Last Friday, our incredible math teachers Darryl Yong and Bowen Kerins asked us to reflect about our first five days of problem solving. Let me begin by saying that this part of PCMI is the most mathematically satisfying thing in my memory, and follow that up with three sections:
I. Mathematical ideas that I was supported in discovering on my own:
Sequences can be added together piecemeal like linear combinations to create new sequences. I (think I) can think of sequences like elements of a group under addition or multiplication (this is an idea I haven't explored, but it seemed like one worth exploring...sequences as generators, etc.) What happens when you multiply the terms of a sequence by the terms of another sequence. A new way of finding two numbers given their sum and product, which deserves it's own blogpost. How to express a sequence recursively given it's closed rule (like the sum of two different bases both raised to the x power). What kinds of starting values will make a recursive function (like J(n) = 7J(n-1) - 10J(n-2)) exponential and why. How to express a sequence given recursively in closed form (over and over again, which was amazing every time.) How to use a TI-Nspire. The rational representation of .001001002003005008013... which is really neat. Finally, the closed form of the Fibonacci sequence. This literally made me cry. I did it. I wrote it down. I started crying. I stopped writing and looked around the room. Nothing had changed that I could see. But I felt like someone had just opened up a window to God. It's one of the most beautiful things I've ever seen. I have never cried about math before.
Rich and rewarding mathematical environments arise in groups when problems have a low threshold and a high ceiling. Humor helps. I like to work near people but be able to move at different paces. When I feel satisfied in my own mathematical exploration, I don't care whether I'm ahead of or behind those around me. I don't want to be told the wow: I want to find it on my own.
My peers are superlatively hard on themselves when they feel behind or make mistakes or don't understand. It is of supreme value to be generous and loving with myself, (even if I didn't deserve it!) because then I am able to put my full attention on the mathematics, rather than the distraction of berating myself. I have had a supremely enjoyable time doing math here this week, and feel really blessed. It's really hard (so hard!) to continue doing math when you're feeling stupid and behind. My admiration soars for those people who are willing to be here and keep trying even when they feel stupid. My admiration soars for my students who are willing to keep trying even when they feel stupid. It's got to be one of the most painful day to day kinds of experience we humans can have.
My dance teacher Nancy Stark Smith says this wonderful thing about a way to approach improvisation, and of late it has become my life mantra in every context I'm in: "Replace ambition with curiosity." I have been practicing this for about a decade, and it's freaking amazing to see some concrete results. Life is so much more fun this way.
I wish for everyone on the planet to experience the freedom and relief of being let off every hook they hang themselves on. I want to tell them, "Enjoy it. Be as good as you can, but enjoy it. For goodness sakes. You are all so bright, so deserving. Be easy on yourselves."
My big question:
What are the brilliant Darryl Yong and Bowen Kerins really up to? What genius is required to make problem sets like this? What do they think about when they're planning? What will I have to keep in mind as I try to follow their example? Because that's why I'm here, and I intend to bring their model to my classroom or fail trying.