Catherine Castillo, Sr. Implementation Specialist
Brea Jimenez, Sr. Specialist, Facilitator Certification and Quality Assurance
What do you notice about this NCTM framework for mathematics teaching? What do you wonder?
You likely noticed that four of these teaching practices happen in service of discourse, which indicates that discourse is a powerful practice for student learning. In the National Council of Teachers of Mathematics (NCTM) publication Principles to Actions, discourse is defined as “the purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication” (p. 29).
Discourse is what many consider the pinnacle of problem-based teaching and learning–the point at which a teacher has established the conditions for students to lead and sustain classroom discussions with peers.
The Levels of Discourse and Collaboration, developed by Hufferd-Ackles et. al (2014), describe observable shifts in classroom behaviors as teachers begin to relinquish control and transfer the mathematical authority from teacher to student. However, the groundwork required to create the conditions for meaningful student discourse has often remained elusive (Hufford-Ackles, Fuson, and Sherin, 2014).
For many educators, facilitating mathematical discourse that doesn’t rely heavily on teacher output can be challenging. To demystify this process, this blog post will discuss using classroom norms, leaning into productive struggle, promoting student agency, setting up peer collaboration, and valuing student thinking. By leveraging these strategies, we can shift mathematical authority from the teacher to the students and unlock the “our” in discourse (Castillo, 2024).
Establish Norms for Doing Math
At IM, we believe in co-creating classroom norms for doing math together, and we’ve even embedded a process for this in our lessons. Norms can be a powerful artifact to anchor classroom discussions. Regularly revisiting them is incredibly important—not only to celebrate when students engage in the agreed-upon practices but also to prompt reflection and discussion when actions diverge from those norms.
Regularly revisiting norms provides transparency for learners about what is expected of them as part of the mathematical community. This practice is especially beneficial for students with diverse learning needs who benefit from a consistent and predictable structure of expectations. When referencing norms becomes routine, students begin to internalize them, using the norms to guide their interactions with peers and the teacher.
We can build on revisiting norms by creating opportunities for students to self-assess their engagement with classroom norms. For example, teachers might provide a rubric focused on norm-related behaviors for students to complete and receive feedback from a peer. To support access, sentence frames can be offered to help students get started. Dedicating time to practice routines and procedures, reflect on progress, and receive feedback on an ongoing basis is pivotal for cultivating an environment ripe for discourse.
Make Time for Productive Struggle
Productive struggle is another practice that is difficult to quantify. Educators often ask, how do I know when struggle is productive and when it isn’t? How do I teach students that struggle is expected and important to their learning? How do I ensure that all my students have their needs met?
Teachers who model the process of struggle through think alouds provide an example of what it looks and sounds like to engage in struggle without giving up. Another way teachers support productive struggle is by providing opportunities for students to grapple with problems that remain unsolved. Students learn that there isn’t always a direct approach to solving a problem—and that mathematicians often persist for years, or even centuries, in seeking solutions.
The actions we take as educators—including how we allocate classroom time—communicate to students what we value in our learning community. It’s important to give students time to think and talk through the problem-solving process without stepping in immediately to rescue them. Employing discourse structures like Think-Pair-Share can help students build stamina for productive struggle. Teachers can also create norms that clarify how support will—and won’t—be offered during math time.
For instance, a teacher may have a norm prohibiting “yes or no” questions during math time to encourage higher-level thinking. Or they might post an anchor chart outlining general steps for approaching a problem, such as Math Language Routine 6: Three Reads. Some teachers find that explicitly teaching decision-making strategies—like gathering relevant information or identifying alternative solutions—gives students confidence to make decisions during the problem-solving process and empowers them to dismiss an approach that isn’t beneficial for the context.
Promote Student Agency
Discourse is much more likely to occur when students are given agency in their learning. In the traditional classroom of years past, school was a place of conformity. Teachers were centered as experts and students as passive recipients of information. Students were not encouraged to look at each other’s work, and it was often considered cheating to talk during class or to use another student’s ideas.
However, a growing body of research suggests that students achieve deeper understanding through discussion and benefit from talking through their ideas with peers. This research is what IM’s problem-based teaching and learning model is built on. To support students in taking ownership of their learning, teachers can encourage them to “borrow” ideas from peers or peer groups and try them out. For students still accustomed to more traditional pathways, teachers may use annotation to model the process of borrowing ideas from peers.
In a problem-based classroom, students are encouraged to roam the room to discuss ideas with other groups—without needing the teacher’s permission. The classroom is set up and facilitated like an ocean of ideas flowing from one group to another.
Students are also given agency in how they receive feedback and support, whether from their teacher or their peers. They can opt in or out of discussion, with the teacher using warm calling as opposed to cold calling on students. Warm calling can take many forms and is a great strategy for differentiation among learners. For example, students could be given a unique hand signal, a color card, or self-select questions they answer, ensuring all students are invited in on terms that align best with their unique needs.
Introduce Structures for Peer Collaboration
Structures like Think-Pair-Share ensure students have time to think to themselves as well as time to discuss their ideas with a peer, which can boost their confidence before sharing with the whole class. Many teachers also find it helpful to use nonverbal signals to support interactions between groups and with the teacher. Examples include hand signals for “I have a strategy” or “me too.”
Other structures—such as silent team building, paraphrasing, or role playing—can help students build stronger relationships with peers and develop key skills needed for discourse. Strategic pairing is another effective approach, allowing teachers to leverage students’ unique strengths and areas for growth. Sentence frames, like those used in the Math Language Routine 8: Discussion Supports, can also be provided to help students begin and extend meaningful conversations.
Check out these printable posters and student reference sheets that can be downloaded for use in your own classroom.
Set Up a Culture that Values Student Thinking
Finally, if we want students to share their thinking, we have to create a space where student thinking is valued. Teachers can do this by demonstrating the value of incomplete ideas and positioning wrong answers as opportunities for discussion. Rather than focusing on answer-getting, teachers can draw attention to the connections between strategies as well as the importance of problem solving and productive struggle.
This means responding neutrally to student thinking, offering soft landings for non-formalized ideas, and inviting peers to build on or respond to one another’s thinking. Teachers may also engage students in metacognitive activities to reflect on their problem-solving process.
Conclusion
When teachers are intentional about setting up a classroom community that promotes mathematical discourse by fostering student agency and valuing student thinking, all learners are given opportunities to develop confidence, critical thinking, and problem-solving practices that transcend the walls of the classroom.
Next Steps/Call to Action
Look back at the graphic of NCTM’s framework for mathematics teaching and reflect on your journey toward facilitating meaningful mathematical discourse. Where are you? Where are your students? Remember that discourse takes time to develop. Consider taking a next step of launching each task with a Think-Pair-Share or by being intentional about revisiting norms during or at the end of each class.
Catherine Castillo
Sr. Implementation Specialist
Catherine Castillo (she/her) has spent her career supporting students and educators as a teacher, instructional math coach, math recovery intervention specialist, and district math coordinator. Catherine now serves as a senior specialist on the Implementation Portfolio team where she creates resources that support coaches and instructional leaders with IM implementation. Catherine is passionate about cultivating positive math identities in students and teachers and supporting the implementation of problem-based teaching and learning.
Brea Jimenez
Sr. Specialist, Facilitator Certification and Quality Assurance
Brea J. Jimenez (she/her/hers) received a BA in English from Ottawa University in Ottawa, Kansas. She received an MA in Special Education from the California State University of Bakersfield. She is dual-certified in K–12 moderate to severe special education (exceptional children) and early childhood special education. She holds authorizations for English learners and autism populations. She has spent her classroom and administrative career working with Pre-K to 22-year-old students with exceptionalities in a wide variety of inclusion, self-contained, and transition settings. Brea is currently the Senior Specialist of Facilitator Certification and Quality Assurance at IM. She has a passion for universally accessible classroom design and empowering all learners to have agency over their educational experiences.
