How to solve it : a new aspect of mathematical method

<h3>Contents</h3> <pre> From the Preface to the First Printing...................................v From the Preface to the Seventh Printing.................................viii Preface to the Second Edition............................................ix "How to Solve It" list...................................................xvi Foreword.................................................................xix Introduction.............................................................xxv 1. Helping the student...................................................1 2. Questions, recommendations, mental operations.........................1 3· Generality.......................................................2 4· Common sense.....................................................3 5· Teacher and student. Imitation and practice......................3 6. Four phases...........................................................5 7· Understanding the problem........................................6 8. Example...............................................................7 g. Devising a plan.......................................................8 10. Example..............................................................10 11. Carrying out the plan................................................12 12. Example..............................................................13 13· Looking back....................................................14 14· Example.........................................................16 15· Various approaches..............................................19 16. The teacher's method of questioning..................................20 17· Good questions and bad questions................................22 18. A problem of construction............................................23 19. A problem to prove...................................................25 20. A rate problem.......................................................29 A dialogue...............................................................33 Analogy..................................................................37 Auxiliary elements.......................................................46 Auxiliary problem........................................................50 Bolzano..................................................................57 Bright idea..............................................................58 Can you check the result?...............................................59 Can you derive the result differently?...................................61 Can you use the result?..................................................64 Carrying out.............................................................68 Condition................................................................72 Con tradictory...........................................................73 Corollary................................................................73 Could you derive something useful from the data?.........................73 Could you restate the problem?...........................................75 Decomposing and recombining..............................................75 Definition...............................................................85 Descartes................................................................92 Determination, hope, success.............................................93 Diagnosis................................................................94 Did you use all the data?................................................95 Do you know a related problem?...........................................98 Draw a figure............................................................99 Examine your guess.......................................................99 Figures..................................................................103 Generalization...........................................................108 Have you seen it before?.................................................110 Here is a problem related to yours and solved before.....................110 Heuristic................................................................112 Heuristic reasoning......................................................113 If you cannot solve the proposed problem.................................114 Induction and mathematical induction.....................................114 Inventor's paradox.......................................................121 Is it possible to satisfy the condition?.................................122 Leibnitz.................................................................123 Lemma....................................................................123 Look at the unknown......................................................123 Modern heuristic.........................................................129 Notation.................................................................134 Pappus...................................................................141 Pedantry and mastery.....................................................148 Practical problems.......................................................149 Problems to find, problems to prove......................................154 Progress and achievement.................................................157 Puzzles..................................................................160 Reductio ad absurdum and indirect proof..................................162 Redundant................................................................171 Routine problem..........................................................171 Rules of discovery.......................................................172 Rules of style...........................................................172 Rules of teaching........................................................173 Separate the various parts of the condition..............................173 Setting up equations.....................................................174 Signs of progress........................................................178 Specialization...........................................................190 Subconscious work........................................................197 Symmetry.................................................................199 Terms, old and new.......................................................200 Test by dimension........................................................202 The future mathematician.................................................205 The intelligent problem-solver...........................................206 The intelligent reader...................................................207 The traditional mathematics professor....................................208 Variation of the problem.................................................209 What is the unknown?.....................................................214 Why proofs?..............................................................215 Wisdom of proverbs.......................................................221 Working backwards........................................................225 Problems.................................................................234 Hints....................................................................238 Solutions................................................................242 </pre>