During the past few weeks, I’ve had a teaching intern in my class. This has given me the opportunity to observe and listen to how my students talk about math. It has been a real gift to be able to listen. What have I heard?

First, let me say that students in my classroom typically work in groups. They are accustomed to working as teams and talking about math. I just usually don’t get a chance to listen to them all.

Two of my classes have begun a short unit on circles and properties of circles. They were investigating the relationship between central angles and their corresponding chords. The problem in the book asked them to prove that if AB = CD, then the measures of arcs AB and CD are also equal. Keep in mind that we use an integrated curriculum, so the last time kids had to prove anything geometric was in January. They were using whiteboards and drawing diagrams and pointing and talking math. While they worked the problem, I walked around and listened to what they had to say.

Here’s a sample from one group:

M: Oh, wait, those sides are all equal. E: Why? M: They’re all radiuses. K: Okay. So if those are all equal and these two are equal, then the triangles are congruent from side-side-side. M: Right. E: But how does that help us prove that the arcs are equal? M: These two central angles are the same because the triangles are congruent. E: And if the central angles are the same then the arcs are the same. K: Right. Good.This was a typical interaction for this class. A couple of groups needed some teacher questioning to point them in the right direction, but most of them were talking about math and reasoning their way through the problem. Isn’t that the ultimate goal?