Mathematical reasoning : patterns, problems, conjectures, and proofs

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書誌事項

Mathematical reasoning : patterns, problems, conjectures, and proofs

Raymond S. Nickerson

Psychology Press, c2010

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注記

Includes bibliographical references and indexes

"This book explores when and why the rudiments of mathematical capability first appeared among human beings, what its fundamental concepts are, and how and why it has grown into the richly branching complex of specialties that it is today. It discusses whether the 'truths' of mathematics are discoveries or inventions and what prompts the emergence of concepts that appear to be descriptive of nothing in human experience. Also covered is the role of esthetics in mathematics: What exactly are mathematicians seeing when they describe a mathematical entity as "beautiful"? There is discussion of whether mathematical disability is distinguishable from a general cognitive deficit and whether the potential for mathematical reasoning is best developed through instruction." "This volume is unique in the vast range of psychological questions it covers, as revealed in the work habits and products of numerous mathematicians. It provides fascinating reading for researchers and students with an interest in cognition in gener

収録内容

  • What is mathematics?
  • Counting
  • Numbers
  • Deduction and abstraction
  • Proofs
  • Informal reasoning in mathematics
  • Representation in mathematics
  • Infinity
  • Infinitesimals
  • Predictions, presumptions, and personalities
  • Esthetics and the joy of mathematics
  • The usefulness of mathematics
  • Foundations and the "stuff" of mathematics
  • Preschool development of numerical and mathematical skills
  • Mathematics in school
  • Mathematical problem solving
  • Final thoughts

内容説明・目次

内容説明

The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition. This book explores when and why the rudiments of mathematical capability first appeared among human beings, what its fundamental concepts are, and how and why it has grown into the richly branching complex of specialties that it is today. It discusses whether the 'truths' of mathematics are discoveries or inventions, and what prompts the emergence of concepts that appear to be descriptive of nothing in human experience. Also covered is the role of esthetics in mathematics: What exactly are mathematicians seeing when they describe a mathematical entity as 'beautiful'? There is discussion of whether mathematical disability is distinguishable from a general cognitive deficit and whether the potential for mathematical reasoning is best developed through instruction. This volume is unique in the vast range of psychological questions it covers, as revealed in the work habits and products of numerous mathematicians. It provides fascinating reading for researchers and students with an interest in cognition in general and mathematical cognition in particular. Instructors of mathematics will also find the book's insights illuminating.

目次

1. What Is Mathematics? 2. Counting. 3. Numbers. 4. Deduction and Abstraction. 5. Proofs. 6. Informal Reasoning. 7. Representations. 8. Infinity. 9. Infinitesimals. 10. Predilections. 11. Esthetics in Mathematics. 12. The Usefulness of Mathematics. 13. Foundations. 14. Preschool Development. 15. Math in Schools. 16. Math Problem Solving. 17. Final Thoughts.

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