Geometrical Reasoning in the Teaching and Learning of Integrals of Multivariable Functions

This study aims to address the teaching of integrals in multivariable calculus concerning the role taken by geometry, specifically, geometrical content dealing with boundaries in integrals that appear as curves and surfaces in R[superscript 2] and R[superscript 3]. Adopting the framework of the Anthropological Theory of the Didactic, we approached the problem by performing a praxeological analysis of typical tasks in a multivariable calculus course to describe the geometry involved, and to support the transition from textbook knowledge where geometry seems to be hidden. Our analysis reveals a collection of geometrical praxeologies that are used in the tasks but are not explicitly taught. Interviews with students show that we cannot consider that students are equipped with these praxeologies prior to taking the course on multivariable calculus.