Mathematical discourse is the way students communicate and share their understanding about math in the classroom through words, symbols, diagrams, and technology. Through these discussions, students reveal their understanding of mathematical concepts and processes. Areas of confusion expressed by one or more students can help spur conversation and engage students in discussions and debates as they question, negotiate, and rethink or confirm their ideas.

Allow students to struggle some with the math problems; resist the urge to assist or to give the answer. This encourages the students to attempt to solve the problems themselves. It may help to prompt students with questions like the following if they get stuck.

• What do you already know? (Write it down.)
• What do you need to know? (Write it down.)
• What strategies could you use to find solutions? (Start working on it.)
• What do you think would happen if . . . ? (Try it.)

Watch the following video by The College Transition Program: CUNY (runtime: 5:52) to see how using questions can help students think critically and solve problems.

Mathematical Discourse Example

Read the following example of a teacher facilitating mathematical discourse in a CTE construction course for a lesson on calculating the pitch of a roof.

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Teacher: Let's review what we talked about yesterday. What two things do you need to know to calculate the pitch of a roof?

Marcus: Is one thing the slope?

Teacher: Using thumbs up or thumbs down, let me know if you agree with Marcus that you need to know the slope to calculate the pitch. [Several students give a thumbs down response.] Some of you agree and some disagree. Who will explain why you disagree?

Amber: Well, the pitch and the slope are the same thing, right? So if you know the slope, you know the pitch already.

Teacher: Is there anyone who disagrees with Amber? [No students raise their hand.] Okay, who has a different answer than slope?

Lindsay: The rise is one thing you need.

Teacher: And what does the rise tell us?

Lindsay: How tall you want the top of the roof to be from the top of the side wall?

Teacher: Can someone else say that differently using the vocabulary we have been learning?

Mario: It's the vertical change in height per unit of horizontal run, which is the other thing we need to know to calculate the pitch—the run.

Teacher: Thank you, Mario. So it sounds like rise and run are the two things we need to know. Thumbs up or down to show if you agree or disagree. [All students give a thumbs up.] It looks like we all agree. [Writes "P=" on the board.] So, how do we calculate the pitch using rise and run where pitch is expressed as a percentage?

Amber: Divide rise by run?

Teacher: Thank you, Amber. If I have a rise of 4 inches and a run of 12 inches, how would I write that as a ratio? Who will come up and write it on the board?

[Trang writes 12:4 on the board.]

Teacher: Thumbs up or thumbs down, tell me whether you agree or disagree with Trang. [Half the class gives a thumbs down.] Who will come up to the board and show what you think it should be?

[Amber writes 4:12 on the board.]

Teacher: Does anyone have a different answer? [No one raises their hand.] So we all agree that the ratio would be 4 to 12. Thank you, everyone. I think we have it figured out! Good job!

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In this example, notice that the teacher didn't just tell the students that rise over run is how you calculate pitch. Instead, she allowed the students to engage in mathematical discourse to figure it out for themselves. The teacher discovered where the gaps in students' understanding was, and the students learned to express mathematical concepts more precisely within the context of the CTE content.

If the questioning technique is new to you, it may be a bit intimidating at first. It may not feel natural, and you may find yourself slipping back into the mode of providing answers when students get stuck. Stick with it, and keep practicing! 

The National Council of Teachers of Mathematics suggests recording yourself teaching and playing it back to see where you could improve. Another idea is to invite another teacher or faculty member to observe you in your classroom and provide honest feedback. Soon, it will become second nature to ask questions to guide your students to success.

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The Curriculum Associates© website provides several questions you can ask students to promote mathematical discourse. Take a moment to explore the questions or bookmark it to explore later.