This past Saturday morning my girlfriend (whom I’ll call A because she likes anonymity) and I participated in an awesome workshop led by Sam Coskey at The School of Mathematics in Brooklyn. At some point soon I need to do an appreciation post about Sam and School of Mathematics founder Avital Oliver because they are both rad. In the meantime:
School of Math workshops are always fun. They assume virtually no knowledge from the participants; they start with a simple question about some simple thing; and they uniformly turn into deep and rigorous arguments about fundamental questions. On this occasion we started with a simple question about perimeters and ended up proving that triangles with the same angles really do have proportional sides, and then using this result to prove the Pythagorean theorem. None of us was especially expecting this to happen.
But the best part for me was hearing from A afterward about what the experience was like for her. Describing a moment where one of the participants decided to cut up a rectangle –
and rearrange the pieces –
proving that ab = cd, she said –
"It's like, you look at something, and you try to think what else it could be, in order to bring you closer to what it was in the first place."
I love this. It captures something at the heart of how I experience math and why I love it so much. Math as the opportunity to
a) behold something; entertain a question about it
b) speculate wildly about how it could be transformed
c) and by this act of creative imagination, manage to answer the question; the altered, manipulated, crafted, reimagined form reflecting light from another galaxy back onto what you began with, illuminating it in a new way.
So, a challenge to you, me, all of us:
When was the last time you gave your students this opportunity? When will be the next time?
Research in Practice 
Hmm. All the way there? Rarely. Very rarely.
But I demand assistance as I develop topics. I do a lot of (a) to (c) with discussion taking the place of the wild speculation for (b). But with them talking, and me talking. Frequently.
Last time? There was a good example 5 lessons ago (that’s the Friday before last). Developing the law of cosines (coordinate approach) giving kids a chance to notice “hey, we could use the distance formula there”
It’s not as big as ab = cd, but the habit becomes strong.
Trad: Let me prove something, and then you guys can use it
vs.
jd: Let’s look at this, and together see where it can go (and then I won’t give you a whole lot of time to use it, but the figuring it out part is more interesting, anyhow).
There’s a lovely line in Zen:
“Before I began practicing Zen, the mountains were mountains, and the river was a river. Then I started practicing, and the mountains were no longer mountains, and the river was no longer a river. Now I am enlightened, and the mountains are mountains, and the river is a river.”
Or something to that effect. I don’t know if there’s any way to get closer to this feeling than mathematics (short of some kind of meditation, perhaps. Though there’s evidence that they would meditate on geometry problems in Zen monasteries, back in the day).
Thanks for the post. I’m looking forward to reading your blog.
gorgeous call to action, benjamin. thank you, as always, for your wit and wisdom, your heart and hope. you inspire me.