The theory of calculus for calculus teachers

The aim of this paper is to narrate a significant part of the experience of a professor over several semesters teaching fundamental ideas of the Theory of Calculus. The narrative explains how his didactic and mathematical mentality evolved as he witnessed students' conceptual difficulties, as they were confronted with situations that generated tensions between their intuitions and the formal rigor of the theory of calculus. The evolving atmospheres in the classroom, the efforts to teach, and the deep obstacles to learning, indicate the existence of an epistemic fissure as one leaves the dynamic and geometric way of thinking and tries to adopt a mode of thinking developed from arithmetical rigor and the presence of infinity.