I recently worked with a 6^{th} grade teacher, Miss T, as she led a string for the very first time. Twenty-eight middle school students quietly re-arranged desks and chairs and situated themselves at the front of her room — itself, no small feat — as she prepared to facilitate a messy multiplication string:

17 x 10 =

17 x 2 =

17 x 12 =

17 x 20 =

17 x 19 =

17 x 21 =

Miss T started slowly, giving all of her students lots of “think time.” She worked really hard to get more and more students into the conversation — “Ronald, did you hear what Georgiana just said? Well, good. Say it for all of us in case we missed it.” The kids stayed engaged, and she generated some good conversation — it was a really nice first string.

As we debriefed the experience, I asked Miss T what she was thinking or wondering about. “I’m not always sure how to model their answers,” she said. I wanted to clarify what she meant, so I returned to the string to have her illustrate. “Once they’ve said their answer and the class agrees, how do I make a picture of it?” This was really interesting to me, because I knew I could support a shift in her thinking, so I made a quick sketch of 17 x 19.

We then talked about the very subtle difference between representing an answer and representing a strategy (e.g., students’ thinking). When a student says, “First I did 17 x 20, which I knew was 340, and then I subtracted one group of 17 to make 323,” we want to capture that thinking, shown on the right. Our goal, we decided, was not to make a picture of their answer but to create a visual model of their strategies — that is, to take students’ thinking (which is not visible to the rest of us) and give it a form that could be seen and studied by the class. This made a lot of sense to Miss T: “Right, because I can’t really generalize from the first picture you drew, but I can from the second one. It shows me what you did and how you solved it. Both your process and your final result.” We also agreed that hearing a strategy and then seeing a corresponding model gave the mind two forms of input, and therefore, two ways to make sense of it.

I am a K-12 math educator, primarily working with teachers and instructional teams in and near New York. In addition, I enjoy writing with and for teachers about current issues of mathematics teaching and learning. Currently I work for Math in the City (City College), with ongoing collaborations with Math for America and The Urban Assembly.
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