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Interpreting a student's comment

The situation

In the previous episode of Cara's session with Judy, they started examining the Discussion Sketch that Judy had made of Cara's attempt to lead a discussion in class. They had looked at some tools that they might use to analyze the Sketch. And Judy is about to call Cara's attention to the following bit of interaction with Tyrell, and then with other students. Here's a transcript of the portion of the lesson they will talk about:

Discussion Sketch, Side 2*


Teacher's question or statement

Student's statement or question

Teacher's response


Is there any other way I can think about writing this in a number sentence? Teacher records 6+6=12 on overhead.


T: You could do 3+3+3+3.

I don’t have three of any groups though, but that would equal twelve. You’re right.


J: (calling out) I know six times four.


I’m wondering not how I could make 12, but how I could use this tile pattern to make a number sentence. Carole?

C: You could do 4+4+4

I could break this up into groups of four. I guess I could. I could do 4+4+4 equals 12.


And I guess by the same account, I could also take Tyrell’s idea and break this off. I could make equal groups of 3, couldn’t I?. 1,2,3 ... 1,2,3 ... 1,2,3 ... yeah. I would write 3+3+3+3.


Megan, do you have another way?

M: 2+2+2+2+2+2

Oh, I could do 2 six times.


Are there still more ways I could do this?

Jason: I know, I know. I got a really good way.

Jason, what’s another way we could do this?


J: 6 times 4

We don’t have 6, four times.


Sasha: Oh, I know, I know.

Hmmm? Sasha?


Sasha: 6 times 2.

Thank you. Do we have 6 two times. We do. So we can do 6 times 2 equals 12.


Peter, do you have another idea?

P: We could do 2 times 6.

Two times six. Remember we talked about this. Two times six is the same as six times two. They’re going to be the same thing. And these all equal twelve. We have so many ways to make 12.


[Teacher asks students to take out math books. The page shows a variety of tile arrays. As a class, they write a number sentence for the first two. Then the children complete the third problem independently.]


Cara and Judy talk

JUDY: Let’s skip down a little bit here. I really wanted to talk about what happened when you asked the kids to give you number sentences for that tile array up on the overhead.

CARA: Yeah. I got confused there. Tyrell said 3+3+3+3 and I didn’t know what to do at the moment. He was right because that is equal to 12, but it didn’t match the tile array, which had two rows of six.

JUDY: So the task you set was to give a number sentence that described the picture. But what Tyrell actually did was give you a number sentence where the answer matched the number of tiles on your drawing.

CARA: That's why I told him it wasn't right. That it got the right answer, but it didn't show the right number of groups.

JUDY: And do you think your response mattered?

CARA: I don't know, maybe. Maybe I should have asked him why he said that, instead of just telling him it was wrong.

JUDY: Why would that have mattered?

CARA: Well, I'm not really sure what he was thinking. I don't know if he was just trying to get the answer 12 or if he really pictured the tiles as divided into groups of three. He might have. Also, I think I kind of got into a pattern right there. Once I answered Tyrell like that, I just kept evaluating all the answers. I was the one saying whether they were right or wrong, and I really wanted to have the kids try to do that.

Tools in use

Something that Judy saw in the Discussion Sketch made her "really want" to discuss Cara's exchange with Tyrell, about how to compose number sentences matching an array of tiles on the overhead. Cara replies that she "got confused there," and later describes her response to Tyrell as a turning point in the lesson, after which she found herself evaluating all the students' answers, when what she wanted was for the students to evaluate their own answers. At this juncture, Cara might have gotten value from our tool on responding to students' comments. [Responses to students]

Cara now sees the exchange as a fork in the road, where the discussion could go in quite different directions. One of the tools in the set invites teachers to look at the Discussion Sketch in just that way. [Forks in the road]

Or, this event might amount to a "critical incident," a moment in teaching that deserves close analysis and careful thought because there might be much to learn about how to teach well. For several days, Judy and Cara will have the option to treat the interaction with Tyrell as a critical incident: Such incidents tend to be memorable, and Judy has made the lesson sketch to support memory. If they do elect to dig into the event more closely, they might find some help in our discussion of critical incident techniques. [Using critical incidents]

About the partnership

To dig into Cara's interaction with Tyrell to the degree that the critical incident technique suggests, Cara and Judy are likely to need some shared understandings about what they are doing and what they mean by what they say.

In any such discussion, Cara's practice is exposed; she is vulnerable. Knowing that, Judy might be inclined to paint a rosy picture, or to avoid digging into what might turn out to look like a problem. But notice where that leaves Cara: perhaps in some kind of trouble, but without useful help.

Perhaps the partners can dig into the teaching more freely if they share some understanding like this: They are NOT discussing Cara and her competence; rather, they are discussing teaching practices and their consequences. Not persons and competence, but practices and consequences. If they just discuss the teaching as though it were Cara's or Judy's or anyone's, then they will not be saying anything about Cara's competence. They will only be relating practices to consequences, and that can be helpful to them both.

Cara can promote this sort of understanding by taking an objective stance toward her teaching. Judy can promote it by speaking of the teaching, as distinct from speaking of Cara. And both might be helped by proposing the understanding suggested just above.

Using the story and tools

Like all other tools, the tools offered here require skills from their users. In collegial mentoring, the users are a pair of colleagues, and the needed skills include not only understandings and skills in working with the tools, but also understandings and skills in working with each other.

The more your partnership shares understandings that enable you to dig into each other's teaching, and the more skillful you are in working together to do just that, the more value you are likely to get from your partnership and from using the mentoring tools.

Like other understandings and skills, collegial skills can be improved deliberately. To do that, you could consult the Partnership section: Agree on your idea of partnership, Get acquainted professionally and Decide how you will work together.

The next episode: Responding to students