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Geometry is the science of correct reasoning on incorrect figures.

George Polya

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George Polya
Math
Mathematics
Figures
Education
Incorrect
Science
Geometry
Reasoning
Correct
Mathematical

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The open secret of real success is to throw your whole personality into your problem.
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Beauty in mathematics is seeing the truth without effort.
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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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It may be more important in the mathematics class how you teach than what you teach.
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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 consists in proving the most obvious thing in the least obvious way.
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The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
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A mathematics teacher is a midwife to ideas.
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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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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.
George Polya
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If you cannot solve the proposed problem try to solve first some related problem.
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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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When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
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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.
George Polya
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Look around when you have got your first mushroom or made your first discovery: they grow in clusters.
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.
George Polya
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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.
George Polya
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The principle is so perfectly general that no particular application of it is possible.
George Polya
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My method to overcome a difficulty is to go round it.
George Polya
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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
George Polya