A professor … may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. … The framework is lacking; students do not know where the subject fits in, and this has a paralyzing effect on the mind.

Education consists in co-operating with what is already inside a child's mind … The best way to learn geometry is to follow the road which the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things.

If a child left school at ten, knowing nothing of detailed information, but knowing the pleasure that comes from agreeable music, from reading, from making things, from finding things out, it would be better off than a man who left university at twenty-two, full of facts but without any desire to enquire further into such dry domains.

In a class I was taking there was one boy who was much older than the rest. He clearly had no motive to work. I told him that, if he could produce for me, accurately to scale, drawings of the pieces of wood required to make a desk like the one he was sitting at, I would try to persuade the Headmaster to let him do woodwork during the mathematics hours—in the course of which, no doubt, he would learn something about measurement and numbers. Next day, he turned up with this task completed to perfection. This I have often found with pupils; it is not so much that they cannot do the work, as that they see no purpose in it.

In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears.

It has been proposed (in despair) to define mathematics as “what mathematicians do.” Only such a broad definition, it was felt, would cover all the things that might become embodied in mathematics; for mathematicians today attack many problems not regarded as mathematics in the past, and what they will do in the future there is no saying.

Many people regard mathematicians as a race apart, possessed of almost supernatural powers. While this is very flattering for successful mathematicians, it is very bad for those who, for one reason or another, are attempting to learn the subject.

Mathematics is the classification and study of all possible patterns.

Nearly every subject has a shadow, or imitation. It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When it played a wrong note, it would see the frown of its teacher, and try again. But it would obviously have no idea of what it was doing, or why anyone should devote hours to such an extraordinary exercise. It would have learnt an imitation of music. and it would fear the piano exactly as most students fear what is supposed to be mathematics.

One can learn imitation history—kings and dates, but not the slightest idea of the motives behind it all; imitation literature—stacks of notes on Shakespeare’s phrases, and a complete destruction of the power to enjoy Shakespeare.

The fear of mathematics is a tradition handed down from days when the majority of teachers knew little about human nature and nothing at all about the nature of mathematics itself. What they did teach was an imitation.

The history of mathematics is exhilarating, because it unfolds before us the vision of an endless series of victories of the human mind, victories without counterbalancing failures, that is, without dishonorable and humiliating ones, and without atrocities.

The main difficulty the student of groups meets is not that of following the argument, which is nearly always straightforward, but of grasping the purpose of the investigation.

The main sources of mathematical invention seem to be within man rather than outside of him: his own inveterate and insatiable curiosity, his constant itching for intellectual adventure; and likewise the main obstacles to mathematical progress seem to be also within himself; his scandalous inertia and laziness, his fear of adventure, his need of conformity to old standards, and his obsession by mathematical ghosts.

The professor may choose familiar topics as a starting point. The students collect material, work problems, observe regularities, frame hypotheses, discover and prove theorems for themselves. … the student knows what he is doing and where he is going; he is secure in his mastery of the subject, strengthened in confidence of himself. He has had the experience of discovering mathematics. He no longer thinks of mathematics as static dogma learned by rote. He sees mathematics as something growing and developing, mathematical concepts as something continually revised and enriched in the light of new knowledge. The course may have covered a very limited region, but it should leave the student ready to explore further on his own.

To keep pace with the growth of mathematics, one would have to read about fifteen papers a day, most of them packed with technical details and of considerable length. No one dreams of attempting this task.

To master anything—from football to relativity—requires effort. But it does not require unpleasant effort, drudgery.

To see the clear, logical ideas gradually being disentangled from vagueness and confusion is vastly more instructive than simply starting with the logical ideas.

To solve a problem means to reduce it to something simpler than itself.

Very few people realize the enormous bulk of contemporary mathematics. Probably it would be easier to learn all the languages of the world than to master all mathematics at present known. The languages could, I imagine, be learnt in a lifetime; mathematics certainly could not. Nor is the subject static.