Getting Students Talking to Each Other About Math

In this Elementary School Journal article, Megan Franke, Angela Turrou, Noreen Webb, Jacqueline Wong, Nami Shin, and Cecilia Fernandez (University of California/Los Angeles) and Marsha Ing (University of California/Riverside) examine ways to get students to engage with each others’ mathematical ideas. “Researchers increasingly recognize,” say the authors, “that promoting mathematical learning requires teachers to engage students in ‘productive struggle,’ where students expend effort to make sense of mathematics and figure out something that is not immediately apparent. One way students can productively struggle with the mathematics is through their communication with others – both through explaining one’s own thought processes (e.g., reasoning about mathematical concepts and how to solve problems) and discussing other students’ reasoning process.” 

This sounds good in theory, but implementing it in classrooms is not a simple matter. It’s relatively straightforward to get students talking about math problems, say Franke and her colleagues, but getting classrooms to the level of “productive struggle” is quite challenging. Here is a continuum of students’ degree of engagement with other students’ ideas, from low to high:

• Saying “I agree” or “I disagree” with an idea that was shared.
• Pointing to the strategy that most closely resembles their own strategy.
• Repeating the details of what a student shared.
• Explaining another student’s strategy after it was written on the board.
• Adding further detail to another student’s strategy.
• Providing a correction to an problematic portion of a student’s solution.
• Proposing an alternative solution and explaining how it differs from the idea already posed.
• Co-constructing a solution with another student.

The researchers observed a number of teacher “invitations” designed to elicit higher-level mathematical discourse:

• Asking a student to explain someone else’s solution – “Joey, can you explain what Natalia did?”
• Discussing differences between solutions – “Let’s look again at what Dylan said. Dylan said it is a whole number. Stella, do you want to respond to that, given what you said to start with?”
• Making a suggestion to another student about his or her idea – “What is he going to have to do with that set of numbers, with 387? What does he have to do, Grayson?”
• Connecting students’ ideas to other’ ideas – “Joaquin, can you see what Enrique is doing or what Natalia is doing and see if it looks like yours? Or if it’s different?”
• Getting a student to create a solution with another student – “Griffin, why don’t you sit down and work on the problem together with Easton?”
• Using a solution that was shared by another student – “See how Paige counted? Could you take this problem and count like her?”

As they observed classrooms in a California elementary school, Franke and her colleagues noticed three challenges that teachers faced as they tried to orchestrate good mathematical discussions: 

• Students sometimes seemed unable to engage with each others’ ideas.
• Students sometimes provided little or no detail about others’ thinking;
• At times, students provided details but didn’t address the mathematical ideas underlying other students’ strategies.

In other words, say the researchers, “just inviting students to engage with others will not guarantee that students will, in fact, engage with each other, nor necessarily engage in ways that are supportive of mathematical learning.” 

When discussions fizzled, there were big differences in how teachers reacted. Some provided their own solution. Some moved on to another topic. But some teachers had a broader repertoire of in-the-moment strategies: probes (pressing students to engage further); scaffolding (providing some information or clarification); and positioning (interacting with students in ways that acknowledge the students’ connection with the math idea being discussed – for example, “What Aaron’s saying is that four-fourths is a whole, or one. That’s what he says. What do you say to that?”). These teacher moves, say the authors, “require not only pedagogical skill and knowledge, but also pedagogical content knowledge and mathematical content knowledge, and well as identities as teachers who see each of their students as capable. We need to better understand how teachers draw on their knowledge and identities as they make their in-the-moment decisions.” 

“We never saw a teacher use the same series of support moves more than once,” say the researchers, “even in response to the same kinds of challenges. This implies that the teacher support moves were not a set of fully planned actions that could be applied repeatedly in the same way, but rather served as a repertoire of pedagogical moves that teachers drew upon in the moment. Our findings resonate with those of previous researchers and suggest that understanding the teacher moves that support student thinking requires looking beyond the first move a teacher makes and toward how teachers extend their interactions with students to support opportunities for productive struggle.” This involves a sophisticated knowledge that takes into account the student, the math, and the context – something teachers develop with years of experience, interaction with colleagues, and high-quality professional development. 

The researchers close with a description of what happens when classroom mathematics discussions are at their best: “Teachers learn about content, about the development of student thinking, about their students as mathematics learners and people, and about how to support their students. The students, while learning mathematical content, learn how to listen to one another, how to ask a question that moves the mathematics forward, and how to position their ideas in relation to others’ ideas. The interaction among the teacher and students supports students to learn to persevere as they communicate with each other and productively struggle to understand and articulate each others’ ideas.”

“Student Engagement with Others’ Mathematical Ideas: The Role of Teacher Invitation and Support Moves” by Megan Franke, Angela Turrou, Noreen Webb, Marsha Ing, Jacqueline Wong, Nami Shin, and Cecilia Fernandez in The Elementary School Journal, September 2015 (Vol. 116, #1, p. 126-148), available for purchase at http://bit.ly/1NezQjd 

From the Marshall Memo #615