George Polya Quotes

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

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

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.

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

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.

An idea which can be used only once is a trick. If one can use it more than once it becomes a method.

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

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.

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.

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.

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

Beauty in mathematics is seeing the truth without effort.

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.

Mathematics is not a spectator sport!

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.

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

The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.

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.

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

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.

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.

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

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.

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

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