Mathematics for Tomorrow’s Young Children

334 Pages·1996·21.603 MB·English

by  H. M. Mansfield (auth.), Helen Mansfield, Neil A. Pateman, Nadine Bednarz (eds.)

MATHEMATICS FOR TOMORROW'S YOUNG CHILDREN
Mathematics Education Library
VOLUME 16
Managing Editor
A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board
H. Bauersfeld, Bielefeld, Germany
J. Kilpatrick, Athens, U.S.A.
C. Laborde, Grenoble, France
O. Leder, Melbourne, Australia
S.~,Krakow,Po~nd
The titles published in this series are listed at the end oft his volume.
MATHEMATICS
FOR TOMORROW'S
YOUNG CHILDREN
Edited by
HELEN MANSFIELD
Curriculum Research & Development Group,
University ofH awaii, U.S.A.
NEIL A. PATEMAN
Department ofC urriculum & Instruction,
University ofH awaii, U.S.A.
and
NADINE BEDNARZ
a
CIRADE, Universite du Quebec Montreal, Canada
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Mathemat1cs for tOlorrow's young ch1ldren : 1nternational perspect1ves
on curr1culum / edlted by Helen Mansf1eld and Nei1 A. Pateman and
Nad1ne Bednarz.
p. cm. -- (Mathelatics educat10n 11brary ; v. 18)
Papers from meet1ngs of a work1ng group at the 7th Internat10nal
Congress on Mathe_at1cal Educat1on. held 1n Quebec. Canada. 1n 1992.
Includes b1bl1ograph1cal references.
ISBN 978-90-481-4690-1 ISBN 978-94-017-2211-7 (eBook)
DOI 10.1007/978-94-017-2211-7
1. Mathemat1cs--Study and teach1ng (Elementary)--Congresses.
1. Mansf1eld. Helen (Helen M.) II. Pate_an. Ne11 A. III. Bednarz.
Nad1ne. IV. International Congress on Mathematical Education (7th :
1992 : Quebec. Quebec) V. Serles.
QA135.5.M38924 1998
372.7--dc20 96-10933
ISBN 978-90-481-4690-1
Printed on acid-free paper
AlI Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying. recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
PART ONE
CHAPTER 1.1: YOUNG CHILDREN'S MATHEMATICAL LEARNING:
COMPLEXITIES AND SUBTLETIES ......................... H. M. Mansfield 1
I.THE THEORETICAL CONTEXT .................................................................................................... 1
2. THE THEMES .................................................................................................................................. 5
3. THE CONFERENCE ........................................................................................................................ 6
4. USING THIS VOLUME ................................................................................................................... 7
PART TWO
CONTRIBUTIONS TO PART TWO .................................................................... 9
CHAPTER 2.1: CONSTRUCTIVISM AND ACTIVITY THEORY: A CONSID
ERATION OF THEIR SIMILARITIES AND DIFFERENCES AS THEY
RELATE TO MATHEMATICS EDUCATION .................. P. Cobb et al. 10
I. ACTIVITY THEORY ...................................................................................................................... 11
1.1 Vygotsky and Leont' ev ............................................................................. 11
1.2 The rehabilitation of Vygotsky ................................................................. 13
1.3 Davydov's arithmetic program ................................................................ 15
1.4 Scientific and empirical concepts ........................................................... 20
1.5 Idealforms and mathematical structure ................................................. 21
1.6 Exposition and Investigation ................................................................... 24
1.7 The "activity" in activity theory ............................................................. 25
1.8 The mind-in-social-action ....................................................................... 28
1.9 Glasnost .................................................................................................. 29
2. CONSTRUCTIVISM ...................................................................................................................... 30
2.1 Basic assumptions ................................................................................... 31
2.2 An approach to arithmetical computation compatible with
constructivism ......................................................................................... 33
2.3 The concrete and the abstract ................................................................. 36
2.4 The teacher's role .................................................................................... 38
2.51nstructional development ....................................................................... 41
2.6 Cultural tools and mathematical symbolizing ........................................ 44
2.7 Theoretical concepts and conceptual actions ......................................... 46
3. THE INDIVIDUAL AND THE SOCIAL ....................................................................................... 48
3.1 Individual construction and social acculturation ................................... 50
3.2 School mathematics and inquiry mathematics ........................................ 51
3.2 Reform in mathematics education ........................................................... 53
4. A CHALLENGE FOR THE FUTURE ............................................................................................ 55
vi TABLE OF CONTENTS
CHAPTER 2.2: A SOCIOCULTURAL VIEW OF THE MATHEMATICS
EDUCATION OF YOUNG CHILDREN ............................... P. Renshaw 59
1. INTRODUCTION ........................................................................................................................... 59
2. CONCEPTUAL DEVELOPMENT: VYGOTSKY'S VIEW ......................................................... 60
2.1 The underlying form ofchildren:S concepts ............................................ 60
2.2 The pseudoconcept: A vehicle for instructional dialogue ....................... 61
2.3 Spontaneous and scientific concepts ....................................................... 63
2.4 The institutional context of conceptual development .............................. 63
2.5 Relevance of\ygotsky's theory to Davydov:S early mathematics curricu-
lum ......................................................................................................... 64
3. DAVYDOV'S INTERPRETATION OF SOCIOCULTURAL THEORy ....................................... 64
3.1 Number and the relations of quantity ...................................................... 65
4. THE TEACHING EXPERIMENTS OF DAVYDOV ..................................................................... 67
4.1 Connecting spontaneous and mathematical concepts ............................ 67
4.2 Teaching strategies used to connect everyday and mathematical
concepts ................................................................................................... 67
4.3 Intermediate means of representing relations of quantity ....................... 69
5. REFLECTING ON THE TEACHING EXPERIMENTS ............................................................... 71
5.1 Encouraging the movement to self-regulation ........................................ 71
5.2 Peer interaction and self-regulation ....................................................... 72
5.3 Avoiding empty formalism ....................................................................... 72
6. CONCLUSION ............................................................................................................................... 73
6.1 Learning to employ the mathematical voice ........................................... 73
6.2 Synthesizing the individual and the social .............................................. 76
CHAPTER 2.3: SOCIAL-CULTURAL APPROACHES IN EARLY CHILD-
HOOD MATHEMATICS EDUCATION: A DISCUSSION ..... L P. Steffe 79
1. INTERACTION AS A HARD-CORE PRINCIPLE ....................................................................... 79
1.1 An interpretation of constructivist learning ............................................ 80
1.2 Interaction in genetic epistemology ........................................................ 81
2. TWO VIEWS OF LEARNING ....................................................................................................... 82
2.1 Learningfrom a \ygotskian perspective ................................................. 82
2.2 Learning in social constructivism ........................................................... 84
2.21 Learning as accommodation. . ............................................................. 85
2.22 Interaction and construction. .. ............................................................. 86
2.23 Learning as a process of acculturation ................................................. 88
3. MATHEMATICAL REALITIES OF CHILDREN ......................................................................... 91
3.1 First and second order models ................................................................ 91
3.2 The Candy Factory .................................................................................. 92
3.3 The mathematics of education ................................................................. 93
3.4 Enculturation vs. acculturation ............................................................... 94
3.5 Realistic mathematics vs. mathematical realities of children ................. 95
4. FINAL COMMENTS ...................................................................................................................... 96
TABLE OF CONTENTS vii
PART THREE
CONTRIBUTIONS TO PART THREE ............................................................ 101
CHAPTER 3.1: THE PSYCHOLOGICAL NATURE OF CONCEPTS
.............................................................................................. E. Fischbein 102
I. INTRODUCTION ........................................................................................................................ 102
2. CONCEPT FORMATION ........................................................................................................... 104
2.1 Inductive concept/ormation .................................................................. 105
2.2 Deductive concept/ormation ................................................................ 106
2.3 Inventive concept/ormation .................................................................. 106
3. CONCEPTS, MEANINGS AND LANGUAGE .......................................................................... 107
4. PROTOTYPES AND EXEMPLARS .......................................................................................... 107
5. THE PARADIGMATIC VIEW .................................................................................................... 108
6. EXPLICIT AND IMPLICIT MODELS ....................................................................................... 110
7. MATHEMATICAL CONCEPTS ................................................................................................. 112
7.1 Empirical and mathematical concepts .................................................. 112
7.2 The duality process-object .................................................................... 113
7.3 Levels o//ormalization .......................................................................... 114
8. SUMMARY AND CONCLUSIONS ........................................................................................... 117
CHAPTER 3.2: WHAT CONCEPTS ARE AND HOW CONCEPTS ARE
FORMED ..................................................................................... J. Brun 120
I. INTRODUCTION ........................................................................................................................ 120
2. THE FORMATION OF CONCEPTS AND THE THEORY OF CONCEPTUAL FIELDS ....... 120
2.1. The Piagetian question on the trans/ormation o/knowledge .............. 120
2.2 The question o/mathematical content: Logico-mathematical structures
and con ceptualfields .......................................................................... 123
3. SCHEMES, ALGORITHMS, AND COMPUTATIONAL ERRORS3 ........................................ 126
4. CONCLUSIONS .......................................................................................................................... 134
CHAPTER 3.3: YOUNG CHILDREN'S FORMATION OF NUMERICAL
=
CONCEPTS: OR 8 9 + 7 .................................................... K. C. Irwin 137
I. INTRODUCTION ........................................................................................................................ 137
2. A FRAMEWORK FOR NUMERICAL CONCEPTS .................................................................. 137
2.1 Counting ................................................................................................ 138
2.2 Protoquantitative concepts .................................................................... 142
2.3 Quantitative concepts: Integrating counting and partlwhole schemas 144
3. CONCLUSIONS .......................................................................................................................... 148
CHAPTER 3.4: CONCEPT FORMATION PROCESS AND AN INDNIDUAL
CHILD'S INTELLIGENCE .................................... E. G. Gelfman et al. 151
I. INTRODUCTION ........................................................................................................................ 151
2. A STORY ENGENDERING CONCEPT DEVELOPMENT ...................................................... 152
viii TABLE OF CONTENTS
3. CONTINUING WITH POSITIONAL NUMERATION .............................................................. 158
4. TASKS FOR LATER STAGES .................................................................................................... 160
5. CONCLUSION ............................................................................................................................ 162
PART FOUR
CONTRIBUTIONS TO PART FOUR .............................................................. 165
CHAPTER 4.1: INTERACTIONS BETWEEN CHILDREN IN MATHEMAT
ICS CLASS: AN EXAMPLE CONCERNING THE CONCEPT OF
NUMBER .............................................................. L Poirier & L. Bacon 166
1. FOUNDATIONS OF THE APPROACH: THE SOCIO-CONSTRUCTIVIST PERSPECTIVE 166
2. THE PROCEDURE ...................................................................................................................... 167
2.2 Social interactions and the construction of counting schemes by
children.................................................................................................. 169
2.21 Counting tasks with a collection of real objects ........................... 169
2.22 The task of counting a collection drawn on paper. ....................... 170
2.23 Comparison task of two drawn collections ................................... 173
3. CONCLUSION ............................................................................................................................ 174
CHAPTER 4.2: WHAT IS THE DIFFERENCE BETWEEN ONE, UN
AND YI? .................................................................................... T. Nunes 177
1. INTRODUCTION ........................................................................................................................ 177
2.WHAT IS THE DIFFERENCE BETWEEN COUNTING AND UNDERSTANDING A
NUMERATION SYSTEM? .................................................................................................... 178
2.1 Conceptual problems in understanding a numeration system with
a base ................................................................................................... 178
3. DOES THE REGULARITY OF THE NUMERATION SYSTEM FACILITATE ITS UNDER-
STANDING? ........................................................................................................................... 181
4. CONCLUSIONS .......................................................................................................................... 184
CHAPTER 4.3: HOW DO SOCIAL INTERACTIONS AMONG CHILDREN
CONTRIBUTE TO LEARNING? ............... A. Reynolds & G. Wheatley 186
1. INTRODUCTION ........................................................................................................................ 186
2. THE LEARNING OF MATHEMATICS ..................................................................................... 186
2.1 The scene ............................................................................................... 187
2.2 The activity ............................................................................................ 187
2.3 The social climate of the mathematics class ......................................... 187
2.4 What we observed as the students collaborated on these tasks ............ 191
3. FINAL THOUGHTS .................................................................................................................... 196
CHAPTER 4.4: CULTURALAND SOCIAL ENVIRONMENTAL HURDLES A
TANZANIAN CHILD MUST JUMP IN THE ACQUISmON OF MATH-
EMATICS CONCEPTS ................................................ V. G. K. Masanja 198
1. INTRODUCTION ........................................................................................................................ 198
TABLE OF CONTENTS ix
2. LEARNING MATHEMATICS IN THE TANZANIAN ENVIRONMENT ............................... 200
2.1 Education before foreign intervention .................................................. 200
2.2 Mathematics education in the wake offoreign intervention ................. 200
3. KIHAYAAND KISWAHILI ........................................................................................................ 201
3.1 Counting and numeration in Kihaya and in Kiswahili ......................... 201
3.2 Reckoning time in Tanzania .................................................................. 204
3.3 The general school process ................................................................... 204
3.4 Any improvement after attaining political independence? ................... 205
3.5 The current problems of conceptualization ........................................... 206
3.51 Classroom social-cultural environment problems ........................ 206
3.52 School location .............................................................................. 207
3.53 Other sources ................................................................................ 207
3.6. Can something be done to alleviate the problem? ............................... 207
3.7. What about the cultural gap? ............................................................... 208
4. MATHEMATICS IN AFRICAN CULTURE ............................................................................... 208
5. CURRICULUM REFORM FOR THE 21ST CENTURY IN AFRICAN CULTURE ................. 213
6. CONCLUSION ............................................................................................................................ 214
PART FIVE
CONfRIBUTIONS TO PART FIVE ................................................................ 217
CHAPTER 5.1: LIMITATIONS OF ICONIC AND SYMBOLIC REPRESENTA
TIONS OF ARITHMETICAL CONCEPTS IN EARLY GRADES OF
PRIMARY SCHOOL ........................................................... Z. Semadeni 218
1. INTRODUCTION ........................................................................................................................ 218
2. THE PROBLEM OF OPERATIONAL REASONING AT THE BEGINNING .......................... 218
3. LEARNING TO HANDLE SYMBOLS ...................................................................................... 220
4. SEVERAL OVERSIMPLIFICATIONS OR MISCONCEPTIONS ............................................ 222
4.1 First oversimplification: representations form a one-way sequence:
enactive, iconic, symbolic .................................................................... 222
4.2 Second oversimplification: iconic representations are easy for
children ................................................................................................ 223
4.3. Third oversimplification: if only small numbers are involved, the task is
easy. ...................................................................................................... 226
CHAPTER 5.2: LANGUAGE ACTIVITY, CONCEPTUALIZATION AND
PROBLEM ............................................................................. N. Bednarz 228
1. INTRODUCTION ........................................................................................................................ 228
2. THE ROLE OF LANGUAGE IN THE DEVELOPMENT OF CHILDREN'S
MATHEMATICAL THOUGHT .................................................................................................. 229
2.1 Procedure .............................................................................................. 231
3. FORMULATION, CONCEPTUALIZATION AND SOLVING OF A MATHEMATICS
PROBLEM ................................................................................................................................... 232
3.1 Discussion ofp roblems ......................................................................... 232
x TABLE OF CONTENTS
s
3.2 Children rewording of a problem presented to the whole class .......... 233
s
3.3 The children formulation of problems for the other children to solve 233
3.4. What was the nature of the children's first problem formulations? ..... 234
4. CONCLUSION ............................................................................................................................ 237
CHAPTER 5.3: CHILDREN TALKING MATHEMATICALLY IN MULTILIN
GUAL CLASSROOMS: ISSUES IN THE ROLE OF LANGUAGE
................................................................................................ L. L. Khisty 240
1. INTRODUCTION ........................................................................................................................ 240
2. ISSUE 1: JUNKY INPUT ............................................................................................................ 241
3. ISSUE 2: OPERATING IN A SECOND LANGUAGE ............................................................... 243
4. ISSUE 3: COMING TO KNOW .................................................................................................. 244
5. CONCLUDING REMARKS AND RECOMMENDATIONS ..................................................... 245
CHAPTER 5.4: USE OF LANGUAGE IN ELEMENTARY GEOMETRY BY
STUDENTS AND TEXTBOOKS ............................................. A. Jaime 248
1. INTRODUCTION ........................................................................................................................ 248
2. THE STUDY ................................................................................................................................ 248
2.1 Students' uses of language .................................................................... 248
2.2 Ways to Use Language in Textbooks ..................................................... 252
3. CONCLUSIONS .......................................................................................................................... 254
PART SIX
CONTRIBUTIONS TO PART SIX .................................................................. 257
CHAPTER 6.1: CONCEPT DEVELOPMENT IN EARLY CHILDHOOD
MATHEMATICS: TEACHERS' THEORIES AND RESEARCH
................................................................................................... R. Wright 258
I. TEACHERS' THEORIES ON THE FORMATION AND DEVELOPMENT OF NUMBER
CONCEPTS ............................................................................................................................. 258
2. STUDIES OF CONCEPTUAL DEVELOPMENT IN EARLY CHILDHOOD
MATHEMATICS ..................................................................................................................... 263
3. COMPARING TEACHERS' AND RESEARCHERS' VIEWS OF CONCEPTUAL DEVELOP-
MENT IN EARLY CHILDHOOD MATHEMATICS ............................................................ 264
4. SOME RECENT RESEARCH INTO YOUNG CHILDREN'S NUMBER LEARNING .......... 265
5. CURRENT INVESTIGATION .................................................................................................... 266
6. ASSESSING YOUNG CHILDREN'S ARITHMETICAL KNOWLEDGE ................................ 267
CHAPTER 6.2: TEACHERS' BELIEFS ABOUT CONCEPT FORMATION
AND CURRICULUM DECISION-MAKING IN EARLY
MATHEMATICS ........................................................... M. Hughes et ai. 272
1. INTRODUCTION ........................................................................................................................ 272