Mathematical Enculturation: A Cultural Perspective on Mathematics Education

209 Pages·1988·10.083 MB·English

by  Alan J. Bishop (auth.)

MATHEMATICAL ENCULT URAn ON
Mathematics Education Library
VOLUME 6
Managing Editor
A.J. Bishop, Cambridge, U.K.
Editorial Board
H. Bauersfeld, Bielefeld, Germany
J. Kilpatrick, Athens, U.S.A.
G. Leder, Melbourne, Australia
S. Turnau, Krakow, Poland
G. Vergnaud, Paris, France
The titles published in this series are listed at the end 0/ this volume.
ALAN J. BISHOP
Department of Education,
University of Cambridge
MATHEMAT ICAL
ENCULT URAT ION
A Cultural Perspective on Mathematics
Education
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
Bishop, Alan J.
Mathematical encuIturation.
(Mathematics education library)
Bibliography: p.
Includes index.
1. Mathematics-Study and teaching. I. Title.
II. Series.
QA11.B545 1988 507 87-32329
ISBN-13: 978-0-7923-1270-3 e-ISBN-13: 978-94-009-2657-8
DOl: 10.1007/978-94-009-2657-8
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
03-0397-300 Is
Third prinlig 1997
Printed on acid-free paper
All Rights Reserved
© 1991 Kluwer Academic Publishers
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
PREFACE xi
ACKNOWLEDGEMENTS xiii
CHAPTER 11 Towards a Way of Knowing 1
1.1. The conflict 1
1.2. My task 3
1.3. Preliminary thoughts on Mathematics education and
culture 3
1.4. Technique-oriented curriculum 7
1.5. Impersonal learning 9
1.6. Text teaching 10
1.7. False assumptions 12
1.8. Mathematical education, a social process 13
1.9. What is mathematical about a mathematical education? 16
1.10. Overview 18
CHAPTER 21 Environmental Activities and Mathematical Culture 20
2.1. Perspectives from cross-cultural studies 20
2.2. The search for mathematical similarities 22
2.3. Counting 23
2.4. Locating 28
2.5. Measuring 34
2.6. Designing 38
2.7. Playing 42
2.8. Explaining 48
2.9. From 'universals' to 'particulars' 55
2.10. Summary 59
CHAPTER 31 The Values of Mathematical Culture 60
3.1. Values, ideals and theories of knowledge 60
3.2. Ideology - rationalism 62
3.3. Ideology - objectism 65
3.4. Sentiment - control 69
3.5. Sentiment - progress 72
3.6. Sociology - openness 75
3.7. Sociology - mystery 77
v
vi TABLE OF CONTENTS
CHAPTER 4/ Mathematical Culture and the Child 82
4.1. Mathematical culture - symbolic technology and values 82
4.2. The culture of a people 84
4.3. The child in relation to the cultural group 87
4.4. Mathematical enculturation 89
CHAPTER 5/ Mathematical Enculturation - The Curriculum 92
5.1. The curriculum project 92
5.2. The cultural approach to the Mathematics curriculum -
five principles 95
5.2.1. Representativeness 95
5.2.2. Formality 95
5.2.3. Accessibility 96
5.2.4. Explanatory power 96
5.2.5. Broad and elementary 97
5.3. The three components of the enculturation curriculum 98
5.4. The symbolic component: concept-based 99
5.4.1. Counting 100
5.4.2. Locating 100
5.4.3. Measuring 101
5.4.4. Designing 102
5.4.5. Playjng 102
5.4.6. Explaining 103
5.4.7. Concepts throl:lgh activities 103
5.4.8. Connections between concepts 108
5.5. The societal component: project-based 110
5.5.1. Society in the past 111
5.5.2. Society at present 112
5.5.3. Society in the future 113
5.6. The cultural component: investigation-based 114
5.6.1. Investigations in mathematical culture 116
5.6.2. Investigations in Mathematical culture 117
'5.6.3. Investigations and values 117
5.7. Balance in this curriculum 119
5.8. Progress through this curriculum 120
CHAPTER 6/ Mathematical Enculturation - The Process 124
6.1. Conceptualising the enculturation process in action 124
6.1.1. What should it involve? 124
6.1.2. Towards a humanistic conception of the process 125
6.2. An asymmetrical process 128
T ABLE OF CONTENTS vii
6.2.1. The role of power and influence 128
6.2.2. Legitimate use of power 130
6.2.3. Constructive and collaborative engagement 131
6.2.4. Facilitative influence 132
6.2.5. Metaknowledge and the teacher 135
6.3. An intentional process 135
6.3.1. The choice of activities 135
6.3.2. The concept-environment 139
6.3.3. The project-environment 142
6.3.4. The investigation-environment 147
6.4. An ideational process 151
6.4.1. Social construction of meanings 151
6.4.2. Sharing and contrasting Mathematical ideas 154
6.4.3. The shaping of explanations 157
6.4.4. Explaining and values 159
CHAPTER 7/ The Mathematical Enculturators 160
7.I. People are responsible for the process 160
7.2. The preparation of Mathematical enculturators -
preliminary thoughts 161
7.3. The criteria for the selection of Mathematical
enculturators 164
7.3.I. Ability to personify Mathematical culture 164
7.3.2. Commitment to the Mathematical enculturation
process 165
7.3.3. Ability to communicate Mathematical ideas and
values 166
7.3.4. Acceptance of accountability to the Mathematical
culture 167
7.3.5. Summary of criteria 168
7.4. The principles of the education of Mathematical
enculturators 168
7.4.1. Mathematics as a cultural phenomenon 169
7.4.2. The values of Mathematical culture 170
7.4.3. The symbolic technology of Mathematics 171
7.4.4. The technical level of Mathematical culture 172
7.4.5. The meta-concept of Mathematical enculturation 173
7.4.6. Summary of principles 175
7.5. Socialising the future enculturator into the Mathematics
Education community 176
7.5.I. The developing Mathematics Education
community 176
7.5.2. The critical Mathematics Education community 178
viii TABLE OF CONTENTS
NOTES 180
BIBLIOGRAPHY 184
INDEX OF NAMES 192
APPENDIX 195
To Jenny
with grateful thanks for her patience and support
PREFACE
Mathematics is in the unenviable position of being simultaneously one of the
most important school subjects for today's children to study and one of the
least well understood. Its reputation is awe-inspiring. Everybody knows how
important it is and everybody knows that they have to study it. But few
people feel comfortable with it; so much so that it is socially quite acceptable
in many countries to confess ignorance about it, to brag about one's incompe
tence at doing it, and even to claim that one is mathophobic!
So are teachers around the world being apparently legal sadists by inflicting
mental pain on their charges? Or is it that their pupils are all masochists,
enjoying the thrill of self-inflicted mental torture? More seriously, do we
really know what the reasons are for the mathematical activity which goes on
in schools? Do we really have confidence in our criteria for judging what's
important and what isn't? Do we really know what we should be doing?
These basic questions become even more important when considered in the
context of two growing problem areas. The first is a concern felt in many
countries about the direction which mathematics education should take in the
face of the increasing presence of computers and calculator-related technol
ogy in society. The second problem area concerns children whose home and
family culture does not fully resonate with that of the school and the wider
society, be they in London, in Aboriginal Australia or in a Navajo reserva
tion.
These problem areas are not unrelated of course. The first provokes many
thoughts about educational values, about the importance attached by society
to different kinds of knowledge, and about the relationship individuals have
with that knowledge. And so does the second.
I therefore felt the need to explore the relationship between developments
in these two problem areas, and to do this through their common denomina
tor - culture.
This book is all about Mathematics as "a way of knowing". It takes a
cultural look at this supposedly familiar subject, and it analyses the educa
tional consequences of the cultural perspective. In the first half of the book I
explore a range of anthropological, cross-cultural and historical literature
concerning Mathematics and culture. My aim is to create a new conception of
Mathematics which both recognises and demonstrates its relationship with
culture - the notion of mathematics as a cultural product, the environmental
and societal activities which stimulate mathematical concepts, the cultural
values which mathematics embodies - indeed the whole cultural genesis of
mathematical ideas.
Xl