Jacob Disston and his students share their insights into the day's lesson and reflect on what they learned.

7th Grade Math - Algebraic Equations, Inequalities, & Properties*Jacob Disston, Willard Middle School, Berkeley Unified School District, Berkeley, California*

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TEACHER: I love watching you teach and it gives me a sense of satisfaction. It's really exhilarating. I think that you referred to the beginning... as a whole lesson, did it work, as opposed to just a research lesson? But one of the things that I was reminded of, in terms of when you have a class where you have all these different formats -- you were in the huddle, you were in one group you came in with, and you were in a different group later -- is that it builds relationships between students, that I think become really valuable on the other days when you have a more traditional, like, let's do the book, lets do these problems... I think that those flexible groupings and being comfortable with, like, "Now we're all here, now we're there, now we're going to be over there with different people," I just felt like there's this unspoken bonding that happened, a comfort of sitting together and talking and engaging with each other, with a different group than it was twenty minutes ago...

You just kind of let people throw out their ideas, but just left it there, versus at the end where you tried to pull things together more... I like that because I felt like you didn't...it wasn't really driven in the beginning, and then you sort of just let people show out different ways and didn't say if anyone's way was right or wrong, but got them to this place of these three big groups, where then, when they sat down to better analyze... one of the three groups, they didn't have a direction, you hadn't told them what direction to go in so it allowed them to...the group I was looking at was the Maria, Sara, Angela, Alex group and they started with expressions, but it allowed them to come at it with a blank slate but yet have some things they had heard being said earlier...

What was neat about Maria is she really liked when people contradicted...she accepted when people contradicted. She would say "I think these are like terms and unlike terms." I liked it because there was some contradiction there, and she was the one who finally threw out the y=2. People were saying "that's an equation, it has to be an equation" and she was like "yeah it has an equal sign but I think it's really different." I think she sort of made it okay not to have an answer, that the discussion was really their thing.

JACOB DISSTON: Know that when you are teaching and you are giving kids an opportunity to talk that they are surfacing all these conceptions and misconceptions and partially formed ones and that that's really important for them putting all these ideas together. And if we're not giving those opportunities in lessons, and just saying "do this and then do this and this," and just assuming that they'll learn kind of a coherent...they'll have a coherent understanding of math in the end, I think is... you know, because they still have these questions...

TEACHER: Thank you Jacob, it was really wonderful watching you teach. It really is exhilarating and I'll say I feel great pride watching you teach as being a colleague of yours, and just another teacher and watching what marvelous things you do with the kids, and when things happen in the classroom when you're up in front. I was especially watching this group of Srikar and Maya and Noah, although I watched some other groups too. To your point or question of whether or not you were teaching them what to focus on, as opposed to finding out what it is that they were focusing on, you know, were you leading them too much or were you able to mine and find out what it is they knew about this already and were figuring out for themselves... I think it was a perfectly good balance. I was afraid going in that there would be more kids kind of saying "I don't know what to do" or "I don't know," or focusing on things that... that trivial things would dominant as much as what we would think would be more substantial and more important mathematical distinctions. And some of those came up, I mean, at one point they said, "Well this has all A's and this one doesn't..." But for the most part I thought that there were disagreements that were substantial, especially doing the inequalities -- that's the first one they focused on -- once they had to break the inequalities into sub-groups, Maya said... no, Noah said, "Well, these are all 'greater than' and these are all 'less than.'" And you happened to come along then and you asked, "But why is that an important distinction, what makes that important?"

Overall I thought they stuck with it, especially for ninety five minutes of confronting these, and you know, the variety of groupings helped with that. But still, they were basically the same kind of questions that they were being asked to tackle for a long period of time and they stuck with it...

COMMENTARY BY COACH LINDA FISHER: There seem to be two big threads that run through the discussion in the lesson debrief, the idea of a research lesson and the mathematics being tackled by the students. The first is the issue of the research lesson. How did all the new features and structures work in this lesson to allow us insight into student thinking? Would these techniques work in a real classroom, when there aren’t cameras and observers? I have conjectures that we, as teachers, often try to give too many instructions. I think the structures made the mathematics interesting enough for students that they didn’t need lengthy instructions to find ideas to discuss. I think the change of pace throughout the lesson kept them involved and gave them new inputs for discussion. How could you test this out? What conjectures do you have about the lesson structure?

The second discussion is about the depth of mathematics students and teachers are thinking about. What does it mean to “solve” something? How is a solution with a single answer different from a solution with an infinite number of possibilities? As you watch these pieces, try to write down all the big mathematical ideas brought up by the lesson. What ideas are being surfaced that are different from those in more traditional lessons? Why are these ideas significant?