Samspel mellan intuitiva idéer och formella bevis : en fallstudie av universitetsstudenters arbete med en analysuppgift

Sammanfattning: The aim of this study is to illuminate the interplay between intuitive ideas and formal justification. In a perspective where we focus on the students’ competences we describe the interaction between intuitive ideas and formal structures, as they appear especially in relation to concept definitions and formal proofs. A group of university students have been working on a task in calculus. The task includes the concepts function and derivative and requires of them to use a proof by induction. The discussion between the members of the group has been analyzed in accordance with the principles of intentional analysis, a method by which we regard the students’ activities as intentional. This method of analysis makes the process of interpretation visible. It also makes it possible to make explicit the competences of the students’ and the mathematical content of their actions. In the study examples are given of the interplay between the intuitive ideas and the formal structures and also of the function for this interplay. The students have rich concept images and access to intuitive ideas relevant to the concepts brought to the 4 fore by the task. During the group discussion all components of a complete proof are included in the students’ reasoning. The students create a proof by induction which matches the ordinary pattern for such a proof, but they do not themselves regard it as a proof fitting into the ordinary scheme of argumentation, as they remember it from text-books and teaching. The students put heavy demands upon the formalization of their ideas and these demands are sometimes hampering the problem solving process, but they also encourage the students to expand their search for a solution to the problem at hand.