Discourse: The ways in which knowledge/ideas are constructed and exchanged in the classroom.
In terms of discourse, my goal in teaching the lesson was to put as much of the authority in the students’ hands as possible. As a result, I tried to turn many of the questions, especially those involving a correct answer, back towards the students, encouraging them to offer their own insights and ideas. Throughout the lesson, the students talked both with me and with their peers to make sense of the array method.
The original pedagogical focus of my lesson was facilitating mathematical discussion around mathematical ideas by eliciting, clarifying and following up on student solution strategies and explanations. Although I decided to instead focus on selecting and using representations, I still wanted to make the lesson discussion-based because I felt that a discussion-based lesson would assist student in grappling with the many complex concepts involved in the array method.
Depending on the complexity of a step, I would either immediately ask students what they thought we should do or model the step and then ask how we can do the same thing with a different number. For example, when setting up the array, I asked students how they thought we could draw a box to represent our Thanksgiving problem, and then asked them if that box actually represented our problem. However, once we got to using the arrays to solve the multiplication problems, I modeled how to find how many blocks were in the first rectangle (10 x 5) and then asked the students how they thought we could find the number of blocks in the second rectangle (5 x 5).
Furthermore, in addition to these guiding questions, I also encouraged students to talk to each other and answer each other’s questions. For instance, when initially discussing whether the array we drew represented the Thanksgiving problem, a few of the students seemed to be confused. As a result, I decided to do a turn and talk. Here, I was hoping that by talking with each other, they would be able to arrive and the correct conclusion without me giving them the answer. Moreover, when students asked me questions, I frequently asked one of the other students to explain instead. Although I would often restate, and occasionally clarify, these explanations in order to “help the class make sense of what another student is trying to say” (Shindelar, 2009, p. 175), I felt that these student interactions still helped to keep all students engaged and give them authority in their learning.
One thing I wish I had done differently was given students time to make sure their answers made sense and to discuss that with each other. Although I often turned the authority back towards the students and often responded to students’ ideas by asking “why” and “show me how” questions and by affirming they were using “good strategies,” the conversations still frequently revolved around correct answers rather than answer that did or did not make sense. Also, although I tried to avoid doing so, I found that I would occasionally give students the answer. For the most part I usually caught myself midway through the sentence and tried to rephrase it as a question, but the students likely caught on.
Herbel-Eisenmann & M. Cirillo (Eds.), Promoting purposeful discourse, (pp. 165-178). Reston, VA: NCTM.
Kazemi, E., & Hintz, A. (2014). Intentional Talk: How to Structure and Lead Productive Mathematical Discussions. Portland, ME: Stenhouse.
Shindelar, A. (2009). Maintaining Mathematical Momentum through “Talk Moves.” In B.