Calculus students’ quantitative reasoning in the context of solving related rates of change problems

Abstract

ABSTRACTDespite the increasing amount of research on students’ quantitative reasoning at the secondary level, research on students’ quantitative reasoning at the undergraduate level is scarce. The present study used task-based interviews to examine 16 high-performing undergraduate calculus students’ quantitative reasoning in the context of solving three related rates of change problems-two geometric and one non-geometric. A qualitative analysis of verbal responses and work written by the students when solving the three problems revealed that 15 students created and used diagrams to support their quantitative reasoning in the geometric problems, and that the creation of these diagrams helped the students to solve the two problems. In addition, seven students exhibited lack of facility with the product or quotient rule of differentiation. We argue that the use of diagrams in calculus instruction at the undergraduate level could play an important role in supporting students’ quantitative reasoning in topics that require students to make sense of relationships between quantities such as related rates of change problems. Furthermore, we argue that helping students develop greater facility with rules of differentiation such as the product or quotient rule could increase students’ success in many calculus application problems, including related rates of change problems. Directions for future research are included.