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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Faculty Debrief Part C:

PHIL TUCHER: Let's shift over to questions.

BARBARA SHREVE: Okay.

PHIL TUCHER: What you got on your list?

BARBARA SHREVE: Um, well, the first piece that we talked about together with the three examples for solving the equation, and which was a correct way to start, um, didn't go quite as I anticipated. It took a bit longer and it was a little bit harder for kids to access than I had expected. Um...

PHIL TUCHER: What's your question about it?

BARBARA SHREVE: Um, I think I have a question on how I structured it, of how did I kind of give them access to parsing through what those different things were, and was this the right way to start to move them into it and build their confidence? Um, I had moments where I wasn't sure what question to ask to help them keep thinking. I felt like I had to insert more information and I'm not sure if there's another way around that and I just wasn't finding the questions in the moment or different ways to look at it. And I had a question of, is there another representation here that if they were able to look at it in a different way where these three are still fairly symbolic, if that would've been a different way to help them start accessing it?

PHIL TUCHER: So the "Who's Right?" the first one that didn't...the conversation didn't go the way you had expected, you said. You're wondering about whether that's the right start, what information did they need, what information might have you provided as well, are there other representations besides these three that would've completed the picture for them. Let's look at that in a minute. What else were you...do you have anything else on your questions list?

BARBARA SHREVE: Um, overall on the lesson I think because this took a little bit longer, because they really had these quite deep conversations – they had deeper conversations or lengthier conversations about the matching than I expected -- they didn't get to the piece following that where they actually would've had to do some problems on their own. And they would've had answers and problems but actually had needed to do the work to see what went together. And so in terms of individual information that I have about where they are individually, and how they were making sense is really only from the pieces of their conversations that I was able to overhear, or the questions that I was able to ask them when I was at their groups. And so in that respect, I don't have as much information as I would like about their individual understanding. So I'm still wondering how well they'll be able to apply these ideas of first steps and what the answer should look like.

PHIL TUCHER: I thought about a similar question in terms of closure and I wondered, um, so the bell surprised you, slightly shorter day, and not an issue at all except that being Friday, wondering what can you do next time you see them that comes close to the wonderful way that you started today. Here's where we were when we're last together, I wondered were there any things you could do for closure beyond the acknowledging that you did - the solid work, the good work is going to get you through when we meet next time, you'll be in a good place. Was there something that you could've done in terms of individual, or in terms of summary statement? Or similar kind of question. What else did you have?

BARBARA SHREVE: Um, I think by nature again, this day was really symbolic and as we're having the conversations in the warm-up and thinking about what is an x-intercept, I wondered if there were pieces where another representation or saying something graphically could've been brought in. I don't know if that's within this context or in a next step, but it‘s definitely something I'm curious about.

PHIL TUCHER: You asked at the beginning whether or not students would be able make some generalization statements, "Oh, I get it. You use a generic rectangle when..." I want to ask...and If we have time, I'd like to look at, did you hear those kinds of statements? If so what helped bring those to the surface? And if you didn't hear them, what do you think could happen differently so the kids would be asking those...making those generalization statements?

BARBARA SHREVE: Okay.

PHIL TUCHER: Well, that's more than enough.

BARBARA SHREVE: Do we want to dig into one of those?

PHIL TUCHER: Let's do that.

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.