## Abstract

This case study investigated the development of group cognition by tracing the change in mathematical discourse of a team of three middle-school students as they worked on a construction problem within a virtual collaborative dynamic geometry environment. Sfard’s commognitive framework was employed to examine how the student team’s word choice, use of visual mediators, and adoption of geometric construction routines changed character during an hour-long collaborative problem-solving session. The findings indicated that the team gradually moved from a visual discourse toward a more formal discourse—one that is primarily characterized by a routine of constructing geometric dependencies. This significant shift in mathematical discourse was accomplished in a CSCL setting where tools to support peer collaboration and pedagogy are developed through cycles of design-based research. The analysis of how this discourse development took place at the group level has implications for the theory and practice of computer-supported collaborative mathematical learning. Discussion of which features of the specific setting proved effective and which were problematic suggests revisions in the design of the setting.

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## Notes

The full log for Session 3 is available at: http://gerrystahl.net/vmt/icls2014/Topic3.xlsx. The VMT Player is available at: http://gerrystahl.net/vmt/icls2014/vmtPlayer.jnlp. The replayer file for Session 3 is available at: http://gerrystahl.net/vmt/icls2014/Topic3.jno.

The instructions specified that, “point H is an arbitrary point on line FG.” In Euclidean geometry, that would mean that even though H can be any point on line FG, it is not something that moves. Thus, although one looks for a solution that would work for any point H, any treatment of H would be static. In dynamic geometry, however, an arbitrary point H is a free point that can be dragged along line FG. Thus, there is some legitimacy to the students’ solution. Ultimately, however, the solution fails the drag test of dynamic geometry. If one properly constructs the perpendicular through point H, then one should be able to drag point H along line FG and have the perpendicular to FG move with it so that it always passes through H and remains perpendicular to FG. Cheerios, however, had only dragged their final construction by moving point G.

In a similar analysis of all eight sessions of the Cereal Team, Stahl (2016) conceptualizes the development of the group’s mathematical cognition in terms of the successive adoption of

*group practices*, rather than*routines*, in order to emphasize that they are being theorized as group-level rather than individual phenomena. As illustrated in the six episodes here, the Cereal Team questions, negotiates, and adopts new practices through their discourse (including shared GeoGebra actions). This meaning-making process creates a shared understanding within the team. Once the team agrees to use a routine, it may become a group practice, which can be used in the future without further discussion.

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## Acknowledgments

The author would like to thank the Fulbright Core Program for funding her sabbatical, and Gerry Stahl for welcoming her to his research team and sharing the VMT data analyzed in this study.

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Oner, D. Tracing the change in discourse in a collaborative dynamic geometry environment: From visual to more mathematical.
*Intern. J. Comput.-Support. Collab. Learn* **11**, 59–88 (2016). https://doi.org/10.1007/s11412-016-9227-5

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DOI: https://doi.org/10.1007/s11412-016-9227-5

### Keywords

- Mathematical discourse development
- Mathematical routines
- Group cognition
- Collaborative dynamic geometry
- Dependencies