Of all I saw and learned this past half year, one thing stands out. What goes on in the class is not what teachers think — certainly not what I had always thought. For years now I have worked with a picture in mind of what my class was like. This reality, which I felt I knew, was partly physical, partly mental or spiritual. In other words, I thought I knew, in general, what the students were doing, and also what they were thinking and feeling. I see now that my picture of reality was almost wholly false. Why didn’t I see this before?
-John Holt, How Children Fail
I was fooling around on Youtube and ended up watching Beyonce’s Irreplaceable. It’s a song I’ve liked for a while, though I’m only a casual fan. Watching it last night I realized I’d been both hearing and singing it wrong for years. I had been singing, “everything I own in a box to the left” when the actual lyric is “everything you own in a box to the left”. And that’s a pretty significant distinction. Here’s a context clue from the video to help:
I had been listening to and enjoying the song for a long time — but in all that time, I had managed not to change a significant misconception or probe beyond the surface of my understanding of what was happening.
I’m curious how many of my students experience my teaching in this way, spending their time in class thinking about surface features of the mathematics we are studying without putting significant cognitive work into the underlying meaning of the content.
At NCTM a few weeks ago, I attended a talk by Skip Fennell, Beth Kobett, and Jon Wray on formative assessment. You can check out their slides here. One strategy that stuck with me was using an interview to explore student thinking after a task, asking the student how they solved the problem, why they solved it that way, and what else they can tell me about their thinking. It’s obviously impractical to do this every day or with every student. But it’s also a strategy I’ve never used to explore student thinking in depth, and with the premise that students often know less than I think they do I’m sure I would get some great insights out of it. One more point the presenters made was that the interview doesn’t have to be with a student who is struggling; talking with a student who is effectively using certain strategies could be useful in figuring out what moved their thinking forward and how to help other students with that thinking.
Here’s a final thought:
There do appear to be cognitive differences in how we learn. … One of these differences is the idea … that psychologists call structure building: the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them. These frameworks are sometimes called mental models or mental maps. High structure-builders learn new material better than low structure-builders. The latter have difficulty setting aside irrelevant or competing information, and as a result they tend to hang on to too many concepts to be condensed into a workable model (or overall structure) that can serve as a foundation for future learning.
-Brown, Roediger & McDaniel, Make It Stick
I see structure building as the biggest difference between successful students and students who struggle. The most important piece of the research that the authors present on structure building is that guidance toward the key elements of a problem that makes explicit the essential relationships can support all students in structure building and making sense of the mathematics.
I’m not sure how well these ideas are connected — Holt, Beyonce, formative assessment interviews, and structure building. But it’s been some good food for thought in probing more meaningfully into student thinking, and constantly asking myself whether students are actually doing the thinking that I hope they are doing.