Part I. Reflections on the Nature and Teaching of Proof
Chapter 1. The Conjoint Origin of Proof and Theoretical Physics
Hans Niels Jahnke
Chapter 2. Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity.
Chapter 3. Pre-Axiomatic Mathematical Reasoning: An Algebraic Approach
Mary Catherine Leng
Chapter 4. Completions, Constructions and Corrollaries
Chapter 5. Authoritarian vs. Authoritative Teaching: Polya and Lakatos
Chapter 6. Proofs as Bearers of Mathematical Knowledge
Gila Hanna & Ed Barbeau
Chapter 7. Mathematicians' Individual Criteria for Accepting Theorems and Proofs: An Empirical Approach
Part II. Proof and Cognitive Development
Chapter 8. Bridging Knowing and Proving in Mathematics: A Didactical Perspective
Chapter 9. The Long-term Cognitive Development of Reasoning and Proof
David Tall & Juan Pablo Mejia-Ramos
Chapter 10. Historical Artefacts, Semiotic Mediation and Teaching Proof
Chapter 11. Proofs, Semiotics and Artefacts of Information Technologies
Part III. Experiments, Diagrams and Proofs
Chapter 12. Proof as Experiment in Wittgenstein
Chapter 13. Experimentation and Proof in Mathematics
Michael D. de Villiers
Chapter 14. Proof, Mathematical Problem-Solving, and Explanation in Mathematics Teaching
Chapter 15. Evolving Geometric Proofs in the 17th Century: From Icons toSymbols
Chapter 16. Proof in the Wording: Two modalities from Ancient Chinese Algorithms
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
Critically examines recent literature in the philosophy of mathematics on mathematicians' methods for devising and judging proof
Creates a much needed bridge between the discipline of philosophy of mathematics and mathematics education
Demonstrates that mathematical practice has lessons for instructional practice
Stresses the relevance of pragmatic dimensions of mathematics for current philosophy of mathematics