Windows on Mathematical Meanings: Learning Cultures and Computers

287 Pages·1996·5.018 MB·English

by  Richard Noss, Celia Hoyles (auth.)

WINDOWS ON MATHEMATICAL MEANINGS:
LEARNING CULTURES AND COMPUTERS
Mathematics Education Library
VOLUME 17
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
G. Leder, Melbourne, Australia
S. Tumau, Krakow, Poland
The titles published in this series are listed at the end o/this volume.
WINDOWS ON MATHEMATICAL MEANINGS
LEARNING CULTURES AND COMPUTERS
by
RICHARD NOSS & CELIA HOYLES
University of London, u.K.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
Noss. R1ch.rd.
W1ndows on mathemat1cal mean1ngs : learn1ng cultures and computers
I by R1ehard Noss & Cel1a Hoyles.
p. ca. -- (Matheaat1cs educat10n 11brary ; v. 17)
Inc I udes 1n dex.
1. Matheaat1es--Study and teach1ng. I. Hoyles. Cel1a. 1946-
II. T1tle. III. Ser1es.
CA11.N66 1996
610' .71--dc20 96-17822
ISBN-13: 978-0-7923-4074-4 e-ISBN-13: 978-94-009-1696-8
001: 10.1007/978-94-009-1696-8
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
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In all other countries, sold and distributed
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Printed on acid-free paper
All Rights Reserved
© 1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t edition 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
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CONTENTS
Foreword Vlll
Chapter 1. Visions of the mathematical 1
1. Reshaping mathematics, revisioning learning 2
2. Computers ... 5
3 ....a nd cultures 7
4. Windows on methodologies 9
Chapter 2. Laying the Foundations 12
1. Introduction 12
2. Meanings in mathematics education: a brief survey 14
2.1. Meanings from mathematical objects 14
2.2. Meanings from problem solving 15
2.3. Meanings constructed by the individual learner 17
2.4. From enactive to reflective mathematical knowledge:
spirals of abstraction? 19
3. Vignette: The N-task 22
4. The influence of setting on mathematical behaviour 30
4.1. Problematising children's performance 31
4.2. Into the cul-de-sac: situated mathematical expression 33
5. Street mathematics 36
5.1. Meaning mathematically in the street and in school 36
5.2. The role of representations 38
6. A way out of the cul-de-sac? 41
6.1. Language and tools 42
6.2. Abstract versus concrete? 43
6.3. Returning to the vignette 47
7. Rethinking abstraction 48
Chapter 3. Tools and technologies 52
1. Computers and educational cultures 52
2. A preliminary case for programming 54
2.1. Computers and conviviality 57
3. The development of a programming culture 59
3.1. The development of a literature 59
3.2. The development of a research community 60
3.3. Summing up the case for programming 61
4. Microworlds: the genesis of the idea 63
vi CONTENTS
5. Opening windows on microworlds 65
5.1. The issue of transfer 65
6. Objects and structures 67
6.1. The question of pedagogy 69
Chapter 4. RatioWorld 74
1. What do we - and learners - know? 74
2. Building an alternative methodology 77
2.1. The written tasks 78
2.2. The computational objects 80
3. The data set 82
3.1. The students 83
3.2. Data collection 83
4. The microworld activities and student responses 84
4.1. Meanings of similarity 84
4.2. Broadening the notion of multiplication 85
4.3. The imperative offormalisation 87
4.4. Towards a consistent approach 90
5. Quantitative windows on learning 92
5.1. Gross trends 92
5.2. A step towards the individual 95
5.3. The baseline group 97
5.4. Individual student response profiles 97
6. Reflections 100
Chapter 5. Webs and situated abstractions 105
1. Reviewing the foundations 105
2. Webs of meaning 107
2.1. Flagging 110
2.2. Adjusting 112
2.3. Sketching 115
2.4. Patterning 117
3. Domains of situated abstraction 119
3.1. Situated abstraction 122
3.2. Webs and abstractions 125
3.3. Connecting drawing to geometry 126
3.4. Towards a theory of mathematical meaning 129
Chapter 6. Beyond the individual learner 134
1. Extending the web 134
1.1. Some theoretical background 135
2. Collaborative activity in mathematics 137
2.1. Knowledge domain, task and medium 138
2.2. The role of language in mathematical learning 141
CONTENTS vii
3. A study of groupwork 143
3.1. The tasks 144
3.2. A complementarity between quantitative
and qualitative data 145
3.3. Gross trends 146
3.4. What was happening inside the groups? 147
3.5. Individualleaming 148
3.6. Summing up the study 150
4. Opening new windows 152
Chapter 7. Cultures and change 156
1. Innovation and inertia 156
2. Visions of Logo 159
2.1. Snapshots of Logo 159
2.2. Logo in the U.K. 161
3. Myths and methodologies: a case study of Logo research 166
3.1. Pea and Kurland's approach 167
3.2. Methodologies and the Papert-Pea debate 173
3.3. Transformations of meaning: what is
Logo pedagogy? 175
3.4. The creation of conventional wisdom 177
4. The struggle for meanings 179
Chapter 8. A window on teachers 184
1. Investigating teachers' attitudes and interactions 186
1.1. Mary - the frustrated idealist 188
1.2. Rowena - the confident investigator 190
1.3. Denis - the controlling pragmatist 190
1.4. Fiona - the anxious traditionalist 192
1.5. Bob - the curriculum deliverer 193
2. Teachers making meanings 195
2.1. Reshaping school mathematics 196
2.2. Reshaping the microworlds innovation 197
2.3. Reshaping pedagogy 198
3. Connections and cultures 199
3.1. The social construction of change 200
Chapter 9. A window on schools 203
1. The background to the case study 204
2. The teachers' voices 206
2.1. The head teacher 206
2.2. Two teachers: Colin and John 208
2.3. The advisory teacher 211
3. The students' voices 214
viii CONTENTS
3.1. Meanings of programming 215
3.2. Meanings of LogoMathematics 216
3.3. Teaching and the trajectory of Logo learning 218
3.4. Classroom culture 219
4. Meanings in conflict 222
Chapter 10. Re-visioning mathematical meanings 226
1. Reviewing the scene 226
2. Epistemological revisions: six examples 229
2.1. Small changes in tools afford large changes
in meanings 230
2.2. Programming to reconnect with mathematics 231
2.3. New geometry with new tools 232
2.4. Revisioning number patterns 236
2.5. Convivial mathematics 240
2.6. Connecting mathematics with practice 245
3. Reconnecting mathematics and culture 251
3.1. The invisibility of mathematical meanings 253
3.2. Re-visioning learning cultures 255
References 259
Index 272
FOREWORD
This book is the culmination of some ten years' theoretical and empirical
investigation. Throughout this period, we have come into contact with many
who have stimulated our thinking, some of whom belong to the community of
Mathematics Educators. Our membership of that community has challenged
us to make sense of some deep issues related to mathematical learning,
especially the cognitive and pedagogical faces of mathematical meaning
making.
Alongside this community, we are privileged to have been part of another,
whose members are centrally concerned both with mathematics and educa
tion. Yet many of them might reject the label of Mathematics Educators. This
community has historically been clustered around what is now called the
Epistemology and Learning Group at the Massachusetts Institute of Technol
ogy. Their work has focused our attention on cognitive science, ethnography,
sociology, artificial intelligence and other related disciplines. Crucially, it has
forced our awareness of the construction of computational settings as a crucial
component of the struggle to understand how mathematical learning happens.
We have sometimes felt that few have tried to span both communities.
Indeed, an analysis of the references in the literature would, we are sure,
reveal that the two communities have often ignored each other's strengths.
One reason for writing this book is born of our hope that we might draw
together Mathematics Educators and mathematics educators, and assist both
communities in recognising that there are insights that might be derived from
each other.
One group of scholars who have consistently shared this dual aim has been
the Logo and Mathematics Education group, founded in London in 1985, and
a constant source of inspiration and insight. To them, our thanks.
We owe our greatest intellectual debt to Seymour Papert, whose ideas have
provided continual inspiration, and whose support and friendship has been
invaluable throughout the last fifteen years.
Others who have, at various times, formed part of the MIT community,
have similarly informed our approach, and generously - sometimes unknow
ingly - helped us towards whatever pretence to theoretical clarity we now
claim. Prime among these are Andy diSessa, Brian Harvey, Edith Acker
mann, Sylvia Weir and Uri Wilensky. We especially thank Sylvia Weir for
first drawing our attention, more than a decade ago, to the power of the
window metaphor. Not far from MIT, either geographically or intellectually,
others who have provided us with a rich vein of ideas include Al Cuoco, Paul
Goldenberg, Wally Feurzeig and many others at TERC, EDC and BBN. In
x FOREWORD
these communities, and especially in the work of Cuoco, Goldenberg, Papert
and Wilensky, we have found a sense of intellectual resonance which has
helped to clarify many of our own ideas.
We thank the following for their help and invaluable suggestions for
improvement on an earlier draft of the manuscript: Al Cuoco, Sandy Daw
son, Brian Harvey, Lulu Healy, Colette Laborde, Seymour Papert, and Stefano
Pozzi. Our thanks too, to an anonymous reviewer who pointed out the need
for some revisions, and whom we hope will now come out of the closet so
that we can thank him or her personally.
We would like to thank all our colleagues and students in the Mathematical
Sciences Group at the Institute of Education, University of London, for help
ing to create a scholarly, supportive, and collaborative milieu. We must single
out for special thanks, Lulu Healy and Stefano Pozzi, for their help with some
of the studies referred to in the book, and for their generosity in sharing their
insights with us over many years. Thanks too, to Rosamund Sutherland, for
her collaboration in the work on the Microworlds project.
We have worked individually and jointly on a large number of projects
over many years, and part of the corpus of data we have collected has found a
place in this book. It is our sense that the theoretical fruit of this personal and
intellectual collaboration has been greater than the sum of its constituent parts.
Celia Hoyles, Richard Noss
London
March 1996