From visual reasoning to logical necessity through argumentative design

Abstract

Our main goal in this study is to exemplify that a meticulous design can lead preservice teachers to engage in productive unguided peer argumentation. By productivity, we mean here a shift from reasoning based on intuitions to reasoning moved by logical necessity. As a subsidiary goal, we aimed at identifying the kinds of reasoning processes (visual, inquiry-based, and deductive) pre-service teacher's students adopt, and how these reasoning processes are interwoven in peer-unguided argumentation. We report on a case study in which one dyad participating in a pre-service teachers program solved a mathematical task. We relied on three principles to design an activity: (a) creating a situation of conflict, (b) creating a collaborative situation, and (c) providing a device for checking hypotheses/conjectures. We show how the design afforded productive argumentation. We show that the design of the task entailed argumentation which first relied on intuition, then intertwined the activities of conjecturing and checking conjectures by means of various hypotheses-testing devices (measurement, manipulations, and dynamic change of figures with Dynamic Geometry software), leading to a conflict between conjectures and the outcome of the manipulation of DG software. Peer argumentation then shifted to abductive and deductive considerations towards the solution of the mathematical task. These beneficial outcomes resulted from collaborative rather than adversarial interactions as the students tried to accommodate their divergent views through the co-elaboration of new explanations.

Analyzing effective communication in mathematics group work: The role of visual mediators and technical terms

“…Radford, 2011;Valero & Stentoft, 2010), others examine how learning opportunities of content are established in interactions (e.g., Kieran, 2001; Nilsson & Ryve, 2010;Weber et al, 2008). Furthermore, several studies show positive effects of mathematical group work (e.g., Amigues, 1988;Coleman, 1998;Prusak, Hershkowitz, & Schwarz, 2012) while others find that students have substantial difficulties establishing productive group discussions (e.g., Barron, 2003McCrone, 2005Ryve, 2006;Sfard, 2001). In this article we focus on how students manage to establish effective group discussions about mathematical content.…”

Acting on Work Designs: Providing Support for Validation and Implementation of Envisioned Changes

“…As argued earlier, augmenting the design of stakeholder-centric modeling activities with a scaffolding perspective could help improve the understanding and creation of enterprise models in the target group. In the following, we therefore review scaffolding approaches proposed in other disciplines, which require skills similar to stakeholder-centric conceptual modeling, in particular with respect to articulation of abstract concepts describing real-world phenomena (Frederiks and van der Weide 2006) and the support of developing a common understanding about these phenomena (Prusak et al 2012). Based on these approaches, we develop a framework for scaffolding in enterprise modeling, which constitutes our nascent design theory.…”

“…They also suggest providing conceptual scaffolds in the form of additional resources on the topic of interaction and procedural scaffolds in the form of guiding structures (such as writing stems or diagram templates). Finally, the authors propose to work with heterogeneous groups, where participants have different views, to facilitate confronting argumentation (Prusak et al 2012). Ge and Land (2004) focus on the development of domain-specific question prompts to scaffold problem-solving in peer interaction settings.…”

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

“…Toulmin's model has been usefully adapted for analyzing argumentative discourse in mathematics (e.g., Boero, Douek, Morselli, & Pedemonte, 2010;Inglis, Mejia-Ramos, & Simpson, 2007;Krummheuer, 1995;Pedemonte, 2007;Prusak, Hershkowitz, & Schwarz, 2012;Rasmussen & Stephan, 2008). In our study, we use selected elements of Toulmin's model, namely, claim, data and warrant.…”