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

Beauty in mathematics is seeing the truth without effort.

John von Neumann was the only student I was ever afraid of.

A GREAT discovery solves a great problem but there is a grain of discovery in any problem.

The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

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.

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