Teaching and Learning of Geometry as a process of Objectification: conditions and obstacles to argumentation and proof. The role of natural language, specific language, and figures

This paper examines some examples (taken from research conducted over the years) that show students’ linguistic attitudes in geometry tasks. The examples are framed within the Theory of Objectification with reference to the notion of sensuous cognition, semiotic means of objectification and levels of generality. We show the struggle students live, at higher levels of generality, in intertwining natural language, specific language and the spontaneous use of geometrical figures, bound to perception and kinaesthetic activity. Within the networking paradigm, we coordinate the Theory of Objectification and Duval’s semio-cognitive approach to frame the interplay between the ideal and the material that occurs in geometrical argumentations and proofs.

Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students.

Relationship between drawing and figures on students’ argumentation and proof

In this article, we wish to explore the influence of the figure and the drawing on students’ argumentation and proof during the solving problem. Our research is based on both Toulmin’s model and Vinner’s concept image and concept definition. After analysing the arguments with the Toulmin model, we analyse the personnel concept definition, concept image evoked about the figure and the effect of drawing which intervene in students’ arguments. Our data suggest that wrong student’s arguments seem to be based on their concept image evoked on the figure manipulated which is in contradiction with the formal axiomatic system. Moreover, the data of the arguments seem to come from an abusive interpretation of the drawings. Observations that emerge from students’ oral and written speeches reveal continuities and gaps between their argumentation and proof.