Analyzing Mathematicians' Concept Images of Differentials

Summer

2019

Dissertation

PhD

College

Eberly College of Arts and Sciences

Mathematics

Vicki Sealey

Jessica Deshler

Jessica Deshler

John Goldwasser

Committee Member

Nicole Engelke Infante

John Stewart

Abstract

The differential is a symbol that is common in first- and second-year calculus. It is perhaps expected that a common mathematical symbol would be interpreted universally. However, recent literature that addresses student interpretations of differentials, usually in the context of definite integration, suggests that this is not the case, and that many interpretations are possible. Reviews of textbooks showed that there was not a lot of discussion about differentials, and what interpretations there were depended upon the context in which the differentials were presented. This dissertation explores some of these issues. Since students may not have the experience necessary to build their own interpretations totally free of their instructors’ influences, I chose to interview experienced mathematicians for their differential interpretations. Most of the current literature involves the differential within the context of definite integrals; my work expands on this literature by exploring additional expressions that contain differentials. The goal was to build a dataset of multiple instructors’ interpretations of multiple differentials to see how uniform those interpretations were.

Initial interviews discussing five expressions which contained differentials, three contexts in which most of these expressions were used, and auxiliary questions that asked the meaning of “differential,” the differences between and , and the interpretation of phrases used to describe infinitely small quantities were conducted with seven expert mathematicians from a large research university. By analyzing the responses given by these mathematicians, two lists of themes were created: one based on remarks that address the quality of the differential directly, and one based on remarks that address one’s feelings about differentials. In addition, for the responses that address differentials directly, a flowchart was created to guide each of these responses to its proper theme. After the creation of these lists, three more mathematicians were interviewed to ensure that the theme lists would still be valid outside of the interviews used to create them.

Not only was no overall formal concept image for the differential found, but many different and sometimes contrasting themes were found within each interview subject’s personal concept image. A framework for categorizing the multiple conceptualizations that were found for the differentials themselves was created, as well as a beginning list of ancillary themes that address possible thoughts about and uses of differentials. The dissertation concludes with a list of possible teaching implications that might arise from the existence of multiple differential conceptualizations, as well as some suggested future research that might expand upon this work.

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