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

— Paul Lockhart

And I'll go even further and say that mathematics, this art of abstract pattern-making — even more than storytelling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.

— Paul Lockhart

Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.

— Paul Lockhart

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

— Paul Lockhart

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

— Paul Lockhart

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?

— Paul Lockhart

If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.

— Paul Lockhart

IMPLICIT: ... You have to [learn to] walk before you can run.SALVAGE: No, you have to have something you want to run toward.

— Paul Lockhart

In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world.

— Paul Lockhart

It is the story that matters not just the ending.

— Paul Lockhart