Students are presented with three possible ways to begin solving the problem, “Solve for x: 2x2-14x+20=0,” and asked to decide together which way(s) of starting the problem are correct.

9th-11th Grade Math - Quadratic Functions*Barbara Shreve, San Lorenzo High School, San Lorenzo Unified School District, San Leandro, California*

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BARBARA SHREVE: I'm so proud of you, and thank you so much for being willing to do this because I get to be here with you guys and watch how hard you're working every day. And I'm excited about what we've accomplished this year as a group, and I'm looking forward to seeing what we still do in our last weeks together. Okay? So should we get back into some math? Good! You're going to need a clear, clear space. You can leave your notebook off to the side. If your backpack’s on the table, will you clear it because you're going to need to be able to work together with the person next to you.

All right, are we ready? Okay. On Wednesday, we were working together and looking together at some problems about quadratics where you guys had some amazing conversations in your teams about what – and how to start different problems. So if you remember, I was taking notes on what you were saying. And some of the things that I saw that really helped you guys, were I saw you sticking together, I saw people asking questions and thinking "Oh, I can look back in my notebook to find information about that." At table 2, I heard people having really specific step-by-step discussions where they kind of gave a couple of steps, and gave people time to think, and then went back and did some more explaining. I heard people not just having questions themselves but checking with everybody, and then calling me over for questions, so that everybody together was kind of stuck in the same place and you really had a chance to think about something before you went for extra help. But I also saw you guys reaching across the table, putting ideas in the middle of the table even when you weren't quite sure yet. And that's the kind of thing we want to work on today. So today, some of those things that you were thinking about on Wednesday, we're going to look at really carefully. We're going to look at, first of all when you get all these directions about quadratic equations, where do you start? So on your table there's this piece of upside-down paper. Will you get to where you and the person next to you can see it? So you should only need like two because we're not doing a lot of writing on it but you can absolutely have another one. Here you go. Okay, does everybody have one they can see?

So I have geometry students in other years, right? I don't have them this year, but one of the things I notice a lot is that when they go back to work on these quadratics, they forget sometimes how to start. So they had this problem like the ones you've been seeing, solve for x, where the equation is 2x2-14x+20=0. And three different people started that in different ways, and you can see that. I'm going to give you two minutes as a table to decide who is correct. So look at those three ways to start and you don't have to finish the problem, but whose first step makes sense? Okay? I already see table 6 leaning in to talk. Can you guys talk together, and in two minutes I'm going to ask you to decide as a table who is correct.

STUDENT: Hold on.

STUDENT: This is a, b, and c is 20 but...

STUDENT: Okay, that's right.

STUDENT: That's right and that's right.

STUDENT: Yeah, this is right.

STUDENT: This one?

STUDENT: Yeah.

STUDENT: Dulce, get me the list that’s right here.

STUDENT: This one?

STUDENT: Mm-hmm, but why do they give you all of the things correct?

STUDENT: Yes, but we need to find the value only of x [inaudible].

STUDENT: If we do this one, it's going to be like different answers. And if we do this one...

STUDENT: This is right?

STUDENT: Yeah. Because that's how it started off with.

STUDENT: Okay.

I put examples of common mistakes in front of students so that they can confront them when they are not part of the work the student has created him or herself. I find that the distance created when students analyze another person’s work – especially when that person is not immediately present – allows students to be more analytical. In this case, it also presents the challenge for students of trying to make sense of someone else’s work when that person is not present to answer questions about it. In pushing students to take a stand or make a choice about who is correct, I am trying to get them invested in either confirming their answer or understanding why someone disagrees with it so that they are less passive listeners. In this case students were reluctant to do so, leaving me unsure whether they had been able to understand the students’ work or were not sharing their opinions for other reasons. It was helpful to hear how students were interpreting factoring as a tool that is used to rewrite an expression when the rewritten expression is the goal, rather than as a tool that can help you work toward a solution to an equation. Through the discussion I recognized that students did have confusion about what question they were answering, and what the answers meant. I was faced with decisions about how much more to explore in a class discussion and what to leave for them to discuss in pairs and teams, and I opted to move students to work in pairs rather than continuing to pursue the discussion whole class, so that more students would have the chance to participate in articulating questions and trying to answer them.