George Polya Quotes

To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.

[Hilbert] once had a student in mathematics who stopped coming to his lectures, and [he] was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.'

If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. 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.

When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

There are many questions which fools can ask that wise men cannot answer.

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

Beauty in mathematics is seeing the truth without effort.

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.