Communicating in the Math Classroom, Part 2

This blog is a repost of a four-part series I wrote for NCTM’s Blogarithm in 2016. Read the original post here.

Middle school students desperately want to be heard, but on their own terms. Several years ago, I was introducing the Pythagorean theorem to my students. From the back of the room, I heard a student repeatedly interrupting; loud enough to be heard by his classmates, but not by me (or so he thought). “A squared plus B squared equals C squared.” Over and over again. Finally, I stopped and asked him to elaborate.

“That’s it. That’s the Pythagorean theorem.”

“Okay, but what does it mean?”

“The sum of the legs squared equals the sum of the hypotenuse squared.”

“Okay, but what does that mean?”

“…That’s it.”

To him (and in all fairness, to many people), that was the extent of it. What more could I possibly want? He was eager to be heard, and wanted to contribute his knowledge. This is about as far as many of our students go because too often we don’t ask them to dive deeper.

So, what does discussion sound like in the math classroom? I make a point to say “discussion,” because the kind of communication we want among students is so much more than simply answering questions or reciting procedures. Are these important? Absolutely! However, there’s no engagement. Just like in any other context, discussion involves the exchange of ideas and challenging preconceived notions. For this to happen, teachers need to take a giant step back and let students get their hands dirty with math. One strategy is to give students a difficult problem that is just out of their reach. Give sufficient independent struggle time, and encourage students without giving anything away. Next, allow small groups to talk about the problem. Many students are often reluctant to participate in small-group discussions because they fear embarrassment at being wrong, or not getting as far as their peers. The important thing here is to let students know that you’re not asking them to discuss answers: They are just discussing the problem itself, including any struggles or difficulties they encountered. That way, students aren’t discouraged from contributing from the conversation.

I know that there are times when it’s especially hard for me to resist the urge to swoop in and rescue my students from their mistakes. Instead, I redirect that feeling back onto my other students, and make a habit to regularly ask the class whether they can support or disprove a claim. At the same time, many of my students think “debate” is synonymous with “argue,” and I don’t want them shouting each other down. My role as facilitator includes establishing boundaries and reminding students that disagreements should be rooted in facts and logic.

The easiest way to get students talking is with four simple questions.

  1. Why?
  2. How do you know?
  3. Can you prove that?
  4. Can someone else disprove what’s been said?

Repeat these questions, and hold students responsible for answering. With time, students will slowly start to include explanations in their answers without prompting, and respond to one another in meaningful ways. 

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