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.

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

[Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on.

[Math is] not at all like science. There's no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations.

Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.

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.

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