Application of the complementarities of two theories, APOA and OSA, for the analysis of the university students' understanding on the graph of the function and its derivative

The goal of this paper is the combined use of two theories, APOS and OSA, for the analysis of the university students' understanding on the graph of the function and its derivative. For this, we study the students' understanding to solve one graphing problem in relation to the first derivative and characterize their schemas in terms of levels (intra, inter and trans) of development of the schema for sketching f' when given the graph f. We present a multiple case study in which 14 students of the first course of Calculus in one university of Iran participated voluntarily. Results show that most of the students in our study had major problems in developing mental constructions and doing the practical work needed to solve the problem, particularly those mental constructions that have to be made to calculate the derivative at the critical points and to determine the speed of the variation of the inclination of the tangent lines to f, which is why most of them have constructed a schema at the intra level of development of the schema for sketching f' when given the graph f. We finish with some final conclusions