"No offense, Ms. Hopkinson, but I learn better from my friends' thinking than I do from you."
Christine Hopkinson wasn't offended. She smiled. Because this type of moment—a student so invested in peer conversation that they'd say this directly to their teacher—is exactly what great math discourse looks like.
Meaningful classroom discussion doesn't just happen. It has to be designed. And when it is, the results are eye-opening: students reason out loud, push back on each other's thinking, and arrive at deeper understanding than any lecture could produce.
Research by Margaret S. Smith and Mary Kay Stein provides an important framework for how to effectively bring student discourse into the classroom. Their book Five Practices for Orchestrating Productive Mathematics Discussions (2011) helps teachers become strategic facilitators of math conversations.
Below is a summary of each of the practices.
Anticipating occurs early in lesson planning. During this stage, teachers are asked to put themselves in the students' shoes and predict what types of approaches students may take with the math. Because every student brings a different thought process, this stage helps teachers prepare to respond to a variety of potential strategies, solution paths, misconceptions, and misapplications.
Smith and Stein recommend utilizing the chart below to record information during the five practices. This chart is built out stage by stage. During the anticipating stage, teachers predict and document various strategies. They can also include “Other” in the Strategy column to account for strategies that they didn’t predict.
Monitoring occurs through two approaches: collecting information about students’ solution paths and asking questions that help students navigate and explain their chosen strategy.
During student work time, teachers monitor students’ approaches by asking questions that reveal their reasoning. Asking students the right questions is a balance between teacher authority and student autonomy (Smith and Stein 2011, 61–74). These questions can help guide students toward more productive solution paths and give them a chance to revise their answers (Smith and Stein 2011, 10) prior to discussion time.
Monitoring is used as a foundation for shaping discourse, allowing teachers to identify common themes appearing in student work, whether a strategy or a mistake. This directly informs the next two practices—selecting and sequencing.
While monitoring, the teacher records which students or groups are using each strategy in the second column of the chart, along with notes on how they applied those strategies to the problem.
When shared student work is selected with intention, classroom discussion can advance student thinking while remaining focused on the lesson objective. Calling on student volunteers without structure can lead to disorganized classroom discussions and disconnected solution paths, often resulting in more confusion.
Choosing the right example work is an important part of inspiring classroom discussion, but the real impact comes from how teachers use these examples to drive discourse and support the goal of the lesson. True impact comes from intentionally sequencing selected student work.
As the chart is filled out in each stage, teachers can refer to it, which helps them prioritize which strategies best support lesson goals.
A sequence of selected student work should improve the accessibility of math by creating a logical flow of presented examples (Smith and Stein 2011, 11, 44).
As a baseline, teachers should order student example work from simple to complex—providing an entry point to the lesson that builds with time. However, just as there are many ways to tell a story, there is no one way to sequence student responses (Smith and Stein 2011, 49). Each class will look different based on the teacher’s style and the makeup of the class.
The secret power of sequencing is guiding students to tell a particular math story in their own way. When students explain their math understanding in a way that makes sense to them, they grasp the concepts more deeply and clearly. When students hear their classmates explain math ideas in their own way and then respond to those explanations, they not only hear their classmates' stories but also learn about cooperation, argumentation, and reasoning (Chapin, O’Connor, and Anderson 2009, 7).
Sequencing transforms a “show and tell” discussion into a focused discussion that gives students agency in their learning. Once the flow of student example work is finalized, teachers can guide discourse toward connecting the dots across math ideas.
Connecting gives students the opportunity to make meaning of math collaboratively.
In the connecting stage, the teacher’s duty is not to reveal math connections, but instead to ask questions that can help students discover connections through their own reasoning. These questions must be grounded in student work and begin with what students already know.
During this stage, students compare their solution pathways with those of their classmates, explain their reasoning, and engage in discourse that highlights similarities and differences between solution paths. This builds upon the previous practices by using selected and sequenced student work to shape meaningful discussions.
Through these conversations, students begin to recognize relationships among math ideas and move beyond focusing on whether an answer is right or wrong. They develop a deeper understanding of how different approaches contribute to meaning, leading to stronger conceptual understanding.
Student discourse is more than just classroom conversation—it’s a tool for learning. When used together, each stage of the five practices enables teachers to facilitate class discussions in strategic ways that support student agency. Beyond mathematical understanding and achievement, productive classroom discussion can build many other skills, such as collaboration, self-confidence, and argumentation (Chapin, O’Connor, and Anderson 2009, 8–9, 165–7). Focused discussion helps students write their own mathematical story, see math through many different lenses, and think about themselves and the world around them in a new way.
With the five practices woven into Eureka Math², teachers are equipped with the support they need to facilitate meaningful student discourse that expands understanding.