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.

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.

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.

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

In order to solve this differential equation you look at it until a solution occurs to you.

In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.

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