Collective Mathematical Progress in an Introductory Calculus Course during the Treatment of the Quadratic Function

Abstract

This study characterizes the classroom mathematical practices that support the collective mathematical progress during the treatment of a quadratic function in an Introductory Calculus course comprised of 15 students from eleventh grade (from 16 to 18 years old). A teaching experiment that discussed two variational situations was designed for the treatment of the quadratic function. This paper only reports data of one these situations. The analysis of the empirical data involved an analytical approach that used Toulmin’s model of argumentation and two criteria to inform the classroom mathematical practices. Results correspond with two mathematical practices that supported a collective mathematical progress: 1) analyze and quantify what changes in a variational situation; and 2) generate and interpret representations of how the variable magnitudes change in a variational situation. Three connected elements were identified in the evolution of the mathematical practices: time, structure and mathematics. These elements enabled the establishment of a model of collective mathematical progress based on empirical evidence; this model documents a path students could follow to learn quadratic function. The proposed model has the quality of being explicative and the ability to adjust to interact with other students.