Linda Fisher asks the observing teachers what they thought of using "wrong answers" as fodder for a lesson, and suggests some things to keep in mind as teachers "strip away" examples of student work to use for re-engagement.

5th Grade Math - Proportions & Ratios*Hillary Lewis-Wolfsen, Forest Park Elementary School, Fremont Unified School District, Fremont, California*

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LINDA FISHER: A different question, you know, they always talk about “using wrong answers” as part of the learning process. We got to see a really nice example of that. How do you think students reacted to seeing those wrong answers being put up there?

JEAN LIU: Well, this class loves to point out (laughter) wrong answers, or correct me, so I think this reall, they loved that. They loved correcting the teacher, and correcting. I think it helps them try to explain how one person got this, and “Let me tell you how it’s supposed to go!” It corrects, and then it correctly teaches. Also, I noted that the kids who, when they see the wrong answer, it’s like “Oh, maybe mine’s wrong.” And they’ll correct it. They’ll think, and it confuses them, and it challenges them.

LINDA FISHER: It puts them into that disequilibrium?

JEAN LIU: Mmm hmmm.

LINDA FISHER: What specific features helped to sort of promote student thinking (bell). Oh – did it get lost in the bell?-- what features of the design of the lesson do you think helped to promote student thinking or the types of discussion that went on?

TEACHER 1: I think the going back and looking at different solutions! Like you said, going through the student’s papers and grouping them to “like” answers, and going back and looking at those different answers, and where those students might have come up with that answer.

LINDA FISHER: This probably isn’t evident to you, but one of the key things about picking out student work is stripping enough of their work away so that everybody has to think about it. A lot of times, there were labels, or things on their work, so when you want to sit down and try this on your own, you need to give them only the bare bones of the answer so it’s interesting enough to think about.

TEACHER 4: And I think too that nobody was pointing fingers, like “You got it wrong!” And the kids weren’t—“Are they looking at me? Am I the one with the wrong answer?” I think they need that chance to see that maybe there are other possibilities. Is this a right possibility, or a wrong possibility? Then they were able to discuss that—it wasn’t the teacher going “You got it wrong, because…” or “This is the right way to do it, because…” I think they felt like they were part of the process, and that was really good for them. They really enjoyed that.

JEAN LIU: And it helped to have just a blank sheet of paper in front of them, not the actual ones that they did. It wasn’t “Oh!” it was “Nobody knows what I got!” so they felt comfortable, it gave them a blank—“I’m ready to learn something new, and I’m ready to contribute. I may not have done this on my paper, but I’m not worried about that.” It gave them a chance to …

LINDA FISHER: So there were some safety nets built into the lesson.

JEAN LIU: Mmm hmmm.

TEACHER 6: It also expanded the boys who were thinking mathematically first, when the second chart was shown, the simplified …not the diagonal but the next one, they go “Oh!” and then they discussed “Oh, that makes it easier, more simplified.” They hadn’t thought of that.

LINDA FISHER: I know because the group that I was listening to, as soon as that second one went up there, because she even posed the question, it was like “Oh! That is so much clearer!” They were making that choice between strategies that you want them to make without—even before you posed that question. That was nice.

LINDA FISHER: I’m out of questions, so-- anybody have any wrap-up comments that you want to make?

TEACHER 6: Not wrap-up yet, but just to add to what you said earlier, I think the discussion brought out a lot, too.

TEACHER 2: And I think that for the early finishers, or for a leson to have a time to wrap it up, or to pose a question, “Have you changed your thought process in terms of each of the questions, or did you use different strategies?” Trigger some metacognitive thinking about their thinking about their own thinking while they’re doing the math problems over again. Maybe just write down, “Did you change your thought process?” Just explicitly ask them, instead of, in addition to our interpretation of them changing their thought process. Do they know they’re changing their thought process at all.

LINDA FISHER: It would be really interesting to see what they put on with the red pen. But I’d give them a couple of days rest first. (laughter) They worked really hard today, it was great.

VARIOUS: thank you!

TEACHER 6: I enjoyed seeing the different ways they think, thank you!

LINDA FISHER: And it is a luxury, just to watch a couple of students, instead of worrying about a whole class, isn’t it?

TEACHER 2: In addition to that, I just think that it worked well, one of the questions was “How do you think the lesson worked in terms of just one teacher being up in front?” I think it works out great. I love the think-pair-share, and encouraging questions, and feedback in terms of other people’s answers. I think that it’s totally feasible with one person, one teacher.

LINDA FISHER: Good! So I want to be invited back when you guys have designed your own lessons to try!

TEACHER: We’re gonna need your help!

LINDA FISHER: Okay, well, Valerie has my number.

HILLARY LEWIS-WOLFSEN: The observing teachers were all from our school, Forest Park. They represented grades 1-6, as well as both administrators and a long-term substitute (who was hired on full time for our site the following school year). This was the first lesson study experience for most of the people there. As a result of this experience, we have a lesson study team based at Forest Park this year.

LINDA FISHER: In my years teaching mathematics, I've learned that kids need to have their misconceptions confronted head-on. With re-engagement, we thought, let’s take that and pose them as dilemmas for kids to think about – get them talking about why some of these common things don’t make sense. That way, we can bring a focus on the mathematics and the concept, rather than solely on the solution and the answer.

When I work with teachers, they want to know what to do in terms of remediation. Teachers usually confront student mistakes by going back to a clean slate and start at ground zero. But there’s something profoundly different about reteaching than teaching it for the first time. When you go back to do a re-teaching leson, you don’t want to start as if people have never learned things. You want to get students to let go of why what they’re doing doesn’t work. Teachers need to find a way in to facilitating the conversation, to helping students see why what they’re doing doesn’t work. That’s one kind of reengagement: having kids reach the conclusions.

One of the critical things is that kids have a lot of mathematical ideas that teachers don’t see. My “hidden” agenda is to get teachers to be able to read student work and make sense of what they’re not understanding. Everyone with the wrong answer doesn’t need the same kind of help. In learning to classify student errors, we want teachers to tease out what are the different kinds of help, and why are they different, because they’re based on different mathematical ideas.