Paul Lockhart said this quote

“ 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’s not at all like science. There’s 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. ”

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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.

~ Paul Lockhart

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?

~ Paul Lockhart

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

~ Paul Lockhart

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