Mathematical discourse is the way students think, discuss, question, and represent their ideas to reveal their level of understanding of concepts. Teachers must ask strategic questions that will guide students to share how a particular problem was solved and why that specific method was chosen to build mathematical discourse. During this process, students can learn effective ways to address problems by critiquing their ideas as well as solutions from peers.

The constructivist view of learning states that humans make meaning or develop knowledge through interaction between their experiences and their thoughts. According to Vygotsky's sociocultural theory of learning, social interaction plays a fundamental role in fostering comprehension. As a result, students can learn more through collaboration which allows them to build upon the ideas of peers versus working alone.

Learning math is not about memorization and recall. As a math teacher, I would say "I taught this last week. I'm not sure why my students don't know how to do this." Perhaps, I showed my students a quick trick to solve one type of problem, and they didn't develop the critical thinking skills to transfer that knowledge to tackle a new challenge. The use of discourse in math classrooms helps students explain the thinking processes used to reach solutions. As a result, students can utilize previous knowledge and apply it to new situations.

### What does mathematical discourse look like in classrooms?

There are four modes of discourse:

- Student to teacher
- Student to student
- Student to group or class
- Individual reflection (which students can use as conversation starters)

Often, the most common type of exchange that I observe in classrooms is when students communicate with the teacher. The teacher asks a question, and the student states the answer. It’s beneficial to utilize multiple styles of discourse to keep students engaged. Implementing various types of discourse in a single lesson is feasible. A simple way to make each type of dialogue visible in your classroom is to note when and how you will implement each mode in daily lesson plans.

Discourse has proven to be a useful strategy for developing a deep understanding of mathematical concepts, yet it's missing from many math classrooms. Based on conversations with math teachers, there are an array of challenges that make facilitating math talk difficult. Here are some concerns that math teachers have shared with me.

- Time: How do I find the time to allow discussions for an extended period? I need to cover all of the standards.
- Confusion: I’m unclear on what students should talk about during discussion. I’m also unsure about how to guide a math conversation.
- Doubt: What if no one will talk and I have to experience an awkward silence in class?
- Worry: I don't want to embarrass students who may not know the answer by forcing them to talk.
- Fear: What if I lose control of my classroom? I’m concerned that students will “take over” the lesson and begin off topic debates.

These concerns are real for teachers new to the idea of math discourse. However, by using five standard teaching practices, it's possible to improve the quality of discussion in math classrooms.

- Utilize talk moves that engage students in discourse.
- Focus on the art of questioning.
- Use student thinking to propel discussions.
- Create a supporting environment.
- Facilitate the discourse.

To find out more about each teaching practice, check out the book *Classroom discussions: Using math talk to help students learn (Chapin, O’Connor & Anderson, 2009)* .

It's important to mention that it takes time and practice to get students to begin discussing math concepts freely. Start utilizing talk moves and participation moves to make discourse a part of your classroom culture. Talk moves are teacher habits that encourage student participation, clarify misunderstandings, and ask students to engage with the content deeply. Participation instructions are given by the teacher to tell students how they should engage in the lesson. I suggest becoming familiar with a few moves and then add more to your teaching toolkit over time.

Here are some examples of talk and participation moves that teachers can become familiar with to create an environment of risk-taking in their classroom using math discourse.

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Next steps: Reflect on your current classroom environment. What is one way that you are encouraging students to talk about mathematical concepts? What is one new strategy that you plan to try to increase your efforts?