Establishing links between conceptions, argumentation and proof through the ck¢-enriched Toulmin model

• We analyze the interaction between students’ conceptions and their argumentation and proof.
• Students’ conceptions impact argumentation and proof
• The enrichment of cK¢ model in Toulmin model is a useful methodological tool to analyze students’ knowledge system
• There is a strictly connection between conception's components and the warrant and the backing in the argument

In this article we analyze students’ conceptions in geometrical problem-solving and their relations to proving. The main purpose of this paper is to show how students’ conceptions strongly impact the argumentation activity and the construction of a proof. This is illustrated by analyzing two pairs of students’ argumentations and proofs taken from a set of data collected from a teaching experiment. Through the use of Toulmin's model enriched with the ck¢ model, we show the complexity of a cognitive analysis of argumentation and proof that accounts for the students’ knowledge system. Toulmin's model is useful to select those elements in the argumentation that are part of students’ conceptions while ck¢ allows us to see the role they have inside the argumentation.