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SYMBOLIC FORMS of the INTEGRAL Steven Robert J ABSTRACT Title of Dissertation: APPLYING MATHEMATICS TO PHYSICS AND ENGINEERING: SYMBOLIC FORMS OF THE INTEGRAL Steven Robert Jones, Doctor of Philosophy, 2010 Dissertation directed by: Professor Patricia F. Campbell Department of Curriculum and Instruction A perception exists that physics and engineering students experience difficulty in applying mathematics to physics and engineering coursework. While some curricular projects aim to improve calculus instruction for these students, it is important to specify where calculus curriculum and instructional practice could be enhanced by examining the knowledge and understanding that students do or do not access after instruction. This qualitative study is intended to shed light on students’ knowledge about the integral and how that knowledge is applied to physics and engineering. In this study, nine introductory-level physics and engineering students were interviewed about their understanding of the integral. They were interviewed twice, with one interview focused on and described as problems similar to those encountered in a mathematics class and the other focused on and described as problems similar to those found in a physics class. These students provided evidence for several “symbolic forms” that may exist in their cognition. Some of these symbolic forms resembled the typical interpretations of the integral: an area, an addition over several pieces, and an anti-derivative process. However, unique features of the students’ interpretations help explain how this knowledge has been compiled. Furthermore, the way in which these symbolic forms were employed throughout the interviews shows a context-dependence on the activation of this knowledge. The symbolic forms related to area and anti-derivatives were more common and productive during the mathematics interview, while less common and less productive during the physics interview. By contrast, the symbolic form relating to an addition over several pieces was productive for both interview sessions, suggesting its general utility in understanding the integral in various contexts. This study suggests that mathematics instruction may need to provide physics and engineering students with more opportunities to understand the integral as an addition over several pieces. Also, it suggests that physics and engineering instruction may need to reiterate the importance, in physics and engineering contexts, of the integral as an addition over several pieces in order to assist students in applying their knowledge about the integral. APPLYING MATHEMATICS TO PHYSICS AND ENGINEERING: SYMBOLIC FORMS OF THE INTEGRAL by Steven Robert Jones Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2010 Advisory Committee: Professor Patricia F. Campbell, Chair Professor Ann Edwards Professor David Hammer Professor Edward Redish Professor Wesley Lawson © Copyright by Steven Robert Jones 2010 ii TABLE OF CONTENTS List of Tables.......................................................................................................................v List of Figures.....................................................................................................................vi Chapter I: Rationale and Significance ............................................................................1 What are possible difficulties in applying mathematics to physics and engineering courses?............................................................................................1 What is known about the knowledge and understanding students activate when applying mathematics to physics and engineering problems? ......................4 What is the significance of studying student understanding of mathematics in physics and engineering contexts? ...............................................6 Conclusion and Research Questions.....................................................................8 Implications and Limitations..................................................................................10 Chapter II: Literature Review ......................................................................................11 Domain and Definitions .....................................................................................11 Domain of the Study..................................................................................11 Definitions..................................................................................................11 Mathematics in Physics and Engineering............................................................13 Mathematics, Physics, and Engineering as Symbolic Systems..................13 Mathematics is Fundamental to Physics and Engineering.........................13 The Problem to be Addressed: The Gap between Math and Physics/Engineering...................................................................................14 Theoretical Perspective: Cognitive Resources ....................................................14 The "Predecessors" to Resources................................................................14 Definition and Explanation of Resources...................................................15 Epistemological Resources.........................................................................16 Framing.......................................................................................................18 Resources and Transfer...............................................................................18 Resources as a Way to Understand the Gap between Mathematics and Physics/Engineering.............................................................................19 Symbolic Forms.................................................................................................20 Chapter III: Methodology ............................................................................................23 Background Information of Participants.............................................................23 Student Population......................................................................................23 Brief Student Profiles.................................................................................26 Framework: Activating Symbolic Forms of the Integral .....................................27 Data Collection, Resource Activation, and Coding.............................................29 Interview Data............................................................................................29 Potential Symbolic Forms and Interview Items.........................................30 Looking for Symbolic Forms: Grounded Theory.......................................31 Interview Items and Analysis .............................................................................32 Chapter IV: Results .....................................................................................................39 The “Area” Symbolic Form................................................................................39 iii The “Adding Up Pieces” Symbolic Form...........................................................44 The Infinite Addition..................................................................................46 The “Function Mapping” Symbolic Form...........................................................49 The Problematic “Add Up Then Multiply” Symbolic Form................................53 Contrasting the Four Major Symbolic Forms......................................................57 Conceptual Schema Blended with "[]d[]"...................................................58 Conceptual Schema Applied to the " " Symbol......................................63 [] Conceptual Schema Applied to the Limits, " "......................................67 [] Other Symbolic Forms Pertaining to the Integral Symbol Template ...................71 Two Interpretations of the " " Symbol with No Limits..........................71 [] Symbolic Form for the Template: [1] [2] d [].....................................73 [] The Front Multiplier: "[] "......................................................................74 The Dependence of the Differential "d[]" on either the Integrand or the Domain.............................................................................................79 The Symbol " " as Representing a Region in Space...............................83 [] Other Cognitive Resources of Interest ................................................................84 [] The Symbol " " as Shorthand for " ".................................................84 [] [] Facing the Other Way and Negative Area...................................................86 Summary of Results ...........................................................................................88 Summary of Symbolic Forms Activated during the Interviews..................92 Chapter V: Discussion and Conclusions.......................................................................95 Discussion of Resource Activation during the Interviews ...................................95 Resource Activation in the Mathematics-Framed Interview......................95 Resource Activation in the Physics-Framed Interview..............................97 Intersection and Disjunction between Resource Activation.................................98 Characteristics of the Tasks and Symbolic Form Activation......................99 The Function Mapping Symbolic
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