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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.

George Polya

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George Polya
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More quotes by George Polya

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Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.
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Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself
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The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
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The principle is so perfectly general that no particular application of it is possible.
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There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
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I am intentionally avoiding the standard term which, by the way, did not exist in Euler's time. One of the ugliest outgrowths of the new math was the premature introduction of technical terms.
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My method to overcome a difficulty is to go round it.
George Polya
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The open secret of real success is to throw your whole personality into your problem.
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You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it.
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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.
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Mathematics consists in proving the most obvious thing in the least obvious way.
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The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving...The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
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It may be more important in the mathematics class how you teach than what you teach.
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The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
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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.
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Geometry is the science of correct reasoning on incorrect figures.
George Polya
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The result of the mathematician's creative work is demonstrative reasoning, a proof but the proof is discovered by plausible reasoning, by guessing.
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Beauty in mathematics is seeing the truth without effort.
George Polya
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Look around when you have got your first mushroom or made your first discovery: they grow in clusters.
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If you cannot solve the proposed problem try to solve first some related problem.
George Polya