In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles).A sampling of the coverage:The conjoint origins of proof and theoretical physics in ancient Greece.Proof as bearers of mathematical knowledge.Bridging knowing and proving in mathematical reasoning.The role of mathematics in long-term cognitive development of reasoning.Proof as experiment in the work of Wittgenstein.Relationships between mathematical proof, problem-solving, and explanation.Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

Directs attention of educational researchers to newest developments in the philosophy and practice of mathematics and their relevance

Part I. Reflections on the Nature and Teaching of ProofChapter 1. The Conjoint Origin of Proof and Theoretical PhysicsHans Niels JahnkeChapter 2. Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity.Teun KoetsierChapter 3. Pre-Axiomatic Mathematical Reasoning: An Algebraic ApproachMary Catherine LengChapter 4. Completions, Constructions and CorrollariesThomas MormannChapter 5. Authoritarian vs. Authoritative Teaching: Polya and LakatosBrendan LarvorChapter 6. Proofs as Bearers of Mathematical KnowledgeGila Hanna & Ed BarbeauChapter 7. Mathematicians¿ Individual Criteria for Accepting Theorems and Proofs: An Empirical ApproachAiso HeinzePart II. Proof and Cognitive DevelopmentChapter 8. Bridging Knowing and Proving in Mathematics: A Didactical PerspectiveNicolas BalacheffChapter 9. The Long-term Cognitive Development of Reasoning and ProofDavid Tall & Juan Pablo Mejia-RamosChapter 10. Historical Artefacts, Semiotic Mediation and Teaching ProofMariolina Bartolini-BussiChapter 11. Proofs, Semiotics and Artefacts of Information TechnologiesAlessandra MariottiPart III. Experiments, Diagrams and ProofsChapter 12. Proof as Experiment in WittgensteinAlfred NordmannChapter 13. Experimentation and Proof in MathematicsMichael D. de VilliersChapter 14. Proof, Mathematical Problem-Solving, and Explanation in Mathematics TeachingKazuhiko NunokawaChapter 15. Evolving Geometric Proofs in the 17th Century: From Icons toSymbolsEvelyne BarbinChapter 16. Proof in the Wording: Two modalities from Ancient Chinese AlgorithmsKarine Chemla