# On Learning

I’ve done a good bit of math the last two days, and plan to continue to do so over my break. I’m trying both to keep my skills sharp and to learn some new things about the math I currently teach to add to my toolbox. I’m focusing for now on the Mathematics 4-5 problem sets from the Exeter curriculum and the PCMI 2014 problem sets. Both have been enjoyable. Here are a few problems from the Exeter sets I found to be particularly interesting:

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Page 5

I learned a bit of math and got a new perspective on topics I teach from each problem. I was particularly interested in the final problem, as I think it would fit well into a unit on the Binomial Theorem to see if students can apply what they know in a new context and to reinforce a range of relevant mathematical ideas.

But then I realized that I may solve several hundred math problems over the coming days. How would I remember that particular problem, during the particular unit when it was relevant? I could toss it into a document in the appropriate folder in my file system, but even then I would be unlikely to stumble across it at the right moment.

Imagine that. Solving a problem, and remembering some things about it to be retrieved at the appropriate moment in the future. Sounds like what I want for my students — and what I often fail to achieve for them.

I’m going to take a few steps to try to reinforce my learning. I don’t want to try to exhaustively catalog everything I learn in the perfect place so I will remind myself of it. Instead, I want to learn by doing — to reinforce key ideas so that I remember them and am likely to return to them when they are useful in the future. Here are some things I’m trying to make this more likely:

1. Summarizing key ideas. Every 5 pages (~50 problems) I am looking over both the problems and my notes and summarizing, in writing, the key ideas of problems and sequences of problems that I’ve found most useful.
2. Generating additional examples. When I find a problem I particularly like, I try to come up with an extension problem, a different problem that gets at the same idea, or a similar problem with different numbers to play with the concept and look at additional examples.
3. Spacing over time. I’m not working very fast and I’m working in a few short, focused sessions a day. The sets are designed so problems getting at big ideas come up repeatedly over time so that I’m retrieving that knowledge, interleaved with other ideas, in multiple sessions over multiple days.

This has been a fascinating metacognitive experience; I am trying very hard both to learn something, and to use some of what I know about teaching and learning to make sure I am making an effective effort to do so. I’m not sure how I can extrapolate this to math class, but I’d like to linger on these ideas of teaching and learning, in addition to the content I’m exploring, to try to come up with some new structures I can use with my students.