Monday, July 5, 2010

Math Reflections: What I've learned so far...

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.

Other ideas:
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.


  1. Truly, 99% of the brilliance comes from Bowen. He has such a great sense for how to sequence and write problems carefully so that the mathematics unfolds organically for participants. I think it takes a huge amount of pedagogical content knowledge to do this...

    So glad that you're enjoying yourself. And I love your wish for everyone on the planet.

    Look forward to talking with and learning from you. Darryl

  2. Jesse, I'm feeling really, what's the word, into, taken with, the close relationship of what you're talking about here with what you were talking about in the previous post. The teacher/teaching distinction feels like a perfect second example of the ambition/curiosity thing.

    This is hooking into a new experience I had this past year by being a teacher trainer. I watched more different math teachers' classes in the last year than I maybe ever had since becoming a teacher. (This may not be literally true but it certainly feels like it.) Anyway one of the big lessons of the year was a kind of demystification of a number of teacher skills. Maybe you're having a similar experience watching all these videos?

    Classroom management skills in particular. I found out (okay, this is obvious, and I probably "knew" it, but I definitely "felt" it in a new way) that smooth classroom flow and teacher authority are affected by an extremely complicated but very concrete set of teacher behaviors, actions, postures and tones, each of which is individually very small, and that are susceptible to being thought about and talked about concretely and worked on one at a time. Most crucially, that these behaviors, actions, postures and tones were all too small and concrete to be something that the teacher would have any reason to feel bad about. They were simply things to think about and work on. I recognized how my own concern with my worth as a teacher, whenever I reflected on classroom management, had distracted me completely from thinking about most of them.

  3. For example, in one class I saw, the teacher spent the first few minutes of class bustling about the room attending to small administrative tasks. His body language communicated to me "don't bother me, I'm busy." During this time the students were supposed to be working on a warmup but they weren't focused. It seemed to me that this was partly because they didn't feel supervised. Although the teacher was in the room, he was giving off the vibe that his attention was not on them but on the little tasks he was attending to.

    I wondered how many times I'd done this; i.e. convey "I'm busy with these tasks" at a time when it probably would have been much more useful to convey "I'm watching you," in particular at the opening of class. I was also struck by the fact that this question felt more fine-grained and concrete than almost any of the questions that had obsessed me about my management when I had a classroom. My questions were always more like "how do I convey strength?" or "what do I do when a kid openly defies me?" or, on bad days, "what's wrong with me that they don't listen to me (and how do I fix it)?" Big, abstract questions. Too big and too abstract. The answer doesn't live at that broad and abstract a level. It lives in the immediate moment-to-moment, where what you do is too small and too concrete to be a matter of self-worth: "should I convey business or watchfulness in this moment?" (This reminds me of that Doug Lemov thing about "always stand still when you issue instructions." Similarly fine-grained and concrete.)

    The big point: where the actual work lies is on a different planet from the conversation you have with yourself when your self-worth is at stake. This is just like what you were saying about doing math: "...then I am able to put my full attention on the mathematics, rather than the distraction of berating myself." Both teaching and doing math are areas where lots of us have trauma because our cultural context teaches us to understand struggle as failure as a person. But the actual work of teaching and doing math (in particular, growing as a teacher or mathematician) centrally involves being concretely aware and engaged with the process during struggle. In both arenas, this awareness is incompatible with concern with questions of self-worth.

    (Just to illustrate the parallel and perhaps flesh out what you were saying above: in math it looks like,

    "What's going on here?"
    "What's this thing's effect on this other thing?"
    "What are the consequences of this thing?"
    "Where else in this situation could I find information about such-and-such?"
    "What would this idea look like from different angles?"

    Enter questions of self-worth and it becomes:
    "Do I know this?"
    "Am I supposed to know this?"
    "Why is this so hard for me?"
    "What's wrong with me that I don't know this (and how do I fix it)?"
    "What will they think of me when they see I don't know it?"

  4. Jesse, your recent posts have been deep and thought-provoking. I think that one of the things that make you a great teacher is how deeply you care both about the mathematics and your students.

    Just to quickly respond to Ben's comment about classroom management, I think that students respond positively (even if it's just on the inside) to being supervised or knowing that you're always watching what's going on, because it communicates the message that you care about their work and what they're doing. One small thing I do is when I remind them to clear off their tabletops except for their working materials, I add that it's because I want to quickly be able to see what they're working on and where they are without having to wade through all the unnecessary stuff.