Discourse

Tasks can be vehicles for productive mathematical discussions. How to support such discourse in collaborative digital environments is the focus of our theorization and empirical examination of task design that emerges from a larger research project. We present our task design principles that developed through an iterative research design for a project that involves secondary teachers in online courses to learn discursively dynamic geometry by collaborating on construction and problem-solving tasks in a cyber learning environment. In this study, we discuss a task and the collaborative work of a team of teachers to illustrate relationships between the task design and productive mathematical discourse. Implications suggest further investigations into interactions between characteristics of task design and learners mathematical activity.

To contribute to understanding how teachers can develop geometrical understanding, we report on the discursive development of teachers’ geometrical reasoning through instrument appropriation while collaborating in an online dynamic geometry environment (DGE). Using the theory of instrument-mediated activity, we analysis the discourse and DGE actions of a group of middle and high school mathematics teachers who participated in a semester-long, professional development course. Working in small teams, they collaborated to solve geometric problems. Our results show that as teachers appropriate DGE artifacts and transform its components into instruments, they develop their geometrical knowledge and reasoning in dynamic geometry. Our study contributes to a broad understanding of how teachers develop mathematical knowledge for teaching.

Rich tasks can be vehicles for productive mathematical discussions. How to support such discourse in collaborative digital environments is the focus of our theorization and empirical examination of task design that emerges from a larger research project. We present the theoretical foundations of our task design principles that developed through an iterative research design for a project that involves secondary teachers in online courses to learn discursively dynamic geometry by collaborating on construction and problem-solving tasks in a cyberlearning environment. In this study, we discuss a task and the collaborative work of a team of teachers to illustrate relationships between the task design, productive mathematical discourse, and the development of new mathematics knowledge for the teachers. Implications of this work suggest further investigations into interactions between characteristics of task design and learners mathematical activity.

We report on a case study that seeks to understand how teachers’ pedagogical interventions influence students’ instrumentation and mathematical reasoning in a collaborative, dynamic geometry environment. A high school teacher engaged a class of students in the Virtual Math Teams with GeoGebra environment (VMTwG) to solve geometrical tasks. The VMTwG allows users to share both GeoGebra and chat windows to engage in joint problem solving. Our analysis of the teacher’s implementation and students’ interactions in VMTwG shows that his instrumental orchestration (Trouche, 2004, 2005) supported students’ instrumentation (Rabardel & Beguin, 2005) and shaped their movement between empirical explorations and deductive justifications. This study contributes to understanding the interplay between a teacher’s instrumental orchestration and students’ instrumentation and movement towards more deductive justifications.