High school students’ learning trajectories of the concept of tangent line

The goal of this research is to characterize high school students’ learning trajectories of the concept of tangent line in a teaching experiment. The experiment considers a model of learning progression of the concept of tangent line using the local linearity of a function (Leibnizian model) to support the transition from the Euclidean conception to the Cartesian one. We identify three learning trajectories characterized by how high students manage both: i) the relation between the graphical and analytical registers to support the progression from the Euclidean conception to the Cartesian one through the Leibnizian model; and, ii) the approximation to the value of a function by the tangent line in the neighbourhood of the point of tangency. The findings suggest that the internalization of the Leibnizian conception is necessary to overcome the epistemological obstacle that the Euclidean model entails for the learning of the concept of tangent line.