Discovering Dependencies: A Case Study of Collaborative Dynamic Mathematics

The Virtual Math Teams (VMT) Project is exploring an approach to the teaching and learning of basic school geometry through a CSCL approach. As one phase of a designbased-research cycle of design/trial/analysis, two teams of three adults worked on a dynamicgeometry task in the VMT online environment. The case study reported here analyzed the progression of their computer-supported collaborative interaction, showing that each team combined in different ways (a) exploration of a complex geometric figure through dynamic dragging of points in the figure in a shared GeoGebra virtual workspace, (b) step-by-step construction of a similar figure and (c) discussion of the dependencies needed to replicate the behavior of the dynamic figure. The teams thereby achieved a group-cognitive result that most of the group members might not have been able to achieve on their own. Based on a Vygotskian perspective, our CSCL approach to the teaching of geometry involves collaborative learning mediated by dynamic-geometry software—such as Geometer’s Sketchpad or GeoGebra—and student discourse. During the past decade, we have developed the Virtual Math Teams (VMT) environment and have recently integrated a multi-user version of GeoGebra into it (Stahl, 2009; Stahl et al., 2010). Our environment and associated pedagogy focus on supporting collaboration and fostering significant mathematical discourse. In developing this system, we have tested our prototypes with various small groups of users. Recently, two small groups worked together on a problem based on the construction of inscribed equilateral triangles (see Figure 1). The geometry problem is adapted to the VMT setting from (Öner, 2013). In her study, two co-located adults were videotaped working on one computer screen using Geometer’s Sketchpad. We have “replicated” the study with teams of three adults working on separate computers with our multi-user version of GeoGebra in the VMT environment, allowing them to construct, drag, observe and chat about a shared construction. Öner chose this problem because it requires students to explore a dynamic-geometry figure to identify dependencies in it and then to construct a similar figure, building in such dependencies. We believe that the identification and construction of geometric dependencies is central to the mastery of dynamic geometry (Stahl, 2012b; 2013). In this study, we analyzed the processes through which the two groups (A and B) identified and constructed the dependencies involved in an equilateral triangle inscribed in another equilateral triangle. We were able to replay the entire sessions of the groups in complete detail, observing all group interaction (text chat and dynamic-geometry actions) that group members observed—for logs and analysis, see (Stahl, 2013, Ch. 7). Group A went through a collaborative process in which they explored the given figure by varying it visually through the procedure of dragging various points and noticing how the figure responded. Some points could move freely; they often caused the other points to readjust. Some points were constrained and could not be moved freely. The group then wondered about the constraints underlying the behavior. They conjectured that certain relationships were maintained by built-in dependencies. Without having figured out the constraints completely, they began trying to construct the figure as a way of exploring approaches experimentally using trial and error. Finally, the group figured out how to accomplish the construction of the inscribed equilateral triangles by defining the dependencies into their figure using the tools of GeoGebra. Team B went through a similar process, with differences in the details of their observations and conjectures. Interestingly, Team B made conjectures leading to at least three different construction approaches. Like Group A, they initiated a collaborative process of exploring the given diagram visually with the help of dragging points. They developed conjectures about the constraints in the figure and about what dependencies would have to be built into a construction that replicated the inscribed equilateral triangles. They decided to Figure 1. Discussion of the inscribed triangles problem. CSCL 2013 Proceedings Volume 2: Short Papers, Panels, Posters, Demos, & Community Events