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. ● Followup 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. 
Engaging with others’ ideas can benefit students’ learning in many ways. Such interaction can help students make sense of mathematics ideas and deepen their understanding. For example, they can recognize and correct their own and others’ misconceptions, fill in gaps in their understanding, and develop new ideas and problemsolving strategies. Consistent with these benefits, many studies have found higher achievement among students who engage with others’ ideas at a high level (as Gus and Dante did) than among students who do so only at a lower level – for example, saying they agree with another student without saying why – or do not engage at all.
How can teachers support student engagement with each other’s ideas? Our observations of teachers’ interaction with students in studentdirected pairs or small groups, in wholeclass discussions, and in turnandtalk to your neighbor private conversations amidst wholeclass discussions, have between them revealed a variety of moves teachers can make that help students engage with other students’ ideas at a high level. Moreover, these teacher moves (actions that are decided by individual teachers in their classrooms) can spur students to engage with each other’s thinking, even when the teacher is not present.
Invitation to engage: Important but not enough
Teachers can use a range of moves to invite students to engage with each other’s ideas. For example, the teacher might encourage students to compare their strategies with those their classmates generated, explain problemsolving strategies that other students used, ask questions of each other, and work together with other students to create a solution strategy.
Teachers can make multiple invitation moves in a single exchange with a student or group of students. We have found that initial invitation moves sometimes yield a high level of student engagement with others’ ideas. But often initial invitations do not result in students engaging with others’ ideas, or not in ways that address important mathematical details. So, invitations to engage often require teacher follow up.
Followup moves in interaction with students
Students do not always respond to initial teacher invitations in a detailed way. Students may not know how the teacher expects them to engage, may not know they can, or may not be able to distinguish between important mathematical ideas and less central details of another student’s work. Teachers can follow up in multiple ways to address these challenges.
Press students to detail another student’s mathematical thinking. Teachers can press students to engage further with another student’s idea by asking probing questions. For example, teachers may ask students to explain a specific detail in another student’s strategy – “Why did he divide each whole into five pieces?” – or ask students to explain whether they agree with a step in another student’s strategy – “Here she added the 40 10s and got 400. Do you agree? Why?” Such probing questions can help draw students’ attention to important mathematical ideas and give them a starting point to engage with the ideas of others.
Provide scaffolding using language and tools. Teachers can carry out a part of the engagement with a student’s idea to spur other students to do the same. For example, teachers might restate something a student said or ask questions that help other students make sense of the important ideas in the student’s strategy and enable them to build onto the strategy – “What did Jessica just say? If we only take away 400, we are only going to be in the what range? The 1,000’s range. That gives us a clue that we have not taken away enough yet.” Teachers can also suggest particular tools, such as manipulative materials, or mathematical representations or terms, to help students understand and engage with others’ ideas.
Enable students to see that they can engage with others’ ideas. Teachers can encourage engagement with others’ ideas by explicitly acknowledging students’ ability to do so, especially while simultaneously drawing connections between different students’ ideas – “You can explain his picture. Yours is very much like it.”
Supportive classroom features for engagement with each other’s ideas
Teachers can lay the groundwork for high levels of student engagement with others’ ideas by establishing certain expectations for participation in their classrooms:

Students solve problems in ways that make sense to them.
 Multiple classroom contexts, such as problemsolving time and wholeclass discussions, give students opportunities to share their thinking and make sense of their classmates’ ideas.
 Teacher and students alike attend to the mathematical details of students’ ideas.
 Confusion, disagreements, and alternative perspectives are viewed as legitimate and respected reasons for questioning others.
Invite and follow up 

The teacher uses invitation and followup moves to help Julian engage with Melissa’s mathematical idea. Teacher: (whispering to Julian) You still look confused. Do you want to ask another question? Ask her. Julian: Why did you do that? Teacher: Can you be more specific with your question? Did what part? Julian: Why did you put 1… it’s just confusing. I don’t know why you put the 1. Teacher: That’s it! Nice question Julian. Melissa, can you answer Julian’s question? 
Principles for practice
Draw upon your inclass moves flexibly, in the moment. Teachers can select from followup moves of various kinds, such as asking probing questions about mathematical details that a student mentioned but did not elaborate on; describing some ways in which one student’s ideas were similar to or different from another student’s ideas to guide further engagement; and suggesting particular steps in a problemsolving strategy a student could use to compare or contrast his and his neighbor’s strategies. These moves are not a set of fullyplanned actions, but rather serve as a repertoire of moves that teachers can draw upon in the moment.
Continuously adjust followup moves according to how students respond. Teachers can customize their moves to help students engage with others’ ideas. For example, in an interchange to help one student explain another student’s idea, the teacher may at different times ask probing questions, suggest drawing pictures, revoice important details, and acknowledge the student as a competent explainer, depending on what the student said.
Connect to the details of key mathematical ideas. Teachers can call students’ attention to specific steps or central quantities in another student’s problemsolving solution, and ask students to engage with those specific mathematical elements. For example, teachers may ask students whether they agree with, have a question about, or have an alternative to a key mathematical detail of another student’s idea.
About the authors
Lead authors Noreen M Webb and Megan L Franke are Professors of Education at the University of California, Los Angeles. Noreen conducts research on learning and assessment in mathematics and science classrooms, and Megan studies teacher and student learning in mathematics classrooms.
Marsha Ing is an Assistant Professor of Education at the University of California, Riverside. She studies methods for measuring student performance and instructional opportunities.
Nicholas C Johnson is an Educational Research Assistant in the University of California, Los Angeles. He works to support teachers to make sense of, and build from, students’ mathematical thinking.
Angela C Turrou is a Senior Researcher of Education at the University of California, Los Angeles. She studies the interaction between teachers, students, and children’s mathematical thinking in preschool and elementary classrooms.
Joy Zimmerman is an Educational Research Assistant in the University of California, Los Angeles. She studies interactions in the mathematics classroom.
Further reading
Franke ML, Webb NM, Turrou AC, Ing M, Wong J, Shin N, and Hernandez C (in press), Student Engagement with Others’ Mathematical Ideas: The Role of Teacher Invitation and Support Moves. Elementary School Journal.
Webb NM, Franke ML, Ing M, Wong J, Fernandez CH, Shin N, and Turrou AC (2014), Engaging With Others’ Mathematical Ideas: Interrelationships Among Student Participation, Teachers’ Instructional Practices and Learning. International Journal of Educational Research, 63, 79–93.
Carpenter TP, Fennema E, Franke ML, Levi L, and Empson S (2014), Children’s Mathematics: Cognitively Guided Instruction, 2nd Edition. Portsmouth, NH: Heinemann.