How Humans Learn to Think Mathematically

How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.Part I. Prelude: 1. About this book; Part II. School Mathematics and its Consequences: 2. The foundations of mathematical thinking; 3. Compression, connection and blending of mathematical ideas; 4. Set-befores, met-befores and long-term learning; 5. Mathematics and the emotions; 6. The three worlds of mathematics; 7. Journeys through embodiment and symbolism; 8. Problem-solving and proof; Part III. Interlude: 9. The historical evolution of mathematics; Park IV. University Mathematics and Beyond: 10. The transition to formal knowledge; 11. Blending knowledge structures in Canada; 12. Expert thinking and structure theorems; 13. Contemplating the infinitely large and the infinitely small; 14. Expanding frontiers through mathematical research; 15. Reflections; Appendix. Where the ideas came from.