George Polya

A unique, heuristic approach to mathematical discovery and problem solving This combined edition of Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving is unique among mathematics texts. Espousing a heuristic approach to mathematical problem solving, the text may be followed sequentially or according to instructors' individualized curricula. Beginning with a discussion of patterns and practical approaches to problem solving, the book then presents examples from various branches of math and science to help students discover how to solve problems on their own – an invaluable skill for the classroom and beyond.

Mathematical Discovery on Understanding, Learning and Teaching Problem Solving, Volumes I and II

Stefano Zorzi

Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which was not immediately attain- able. Solving problems is the specific achievement of intelligence, and intelligence is the specific gift of mankind: solving problems can be re- The aim of this garded as the most characteristically human activity.

3. I wish to call heuristic the study that the present work attempts, the study of means and methods of problem solving. which was used by some philosophers in the past, is half-forgotten and half-discredited nowadays, but I am not afraid to use it. 

In fact, most of the time the present work offers a down-to-earth practical aspect of heuristic: I am trying, by all the means at my disposal, to entice the reader to do problems and to think about the means and and methods he uses in doing them. 

In most of the following chapters, the greater part of the text is devoted to the broad presentation of the solutions of a few problems. The pre- sentation may appear too broad to a mathematician who is not interested in methodical points. In fact, what is presented here are not merely solutions but case histories of solutions. Such a case history describes the sequence of essential steps by which the solution has been eventually discovered, and tries to disclose the motives and attitudes prompting these steps. 

Same as the teaching of history as “exempla”. 

The teacher should know what he is supposed to teach. He should show his students how to solve problems--but if he does not know, how can he show them? The teacher should develop his students' know-how, their ability to reason; he should recognize and encourage creative think- ing-but the curriculum he went through paid insufficient attention to his mastery of the subject matter and no attention at all to his know-how, to his ability to reason, to his ability to solve problems, to his creative think- ing. Here is, in my opinion, the worst gap in the present preparation of high school mathematics teachers.

Mathematical Discovery on Understanding, Learning and Teaching Problem Solving, Volumes I and II quotes

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