How to Solve It : A New Aspect of Mathematical Method

Description

Cover Title Copyright Contents From the Preface to the First Printing From the Preface to the Seventh Printing Preface to the Second Edition "How to Solve It" list Foreword Introduction PART I. IN THE CLASSROOM Purpose 1. Helping the student 2. Questions, recommendations, mental operations 3· Generality 4· Common sense 5· Teacher and student. Imitation and practice Main divisions, main questions 6. Four phases 7· Understanding the problem 8. Example 9. Devising a plan 10. Example 11. Carrying out the plan 12. Example 13· Looking back 14· Example 15· Various approaches 16. The teacher's method of questioning 17· Good questions and bad questions More examples 18. A problem of construction 19. A problem to prove 20. A rate problem PART II. HOW TO SOLVE IT A dialogue PART III. SHORT DICTIONARY OF HEURISTIC Analogy Auxiliary elements Auxiliary problem Bolzano Bright idea Can you check the result? Can you derive the result differently? Can you use the result? Carrying out Condition Contradictory† Corollary Could you derive something useful from the data? Could you restate the problem?† Decomposing and recombining Definition Descartes Determination, hope, success Diagnosis Did you use all the data? Do you know a related problem? Draw a figure† Examine your guess Figures Generalization Have you seen it before? Here is a problem related to yours and solved before Heuristic Heuristic reasoning If you cannot solve the proposed problem Induction and mathematical induction Inventor's paradox Is it possible to satisfy the condition? Leibnitz Lemma Look at the unknown Modern heuristic Notation Pappus Pedantry and mastery Practical problems Problems to find, problems to prove Progress and achievement Puzzles Reductio ad absurdum and indirect proof Redundant† Routine problem Rules of discovery Rules of style Rules of teaching Separate the various parts of the condition Setting up equations Signs of progress Specialization Subconscious work Symmetry Terms, old and new Test by dimension The future mathematician The intelligent problem-solver The intelligent reader The traditional mathematics professor Variation of the problem What is the unknown? Why proofs? Wisdom of proverbs Working backwards PART IV. PROBLEMS, HINTS, SOLUTIONS Problems Hints Solutions