Advanced Mathematical Thinking (Mathematics Education Library)

✍ Scribed by D. Tall

• Publisher: KLUWER ACADEMIC PUBLISHERS
• Year: 1991
• Tongue: English
• Leaves: 310
• Series: Mathematics Education Library
• Edition: 1
• Category: Library

✦ Synopsis

This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Its three main parts focus on the nature of advanced mathematical thinking, the theory of its cognitive development, and reviews of cognitive research. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. The reviews of recent research concentrate on cognitive development and conceptual difficulties with the notions of functions, limits, infinity, analysis, proof, and the use of the computer. They provide a wide overview and an introduction to current thinking which is highly appropriate for the college professor in mathematics or the general mathematics educator.

✦ Table of Contents

TABLE OF CONTENTS......Page 6
PREFACE......Page 14
ACKNOWLEDGEMENTS......Page 18
INTRODUCTION......Page 20
CHAPTER 1 : The Psychology of Advanced Mathematical Thinking......Page 22
1. Cognitive considerations......Page 23
2. The growth of mathematical knowledge......Page 33
3. Curriculum design in advanced mathematical learning......Page 36
4. Looking ahead......Page 39
I: THE NATURE OF ADVANCED MATHEMATICAL THINKING......Page 42
CHAPTER 2 : Advanced Mathematical Thinking Processes......Page 44
1. Advanced mathematical thinking as process......Page 45
2. Processes involved in representation......Page 49
3. Processes involved in abstraction......Page 53
4. Relationships between representing and abstracting (in learning processes)......Page 57
5. A wider vista of advanced mathematical processes......Page 59
1. The stages of development of mathematical creativity......Page 61
3. A tentative definition of mathematical creativity......Page 65
5. The motive power of mathematical creativity......Page 66
6. The characteristics of mathematical creativity......Page 68
7. The results of mathematical creativity......Page 69
9. Consequences in teaching advanced mathematical thinking......Page 71
CHAPTER 4 : Mathematical Proof......Page 73
2. More recent views of mathematics......Page 74
3. Factors in acceptance of a proof......Page 77
4. The social process......Page 78
6. Teaching......Page 79
II: COGNITIVE THEORY OF ADVANCED MATHEMATICAL THINKING......Page 82
1. Definitions in mathematics and common assumptions about Pedagogy......Page 84
2. The cognitive situation......Page 86
3. Concept image......Page 87
6. Concept image and concept definition - desirable theory and practice......Page 88
7. Three illustrations of common concept images......Page 92
8. Some implications for teaching......Page 98
CHAPTER 6 : The Role of Conceptual Entities and their symbols in building Advanced Mathematical Concepts......Page 101
1. Three roles of conceptual entities......Page 102
2. Roles of mathematical notations......Page 107
3. Summary......Page 112
CHAPTER 7 : Reflective Abstraction in Advanced Mathematical Thinking......Page 114
1. Piaget’s notion of reflective abstraction......Page 116
2. A theory of the development of concepts in advanced mathematical thinking......Page 121
3. Genetic decompositions of three schemas......Page 128
4. Implications for education......Page 138
III: RESEARCH INTO THE TEACHING AND LEARNING OF ADVANCED MATHEMATICAL THINKING......Page 144
CHAPTER 8 : Research in Teaching and Learning Mathematics at an Advanced Level......Page 146
1. Do there exist features specific to the learning of advanced mathematics?......Page 147
2. Research on learning mathematics at the advanced level......Page 152
3. Conclusion......Page 158
1. Historical background......Page 159
2. Deficiencies in learning theories......Page 161
3. Variables......Page 163
4. Functions, graphs and visualization......Page 164
5. Abstraction, notation, and anxiety......Page 167
6. Representational difficulties......Page 170
7. Summary......Page 171
CHAPTER 10 : Limits......Page 172
1. Spontaneous conceptions and mental models......Page 173
2. Cognitive obstacles......Page 177
3. Epistemological obstacles in historical development......Page 178
4. Epistemological obstacles in modem mathematics......Page 181
5. The didactical transmission of epistemological obstacles......Page 182
6. Towards pedagogical strategies......Page 184
CHAPTER 11 : Analysis......Page 186
1. Historical background......Page 187
2. Student conceptions......Page 193
3. Research in didactic engineering......Page 205
4. Conclusion and future perspectives in education......Page 215
CHAPTER 12 : The Role of Students’ Intuitions of Infinity in Teaching the Cantorian Theory......Page 218
1. Theoretical conceptions of infinity......Page 219
2. Students’ conceptions of infinity......Page 220
3. First steps towards improving students’ intuitive understanding of actual infinity......Page 224
4. Changes in students’ understanding of actual infinity......Page 228
5. Final comments......Page 233
1. Introduction......Page 234
2. Students’ understanding of proofs......Page 235
3. The structural method of proof exposition......Page 238
4. Conjectures and proofs - the scientific debate in a mathematical course......Page 243
5. Conclusion......Page 248
2. The computer in mathematical research......Page 250
3. The computer in mathematical education - generalities......Page 253
4. Symbolic manipulators......Page 254
5. Conceptual development using a computer......Page 256
6. The computer as an environment for exploration of fundamental ideas......Page 257
7. Programming......Page 260
8. The future......Page 262
Appendix to Chapter 14 ISETL : a computer language for advanced mathematical thinking......Page 263
EPILOGUE......Page 268
CHAPTER 15 : Reflections......Page 270
BIBLIOGRAPHY......Page 280
B......Page 294
C......Page 295
D......Page 297
E......Page 298
F......Page 299
H......Page 300
I......Page 301
L......Page 302
N......Page 303
P......Page 304
R......Page 305
S......Page 306
U......Page 307
Z......Page 308