George Polya
It is better to solve one problem five different ways, than to solve five problems one way.Collection: Education
Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.Collection: Art
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.Collection: Science
If there is a problem you can't solve, then there is an easier problem you can't solve: find it.Collection: Education
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.Collection: Mean
An idea which can be used only once is a trick. If one can use it more than once it becomes a method.Collection: Math
It may be more important in the mathematics class how you teach than what you teach.Collection: Class
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.Collection: Education
Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.Collection: Sides
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.Collection: Science
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.Collection: Swimming
Beauty in mathematics is seeing the truth without effort.Collection: Education
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.Collection: Teacher
Mathematics is not a spectator sport!Collection: Sports
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.Collection: Teacher
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.Collection: Teaching
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.Collection: Truth
The best of ideas is hurt by uncritical acceptance and thrives on critical examination.Collection: Hurt
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 yourselfCollection: Hard Work
I am too good for philosophy and not good enough for physics. Mathematics is in between.Collection: Philosophy
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.Collection: Two
The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.Collection: Science
Mathematics consists in proving the most obvious thing in the least obvious way.Collection: Knowledge
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.Collection: Education
What is the difference between method and device? A method is a device which you use twice.Collection: Math
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.Collection: Mean
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.Collection: Expression
There are many questions which fools can ask that wise men cannot answer.Collection: Wise
The open secret of real success is to throw your whole personality into your problem.Collection: Success
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.Collection: Trying
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.Collection: Good Luck
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.Collection: Appreciation
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.Collection: Attitude
In order to solve this differential equation you look at it until a solution occurs to you.Collection: Math
Look around when you have got your first mushroom or made your first discovery: they grow in clusters.Collection: Science
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.'Collection: Creativity
If you cannot solve the proposed problem try to solve first some related problem.Collection: Motivational
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.Collection: Education
My method to overcome a difficulty is to go round it.Collection: Overcoming
A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.Collection: Real
A mathematics teacher is a midwife to ideas.Collection: Teacher
The principle is so perfectly general that no particular application of it is possible.Collection: Knowledge
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.Collection: Thinking
John von Neumann was the only student I was ever afraid of.Collection: Students
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.Collection: Independent
There exist a lot of questions that the fools can ask, and the intelligent cannot answer.Collection: Intelligent
Geometry is the science of correct reasoning on incorrect figures.Collection: Education
A GREAT discovery solves a great problem but there is a grain of discovery in any problem.Collection: Education
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.Collection: Teacher
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.Collection: Teaching