George Pólya (1887-12-13–1985-09-07) was a Hungarian mathematician and professor of mathematics at ETH Zürich and at Stanford University. His work on heuristics and pedagogy has had substantial and lasting influence on mathematical education, and has also been influential in artificial intelligence.
How to Solve It (1945)
Page references from Expanded Princeton Science Library Edition (2004), ISBN 0-691-11966-X
- Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
- Page 37
- Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.
- Page 70
- [...] the best of ideas is hurt by uncritical acceptance and thrives on critical examination.
- Page 100
- We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.
- Page 113
- Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. [...] To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.
- Page 148
- To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument....
- Page 207
- The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. [...] Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.
- Page 219
About George Pólya
- For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya.
- A. H. Schoenfeld, "Polya, Problem Solving, and Education" in Mathematics Magazine (1987)