How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics

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 Part IV. University Mathematics and Beyond: 10. The transition to formal knowledge 11. Blending knowledge structures in calculus 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.

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