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.

Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.

It is the story that matters, not just the ending.

So how does one go about proving something like this? It's not like being a lawyer, where the goal is to persuade other people; nor is it like a scientist testing a theory. This is a unique art form within the world of rational science. We are trying to craft a "poem of reason" that explains fully and clearly and satisfies the pickiest demands of logic, while at the same time giving us goosebumps.

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

Mathematics is the purest of the arts, as well as the most misunderstood.

I don't see how it's doing society any good to have so many members walking around with vague memories of algebraic formulas and geometric diagrams and clear memories of hating them.