George Pólya Quotes

If you cannot solve the proposed problem...try to solve first some related problem.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

Analogy pervades all our thinking,

In order to solve this differential equation you look at it till a solution occurs to you.

Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically — he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with

Problems “in letters” are susceptible of more, and more interesting, tests than “problems in numbers

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

Pedantry and mastery are opposite attitudes toward rules. 1. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally. 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.

Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice

"Now and then, teaching may approach poetry, and now and then it may approach profanity. May I tell you a little story about the great Einstein? I listened once to Einstein as he talked to a group of physicists in a party. "Why have all the electrons the same charge?" said he. "Well, why are all the little balls in the goat dung of the same size?" Why did Einstein say such things? Just to make some snobs to raise their eyebrows? He was not disinclined to do so, I think. Yet, probably, it went deeper. I do not think that the overheard remark of Einstein was quite casual. At any rate, I learnt something from it: Abstractions are important; use all means to make them more tangible. Nothing is too good or too bad, too poetical or too trivial to clarify your abstractions. As Montaigne put it: The truth is such a great thing that we should not disdain any means that could lead to it. Therefore, if the spirit moves you to be a little poetical, or a little profane, in your class, do not have the wrong kind of inhibition." - George Polya's Mathematical Discovery, Volume 11, pp 102, 1962.

It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire.

The worst may happen if the student embarks upon computations or constructions without having understood the problem.

Mathematics consists in proving the most obvious thing in the least obvious way.

Look around when you have got your first mushroom or
made your first discovery: they grow in clusters.

The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.