Graduate Theses, Dissertations, and Problem Reports 2019 Analyzing Mathematicians' Concept Images of Differentials Timothy Shawn McCarty [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Other Mathematics Commons Recommended Citation McCarty, Timothy Shawn, "Analyzing Mathematicians' Concept Images of Differentials" (2019). Graduate Theses, Dissertations, and Problem Reports. 4062. https://researchrepository.wvu.edu/etd/4062 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. ANALYZING MATHEMATICIANS' CONCEPT IMAGES OF DIFFERENTIALS Tim McCarty Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Vicki Sealey, Ph.D., Chair Jessica Deshler, Ph.D. John Goldwasser, Ph.D. Nicole Engelke Infante, Ph.D. John Stewart, Ph.D. Department of Mathematics Morgantown, West Virginia 2019 Keywords: Differentials, Concept Image, Calculus, Derivative, Integral Copyright 2019 Tim McCarty ABSTRACT Analyzing Mathematicians' Concept Images of Differentials Tim McCarty 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 푑푥 and Δ푥, 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. iii DEDICATION To my mother, Betty McCarty, whose love, support, and help has been a precious gift, not just during the creation of this work, but throughout my entire life. This dissertation is as much a tribute to her as it is a requirement for me. iv ACKNOWLEDGEMENTS Thanks to my family: Dad, Donna, Terry, Jereme, John, Cindy, Colsen, and a myriad of aunts, uncles, and cousins. Having your love and support throughout the years has helped me get this far – may this love and support continue beyond this dissertation! A very special thanks to my advisor, Dr. Vicki Sealey, who was a constant source of positivity and encouragement through my ever-changing, sometimes-depressed-and-sometimes- euphoric moods, through numerous meetings and discussions, through countless permutations of the data set that were always “almost, but not quite correct”, and through many, many gold stars. May I be as helpful to my future students. Thanks to the other members of my dissertation committee: Dr. Jessica Deshler, for her support during my early semesters of research and her work with the GTA’s, Dr. John Goldwasser for his helpful comments and for stepping in at the last minute, Dr. Nicole Engelke Infante, for allowing me into some of her research, teaching me a lot about everything, and always having answers to my many and varied questions, and Dr. John Stewart, for bringing to our committee discussions a “Physics” perspective and many thoughts I would never have otherwise considered. A special thank you to the wonderful people at Scott’s Run Settlement House: Deb, Ira, Shay, Courtney, Becca, Pete, and Jen. All I needed from you was somewhere to go get volunteer hours for my fellowship – but in addition to that, you gave me a safe haven full of love, support, and friendship and the opportunity to change myself and make me a better man by helping you in your charitable work. You all saved my life and I will never forget it. Additional thanks to my volunteer buddies Erina, Julie, Kristen, Maverick, P.J., Chuck, Vicki, Allyson, Allisynne, Isabella, and Juliana – you made this past year of volunteering so much better! v My thanks to my WVU RUME cohort: the “integration gang” Krista, Cody, Abby, and Olga, Keith and Ming for being conference roommates and sounding boards, Jenny, Samantha, Jaylyn, Heather, and Morgan for being parts of great RUME classes and/or research groups, and special thanks to Josh, who was a good friend to me, and whose intelligence and enthusiasm and excitement for my work (which sometimes dwarfed my own!) inspired me to always do better. My thanks to all those who helped me begin and continue my climb from the lowest places: Jimmy for his initial love, help, and support, and the members of my anonymous support group who gave me brotherhood (and sisterhood), advice, and a judgement-free space in which I was safe to open myself up completely so that true healing could begin. I love you all. To anyone whom I have ever called “friend”: I can’t list all of you, because I know I would forget some people, but I love you all as well. To my shmoo … my Baby Doll … my Lucky Charms … you know I almost always don’t have the correct words to express my feelings, but I hope that you always know these feelings anyway. You helped me complete my climb by trying the crazy idea of loving and accepting me as I am, warts and all. It is truly humbling, and I hope that I have loved you the same way. Sure, I win at the game of “creating a dissertation,” but you win at everything else! And all the Glory to my Father in Heaven. If a middle-aged, barely-mathematically-literate goofball who spent his formative years studying music could actually survive graduate school, earn a master’s degree, gain admission to a Ph.D. program, and successfully complete this dissertation, than I guess I can do all things1, indeed! 1 Referencing Philippians 4:13: I can do all things through Christ who strengthens me. vi TABLE OF CONTENTS ABSTRACT . ii DEDICATION . iii ACKNOWLEDGEMENTS . iv LIST OF TABLES . ix LIST OF FIGURES . x 1. INTRODUCTION . 1 Research Questions . 3 2. LITERATURE REVIEW . 4 Infinitesimal-Based and Limit-Based Calculus: A Brief History . 4 Some Conceptualizations of Differentials Found in the Literature . 6 Differentials in Definite Integrals . 6 Differentials in Leibniz Derivative Notation . 9 Other Contexts . 11 Differences in “Mathematics” and “Physics” differentials . 11 Notable conceptualizations in some textbooks . 14 Differentials as algorithmic tools . 17 Differential Conceptualizations in Randomly-Selected Textbooks . 17 Summary of the Literature Review . 21 3. THEORETICAL PERSPECTIVE . 22 4. PREVIOUS WORK . 26 Exploratory Study . 26 Methods . 26 Data . 28 Results . 30 Pilot Study . 32 Methods . 32 Data and Results . 34 Summary of the Previous Work . 39 5. METHODS . 40 Data Collection . 40 Interview subjects and consent . 40 Interview protocol . 43 Introductory questions . 43 The questions containing differentials . 44 Two ancillary sets of questions . 45 vii Data Analysis . 46 Thematic analysis . 46 Coding data points . 47 Data analysis for the second round of interviews . 54 Visual Representations of Data . 56 6. DATA AND RESULTS . 61 The Flowchart and the Final Tier 1 Themes . 61 Descriptions of Each Tier 1 Theme . 61 Themes N.4 and L.4 . 63 Themes N.4a and L.4a . 63 Themes N.4b and L.4b . 64 Themes N.4c and L.4c . ..