George Polya Quotes
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A mathematics teacher is a midwife to ideas.
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Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
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Mathematics consists in proving the most obvious thing in the least obvious way.
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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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If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.
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Epitaph on Newton: Nature and Nature's law lay hid in night: God said, "Let Newton be!," and all was light. [added by Sir John Collings Squire: It did not last: the Devil shouting "Ho. Let Einstein be," restored the status quo] [Aaron Hill's version: O'er Nature's laws God cast the veil of night, Out blaz'd a Newton's soul and all was light.
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A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
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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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In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
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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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Mathematics is not a spectator sport!
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There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
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Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
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If you cannot solve the proposed problem try to solve first some related problem.
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The teacher can seldom afford to miss the questions: What is the unknown? What are the data? What is the condition? The student should consider the principal parts of the problem attentively, repeatedly, and from from various sides.
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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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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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An idea which can be used only once is a trick. If one can use it more than once it becomes a method.
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In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.
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Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.
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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.
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What is the difference between method and device? A method is a device which you use twice.
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To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument....
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John von Neumann was the only student I was ever afraid of.
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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.'
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Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
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Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.
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Beauty in mathematics is seeing the truth without effort.
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There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.
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The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.
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Related Authors
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Ralph P. Boas, Jr. Mathematician -
Rudolf Carnap Philosopher -
Leonhard Euler Mathematician -
G. H. Hardy Mathematician -
Thomas Hardy Novelist -
Paul Halmos Mathematician -
Paul Erdos Mathematician -
Imre Lakatos Philosopher -
Herbert Simon Scientist -
Bertrand Russell Philosopher -
Srinivasa Ramanujan Mathematician -
John von Neumann Mathematician -
David Hilbert Mathematician -
John Edensor Littlewood Mathematician -
Euclid Mathematician -
Henri Poincare Mathematician -
Jacques Hadamard Mathematician -
Hermann Weyl Mathematician -
Thomas Kuhn Physicist -
Richard Courant Mathematician