Students’ Smooth Continuous Covariational Reasoning: A Comparative Case Study

Abstract

Research results from this study reveal students have difficulties understanding and using of the concepts of average rate of change and the derivative function. Students in this study held multiple approach to understand the concepts that made it difficult to develop a strong understanding of the average rate of change and derivative function. In particular, students struggled to visualize or imagine a continuously varying rate of change and had difficulties in making meaning and interpreting concepts of average rate of change and derivative function. This dissertation presents research on how first-year calculus students develop smooth continuous covariational reasoning abilities in the context of the concepts of rate of change and derivative functions. This study utilizes a comparative case study methodology to explore each research participant’s construction of understanding and reasoning pattern development. An initial instructional sequence was designed to support Calculus I students in constructing understandings of average rate of change and derivative function. Students were then supported in reasoning about how two quantities vary and co-vary dynamically. The instruction supported students’ reasoning abilities when solving problems related to the concept of average rate of change and derivative function in linear and nonlinear function situations. The research findings show that the study participants demonstrated different types of reasoning to conceptualize the concept of quantity, variation, and covariation when solving mathematical problems related to the concept of average rate of change and derivative function. Sam, one of the study’s participants, demonstrated strong concrete object-oriented reasoning to conceptualize the average rate of change and derivative function. Another study participant, Ruby, engaged in procedure-oriented reasoning to conceptualize the average rate of change and derivative function. Chris, the third study participant, engaged in terminology-oriented reasoning to conceptualize the average rate of change and derivative function. The analysis of the data results of this study shows in detail how these three types of reasoning were a limitation for the participants’ mathematical problem-solving ability and conceptualizations of covariation, average rate of change, and the derivative function. This study uncovered the above three types of problematic reasoning orientations as it relates to covariational reasoning and learning average rate of change and derivative, but these types of reasoning orientations are most certainly not the only types of problematic reasoning orientations for Calculus I students —there are likely other problematic reasoning orientations that might be discovered in future studies.