In this clip, Margie Trainer reflects on her role as a coach during the lesson and the teaching and learning that occurred.

5th & 6th Grade Math – Multiple Representations of Numeric Patterning*Fran Dickinson, San Carlos Charter Learning Center, San Carlos School District, San Carlos, California*

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| MARGIE TRAINER: As a coach, I see my role mostly as ton of listening and asking a lot of questions; it's my job to guide the process. So I get involved with helping the teacher identifying the goals of the lesson and we do a lot of exchange back and forth; the long and the short term goals. Always keeping the focus on the mathematics. Posing those questions throughout the process of the lesson design. Um, and we get involved in collaborating totally on the lesson, and I bring in whatever materials that I think might help, suggest some ideas, and so on. But ultimately it is the teacher's decision on all of those steps. We just work together to talk it out, evaluate the pros and cons, and um, it might be the questions like "Well, what do you really want the students to understand at this part of the lesson or with this particular activity?" or "What do you think about perhaps doing it this way?" might be a way of making a suggestion.

One big part of my job is to work with the teacher and whoever else we're collaborating on the mathematics. So we actually do the problems and try all kinds of different ways of...in this particular case we got the manipulatives out and we tried to find as many different ways that we could see this pattern growing, and we color coded, and we recorded those, and trying to anticipate where the students might go with this. Um, and so when we get totally grounded in the mathematics that way, and it helps to guide the direction of the lesson, and make those decisions, okay, "Where do you think the students might go with this? What might they be able to do? What might be pushing it too far?" And then anticipating student responses; that was a big part of it and especially potential problems. We had a lot of discussions about that. The most important thing that I see my role is to empower the teacher to own the process because the change process is not going to happen unless it's put in their hands. And if we really want mathematics education to change, the teacher is the most important person in this process.

I had planned to have learners explore all 5 representations in this lesson. We started with the numerical representations of Learners A and B. It turned out to be challenging for many of my learners to represent these numeric representations mathematically. We never made it to the other representations today. Earlier in my career, I may have been tempted to push through and “finish” the lesson I had planned. I made the call to stay on pictorial representations because there were so many great conversations happening in the classroom and there was also a considerable amount of disequilibrium in the room. Had I pushed through to quickly get to the graphical representation, then we would have missed out on some great discourse and reasoning. I really like Stacy Emory’s description of how learners might gain a deeper understanding of pictorial representations and how to model a function by looking more closely at the relationship between the numerical and pictorial relationships.