The linear motion as a scenario for addressing relations between a function and its derivative

Abstract

In this paper we present some results of a study conducted with 65 engineering students in a first Calculus course. SimCalc MathWorlds® software was included in the design of a didactic sequence to incorporate the scenario of the motion of an object over a horizontal straight line. The software document that was designed to work in classroom, plays an important role for analyzing the graphical behavior of position and velocity functions. By dragging the velocity graph, the software acts with the corresponding change in the position graph. The students used the software in their laptops by pairs during the two weeks didactic sequence. The aim of the sequence is to set up some relationships between the behavior of velocity graph and position graph. In a conventional curriculum those relations refer to the positive (negative) sign, and increasing (decreasing) behavior of derivative function, corresponding to the increasing (decreasing) and concave upward (downward) behavior of the function. Software brings the scenario for learning those facts analyzing the real context of linear motion. As part of the study, an assessment instrument was designed in order to appreciate the students' appropriation of those relations. The instrument' items are classified by corresponding to the linear motion context, or corresponding to different real contexts (no motion), or without including any real context. They also consider the posing information of the item and of the answer, being described by text or by a graphic. Application of the instrument lead us to reflect that, once the appropriation is achieved through the motion context, it could be easier for students to apply it without connection with a real context. It also reveals the difficulties for interpreting graphical information based on the derivative function. These findings are part of the overall results of a doctoral dissertation concerning with the use of digital technologies for the learning of Calculus. Keywords: Calculus learning, digital technologies, linear motion, real context, mediation.