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
The most unusual George Polya quotes that are life-changing and eye-opening
In order to solve this differential equation you look at it until a solution occurs to you.
It is better to solve one problem five different ways, than to solve five problems one way.
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.
Beauty in mathematics is seeing the truth without effort.
If there is a problem you can't solve, then there is an easier problem you can't solve: find it.
Geometry is the science of correct reasoning on incorrect figures.
Mathematics is not a spectator sport!
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.
Mathematics consists in proving the most obvious thing in the least obvious way.
Look around when you have got your first mushroom or made your first discovery: they grow in clusters.
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.'
John von Neumann was the only student I was ever afraid of.
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.
A mathematics teacher is a midwife to ideas.
The open secret of real success is to throw your whole personality into your problem.
The principle is so perfectly general that no particular application of it is possible.
A GREAT discovery solves a great problem but there is a grain of discovery in any problem.
What is the difference between method and device? A method is a device which you use twice.
The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.
I am too good for philosophy and not good enough for physics. Mathematics is in between.
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.
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.
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
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
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.
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
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.
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.
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.
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.
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.
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.
Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.
There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
An idea which can be used only once is a trick.
If one can use it more than once it becomes a method.
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.
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.
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.
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.
My method to overcome a difficulty is to go round it.
If you cannot solve the proposed problem try to solve first some related problem.
Euclid 's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.
Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.
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.
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.
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.
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.
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....
It may be more important in the mathematics class how you teach than what you teach.