Exemplar: on the limits of teaching as explaining

From a recent classroom observation:

T: Some of the things that we need to remember [gestures to screen]. Parallelograms are quadrilaterals that have opposite sides equal and parallel. So remember that on a parallelogram, the sides that are opposite of each other [gestures to parallelogram on screen] will be the same length and will be parallel to each other. On both sides. That's how you can identify all that's a parallelogram. So is a square a parallelogram? Would a square be considered a parallelogram?
SS: [silence]
T: Opposite sides are equal, and they're the same length, and they're parallel to teach other. So, on a square, are their opposite sides not parallel? [silence] Just answer that. Are their opposite sides parallel?
S: [timidly] Yes?
T: They are. Are they the same length?
SS: [confidently] Yes.
T: Yes, because a square is the same all the way around, correct? So is a square a parallelogram?
SS: Yes.
T: How about a rectangle?
SS: No.

Granted, this is a short excerpt from a longer class in which lots of good teaching and learning happened, but this little exchange jumped out at me as a great example of what seems a typical teaching-as-explaining moment. It's difficult to infer any evidence that the students have understood the teacher's explanation; to the contrary, the students seem to be playing the Yes/No Guessing Game. I'll also add that though it may not be apparent in the transcript, the teacher's questioning becomes rather aggressive. Starting when she asks "So is a square a parallelogram?" the first time, when students respond with silence, she repeats her question in a more aggressive tone, signalling that, in her view, the answer to her question should be obvious. I wonder how and to what extent this influences the ways in which students respond to her line of questioning.

One final curricular thought: it seems that her students have learned about squares and rectangles before moving up the hierarchy to the more general case of parallelogram (and earlier, she makes a reference to moving sequentially through a curriculum). To my way of thinking, it makes more sense (particularly in the context of thinking about area, which is what this unit is about), to start with the general case (parallelogram) and then to understand squares and rectangles as instances of a parallelogram. Can anybody make an argument for the ordering this teacher was following?

Establishing a discourse-rich classroom

I watched a teacher this week setup an activity in which one of the "rules" (her word) for the activity was that students should challenge each other's thinking. She went on to explain that you shouldn't just let your partner write down their answer, but say things like "Why did you do that?" and "Are you sure? How do you know you're sure?" and "I don't agree with you. This is what I think." She admonished the students that this should be "good discussion time." So this is great . . . these are all things I'm sure any teacher would love to hear while listening to student discourse.

After the teacher finished with her rules for the activity, the students got to work. Except there was none of the kinds of discourse that the teacher said she wanted to hear. So what went wrong? I can only speculate (since I only got to see a single class period that was in the middle of the school year), but there are two things I'd like to suggest.

The first is that classroom discourse is a reflection of the norms (both explicit and implicit) that you establish in your classroom. Establishing those norms early in the school year is important, but maintaining them over the course of the school year takes work. Perhaps if those norms have been firmly established, then establishing rules for an activity is one way to reinforce those norms. However, if the norms were never established in the first place, setting up rules for discourse is setting yourself up for failure.

Second, what the teacher was really wanting her students to do is engage in mathematical argument. This is a disciplinary practice, and sociocultural theories of learning tell us that these kinds of social practices are learned through legitimate peripheral participation in a community that is engaged in those practices. In this case, I would argue that if the teacher wants her students to engage in this type of argumentation, then she herself, as the more knowledgeable person in the room, needs to engage in that type of argumentation. By doing so, she creates a classroom culture in which that kind of discourse is normative. I recognize the limitations of my data, but given the kinds of student discourse I heard, I would bet that what I saw in the single class period is fairly representative of the classroom. In particular, the teacher's discourse was what we hear in the typical American classroom - lot's of IRE questioning, though there were some superficial attempts to press students to explain their reasoning. For example, when she asked a student how they knew the expression they'd chosen correctly represented the area of a compound shape, the student responded "order of operations" and the teacher moved on to the next problem, accepting the student's response as an adequate warrant, even though "order of operations" by itself doesn't tell us anything about why the student's expression was an accurate representation of the area of the shape.

So, how do you achieve the kinds of student discourse in your classroom that you want? What early-year activities do you engage in that help establish norms? Do you model disciplinary practices for your students? How?