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.

Preface.- Introduction.- 1. The Psychology of Advanced Mathematical Thinking.- I: The Nature of Advanced Mathematical Thinking. 2. Advanced Mathematical Thinking Processes. 3. Mathematical Creativity. 4. Mathematical Proof.- II: Cognitive Theory of Advanced Mathematical Thinking. 5. The Role of Definitions in the Teaching and Learning of Mathematics. 6. The Role of Conceptual Entities and their Symbols in Building Advanced Mathematical Concepts. 7. Reflective Abstraction in Advanced Mathematical Thinking.- III: Research into the Teaching and Learning of Advanced Mathematical Thinking. 8. Research in Teaching and Learning Mathematics at an Advanced Level. 9. Functions and Associated Learning Difficulties. 10. Limits. 11. Analysis. 12. The Role of Students' Intuitions of Infinity in Teaching the Cantorial Theory. 13. Research on Mathematical Proof. 14. Advanced Mathematical Thinking and the Computer. Appendix to 14. ISETL: A Computer Language for Advanced Mathematical Thinking. Epilogue. 15. Reflections.- Bibliography.- Index.