This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.
Lampert has a fascinating classroom routine. Each class, she begins with a problem on the board. Students work to answer it, alone or with other students at their table. If they finish early, Lampert asks them to make more conjectures and try to answer them. At some point during the class, she brings the whole group together, and they discuss the problem and its connections with other mathematics.
In this chapter, Lampert examines the discussion that followed student work on this problem:
Here are some of my takeaways thinking about discussions in my classes:
Call on students for a reason
The Five Practices advocated this years ago, but Lampert’s writing is a good reminder. Here’s a great quote:
I called on Richard because I wanted to teach him and others in the class that everyone would indeed by asked to explain thier thinking publicly. I also wanted to teach everyone that what they said would be expected to be an effort to make mathematical sense (146).
As with my choice of Richard to begin talk about problem A, the choice of Anthony here is a deliberate teaching act, meant to get at a particular piece of mathematics and a particular aspect of my relations with students (165).
Lampert makes explicit that she calls on students with knowledge of what mathematics they may share with the class. She also makes explicit that her choice of who to call on sends a message to the class about the nature of participation, and the nature of having mathematical ideas. I like this:
Many would no doubt make their own conjectures about why I called on Richard and why I did not call on someone else. They would continue to conduct experiments to learn more about how to get called on or not, depending on their purposes (147).
Lampert gives a searing glimpse into the world inside students’ heads that I too often don’t see in my own classroom, and the consequences my decisions have for students’ beliefs about what it is we do in math class.
Represent mathematics deliberately
Principles to Actions makes an interesting statement about the ideal teacher role in discussions:
Students carry the conversation themselves. Teacher only guides from the periphery of the conversation. Teacher waits for students to clarify thinking of others (32).
I’m pretty skeptical of this assertion. Let’s juxtapose it with a teacher move of Lampert’s. Richard shares an incorrect assertion — that ten times twelve is twenty-two. It becomes clear that he did not misspeak, and has a misconception about multiplication, one that many other students in this fifth grade class are likely to have as well.
One of the things that I came around and did with some people is to draw a picture that would help you to reason about these problems. Twenty-two groups of twelve, you could draw as a twelve, a twelve, a twelve, and so on until you got to twenty-two of them [drawing circles around 12s as I talk] (151).
But let’s look at ten groups of six for a minute [drawing on the board, next to what I had already done] (153).
Lampert’s representation adds information, clarifies student statements, yet still leaves room for thinking and learning. Those are great criteria for teacher input, and are almost always useful in making a discussion more productive.
Students may at points be able to carry the conversation themselves — but that seems to me better suited to a discussion where students are already confident with the material. Much more typical of my classroom is a discussion where students are still working through their ideas about a piece of mathematics, and a teacher taking an active role to clarify and represent student thinking to create more opportunities for learning seems to me an essential part of discussion.
Give hints to create better opportunities for further discussion
Michael Pershan and I exchanged ideas earlier this year about hints. One goal of a hint that he proposed was to prepare for future learning — if students notice certain problem features, they will be better able to learn from a discussion. Lampert is unafraid to propose a representation or present a mathematical idea if she feels it is an opportunity for learning. At the same time, she is acutely aware of the messages that she sends with these choices — and that when students are generating mathematical ideas, other students learn that it is their role in the classroom to share what they think about a problem. A useful teacher role here is to make the most of independent work time by giving hints when necessary that create a discussion more focused on essential mathematical ideas.
An instance of my teaching practice
I want to zoom in on a moment in my classroom where I could have benefited from this type of thinking. I started an Algebra-II class with this visual pattern as a warmup:
Students were trying to write an expression for the number of pink nubs in the nth step. After some work time, I called on a student with a raised hand to share his expression and reasoning. I did not know what he was going to say, though it turned out to be a useful starting point:
He shared his reasoning — he saw it as two squares growing, and then subtracted the overlap, which was a square one unit length smaller. His reasoning and explanation were sound, though he was counting Lego pieces rather than pink nubs. I hoped to have another student make this observation, and called on a second student with a raised hand. Instead of commenting on the first student’s expression, she shared a second expression:
She explained that she saw it as a single square, and then an L-shaped group of squares on the outside, which started at 1 and then increased by 2.
This was a useful alternate way to look at the problem, but still did not get at the point I wanted to make. Returning the focus to the first equation, I changed it to read:
and asked the class what that might mean. A student volunteered that the expression was now counting the number of pink nubs, rather than the number of Lego pieces.
At this point, the energy in the room seemed to have hit a dead end. I didn’t feel like I had gotten everything I wanted out of the problem, and decided to share one additional interpretation. I wrote this on the board, and asked students what it might mean:
After a short chance to chat with someone next to them, one group realized that this expression interpreted the pattern as one large square, with two pieces missing in either corner, then multiplying by 4 to account for the transformation from Lego pieces to nubs.
I think I made a useful choice in sharing an additional interpretation of the problem. Given the lack of focus in the first part of the discussion, that example was a worthwhile way to cement some of the understanding about the structure of the expressions and its relationship with the pattern — and a number of students in that class have struggled with representing quadratic patterns over the course of the year.
I came up pretty short in anticipating what students were going to see. Students generated the first three expressions, but I did not sequence them effectively because I did not know what students were going to share. I was reacting to what they said, rather than following a plan I developed while students worked, and too focused on one possible avenue forward. As a result, I missed an opportunity to more effectively connect between expressions, and to create dialogue between students, rather than constantly bouncing off of me.
I also think that the difference between Lego pieces and pink nubs is a less important feature of this task, and I could have used the time more effectively by pointing out that difference to students individually, in their groups, or by being more explicit about it from the beginning.
Lots left to improve here. Back to work.