Interactions between teachers' use of collaborative, dynamic geometry environment and their geometrical knowledge

Dynamic geometry environments allow learners to manipulate mathematical objects and to explore their properties and relations among them. In such environments, learners can observe mathematical objects in different forms and observe changes to each form while manipulating the objects. These affordances can significantly support learning of geometry, especially when coupled with collaborative problem solving. Using Vygotsky’s (1978) notion of mediated activity and Rabardel and Beguin’s (2005) theory of instrumental genesis, this dissertation explores how such an environment influences the learning of geometry. It investigates how middle and high school teachers appropriate a collaborative dynamic geometry environment called Virtual Math Teams with GeoGebra (VMTwG) and how their appropriation shapes their geometrical knowledge. Over the last three years, 23 middle and high school mathematics teachers collaborated synchronously in VMTwG to solve open-ended geometrical tasks. In each year, different groups of mathematics teachers engaged in 15-week course and worked in small teams (2-4 teachers) to solve open-ended problems and construction tasks that a research team, including the researcher, from Rutgers University and Drexel University designed and revised. The data used for this dissertation include discursive and inscriptive interactions of teams of mathematics teachers in VMTwG. The VMTwG environment records users chat messages, the dynamic geometry software (GeoGebra) actions, and system actions. To understand how mathematics teachers appropriate a collaborative, dynamic geometry environment and how their appropriation shapes their geometrical understanding, the researcher conducted three interrelated studies to investigate the following: (1) how teachers appropriate the dragging feature of dynamic geometry environments and how their appropriation influences the discursive development of their understanding of dependencies among geometric objects; (2) longitudinally, how teachers appropriate a collaborative, dynamic geometry environment and how this appropriation shapes their actions when solving geometrical tasks; and (3) what mediational roles of the VMTwG environment are evident as teachers solve geometrical tasks. Analyses show the teachers’ appropriation and application of the dragging feature of VMTwG allowed them to understand and identify dependencies among geometrical objects. Relying on their instrumentation of technological and mathematical affordances of the environment, teachers explored geometrical objects and relations, conjectured about them, and justified their conjectures. Finally, analysis also shows that in addition to Rabardel and Beguin’s (2005) epistemic and pragmatic mediations, a third mediation, pedagogic mediation, was evident in teachers’ mathematical activities. For the mathematics education community, this study provides understanding of how teachers use technological tools. It also informs the design and implementation of instructional programs that engage learners with such tools to extend their mathematical knowledge. This study suggests that further research is needed to understand how to orchestrate learners’ instrumentation of collaborative technological tools in which teachers do not have a conventional instructional role in the classroom.Ph.D.Includes bibliographical referencesby Muteb M. Alqahtan