From Interpretations to Graphical Representations: A Case Study Investigation of Covariational Reasoning

Abstract

Problem Statement: The idea of change - both how things change and at what rate things change with respect to each other - is fundamental to a study of calculus, which is a critical course for students majoring in mathematics, sciences, engineering, business and several other majors. A lack of ability to reason about change in continuously changing functional relationships may cause difficulties in learning basic calculus concepts such as limits and derivatives. Despite a variety of research studies that emphasized the effects of students' understanding of rate of change on understanding of calculus concepts such as limits, derivatives, and integrals, there is little information about how college students reason about continuously changing functional relationships. Purpose of Study: The aim of this study is to explore, describe and analyze college students' covariational reasoning abilities. More specifically, this study investigates and provides a "thick" description, or which explains a behavior and its context, of how a college student uses understanding and reasoning to interpret a functional situation and uses these interpretations to demonstrate the covariation of two variables in graphical representation.

Methods: Case study design and techniques were used in this study to provide a thick description of Karl's thinking and reasoning processes in order to comprehensively understand his covariational reasoning. Data were obtained from a detailed examination of student's thinking and reasoning processes through the task based in-depth clinical interviews. Data obtained from students' verbal expressions and graphical representations were analyzed in light of the theoretical lens. The "Covariation Framework" provided a skeletal structure for the description and interpretation of findings.

Findings and Results: Analysis of data disclosed that conceiving of a functional situation statically leads to difficulties in coordinating the continuously changing rate of change over the entire domain. Students' strong procedural tendency hinders meaningful interpretations and reasoning. Lack of transformational reasoning appears to prevent forming an image of the dynamically changing event and foster dependence on the procedural steps. Reasoning based on irrelevant details and arguments leads to either erroneous or pseudo-analytical conclusions about the simultaneous changes of two variables.

Conclusion and Recommendations: Instead of introducing the concept of function as correspondence, which is more traditional and concentrated around the application of certain rules and formulas, introducing the functions as covariation will be more helpful for students to develop a better conceptual understanding. Utilizing computer technology such as dynamic software in classroom instruction may provide more visual representations in order to enhance students' conceptualization of the changing nature of functions.