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Analyzing Applied Calculus Student Understanding of Definite Integrals in Real-Life Applications

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Analyzing Applied Calculus Student Understanding of Definite Integrals in Real-Life Applications Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation HOOD, CODY, "ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS" (2021). Graduate Theses, Dissertations, and Problem Reports. 8260. https://researchrepository.wvu.edu/etd/8260 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]. Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS Cody Hood 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 Harvey Diamond, Ph.D. Erin Goodykoontz, Ed.D. Nicole Engelke Infante, Ph.D. Virginia Kleist, Ph.D. Department of Mathematics Morgantown, West Virginia 2021 Keywords: Applied Calculus, Calculus, Definite Integrals, Guided Reinvention Copyright 2021 Cody Hood ABSTRACT Analyzing Applied Calculus Student Understanding of Definite Integrals in Real-Life Applications Cody Hood An individual’s knowledge of definite integrals can range from rote memorization to a strong foundational connection harkening back to its Riemann sum limit definition, ! # ∫ �(�)�� = lim ∑&'( �(�&)Δ�. In my research, I conducted seven task-based face-to-face " #→% interviews with Applied Calculus students. Through the use of real-life examples and guided reinvention, I analyzed ways in which these students, who all initially demonstrated rote memorization, could exhibit a Riemann sum based level of comprehension. This research was conducted in the confines of a student population with definite integral experience, but no formal instruction on limits. My results show that the lack of computational emphasis in class produced little to no restrictions in student development of a Riemann sum based understanding of the definite integral. Emphasis on units facilitate this evocation of a Riemann sum definition while attempting to rationalize the use of a definite integral in an application context. The use of units also aided in the assigning of meaning to the individual symbols that construct the definite integral. By applying meaning to each symbol in the decomposition of the definite integral, students were able to better rationalize the connections the integrand, differential, and their multiplicative underpinnings have to a Riemann sum. I also demonstrate that the limit’s role in student understanding of the definite integral should be dichotomized into approximation and exact sublayers. My work culminates with a sample lesson on definite integrals, which places primary focus on using population growth to elicit connections between the definite integral and its Riemann sum definition through the use of unit transformation and graphical representations. iii DEDICATION To my daughters, Kiley and Zoey Without them, this would have been finished much sooner iv ACKNOWLEDGEMENTS Thank you, Dr. Vicki Sealey, for your impeccable balance of support, motivation, advice, and encouragement. Even though there were times where my progress felt like it slowed to a halt, you were always more than willing to provide feedback and were the beacon of optimism I needed to push through to the finish. You may have felt at times like you provided me nothing but criticism, but from where I’m sitting, you gave me nothing but commendation. There is no question I wouldn’t have been able to complete this journey without you and for that I am forever grateful. To the greatest person in my life, Jess, I will never be able to repay you for your patience over the past few years while I have been taking classes, conducting research, writing, writing, and writing. I am so thankful that we are still only at the beginning of our journey together. I can’t wait to see what the rest of our years together have in store for us. Mom and Dad, thank you for the lifelong support you have given me through my personal and professional life. If I ever needed an encouraging word, you all were always the first people I turned to. And Jesse, if it weren’t for you, I would have never begun my academic journey at WVU. I hate to admit it, but the role model and big brother you have been for me has led me to accomplish what I have in life. Thank you to all the members of my dissertation committee: Dr. Harvey Diamond, for your welcoming guidance as my professor and advisor during every level of my undergraduate and graduate career. Dr. Erin Goodykoontz, for never setting me up to fail and for being a friend beyond all else. Dr. Nicole Engelke Infante, for playing such a strong role in introducing me to the RUME community and teaching me so much in the early stages of my graduate work. Dr. Virginia Kleist, for being a sounding board outside the confines of my math community. The insight and perspective you brought to my committee helped shape my research to be accessible to everyone. v Thank you to all my classmates and colleagues, Krista, Tim, Keith, Renee, Matt, and Josh, that were always willing to share ideas and resources to make this process more manageable for everyone. And a special thanks to Doug Squire, for taking me under your wing early on. Your true love for teaching has made you my biggest role model for how I want to be as an educator. vi TABLE OF CONTENTS ABSTRACT ................................................................................................................................... ii DEDICATION ............................................................................................................................. iii ACKNOWLEDGEMENTS ........................................................................................................ iv LIST OF TABLES ....................................................................................................................... ix LIST OF FIGURES ...................................................................................................................... X INTRODUCTION ........................................................................................................................ 1 PAPER 1: STUDENT PROGRESS THROUGH THE LIMIT LAYER OF DEFINITE INTEGRALS IN THE CONTEXT OF AREA UNDER A CURVE ........................................ 7 Abstract ...................................................................................................................................... 7 Introduction ............................................................................................................................... 8 Literature Review ................................................................................................................... 10 Riemann Sum Concept Image .............................................................................................. 11 Two Common Images ........................................................................................................... 14 The Need for Multiple Images .............................................................................................. 16 Theoretical Perspective .......................................................................................................... 17 Understanding Concept Image .............................................................................................. 17 Realistic Mathematics Education (RME) ............................................................................. 19 Methods .................................................................................................................................... 20 Interview Subjects ................................................................................................................. 20 Data Collection ..................................................................................................................... 21 Interview Protocol ................................................................................................................. 22 Data Analysis ........................................................................................................................... 25 Thematic Analysis ...............................................................................................................
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