George Pólya

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.

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.

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

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

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.

"Now and then, teaching may approach poetry, and now and then it may approach profanity. May I tell you a little story about the great Einstein? I listened once to Einstein as he talked to a group of physicists in a party. "Why have all the electrons the same charge?" said he. "Well, why are all the little balls in the goat dung of the same size?" Why did Einstein say such things? Just to make some snobs to raise their eyebrows? He was not disinclined to do so, I think. Yet, probably, it went deeper. I do not think that the overheard remark of Einstein was quite casual. At any rate, I learnt something from it: Abstractions are important; use all means to make them more tangible. Nothing is too good or too bad, too poetical or too trivial to clarify your abstractions. As Montaigne put it: The truth is such a great thing that we should not disdain any means that could lead to it. Therefore, if the spirit moves you to be a little poetical, or a little profane, in your class, do not have the wrong kind of inhibition." - George Polya's Mathematical Discovery, Volume 11, pp 102, 1962.

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

The worst may happen if the student embarks upon computations or constructions without having understood the problem.

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

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

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

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.

It so happens that one of the greatest mathematical discoveries of all times was guided by physical intuition.

The advanced reader who skips parts that appear too elementary may miss more than the less advanced reader who skips parts that appear too complex.