SUPPORTING GROUP COGNITION, INDIVIDUAL LEARNING AND COMMUNITY PRACTICES IN DYNAMIC GEOMETRY Gerry Stahl The Math Forum and the iSchool Drexel University Philadelphia, USA Gerry@GerryStahl.net Abstract—Group cognition is analyzed at the small-group unit of analysis. It involves the semantics, syntactics and pragmatics of natural language, gestures, inscriptions, etc. The meaning- making processes involve inputs from individuals, based on their interpretation of the on-going context. They are also responses to the on-going social/historical/cultural/linguistic context, which they can reproduce and modify. Technologies play a central role in mediating the multi-level, intertwined processes. Emergent technologies should be designed to support this mediation. Collaboration environments should be designed to prepare groups, individuals and communities to take advantage of the technical functionality and to promote learning at all levels. This paper reports on the design of a curriculum in dynamic geometry to support group cognition, individual learning and community practices in a coordinated way. Keywords—Computer-supported collaborative learning; group cognition; math discourse, dynamic geometry I. INTRODUCTION Group cognition is analyzed at the small-group unit of analysis. It involves the semantics, syntactics and pragmatics of natural language, gestures, inscriptions, etc. The meaning- making processes involve inputs from individuals, based on their interpretation of the on-going context (Stahl, 2006, esp. Ch. 16). They also take into account the larger social/historical/cultural/linguistic context, which they can reproduce and modify (Stahl, 2013). Applying this perspective to the learning of mathematics, we adopt a discourse-centered view of mathematical understanding as the ability to engage in significant mathematical discussion (Sfard, 2008; Stahl, 2008). Here, “discourse” includes gesture, inscription, representation and symbol, as well as speech and text; these are often closely interwoven in effective interactions (Çakır & Stahl, 2012; Çakir, Zemel & Stahl, 2009). Technologies play a central role in mediating the multi- level, intertwined problem-solving, learning and knowledge- building processes. Emergent technologies should be designed to support this mediation. This involves considering within the design process of collaboration environments how to prepare groups, individuals and communities to take advantage of the designed functionality and to promote mathematical thinking at all levels. This paper reports on the design of a curriculum in dynamic geometry to support group cognition, individual learning and community practices in a coordinated way. We have been developing a collaboration environment for small groups of students to explore mathematics – especially dynamic geometry – together online (Stahl, 2009). Our Virtual Math Teams (VMT) environment now includes a multi-user version of GeoGebra, an open-source dynamic- geometry tool (Stahl et al., 2010). Shared chat rooms in this VMT environment can include: • Personal GeoGebra tabs for an individual to experiment with dynamic-geometry explorations and constructions. • Group GeoGebra tabs for a team of students to experiment together with dynamic-geometry explorations and constructions. • A text-chat window for a team to discuss its collaborative explorations, while it is working together or to ask questions when team members have problems in their individual work. • A shared whiteboard and a group wiki page for the group to summarize its findings. • The wiki can be used by a whole class or a community of teams to view and comment on what each team has accomplished. • Logs of the text chat and a replayer, which allows anyone to replay a collaboration session in complete detail for purposes of reflection and/or analysis. We have conducted pilot trials of the VMT-with- GeoGebra environment and have found that this relatively complex system requires some preparation and training for students, student groups and classes to use effectively without encountering frustration. In response to issues identified in the analysis of the multi-user GeoGebra use sessions, we have drafted a set of dynamic-geometry curricular activities, interspersed with tutorial tours of the technology features (Stahl, 2012a). These materials are designed for use both by teachers in professional- development contexts and by students in online-classroom or after-school settings. The curriculum activities have been designed to promote collaborative learning, particularly as it occurs in significant mathematical discourse about geometry. Collaborative learning involves a subtle interplay of processes at the individual, small-group and classroom levels of engagement, cognition and reflection. Accordingly, the activities are