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

George Polya

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George Polya
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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.
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If you wish to learn swimming you have to go into the water and if you wish to become a problem solver you have to solve problems.
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The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.
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Beauty in mathematics is seeing the truth without effort.
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When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
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The open secret of real success is to throw your whole personality into your problem.
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The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
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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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It may be more important in the mathematics class how you teach than what you teach.
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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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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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There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
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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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A mathematics teacher is a midwife to ideas.
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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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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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My method to overcome a difficulty is to go round it.
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Geometry is the science of correct reasoning on incorrect figures.
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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.
George Polya