Engaging students in others’ mathematical ideas

Students can learn a great deal by engaging with other students’ mathematical ideas. Noreen Webb, Megan Franke, and colleagues explore what teachers can do to help

A pair of students is working together to confirm that driving 21 miles each day for five days yields a trip length of 105 miles.

Dante: [looks over at Gus’s paper] What are you counting by?

Gus: You can count by 21s.

Dante: I don’t get it.

Gus: I went 21, 42, 64.

Dante: Oh, okay.

Gus: I kept counting and got to 108.

Dante: But wait, how can it be 108? She only drove 105 miles.

Gus: Oh, yeah. Hold on… There’s no possible way we can get to 5 [in the 105].

Dante: But, if we count by 20, we can get to 100. 20, 40, 60, 80, 100. See?

Gus: Oh yeah. So then we just need 5 more to get to 105 miles!

As Gus and Dante interact with each other they demonstrate a high level of engagement in each other’s mathematical ideas. Dante asks about the details of Gus’s strategy, challenges it, and offers another idea. Similarly, Gus also contributes to Dante’s suggestion. Throughout this interchange, these students are asking about or referring to the details of each other’s ideas, as well as describing details of their own thinking.

What we know
● Students benefit by engaging with other students’ mathematical ideas.
● Inviting students to engage is a useful first step; following up on invitations is more important.
● Follow-up moves are not a set of fully planned actions, but form a repertoire of moves that teachers can draw on in the moment to address details in students’ mathematical ideas.

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December 2015