Conversations with Materials: Contributions to Collective Argumentation

In this case study, we explored how a team of two students engaged in collective argumentation (Krummheuer, 2023), using a life-size square, to determine partitions of the group 𝐷₄ (symmetries of the square) via group actions. A micro-analysis of conversations with materials (Bamburger & Schön, 1983) allowed us to document the nature of the processes that students adopted as they engaged in collective argumentation. Specifically, we were able to determine how the students arrived at their data, to make a claim, and to state a warrant. Our results suggest that when working with rotation elements, the students required minimal verbal conversation with one another. Conversations with inscriptions via gestures were sufficient for the students to determine two of the cosets. On the other hand, the students required conversations with the life-size square, inscriptions, and their own gestures to arrive at the remaining two cosets. Surprisingly, minimal verbal conversations took place even with the reflection elements and data for their argumentation manifested as the aforementioned conversations. As part of this presentation, I will offer teaching and research implications as they relate to abstract algebra and collective argumentation.