G 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

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

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

In order to solve this differential equation you look at it till a solution occurs to you. — 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 open secret of real success is to throw your whole personality into your problem. — 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

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

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

Mathematics is not a spectator sport! — 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

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

Beauty in mathematics is seeing the truth without effort. — 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

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

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

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 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

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

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

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

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

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