Paul Lockhart

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

So [in mathematics] we get to play and imagine whatever we want and make patterns and ask questions about them. But how do we answer these questions? It__ not at all like science. There__ no experiment I can do ... The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.

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?

... This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.On the other hand, once you have made your choices then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back!

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.

... That little narrative is an example of the mathematician__ art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it__ fascinating, it__ fun, and it__ free!

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.

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

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

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

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?

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

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

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?

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.

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.

Mathematics is the art of explanation.

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.