area. Several of
the misconceptions they observed involve counting units without awareness of the unit's spacecovering character. Analogously, children also counted lengths by counting marks between units of
length, this time showing no awareness of linear units. These authors include a graph depicting the
growth of the concept of area showing that children up to fourth grade gradually acquire a concept of
the unit counting approach to determining area of a figure and children above sixth grade develop a
multiplicative approach, but that there is a gap in between with no apparent connection made between
these methods. Our work with prospective primary teachers indicates that this gap still exists. This then
is a challenge for researchers and teachers - to understand the apparently different mental processes
involved in these stages of dealing with area, and to devise problem solving situations that help students
connect the actions and images of covering, cutting and pasting, and counting with the formulas for the
various figures.
References
Ball, D. (1988) I haven't done these since high school: prospective teachers' understandings of
mathematics. In Behr, M. J., LaCampagne, C.B., and Wheeler, M.M. (Eds.) Proceedings of the
Conference of the North American Group for the Psychology of Mathematics Education 268 274.
DeKalb, Illinois.
Hart, K. M. (1984) Children's Understanding of Mathematics: 11-16. London: John Murray.
Hirstein, J.J., Lamb, C.E. and Osborne, A. (1978) Student misconceptions about area measure. The
Arithmetic Teacher, March, 10-16.
Izzard, J. (1979) An investigation of the effects of spatial and other abilities on children's performance
in area ,volume, and related aspects of school mathematics curriculum, Ph D Thesis Melbourne:
La Trobe University
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Presenting author: Cornelia TIERNEY, Lesley College Graduate School, 14 Wendell Street,
Cambridge MA 02138, USA.
Telephone: (0011 1) 617 868 9600 (work) and (0011 1) 617 864 5317 (home)
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Probability
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PROBABILITY CONCEPTS AND GENERATIVE LEARNING THEORY
Ole Bjorkqvist
Abo Akademi, Vasa, Finland
Earlier research findings on probability concepts are interpreted
within the framework of the generative learning model. The
interpretation is made with reference to the processes of the model,
to the function of long term memory, or to compartmentalization of
long term memory. The social constructivist philosophy is adhered to
through emphasis on viable knowledge.
Introduction.
The generative learning model (Osborne & Wittrock, 1983) is an attempt to
express constructivist views of learning, including some general aspects of the
information processing models of the brain. The emphasis is on the meanings
children (learners) actively construct for words and phenomena. The process of
construction, located in short term memory, involves the generation of links to
long term memory, which is a store of images, episodes, propositions, and skills.
After successful testing of tentative links between sensory information and
memory, meaningful understanding is reached, and the results are subsumed
into long term memory.
The most significant use of the model is in science education where it proposes a
frame for the study of alternative concepts of phenomena in science. It also links
the study of conceptual structures to real world teaching.
In mathematics the subject area of probability shows similarity to elementary
science. It involves concepts that have been constructed in an informal way
before systematic teaching, common-sense thinking, counter-intuitive
phenomena, a mixture of induction and deduction, and, above all, quantification
of properties and relations that present themselves qualitatively in much of
everyday life ("impossible", "more likely than", "good chance", etc). In most
countries the mathematization of the concepts of probability belongs to the
secondary school curriculum, and many students will not experience it
systematically within the school buildings.
In the absence of a precise language in which the learner can describe his
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personal constructs, the alternative concepts of probability constitute a rich world
to explore. One may start from the corresponding mathematical structures and a
theory of their development (Piaget Sr Inhelder, 1975), from the application of
generalized cognitive processes to probability (Fischbein, 1975), or from the
application of probability as compared with other influences in human decisionmaking (Tversky Sr Kahneman, 1974, 1981). It would seem that the framework of
the generative learning model is just as natural.
Some philosophy.
According to radical constructivism (Lerman, 1989) it is beyond the power of an
outsider, such as a teacher, to know that a learner has come to an understanding
of a concept. Understanding is subjective, a sense of freedom from
contradictions sometimes coupled to a sense of completeness of knowledge. A
learner therefore needs a way of comparing his knowledge with standards that he
accepts, i.e., objectivity is located in the social domain rather than the
transcendental. The process of teaching is essentially communication of world
views and appropriate organization of the environment of the learner, to make
active construction of meaning possible.
The remarkable homogeneity of world views in areas like science and
mathematics can be explained with a darwinistic argument. The world views
that have survived are judged to consist of viable knowledge (von Glasersfeld,
1987). "Viability can be construed as a continuous variable, which in science and
mathematics is almost dichotomized (viable knowledge equalling truth or a good
model). The difference in terminology may be of less consequence to a scientist
who is not philosophically inclined, but, interpreting alternative concepts in
elementary science and mathematics, "viability", as assessed by the teacher,
provides the criterion of quality sometimes needed to justify educational
decisions.
On an individual scale, the generative learning model locates knowledge in long
term memory. Viable knowledge is a set of images, episodes, propositions, and
skills that will survive repeated testing when the individual actively generates
links between sensory information and long term memory.
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Interpretations of earlier findings.
The study of probability concepts has produced a vast amount of information in
the form of test results and observational records, very often in connection with
gambling. The analyses, even though starting from different theoretical
viewpoints, have produced some similarities in the form of heuristics and biases
that are repeatedly encountered (Hawkins & Kapadia, 1984; Hope & Kelly, 1983;
Wagenaar, 1988; Walter, 1983). Many of those findings, even if paradoxical, can
be given quite satisfactory interpretations within the framework of the generative
learning model. The analysis will here be divided into interpretations that refer
to the processes of the model itself, to the function of long term memory, and to
compartmentalization of long term memory.
It can be noted that the generative learning model implies an intrinsic search for
coherence, if meaningful understanding is to include freedom from
inconsistencies at an individual level. This is akin to the "quest for certainty"
basic to science itself. The possibility of reaching a state of coherence or certainty
seems evident to anyone who has had intuitive cognitions, which have intrinsic
certainty as one characteristic (Fischbein, 1987).
Probability concepts are sometimes classified as either intuitive or formal. This
is not appropriate, as even highly formal concepts share many of the
characteristics of intuitions, primarily the factors contributing to immediacy,
namely visualization, availability, anchoring, and representativeness. However,
probability concepts are not intuitive unless they exhibit intrinsic certainty, selfevidence, perseverance, and coerciveness (if you accept the definition of
Fischbein), and this is rarely the case.
The immediacy of probability concepts is related to the process of testing
tentative links between sensory information and long term memory. The
extensive use of heuristics, whatever their nature, is similar in the sense that it
speeds up the processing. Heuristics also tend to reduce uncertainty and thus
give partial coherence. Obviously probability concepts are in no way unique here,
neither with respect to immediacy nor to the function of heuristics.
Some of the specific biases discussed by Wagenaar (1988) in connection with
gambling behavior can be interpreted with reference to the processes of the
model.
Confirmation bias, the preference for information that is consistent with one's
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own views or even disregard of evidence to the contrary, exemplifies the search
for coherence at the stage when sensory information is only being received. In
general, a biased learning structure, the tendency not to analyse the existing
alternatives according to their statistical or theoretical weight, reflects parsimony
in the generation of links. Gamblers often develop their strategies from
incomplete analysis of information that maximizes the hope for confirmation of
conjectures.
In retrospect, people are not surprised about what happened, and even believe
that they did predict the outcome. This is the hindsight bias, which can be
associated with the last phase of subsumption in the generative learning model.
One of the best known common characteristics of probabilistic reasoning is the
view that prediction of an event cannot be detached from the outcome of similar
independent events in the recent past. In gambling this is the sequential bias or
the negative recency effect. The dependency on recent empirical evidence is
connected with the temporal correlations between sensory experiences which are
so often found in physical science and in everyday life, and which also constitute
the basis for causality. The tests of tentative links to long term memory follow
time-dependent patterns.
Among the factors contributing to immediacy, representativeness, or the
tendency to judge single events as exemplars of categories of events, can be seen
as crucial to the generation of new links. Representativeness can thus be a
positive heuristic. It can, however, be used to infer unwarranted properties in
the hypothesized category of events.
Turning now to the function of long term memory, both representativeness and
the other factors of immediacy, availability, anchoring, and visualization,
imply the superiority of a rich store of images, episodes, propositions and skills.
For example, a formal conceptualization of probability requires propositions
usually acquired in school. In other cases, skills may have been acquired without
formal education (Acioly & Schliemann, 1986). The specific linguistic problems
often associated with probability concepts (Carpenter et al., 1981) may be seen as
deficiencies in the extent or organization of the memory store, as may the
concrete information bias, characterized by vivid or conspicuous incidents
dominating abstract information, i.e., the attribution is to specifics rather than
generalities.
The problem framing often has decisive influence when one is chooses a
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strategy for solution.The images and episodes saved in long term memory serve
as sources of meaning when similar problems arise. Reliance on habits can be
interpreted with reference to skills or episodes. Episodic memory also has a
central role in the establishment of illusory correlations, exemplified in
superstition, which is a frequent element in gambling situations or in layman
weather forecasting.
Much of the memory store may be non-efficient, in the sense that the modeling
of probability is mathematically deficient. For example, people tend to confuse
conspicuous events with low probability events, and also to make probability
estimates on the basis of absolute rather than relative frequencies. However, the
memory store may represent viable knowledge on a personal level in spite of its
mathematical shortcomings.
The compartmentalization of long term memory, finally, can be seen as a key
factor in the interpretation of some other phenomena in accordance with the
generative learning model.
In gambling, there is a tendency for people to have an illusion of control over
the game, even if they on another level are completely aware of the negative
mathematical expectations. Others exhibit flexible attribution, with successes
due to their own skill and failures due to chance or bad luck. It has been shown
(Wagenaar,1988) that many people regard skill, chance, and luck as three quite
different concepts which together determine the outcome of gambling. As a
consequence of this, people accept unfavorable bets or continue gambling after
losing, which constitutes paradoxical behavior.
The switching of problem solving strategies, not specific to probability but very
much in evidence, can be seen as the result of the attempts to generate links
searching through one compartment after another in situations where the
problem type is unfamiliar.
As a mathematical concept with three different theoretical starting-points
(classical probability, empirical frequencies, and axiomatic probability) it seems
logical that probability concepts should refer to more than one compartment in
long term memory. There have been interesting test designs to study the
interconnections of the conceptual aspects (Koops, 1981), but remarkably little is
still known. More basic research in this area would be beneficial to the
development of curricula and teaching methods.
327
323
Conclusion.
This attempt to interpret earlier findings about probability concepts within the
framwork of the generative learning model aims at a natural description in the
spirit of constructivism with a social twist.
It is only applicable as far as the earlier research findings apply to a given person.
In particular, many children do not have a rich memory store with which to link
sensory information. The immediacy they presumably exhibit when interviewed
about probability concepts may easily be forced immediacy, and in such case any
tendencies observed are likely to be generalities with little specific connection to
probability.
Alternative concepts are somewhat easier to accept in science than in
mathematics. Alternative concepts in mathematics have primarily been studied
to establish the nature of "misconceptions" and ways to overcome them, which
in effect has become error analysis. To accept truly alternative concepts one must
know enough about them to judge how viable they are. One needs a model
showing how they are constructed. It is suggested that the generative learning
model is a good one for concepts like probability, if and when the empirical
foundation is comparable to the empirical basis for elementary concepts in
science.
References.
Acioly, N., & Schliemann, A. (1986). Intuitive mathematics and schooling in a
lottery game. Proceedings of the tenth PME-conference, London, England, 223228.
Carpenter, T., Corbitt, M., Kepner, Jr., H., Lindquist, M., & Reys, R. (1981). What
are the chances of your students knowing probability? Mathematics Teacher,
74, 342-344.
Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children.
D. Reidel, Dordrecht, Holland.
Fischbein, E. (1987). Intuition in science and mathematics. An educational
approach. D. Reidel, Dordrecht, Holland.
328
324
von Glasersfeld, E. (1987). Learning as a constructive activity. In: Problems of
representation in the teaching and learning of mathematics, ed. C. Janvier.
Lawrence Erlbaum Associates, Hillsdale, New Jersey.
Hawkins, A., & Kapadia, R. (1984). Children's conceptions of probability - a
psychological and pedagogical review. Educational Studies in Mathematics,
15, 349-377.
Hope, J., & Kelly, I. (1983). Common difficulties with probabilistic reasoning.
Mathematics Teacher, 76, 565-570.
Koops, H. (1981). Zum Wahrscheinlichkeitskonzept bei Kindern im
Grundschulalter - Darstellung und erste Ergebnisse einer Untersuchung. In:
Stochastik im Schulunterricht. Holder-Pichler-Tempsky, Wien.
Lerman, S. (1989). Constructivism, mathematics and mathematics education.
Educational Studies in Mathematics, 20, 211-223.
Osborne, R., & Wittrock, M. (1983). Learning science: A generative process.
Science Education, 67, 489-508.
Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children.
Rout ledge & Kegan Paul, London.
Tversky, A., & Kahneman, D. (1974). Judgement under uncertainty: heuristics
and biases. Science, 185, 1124-1131.
Tversky, A., & Kahneman, D. (1981). The framing of decisions and the
psychology of choice. Science, 211, 453-458.
Wagenaar, W. (1988). Paradoxes of gambling behavior. Lawrence Erlbaum
Associates, Hillsdale, New Jersey.
Walter, H. (1983). Heuristische Strategien und Fehlvorstellungen in
stochastischen Situationen. Der Mathematikunterricht, 29, 11-23.
325
329
SOME CONSIDERATIONS ON THE LEARNING OF PROBABILITY
CAges 10-11 and 14 -16 years)
Ana Merle Oj eda Salazar
King's College. London
Centre for EducatiOnal Studies, England
Seccien Matematica Educative del
CINVESTAV del IPN, PNFAPM. Mexico
In this work we examined the consistency of answers to
groups of questions along the analysis of a discrete random
situation posed to British children aged 10-11 and 14-16 years.
In the experimentation, the situation was presented to the
pupils in two slightly different contexts:
For the older
children, another variable was whether or not they had had a
previous
introduction to
Probability.
Among our general
hypotheses there was the influence of the context in the
answers to groups of questions concerning impossible, certain,
.complementary and compound events and conditional probability.
This report concerns part of a study carried out in order
to throw light on some aspects involved in the learning of
Probability at pre-university levels.
Usually,
the teaching of Probability is based on the
solution of specific problems demanding an interpretation using
the techniques taught, mainly in the basic grades of education.
In this part of the study we did not ask the students different
questions referred to different problems,
and these last
referred in their turn to different contexts. Instead, we posed
a random situation, and through a Calthough very restricted)
set of questions, pupils were led to its analysis.
One of the variables included in this study was the
context in which the situation is presented Cthat is how the
is
referred). We proposed two discrete contexts
differing in what one of them can be considered more familiar
to pupils than the other. The contexts posed were the throw of
two dice, usually known from everyday games, and the other
situation
This
study
VOA
supervised
by
Dr.
Kathleen
327
BEST COPY MAMA LE
ID
330
Hart.
based on the urn model, which is generally introduced at school
for didactical purposes. In particular we wanted to see whether
pupils' preference for one result dominated their prediction of
the most probable event.
We were also interested in pupils' answers according to
a
previous
instruction
on
they
had
received
Probability. Answers to questions about impossible, certain and
complementary events were of interest to us.
whether
Along the questions posed we were interested in verifying
some results obtained from other
research having different
and
concerning causal
diagnostic reasoning studied by Kahneman and Tversky [1982].
These authors found that people judging conditional probability
assign higher probabilities to the conditioning event when this
concerns the causes of a random result (causal reasoning) than
when it refers to its effect (diagnostic reasoning, that is
characteristics,
such
as
the
one
reasoning about a posteriori probability).
Some
questions
were
posed
about
the
idea
of
exchangeability Obchangeabilita) for the analysis of compound
events. This idea was studied by Lecoutre and Durand [1988],
without the intervention of random variable.
Unlike the last study quoted, in this pupils were asked to
consider the situation posed through a random variable. The
main purpose was to look at the aspects pointed out above, not
in a local perspective, but from a more general point of view.
We
wanted
several
interrelations
to
be
notions
of
considered
Probability
and
their
in the analysis of
the
situation. Our framework was Heitele's [1975] proposition. This
author considered ten fundamental ideas to be the guide for a
curriculum in stochastics. These ideas are
* To assign a number from (0,1] to a random event to
express its probablity of occurrence
* Sample space
* The addition rule
* Independence
* Equiprobability and symmetry
* Combinatorics
* The urn model and simulation
* Random variable
* The law of large numbers
* Sample.
331
328
Heitele suggested the introduction of these ideas by posing
complex examples in which not only basic notions could be
studied, but their interrelations as well Csee Ojeda, (1985]).
Heitele's ideas constituted the framework for our study of
the variables already mentioned involved in the learning of
Probability.
THE POPULATION AND THE METHODOLOGY USED
A questionnaire was given to 63 students aged 14 -16 years
and 23 pupils aged 10-11 from three different British schools.
29 of the older pupils had already received an introductory
course
Probability
of
according to their
The
Probability
school.
and
were
considered
as
a
top
set
mathematics performance at the
had
not
yet
been taught
pupils
general
remaining
The questionnaire consisted of two parts, but the aspects
concerning this report correspond to only the first one.
Two forms of the questionnaire, called A and B, were
designed by only varying the context of the situation proposed.
In Form A the context was
Two
ordinary dice. one white and one red, are thrown
at
the same time.
The
number of
dote
on the
top faces are
added.
The situation in Form B was stated as
Two
cloth
bags,
one
white
and
six squat sized marbles, labelled i to O.
In each bag the marbles are vett mixed.
the
other
red,
each
hold
Without
looking, two marbles
are taken out, one
from
each
bag. The numbers drawn are added And then, each marble is
put back into its bag.
Within each form, all the questions we referred to corresponded
to the same context. Other than the context, the situations,
questions and their order were exactly the same for the same
age.
Most of the questions were multiple choice although some
open questions were posed for getting more information. The
questionnaire was a little different for the younger pupils as
329
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ULl.41
332
we wanted to avoid additional difficulties duo to fraction
numbers in the options proposed rather than to probability
notions. So, some questions posed to the older pupils were not
included in the questionnaire for the younger children, who
were posed 25 questions instead of 27. As the vocabulary used
did not include technical terms it was expected to
be
understood by children aged 10-11 years.
The questionnaire was passed during mathematics class time
in the Spring of 1989. The two forms were given in alternate
form according to the registration list of the class.
GENERAL STRUCTURE OF THE QUESTIONNAIRE
Although the situation posed in the questionnaire may seem
in general its understanding requires a method for
organising the ideas.
So after a few questions with a
simple,
qualitative approach, pupils were asked to fill in a table to
produce the sample space so as to have it at hand. This was
followed for a quantitative approach.
There were several
relationships between the different questions in order to see
the consistency of the performance.
The distribution of the questions according to the
fundamental ideas and some of the objectives are shown in Table
1.
QUESTIONS
FUNDAMENT.
IDEAS
Norm elief
Samplespace
Additicgt.
Independ.
QUESTIONS
14-16
10-11
14-16
2,3,4,5
2,3,4,5
Impossible
event
8,12
3,12
6
Certain
event
2,19
2,19
S.P
C"PLOTNI
9,16
9.10.20
6
10
11,14
11,14
gA3'4W,,:.
14
14
COMbi.n010.
Ileiel4L,
= Ttgn.
FORM B
St,ntilti:
0,0,10
LawnLACM
29,24
Exchange
9.1617 17:13,
10
cign5ttign: 19,20
21,22
Wiral,
io
_______on.
..=17-
28,24
FORM 8
IgTIflOnC
0,9,17f.18
TABLE 1. Distribution of
the Questions according to
the Objectives!
25
330
333
10-11
21,22
20
22
RESULTS
Younger children did better on eighteen questions in Form
B Curn model) than in Form A (dice), although the difference
Only
was bigger than 20% in questions 8, 11, 14, 17 and 22.
the question 13 (certain event) was correctly answered by more
than 20% of the pupils with Form A than with. Form B.
who were not taught Probability
The older children
performed better in questions 8, 0. 10 and 11 (complementary
event and sample space) with Form A than with Form B. in more
than 20%. There was a significant difference in performance,
better with Form B than with Form A. in questions 7 (sample
space) and 13 Ccertain event).
Pupils who received an introduction to Probability did not
show, in general, a difference in their answers according to
the context, except to questions 10 and 13 concerning with the
sample space and the certain event, which were better succeeded
with the urn model.
The questionnaire began asking pupils for their preferred
Question 27 C25 for the younger pupils), at the end of the
analysis proposed, asked pupils to predict the sum of one trial
of the situation posed. About 20% of the younger children
answered this question correctly and almost all those with a
previous instruction, both with Form A and B. There was a
difference of 15% in the correct answering of this question,
better Form A than B, in the older children without a course in
Probability.
its
Table 2 shows the coincidence of the correct prediction,
253
and
the
prefered
sum:
justification (question
AGE 4
FORM A
FORM B
TABLE 2.
14-15
10-11
NO COURSE
COURSE
6%
21%
50%
36%
15%
0%
Coincidence between the most probable sum, its
justification and the prefered sum.
BEST COPY AVAILABLE
331
334
Table 3 shows the success for the sets of questions
concerning the concepts we were interested in. The changes in
the numeration of questions according to the age of pupils were
put into brackets. COND- stands for conditional.
FORM
QUESTIONS'
AGE4
4
410-11
FORM
A
B
14-16
14-16
10-11
NO-COUR COORS
NO-COUR COURSE
4
4
4
63%
100%
50%
'23%
6%
29%
0%
29%
50%
COMPLEMENT le,te,zo
(0, 10)
31%
25%
79%
40%
20%
56%
EXCHANGE
31%
25%
64%
50%
14%
79%
0%
14%
7%
7%
8%.
13%
57%
0%
14%
64%
23%
44%
03%
60%
50%
100%
23%
25%
71%
10%
43%
64%
IMPOSSIBLE
CERTAIN
a3
2,
EXCHANGE
46%
3,12
i15,16,17
1
10,20
EXCHANOECOND 21,22
(10,20)
CAUSAL-corm!i2a,24
(21.22)
71%
93%
.
DIAONOSTICi
22 (20)
COND:
TABLE 3. Comparison of the success in some of
the objectives.
There is a great difference in the results of questions
concerning impossible events and those referring to certain
events. Questions 2 and 13 asked about certain events. Question
2 posed a qualitative approach, whereas Question 13 was
13.
What io the chance of getting a slum between 1 and 13?
a)
0
b)
c)
.1
13
d) Non, of these
(
[
This question was expected to be difficult for younger pupils
and those without an introduction to Probability, mainly
because of the assignation of number 1 to the certain event
might not seem natural to them. Nevertheless, the results from
pupils who had been taught Probability would seem to suggest
that the difficulty is rather the recognition of the certain
335
332
event.
For the older pupils, questions 9, 19 and 20 concerning
the complementary events of those introduced in questions B,
17, and 19, respectively, were better answered than these last.
In particular, question 19, which posed an event defined as
. .
. getting only one 8 from any of the two dice (bags.. .
.
was as difficult for pupils who had done a course C14% correct
answers with Form A and 21% with Form B) as for those who had
not C19% correct answers with Form A and 14% with Form M.
exchangeability
the
idea
of
requiring
Questions
(permutation between the elementary events composing the event
under consideration) were difficult for children without an
introduction to Probability.
Unlike the younger pupils, the children aged 14-16 showed
an ordered; gradual improvement in performance, which referred
to conditional probability, that is, questions 21, 22, 23 and
24 (corresponding to 19, 20, 21 and 22 for the younger
children).
Finally, questions appealing to causal reasoning were
better answered than those requiring diagnostic reasoning by
all the population.
FURTHER RESEARCH
Although there appears to be a better performance of
children (mainly the younger) with the questionnaire using the
urn model than the context of dice, the difference was not
significant, despite the length of the context with urns, which
could be a reason for a poorer performance. Nevertheless, therie
is still the doubt of whether they can transfer Cor rather
repeat) a correct answer given to a question posed in one of
the contexts to a similar question posed in the other context.
In spite of
the success shown by children previously
BEST COPY AVAILABEE
336
taught
the review of
Probability,
answers
to
given
groups
of
consistency of
the
concerning
questions
the
same
the
appears to reveal that even after an introductory
course, elementary notions of Probability, such as certain and
compound events, are difficult to handle.
notions
REFERENCES
Heitele,
D.:
1975,
epistemological view on fundamental
Educational Studies in Mathematics 6,
'An
stochastic ideas',
187-205.
Lecoutre,
M.
and
P.
Durand,
J.
L.:
1988,
'Jugements
probabilistes et Modeles Cognitifs: Etude d'une situation
aloatoire', Educational Studies in Mathematics 19. 357-368.
A.
M.: 1987, 'Ideas Fundamentales y Actividades Modelo
en la EnseHanza de la Probabilidad. Nivel Medio Superior'.
Cuadernos de Investigation, PNFAPM, Mexico.
Ojeda,
Tversky,
A.
and
judgments
Kahneman,
under
D.:
'Causal
1982,
uncertainty',
in
schemes
"Judgment
in
under
Heuristics and biases" CKahneman, Slovic and
Tversky, Eds.), Cambridge University Press, 117-128.
uncertainty:
337
334
RESEARCH PAPER FOR PRESENTATION AT P.M.E. 14 - MEXICO
AUTHOR: ROBERT PEARD
TITLE: GAMBLING AND ETHNOMATHEMATICS IN AUSTRALIA
Summary
Thus
The phenomenon of gambling is widespread throughout Australian society.
we have an identifiable subgroup of the population for whom gambling,
particularly on horse-racing, constitutes a form of "ethnomathematics".
Children
from this group bring probabilistic knowledge with them to the school
environment.
This study researches what this knowledge is, how it is
constructed and used, and the implications of this in the school system.
Introduction and Rationale
This study will involve research in two broad, independent, yet interrelated
fields of mathematics education.
The first of these is what may be ldosely
defined as "ethnomathematics" within Australian culture.
This part of the
study will involve the exploration of culturally-based mathematical knowledge in
probabalistic and related concepts of a segment of the society for whom the
phenomenon of gambling is inherent in their culture.
The second field employs the ideas of "constructivism" in the learning of
mathematics.
In this section, the research will explore how this knowledge is
used to construct mathematical procedures and concepts.
In addition, the
relationship between these constructs and present classroom instruction in the
topics will be researched in order to determine how such knowledge may be
meaningfully incorporated into classroom practices.
If we view Australian society as a changing, developing multicultural mixture
any ethnomathematical concepts will of necessity be confined to various subgroups.
In this study the term "ethnomathematics" will refer to the inherent
335
338
Those who reasoned in this manner effectively constructed their own "common
denominator" algorithm and employed this to questions of the type of number 1.
Others reasoned:
"3 gives 4 or 7 gives 9 ", so 6 gives 8, 7-6 =1,
1 gives 8/6, so 7 gives 8 +
8/6 = 9 2/6 which is greater than 9 so 4:3 is better ".
It is interesting to note that although those using this procedure sometimes
made errors
both computational and procedural - none was consistently
incorrect and none used the incorrect "additive" algorithm reported to be common
amongst children 12-16 by researchers such as Hart (1984) "erroneous reasoning referred to as the incorrect addition strategy" (p.4).
Such reasoning here would go
"3 gives 4 or 7 gives 9, if 3 gives 4, 7 gives 4 + 4 = 8 which is less
than 9"
Non-gamblers employed a variety of techniques including'the finding of a common
.denominator and conversion to decimals for the first type of question and were
generally unable to attempt the second.
CO
The concept of equal likelihood of occurrences.
Sample Questions;
1.
When a single die is rolled are all numbers 1 to 6 equally likely?
Is,
say, a "six" harder to get than any other?
(4)
The concept of independence.
Sample questions
1.
When a fair coin is tossed it is just as likely to land heads as tails.
If a fair coin is tossed three times and lands heads each time, is it
still just as likely to be heads as tails on the next (fourth) toss?
339
336
(a)
2.
2:9 = 10
:
x
(b)
4:7 = 5
(a)
If $10 is bet at odds of 9:2, how much can be won?
(b)
If $5 is bet at odds of 7:4, how much can be won?
Analysis:
:
y
The "gamblers" consistently employed the same algorithm in all
questions, in nearly all cases successfully. Typical reasoning was:
"2 gives 9, 10 is S x 2, so 10 gives 5 x 9 = 45"
"4 results in 7
4 is one less than 5
This additional 1 gives
x 7 = 7/4
So 5 results in 7 and 7/4 = 8 3/4"
"Non-gamblers" were generally unable to attempt the second question but answered
the first using more traditional school-taught algorithms, though not always
correctly. It would appear that in this situation the gamblers do construct
their own algorithm and apply it in traditional non-gambling situations.
Further research will be conducted in this area.
(2)
Equivalence of fractions and comparison of "odds".
Sample Questions
1.
Which is the larger, 3/5 or 5/8?
2.
Which are the better "odds" 4:3 or 9:7? Why?
Analysis:
Why?
Again, the gamblers constructed their own algorithm, though not as
consistentlk, as in the first instance.
Some reasoned:
"3 gives 4 or 7 gives 9
'So 7 x 3 gives 4 x 7 or 3 x 7 gives 3 x 9
21 gives 28 or 21 gives 27
28 is greater than 27, 4:3 is better".
337
340
This research will therefore employ the use of a number of case studies as the
dominant research methodology.
Research Questions
Twenty case studies are being conducted
This research is currently in progress.
with upper secondary pupils.
Ten of these come from a social background in
which gambling practices are commonly accepted.
The most prevalent of these
practices is betting on horseracing and trotting, though gambling on card games
is also evident.
The other ten pupils are from a similar socio-economic
background but are unfamiliar with such practices.
The research questions include:
What mathematical knowledge (both concepts and processes) do pupils from a
gambling background bring to school with them?
How is this knowledge acquired?
Does this knowledge transfer to use in classroom situations to perform
traditional, related mathematics?
To what extent do these procedures parallel or differ from those employed
by non-gamblers and traditional classroom practices.
Students from both groups were each asked a number of questions in a clinical
interview situation.
The full report will contain the questions, responses,
analysis and implications of the results to the date of the presentation but
some preliminary results from five areas of study are presented here for
discussion.
Results and Implications to Date
CO
Algorithms employed in calculations involving proportions.
Sample questions:
1.
Complete the following proportions:
341
338
Clements.(1988) "it needs to be remembered that often in
Australia there are
unique. factors influencing how children learn mathematics" (p. 5).
Secondly, that the phenomenon is inherently mathematical in nature.
Bishop
(1988a) notes that developing ideas about chance and prediction are important
mathematical activities (p. 106) and that gambling games are part of modern
western society at present (p. 112). Investigations in our mathematical culture
include experimental probabilities (p. 117).
D'Ambrosio (1985b) has stressed the
need for incorporation of ethnomathematics into the curriculum in order to
avoid the "psychological blockade" that is so common in mathematics.
The interrelation between ethnomatics and constructivism results from the use
of the learners' experiences which are culturally determined, to construct
mathematical concepts.
As Davis (1989) notes:
It is now far from a new idea'that mathematical ideas and concepts are
actively constructed by individual children and older people alike (p. 32).
This research will explore how probabilistic ideas and concepts are actively
constructed by individuals in, as Davis says as "intelligent responses to their
environment" (p.32).
Leder (1989) refers to "the growing adoption by contemporary mathematics
educators of constructivist perspectives" (p. 2).
Higginson (1989) refers to constructivism as a conception of knowing and
learning with its emphasis on the active involvement of the learner (p.
11).
Harris (1989) argues that:
The mathematical meanings acquired in the classroom are the personal
constructs of each individual learner and that they are strongly
influenced by past experiences and the social context of the school (p. 81).
339
342
mathematical ideas related to the probabilistic concepts in gambling that some
of the population possess and that might be reasonably expected to be brought to
school by the children of this segment.
This working definition is in keeping
with that employed by other researchers in the field.
Beth Graham (1988) in
researching the ethnomatics of Aboriginal children of Australia uses the term
to refer to "the mathematical understandings that the Aboriginal children bring
to the educational encounter ... the mathematical relationships inherent in their
own culture". (p. 121).
Gerdes (1988) uses the same definition to describe the
intuitive mathematics of the native culture in a post-colonial society.
Carraher (1985) in Brazil uses the term to refer to the knowledge of "the
everyday use of mathematics by working youngsters in commercial transactions"
(p. 21).
D'Ambrosio (1985a) first coined the term to refer to "mathematics which
is practiced amongst identifiable cultural groups" (p. 45) and it is in this
context that the term is employed in this study.
The rationale for the selection of the phenomenon of gambling relies on two
major factors:
Firstly, that the phenomenon is widespread within the culture and is related to
the culture in a unique way.
Award winning Australian author Peter Cary (1987)
in refering to the social history of Australia commented "It was as if the
colony were founded on gambling" (p.263).
In the Australian Newspaper (2613(89)
Phillip Adams refers to Australia as "the gold-medal country of gambling" (p.42).
Statistical evidence showing the monies bet on legal gambling - TABS and
casinos per capita of population supports this, as does the observation of social
phenomenon such as "Melbourne Cup Day".
Thus it may be that in researching the concepts employed in gambling the study
will add to factors unique to our culture.
340
343
The need to do this is supported by
2.
Which
If it is equally likely for a new-born child to be a boy or a girl.
sequence is more likely when a family has four children.
(a)
BBBG, (b)
BGBG, (c)
both equally likely
Shaughnessy (1981) refers to the use of "representativeness" in this estimation
(p.91).
This study will explore the comparative use of "representativeness"
amongst the "gamblers" and "non- gamblers".
(5)
Combinations and Permutations in Probability Estimations.
Sample Questions
1.
A student has eight books at school and decides to take two home.
In how
many ways can this be done?
2.
There are .eight horses in a race.
To win "the double" you must select
the first two (i.e. first and second but not necessarily in the right
order). How many selections can be made?
3.
A committee is to' be selected from ten students.
Which of the following
committee sizes would result in more possible committees:
(a)
8, (b) 2, (c) no difference between 2 and 8.
Shaughnessy (1981) refers to the use of "availability" in making this type of
estimation (p.93).
Again, this study will compare the use of this by "gamblers"
and "non-gamblers".
The results of the questions of item (3) to (5) are presently incomplete.
and other data are to be presented for discussion and reaction.
341
344
These
References
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D'ambrosio, U. (1985a). Ethnomathematics and its place in the history and
pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44-48.
D'Ambrosio (1985b). Sociocultural Bases for Mathematics Education, UNICAMP,
Campinas.
Davis, G.E. (1989). Attainment of rational number knowledge: A study in the
constructive evolution of mathematics. School Mathematics The Challenge
for Change, Deakin University.
Gay, J. and Cole, M. (1967). The New Mathematics and an Old Culture: A Study of
Learning Among the Kpelle of Liberia, Holt, Rinehart and Winston, New
York.
Gerdes, P. (1988). On Culture, Geometrical Thinking and Mathematics Education.
Educational Studies in Mathematics, 19(2), 147 -162.
Graham, B. (1988). Mathematical Education and Aboriginal Children. Educational
Studies in Mathematics, 19(2), 119-135.
Harris, Pam (1989), Context for Change in Cross Cultural Classrooms. School
Mathematics: The Challenges to Change, Deakin University.
Hart, K.M. (1984). Ratio: Children's Strategies and Errors. NFER-Nelson.
Higginson, W. (1989). Beware the Titanic Deckchair: Remarks on Desirable
Directions for Change in Mathematics Education, School Mathematics: The
Challenge to Change, Deakin University.
Leder, G.C. (1989). Mathematics Education: Philosophical Perspectives, School
Mathematics: The Challenge to Change, Deakin University.
Popkewitz, T.S. (1988). Institutional Issues in the Study of School Mathematics:
Curriculum Research. Educational Studies in Mathematics, 19(2), 221-247.
Misconceptions of Probability. Teaching Statistics and
Probability, NCTM 1981 Yearbook.
Shaughnessy, J.M. (1981).
345
342
A Mathematization Project in Class as a
Collective Higher Order Learning Process
Hans-Georg Steiner
The Institute for the Didactics of Mathematics.(IDM)
University of Bielefeld, F R Germany
Various positive classroom experiences with a mathematization program related to mathematical modelling of situations and problems in the social-political domain of decision
making by voting (voting bodies) provide us with a learning context for students in grades
11 12 and an educational setting in which social cognition and social learning, metacognition, learning about learning, communication about communication play a significant role
and can be made a matter of in depth didactical investigations. It is considered important that, in the context and setting specified, these factors are not seen independent from
the content and its epistemological structures. Rather they are viewed as being profoundly
connected with the constitutive interrelation between theoretical concepts, applications,
knowledge development and social interactions in the broad field of mathematics-related
activities in science, education and practice, understood as a socio-historical reality. The
paper is concerned with interpreting the observed classroom processes from these epistemological and socialcognitive points of view and identifies further research questions.
1. The Learning Context
The learning context is a developing and expanding one and grows out of discussions in
class about situations and problems related to the role and functioning of bodies in various
domains of society that reach decisions by voting (Steiner 1986a, 1988). The students
immediately provide a number of examples and find more of them
and more specific
information
by searching in the library or talking to people inside and outside of school:
the Federal House of Representatives, a city council, a jury, an examination board, the
Security Council of the United Nations, a stockholders' meeting, their own class when e.
g. electing a class-speaker etc. Problematic situations and special concerns among the
students come up when trying to describe such voting bodies in general terms like the
given vote distributions (v) on the set of voters (V) and the majority quotient (q), or when
interpreting particular situations like the position of a chairperson in case of a tie, the very
unusual regulations for the Security Council etc.
Stimulating points and questions of debate and personal involvement in the beginning phase
are e. g.: Meaning of and reasons for unequal vote distributions; can there meaningfully be
a member with 0 votes?, meaning of and reason for majority quotients like 2/3, 3/4, i. e.
different from 1/2 (simple majority) or 1 (unanimity); does the generalization 1/2 < q <
1 make sense?, can we describe the decisive role of a chairperson in case of a tie by
giving him or her a bigger number of votes?, can the number of votes be meaningfully
used as a measure of power?, why does the Security Council SC = F v T, consisting of
the Big Five F and the Small Ten T, have such strange by-laws, saying that in nonprocedural matters a proposal is carried if at least 9 of the 15 members, including all big five,
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vote for it?, can we replace these unusual regulations, which do not apply any vote distribution and majority quotient, by assigning a certain vote distribution to the members and
determing an appropriate majority quotient?, what power do the Big Five and the Small
Ten have?, can we explain or define power positions like veto-power, powerlessness,
dictatorship?
Dealing with these questions and problems in the collective learning process goes together
with the development of an enriched theoretical framework starting out from empirical
conceptualizations, based on concrete voting sets, vote distributions and majority quotients and the definition of the (normal) voting body to be a triple (V, v, q) with v being a
mapping from V into the set N of natural numbers, including 0, different from the constant 0-mapping, and q being a rational number with 1/2 < q < 1. The understanding of
V, v, q as variables is the first step to creating a theoretical model which can be put in a
dynamic relation to concrete cases and related problems.
The next step consists in the introduction of concepts like winning-, losing-, and blocking
coalitions, first described as derived concepts with respect to normal voting bodies (V, v,
q) by saying that a subset C of V is called a winning coalition, if it is strong enough to
carry any proposal, i. e. if v (C) > q v(V) in case of q = 1/2 and v (C) > q v(V) in
case of q > 1/2 (and for the losing- and blocking-coalition correspondingly).
By flexibly using the term winning coalition also with respect to non-normal voting bodies
like SC, by e. g. saying that the above identified subsets with 9 elements are the minimal
winning coalitions in SC, the concept winning coalition is also made a variable for a
possible richer theory and a larger domain of applications. In this anticipating way the
concept is creatively used by the students, e. g. to define concepts like "x is a powerless
member" by means of "x does not belong to any minimal winning coalition" or "x has
veto-power" by means of "x belongs to all minimal winning coalitions". The need to
further elaborating the theory and giving it firm foundations appears in connection with the
problem whether all voting bodies of the SC-type can always be weighted, as was found
by the students to be the case for SC itself (by e. g. putting v (I) = 7, v (t) = 1 for all f
a F and t e T, and q = 13/15). This leads to an axiomatic definition of a voting body (V,
IV) of the Security Council type in which W is a non-empty set of subsets of V, called
winning coalitions, and in which the concept of winning coalition W is implicitly defined.
A big breakthrough and an expansion of the context is then made by proving that there
really exist voting bodies of the SC-type (from 5 elements up) which cannot be weighted.
Thus the new theory turns out to be the more general one and gives reason for further
applications and for total reorganization of the developed body of knowledge by embedding
the old knowledge into the new conceptual framework.
We can only mention here that in the course of further explorations some situations appear
which seem to be covered by the theory developed thus far but actually create a kind of
paradox and crisis which lead to an even more extended theory that we called the theory of
a-, 13-voting bodies (Steiner 1969b). We should also mention that the core of the learning
context described can be extended into various directions which have been pursued by
specials groups of students on their own: different theories of power, relations between
non-measurable voting bodies and finite geometries, relations to the theory of games
(Steiner 1976b, 1986a, 1986b, 1988).
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347
2. The Setting, Goals, Activities, and Processes
The classroom experiences were made over several years in gradses 11 and 12 of
West-German high schools (Gymnasia and comprehensive schools), and in grades 9 and 10
within the gifted student program of the Comprehensive School Mathematics Program
(CSMP, at Carbondale Ill.) and the ongoing project Mathematics Education for Gifted
Students in Secondary Schools (MEGSSS at Ft. Lauderdale, Fl.) (see also Steiner 1966,
1969a, 1969b, 1976a). This present report is referring to specially arranged settings in
West-German classrooms in which the mathematization program was made a matter of a
project-type teaching-learning activity lasting for about 3 weeks with a total number of
about 15 classroom-meetings of 45 minutes each. The social organization of students' work
changed between all class discussions, group work and individual work. Each project was
taught by one teacher (including the author) and observed by another; the explorative
teaching was done by the author. The analysis and interpretatiOns are based on observations, related notes, papers produced by students, a kind of individual diary kept by the
students and two kinds of essays written by the students at home at the end of the whole
process when the students had to choose between a systematic deductive presentation of the
knowledge developed in the project and a genectic description of the actual processes in the
course of creating the theory.
As for the goals, the project was explored and designed to give students a special opportunity to experience how in a collective activity a mathematical model related to a relevant
problem domain originally situated outside of mathematics can be developed. They should
learn by doing, that such a model is not to be taken as ready-made mathematics but can be
created in a process of mathematization in which the students themselves can actively play
different roles. First, they are people concerned with the situations and problems in the
political domain of voting and decision making, with related values, interests and expectations they may hold themselves and which via social interaction and communication should
go into criteria for acceptability and adequacy of the model, thus representing an important
(the external) aspect of the social dimension of mathematics. Second, they are having the
role of mathematizing mathematicians, i. e. of experts, as which all of them should be
accepted and respected, though at the same being learners, yet somehow sharing this with
searching mathematicians. Especially in this role they have to take into account the concerns related to the problem domain and be in communication with others. But they should
also experience that the more internal mathematical problems of the model construction are
often matters of social interactions and negotiations among mathematicians, which repre-
sents another (the internal) aspect of the social dimension of mathematics (see Steiner
1988).
As an example of the second aspect experienced in class the following phenomenon can
count which happened in almost all relatizations of the project: When trying to define the
concept of dictator within the conceptual framework of the not yet fully elaborated theory,
usually several suggestions are made, among which the following three almost always
appear: (1) a dictator is making all other people powerless, (2) a dictator is by himself a
winning coalition, (3) a dictator belongs to all minimal winning coalitions. Reactions are:
that only one of the definitions can be "true" which causes debates about the nature of
definition: not being true or false, but useful, adequate etc.; that the suggested definitions
may be logically equivalent, which would be an indication of adequacy since different
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experts have different intuitive ideas but are logically saying the same. The first suggestion
can be interpreted as expressing a kind of pitiful attitude towards what a dictator does to
other people, the second one as representing a kind of egoistic position. The students then
get very much involved in trying to prove or disprove logical equivalences, and in doing
so, proof becomes a matter of social interest and communication and is no longer some-
thing the teacher is giving to them as an order. They find that (1) and (2) are indeed
logically equivalent and that (3) is actually saying something different, which they identify
as veto-power. When standardizing the definitions, again arguments are being exchanged
and the preference of (2) is usually based on the agreement about its simplicity in directly
referring to the fundamental concept of winning coalitions, whereas (1) is rejected as
using powerlessness as a more complex derived concept. The equivalence of (1) and (2) is
then turned into a theorem saying that x is a dictator if and only if x makes all other
members powerless. This again is a matter of social acceptance in relation to building up a
collectively owned theory.
Social interactions also play an important role in coping with epistemological obstacles (see
Sierpinska 1989). A profound obstacle is coming up for the students in connection with the
implicit definition of winning coalitions within the axiomatic definition of voting bodies of
the SC-type because of the circularity involved and the contrast to the concrete empirical
meaning given to this concept in normal voting bodies. The broad bases laid in the total
learning context together with intensive social interaction and communication among the
students in which many aspects and views of the problem are expressed, contributes to a
dynamic attitude and a flexible relation to possible applications which is the adequate way
to handle and develop theoretical concepts (see Jahnke 1978, Steiner 1990).
The content of the project does not belong to the obligatory normal school topics and is
not meant to have this status, in particular it is not thought of as learning material to be
spread in bits and pieces over longer periods of time. Its strength and potential lies in the
concentration during a limited project time on a dynamically coherent and surveyable
context which is rich in different kinds of activities, interactions and reflexions and is put
by the students themselves into a dynamic relation to their previous learning experiences in
mathematics and to other mathematics-related contexts they may meet in the future. Because of this intended biographic role, the project is purposefully placed at grade level 11 or
12.
In this way the project creates a kind of distance to the normal mathematical classroom
which causes reflexions and transfer. The role of definitions, proofs, theorems, problems,
applications etc. is comparatively discussed by the students, sometimes in a spontaneous
way and at a local level: definition, theorems, proofs in geometry, applications of algebra
and calculus etc.
In describing the overall project work at the end in two different ways, the systematic
reorganization of all knowledge gained from an axiomatic point of view on one hand, and
the genetic reconstruction of the developmental processes and events on the other hand, the
students become aware of several styles to talk about and to present mathematics. They are
trying, now at a more global level, to characterize how other parts of school mathematics
have been taught to them and they are wondering why these parts have not been developed
in the same genetic and inductive way as experienced in the project. They want to know
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349
how mathematics is taught at the university and they begin to make differentiations between
research and learning, between different learning styles, they bring in aspects of time and
time economy etc.
Already during the project it could be observed that most of the participating students
deeply changed their attitude towards mathematics in a positive direction. This particularly
holds for girls, a matter which deserves special attention and further analysis. It has also
been found in later conversation and interaction with students who participated in the
project that they had very much internalized their project experiences and used them as
guiding references for their appreciation of mathematics. Interestingly several of them
declared that becoming mathematics teachers might be the best way to learn more about
and to professionally enjoy unexpected aspects and dimensions of mathematics they had
experienced in the project.
3. Theoretical Concepts, Applications, Social Interactions, Knowledge Development, and
Higher Order Learning
In order to design and execute more specified research in relation to teaching and learning
in a context and setting as has been indicated with respect to the mathematization project, it
seems important to have a sufficiently developed theoretical background. The author sees
substantial and relevant components for this background in socio-historical and epistemological studies in mathematics and the empirical sciences (Sneed 1971, Jahnke 1978, Steiner
1989, 1990) on one hand and in the presently very dynamically growing research charac-
terized by terms like "Zone of Proximal Development" (Vygotsky 1978), "Construction
Zone" (Newman et al.), "Learning by Expanding" (Engestrom 1987) on the other hand,
both components being essentially interrelated. The interrelation consists in the analysis of
the mutual interdependence between theoretical concepts, applications, social interactions
and communications, knowledge development, and higher order learning (see also Seeger
1990).
Engestrom (1987) and Newman et al. (1989) are both referring to the so-called "learning
paradox", formulated by Fodor (1980), as a challenge to cognitivists, as follows: "There
literally isn't a thing as the notion of learning a conceptual system richer than the one that
one already has" (p. 149). They both are criticizing Bereiter's (1985) interpretation of the
paradox as basically being a problem to understand how a learner can internalize more
complex cognitive structures located in the culture while not knowing how internalization
actually takes place, which means dismissing Vygotsky's cultural-historical position as a
solution.
Newman et al. (1989) are responding: "Internalization need not be the construction process
which creates the more powerful structures. We are pointing to the social interaction in the
zone of proximal development as the more central locus for constructive activity in the
Vygotskian framework" (p. 68). Engestrom (1987) who is reacting from his concept of
learning by expansion as transcending given contexts, based on Leont'ev's (1978) activity
theory (see also Steiner 1987), linked with Bateson's (1972) complex hierarchy of learning
processes, is also considering the general problem of how the new is generated from the
old as well as Davydov's searching paradox. He points out that "the new is not generated
from the old but from the living movement leading away from the old" (p. 164).
la
EST COPY AVAILABLE
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Jahnke (1978) who is concerned with the problem how justification and development of
knowledge, especially in mathematics, are related, is referring to the paradox of the proof
and the dilemma of the theorist which have a structure similar to that of the learning
paradox and can also be related to the problem of the relation between reflective and
simple abstraction in Piaget's genetic epistemology or the dual-control-problem in artificial
intelligence research (see also Otte 1980). From his profound historical and epistemological
investigations applying Sneed's (1971) clarification of the nature of empirical theories and
theoretical concepts as well as Churchman's (1968) systems approach and philosophy of the
maximal loop, he makes clear that the kernel of the problem lies in the dialectic relation
between sign and signified which turns out not only to be the crucial point in relating
theory and applications from a developmental point of view but also to be deeply connected
with social contexts and communication as indispensable components of understanding the
problem. With respect to the dynamically inseparable connections between justification and
application, Jahnke comes to the conclusion that "justification (evidence) is, so to say,
placed into the future. The more general, more extended, more developed theory is
founding and justifying the less general theory" (pp. 108-109).
Engestram (1987), Newman et al. (1988) and others have designed research methodologies
to study learning in the construction zone and learning by expansion. At the IDM in
Bielefeld research on classroom interactions (Bauersfeld et al. 1988) and on the epistemology of school mathematics (Steinbring 1984) are now being put into closer relation (see
Seeger 1990). It seems important to do more detailed empirical research based on these
theoretical. and methodological developments particularly in contexts and settings with
respect to existing observations as have been sketched in this paper.
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Bauersfeld, H., Krummheuer, G., Voigt, J. (1988). Interactional Theory of learning and
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174 188.
Churchman, C. W. (1968). Challenge to Reason. New York: Mc Craw-Hill.
Engestrom, J. (1987). Learning by Expanding
An Activity-Theoretical Approach to
Developmental Research. Helsinki: Orienta-Konsultit Oy.
Fodor, J. A. (1980). Fixation of belief and concept of acquisition. In M. Piatteli-Palmerini
(Ed.), Language and Learning: The debate between lean Piaget and Noam Chomsky.
Cambridge, MA: Harvard Univ. Press, 142-149.
Jahnke, H.-N. (1978). Zum Verhaltnis von Wissensentwicklung and Begrundung in der
Mathematik
Beweisen als didaktisches Problem. Bielefeld: IDM
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Leont'ev, A. N. (1978). Activity, Consciousness and Personality. Engelewood Cliffs:
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Newman, D., Griffin, P. & Cole, M. (1989). The Construction Zone
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die Grenzen der Lehrbarkeit. Journal fiir Mathematik-Didaktik (forthcoming)
Sierpinska, A. (1989). On the concept of epistemological obstacle in research on teaching
and learning of mathematics. In Steiner, H.-G. & Hejny, M. Proceedings of the First
Bratislava International Symposium on Research and Development in Mathematics Education. Bratislava: Faculty of Mathematics, 21-36.
Sneed, J. D. (1971). The Logical Structure of Mathematical Physics. Dordrecht: Reidel
Steinbring, H. (1984). Mathematical concepts in didactical situations as complex systems:
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ALICE ALSTON
Rutgers University
Center for Maths,
Science & Comp. Educ.
New Brunswick, New Jersey 08903
USA
MIRIAM AMIT
Technion-Israel Inst.of Technology
Dept. of Ed. in Sc. & Technology
Technion, Haifa 32000
ISRAEL
ABRAHAM ARCAVI
Science Teaching Weizmann
Institute, Rehovot 76100
ISRAEL
MICHELE ARTIGUE
Irem Paris VII 2 Place Jussieu
Paris 75005
FRANCE
ALFONSO AVILA
Unidad de Apoyo Didactico
Srla.de Educ.Cultura y Bienestar
Social del Estado de Mexico.
MEXICO
CARMEN BATANERO
Escuela Univ. Profesorado Egb.
Campus de Cartuja
Granada 18071
SPAIN
LUCIANA BAZZINI
University di Pavia
Strada Nuova 65
Pavia 27100
ITALY
CANDICE BEATTYS
Rutgers University
Center for Maths,
Science & Comp. Educ.
New Brunswick, New Jersey 08903
USA
GERHARD BECKER
Modersohnweg 25 D-28
Bremen 33, 2800
GERMANY
MERLYN BEHR
Mathematics Department
Northern Illinois University
Dekalb, ILL 60115
USA
ALAN BELL
Shell Centre Math. Educ.
Univ. Nottingham
Nottingham NG7 2RD
ENGLAND
JACQUES C. BERGERON
Universite de Montreal Fac. des
Sciences de I'Education C.P.6128
Succ "A" Montreal, P.Q. H3C 3J7
CANADA
KATHRYN BERTILSON
Elem /Sec. Education,
Washington State University
Pullman, WA 99164-2122
USA
NADINE BEZUK
Center for Research Math. &
Science Educ.
San Diego State University
San Diego, California 92182-0315
USA
VICTORIA BILL
Learning Research & Develop.Center
University of Pittsburgh
USA
DERRICK BIRKS
Schell Centre Math. Educ.
Univ. Nottingham
Nottingham NG7 2RD
ENGLAND
ALAN J. BISHOP
Cambridge Univ. Dept.of Education
Cambridge CB1 1QA
ENGLAND
OLE BJORKQVIST
Faculty of Education,
Abo Akademi Box 311
Vasa 65101
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JOAN BLISS
552 King's Road
London SW10 OUA
ENGLAND
DAVID CARRAHER
Fygenho Poeta 66-202
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PAOLO BOERO
Univ. Genova Via L.B. Alberti 4
Genova 16132
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TEREZINHA CARRAHER
R. Mendes Martins 112 Varzea
Recife Pernambuco 50741
BRASIL
ANDRE BOILEAU
Department of Maths. & Inform.
Univ. du Quebec a Montreal
CP 8888, Montreal Quebec H3C 3P8
CANADA
IVANA CHIARUGI
Univ. Genova Via L.B. Alberti 4
Genova 16132
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LESLEY R. BOOTH
James Cook University
School of Education
Townsville, Queensland 4811
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GILLIAN BOULTON-LEWIS
Brisbane CAE
School of Early Childhood Studies
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Phillip Institute of Technology
Melbourne
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YOLANDA CAMPOS CAMPOS
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VICTOR CIFARELLI
Univ. of California
Office of Academic Support, B-036
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McKENZIE A. CLEMENTS
School of Educ. Deakin Univ.
Geelong Victoria 3217
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ANIBAL CORTES
Lab. Psydee 46 Rue St. Jacques
Paris 75005
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MARTIN COOPER
Po Box 1 Kensington School
Educ. Univ. New South Wales
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CATHLEEN CRAVIOTTO
Elem/Sec. Education
Washington State University
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REGINA DAMM
4 Rue Charles Appell.
Univ. Louis Pasteur. Irem
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School of Math. & Inform.
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Rutgers University
Center for Maths, Science &
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U.C.Berkeley-Tolman H. 4533
Berkeley, California
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Escuela Univ. Profesorado Egb.
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Curric. Research & Developm.Group.
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Equipe de Didact.des Maths.
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Math Dept Ben Gurrion Univ.
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College of Educ. Menoufia Univ.
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Dipartimento di Matematica
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Northeastern Illinois Univ.
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Div. of Science, Business & Math
Univ. of Minnesota
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Department of Maths. & Inform.
Univ. du Quebec a Montreal
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Dept. Didact. Matem.
Univ. Valencia
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Irem de Strasbourg 10 Rue
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Univ. of Haifa, Oranim Shool
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Concordia University
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Univ. Valencia
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Florida State University
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Universitat Klagenfurt IFF
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Louisiana State University
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LRDC, University of Pittsburgh
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Univ.of Georgia Dept.Math.Educ.
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Penn. State University,
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