Figuring out a problem yourself is better than copying someone else’s down because then it stays processed in your brain. And once it’s processed in your brain, you can’t lose it because you did it on your own.

-A student, reflecting on her first term.

Since September I’ve been dealing with two things that are forcing me to grow. First, I’ve been doing what I’ve done for a year or two, only this year, the response is different. Students are more restless, with shorter attention spans, so giving them long periods of independent work time results in little work from them and little coaching from me because I am spending all my time redirecting. So it’s back to table groups and roles and more, shorter tasks and putting things in front of them to complete instead of writing a broad direction on the board.

Second, I’ve been trying to learn how to develop fluency with some basic skills. My current experiment is in learning how to teach graphing quadratics. There’s lots of great material out there. (Check out James Tanton’s course at G’Day Math). But I thought they would be able to do it as easily as me walking them through it. The skill that common core calls “Seeing Structure in Equations” seems to take time and repetition to build. It’s something I do without realizing I’m doing it. How did I not know this? Seriously, I thought we’d do one day on graphing from vertex form and maybe one additional day to review, then on to the next standard. What’s required, at least for this group this year, is one day on where’s the vertex. One day on width. One day on finding intercepts. One day on synthesis of the above. So good: that’s what it takes. But I’m inventing as I go. It feels a lot like I am just now figuring out how to do what every other teacher already knows how to do.

The student quoted above came to see me after school to get coaching on a problem because she didn’t present any original solutions last term and wants to start 2nd term off by presenting. She announced to me as she arrived that she’s “really horrible at math.” As we worked through some background problems and re-discovered what her classmates had discovered, she lit up (“Whoa! That’s really smart!”), and as she got into the new problem she wants to present, she was really enjoying it. She told me she’s been doing what a lot of students do: waiting for other people to present and then just writing reports of their solutions (which I allow, though I will taper that off as winter turns to spring). Then out of the blue, she said what’s quoted above. I said, “Wait, I want to write that down. You just reminded me what I’m doing here.”

Nice insight by your student. Tough sell to those who haven’t figured that out, for the most part. I wonder how this connects with what you described in the previous post, about equitable solutions to a tough classroom ‘management’ issue.

Is it curmudgeonly for me to disagree? Apologies, if so, but I’ve lost oodles of math that I slaved over on my own.

Of course, of course, we’re way more likely to remember it if we’ve done it on our own. But sometimes, we just forget stuff.

I tend to forget math I rarely or never use. I tend to remember the general experience of having worked something through, however, even if I at least temporarily forget the answer, the proof, the method, or what-have-you. And if I have worked through something I didn’t necessarily solve on my own to the point where I really grasp what’s going on, for me that can be just about as good as having done all the work myself the first time I see the problem.

So yes, nothing is guaranteed to produce memory “immortality,” and it’s possible to recall quite vividly non-original/discovered solutions (naturally, someone else might have solved the problem in the past the same way, but it’s new FOR ME). But at the level of math the student was talking about, I think she’s onto to something rather important.

… and if it’s curmudgeonly, is that bad?

I’m excited about the combination of the student’s insights, Dan’s, Michaels, and Brian Frank’s idea about “U-shaped development.” Michael is pointing out that we do sometimes lose things. Brian is pointing out that declining performance is sometimes a sign that we had the courage to try something we’re novice at instead of sticking to our previous strategies. The student is pointing out that *something* different happens when you think things through yourself.

She says “you can’t lose it,” and that may not be true of the mathematical insight. Is there *something* that you “can’t lose”? If so, what might it be?

I’m thinking of students who painstakingly gather evidence that current *doesn’t* always take the path of least resistance, but then revert to that way of thinking when each time they encounter a new context. It sure looks like the are “losing” their previous discovery. If you *do* lose it, is that a pre-condition for achieving a new, higher level of skill?

If it is a pre-condition, how do we help student navigate the disappointment of “losing” something they thought they couldn’t lose because they figured it out themselves? (Or, is this a real problem? Am I worrying unnecessarily here?)

Like Dan, this is why I do what I do. So, how do I tell the difference between a constructive, necessary decline in performance, and one that isn’t? Also, how do I help myself navigate my *own* disappointment when I am reminded that I should imagine good-quality thinking less like a ratchet and more like something we need to form, deform, and reform in order to make progress… that maybe the process of deforming and reforming our thinking *is* the progress?