Quantitative and Covariational Reasoning in Calculus Exploring Students’ Thinking About Functions and Derivatives

Abstract

This qualitative study investigated calculus students’ quantitative reasoning and covariational reasoning about functions that are situated in real-world non-kinematics contexts. Specifically, the study examined students’ thinking about the relationship between a function and its first derivative, second derivative, or the combination of the two derivatives. Furthermore, the study examined students’ thinking about open intervals where a function is positive or negative. Analysis of student work and verbal responses given by students while working on two tasks revealed that a majority of the 10 students who participated in the study were able to make sense of the relationship between a function and its first derivative. However, most of the students demonstrated a limited understanding of the relationship between a function and its second derivative. Additionally, most of the students exhibited a limited understanding of what it means for a function to be positive or negative on an open interval. Taken together, the students exhibited weak quantitative and covariational reasoning abilities. This is particularly concerning because calculus often serves as a gateway course for many STEM (Science, Technology, Engineering, and Mathematics) majors. Specifically, weak quantitative and covariational reasoning abilities such as those exhibited by the students in the current study could not only lead to high failure rates in calculus, but also to attrition from STEM majors. Implications for calculus instruction are included.