Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.

The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was "out there" all along I just couldn't see it. And now I can! This is really what keeps me in the math game- the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.

The only thing I am interested in using mathematics for is to have a good time and to help others do the same.

Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself.

Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.

A good problem is something you don't know how to solve. That's what makes it a good puzzle and a good opportunity.

If teaching is reduced to mere data transmission, if there is no sharing or excitement and wonder, if teachers themselves are passive recipients of information and not creators of new ideas, what hope is there for their students?