Mathematical reasoning required when students seek the original graph from a derivative graph

Background: Finding the original graph when given the derivative graph is not a trivial task for students, even though they can find the derivative graph when given the original graph. Objective: In the context of qualitative research, this paper presents and analyses the mathematical reasoning that comes to light when the students seek the original graph from a derivative graph. Design: The research is assigned as a qualitative study, where the analyses of cases aim to extend understanding with respect to some phenomena or theory. Setting and participants: The study was conducted with 86 students from a State University in East Java. We conducted clinical interviews, and present data highlighting the reasoning participants used when solving tasks. Data collection and analysis: Task-based interviews were used to collect data, and data analysis was used to analyse interpretations of the graphs that emerged as mathematical reasoning models. Results: From our data analysis, we found that three mathematical reasonings were rooted in students’ awareness of problem-situations on graphs we provided, consisting of direct reasoning, reversible reasoning, and combined direct-reversible reasoning. Conclusions: We suggest that there are different mathematical reasonings in the construction of the original graph, due to the mental activity in which students use the relation between a function and its derivative. We suggest that future projects continue this inquiry with rigorous single-subject experiments with students