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An Abstract of the Dissertation Of AN ABSTRACT OF THE DISSERTATION OF Gregory Mulder for the degree of Doctor of Philosophy in Physics presented on February 26, 2021. Title: Coordinatization Activated: How Students Understand and Apply Coordinate- Independent Integral Equations to Physics Situations Abstract approved: ______________________________________________________ Elizabeth E. Gire Physics problems that require integration, such as finding the center of mass of an object or finding the electric field from a continuous distribution of charge, are one type of problem that is difficult for introductory physics students to solve. These problems require students to define a coordinate system in a physical situation in order to apply coordinate-independent equations. This process of converting coordinate-independent quantities into coordinate-dependent quantities I call coordinatization. In this dissertation, I explore how junior-level physics students coordinatize physics problems that involve integrating vectors. Using a resources theoretical perspective, I perform a qualitative thematic analysis on student written work and clinical problem-solving interviews. I identify 32 types of acts of coordinatization and organize these into 5 categories: drawing an axis, manipulating the infinitesimal, dealing with the vector nature of the integrand, including an infinitesimal in a sketch, and indicating limits of integration. Then, I examine a co- occurrence between indicating an infinitesimal and manipulating the infinitesimal. I also identify symbolic forms that students use while coordinatizing an integral. This coordinatization framework is a productive way to look at how students are using mathematics to understand physical systems. ©Copyright by Gregory Mulder February 26, 2021 All Rights Reserved Coordinatization Activated: How Students Understand and Apply Coordinate- Independent Integral Equations to Physics Situations by Gregory Mulder A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented February 26,2021 Commencement June 2021 Doctor of Philosophy dissertation of Gregory Mulder presented on February 26, 2021 APPROVED: Major Professor, representing Physics Head of the Department of Physics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Gregory Mulder, Author ACKNOWLEDGEMENTS The author expresses sincere appreciation to Dr. Elizabeth Gire and Dr. Corinne Manogue for the guidance, encouragement, and personal attention they have given to this study and the preparation of this thesis. Dr. David Roundy and Dr. Tevian Dray additionally provided great insights and encouragement. Additional appreciation is held for the Linn-Benton Community College community that supported me with a year sabbatical for this project. Bridgid Backus provided the initial peer pressure for me to go back to school. Ralph Tadday and Faye Melius took great care of our students while I was gone. The author is grateful to the inviting, inclusive and stimulating Oregon State University physics department. Emily van Zee, Paul Emigh, and Michael Vignal all provided excellent feedback while writing this dissertation. MacKenzie Lenz, Kelby Hahn, and Jonathan Alfson were instrumental in helping narrow my research question, providing insightful observations and being willing to collaborate on classroom activities – I appreciate their courage in using findings from this research in the classes that we have taught together. Finally, I am especially appreciative of my large supportive family that values knowledge and learning. One family member, my brother Eric Mulder, graciously provided formatting advice and assistance on this document. Also, my loving husband helped by reading drafts, providing ink and paper for the printer, and making many trips to Le Patissiér for pan au chocolat. TABLE OF CONTENTS Page Introduction ........................................................................................................................... 1 Math and concepts in word problems ................................................................................... 3 1.3 Math and concepts in physics problems ............................................................................... 4 1.4 Definition of coordinatize .................................................................................................... 5 1.5 Sophomore-level physics to junior-level physics transitions in starting-point equations .. 10 2.1 Introduction ........................................................................................................................ 12 2.2 Constructivism ................................................................................................................... 13 2.3 Knowledge in Pieces — diSessa ........................................................................................ 13 2.4 The Resource Framework — Elby and Hammer, Bing and Redish .................................. 14 2.5 Symbolic Forms – Sherin ................................................................................................... 18 2.6 Composite symbolic forms of integrals ............................................................................. 21 2.7 Graphical forms .................................................................................................................. 25 2.8 Summary ............................................................................................................................ 26 3.1 Introduction ........................................................................................................................ 27 3.2 Coordinatization of a scalar starting-point equation .......................................................... 27 3.3 Coordinatization of a vector starting-point equation.......................................................... 29 3.4 The Coordinatization Framework ...................................................................................... 32 4.1 Principle investigators ........................................................................................................ 33 4.2 Participant backgrounds ..................................................................................................... 33 4.3 Methodology ...................................................................................................................... 34 4.4 Methods .............................................................................................................................. 35 4.4.1 Written data protocol ................................................................................................... 36 4.4.2 Written data coding ..................................................................................................... 37 TABLE OF CONTENTS (Continued) Page 4.4.3 Video data protocol ..................................................................................................... 41 4.4.4 Video data coding ........................................................................................................ 43 5.1 Introduction ........................................................................................................................ 44 5.2 Acts of Coordinatization .................................................................................................... 44 5.2.1 Coordinatization category: drew axes ........................................................................ 44 5.2.2 Coordinatization category: limits of integration ......................................................... 46 5.2.3 Coordinatization category: equation manipulation of infinitesimal ........................... 46 5.2.4 Coordinatization Category: infinitesimal in sketch .................................................... 48 5.2.5 Coordinatization category: methods of dealing with vector nature of E and/or r. ..... 49 5.3 Looking at one category of coordinatization ...................................................................... 50 5.3.1 An example of a “limits of integration” act as a discrete resource .............................. 51 5.3.2 An example of “limits of integration” as a resource tightly bound to an infinitesimal 52 5.3.3 An example of “limits of integration” not always matching the infinitesimal ............ 53 5.3.4 A possible shift in framing helps one student as another hits a wall ........................... 54 5.4 Summarizing these limits of integration acts as cognitive resources ................................. 55 6.1 Introduction ........................................................................................................................ 56 6.2 Co-occurrence of acts of coordinatization ......................................................................... 56 6.2.1 Vector consistency ....................................................................................................... 58 6.2.2 Infinitesimal in sketch and correct manipulation of the infinitesimal ......................... 59 6.2.3 Pre-junior vs post-junior acts of coordinatization did not change ............................... 61 6.2.4 r-hat and words of grief ............................................................................................... 61 6.2.5 Other fun observations that I simply state ..................................................................
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