The role of representations in learning the derivative

Markus Hahkioniemi The role of representations in learning the derivative. University of Jyvaskyla. Department of Mathematics and Statistics. Report 104. It has been proposed that there are two ways how learning a mathematical concept may develop. A new object may be abstracted from actions performed on already existing objects. For example, the derivative concept may be abstracted from calculating values of the derivative. Another way is to act with the concept to be learnt and perceive it as an object. For instance, some properties of the derivative may be learnt by perceiving the derivative of a function from the graph of the function. These two ways correspond to learning in the symbolic and in the embodied worlds in the theory of the three worlds of mathematics. Several studies have suggested that the learning results of the derivative are enhanced if teaching takes into account, for example, working with several representations including the graphical ones, considering the limiting process inherent in the derivative thoroughly, supporting the process-object development and, in general, emphasizing problem solving. However, there is still need for a detailed analysis on how students are thinking about the derivative in approaches which takes into account the above-mentioned suggestions. The aim of this study is to find out how students may use different kinds of representations for thinking about the derivative in a specific approach. To achieve this, I designed and implemented a five-hour teaching-learning sequence introducing the derivative concept in a Finnish high school (grade 11). The above-mentioned aspects of learning were taken into account in the design. After the teaching-learning sequence, I selected five students into carefully designed task-based interviews. From the interviews I analyzed what kind of representations the students used for thinking about the derivative and for which purpose and how they used these. Especially, using limiting processes and perceiving the derivative from the graph of a function were taken into account in the design of the interviews as well as in the analysis. I found that the embodied world offered powerful thinking tools for the students. They used the increase, steepness, horizontalness and tangent of the graph for thinking about the derivative qualitatively without calculating anything. These representations were accompanied by gestures which were an essential part of thinking. At this very early stage of learning the derivative the students seemed to consider the derivative as an object, which has some properties, in the embodied world. Using the above-mentioned representations, they, for example, considered when the derivative is positive/negative, and what the maximum/minimum point of the derivative is. Therefore, this study supports the claims that learning may begin by considering the derivative as an object. The study also suggests that in the embodied world students may learn as the representations become more and more transparent allowing seeing the derivative. The students used various kinds of limiting processes and connected them in different ways to the limit of the difference quotient. Some of the students changed from one of these two representations to the other, and some of them explained one with the other. I named the two connections associative and reflective connections, respectively. One of the students, who made the reflective connection, had major difficulties in using the limit of the difference quotient. This suggests that a student may have conceptual knowledge of this notion without being able to use it for calculating the derivative. On the basis of the analysis of the students’ use of representations, I constructed a hypothetical learning path to the derivative. According to the learning path, the representations of the tangent, increase, steepness and horizontalness of the graph as well local straightness, moving a hand along the graph and placing a pencil as a tangent may be used to perceive the rate of change in the embodied world. In the symbolic world, students may calculate the average rate of change over different intervals. This way, students may build knowledge of the derivative even before being introduced to its definition and they have readiness to investigate the problem of the value of the instant rate of change.