Advanced Mathematical Thinking
Advanced Mathematical Thinking has played a central role in the development of human civilization for over two millennia. Yet in all that time the serious study of the nature of advanced mathematical thinking – what it is, how it functions in the minds of
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Mathematics Education Library VOLUME 11
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. Tumau, Krakow, Poland G. Vergnaud, Paris, France
The titles published in this series are listed at the end of this volume.
ADVANCED MATHEMATICAL THINKING
Edited by
DAVID TALL Science Education Department, University of Warwick
KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW
eBook ISBN: Print ISBN:
0-306-47203-1 0-792-31456-5
©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow
All rights reserved
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher
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TABLE OF CONTENTS
PREFACE
xiii
ACKNOWLEDGEMENTS
xvii
INTRODUCTION CHAPTER 1 : The Psychology of Advanced Mathematical Thinking - David Tall
1.
2.
3.
4.
Cognitive considerations 1.1 Different kinds of mathematical mind Meta-theoretical considerations 1.2 1.3 Concept image and concept definition 1.4 Cognitive development 1.5 Transition and mental reconstruction 1.6 Obstacles 1.7 Generalization and abstraction 1.8 Intuition and rigour The growth of mathematical knowledge 2.1 The full range of advanced mathematical thinking 2.2 Building and testing theories: synthesis and analysis 2.3 Mathematical proof Curriculum design in advanced mathematical learning 3.1 Sequencing the learning experience 3.2 Problem-solving 3.3 Proof 3.4 Differences between elementary and advanced mathematical thinking Looking ahead
3
4 4 6 6 7 9 9 11 13 14 14 15 16 17 17 18 19 20 20
vi
TABLE OF CONTENTS
I : THE NATURE OF ADVANCED MATHEMATICAL THINKING CHAPTER 2 : Advanced Mathematical Thinking Processes - Tommy Dreyfus 1. 2.
3.
4. 5.
Advanced mathematical thinking as process Processes involved in representation 2.1 The process of representing 2.2 Switching representations and translating 2.3 Modelling Processes involved in abstraction 3.1 Generalizing 3.2 Synthesizing 3.3 Abstracting Relationships between representing and abstracting (in learning processes) A wider vista of advanced mathematical processes
CHAPTER 3 : Mathematical Creativity - Gontran Ervynck 1. 2. 3. 4. 5. 6. 7. 8. 9.
The stages of development of mathematical creativity The structure of a mathematical theory A tentative definition of mathematical creativity The ingredients of mathematical creativity The motive power of mathematical creativity The characteristics of mathematical creativity The results of mathematical creativity The fallibility of mathematical creativity Consequences in teaching advanced mathematical thinking
CHAPTER 4 : Mathematical Proof - Gila Hanna 1. 2. 3. 4. 5. 6.
Origins of the emphasis on formal proof More recent views of mathematics Factors in acceptan
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