Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative

Abstract

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 ′ when given the graph . 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 , which is why most of them have constructed a schema at the intra level of development of the schema for sketching ′ when given the graph . We finish with some final conclusions.

Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

“…If a function is not presented in equation form, and the values of the function are presented in a Table (Question 3) many students cannot estimate the value of the derivative at a point with the use of the data from that table and with the use of the limit of the sequence of difference quotients, Badillo et al, 2011;Sánchez-Matamoros, Fernández, & Llinares, 2015). Furthermore, this difficulty is seen when the function is only presented graphically (Question 4) and students had to sketch the graph of the derivative function (Borji, Font, Alamolhodaei, & Sánchez, 2018). In our case, most students of the control group had fundamental problems in sketching the graph of ′ (Ferrini-Mundy & Graham, 1994).…”

An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran

“…A number of students in the first year of the university have fundamental difficulties in solving graphical and numerical questions related to derivatives and integrals (Borji, Font, Alamolhodaei, & Sánchez, 2018;Baker et al, 2000;Cooley, Trigueros, & Baker, 2007). One of the reasons may be that they did not well understand these concepts in different representations during the high school.…”

“…For this reason, students in the first year of the university may have fundamental difficulties in solving non-algebraic derivatives and integrals' questions. Students in the first year of the university face difficulties in solving the calculus questions where input information is given only through graphical or numerical representations (Borji, Font, Alamolhodaei, & Sánchez, 2018;Baker et al, 2000;Cooley et al, 2007). Many research have recommended that, in order to achieve a conceptual understanding of a mathematical concept, it is better to start learning that concept in its various representations and connections between them (Baker et al, 2000;Borji, Alamolhodaei, & Radmehr, 2018;Dreher & Kuntze, 2015;Goldin & Shteingold, 2001;Kendal & Stacey, 2003;Özmantar et al, 2010;Pape & Tchoshanov, 2001;Ronda, 2015).…”

“…A number of students do not have a conceptual understanding of graphical and numerical representations of calculus concepts (Kendal & Stacey 2003). Students in the first year of the university have difficulty in understanding the relationship between different representations of a mathematical concept (Borji, Font, Alamolhodaei, & Sánchez, 2018;Dreher & Kuntze, 2015). One of the reasons for these difficulties faced by students may be that they have not learned well the relationship between different representations of a calculus concept in the last years of high school (Mao, White, Sadler, & Sonnert, 2017).…”

Una introducción al concepto de derivada en estudiantes de bachillerato a través del análisis de situaciones de variación

“…Existen diversos trabajos de investigación dedicados a explorar el concepto de derivada, en su mayoría, con estudiantes de licenciatura que ya han tenido contacto con este concepto (e.g. Borji, Font, Alamolhodaei y Sánchez, 2018). En Educación Matemática, particularmente, dichas investigaciones analizan y/o promueven la comprensión del concepto a partir de distintos ámbitos según sus intereses (e.g.…”