Abstract
Proof, a key topic in advanced mathematics, also forms an essential part of the formal learning experience at all levels of education. The reason is that the argumentation involved calls for pondering ideas in depth, organizing knowledge, and comparing different points of view. Geometry, characterized by the interaction between the visual appearance of geometric elements and the conceptual understanding of their meaning required to generate precise explanations, is one of the foremost areas for research on proof and argumentation. In this qualitative analysis of the arguments formulated by participants in an extracurricular mathematics stimulus program, we categorized students’ replies on the grounds of reasoning styles, representations used, and levels of generality. The problems were proposed in a lesson on a quotient set based on the similarity among triangles created with Geogebra and the responses were gathered through a Google form. By means a content analysis, the results inform about the reasoning style, the scope of the argumentation, and the representation used. The findings show that neither reasoning styles nor the representations used conditioned the level of generality, although higher levels of argumentation were favored by harmonic and analytical reasoning and the use of algebraic representations.
Highlights
Introduction to the lessonStartS13: harmonic proofR1: visual proof 1Pythagorean theorem in acute and obtuse triangles Drag tool Minute 29 1/group pooling Compare proofsUse Geogebra to visualize proof elementsFamiliarity with Geogebra 2/questionnaires
We explored the effect of reasoning styles and the representations used in argumentation on the level of generality of the justifications put forward by students in proof problems
Reasoning style alone was not observed to determine the level of generality, harmonic and analytical reasoning proved to be more favorable than visual reasoning to generalized argumentation
Summary
Many studies have endorsed the importance and utility of proof in mathematics education, which continues to be deemed a topic meriting further research [2]. Related terms sometimes used include explanation, justification, and argumentation [4,22,50]. They differ in nuance, regarding (for instance) the degree of conviction and rigor, which depends on the community at issue [51]. Argumentation is often found in a broader context of mathematical activity described with terms such as proof or reasoning, which may entail exploring examples or particular instances, formulating or refining conjectures, and putting forward arguments to establish such conjectures as proofs or using them as elements of proofs [20]. One criterion for an argument to be deemed a proof is that it must use statements, types of reasoning and of representation generally accepted in the classroom community’s conceptual environment [49]. The second requirement is that reasoning styles and the representations used by students to support their arguments are consistent with the elements proposed by Stylianides et al [20], to which we added the third criterion, the scope of argumentation/level of generality
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