Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.

- NCTM 2014, p 29

Facilitating Meaningful Mathematical Discourse is a HUGE topic! This first blog dedicated to discourse is focused on building a classroom culture where students are engaged in mathematical discussions.

I’m reflecting on the moments when I knew I was “facilitating meaningful mathematical discourse” with students. How did I know? What were the students doing? And what was I doing? (and not doing?)

Remembering those moments when I knew the magic was happening, some of the things I’ve heard kids say were:
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• I got the same answer as Sophia, but she did it differently. Sophia, why does that work? 
• I see what Mateo did - he multiplied both numbers by the 2 instead of adding first.
• Oh, that’s what I did! Why didn’t that work?
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I especially love the last one, which happened in a classroom we visited last week. We were doing math in Magma and discussing solutions to a problem using the student solution view. One of the solutions we were displaying had an error in it, and the student who raised his hand to share his thinking was excited to both own the solution and to problem solve about his error. This delighted my colleague Natalie and me, because it means that the classroom has a positive classroom culture, where mistakes are expected, inspected, and respected, per Dr. Jo Boaler.

Principles to Action tells us that in a classroom rich in discourse, students are:
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• Presenting and explaining ideas, reasoning, and representations to one another in pair, small-group, and whole-class discourse.
• Listening carefully to and critiquing the reasoning of peers, using examples to support or counterexamples to refute arguments.
• Seeking to understand the approaches used by peers by asking clarifying questions, trying out others’ strategies, and describing the approaches used by others. 
• Identifying how different approaches to solving a task are the same and how they are different. 
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How do we develop the kind of classroom culture where students feel safe and excited to share their thinking and mistakes, and where students are engaged in the behaviors identified by NCTM in Principals to Action. What teacher moves are we enacting so that we have productive mathematical discussions that move our students towards learning targets. And what are we NOT doing? What practices do we need to let go of or abandon?

A few of the practices I have learned from colleagues and reading over the last several decades that support this culture are:
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• Use student names! Students need to be seen and heard. They need to know that they are a part of our learning community and that they have valuable mathematical ideas.
• Our role as teachers is to determine how to build on and honor student thinking while ensuring that the mathematical ideas at the heart of the lesson remain at the center of class discussions (Principles to Action, p. 30). 
• When students share ideas and responses, use talk moves that open discourse rather than position myself as the expert. Elham Kazemi and Allison Hintz share 7 Talk Moves in their book Intentional Talk that support getting classroom discussions going.
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Revoicing: “So you’re saying…”

• Repeat some or all of what the student has said, then ask the student to respond and verify. Revoicing can be used to clarify, amplify, or highlight an idea.

Repeating: “Can you repeat what she said in your own words?”

• Ask a student to repeat or rephrase what another student said. This is a way to restate important parts of a complex idea in order to slow the conversation down and dwell on important ideas.

Reasoning: “Do you agree or disagree, and why?” or, “Why does that make sense to you?”

• After students have had a chance to process a classmate's claim, ask them to compare their own reasoning to someone else’s reasoning.
• Encourages students to engage with each other’s ideas.
• Give students sentence frames such as:
  “I respectfully disagree with that idea because…”
  “This idea makes sense to me because…”

Adding On: “Would someone like to add on to this?”

• Prompt students, inviting them to participate in the conversation or to clarify their own thinking. 

Turn-and-Talk: “Turn and talk to your neighbor…”

• Circulate and listen to partner talk. Use this information to choose who to call on. 
• You can also ask students’ permission to call on them: “Would you please share your thinking with the class in a bit?”
• Allows students to clarify and share their ideas, and specifically supports Multi Language Learners in practicing sharing their thinking.
• Encourages students to orient themselves to each other’s thinking.

Revise: “Has anyone’s thinking changed? Or “Would you like to revise your thinking?

• Allow students to revise their thinking as they have new insights.
• Support students with the sentence stem, “I thought…. But now I think … because…” or “I’d like to revise my thinking.”
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Learning to use these talk moves and resisting the urge to respond with evaluative feedback has been a journey for me. When I remember, magic is more likely to happen. One of my favorite phrases came from one of my mentors, Dr. Gini Stimson, “listen to learn.” I hope that rather than being the source of mathematical knowledge, I can encourage students to listen to each other to learn from and with each other.

Best,
Leslie
leslie@magmamath.com