Construction of the Mathematical Meaning of the Function–Derivative Relationship Using Dynamic Digital Artifacts: a Case Study

This article describes construction processes of mathematical meaning of the function–derivative relationship, as it is studied graphically with a dynamic digital artifact. The discussion centres on a case study involving one student during his interaction with the artifact. He was asked to explain the connection between two linked dynamic graphs: the graph of a function and the graph of its derivative function. The study was guided by the semiotic mediation approach, which treats artifacts as fundamental to cognition and views learning as the evolution from meanings connected to the use of a certain artifact to those recognizable as mathematical, that is, connected directly to the mathematical object. In the course of three rounds of data analysis, the student was shown to progress from a point-specific view to an interval one, and to move toward a construction of the meaning of the derivative as a function. The actions of the student and his interactions with the artifact that enabled him to construct the mathematical meanings of the function–derivative relationship are identified and described.