George Polya Quotes & Sayings

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. — George Polya

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. — George Polya

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

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. — George Polya

John von Neumann was the only student I was ever afraid of. — George Polya

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. — George Polya

If there is a problem you can't solve, then there is an easier problem you can't solve: find it. — George Polya

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. — George Polya

Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side. — George Polya

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. — George Polya

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. — George Polya

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

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

Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. [ ... ] To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery. — George Polya

If you can't solve a problem, then there is an easier problem you can solve: find it. — George Polya

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

Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems. — George Polya

Analogy pervades all our thinking, — George Polya

There exist a lot of questions that the fools can ask, and the intelligent cannot answer. — George Polya

What is the difference between method and device? A method is a device which you use twice. — George Polya

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. — George Polya

Look around when you have got your first mushroom or made your first discovery: they grow in clusters. — George Polya

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

I am too good for philosophy and not good enough for physics. Mathematics is in between. — George Polya

The future mathematician ... should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By this means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and dislikes, his taste, his own line. — George Polya

Geometry is the science of correct reasoning on incorrect figures. — George Polya

It may be more important in the mathematics class how you teach than what you teach. — George Polya

If you cannot solve the proposed problem try to solve first some related problem. — George Polya

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 ... — George Polya

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

Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves. — George Polya

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between. — George Polya

The best of ideas is hurt by uncritical acceptance and thrives on critical examination. — George Polya

The principle is so perfectly general that no particular application of it is possible. — George Polya

In order to solve this differential equation you look at it till a solution occurs to you. — George Polya

Mathematics is not a spectator sport! — George Polya

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. — George Polya

Where should I start? Start from the statement of the problem ... What can I do? Visualize the problem as a whole as clearly and as vividly as you can ... What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. — George Polya

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. — George Polya

My method to overcome a difficulty is to go round it. — George Polya

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. — George Polya

You should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction of a less fortunate one. — George Polya

The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time. — George Polya

If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it. — George Polya

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

Beauty in mathematics is seeing the truth without effort. — George Polya

Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice. — George Polya

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. — George Polya

It is generally useless to carry out details without having seen the main connection, or having made a sort of plan. — George Polya

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. — George Polya

There is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible. — George Polya

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. — George Polya

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. — George Polya

The open secret of real success is to throw your whole personality into your problem. — George Polya

An idea which can be used once is a trick. If it can be used more than once it becomes a method. — George Polya

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

It is better to solve one problem five different ways, than to solve five problems one way. — George Polya

A mathematics teacher is a midwife to ideas. — George Polya

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.' — George Polya

Mathematics consists in proving the most obvious thing in the least obvious way. — George Polya

Good problems and mushrooms of certain kinds have something in common; they grow in clusters. — George Polya