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?

Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.

[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.

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

To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion − not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to.

Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

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.