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Connecting Symbolic Integrals to Physical Meaning in Introductory Physics

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Connecting Symbolic Integrals to Physical Meaning in Introductory Physics Connecting Symbolic Integrals to Physical Meaning in Introductory Physics Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Nathaniel R. Amos, B.S., M.S. Graduate Program in Physics The Ohio State University 2017 Dissertation Committee: Andrew Heckler, Advisor Lei Bao Robert Perry Richard Hughes c Copyright by Nathaniel R. Amos 2017 Abstract This dissertation presents a series of studies pertaining to introductory physics students’ abilities to derive physical meaning from symbolic integrals (e.g., vdt) R and their components, namely di↵erentials and di↵erential products (e.g., dt and vdt, respectively). Our studies focus on physical meaning in the form of interpretations (e.g., “the total displacement of an object”) and units (e.g., “meters”). Our first pair of studies independently attempted to identify introductory-level mechanics students’ common conceptual difficulties with and unproductive interpre- tations of physics integrals and their components, as well as to estimate the frequencies of these difficulties. Our results confirmed some previously-observed incorrect inter- pretations, such as the notion that di↵erentials are physically meaningless; however, we also uncovered two new conceptualizations of di↵erentials, the “rate” (di↵eren- tials are “rates” or “derivatives”) and “instantaneous value” (di↵erentials are values of physical variables “at an instant”) interpretations, which were exhibited by more than half of our participants at least once. Our next study used linear regression analysis to estimate the strengths of the inter-connections between the abilities to derive physical meaning from each of dif- ferentials, di↵erential products, and integrals in both first- and second-semester, calculus-based introductory physics. As part of this study, we also developed a highly reliable, multiple choice assessment designed to measure students’ abilities to connect ii symbolic di↵erentials, di↵erential products, and integrals with their physical interpre- tations and units. Findings from this study were consistent with statistical mediation via di↵erential products. In particular, students’ abilities to extract physical meaning from di↵erentials were seen to be strongly related to their abilities to derive physical meaning from di↵erential products, and similarly di↵erential products to integrals; there was seen to be almost no direct connection between the abilities to derive physical meaning from di↵erentials and the abilities to derive physical meaning from integrals. Our final pair of studies intended to implement and quantitatively assess the effi- cacy of specially-designed instructional tutorials in controlled experiments (with sev- eral treatment factors that may impact performance, most notably the e↵ect of feed- back during training) for the purpose of promoting better connection between sym- bolic di↵erentials, di↵erential products, and integrals with their corresponding phys- ical meaning. Results from both experiments consistently and conclusively demon- strated that the ability to connect verbal and symbolic representations of integrals and their components is greatly improved by the provision of electronic feedback during training. We believe that these results signify the first instance of a large, controlled experiment involving introductory physics students that has yielded sig- nificantly stronger connection of physics integrals and their components to physical meaning, compared to untrained peers. iii Dedicated to Jessica Amos and Nancy McArthur. iv Acknowledgments The number of people to whom I owe gratitude is too large to list here. My wife (Jessica), my parents (Nancy, Hugh, Ronnie, and Paula), my grandparents (Jack, Willena, Harvey, Nancy, Hugh, and Betsy), my siblings (Ben and Jada), my uncle (Rob), and my wife’s family (Frances and Rosel) all shaped who I am today; I was lucky to be guided by their steadfast support, care, and wisdom. I am also grateful to my advisor, Andrew Heckler, for his personal and professional investment in my education and career; it has been an extremely valuable and influential four years. Finally, my thanks are due to my fellow graduate students (most of whom have already gone before me), for their assistance, advice, mentorship, and camaraderie: Brendon, Abby, Ryan, and Dan. Everyone listed above, and many more, deserve to be acknowledged by far more than a single page in this dissertation. In this small way, I hope they recognize their importance to me in my life. v Vita 5May2012.................................B.S.Summa Cum Laude Physics, Uni- versity of Florida, Gainesville, FL 20 August 2014 .............................M.S. Physics, The Ohio State Univer- sity, Columbus, OH August 2013-December 2015 ................Graduate Teaching Assistant, Depart- ment of Physics, The Ohio State University. August 2014-Present ........................Graduate Research Assistant, Depart- ment of Physics, The Ohio State University. Publications Research Publications N. Amos and A. Heckler “Spatial Reasoning and the Construction of Integrals in Physics”. PERC Conference Proceedings,2014. N. Amos and A. Heckler “Student Understanding and Construction of Di↵erentials in Introductory Physics”. PERC Conference Proceedings,2015. Fields of Study Major Field: Physics vi Table of Contents Page Abstract....................................... ii Dedication . iv Acknowledgments.................................. v Vita......................................... vi List of Tables . xi ListofFigures ................................... xiii 1. Introduction . 1 1.1 Current Issues in Integration from Mathematics and Physics Educa- tion Research . 2 1.2 Research Goals & Contributions . 4 1.2.1 Goal 1: Investigating, Discovering, & Estimating Frequencies of Conceptual Obstacles from Unproductive Interpretations of Physics Integrals & Their Components . 4 1.2.2 Goal 2: Investigating Mediating & Moderating Relationships Between the Abilities to Derive Physical Meaning from Dif- ferentials, Di↵erential Products, & Integrals . 5 1.2.3 Goal 3: Designing, Implementing, & Assessing Efficacy of Instructional Intervention Tutorials for Improving Students’ Abilities to Derive Physical Meaning from Symbolic Di↵er- entials, Di↵erential Products, & Integrals . 6 1.3 Thesis Organization . 7 vii 2. Theoretical Motivation . 9 2.1 Student Difficulties Extracting Meaning from Mathematical Symbols inPhysics................................ 9 2.1.1 “WhatDoesthisSymbolMeanPhysically?” . 9 2.1.2 “What Kind of Quantity is this Physical Symbol?” . 11 2.2 Procedural vs. Conceptual Abilities in Integration. 13 2.3 Student Conceptual Difficulties with Physics Integration . 16 2.4 Theoretical Instructional Frameworks . 21 2.4.1 Process-Object Layers of Integration . 22 2.4.2 “Change” and “Amount” Di↵erentials . 23 2.5 Summary . 24 3. Discovery of Common Conceptual Difficulties and Their Frequencies in Connecting Symbolic Physics Integrals and Their Components to Physical Meaning . 26 3.1 Introduction . 26 3.2 Physics Integration Exploratory Study . 27 3.2.1 Study Design & Research Methods . 27 3.2.2 Results & Discussion . 32 3.3 OnlinePhysicsIntegralAssessmentStudy . 37 3.3.1 Study&AssessmentDesign . 38 3.3.2 Results & Discussion . 41 3.4 Conclusions & Outlook . 46 4. The Mediating Relationship of Di↵erential Products in Understanding Physics Integration . 48 4.1 Introduction . 48 4.1.1 Motivation . 49 4.1.2 Regression-Based Statistical Mediation . 50 4.2 Methods & Design . 53 4.2.1 Study Participants . 53 4.2.2 Study&AssessmentDesign . 54 4.2.3 AssessmentValidity&Reliability . 56 4.3 Results & Discussion . 58 4.3.1 DescriptivesStatistics . 58 4.3.2 Mediation from Di↵erential Products . 60 4.3.3 Possible Moderation from Context-Familiarity . 63 4.4 Conclusions . 68 viii 5. Instructional Interventions for Student Understanding of Physics Di↵er- entials and Di↵erentialProducts....................... 70 5.1 Introduction to Di↵erentials Training . 70 5.1.1 Motivation . 70 5.1.2 Instructional Tutorials & Feedback from Computer-Based In- struction . 71 5.2 Experimental Design & Methods . 72 5.2.1 Participant Demographics . 72 5.2.2 EstimatingNecessarySampleSize . 73 5.2.3 Di↵erentials Tutorials . 75 5.2.4 Di↵erentials Post-Test . 79 5.3 Experimental Results & Discussion . 84 5.3.1 The E↵ect of Physical Context . 85 5.3.2 The E↵ect of Feedback . 87 5.4 Conclusions . 92 6. Physics Integration Paths to Understanding Instructional Interventions . 94 6.1 Introduction to Integration Paths to Understanding Experiment . 94 6.1.1 Motivation & Research Questions . 94 6.2 Experimental Methods . 99 6.2.1 Study Participants . 99 6.2.2 Experimental Design . 101 6.2.3 InstructionalInterventionTutorials . 104 6.2.4 Physics Di↵erentials, Di↵erential Products, Integrals Assess- ment............................... 115 6.3 Results & Discussion . 116 6.3.1 Descriptive Statistics . 116 6.3.2 The E↵ect of Instructional Path . 118 6.3.3 The E↵ectofUnits....................... 119 6.3.4 Electronic Feedback vs. Interactive Engagement . 122 6.4 Possible Limitations . 125 6.4.1 Baseline Ceiling E↵ects . 125 6.4.2 Possible Performance from Working Memory . 126 6.4.3 Missing Interaction E↵ects . 126 6.5 Conclusions . 127 7. Conclusions and Implications . 129 ix 7.1 Investigating, Discovering, & Estimating Frequencies of Conceptual Obstacles from Unproductive Interpretations of Physics Integrals & Their Components . 129 7.2 Investigating Mediating & Moderating Relationships
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