Moving Up and Getting Steeper: Negotiating Shared Descriptions of Linear Graphs

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Abstract
This study examines mathematics learning in the context of peer discussions by focusing on students' descriptions of lines graphed on a computer screen. The article describes how these discussions provided a rich context for negotiating shared descriptions, supported conceptual change, and resulted in convergent meanings through reciprocal elaborations and clarifications. The participants in the study used graphing software to explore the connections between linear equations and their graphs with a peer. The article presents the analysis of three case studies examining how students negotiated shared descriptions of lines. These conversations supported students' construction of shared descriptions, but not necessarily by presenting conflicting ideas or through one student guiding another. Rather, negotiation functioned through local conversational resources such as the use of reference objects, spatial metaphors, and coordinated gestures and talk. These case studies also point to an important role for instruction in orchestrating and supporting peer discussions by modeling how to resolve negotiations and maintaining students' focus on mathematically productive learning trajectories.