Ebook: Advanced Mathematical Thinking
- Tags: Mathematics Education, Mathematics general, Interdisciplinary Studies
- Series: Mathematics Education Library 11
- Year: 1991
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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 expert mathematicians, how it can be encouraged and improved in the developing minds of students – has been limited to the reflections of a few significant individuals scattered throughout the history of mathematics. In the twentieth century the theory of mathematical education during the compulsory years of schooling to age 16 has developed its own body of empirical research, theory and practice. But the extensions of such theories to more advanced levels have only occurred in the last few years. In 1976 The International Group for the Psychology of Mathematics (known as PME) was formed and has met annually at different venues round the world to share research ideas. In 1985 a Working Group of PME was formed to focus on Advanced Mathematical Thinking with a major aim of producing this volume. The text begins with an introductory chapter on the psychology of advanced mathema- cal thinking, with the remaining chapters grouped under three headings: • the nature of advanced mathematical thinking, • cognitive theory, and • reviews of the progress of cognitive research into different areas of advanced mathematics.
Content:
Front Matter....Pages i-xvii
The Psychology of Advanced Mathematical Thinking....Pages 3-21
Advanced Mathematical Thinking Processes....Pages 25-41
Mathematical Creativity....Pages 42-53
Mathematical Proof....Pages 54-61
The Role of Definitions in the Teaching and Learning of Mathematics....Pages 65-81
The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts....Pages 82-94
Reflective Abstraction in Advanced Mathematical Thinking....Pages 95-126
Research in Teaching and Learning Mathematics at an Advanced Level....Pages 127-139
Functions and Associated Learning Difficulties....Pages 140-152
Limits....Pages 153-166
Analysis....Pages 167-198
The Role of Studients’ Intuitions of Infinity in Teaching the Cantorian Theory....Pages 199-214
Research on Mathematical Proof....Pages 215-230
Advanced Mathematical Thinking and the Computer....Pages 231-248
Reflections....Pages 251-259
Back Matter....Pages 261-290
Content:
Front Matter....Pages i-xvii
The Psychology of Advanced Mathematical Thinking....Pages 3-21
Advanced Mathematical Thinking Processes....Pages 25-41
Mathematical Creativity....Pages 42-53
Mathematical Proof....Pages 54-61
The Role of Definitions in the Teaching and Learning of Mathematics....Pages 65-81
The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts....Pages 82-94
Reflective Abstraction in Advanced Mathematical Thinking....Pages 95-126
Research in Teaching and Learning Mathematics at an Advanced Level....Pages 127-139
Functions and Associated Learning Difficulties....Pages 140-152
Limits....Pages 153-166
Analysis....Pages 167-198
The Role of Studients’ Intuitions of Infinity in Teaching the Cantorian Theory....Pages 199-214
Research on Mathematical Proof....Pages 215-230
Advanced Mathematical Thinking and the Computer....Pages 231-248
Reflections....Pages 251-259
Back Matter....Pages 261-290
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