*How did you get that?* It’s the quintessential question teachers use to start a math discussion in their classrooms.

But what if the classroom discussion sputters? What other questions and techniques do teachers use to guide their students toward a “productive struggle” with math problems and ideas?

To find out, a team of California researchers collected video of classroom math discussions at an inquiry-focused elementary school in southern California (preK-6th grade). After coding the classroom interactions, the researchers identified 6 invitational moves teachers use to start a discussion and 3 major strategies they use to support continuing dialogue.

The 6 invitational moves identified in the study were asking students to:

- Explain someone else’s solution
- Discuss differences between solutions
- Make a suggestion to another student about his or her idea
- Connect their ideas to other students’ ideas
- Create a solution to a math problem in pairs or teams
- Use a solution to a math problem that was shared by another student
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Of these openers, the two that resulted in the lowest levels of student engagement with other students’ ideas were asking students to connect their ideas to other students’ ideas and to create a solution together with other students, according to the study published in *The Elementary School Journal.*

Teachers face 3 major challenging situations after they open a discussion:

- The discussion never gets off the ground because the students don’t really know how to engage with each others’ ideas
- Students provide little in the way of detail about their ideas or other students strategies
- No one really addresses the major mathematical ideas during the discussion.

“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,” the researchers write.

The researchers identified 3 major follow-up strategies teachers use to support students’ interactions with other students’ ideas or to encourage students to elaborate on their own thinking. The three strategies are:

- Probing—teacher pushes student to engage further in another’s ideas by questioning or revoicing.
*(Did you understand why she divided each whole by 5?*) - Scaffolding—teacher takes on some of the mathematical work such as linking to a representation or context, clarifying an idea or providing more info
- Positioning—Teacher interacts with the students in ways that publicly acknowledges a student’s connection to an idea (e.g. what Aaron is saying is that four-fourths is a one….)

“The type of invitation did not determine how students engaged with each other’s ideas,” the researchers write. “Rather, we found that teachers provided support following the initial invitation in the form of probing, scaffolding, and positioning in order to help students engage in others’ ideas in more detailed ways, engage more deeply in the mathematical idea shared with others, or figure out how and when to engage in others’ ideas,” the researchers write.

To collect data on how teachers support student dialogue, researchers observed math discussions in 12 classrooms across all levels at an inquiry-focused elementary school in southern California (preK-6th grade). Over a 2-week period, each math class was observed 2-3 times for one hour. The discussions were videor ecorded and one session was selected as representative of that teacher’s usual practices. (Prior to formal data collection, researchers informally observed classrooms weekly over a 6-month period to become familiar with the individual students and the participation structures the teacher regularly used, often combinations of whole-group discussion, pair work and small-group work).

The selected videos of student interactions during the math discussions were coded moment-to-moment by at least 2 raters and put on a timeline. Teachers support moves were also coded and put on a timeline, allowing researchers to relate teacher moves to student behavior.

“Our research suggests that planning invitation moves may be beneficial, as they are a way for teachers to begin the work of having students engage with each other’s ideas and create opportunities to learn from the interaction that may unfold,” the researchers write.

“However, our analysis showed that it is the responsive in-the-moment support moves, similar to combinations of ‘instructional actions’ and ‘specialized moves’ mentioned above, that allow students to learn how to engage with each other and build ideas together in mathematically detailed ways.”

*“Student Engagement With Others’ Mathematical Ideas, The Role of Teacher Invitation and Support Moves,” by Megan Franke et al., The Elementary School Journal, 2015, Volume 116, Number 1, pp. 126-148.*