No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a mixed number while 5/2 is an improper fraction. They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade?

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

Teaching is not about information. It's about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can't be real, then you have no right to inflict yourself upon innocent children.

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.

[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 not a language, it's an adventure

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?

If I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern-making, I couldn't possibly do as good a job as is currently being done-I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

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

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?

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.

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.

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

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

There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.

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