Building an understanding of sketching function derivative graphs through the APOS approach

The derivative of a function is a basic concept in calculus with various applications in the STEM fields. A deep understanding of this concept is essential for students to face the challenges of analysis, optimization, and application in science and technology inquiries. However, the understanding of derivatives of functions among the population of prospective mathematics teachers in Indonesia is still low, specifically when it comes to sketching graphs of derivatives of functions and the application of the concepts. The APOS (Action, Process, Object, Schema) theory approach seems to hold a relevant solution needed to improve this understanding. Previous studies have shown that most students still have difficulty understanding the relationship between graphs and derivatives of functions. In the present study, APOS theory was used to analyze the subjects’ understanding of the concept of derivatives, especially related to sketching graphs of derivatives of functions by describing their mental structure and mechanisms completely. The results of the study revealed that the subjects' understanding had reached the Schema stage. However, there was no mental interiorization mechanism since the subjects had difficulty at the Process stage. Consequently, the subjects could not explain the steps to draw a graph of derivatives of functions from the given function graph. Finally, this study highlights the importance of more effective teaching strategies to strengthen students' conceptual understanding of derivatives of functions, especially sketching derivative graphs.