Opportunities for pre-structuralist reasoning at the crossroads from calculus - a case study on the equivalence between the continuity of a function and the path-connectedness of its graph
Résumé
Mathematical reasoning based on definitions and theorems and the attention towards generalisation are important for the transition from calculus to analysis or topology. We analyse proofs of a task about the equivalence of the continuity of a real-valued function on an interval and the pathconnectedness of its graph. In doing so, we examine elements of "pre-structuralist praxeologies" at the crossroads from calculus to more advanced courses. The proofs take pre-structuralist aspects into account in different ways. This holds for the mobilisation of levels of pre-structuralist praxeologies as a means of generalisation. Thus, we contribute to calculus education by focusing on the transition from a more intuitive to a more formal way of reasoning based on underlying structures about functions and graphs.
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