A dozen middle and high school teachers seated around a U-shaped table are scrutinizing stacks of papers spread out before them. The papers include a math problem that one of the teachers’ classes worked on the previous week, copies of student work on the problem, and a transcript of the classroom conversation among one group of students. EDC’s Mark Driscoll stands at the center of the U, leading the teachers through a careful analysis of the “artifacts” before them.

As teachers voice opinions about the students’ understanding of the mathematics, Driscoll gently steers them back to the evidence—fragments of conversation in the transcript, and scrawled calculations in the margins of the worksheets.

“We’re trying to get teachers to see student work as the product of student thinking, rather than simply looking for right or wrong answers,” says Driscoll. “In the middle of teaching a lesson, it’s difficult for teachers to pause and have genuine curiosity about what students are saying and how they are thinking about the problem. We’re encouraging them to take the time to act on that curiosity.”

During this session, for example, Driscoll leads a discussion of the work students did on the Difference of Squares problem: “Can all odd numbers be written as the difference of two perfect squares? Explain why or why not.” In their conversation and on their worksheets, students used several strategies to solve the problem. Some began by calculating the differences between perfect squares—such as 4 – 1 = 3; 9 – 4 = 5; 16 – 9 = 7; 25 – 16 = 9. Another group set up a grid with the perfect squares running across the top and down the left side, and then calculated the difference of every combination of squares.

But there’s one set of calculations on the page that none of the teachers can make sense of at the beginning of the session: “36 – 21 = 15,” with the 15 crossed out. Next to this is written “25 – 21 = 4” and “25 – 4 = 21,” with a check mark next to the 21. Most teachers in the group conclude that the students have misread the problem, since 21 is not a perfect square:

**Teacher 1:** “I think they were thinking that they had to create the perfect squares.”

**Teacher 2:** “Right. They were really confused about the wording of the problem.”

But after several more minutes of conversation about where the students went wrong, a third teacher offers an alternative explanation:

**Teacher 3:**
“She might have been trying to work backward in a way. She knows that
if you subtract a perfect square, it will be an odd number. So maybe if
she has one perfect square and she subtracts an odd number, her answer
will be a perfect square.”

Another teacher in the group then points to a line in the transcript where a student says, “I did it backwards.” Many in the group now begin to realize that what they had dismissed as a misunderstanding of the problem was actually a very sophisticated strategy: If you can subtract every odd number from at least one perfect square and produce another perfect square, then every odd number can be written as the difference of two squares.

That sort of epiphany comes from having a deep curiosity about student thinking, according to Driscoll. And that mindset can help teachers engage in much deeper mathematical exchanges with their students—as the Turning to the Evidence research project suggests. As they wrestle to understand student thinking, teachers gain more insight into how, where, and why students are drifting off track, as well as when they are on the verge of a breakthrough.

*Originally published on May 31, 2006*

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