Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.

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

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.

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....

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.

Mathematics is not a spectator sport!

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.