E-Book Content
REVISITING MATHEMATICS EDUCATION
Mathematics Education Library VOLUME 9
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 of this volume.
HANS FREUDENTHAL University of Utrecht
REVISITING MATHEMATICS EDUCATION China Lectures
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
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0-306-47202-3 0-7293-1299-6
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TABLE OF CONTENTS
PREFACE
ix
APOLOGY ANDEXPLICATION
xi
CHAPTER1:MATHEMATICSPHENOMENOLOGICALLY
1
1.1: WHAT IS MATHEMATICS? 1.1.1: Sure andcertain 1.1.2: Mathematics as common sense 1.1.3: Why mathematics is different 1.1.3.1:Examples 1.1.4: Mathematics as an activity 1.1.5: Mathematics and reality 1.1.6: Concepts ormental objects? 1.2: STRUCTURE ANDSTRUCTURES 1.2.1: Structures --poor and rich 1.2.2: Structures definedby relations 1.2.3: Algebraicstructures 1.2.4: Structures --- from smaller to larger 1.2.5: Generating the number system 1.2.6: Geometric structures 1.2.7: Structure ofmathematics 1.2.8: Structures as viewed from reality 1.2.8.1: Structure of science, and development 1.2.8.2: Structure of science, and instruction 1.2.8.3: Structuringrichcontextmathematically 1.3:MATHEMATISING 1.3.1: The term 1.3.2 Some aspects 1.3.3: Examples 1.3.4: Mathematising -- horizontally and vertically 1.3.5: Examples
1 1 4 9 12 14 16 18 20 20 21 21 22 23 23 24 26 27 28 29 30 30 31 36 41 42
vi CHAPTER 2: DIDACTICAL PRINCIPLES 2.1: GUIDED REINVENTION 2.1.1: Guiding -- where to? 2.1.2: Guiding -- where? 2.1.3: Guiding -- how? 2.1.4: Algorithmising 2.1.5: Reinventing geometry 2.2: BONDS WITH REALITY 2.2.1: Primordial reality 2.2.2: Odd worlds 2.2.3: Rich contexts 2.2.3.1: Contexts versus “material” 2.2.4: Paradigms 2.2.4.1: Paradigms imposed 2.2.4.2: Paradigms reinvented 2.2.4.3: Acts, actions and activities as paradigms 2.2.5: Applications 2.3: LEARNING PROCESSES 2.3.1: “Learning process” as a didactical principle 2.3.2: Teaching/learning processes 2.3.3: Observation as a didactical principle 2.3.4: Levels in the learning process 2.3.5: Reflection 2.3.6: Reflection and observation 2.4: LONG-TERM LEARNING PROCESSES 2.4.1: Learning to forget 2.4.2: Remembering learning processes 2.4.3: Insight 2.4.3.1: Testing insight 2.4.4: Training 2.4.5: Organising the learning process 2.4.6: Acquiring a mathematical attitude
45 45 49 50 55 57 64
66 66 68 73 75 76 77 80 83 84 87 87 87 89 96 102 108 109 109 111 111 113 114 116 121
vii CHAPTER 3: THE LANDSCAPE OF MATHEMATICS EDUCATION 3.1: THEORY OF MATHEMATICS EDUCATION 3.1.1: Theory 3.1.2: Theoretic frameworks 3.1.3: Background Philosophy 3.1.4: Pictures of mathematics and mathematics education 3.1.5: Classifying mathematics education 3.1.6: Philosophies of mathematics education 3.1.7: Use of the classification 3.1.8: A matter of faith 3.1.9: Learning theories and philosophies 3.1.9.1: Gal’perin, and materialism 3.1.9.2: Constructivism, and“Kant” 3.2: RESEARCH IN MATHEMATICS EDUCATION 3.2.1:Research 3.2.2 Educational research 3.2.3: Developmental research 3.3: PRACTICE OF MATHEMATICS EDUCATION 3.3.1: Practice 3.3.2: A background o