Using Common Ground to Understand Innovative Teaching

In recent decades, there have been strong calls for reforms in mathematics education, and in particular for changes in classroom practices. Teachers have been encouraged to organize instruction so that students participate in more collaborative, discussion-based activities with an eye to supporting a community of learners, rather than a set of individuals working on mathematics (NCTM, 1991). Organizing these kinds of collaborative practices, however, has proven to be very challenging (Ball, 1993; Schifter, 2001). Although reform documents often position the teacher as a facilitator in collaborative classrooms, little direction is given for what that looks like or how it is accomplished (Chazan & Ball, 1999). Teachers and researchers have indicated a need to better understand the work of teaching that can support student participation in such practices (Boaler, 2002; Lampert, 2001). In this paper, I report findings of an in-depth case study of a class taught by an experienced teacher, Ms. Nelson, who regularly organized collaborative discussions among her students, across grade levels and course content. Ms. Nelson’s students worked collectively, solving challenging problems and forging new understandings of mathematical concepts. They attended to and extended each others’ ideas thereby engaging in a highly generative kind of intellectual activity not often supported in mathematics classrooms. Indeed there are few analyses of such classrooms and the corresponding teaching practices, particularly on the secondary level (Boaler, 2002). This case study suggests that establishing and monitoring a common ground is an important aspect of the teacher’s role in facilitating students’ mathematical collaboration. I argue that a common ground comprising students’ mathematical ideas is critical to collaborative work and illustrate some of the ways the teacher attended to the development of a common ground among students.