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.

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 ...... 61 6.3 Analyzing a correlation of two acts by six students ...... 62 6.3.1 Student 1 – an unclearly drawn infinitesimal is instantaneously a sliver – triangulating one specific example of “infinitesimal-in-sketch” ...... 63 6.3.2 Student 2 – dq is a small piece of charge, but what is dr? ...... 67 6.3.3 Student 3 – constructing a form where dq is in dx ...... 70

TABLE OF CONTENTS (Continued)

Page 6.3.4 Student 4 – the “functional difference” between physics and engineering classes ..... 73 6.3.5 Student 5 – dq is a small charge and it has stuff...... 76 6.3.6 Student 6 – q is neglected both in the sketch and the equation...... 78 6.3.7 Analyzing 6 students sketch and equation manipulation of the infinitesimal as a group ...... 80 6.3.8 Conclusions from analyzing student acts of including an infinitesimal in a sketch and manipulation of infinitesimals ...... 85

7.1 Why this student and how I will present their work ...... 87 7.2 Examining this student’s work from an instructor’s point of view ...... 87 7.3 Examining this student’s work from the coordinatization framework ...... 90 7.4 Center of mass problem...... 91 7.4.1 Drew axes ...... 91 7.4.2 Coordinate independent infinitesimal equation, but not in sketch ...... 92 7.4.3 Graphical forms revealed while going back to drew axes ...... 93 7.4.4 Limits of integration ...... 96 7.4.5 Equation manipulation of infinitesimal ...... 98 7.4.6 Dealing with the vector nature of r ...... 99 7.4.7 Infinitesimal (maybe) now in Sketch...... 101 7.4.8 Wrapping up the center of mass problem ...... 102 7.5 Bar of charge problem ...... 103 7.5.1 Drew axes and infinitesimals in sketch ...... 103 7.5.2 Vector inconsistency begins ...... 104 7.5.3 The first glimpse of a newly cataloged symbolic form of a vector integral ...... 105 7.5.4 But first, a different problem between r and x ...... 105 7.5.5 dq does not makes an appearance ...... 106 7.5.6 Pointing out the lack of direction ...... 107 7.5.7 A Second glimpse at a newly cataloged symbolic form of a vector integral ...... 109 7.5.8 The missing dq is pointed out and resources are activated ...... 110

TABLE OF CONTENTS (Continued)

Page 7.5.9 The full nature of this student’s symbolic form of a vector integral finally revealed 112 7.6 Conclusions from one student’s actions of coordinatization ...... 115 7.6.1 Student 6 uses graphical forms while problem solving ...... 116 7.6.2 The “integration happens in the component’s direction” symbolic form ...... 117

8.1 Acts of coordinatization and the coordinatization framework ...... 119 8.2 Six students and six new symbolic and graphical forms ...... 119 8.3 Implications for teaching ...... 121 8.4 Pathways for future research ...... 121 8.4.1 More clinical interviews ...... 121 8.4.2 What are students thinking when they sandwich a starting point equation? ...... 122 8.4.3 What role does cognitive load play in problem solving of this type ...... 122

LIST OF FIGURES

Figure Page

Figure 1-1 A sketch of a teeter-totter that lets a person change where the center of mass of the system is situated with respect to the supporting bar underneath. Sketch by Emily Nussdorfer and reproduced here with permission...... 6

Figure 1-2 The ROV Hercules about to be deployed from the Exploration Vehicle Nautilus. Note that connecting point of the ROV to the boom is directly above the ROV’s center of mass. Photo of the ROV Hercules from the E/V Nautilus, Photographer: Kyle Neumann, 2019...... 6

Figure 1-3 A coordinate system used in order to identify the location of the center of mass of a Remotely Operated Vehicle in the design stage. Rendering courtesy of Steven Solders and reproduced here with permission...... 7

Figure 1-4 A baseball bat’s center-of-mass can be found by experimentally finding the place in one dimension where the bat will balance on a person’s finger. Creating a coordinate system with a well-defined origin can allow one to communicate where in space the center of mass is located. The center-of-mass of an object has an x-, y- and z-coordinate...... 8

Figure 1-5 In this figure a “slice” of the baseball bat is singled out in blue shading. The slice of bat is horizontally a distance x from the origin and the slice has a thickness that has been labeled “dx”...... 10

Figure 2-1 This figure was produced by a student and used while explaining their adding-up- pieces symbolic form of an integral that was meant to find the area between the functions f1 and f2 bounded between a and b...... 22

Figure 3-1 This sample problem demonstrates how one textbook introduces integration problems that require coordinatization. KNIGHT, RANDALL D., PHYSICS FOR SCIENTISTS AND ENGINEERS: A STRATEGIC APPROACH WITH MODERN PHYSICS, 4th, ©2017. Reprinted by permission of Pearson Education, Inc., New York, New York...... 27

Figure 3-2 This sample problem comes from the same textbook, eleven chapters later, as the sample problem shown in Figure 3-1. KNIGHT, RANDALL D., PHYSICS FOR SCIENTISTS AND ENGINEERS: A STRATEGIC APPROACH WITH MODERN PHYSICS, 4th, ©2017. Reprinted by permission of Pearson Education, Inc., New York, New York...... 31

Figure 4-1 The center of mass question was used only in the video interviews...... 35

Figure 4-2 This question was used both for the students taking the written diagnostic as well as the video interviews...... 36

Figure 4-3 Question posed to students and an example of one student’s response to the question...... 38

Figure 5-1 Student 2’s act of “limits of integration” as a pre-junior. Although there is no integrand or infinitesimal written after the integral sign, the integral does have limits...... 51

Figure 5-2 The same student as in Figure 5-1 answering the same question a year later. This time the student expresses an equation but does not include limits on the integral...... 52

Figure 5-3 After writing the equation in Figure 5-2, the student then clarifies that they know that there are supposed to be limits of integration but they “don’t remember the bounds” of integration...... 52

Figure 5-4 Although Student 116 incorrectly set up many aspects of the integral, they did include limits of integration that were consistent with the infinitesimals expressed after the integral signs...... 52

Figure 5-5 A year later, the same student has a different, and still incorrect, infinitesimal in the integral, however, once again, the limits of integration match the infinitesimal...... 53

Figure 5-6 In the first line of work the limits of integration do not match the infinitesimal. In the next line of work the student has (correctly) expressed dq as (Q/L)dx and now the original limits of integration match the dx as the infinitesimal...... 53

Figure 5-7 Student 113 starts with limits of integration from zero to L with respect to r with the student asking “what does r =?”. In the next step to the right it becomes clear that the r was a distance dependent upon the constant distance d and the variable of integration L’...... 54

Figure 5-8 Student 110 ends their work without resolving how to evaluate the integral...... 55

Figure 6-1 Student 103 performed both an acts of “infinitesimal drawn in sketch” and “correct infinitesimal manipulation”. The student pointed out a small segment of the bar that was labeled dq. Additionally, the student correctly converted the charge infinitesimal into a spatial infinitesimal with the correct relationship dq = dl...... 59

Figure 6-2 Student 7 did not denote an infinitesimal in their sketch. The only invocation of an infinitesimal is the dy shown here. In this student’s sketch, however, the horizontal axis is clearly defined as the y-axis...... 60

Figure 6-3 These are the two questions posed to students in six video recorded clinical interviews...... 62

Figure 6-4 Student 117 is the only other student whose coded act of infinitesimal-included-in- sketch consisted of a single dot or single line. Like Student 1, the infinitesimal had an r and a coordinate associated with it...... 64

Figure 6-5 The student repeatedly sweeps his hand to the right in a way that seems to be representing many points of charge...... 66

Figure 6-6 A proposed symbolic template to describe the symbolic form held by Student 1. .... 67

Figure 6-7 Student 2's coordinatization of dm happens utilizing dimensional analysis...... 68

Figure 6-8 Drawn when describing an infinitesimal. The little dot in the lower right-hand corner of the front face is described as an “absurdly tiny” small piece. The student explains that “This is easier to draw and explain than (an infinitesimal) mass”...... 68

Figure 6-9 Student 2’s first attempt at an equation for the E-field of the bar of charge...... 69

Figure 6-10 The student modifies their initial equation for the E-field of the bar of the charge. 69

Figure 6-11 Several minutes after drawing the bar seen here, the student adds the unlabeled infinitesimal in this sketch. The Q and the x were both included in the original drawing. The Q describes the charge of the entire bar. The x was used to describe the r in the equation for the electric field of a point charge...... 72

Figure 6-12 A later sketch drawn indicating “… a little bit of charge”...... 72

Figure 6-13 Student 4 includes an infinitesimal in their first sketch – the script above the dot is “dm”. The student then derives an equation for (x)...... 73

Figure 6-14 Student 4’s Equation Manipulation of the Infinitesimal ...... 74

Figure 6-15 The student adds an arrow to point out what they see as “difficult” about the process of coordinatizing dm into dx...... 75

Figure 6-16 This equation was written earlier in the interview but is referred to now as a “… sort of hard conceptual thing to think about.”...... 75

Figure 6-17 Student 4 discusses of dq interacts with different features of their drawing and their equation...... 76

Figure 6-18 Student 5 includes an infinitesimal in the sketch in two different ways. The way in which each infinitesimal is drawn reflects the manner in which the infinitesimal is being used at the time in which it is drawn...... 77

Figure 6-19 The two arrows in this diagram are drawn to the location of various dm’s...... 79

Figure 6-20 The dots are “little bit(s) of charge in the rod”...... 79

Figure 6-21 The student reworks the problem once the lack of “q” in their first attempt is pointed out. In the process, a new picture is drawn...... 79

Figure 6-22 Student 3 draws an unlabeled infinitesimal in their sketch. This lack of commitment to a label coincides with difficulty in deciding what infinitesimal to use in their equation. Additional uncertainty in the meaning held by the infinitesimal included in the sketch might be reflected in the“r” that is drawn disconnected to the infinitesimal...... 84

Figure 7-1 A reproduction of Student 6's solution to the center of mass problem. Pictures of the original work can be found in Appendix B...... 88

Figure 7-2 Reproduction of Student 6's initial work on bar of charge problem. Pictures of the original work can be found in Appendix B ...... 89

Figure 7-3 When it is pointed out to Student 6 that “there's one more thing you forgot up here… what creates the E-field?” the student starts their work afresh this time including charge both in their sketch and in their equations. The complete lack of charge in the initial work followed by charge appearing in multiple types of acts of coordinatization support the notion that the cognitive resource of charge, when activated, will activate in a variety of different acts. Meanwhile the equation written in this figure for dEy is a precursor indicator of a symbolic form of the vector integral held by this student that will be discussed in section 7.5.9. Pictures of the original work can be found in Appendix B...... 90

Figure 7-4 Student’s first written action and first act of coordinatization is to draw unlabeled axes...... 92

Figure 7-5 After drawing the axes, the student draws a bar and labels linear charge densities given for the two sides of the bar. As the axes are unlabeled, we do not yet know whether the axes are spatial or if at the horizontal axis has units of linear mass density...... 92

Figure 7-6 The student’s second act of coordinatization. While writing this the student describes small piece thinking of the integral...... 93

Figure 7-7 The student erases the vertical axis that they had previously centered in the coordinate system and now draws a new axis on the left side of the coordinate system...... 93

Figure 7-8 While attempting to perform a reasonableness check upon the algebraic function used to describe the mass density of the bar, the student blends a conceptual discussion with both an algebraic equation and a graphical representation...... 94

Figure 7-9 A proposed “Line End Points” graphical template...... 95

Figure 7-10 A proposed “Slope Gives Value” graphical symbolic form...... 95

Figure 7-11 The student’s first coordinate dependent infinitesimal in an equation with matching limits of integration...... 96

Figure 7-12 An example of limits of integration consistency seen in the written interviews that demonstrates how closely linked this act is connected to the manipulating the infinitesimal act. 98

Figure 7-13 The student draws the bracket and the dm while explaining how little pieces of mass are proportional to dx based upon where In they are at...... 99

Figure 7-14 Vector inconsistency displayed in the iconic equation for center of mass...... 100

Figure 7-15 This picture was drawn as the student was trying to describe their physics conception of the meaning of center of mass...... 100

Figure 7-16 The vectored r in Figure 7-14 has become a scalar x...... 101

Figure 7-17 The student modifies the already existing picture to draw two position vectors which can count as an act of showing an infinitesimal in a sketch...... 101

Figure 7-18 The coordinate independent “r” has been replaced with an “x” right before the integral is solved...... 102

Figure 7-19 The student begins with both the Drew Acts and Infinitesimal in Sketch acts...... 103

Figure 7-20 Once again, the student demonstrates an equation that serves as a symbolic template for adding up pieces form. As in the previous problem the coordinatization act of Vector Inconsistency has appeared...... 104

Figure 7-21 The student writes down an incomplete version of the starting-point equation for the electric field of a point charge. The charge “q” is missing in this equation. Also missing is the vector notation for the vector nature of the electric field...... 104

Figure 7-22 The student without prompting alters their symbolic template to be vector consistent...... 105

Figure 7-23 The student continues to attempt to put r in terms of x in a form that can be integrated...... 106

Figure 7-24 dr is then (incorrectly) found...... 106

Figure 7-25 The student has come up with a not entirely correct expression for dE to integrate. Note that by this point dq still has not appeared and that the electric field is being treated as a scalar quantity...... 107

Figure 7-26 In the process of considering the vector nature of the electric field, the student draws two “dE’s” for the bar of charge...... 108

Figure 7-27 Pictorially describing how to deal with the vector nature and the integration required to be considered to solve this problem...... 109

Figure 7-28 Expanding upon their symbolic template for a vector integral...... 109

Figure 7-29 The student describing how to find Etotal from the vector some of the dEx’s and the dEy’s...... 110

Figure 7-30 Once reminded of the fact that charge, q, has been missing from all the prior work, the student updates the picture with dq’s, dl’s and the relationship between them...... 110

Figure 7-31 After activation of the cognitive resource of the charge Student 6 relatively quickly produces a correct expression for a scalar dE...... 111

Figure 7-32 This equation for dEx is written. No matching equation for dEy is ever written to accompany this equation...... 112

Figure 7-33 Student 6’s graphical form of the vector integral that had to be solved...... 113

Figure 7-34 The (incorrect) Symbolic Form of a vector integral held by the student...... 115

LIST OF TABLES

Table Page

Table 1-1 Sophomore vs. Junior Level Starting-Point Equations and symbols ...... 11

Table 2-1 Examples of Symbolic Forms in the Literature ...... 19

Table 2-2 Summary of Rodriguez, et al. (2019) Graphical Forms ...... 26

Table 3-1 Possible Cognitive Resources That Could Be Demonstrated in the Center of Mass Sample Problem ...... 28

Table 4-1 Categories of Coordinatization Developed ...... 39

Table 4-2 An Example of Part of the Drew Axes Tranche of Data from Ph 320 ...... 40

Table 4-3 Subcategories Developed from the Fall 2016 Ph 320 Dataset to Describe Student Actions within the “Manipulating the Infinitesimal” Category. (N=39) ...... 41

Table 5-1 Coordinatization Category: Drew Axes ...... 45

Table 5-2 Coordinatization Category: Limits of Integration ...... 46

Table 5-3 Coordinatization Category: Equation Manipulation of Infinitesimal ...... 47

Table 5-4 Coordinatization Category: Infinitesimal in Sketch ...... 49

Table 5-5 Coordinatization Category: Methods of dealing with vector nature of E and/or r ..... 50

Table 6-1 Co-occurrence Table for Pre-Junior Physics Majors Taking at the Beginning of Ph 320 (N=39) ...... 57

Table 6-2 Co-occurrence Table for Post-Junior Physics Majors Taking at the Beginning of Ph 431 (N=32) ...... 58

Table 6-3 Acts of Infinitesimal-In-Sketch produced by Student 1 ...... 64

Table 6-4 Investigating the Correlation Between Student 1’s Acts of Infinitesimal-In-Sketch and Correct-Manipulation-of-Infinitesimal ...... 65

Table 6-5 Time-Line of Infinitesimal-In-Sketch and Manipulation-of-Infinitesimal Acts by Student 3 ...... 71

Table 6-6 Each Student’s Acts of Infinitesimal In Sketch and Equation Manipulation of Infinitesimal ...... 80

LIST OF TABLES (Continued)

Table Page

Table 7-1 Graphical Forms Identified During This Interview ...... 117

Table 8-1 Summary of Symbolic Forms Identified ...... 120

LIST OF APPENDIX FIGURES

Figure Page

Figure B-1 Student 6’s whiteboard work. This is the first board of work on Question 1 which focuses on finding the center of mass of a bar with a linearly increasing mass density...... 169

Figure B-2 Student 6’s second and final board of work for Question 1...... 169

Figure B-3 Student 6’s first board of work for Question 2. Question 2 asks for the student to find the electric field above one side of a uniformly charged rod...... 170

Figure B-4 Student 6’s second board of work for Question 2...... 170

Figure B-5 The bottom of Student 6’s second board of work for Question 2...... 170

Figure B-6 Student 6’s third board of work for Question 2...... 171

Figure B-7 Student 6’s fourth and last board of work for Question 2...... 171

LIST OF APPENDIX TABLES

Table Page

Table A-1 Ph 320 Fall 2016 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization ...... 130

Table A-2 Ph 320 Fall 2016 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r-hat” Observed Acts of Coordinatization ...... 135

Table A-3 Ph 320 Fall 2016 “Limits of Integration” Observed Acts of Coordinatization ...... 140

Table A-4 Ph 431 Fall 2017 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization ...... 145

Table A-5 Ph 431 Fall 2017 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r-hat” Observed Acts of Coordinatization ...... 149

Table A-6 Ph 431 Fall 2017 “Limits of Integration” Observed Acts of Coordinatization ...... 153

Table A-7 Ph 322 Spring 2018 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization ...... 157

Table A-8 Ph 422 Spring 2018 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r-hat” Observed Acts of Coordinatization ...... 160

Table A-9 Ph 422 Spring 2018 “Limits of Integration” Observed Acts of Coordinatization.... 165

– Introduction

Introduction After many years of teaching college physics, I wanted to pursue a dissertation in Physics Education Research with a goal of finding answers to general questions that I face as a physicist and instructor such as “What makes physics hard?” and “How can we teach physics better?” Ever since the widespread inclusion of physics and other natural sciences in the high school and college curriculum in the mid-1800’s, most academic physicists in the United States have been exposed to, if not called to undertake, efforts to improve the way physics is taught at all levels of education (Meltzer & Otero, 2015). However, Physics Education Research as a distinct discipline can trace its relative recent origins to 1979 when the University of Washington Physics Education Group graduated its first doctoral student (McDermott, n.d.). My main motivation for working on this dissertation is my passion for sharing physics with others. Physics is a sector of science that has gone through amazing advances over the past half-century and that will need to work hand-in-hand with other disciplines over what promises to be an exciting next half-century of discovery (Turner, 2009). There never has been such a time when scientific discovery has moved so quickly and scientific investigation was available to so many (Bornmann & Mutz, 2015). My teaching career has included teaching a variety of introductory physics and physical sciences courses at Linn-Benton Community College as well as leading and taking part in workshops and outreach events that include diverse groups local, statewide and global. I, along with many wonderful colleagues with whom I have been able to work, have worked hard to produce what I and we hope are high-quality learning experiences for students. In the process I have used a variety of teaching materials that I felt worked great and other materials that seemed to bomb. But, always in the back of my mind, I wanted to be able to go beyond intuition in determining what students gained from an education experience I desire to gain tools to help me better understand just what it is that students are learning, or having a hard time learning, within this discipline that means so much to me. In order to narrow these goals of finding how physics is hard and how it can better be taught down to something that can produce answers in a reasonable amount of time, I started looking at a few types of problems in which it seemed to me that my students have the most

difficulty. Problems that involved integrals with an infinitesimal that ultimately needs to be converted from a coordinate-independent variable to a coordinate-dependent variable is one such type of problem that stands out to me as being difficult for my introductory calculus-based physics students. Literature in Physics Education Research exists that attempts to analyze student difficulties in these types of problems. Dray and Manogue’s (2009) work in attempting to “bridge the gap” between mathematics and the physical sciences has long been central in my thinking of how to assist students in using the mathematics they learn in math class to solve problems in my physics classes. Pepper, Chasteen, Pollock and Perkins (2012), after citing the dearth of research into student difficulties in upper division physics, discuss some students’ struggles combining mathematical calculations with physics ideas. Wilcox, Caballero, Pepper and Pollock (2013) continue on to describe some specific challenges students face in at this level including expressing the difference vector in appropriate coordinates and reasoning how to associate the differential within an integral to the applied coordinate system. This group develops the ACER framework to “facilitate connecting students’ difficulties to challenges with specific mathematical and physical concepts” (Caballero et al., 2013) to address how mathematical resources are “Activated, Constructed, Executed, and Reflected” upon. From my own experiences as an instructor and my initial foray into the literature, the research question of this dissertation became “How do junior-level physics majors coordinatize a problem that requires integration?” This chapter is written both for physicists and non-physicists. In this introduction I will explain to a broad audience: • what the words coordinatize and integration mean and why this specific research question is important to the instruction of physics • the qualitative research tools, with origins in the social sciences, that I used to address the question • prior research upon which my work is based including “Knowledge in Pieces”, a “Resources Framework”, “Symbolic Forms” and “Cognitive Load”. By the end of this introduction I hope to give the general reader some insight on what makes certain types of problems hard. The future chapters will then get into the details of the theoretical perspective and methodology in which my data is analyzed, a presentation of the

results of how students coordinatize problems that require integration, and then suggestions of future research.

Math and concepts in word problems Readers of this dissertation may remember doing “word problems” in elementary school math class. A word problem presents a situation often in everyday language that requires mathematics to solve. Usually a word problem is presented without formal mathematical notation or an explicit indication of an equation to use and is often presented in the form of a story. For example, a problem in my 2nd grade math book (Nault, 1978) states: “Texas Slim, the leader of the bandit gang, rubbed his hands together happily. He and his men had just robbed the Pecos bank. They had stolen twenty-one thousand dollars!” Slim and his six bandits wanted to split up the stolen goods equally between the seven of them, but when asked how to do that Slim admitted that he did not know and decided “I’ll just start countin’ it out and we’ll see. Ok, here’s a dollar fer Sagebrush Sam, and one fer Deadeye Pete…” until an hour later while Slim was still counting out the money into seven equal piles the deputies rode up and captured the gang. Knowing how to divide $21,000 into seven equal piles quickly is useful not just for bandits but is now an expected ability for most everyone beyond elementary school. However, judging from my memory, and, more importantly from the large amount of research done on the topic as demonstrated by meta-analyses by Hembree (1990,1992) many elementary and high school students find word problems exceedingly difficult. There are a few insights as to what causes word problems to be difficult. Suspension of sense-making is one cause of student difficulty in word problems. Instead of attempting to conceptualize the word problem, many students instead seek a simple application of arithmetic that will produce an answer, with little attention to whether the numerical answer makes sense or not. Schoenfeld (1991, p. 323) describes one student explaining how they solve a word problem. Given the problem “John has 7 apples. He gave 4 apples to Mary. How many apples does John have left?” The student responded: “I look through the problem and circle the numbers. Then I start reading from the back, because the key word is usually at the end.” Thus, in this problem, the student only needs to recognize “7… 4… left?” in order to solve the problem. Schoenfeld attributes this problem-solving behavior to textbooks and educational settings that will reward

correct answers to problems regardless of a student’s potential to not fully understand the problem they have solved. Other students may struggle with word problems because they have difficulty with reading. Abedi and Lord (2001) address issues of this type faced by English language learners and students of low socioeconomic status. Abedi and Lord found that language simplification helped all students perform better on math story problems with English language learners benefiting more than native English speakers. The problem with word problems lies not just with students realizing that sense-making is not necessary to solve some word problems. Chapman (2006) discovers that high school instructors often purposely try to “strip away the stuff that we don’t really need” (p. 219) in order to help students get to the arithmetic of a problem. All the teachers that Chapman interviewed, to differing degrees, attempted to reduce word problem down to numbers and key phrases that indicated mathematical operations. All three of these examples demonstrate that combining real-world situations with a mathematical understanding is something that both students and teachers find challenging. During the twenty years that I have been teaching college level physics, one of my goals has been to help students at a variety of incoming math levels get a better understanding of the physical universe in which we live. I hope as an instructor that I have enabled students to see physics as accessible at many different entry-points and ultimately something everyone can enjoy. The desire to understand what is hard about physics and problems like those word problems many of us experienced in elementary school has remained with me throughout my teaching career.

1.3 Math and concepts in physics problems As a college physics instructor, I have taught a full spectrum of introductory physics courses including physics with calculus for scientists and engineers; physics with algebra and trigonometry; and physics for everybody. Each of these physics sequences covers mostly the same physics but each class is accompanied with a different mathematical framework. Attempting to help students combine their mathematical skills and understanding with their conception of the nature of our physical universe has always been one of my favorite parts of the job.

When trying to come up with a research question that dealt with “why is physics hard?” but narrow enough to pursue with the tools of physics education research, I decided to focus on the types of problems that I think my students find hardest in the year-long calculus-based introductory college physics sequence. I am choosing problems that I think students find difficult as I really want to see if I can tease out ways in which students approach these problems. I focus on problems that involve calculus because the transition from the lower- division to upper-division college level has been a large concern for most of my teaching career. Consequently, for this dissertation, I focus on problems that require students to use integration along the spatial extent of a material object. During the first ten weeks of many calculus-based physics sequences, problems that require differentiation or integration to solve are done with respect to time or space. Problems that require analysis of a coordinate-independent quantity such as mass or charge that eventually results in an integral of a coordinate-dependent expression in space do not appear until later in the year.

1.4 Definition of coordinatize A physics problem I remember having to solve in elementary school was attempting to find the center of mass of a teeter-totter that had two people of different masses sitting upon it. One of my good playground friends had about three times my mass -- our first experience on a teeter-totter together involved me flying over the handlebars and some minor injuries on my part. Within the next few days, however, we discovered that by shifting the teeter-totter bar, as shown possible in Figure 1-1, so that my side of the teeter-totter was longer, made it so that we could nicely balance. I remember my dad mentioning the phrase center-of-mass, although I am not sure how I conceptualized the idea at the time.

A common problem that the Remotely Operate Vehicle (ROV) team at my college needs to solve is to find the center of mass of a more complicated shaped object. When designing an underwater ROV, such as the one shown in Figure 1-2, long before the device is built, calculations are done by hand and by computer so that proper positioning of buoyancy, thrusters and other components happen. I have been the mentor for this team for several years. It turns out that in most years, my students are just learning how to find the center of mass in their physics class at the same time they need to be able to predict the center of mass of their soon-to- be-constructed ROV.

In about the 12th week of many introductory calculus-based physics classes, students are 1 introduced to the equation 푥푐푚 = ∫ 푥푑푚 – the x-position of the center of mass is equal to 푀푡표푡푎푙 the integral of xdm. The dm in this equation stands for an “infinitesimal” piece of mass. The word infinitesimal is defined by Webster’s Ninth New Collegiate Dictionary “immeasurably or incalculably small”. (Merriam, 1983). The ‘dm’ in the above equation represents an infinitesimal piece of mass. The  symbol is an integral symbol and it indicates that we are supposed to sum up every tiny piece of mass of a system times its location in order to find the center of mass of the entire object – this is kind of like finding the position of every atom of the teeter-totter or Remotely Operated Vehicle and multiplying that atom’s mass by its position and then adding all of those values up together. It turns out that for a three-dimensional object like an ROV, you need to perform one of these integrations for each one of the three dimensions. In Figure 1-3 you can see the x-, y- and z- axes of a coordinate system – solving the center of mass integral once for each dimension will give you the x, y and z coordinates for the center of mass of the ROV.

1 For many students in an introductory physics course, this equation 푥푐푚 = ∫ 푥푑푚 푀푡표푡푎푙 presents an integral with an infinitesimal that does not represent a spatial or temporal quantity.

To solve problems using this equation, students need to transform the non-spatial dm into a quantity that includes a spatial infinitesimal such as dx. This process of converting the dm to dx is one act of what I am going to call “coordinatization”. I define coordinatization to be the expression of physical quantities in pictures, words or equations that can be used to determine the relationship between coordinate-dependent and coordinate-independent quantities used in solving a physics problem. For readers who have not necessarily had calculus in the past, I am going to describe some of the purposes for acts of coordinatization. If you want to balance a baseball bat, as shown in Figure 1-4, along its length on your finger you could find experimentally that you would need to place your finger in a position on the bat as shown.

Figure 1-4 A baseball bat’s center-of-mass can be found by experimentally finding the place in one dimension where the bat will balance on a person’s finger. Creating a coordinate system with a well- defined origin can allow one to communicate where in space the center of mass is located. The center- of-mass of an object has an x-, y- and z-coordinate.

A student taking introductory calculus-based physics might be asked to predict the 1 location of the center of mass of the baseball bat using the equation r= rdm which in cm  Mtotal words can be described as: the position, 푟⃗, of the center of mass is equal to the integral of rdm, where r is a vector with components in the x, y, and z directions. One act of coordinatization that needs to occur is to define a coordinate system around the bat – defining the origin allows the problem-solver to assign coordinates to the position value 푟⃗. In Figure 1-4 you can see that an x- and y-axis has been drawn along with a grid system. There is an origin where x=0 meters and y = 0 meters. The center of mass of the bat has an x-coordinate and it has a y-coordinate. The

1 goal of using the xxdm= equation is to figure out where the x-coordinate of the center cm  Mtotal of mass will be in this coordinate system as defined. To find the y-coordinate of the center of 1 mass, you would use the equation yydm= . The symbol 푟⃗ is used by many instructors cm  Mtotal and textbooks to communicate that the quantity we are looking at might have two or more spatial dimensions. Later in this dissertation I will report the acts of coordinatization I observed undergraduate physics majors making when attempting to solve a problem of this type. But, to start, I will describe just one act of coordinatization in order to begin to spell out what a student needs to know and do when coordinatizing a coordinate independent equation.

In both the xcm and ycm equations above, there is a dm. The dm stands for a very tiny piece of mass. Every dm in the object has its own x- and y-extent. In order to find the center of mass of the object, every x- and/or every y-coordinate needs to be looked at to see how much mass of the object is at that location. When integration happens with respect to dx and/or dy, the integral “sweeps” across the coordinate system calculating how much mass there is at every point in space. Discovering how to transform the coordinate-independent dm in the equation above to a spatial differential such as dx is an act of coordinatization that this type of calculation almost always requires. If dm is a very small piece of mass, dx is then the space along the x-axis in which that tiny dm of the baseball bat fits. In Figure 1-5 you can see one dm sitting in a slice of the bat that has a physical thickness of dx. Furthermore, the figure shows that this particular dx is a distance x from the place in our coordinate system where x=0 meters. Ultimately, to find the center of mass of this object, a student will need to find a way to algebraically represent the amount of mass there is in each dx that can be drawn in the bat.

The dm to dx transformation is only one of several different acts of coordinatization that needs to be done in order to use this equation and others like it in order to solve introductory physics problems. The meat of this dissertation in the following chapters will seek to find what other acts of coordinatization physics students use when solving problems that they are likely to face during their junior year at university.

1.5 Sophomore-level physics to junior-level physics transitions in starting-point equations In this section I demonstrate some of the differences junior-level physics students may encounter when transitioning from their sophomore year and how these differences relate to coordinatization. One of the discipline-wide issues that junior-level physics students need to come to terms with in their starting-point equations is a new notation that incorporates a greater attention paid to coordinate-independent features of a problem. I use the term “starting-point equation” to mean an equation that a student would not be expected to re-derive every time it is used and often represents a definition or description of an empirical finding. As the name suggest, starting-point equations are usually equations that students will write down near the beginning of the problem-solving process. An important feature to note in the transition from the sophomore- to junior-level is that the explicit use of coordinate-dependent equations in the sophomore year transition to a more generalized coordinate-independent version of the same equation in the junior year. For example, in many calculus-based introductory physics classes taken during a student’s freshman or sophomore year in college, there are three center of mass equations each of which clearly

denote one of the three spatial axes in a 3-D coordinate system as shown in Table 1-1. In the junior year of a physics major, these three equations are expressed by just one equation by  having a single symbol, r . The symbol is a coordinate-independent symbol that communicates that the equation has a spatial component and implicitly implies that it is up to the formula-user to figure out which coordinate system to use in order to make the calculation easier.

Table 1-1

Sophomore vs. Junior Level Starting-Point Equations and symbols Sophomore Level Junior Level  < x , y , z > r x = xdm  cm     r = rdm  cm  y = ydm cm    z = zdm cm    1 q  1 q   E = rˆ E = (r − r') 4 r 2   3 o 4 o r − r'

This transition is more than just a notational transition. In order to solve junior-level problems, students need to be able to identify the most useful, or at least useable, coordinate system (e.g. rectangular, cylindrical or spherical), employ that coordinate system by choosing an origin and orientation for the axes of the system, and then be able to express various physical quantities in ways consistent with the coordinate system they have chosen. Much of the coordinatization process is somewhat ignored by sophomore level general physics courses by assigning problems where the origin of the system is pre-defined. In the junior year, many standard textbooks give more responsibility to the student to choose the coordinate system appropriate for the problems they are asked to solve.

– Literature Review

2.1 Introduction In creating the coordinatization framework, I draw on several different theoretical perspectives. This chapter describes the foundations upon which I will build. First, I observe students’ acts of coordinatization to be discrete consistent with a knowledge-in-pieces view of students’ ideas. The Knowledge in Pieces Framework (KiP) introduced by Andrea diSessa (1988, 1993, 2016, 2018) is central to my methodology and used to answer my research question “How do junior-level physics majors coordinatize a problem that requires integration?”. KiP has its origins in the Physics Education Research community and has been utilized in other areas such as mathematics, chemistry, computer science and even views of race and racism (Philip, 2011). As the name suggests, KiP focuses on small knowledge elements and the framework builds a theoretical and empirical lattice for understanding knowledge and learning that is fundamentally constructivist in nature. Motivated by KiP, I first identify small- grain-sized knowledge elements of coordinatization that students express while solving problems on paper or in a clinical interview setting. I additionally observe that student acts of coordinatization are subject to activation, suppression and binding that suggest that individual acts of coordinatization function as a set of resources as described by Hammer’s (2000). Some resources are intuitive or based upon experience while others are more formally obtained. The Resources Framework developed by Tuminaro (2004) and Bing & Redish (2009) allow me to include larger, composite acts of coordinatization that include many pieces of knowledge. In the analysis of the data I see evidence of coordinatization occurring through the combining of a student’s conceptual understanding of the physics of the problem with their use of mathematical equations and graphs. Sherin’s Symbolic Forms (2001) provides a framework for understanding how student production and manipulation of equations coincide with their conceptual schemata. This literature review covers the prior work that provides a blueprint for my methodology. I use student acts of coordinatization in order to uncover understandings students have in the coordinatization processes as well as understandings students hold when attempting to use an integral of vectors to solve a problem based upon a physical scenario.

2.2 Constructivism An important starting point is an acknowledgement that the analysis of the data in this dissertation will be done within a constructivist framework. Smith, diSessa, and Roshcelle (1993, p. 116) in their work that reframed thinking about misconceptions describe that in a constructivism framework “all learning involves the interpretation of phenomena, situations, and events, including classroom instruction, through the perspective of the learner's existing knowledge”. When trying to answer the research question “How do junior-level physics majors coordinatize a problem that requires integration?” a constructivist framework acknowledges that in order for a student to get to the junior-level in physics they have already acquired significant “existing knowledge”. The students in this study are at a point in their academic career where they are transitioning from novices to experts as physicists. In a constructivist framework, I expect that students at this level will employ a wide range of problem-solving strategies, many of which may not be fully productive in answering the questions I will be asking of them. The point of this research is to observe what tools are used in the coordinatization process regardless of whether or not those tools lead the student to an answer that matches the solution of an expert.

2.3 Knowledge in Pieces — diSessa diSessa describes a framework in which “intuitive physics consists of a rather large number of fragments” (diSessa, 1988, p. 52). These fragments can be often understood as simple notions taken from everyday experience. For example, from a young age one might recognize that it requires more effort to pick up a big rock than a small rock. Knowledge of the theory of universal gravity is not required to explain this knowledge of effort and relative rock size, it is simply an expected event that the big rock is harder to pick up. diSessa called these very small pieces of knowledge phenomenological primitives, or p-prims. Generally, a p-prim is something that does not require further thinking when being used. Of course, a p-prim can fail in predicting an outcome. Growing up near large stratovolcanoes, I remember my first experience with a piece of pumice about the same size as myself and being delighted that I could pick it up by myself. My p-prim of “larger is harder to move” failed and further thinking ensued. I distinctly

remember enjoying my dad explaining the concept of density and showing that some rocks can float on water. diSessa’s Knowledge in Pieces framework is an epistemological approach that seeks to “develop a modern theory of knowledge and learning capable of comprehending both short-term phenomena – learning in bits and pieces – and long-term phenomena, such as conceptual change, “theory change,” and so on” (diSessa, 2018, p.66). With these goals articulated, the Knowledge in Pieces framework is fundamentally constructivist in nature with a goal of addressing both the smaller bits of learning and expressions of knowledge that happen on smaller and immediate time-scales with the long-term scales of learning and performance phenomena. Knowledge in Pieces is often used as a framework for pursuing understanding in work focused on student pre- conceptions and conceptual change. In this dissertation I focus more on the Resource Framework described in the next section, which has evolved from diSessa’s Knowledge in Pieces framework.

2.4 The Resource Framework — Elby and Hammer, Bing and Redish diSessa’s p-prims form one class of a cognitive structure known as cognitive resources. Elby and Hammer (Elby & Hammer, 2010) define cognitive resources as “fine-grained knowledge elements that a student possesses, the activation of which depends on context”. Whereas p-prims are resources that are based mostly on intuitive knowledge, resources as a broader category include pieces of knowledge that, although fine-grained, might still contain internal structure. As an example of resources and activation of resources, I recall a physics conference I attended, at which a large auditorium of physics instructors and physics students were presented with the question “Which block will a bolt heated to a high temperature melt through first: a block of frozen water or a block of frozen carbon dioxide?” This particular demonstration lead by Toby Dittrich of Portland Community College is one of my favorite not just because of the physical result of empirically determining the answer to the question posed, but also because of the diversity of thoughts and ideas that came from the audience. This presentation was not recorded, so I can only offer excerpts from my and other audience members’ memory.

A group of students sitting near me quickly came to the prediction that the dry ice would melt more quickly due to the heat from the bolt. I was able to ask them to elaborate upon their thinking and I remember one student mentioning that air is “less there” than water. I interpreted the “less there” comment to be a resource that aligned, although maybe not perfectly matched, with the idea of a gas being less dense than a liquid or solid. At one point one student in the group did mention that dry ice was more dense than CO2 in its gaseous state. However, the group decided that even in its frozen state, the CO2 would be impacted more by the hot bolt than frozen water would be due to what CO2 was like in its “natural state” where I took natural state to refer to how CO2 behaves at standard temperature and pressure. When asked by Toby to explain to the auditorium their group’s predictions, other participants activated the resource of specific heat and (correctly) inferred that it would take more energy to sublime the CO2 than an equal amount of water ice – they then inferred that the hot bolt would cause the ice to melt faster. This argument produced a correct prediction of the effect of the bolt upon the two bricks, but the eventual demonstration showed that this was not the full story of what would happen. Another group activated thermal conductivity as an important consideration – but no one in the group knew off the top of their heads the thermal conductivity of water ice versus dry ice and thy reported getting stuck in a discussion of how to conjecture thermal conductivity values of various substances. After a few minutes of discussion, the experiment was run up on stage with a video camera allowing the audience to clearly see how a heated bolt interacted with each type of block. To the delight of the crowd the results were dramatic and easily interpreted. The hot bolt placed on top of the water ice block amazingly quickly cut through the water brick – the melting water ice created liquid water that provided a highly thermally conductive material through which the heat of the bolt could flow to continue to melt the water ice. The bolt on the dry ice quickly sublimed the dry ice creating a protective layer of CO2 vapor upon which the bolt floated in a way analogous to the Leidenfrost effect (Leidenfrost, 1756), well known to a group of physics instructors, and, consequently, very little dry ice was sublimed in a unit of time. From this story, I want to point out some important aspects of cognitive resources:

• Resources are relatively fine-grained pieces of knowledge – in this case different individual groups brought up concepts of density, specific heat and thermal conductivity. • Activation of resources happen often beneath the awareness of the subject and often impacted by the context of the situation. The group of students near me who used the “less there” idea to come up with their prediction did not use the more formal word “density” until I, an obvious professor, asked for elaboration. • Some phenomena might take a large number of resources to correctly predict an outcome to a given problem. In this demonstration concepts of density, specific heat and thermal conductivity were all brought up within my earshot – in the end, all of these concepts did play roles in the final effect although it was thermal conductivity and the Leidenfrost effect that played dominant roles. Bing and Redish (2009) summarize the development of a Resource Framework to describe how students approach solving a physics problem with the resources of knowledge that they have acquired. The Resource Framework seeks to model high-level thinking. It has its foundations in phenomenological education research, cognitive science, and behavioral science. The basic structure of the Resource Framework parallels the image of neurons within a brain. “The activation of the neurons corresponding to one resource or cluster of resources leads to activation of other clusters. Learning is pictured as the establishment of strong connections, so that activation of one resource or cluster of resources leads to the activation of other resources.” (Bing & Redish, 2009) In this framework, clusters of resources that activate together often become strongly tied together. This “binding” action can be strong or loose. Additionally, and importantly, the entire network is very dynamic. On short time-scales, sometimes on the order of seconds, a student can shift from activating one cluster of resources to another. Bing and Redish (2009) describe an upper-division physics student who “shifts” her reasoning from one that was largely based upon “facts” she recites from having heard in class to an idea developed more from conceptual understanding. After the student explains that it is the greater density of electrical insulators that restrict electron flow in an insulator, the interview asks whether or not Styrofoam is an electric insulator. The student responds that it is and when asked “Why?” the student responds that she “memorized it”. A little later to further justify her

views on Styrofoam she makes a blanket statement citing “organic chemistry”. During the first part of the interview the study relies upon authority and the citing of rules and facts gained from these authorities. Later the student is asked to give “any explanation you find”. At this point the student’s reasoning includes a more conceptual discussion of electric conduction by creating a story that discusses “electrons getting torn away from their parent atoms and then being free to move” and explaining “how a battery could perhaps cause this electron-tearing and how a higher temperature wire might also have more energy available to tear electrons off the atoms.” The important conclusion from the story of this student is that a person’s epistemological resources activate and deactivate while a person is reasoning through a problem. An epistemological resource represents a tightly bundled packet of information that, when activated, leads a person to work with information in a certain way. In my recollection of students trying to explain why a hot bolt might melt quickly through a block of dry ice students talking amongst themselves initially talked about how air was “less there” – the students were attempting to utilize their everyday knowledge to answer a question. When I entered the discussion with the status of a professor the students started using and exploring the situation with more formal words such as “density” and “volume” – this same group of students were now framing the question they were trying to address by using concepts to which they had probably been exposed in a science classroom. A person’s mind contains a large amount of knowledge and experiences – thus, the Resource Framework must include a process in which all the possible resources and connections among resources can be pared down to something manageable when a person is asked to confront a particular question. Bing and Redish call this process “framing”. Framing has a well-defined meaning in the behavioral sciences. Framing is the usually subconscious choice the mind makes when trying to figure out the question “what kind of activity is going on here?” Framing is thus the important decision a mind makes when trying to narrow down the set of all possible mental operations that could be made into a more manageable subset. A quick example of framing that is often described due to work done by Schank and Abelson (1977) is how people know how to act when they enter a restaurant. In the U.S., if a patron sees a large menu on a wall, they often know to look for a line and expect to place an

order standing up at a counter with payment happening immediately upon ordering. Meanwhile, if there is not a large menu on the wall, one generally looks for a place to sit down and expects to order from a waiter who will come to your table and pay for the meal after the meal has been eaten. The first meal I ate when traveling in India mixed this routine up a bit: the big menu was on the wall, however the expectation was that we sit down and order from a waiter while reading from the wall menu. Bing and Redish warn that framing should not be thought of activating a long stable instruction list. Rather, framing is better thought of as the cueing of fuzzy adaptable networks of cognitive resources. Thus, with the example of my first meal in India, as my friends and I were traveling in a foreign country we were quite prepared to be able to shift our framing and adapt our responses as needed. The main takeaway from the Resource Framework that I will use in my analysis of data are: • cognitive resources are fine-grained pieces of knowledge, • resources can be activated and associated with other resources in either an excitatory or inhibitory way, • framing can direct resources that are likely to be activated, • and framing and resource instigation are dynamic with associations being activated or inhibited depending upon context.

2.5 Symbolic Forms – Sherin One central aspect of achieving a degree in physics is being able to combine one’s conceptual understanding of how the universe works with knowledge of and ability to use a large set of mathematical tools. To better understand the cognitive processes a physics major faces, in this section I first look to the work of Bruce Sherin in “How Students Understand Physics Equations” (Sherin, 2001). A symbolic form is a combination of a person’s conceptual understanding of a physical phenomenon called a schema with a mathematical symbol template. Conceptual schemata “are relatively simple structures, involving only a few entities and a small number of simple relations among these entities” (Sherin, 2001, p. 491) that Sherin describes as being similar to diSessa’s (1993) p-prims.

Part of Sherin’s work involved having pairs of university physics students work together on problems in which they would eventually create an equation to describe the phenomenon. For example, Sherin presented a problem that involved air resistance where he asked students to consider how terminal velocity differs for two objects with the same shape and size but in which one of the objects is twice as massive as the other. When presented with this problem, some students would at one point write down an equation of the form Fup = Fdown. Sherin’s analysis of work by students who wrote this equation led him to believe that they neither recalled nor derived this equation from a physical principle, but rather constructed this equation to express a particular understanding of the physical situation that two opposing influences were balancing each other. Meanwhile, for the same problem solved by a different student, the student used the terms “upward acceleration” and “downward acceleration”. The equation produced in such a discussion might look more like: 0 = aup – adown. In this form, Sherin believes that the students are instead expressing their understanding that two opposing influences can compete to produce a result. Rather than considering the accelerations as balancing each other, the students saw the accelerations as canceling out with adown taking away all of the aup. Ultimately, Sherin argued that the conceptual content of equations can be understood in terms of a specific vocabulary. In the example above, the notions of “balancing” and “competing” influences are expressed by different equations that are present in a physicist’s vocabulary. Sherin stated “my hypothesis is that successful physics students learn to express a moderately large vocabulary of simple ideas in equations and to read these same ideas out of equations” (p. 482). He calls the elements of this vocabulary “symbolic forms”. Sherin developed examples of symbolic forms that he observed in a series of student interviews. Table 2-1 presents a selection of symbolic forms identified by Sherin and others. Table 2-1

Examples of Symbolic Forms in the Literature

Symbolic Form Symbol Pattern Example

Balancinga  =  Two people balanced on a teeter-totter would have equal torques expressed as 1= 2

Table 2-1 (Continued) In the case of an airplane flying a constant velocity, Cancelinga 0 =  -  the thrust of a airplane’s propellers cancel out the air drag experienced by the airplane to achieve a net force of zero. ...... x The greater the force upon an object the greater the  Prop +a ... F acceleration of that object. a = m The greater a mass of an object, the smaller the a ... Prop -  ...... x acceleration of that object. When looking at two masses falling freely under ...... x mg Canceling (b)a gravity turns into a = . Here, the m’s ...... x m “cancel” indicating that objects of different mass accelerate at the same rate under gravity. Coefficients describe the size of an effect such as comparing the strength of the electrostatic force Coefficienta [x] qQ mM Fk= to the gravitational force FG= . r 2 r 2 3 When finding the value of 4x d x where the “user” 0 b Adding Up Pieces  d indicates that they are using a concept such as a large, or infinite, number of rectangles whose areas are being added up. 3 When finding the value of 4x d x where the “user” 0 c Perimeter and Area indicates that they are using a concept of find the area of a triangle with a base of width “3” and a height of “12”. When finding the value of where the “user” Function Matchingd indicates that they are matching the “4x” to the derivative of their answer. Used to denote a vector expression including the Magnitude Directione = magnitude of a quantity (having units) along with a unit vector to indicate a direction. aSherin (2001, p. 506). bJones (2013, p. 125-126). cJones (2013, p. 128). dJones (2013, p. 129). eSchermerhorn & Thompson (2019, p. 6)

2.6 Composite symbolic forms of integrals Jones (2013) ties in the theoretical framework of cognitive resource with symbolic forms to investigate the possibility of the existence of an integral symbolic form. A symbolic form is considered cognitively “compiled” if the student repeatedly blends the conceptual schemas and the symbols within a given task as well as across many different tasks. Here, conceptual schema “refers to the meaning underlying the arrangement of the symbols” (Jones, 2013, p. 124). From experimental evidence Jones found that there appeared to be at least three main symbolic forms that many students used that were accompanied with the same template  d .These three composite integral symbolic forms are: • The adding-up-pieces symbolic form • The perimeter-and-area symbolic form • The function-matching form All three of these symbolic forms, although differing in schema, utilize the same symbol template:

The adding-up-pieces symbolic form has components of understanding similar to Reimann-sum thinking often introduced in introductory integral calculus. Jones (2013, p. 125) presents an interview he had with a student where the student stated that to understand the integral he had to divide the region of interest into small pieces. The student drew a single rectangle as shown in Figure 2-1 with a width labeled dx. Then, putting his hand in an orientation that lined up with a vertical side of the rectangle explained “Let’s say you slice it this way, and then you add up all the individual lengths [puts hand on left side of shaded region and sweeps hand across to the right side]” (Jones, 2013, p. 126). This hand sweeping motion combined with a dialog that included descriptions such as “from here to there” and the integrands being “starting” and “ending” points support a conceptual schema attached to the adding-up-pieces form.

Figure 2-1 This figure was produced by a student and used while explaining their adding-up-pieces symbolic form of an integral that was meant to find the area between the functions f1 and f2 bounded between a and b.

The perimeter-and-area symbolic form is a case where a person sees an integral as representing an area under a curve. An infinitesimal then describes the domain, or bottom perimeter, that bounds the area. Important to this form is that the area of the region is not subdivided into parts as would be done in a Reimann sum. The function-matching symbolic form resembles the rote procedure of calculating an anti-derivative. Jones argues however that students are doing more than just reciting back memorized procedures but are indeed giving meaning to the symbols of the integral. Jones gives

22 an example where a student clarifies that work displayed as 2x−32− x dx cannot be written 11 down correctly. The student explains “in an integration the dx is always essential, because it shows that this entire thing [waves hand over the integrand, “2/x3–x2”] is a derivative of x.”…“The fact that this entire thing is sitting right next to each other, and dx outside, means that basically this entire function [motions hand over “2/x3–x2”] is the derivative of an original function” (p. 129). And then later the student explains “So I think it’s just for the sake of organization just to have the dx in there, to signify that this is the derivative of the original function” (p. 130). Jones here has used multiple and consistent statements made in different parts of his interview with this student to infer that a) this student does see the dx as a symbol to invoke antidifferentiation and b) function matching in this context really is a symbolic form in that the student appears to see taking an integral as an inverse process that “undoes” a differentiation.

There is an important distinction to make between Sherin’s Symbolic Forms and what I am going to call Composite Symbolic Forms. Sherin discuss that the “…particular schemata associated with forms are relatively simple structures, involving only a few entities and a small number of simple relations among these entities.” He compares Symbolic forms as being similar to diSessa’s p-prims in that they are “presumed to have relatively simple structures.” (Sherin, 2001, p. 491). A student’s conceptualization of integration may not be a single entity – rather, the full symbolic form of integration might include many smaller resources such as ideas of area, boundaries, limits, anti-derivatives, summations and/or differentials. Additionally, each of those smaller units may be made up of even smaller units. At this point in time, in the literature, symbolic forms of integrals are referred to as simply, “symbolic forms”. After very much appreciated conversations with Dr. Corinne Manogue and Dr. Benjamin Schermerhorn on this topic I think that it is important to make clear that the symbolic form of an integral is most likely something with a lot more structure than what Sherin initially proposed. Due to the nature of this internal structure I will adopt the term composite symbolic form to describe the combining of the various conceptual schema each of which might not be fixed in their cognition when used by students when working with a symbol template of an integral. Additionally, the ability to “solve” an integral by writing down a statement such as 5 5x23 dx=+ x C correctly on a piece of paper, does not mean that the same students would be  3 able to correctly interpret a corresponding Riemann sum or explain the meaning of the C that they diligently included in their solutions. In a constructivist framework, calling a student’s conception of integration as either correct or incorrect is an incomplete view of their knowledge and abilities. The “composite” aspect of composite symbolic forms can be seen in Schermerhorn and Thompson (2018) where they have extended Jones’s work by looking at the nature of symbolic forms invoked by students in a vector calculus context. Schermerhorn and Thompson created an unconventional spherical coordinate system and in an interview asked students to construct differential length elements dl , differential volume elements and perform a check that the

volume element created was right. In the process of analyzing the students’ work the researchers identified several previously identified symbolic forms and proposed a few new symbolic forms

including the magnitude-direction symbolic form, = , that was used to denote a vector expression including the magnitude of a quantity (having units) and a unit vector to indicate a specific direction. This form was used to help determine the differential inside an integral when converting from one coordinate system to another. For Schermerhorn and Thompson,

was one component of many within the full integral. In order to see how the presentation of integrals in introductory textbooks align with Jones’ symbolic forms of integrals, Pina and Loverude (2020) investigates sample problems in five calculus-based physics textbooks used commonly in the United States. Four of the five textbooks, including the textbook that the majority of the students in my study were likely to have been assigned as required reading (Knight, 2016), present the majority of their sample problems in the framework of the adding-up-pieces symbolic form. Interestingly, this group also discovered that textbooks presented integral problems in a fourth manner that they called “Procedural” that manifests itself as integration being used as a tool and often focused on deriving an algebraic expression while not paying attention to a conceptual schema. Pina and Loverude (2020) see similarities in the Procedural presentation as a process that Bajracharya and Thompson (2016) describe as the analytical derivation epistemic game: “Playing this game involves establishing the relationships between different variables for physical quantities in hope to evaluate the required physical quantity. The knowledge base students may use in this game are procedural, symbolic, computational, and, less often, conceptual resources” (p. 7). An integral that appears as a “necessary step in the derivation of an equation” is coded by Pina and Loverude as Procedural. For example when attempting to find the decay constant for charge on a capacitor in an RC-circuit Halliday, Resnick and Walker (Halliday et al., 2010, p. 885) starts with the q “appropriate loop equation for the circuit… − −IR = 0 ”. This is followed by the statement C dqq “When we substitute I = dq/dt into this expression, it becomes −=R ”. After one more dtC step of algebraic manipulation Halliday goes on to state “Integrating this expression, using the

qtdq 1 fact that q=Q at t=0, gives =− dt ”. The final step of algebra yields q(t)=Qe-t/RC. Q q RC 0 This use of an integral, Pina argues, is distinct from a symbolic form in that the integral is acting to further a derivation largely divorced from conceptual reasoning. It is unclear how important textbooks are for students in the development of resources, schemas, and symbolic forms. Several studies (Cummings et al., 2002), (Stelzer et al., 2009, p. 185) (French et al., 2015) have found that textbook usage is not particularly high by students in introductory calculus-based physics classes. Podolefsky & Finkelstein (2006), for example, found that even though 97% of their introductory physics students purchased the textbook, only 37% reported that they regularly read the textbook. This dissertation does not, for the most part, address how students acquire the knowledge that they demonstrate when solving calculus problems. The important takeaway from Pina and Loverude’s work is that many textbook authors who are individuals who spend a lot of time thinking about how physics problems in introductory college sequence should be taught, spend a lot of time stressing the small-pieces symbolic form. To summarize the aspects of Jones’s as well as Schermerhorn and Thompsons’ work on which I will build: • Symbolic forms consist of a mathematical template that a person combines with a conceptual understanding of a physical phenomenon. • Currently the literature is exploring what is being called symbolic forms of integrals. Inside an integral template, there can be multiple symbolic forms. I will thus refer to such symbolic forms as composite symbolic forms.

2.7 Graphical forms Graphical forms represent a new line of investigation first and currently being pursued in the Chemistry Education Research community (Rodriguez, Bain, Towns, et al., 2019) (Rodriguez et al., 2018). Similar to symbolic forms, graphical forms consist of a conceptual schema united with a graphical template. Three graphical forms have been identified (Rodriguez, Bain, & Towns, 2019) and are shown in Table 2-2.

Table 2-2

Summary of Rodriguez, et al. (2019) Graphical Forms

Graphical form Conceptual Schema Steepness as rate Varying levels of steepness in a graph correspond to different rates. Straight means A straight line indicates a lack of change/constant rate constant Curve means A curve indicates continuous change/changing rate change

Rodriquez annotates their version of this table with the note that they “…anticipate more nuanced versions of these graphical forms exist where individuals focus on the overall shape of a curve or evoke ideas related to the mathematical definition of the derivative/slope as part of their reasoning.” During the analysis of data in Chapter 7 we will see evidence of students communicating graphical forms along with related symbolic forms.

2.8 Summary There are two main themes in the prior research upon which I will construct my work. First, the “Pieces of Knowledge” framework introduced by diSessa and built upon by Bing, Redish, Elby, Hammer and others when developing the “Resources Framework” offers the concept of cognitive resources as fine-grained pieces of knowledge. Resources can be activated singularly or in tandem with other resources. Framing can influence which resources get activated in a problem-solving process. In this dissertation I will identify and categorize the small pieces of knowledge students communicate while solving physics problems that require integration of vectors. The “Symbolic Forms” framework introduced by Sherin provides a way to understand a student’s combining of conceptual understanding of a physical phenomenon with an equation. Sherin’s work has been extended to describe composite symbolic forms of integrals as well as graphical forms. In the analysis of resources used by students while problem solving, I will identify symbolic and graphic forms used by these students.

– Coordinatization in Problem-Solving and the Coordinatization Framework

3.1 Introduction In this section I introduce a few problems that from my experience as an instructor seem to fit into the “difficult for students” category and require both coordinatization and integration. I then show how the theoretical frameworks discussed in my literature review might help inform an analysis of student difficulties with these problems and the acts of coordinatization that are produced by junior-level physics majors’ attempts to solve them. At the end of this chapter I introduce the Coordinatization Framework.

3.2 Coordinatization of a scalar starting-point equation As discussed in Chapter 1, a common topic covered by students in introductory physics are problems that require calculation of Center of Mass. Figure 3-1 shows a sample problem worked out in a textbook (Knight, 2016) currently used at many colleges and universities in the United States.

A significant feature to notice of this sample solution is the use of the “starting-point 1 xxdm= equation” cm  . A starting-point equation in this discussion is an equation that Mtotal describes a either a fundamental physics concept or an equation that is a common starting point in the problem-solving process. Following Bing and Redish’s discussion (Bing & Redish, 2009) as described in section 2.4 students may frame an starting-point equation as such by “Calculation”, “Invoking Authority” or perhaps “Physical Mapping” – and, important to Bing’s discussion, a student may switch from one framing to another framing in a relatively short period of time during an interview. Additionally, a student may utilize an iconic equation with a symbolic form while trying to combine with it the physical concepts of this problem with the mathematical features of the equation. In Table 3-1 I have listed some possible conceptual understandings of the various terms in the center of mass equation that was used in Knight’s sample problem above. In listing these I do not propose that students will, or even need to, invoke the resources described here in order to solve or understanding the problem – I am only listing possible paths or reasonings a person might take or have when applying the iconic equation to a center of mass problem. Table 3-1

Possible Cognitive Resources That Could Be Demonstrated in the Center of Mass Sample Problem

Piece of the Center of Mass Iconic Equation A possible physical concept of the problem xcm center of mass X position along an x-axis Dx small piece of space Dm the mass of dx (M/L) uniform linear mass density  we will be summing an infinite number of infinitesimal units there is a physical beginning and ending to the physical quantity  we are summing axes drawn a coordinate system is displayed L a length that can also be interpreted as a coordinate is displayed

As described in section 2.6 Jones (2013) discusses the symbolic forms for integrals that he observed for the nine students included in that study. The forms were: adding up pieces, perimeter and area, and function matching. Using any of these forms, students might do very well at trying to integrate a given analytic function or perhaps draw a curve on a graph. Paying attention and attempting to distinguish between possible symbolic form utilization while students are solving center mass will be particularly interesting. Again, as can be seen in Table 3-1, there are at least 5 resources that are part of the starting- 1 xxdm= point equation cm  that are directly displayed in the equation, and then many Mtotal other resources described in the solution presented in this sample problem that a student may need to pay attention to. I propose that the sheer number of resources that a novice needs to juggle in order to solve a problem of this type might be taxing on a student’s working memory and might be one aspect of what makes problems of this type difficult.

3.3 Coordinatization of a vector starting-point equation The starting-point equation for the electric field of a point charge is:  1 q E = 2 rˆ . 4 o r When considering a continuous distribution of charge, integration can be used to find the electric field at a point in space and a possible version of the equation that will need to be 1 dq addressed looks like Er= ˆ when trying to find the electric field at a point in space. In  2 4 o r this vector integral version of the equation for a distribution of charges there are a few more items have been added to the problem to which an individual needs to pay attention: the vector over the E means that there is a direction of the electric field for the overall distribution, and, the hat over the r specifies that not only does the position of a dq need to be paid attention to, but the directional location of each infinitesimal charge within the distribution of charges will need to be considered.

Directionality is an important aspect of both 퐸⃗⃗ and 푟̂ as well as coordinate systems in general. As discussed in section 1.5, a notational shift happens in many introductory calculus- based physics textbooks and upper-division physics textbooks. The lower-division  1 q E = rˆ 4 r 2 o is presented at the upper-division level often as  1 q   E =   3 (r − r') 4 r − r' o . The upper-division equation format of the electric field equation seems to stress the importance of being clear about the starting and ending coordinates involved in the calculations that use the equation. Figure 3-2 is another sample problem from the same textbook as Figure 3-2. This example details a solution for finding the electric field at the center of a uniform bar of charge. Figure 3-2 has some remarkable explanatory differences to it compared to the center of mass sample problem in Figure 3-1. First, a symbol for linear charge density  is clearly denoted in the problem. Then, an impressive amount of space is dedicated to leading the student through a detailed discussion of a process of breaking the bar up into a series of pieces of charge, each of which are going to be treated as a point charge. The total electric field of the bar is then the sum of the electric fields from each piece of charge. Finally, finite pieces of charge represented by Q are fitted inside pieces of bar of length y. Sums are turned into integrals with the explanation “Now we’re ready to let the sum become an integral. If we let N →  then each segment becomes an infinitesimal length y → dy while the discrete position variable y1 becomes the continuous integration variable y.” (Knight, 2016) This problem is given to students typically around 10 weeks after the center of mass problem during which time students have perhaps been expected to have gained more mathematic sophistication. The amount of discussion focused on small pieces of charge and space honors the research described by Jones. As we saw in the literature review, Jones observed three symbolic forms used by students: adding up pieces; perimeter and area; and function matching. The long discussion included in the sample problem in Figure 3-2 suggests that the author is attempting to impress upon students the importance of adding up pieces in this type of problem.

3.4 The Coordinatization Framework Based upon my definition of coordinatization and prior established work on diSessa’s knowledge in pieces and the Resources Framework I will now propose a method of analyzing student work that I am going to call the Coordinatization Framework. The Coordinatization Framework first recognizes that in many physics problems there exist multiple ways in which students can and maybe need to apply a coordinate system. Examples given in this chapter include: converting a coordinate independent infinitesimal to a coordinate dependent infinitesimal (e.g. a dq to a dx); assigning limits of integration to a problem; and figuring out how to incorporate an 푟⃗ to the solution of a problem. The Coordinatization Framework then posits that each “act” of coordinatization that a student performs when solving a problem is a proxy for a cognitive resource or perhaps several cognitive resources being activated by the student. For example, when a student converts a dq to dx while solving a problem on paper, that act of writing dq=dx might represent the student activating the resources of linear charge density, infinitesimals in calculus, and point charges in physics. For another student writing dq=dx might just indicate that they are activating dimensional analysis as a way to derive a relationship. What is important is that just as a cognitive resource might be a superposition of smaller more fine-grained pieces of knowledge, acts of coordinatization have internal structure. Chapter 5 of this dissertation will list and describe many of the acts of coordinatization that will be identified by observing student written responses to a problem that requires vector integration to find the electric field due to a distribution of charge. This chapter will then explore some of the data to see if acts behave in ways similar to resources. Chapter 6 presents the work of students who took part in a clinical interview to solve a center of mass problem and then the same electric field problem seen in Chapter 5. This chapter will focus on just two specific acts that showed a high level of co-occurrence in the previous chapter. This analysis will then be used to infer the composite symbolic forms of integrals used by these students and previously identified by Jones. Finally, in Chapter 7 one student’s clinical interview will be analyzed in full with attention paid to all acts of coordinatization as they happen. From this process I will identify symbolic forms and graphical forms new to the literature used by this student.

– Methodology and Methods

4.1 Principle investigators The Principle Investigators of this research are the Paradigms in Physics team at Oregon State University. The Paradigms in Physics project started in 1997 when Oregon State University’s physics department, with aid from a National Science Foundation grant, restructured the upper-division physics sequence. The project has continued now for over 20 years with continuous research upon student learning occurring throughout. The research team that has grown along the Paradigm in Physics project has a large history of research projects from which to share methodology, practices and finding. The guidelines for the collection and use of this data is spelled out by the team’s protocol that has been approved by the University’s Internal Review Board. In this protocol concerns about possible harm to students and faculty are mitigated by the anonymizing data and by requiring participants to opt-in and to be able to opt-out at any point.

4.2 Participant backgrounds The students who took part in this study are from Oregon State University. The physics department at Oregon State University has invested much time researching the junior year experience in conjunction with the “Paradigms in Physics” project that has been underway for the past 20 years (van Zee & Manogue, n.d.). In a traditional physics sequence, juniors would be taking in parallel three or four courses such as Electricity and Magnetism, Quantum Mechanics, Optics and Thermodynamics. At OSU, themes were identified within these traditionally offered course that now serve to create a single sequence of courses that focus on each theme, one at a time. Three main populations of students took part in the surveys and interviews of this project. “pre-juniors” are students who were in Ph 320 and largely consisted of Physics majors in their first week of their junior year – many of these students transferred in from other institutions and there were a few students who were not physics majors. “post-juniors” are student who were in the first week of Ph 431 — these students are largely Physics Majors in their Senior year most of whom took their Junior level physics classes at OSU. The students who took part in the video

interviews were in their third term of Junior level physics at OSU – all six of these students were physics majors although one of these six students was also a math and engineering major.

4.3 Methodology Creswell (2014) spells out data collection procedures that are appropriate for use in a thematic analysis. • The researcher should identify the purposefully selected sites and individuals for the proposed study. In our case OSU Junior Physics majors were chosen. The Junior level aspect was important as these students are neither novices nor experts in solving this type of problem. While trying to break new ground on the learning process of this topic that students find difficult, it might be useful to take a snapshot of a group of students while they are still in the middle of the learning process. • The number of participants needs to be considered. Saturation is an important concept in a thematic analysis. Charmaz (Charmaz, 2006) says that you stop collecting data when the categories you develop are saturated – this happens when fresh data no longer generates new insights or exposes new properties. • The researcher should be aware of the type or types of data to be collected. Multiple forms of data often need to be collected – in qualitative research this is referred to triangulation. In order to elicit robust data from the students, I, with feedback from members of the Oregon State University Physics Education Research team, focused on choosing two physics problems that: • required use of at least a partially coordinate independent representation • contained an infinitesimal that would require coordinatization • required a definite integral to be solved • contained an integrand that would need to be coordinatized • contained a vector quantity • contained a function that changed with position in space In order to assist with triangulation, two data collection methods were employed. First, the two selected questions were given to six students in a recorded interview. Each student met with

me an as interviewer who presented each question to the student and then would ask probing questions. Triangulation is a qualitative research strategy that tests validity through the convergence of information from different sources and/or methods (Creswell, 2014; Maxwell, 2013). Thus, one of these two questions was later selected and presented to a class of students with students answering the question individually on a piece of paper. I felt that the problem should be challenging, so that rote-memorization or superficial understanding would be insufficient to be able to solve the problem. Additionally, the problem should not be impossibly difficult and be of a nature that this population of students could see that they could be reasonably expected to be able to answer the problem. The guidelines for the collection and use of this data is spelled out by my protocol that has been approved by the University’s Internal Review Board. In this protocol concerns about possible harm to students and faculty are mitigated by the anonymizing data and by requiring participants to opt-in and to be able to opt-out at any point all of which are important aspects of ethical qualitative research.

4.4 Methods In this section I will describe the protocols developed for the collection of both written and video data as well as the coding systems that were developed for the analysis of the data. I chose two problems as diagnostic tools to probe how Junior level physics majors coordinatize problems that requires integration. The first problem, shown in Figure 4-1, asks

Question 1:

This bar (of length L) is made in such a way that the material gets more dense as you move from the left side to the right side. On the far left side the material has a linear mass density of o. On the far right side it has a linear mass density of 2o. The mass density increases steadily as you travel from the left side to the right side.

What is the total mass of this bar?

Where is the center of mass of this bar?    rdm rcenterofmass = M total

Figure 4-1 The center of mass question was used only in the video interviews.

students to find the center of mass of a bar with a linearly increasing mass density – center of mass is covered around week 14 of many introductory calculus-based physics sequences. For the second problem, shown in Figure 4-2, students are asked to find the electric field for a uniform bar of charge at an asymmetric point above one end of the bar. This is a problem that should be similar to common problems encountered by most students in their 20th-25th week in a typical introductory calculus-based physics sequence.

Question 2:

Below is a rod that is positively charged rod. The charges are uniformly distributed and are fixed in place.

What is the E-field at a point a distance d above one end of the charged rod as shown in this picture?

Figure 4-2 This question was used both for the students taking the written diagnostic as well as the video interviews.

Due to time-constraints in the classroom, only question 2 was used for students taking the written diagnostic. Questions 1 and 2 were asked to the six students who participated in the video interviews. Chapter 5, 6, and 7 will present the analysis of the data.

4.4.1 Written data protocol Pilot interviews with members of the OSU Physics Education Research team who also had been Teaching Assistants for many upper-division physics classes were given. These interviews provided feedback on time reasonableness of the diagnostic tool and to elicit ideas on follow-up questions that might be added to the interview protocol. Three sets of data were collected. The first data set came from 39 students on the 2nd day of “Ph 320 Paradigm: Symmetries” near the beginning of class. The majority of these students were now in their first week as a junior-level physics major. Students were told that their work done on the question would be kept anonymous, would not impact their grade in any way, and

was being collected in order to improve physics understanding and advance Physics Education Research. Students sat at circular tables and worked independently on the question shown in Figure 4-2 for a maximum of 20 minutes. A year later, this very same question was then posed in the same manner to 32 students on the 1st day of “Ph 431 – Capstones in Physics: Electromagnetism”. This class largely, although not exclusively, consisted of students from the original Ph 320 class. An additional set of student written solutions (N=26) was collected in Spring 2018 at the beginning of “PH 422: Paradigms in Physics: Static Vector Fields”. This class was a different set of students than the Ph 320 and Ph 431 students. This data was collected as the paradigms sequence had been modified in this year making the data set potentially interesting as a source of new problem-solving strategies by students.

4.4.2 Written data coding After the first data set was collected, all 39 samples were looked over first by me with the goal identifying ways that students demonstrated coordinatization while solving the problem. A list of possible examples of student coordinatization acts was put together. After discussions with Liz Gire, five categories were created in which individual acts could be placed. In line with a thematic analysis, this list of categories was purposely left non-exhaustive with the idea that if new categorization of coordinatization could be found, the list would be expanded.

Figure 4-3 Question posed to students and an example of one student’s response to the question.

Once these categories were determined, I performed one interrater reliability test with the OSU Physics Education Research group. During this 30-minute exercise, members of the team were given the five main coordinatization categories and were each asked to analyze 6-8 student worksheets. Each team member marked acts of coordinatization with colored highlighters with each color representing one of the categories. A separate color of marker was to be used if a possible act was found that could not easily be put into one of these categories, and, the team was also told to use multiple colors on an act if that act could possible fit into more than one category.

Table 4-1 lists the 5 categories of coordinatization what were created along with an example of an act from each category.

Table 4-1

Categories of Coordinatization Developed

Coordinatization Category Example of an Act from this Category

Drew Axes

Infinitesimal Indicated in the Sketch

Equation Manipulation of the Infinitesimal

Dealing with the vector nature of E or r-hat

Methods of dealing with the limits of

integration

At this point I went through all 39 responses one category at a time. For each student in each category I attempted to find all possible incidents of that category within each sample. A clipping was taken of only the parts of the sample that I felt directly related to that particular category and then I wrote a short description of that act. From this I could compile a “category tranche” that contained all examples of acts in each category. Table 4-2 shows a portion of the “Drew Axes” tranche.

Note that in the final count of how many student performed an act of “Drew Axes” examples such as Student 14 where, as seen by my comment, it is not clear if axes were indeed drawn, final counts always error on the side an unclear act being counted as an actual act.

Table 4-2

An Example of Part of the Drew Axes Tranche of Data from Ph 320 Student Student Example of Drew Axes Description of this Act Number

An x and y axis seems to be indicated. A Student 12 zero of x and a zero of y also seem to be indicated.

Student 13 None evident

Student 14 Maybe axes are indicated.

An arrow labeled x is shown that goes from the right side of the bar to dx. I Student 15 suspect that this is not an axis but rather an "integration arrow". In a different place on the page, a coordinate system is drawn with the point P at the origin.

Student 16 There is possibly a vertical axis indicated placed in the middle of the rod.

After all acts had been recorded within a tranche, a new set of subcategories was created within each main category was developed. Table 4-3 shows an example of the subcategories of the “Manipulating the Infinitesimal” categories. The results from this analysis of the written data is presented in Chapter 5. A year later, in Fall 2017, the same question was given to the same cohort of students who were now in Ph 431 – “Capstones in Physics: Electromagnetism”. One more set of data

Table 4-3

Subcategories Developed from the Fall 2016 Ph 320 Dataset to Describe Student Actions within the “Manipulating the Infinitesimal” Category. (N=39) Sub-Category Code N % Conversion from dq to dx (correct) 12 31% Conversion from dq to dx (incorrect) 8 21% Non-horizontal infinitesimal (e.g., dr, dy) 7 18% Source-like infinitesimal (e.g., dE, dq) 16 41% Double infinitesimal (e.g., dxdy, dqdx) 2 5% Infinitesimal in the denominator 4 10% Description of an infinitesimal in words 5 13% was then collected in Spring 2018 on the first day of Ph 422 – “Paradigms in Physics: Static Vector Fields”, this time with a different cohort of students who were at the end of their Junior year and who had taken classes during a year in which a re-organization of curriculum occurred. In both cases, the same procedure was used when analyzing this data. Attention was paid to look for possible new examples of acts of coordinatization and keeping in mind that perhaps even new broader category themes could be observed. The full data sets for these three classes are presented in Appendix A. Later, it was discovered that a sixth category of coordinatization was missed in the original coding. This sixth category of coordinatization can be described as “coordinatization of the variable r”. A common act that fits in this category would be the student production of something like “r2 = x2 + d2”. Another student act that dealt with the coordinatization of the variable r can be seen in Student 13 who wrote “r2 = dl2 + d2”. Student 13’s act described here was included as an act in the “Equation manipulation of the Infinitesimal” and is a reminder that acts have structure and components of one act might also be a component of another act. I decided that adding in the 6th category at this time is not necessary as there currently existed enough data with which to perform meaningful analysis.

4.4.3 Video data protocol Six students who were in the third term of their junior year of physics classes were asked to solve the two problems while being asked probing questions from an interviewer. The first

problem they confronted required the use of a one-dimensional vector integral in order to find the center of mass of a rod with non-homogenous density. The second problem asks student to solve is the same vector integral question that was asked in the written data that was collected and shown in Figure 4-2. An interview protocol was developed using repeated input from the OSU PER group during iterations that came out of pilot testing with graduate students in physics at OSU. An oral request for students interested in taking part in this research project was made to students taking traditional junior-level courses during spring term. All six students who responded were interviewed over a period of 5 days in May of 2016. Students were paid $10 for participation in this study. Interviews were recorded by a single video camera with wireless mic. Students demonstrated their written work on a 61cm by 91cm white boards with access to a variety of colors of fresh whiteboard markers. No eraser was provided, and students were asked to not erase work but rather to cross out anything they thought was wrong – this turned out to be difficult to fully enforce. Before each interview the following statement was read to the student: “Thank you for taking part in this interview. I greatly appreciate you taking your time to help me investigate how people approach physics problems. I want you to know that this interview and in no way will impact your grade in any class. And please don’t discuss what we do with other people as I will be performing this same interview with many people some of whom you might know. I am going to have consider one or two physics problems. My main goal is to understand how you personally go about reasoning as you solve a problem. Thus please feel free to talk out and explain as much as possible what you thinking as you do it. If I ask you follow-up questions, don’t worry that you are “wrong” – I am just trying to make sure that I understand your reasoning.”

The student was then given “Question 1” shown in Figure 4-1 to read quietly to themselves after which they began work on the problem. While working out the problem, the interviewer would occasionally ask questions that generally sought to have the student explain what they had just written. After the student completed Question 1 the same procedure was followed for “Question 2” shown in Figure 4-2.

During the interview, after a whiteboard was filled with work, a blank whiteboard would be provided with the original whiteboard made still visible to the student. After each interview, photos were taken of all the whiteboard with each whiteboard being numerically labeled so that the order of the work could be recalled. During the analysis, it was understood that the final white boards did not necessary contain everything a student had written as the “do not erase” rule was hard to completely enforce – video analysis later helped capture this lost data. Additionally, immediately after an interview, a short discussion of what had occurred was typed up to assist future analysis.

4.4.4 Video data coding Although the video data was collected before the written data, the coding for the written data was applied to the video data with attention paid to the possibility of new categories or subcategories of coordinatization presenting itself. Pseudo transcripts of all videos were made with the intent of identifying parts of the transcripts where coordinatization acts seemed to be occurring. From the results of creating a coding for acts of coordinatization that can be found in Chapter 5, I found that there were two acts of coordinatization that seemed to correlate with each other at a much higher rate than other correlating acts. Chapter 6 looks at the correlation of these two acts. Using the pseudo transcripts as a guide, detailed transcripts were made from all interviews at the points in the interview that were helpful in describing student thinking based around those two specific acts. Chapter 7 focuses on one student and all acts of coordinatization performed by that student. A complete transcript was made of this student’s video – approximately half of the video was transcribed using the software package InqScribe, the other half of the video was produced using the free version of the video software VLC Media Player along with Microsoft Word and two computer screens. Using either InqScribe and VLC Media Player, doing a first run-through of the transcript at about 60% of normal speed seemed to be about the speed at which I could type for most students; a second pass at normal speed was then good at finding errors. Parts of the video that were of particular importance often required many passes through in order to fully capture gestures, facial expressions and/or hard-to-hear words.

– Acts of Coordinatization in students’ written work

5.1 Introduction The Coordinatization Framework treats acts of coordinatization as proxies for cognitive resources. This chapter presents the results of the analysis of 97 students’ written work solving the problem shown in Figure 4-2 where students are asked to find the electric field at a non- symmetric point above one end of a bar of charge. The 97 students collectively produced 489 observed acts of coordinatization. These 489 individual acts of coordinatization were sorted into 32 different classifications acts. The 32 classifications were then sorted into 5 main categories. In section 5.2 I will first present each of the five categories one at a time and describe the specific types of acts that fall inside each category. In section 5.3 I will present just one category of acts and demonstrate how occurrences of acts within that category have several of the attributes of resources.

5.2 Acts of Coordinatization The 32 classifications of acts of coordinatization discovered are listed and described in Tables 5-1 through 5-5. The complete data set of acts of coordinatization that were observed in the three physics classes that were surveyed are presented in Appendix A. As described in Chapter 4, these 32 act classifications were sorted and then grouped into 5 main categories. Each category and the acts within each category are detailed below.

5.2.1 Coordinatization category: drew axes Drawing axes is a method in which a student visually articulates information such as the orientation of the coordinate system, the origin of the coordinate system, and a naming/labeling system for the coordinates used. Table 5-1 describes the methods in which students included axes in their work as well as an example from student work.

Table 5-1 Coordinatization Category: Drew Axes Act of Description of Act Example of an Act Coordinatization

The origin of a coordinate system is Origin Denoted clearly denoted or labeled.

At least one axis labeled with a Axis labeled coordinate such as x or y.

Floating axes indicate just the orientation of the x- and y-axes and Floating Axes not the location of the axes with respect to the physical quantities.

Including the “zero” of at least one End-point shown coordinate is indictated.

Rod as axis Using the rod as an axis without including a separate axis line.

Two possible acts of coordinatization were seen that were difficult to chacterize due to uncertainty about < what the student was trying to Other communicate. In this case, for example, it is unclear if the arrows drawn represent coordinate axes or electric-field vectors

A line that can be interpreted as an Unlabeled Axes axis with no label.

None

5.2.2 Coordinatization category: limits of integration The limits of integration placed on an integral sign can communicate the starting and ending coordinates of the axis along which a student integrates. During the analysis phase I discovered that there is a large number of combinations of various limits that could be used with the a wide selection of differentials during the problem-solving process for the problem analyzed. After several coding systems were created and then cast aside for having too many possible sub-categories to be useful for analysis, I decided upon the coding system shown in Table 5-2. Section 5.3 focuses on the category of “Limits of Integration” and demonstrates that the three acts listed in this category do function like resources in how they may activate or not when a student is problem-solving. Table 5-2 Coordinatization Category: Limits of Integration

Act of Coordinatization Description of Act Example of an Act

The limits of integration Limits consistent with consist of symbols and/or infinitesimal numbers that are related to the infinitesimal.

The limits of integration have different units than and/or Limits inconsistent with consist of symbols or numbers infinitesimal that are not consistant the infinitesimal quantity.

Integral sign with no An integral sign is written limits with no limits.

5.2.3 Coordinatization category: equation manipulation of infinitesimal Many students included coordinate independent infinitesimals such as dE or dq in their problem-solving process. In order to successfully solve the problem, such infinitesimals would need to be converted to coordinate dependent infinitesimal such as dx or dl. I have titled this process “Equation Manipulation of Infinitesimal”. Many students start with a coordinate dependent infinitesimal in their equation without providing explanation – this is considered a “manipulation” process as well.

Table 5-3

Coordinatization Category: Equation Manipulation of Infinitesimal

Act of Coordinatization Description of Act Example of an Act

(Q/L)dx, or variables that Proper (Q/L)dx happens match the intent of (Q/L)dx somehow appear at least once.

The starting-point-equation of the electric field, or an qdx or qdr happens attempt of this equation, is somehow sandwiched between an integral sign and a spatial infinitesimal. A dr appears along with evidence that the dr represents an infinitesimal along the dr hypotenuse that connects a point on the rod and the point

in space at which the integral is to be evaluated. The infinitesimal used indicates an attempt of dy (along vertical) integration along the vertical axis.

Indication that integration of dE or dE-vector the electric field is attempted.

This expression 푞 = ∫ λdl or q=integral(lambda dL) something functionally similar is presented.

Integration or summation with respect to charge is attempted dq (or delta q) with the symbol dq and/or q being used.

Table 5-3 (Continued)

Integration in 2D space is dxdy attempted.

Integration with one spatial dqdx or dqdr and one non-spatial variable is attempted.

infinitesimal in An infinitesimal appears in denominator the denominator

Integration with respect to an dtheta angle is attempted

The d in an infinitesimal is d = differentiate interpreted to mean differentiation

Words that describe Words infinitesimals or use of infinitesimals are presented

5.2.4 Coordinatization Category: infinitesimal in sketch Many students included representations of coordinate dependent and/or coordinate independent infinitesimals in their sketches. Some infinitesimals were indicated without a label. In Chapter 6 I will examine a co-occurrence observed in the data between students who include an infinitesimal in their sketch and students who obtain the correct expression dx in their integral.

Table 5-4

Coordinatization Category: Infinitesimal in Sketch

Act of Coordinatization Description of Act Example of an Act

Lines are drawn from at least multiple lines two points on the bar to the evaluation point.

A piece of, or point on, the dq bar is labeled dq.

A piece of, or point on the bar dx or dL or dtheta is labeled dx or dL. Or, an angle is labled d.

unlabeled infinitesimal A piece of the bar is denoted block but not labeled.

Words are used that indicate Words that many pieces of the bar need to be considered.

At least one vector labeled dE is indicated or many vectors, dEs drawn labeled or not, are drawn clustered together.

5.2.5 Coordinatization category: methods of dealing with vector nature of E and/or r. The electric field is a vector quantity. A common way to solve this problem is to integrate along a one-dimensional coincident to the bar of change in order to find the two- dimensional value of the electric field at a point in space. Students are thus working with the coordinates along one axis in order to find values that have to take two vector-components into consideration.

It turns out that a majority of students both in the written work and in the work produced during clinical interviews did not fully activate all the resources necessary to adequately take into consideration the vector nature of the electric field. To put this a little more bluntly, most students simply forgot at the start or near the start of their problem-solving process that the electric field is a vector quantity. During the clinical interviews when students were asked “What about the vector nature of the electric field?” most students were able to go back to the start of the problem and then correctly solve the problem. This is a clear indication that the vector-nature of the electric field is a concept that the majority of this group of students understand and to which they can apply appropriate mathematical equations. However, this group of students for the most part did not activate this particular resource until they were reminded that this is a consideration that one needs to take into account for this problem. Table 5-5

Coordinatization Category: Methods of dealing with vector nature of E and/or r

Act of Coordinatization Description of Act Example of an Act

Both sides of an equal sign vector consistency are either vector quantities or scalar quantities.

One side of an equal sign as a vector quantity while the vector inconsistency other side has a scalar quantity.

An attempt to manipulate an r-hat to r-vector changes r-hat into a different type of vector quantity

5.3 Looking at one category of coordinatization Acts of coordinatization on paper or in a clinical interview communicate to an observer what a student understands about how coordinate systems are used in the problem-solving process. Each act represents a fine-grained piece of knowledge. In the Coordinatization Framework each act represents an activated resource. Activation of resources can be based upon context – in this case the context for the student is a researcher coming into a classroom asking

the class to solve a particular problem with a request to show all work and elaborate their understanding. The Resources Framework then postulates that activation of one resource “tends to lead to the activation of a cluster of related resources” (2004, p. 16). In the Coordinatization Framework, acts of coordinatization like resources should be: • of small grain-size • sometimes activated, sometimes not • activated in clusters • sensitive to shifts in framing In this section I will discuss my observations in the “limits of integration” category in which I found the results more robust than I anticipated. A goal of this section will be to analyze several acts of coordinatization using the attributes resources described above.

5.3.1 An example of a “limits of integration” act as a discrete resource Student 2 took the written diagnostic both as a pre- and again post-junior. As a pre-junior when they are asked to find the electric field from the presented bar of charge, they do not ever write an equation with an integrand nor do they write a symbol for an infinitesimal. However, this student did write an integral sign with limits.

Figure 5-1 Student 2’s act of “limits of integration” as a pre-junior. Although there is no integrand or infinitesimal written after the integral sign, the integral does have limits.

In Figure 5-1, the student writes: “If  amount of point charge would need to integrate L  ”. There was no integrand or infinitesimal written after this integral sign. 0 Being able to express the limits of integration independent of any equation, drawing or infinitesimal strongly suggests that, at least for this student, the limits of integration is a cognitive resource independent of those particular resources that can be applied to this problem. A year later this student is asked the same question. This time, the student is able to include an equation and infinitesimal after the integral sign as seen in Figure 5-2.

Figure 5-2 The same student as in Figure 5-1 answering the same question a year later. This time the student expresses an equation but does not include limits on the integral.

Included with the equational work in Figure 5-2 is this statement:

In this second attempt at the problem, Student 2 does not put bounds of integration on the integral and explicitly references that the bounds need to be attended to. From this evidence, I claim that writing the limits of integration truly represents a separate cognitive resource from other acts of coordinatization. As is true of cognitive resources, this act is one that can be activated or not separately from other distinct resources involving the integral. Going back to the Resource Framework this could be an example of a person who has multiple resources related to this problem but these resources have not yet been closely enough linked together to form a full stable schema that could be activated as a whole.

5.3.2 An example of “limits of integration” as a resource tightly bound to an infinitesimal Student 116 reveals an example of a tight binding between the limits of integration resource and the infinitesimal in an integral. In Figure 5-4 this student demonstrates difficulties on paper constructing the appropriate integrand. At the beginning of their Junior year, they construct a double integral that they apparently attempt to integrate over both charge and space.

Despite the incorrect construction of the integrand, the limits of integration do match the infinitesimals displayed in their work. A year later at the beginning of this student’s senior year, Student 116 is still having difficulty setting up the integrand. However, this time, as shown in Figure 5-5, it is a different set of difficulties that seem to arise, namely, Student 116 is attempting to integrate along the hypotenuse distance from a point on the bar to the point at which the E-field was to be calculated.

Figure 5-5 A year later, the same student has a different, and still incorrect, infinitesimal in the integral, however, once again, the limits of integration match the infinitesimal.

In both years, Student 116 is not able to correctly construct the integral for this problem and in each year includes different infinitesimals in the integral expression. Despite their difficulty in constructing the integral, Student 116 does carefully and correctly list limits of integration that match the infinitesimals stated.

5.3.3 An example of “limits of integration” not always matching the infinitesimal Student 129, like student 2, produces work that suggests the discrete nature of the cognitive resource of the act “limits of integration correctly match the infinitesimal” and act in the category of “manipulation of the infinitesimal”.

In this case it is unclear if the student notices or feels it necessary to pay attention to the limits of integration written in the first line as being inconsistent with the dq in the integral. Only 10% of pre-junior students use limits of integration inconsistent to the infinitesimal written down. Student 129 both incorrectly and then, immediately after, correctly matches the limits to

the infinitesimal. These observations show that in the Coordinatization Framework the limits of integration are closely bound to the infinitesimal in most, but not all, students’ written work. The activation or suppression of the correctly-matches-infinitesimal act in the Coordinatization Framework parallels the activation and/or suppression of resources in the Resources Framework.

5.3.4 A possible shift in framing helps one student as another hits a wall Four students in the post-junior group attempt to integrate from 0 to L with respect to dr. These students attempt to integrate along the hypotenuse that connects a piece of the bar of charge to the point in space at which they are asked to evaluate the electric field from the bar of charge. This group tended to present work that suggests strongly that they have problems interpreting the meaning of the variable r in the starting point equation. One of the four students eventually manages to convert the 0 to L with respect to dr to a more correct form of 0 to L with respect to dL. In Figure 5-7 Student 113 specifically makes a point that they are having problems with the meaning of r by writing “but what does r = ?” Eventually they shift to the correct infinitesimals to match the limits of integration.

I might surmise that Student 113 experiences a shift in framing that eventually leads the them to correctly identify the meaning of r in the starting-point-equation for the electric field and then produce an almost correct conversion to dL. In Figure 5-8, Student 110 also attempts to integrate with respect to dr but was not able to connect the limits of integration to the infinitesimal in their integral. Student 110 writes the stating-point-equation for the electric field of a point charge as the integrand to be integrated with respect to dr’. The “Q → dq?” written along with the integral suggests that the student is attempting to reframe the question due to some perceived foreboding about what they have

written. The question of what to do with the dq and/or dr’ is not answered by the student in this work with the statement “…hit a wall” ending their work on the problem.

Figure 5-8 Student 110 ends their work without resolving how to evaluate the integral.

5.4 Summarizing these limits of integration acts as cognitive resources The acts of coordinatization of “limits of integration” do seem to behave as cognitive resources as the acts seem to be discrete in their activation and suppression and can be impacted by framing. Student 2, in two different years, activates the limits-of-integration act of coordination once and then the next year the included-infinitesimal act of integration, but never both acts of coordinatization at the same time. Student 116, again over two different years, always links the limits of integration to the infinitesimal in the integral even when the infinitesimal does not appear in a way productive to problem-solving indicating that the student held either: (1) two cognitive resources tightly bound to each other or (2) the integral sign and the infinitesimal work together as one cognitive resource in the manner similar to Jones’s function matching form discussed in section 2.5. Student 129’s work again suggests the discrete nature of the limits of integration as a cognitive resource that can operate independent of other acts of coordinatization. Finally, students 113 and 110 suggest the resource activation can occur, or not occur, depending upon a shift of framing. This analysis, along with the creation of the co-occurrence table in the next chapter, suggests that the acts of coordination behave in the Coordinatization Framework in ways similar to how resources work in the Resources Framework. This feature is important to establish as in future chapters I will attempt to analyze student thinking via the Coordinatization Framework.

— Co-occurrence of acts of coordinatization within a student’s work

6.1 Introduction The identification and coding of the observed acts of coordinatization allow me to track the co-occurrence of individual acts. In the resources framework it is expected some resources will be often bound with other resources. Tables 6-1 and 6-2 present the cooccurrence of acts with respects to all other acts. Creating co-occurrence tables allows for identification acts and co-acts that stand out in ways that might be explained by the theoretical framework discussed above. In Section 6.2 I describe some of the correlations that can be directly observed in the correlation tables and discuss how these results will inform later analysis. In Section 6.3 I focus on one co-occurrence that stands out in the data: students who correctly utilize the expression dx inside the integral during problem solving are also much more likely to include an infinitesimal in the sketch. Through transcripts of the parts of the clinical interviews where the student is performing either acts of including an infinitesimal in a sketch and/or manipulation of an infinitesimal I will investigate why these acts seem to coincide in the way that is observed.

6.2 Co-occurrence of acts of coordinatization Table 6-1 presents the incidence of individual acts of coordinatization and the co- occurrence of these acts for the pre-junior students who took the written diagnostic at the beginning of Ph 320. Table 6-2 presents the same information for the post-junior students who took the written diagnostic at the beginning of Ph 431. Separating the data in this way allows for observation of shifts in the acts of coordinatization students perform as they work out the electric field problem after a year of junior-level physics.

Table 6-1

Co-occurrence Table for Pre-Junior Physics Majors Taking at the Beginning of Ph 320 (N=39)

Table 6-2

Co-occurrence Table for Post-Junior Physics Majors Taking at the Beginning of Ph 431 (N=32)

6.2.1 Vector consistency Post-junior Ph 431 students were significantly more likely to maintain vector consistency in the equations they wrote down (23 consistent vs 7 inconsistent) than pre-juniors Ph 320 were (21 consistent vs 13 inconsistent). This observation fits in nicely within the theoretical framework of symbolic forms. With a year of additional physics and experience in related mathematical tools, one would expect to see a stronger combining of mathematical reasoning with the underlaying physical concepts.

This stronger combining could express itself with more consistency in how these students expressed their thoughts on paper. While examining the transcripts and video data from Problem 2 in this clinical interviews, I will pay special attention of how each of the six students describe their understanding of the physical nature of the electric field with their expressed mathematical notation. I will also pay attention to how the Coordinatization Framework can help identify symbolic forms used by students.

6.2.2 Infinitesimal in sketch and correct manipulation of the infinitesimal In Ph 320, people who correctly expressed the term dx were much more likely to have indicated an infinitesimal in their sketch. This was also true, to a slightly smaller degree, with the Ph 431 students. Analysis of this co-occurrence was published as a Physics Education Research Conference paper (Mulder et al., 2018). In the Ph 320 class, there were 12 students who correctly used  or Q/L in their manipulation of the infinitesimal. Of those 12 students 75% of them also included an infinitesimal in their sketch as did the Student 103 shown as shown in Figure 6-1.

For the eight students who had an incorrect infinitesimal, such as qdx, appear in their symbolic work, only three (38%) also completed an act of drawing an infinitesimal in their sketch. Many of the students who had an incorrect or incomplete symbolic implementation of dx displayed no pictorial representation of an infinitesimal and often just jumped into equations as can be seen in Figure 6-2. …….

Ten of out 39 of the pre-junior students added an integral sign and a spatial infinitesimal sandwiched around the general starting-point equation for the electric field from a point charge as did Student 7 in Figure 6-2. This phenomenon has been observed before (Meredith & Marrongelle, 2008). Meredith and Marrongelle attributed this type of reasoning as a “misapplied symbolic form”. Students who created an integral in this manner for this type of problem were attempting to apply a “dependence” symbolic form, they argued, for the kq/r2 in the integrand while not noticing that the units and thus physical meaning of the overall integral changed when a dr was multiplied at the end of the integral. For these students, the inclusion of the dr may be to convey that r is changing. In a discussion with Dr. Paul Emigh (personal communication, Aug. 6, 2020), we hypothesize that an integral set up in this way might represent a student utilizing a composite symbolic form with a template [ ]= ∫[… 푥 … ] 푑푥. […x…] is Sherin’s symbolic template for dependence where the x between the two ellipses signifies a variable that represents a physical quantity a person is considering. In our case, the x is not just one variable, but rather is the entire iconic equation kq/r2. This symbolic form would represent a conceptual schema in which a student has a conceptual understanding that small piece thinking will need to be utilized, understands that there is an “iconic” equation that has variables of they have come across such as Coulombs Law in this case, and with a desire to integrate over space which is denoted here by an explicit dx. The correlation effect between correct infinitesimal coordinatization and indicating small piece thinking continued to hold at the end of a year of Junior level physics, although perhaps not as strongly. By this point in their academic career, 15 out of 32 students now correctly produced the spatial infinitesimal although only 7 of these students had an infinitesimal drawn in their

sketch. The loss of an act in the process of correctly solving the problem is perhaps an indication that successful chunking and automation has occurred allowing students to skip steps on paper when solving a problem.

6.2.3 Pre-junior vs post-junior acts of coordinatization did not change Post-Juniors did not produce significantly many, if any, acts of coordinatization that were not produced by the Pre-Juniors. This suggests that a standard collection of students who have completed a year of introductory physics have been given access to all the tools necessary to solve problems that require analysis with vector integration. Of course, this says nothing about an individual’s prior exposure to or ability to produce acts of coordinatization. In at least one category, “Infinitesimal in Equation” we can see that there was a general migration of the class in the Junior year toward a correct expression of (Q/L)dx. However, with only one exception, there was still at least one student who used at least one of the various acts of “Infinitesimal in Equation” as a post-junior. Again, this could suggest a stronger combining of conceptual physical reasoning with mathematical reasoning. Both comfortingly and in accordance with Jones’s observation in student reasoning with integrals, the one act that post- juniors did not make was clearly confusing the “d” in an integral to mean differentiation.

6.2.4 r-hat and words of grief 6 students in Ph 320 and 6 students in Ph 431 each use what I call “words of grief” where they indicate that they do not quite know what to do with r-hat. Despite evidence discussed above that students in general were more likely to have more solid symbolic forms developed by the end of their junior year, there were still a number of students who articulated a not full understanding of at least the meaning of r-hat. While observing videos, I will pay special attention to student use of and discussion of their understanding of r-hat.

6.2.5 Other fun observations that I simply state Like most qualitative research, the data is rich. Here are a few additional observations that can be made from the two co-occurrence tables. I am going to state these observations here

without any deeper analysis with the hope that I or someone else will be able to more deeply analyze these results at a later date: • Ph 431 students were more than twice as likely to indicate axes on paper. • Nine Ph 320 students put a dr in an integral where r could be understood as to represent the distance between a point on the bar and the point in space at which E was supposed to be found. Only two Ph 431 students did this. • It was less common for a Ph 431 student to reference a dq or dE than a Ph 320 in an equation statement. In other words, students were more likely to immediately start with an integral written in terms of a spatial infinitesimal.

6.3 Analyzing a correlation of two acts by six students Analyzing written data allows for the identification and classification of acts of coordinatization students perform when solving a problem that requires vector integration. In this section, video recording and transcripts will be used for two main purposes: 1) to triangulate observations made with the written work and 2) to more deeply analyze the correlation seen in the written work with a student who includes an infinitesimal in a sketch to also be more likely to correctly coordinatize a coordinate independent infinitesimal. Figure 6-3 provides a reminder of the two questions the students were asked. All six students who participated in the video interview in one way or another eventually sketched an infinitesimal and correctly utilized the relationships dm = dl and dq = dl. However, one student required prompting before an infinitesimal in the sketch was produced while the other five students produced both acts nearly simultaneously throughout the interview.

Figure 6-3 These are the two questions posed to students in six video recorded clinical interviews.

By the end of this of this chapter I show that the original coding of the 32 acts of coordinatization seems consistent with these additional deeper interviews. Additionally, acts of coordinatization do function within a framework where activation, suppression and frame shifting can explain observed phenomena. Finally, I will describe two symbolic forms for what I call the physical infinitesimal that can help explain at least part of the correlation seen between including a sketch in a picture and the correct use of dx.

6.3.1 Student 1 – an unclearly drawn infinitesimal is instantaneously a sliver – triangulating one specific example of “infinitesimal-in-sketch” Student 1 included an infinitesimal-in-sketch and correctly included dx in their equations. The infinitesimal-in-sketch drawn by this student is shown in Table 6-3 and consists of a single line labeled x that has additional information associated with it such as r and . Of all the acts described from the work of the 97 students analyzed in the past chapter, only one other student drew an infinitesimal similar to the way this student did. A main premise in the coding system used was that if a piece of student work could conceivably be interpreted as an act then it was coded as an act. Observing whether or not this student did indeed intend the vertical line they drew as an infinitesimal is an important piece of triangulation to test the validity of the coding system. In Table 6-3 two examples of the student’s not-often-used method of indicating an infinitesimal-in-sketch are shown. The drawing and equation in the first row is produced when

L the student is directly asked to describe how they came up with the equation mxdx=  () in 0 Question 1. The student explains: This is some continuous function (pointing at (x)) that tells you the mass density as a function of x and then if you have some mass density at some… at some value x then you multiply it by a little sliver of x and you get by definition dm so just by definition dx is dm…some little sliver of mass… and we’re integrating so we are adding up continuously to get total mass.”

Table 6-3

Acts of Infinitesimal-In-Sketch produced by Student 1 Infinitesimal-Included-In-Sketch Evidence that coding matched student intent

Student states: “dm…some little sliver of mass…”

While describing the “sliver” in this drawing the student says: “Now I have dq in terms of something I can actually work with…”

Student 1’s single line labeled x is indeed meant by the student to represent an infinitesimal. Student 117 is the only other student who draws, in their case, just a single dot to represent an infinitesimal as seen in Figure 6-4. The single dot Student 117 draws is, similar to Student 1, connected with a line labeled r to the evaluation point for the electric field. Evidence in the video interview for Student 1 lends support to coding such work as acts of infinitesimal-in- sketch.

Figure 6-4 Student 117 is the only other student whose coded act of infinitesimal-included-in-sketch consisted of a single dot or single line. Like Student 1, the infinitesimal had an r and a coordinate associated with it.

Student 1’s infinitesimal-in-sketch and the inclusion of dx in their equation Table 6-4 compares Student 1’s infinitesimal-in-sketch with their correct-manipulation- of-infinitesimal. In Chapter 5 it is shown that students who include an infinitesimal in their sketches are also more likely include dx in their equation inside the integral.

Table 6-4

Investigating the Correlation Between Student 1’s Acts of Infinitesimal-In-Sketch and Correct- Manipulation-of-Infinitesimal Infinitesimal- Correct-Manipulation- Included-In- Student discussion of-Infinitesimal Sketch “if you have some mass density at some at some value x then you multiply it by a little sliver of x and you get by definition dm so just by definition dx is dm…some little sliver of mass”

While describing the “sliver” in this drawing the student says: “Now I have dq in terms of something I can actually work with” and this dq has an “r-squared” associated with it and that

“this r distance is going is actually going to be equal to, just by Pythagorean theorem…”

The sketch of the infinitesimal in Question 2 is produced immediately after the student produces the equation seen in Table 6-4 and without prompting from the interviewer. The student appears to be using the drawing to better interpret the equation. The student begins the problem solving by noting: This is basically another integration problem. Uh, If I recall correctly... so we know that the E-field is going to be kqq/r2. Um… 2 So what I am going to do is say that dE is instead kq1dq/r so therefore E is going to be equal to the integral of kqdq/r2.

Next, they describe their understanding of the equation dq = dL. The student uses the words “instantaneous” and “constant” in this problem similar to how they used “continuous” in question 1 where they were describing slivers of space and mass. So this is just the total charge divided by... assuming this is a linearly charged rod or uniformly charged then this would hold because this is the average charge density (pointing at the equation =Q/L) and now I am saying that the average change density is basically the instantaneous charge density (pointing at the equation dq=dL). When asked about what is meant by instantaneous density, the student responds: So for you know for the last part we did mass density so you have to say there is a certain mass density at some point in the rod. (with a hand gesture as shown in Figure 6-5 repeated many times

sweeping from left to right that seems to be indicating a lot of small pieces)

Figure 6-5 The student repeatedly sweeps his hand to the right in a way that seems to be representing many points of charge.

The student then connects the r and the limits of integration to the “sliver” in the drawing of the bar: so this is kq… Now I have dq in terms of something I can actually work with or know actually Um r-squared um. yeah And now we just integrate from 0 to… and actually it is constant… so yeah, we are going to integrate from 0 to L. Now we have the issue that our r is, this is d (pointing at the vertical distance of a triangle) r is something here (pointing to the distance between a dq and the evaluation point)… so probably we what we want to do is integrate…. Yeah, so this r distance is going is actually going to be equal to, just by Pythagorean theorem… And the student connects the infinitesimal to the sliver of x. …or, but, right here this is dl..that’s going to be really confusing (Erases the “l” in the “dl” and writes x) “we’re going to say this is ‘x’ so it is ‘x’” (And then erases and re-writes the x in the place where he had originally written it.) “and x is bounded between L and 0 (writing the inequality from right to left) So this is just some x some instantaneous length of the rod. So now we have the r- squared on the bottom which is going to be…

Student 1 explains how each symbol drawn in the sketch is associated to the “sliver” drawn and how that symbol is used in the equation. At least when determining r and the limits of integration, the act of infinitesimal-in-sketch was an important part of his problem-solving method. Jones discovered that the adding-up-pieces symbolic form of the integral was more likely to be productive for a student to be able to achieve a correct final answer. Student 1 clearly holds a conceptual schema of integrals as mathematical devices that added up pieces. By

looking at the acts of coordinatization associated with the creation and description of the integral more possible symbolic forms can be seen to be composite inside this student’s integral. In the example above, the student describes the conceptual schema that leads them to an equation template in which both the equations dm = dx and dq = dx fit. This equation template can be expressed as d = d. Student 1 is displaying additional combining of a conceptual schemas with parts of an equation template. While working through this problem the student discusses that the bar consists of many x’s where “x is bounded between L and 0”. Associated with each dq and every “instantaneous” x is an r that can be found by using Pythagorean’s theorem. The inclusion of the infinitesimal in the sketch assisted the student with the geometry that allowed for a correct use of r in the equation and allowed for the student to articulate how the limits of the integral were determined. An equation template for this mathematical thinking can be described as:

Figure 6-6 A proposed symbolic template to describe the symbolic form held by Student 1.

I argue here that the student constructs integrals in pieces. At least two of those pieces utilize independent symbolic forms: one for the coordinatization of the limits of integration as well as r and the other for the coordinatization of dq. I have proposed two new composite symbolic forms of integrals from this one interview. As I continue through the next five students, these two ways of communicating conceptual understanding combined with a mathematical template will be repeated in many of the students’ work. A different way of correctly producing dx will also present itself.

6.3.2 Student 2 – dq is a small piece of charge, but what is dr? Student 2 is the only student who does not indicate an infinitesimal in a sketch until prompted. For example, when the provided the starting-point-equation for the center of mass,  rdm rcm = , is pointed out to them the student immediately writes down dm = dr followed by a M tot unit check to make sure that the units of this expression work as shown in Figure 6-7.

Figure 6-7 Student 2's coordinatization of dm happens utilizing dimensional analysis.

When asked “How did you come up with this to begin with?” Student 2 points at a unit check they have written and says: “I don’t have dm… that is not something that is given. But I can build dm by taking the components of dm. Which is the mass density time the length. Especially since I am looking… length is something I can integrate over…. I want to turn dm into something that I can integrate over, which is r.” When next asked “What is the dm?” the student replies: “The dm is the tiny infinitesimal mass?” with an inflection at the end of the sentence as if it were a question. It isn’t until asked, “What does that mean, the tiny infinitesimal mass?” that Student 2 responds and attempts a sketch at an infinitesimal:

“It means… that is a really good question… it means, I guess that I usually that I usually think more in terms of volume… which is a little bit easier for me to quantify. If I want to know the volume of this it’s going to be (student draws this box) this is length 1, this length 2 this over here is length 3. The infinitesimal volume which is not that different from the infinitesimal mass except it’s a mass of…. That just taking on these things and making them absurdly tiny (draws the little dot in the lower right hand side of the cube as shown in Figure 6-8) into very very small pieces. So an infinitesimal piece of mass, I mean this (pointing at pictures) but with mass you know, if that makes sense. This is easier to draw and explain than mass.”

When asked about the 푟⃗ in the starting-point-equation the student responds,

The r-vector is the distance from your starting point to a point on the, on the bar. *sigh* so my question, I guess is, if… I understand this of kind of little mark dm is (pointing at the dm in the starting-point-equation), physically is, then I can check to make sure that this makes sense (pointing at the integral he has written in terms of r). Because right now I have r, after I have integrated of r…. I have r in the integral because this has an r in the integral. Because I’m not really too sure what this physically means (pointing at the dm in the starting-point-equation) it is hard for me to say, oh yeah, that this should or should not be in here and give this physical reason for it.

When solving the electric field problem, the student again confronts the issue of r. After re-writing the iconic equation for the electric field of a charge they write the equation shown in Figure 6-9.

Figure 6-9 Student 2’s first attempt at an equation for the E-field of the bar of charge.

Then after some more writing with talking that is hard to follow, the qdr in the initial Q equation is converted into a dq with dq being correctly described as dqdr= : L

Figure 6-10 The student modifies their initial equation for the E-field of the bar of the charge.

At this point, I ask the student to clarify what just happened. The student responds: “I have a Q, and I have an L. But I can integrate across that length. These are going to be the pieces (makes a tap, tap, tap, tap with hand across board) but I don’t want that, I want to cut q up into these little itty bitty pieces because q is what determines E the electric field not L so… um… I made some corrections to my original formula because that wasn’t just wasn’t true, this is actually true. um because I want infinitesimal little charges, and then I want to but I have or can have infinitesimal little lengths. And so if I multiply that by the charge density the same as we did when we multiplied length

or we wanted mass we had length so I multiplied by mass density. We’re going to do the same thing here. Up to this point, Student 2 has not included an infinitesimal in their sketch for the electric field question. However, using dimensional analysis “the same way as we did” in the previous problem causes the student a shift in framing. This leads not only to the correct dq = dr but also to seems to activate a resource of “little itty bitty pieces” of charge in the bar that determine “the electric field”. There are two main points in this interview that I point out: • In order to coordinatize dq Student 2 utilizes a conceptual schema that I call “units must match on either side of an equal sign” that goes along with the symbolic template d = d. This appears to be a symbolic form of the infinitesimal that is distinct from what Student 1 uses but still produces a correct manipulation of the infinitesimal. • The non-use of the infinitesimal-in-sketch act coincides with a still- developing conception of the meaning of dm or dq.

6.3.3 Student 3 – constructing a form where dq is in dx Student 3 spends a significant amount of time in their solution to both problems attempting to discern whether they should be integrating with respect to d or dl. Eventually, like Student 2, it is the utilization of dimensional analysis that allows the student to create the correct relation between the coordinate-independent and the coordinate-dependent infinitesimals. As can been seen in Table 6-5, Student 2 often attempts to match the limits of integration, the differential in the integral, and the infinitesimal drawn in the sketch. In the Coordinatization Framework it appears that these three acts are tightly bound for this student. In the last row of the table, the student’s difficulty in deciding which infinitesimal is the correct one to use in this problem is matched by a difficulty in verbally articulating the meaning of x.

The student starts question 2 by creating a sketch of a bar without an infinitesimal qQ included in the sketch along with an integral expression written as kdx .  xd−

Table 6-5

Time-Line of Infinitesimal-In-Sketch and Manipulation-of-Infinitesimal Acts by Student 3 < Student Work Description Student comments

The student’s first act of coordinatization is writing an integral Silent thinking sign with limits of integration. “So we need to integrate The original limits of integration of 0 over the bar to find each to L are erased and replaced when the change in linear density and differential d is written down. add them up.”

The student’s first act of infinitesimal- “I am trying to integrate over each little linear density in-sketch is the inclusion of d as it changes. So each of matching the d currently in the these is going to be a little equation. d…um, as it changes This image was drawn immediately “Although I should probably after the above image. be adding up dL’s.”

“I did something wrong… After creating an equation that This is a mass density plus a described  as a function of position, ratio times a mass density… this integral is produced. times a mass density... that will not give you mass.”

Student integrates with respect to “L” not “x” in the integrand. The units of the A new integral is written. result make the student aware that something is wrong.

“x is the distance you travel After significant prompting from the along the bar” interviewer about the meaning of x, L, “x is the mass density at a

and . This integral is produced. certain length away from the zero point”

When asked what the difference is between q and Q the student responds: The little q is this whatever point is at this distance (pointing at a dot at which the electric field was to be evaluated) But we need to create this… a dQ because it is just a little sliver of charge at point x” At this point in the discussion the student does add an unlabeled infinitesimal in the sketch they had earlier drawn, as shown in Figure 6-11.

After the above modification to the sketch, the student works on creating an equation for the infinitesimal saying: But the dQ is constant. It’s the charge density of the rod. Because it is a constant charge divided by the length to find the charge at each interval. Q At this point the student writes dQ == L Q Here the student has conflated the infinitesimal dQ with linear charge density. This is remedied a few steps later where, after performing a bit of dimensional analysis the student Q writes a new equation dQdx= . The student then goes back and looks at the original integral L that they wrote and, pointing at the Q says, “I think that this should have been a dQ to begin with” and draws a picture with an infinitesimal in the sketch labeled with a Q.

Figure 6-12 A later sketch drawn indicating “… a little bit of charge”.

The student explains: …you want a little bit of charge times this distance to find the electric field. But mainly it is based upon the charge here and distance here (pointing at the “sliver” in the dot in the picture) that it is dependent upon. We should know what the charge is at each individual point… at each sliver dx… it’s a constant, but that’s how you get there.

In trying to address the correlation seen between students who included an infinitesimal in a sketch and the correct manipulation of the infinitesimal I first point out that this student,

similar to Student 2 has a difficult time incorporating the correct infinitesimal into their equations and this is reflected in the variety of labels attached to infinitesimals that are added in sketches. In both Question 1 and Question 2 the student alternates labeling the slivers that have been included, sometimes belatedly, with descriptors such as d, dl, and Q or sometimes no label at all. Additionally, for both students 2 and 3, dimensional analysis plays a significant role in producing the correct manipulation of the infinitesimal. For both Student 2 and 3 their labeling of the infinitesimals in their sketches reflect their shifting thinking about the infinitesimals in their integrals as they approach solving these problems.

6.3.4 Student 4 – the “functional difference” between physics and engineering classes Student 4 includes a sketch of an infinitesimal in both problems and has a correct expression for the coordinate dependent integrals in both questions. Additionally, the student, a triple major in physics, math and nuclear engineering, goes on to explain what he sees as a difference between physics and engineering classes – in the process, the student describes differences they perceive in problem solving methods used in physics and engineering classes that correlate with Jones’s composite symbolic forms of integrals. Student 4 begins by sketching an infinitesimal and deriving an equation (x) as seen in Figure 6-13.

Figure 6-13 Student 4 includes an infinitesimal in their first sketch – the script above the dot is “dm”. The student then derives an equation for (x).

During this process the student explains: “This is the mass per unit length at a certain point. And so if you integrate that you’re multiplying it by… you are doing like a Reimann sum, so if you think about it in a geometric approach you are multiplying by like a dx, so a small piece of length, to get the amount of mass is in that small piece of length, and then you sum that over the whole thing so that what is basically what the integral

is doing conceptually and that is how you are going to get the full mass because you’re adding up all the mass little pieces. If there was a center of mass of a system like this. You would, you would pick a zero point and then you would say that this is that distance and that is that distance since they are all smeared you have to say that this dm is at this r”.

While saying this the student adds a dm to their drawing of the bar and goes on to explain what they need to do with dm: So you are integrating rdm, and I guess x times dm, which you are going to have to find dm in terms of dx.

Student 4 then comes up with this relationship:

Figure 6-14 Student 4’s Equation Manipulation of the Infinitesimal

At this point, Student 4 has employed both acts of infinitesimal-in-sketch and correct- equation-manipulation-of-infinitesimal. The student’s explanations of their work demonstrate that the x in the sketch describes r in the starting point equation’s rdm. Student 4’s acts of infinitesimal-in-sketch and correct-equation-manipulation-of-infinitesimal are tightly bound. The student goes on to discuss how, in his view as a Math, Physics and Engineering major, the Equation Manipulation of the Infinitesimal is handled differently by different academic disciplines. “In engineering class they would give you that formula. They would say that x times your density function dx the integral over that you would get…the center of mass. It would have no vector language in it… it would say take x and multiply by your (x) or your lambda of x dx over your total mass. And they would just say do that. The student then explained that in a physics class you get “the general formula… r times dm… that this is sort of the general formula form for your center of mass. They would just, they would give you this because this is the functional version (of) what you are asked to do Student 4 then explains that:

“it would be easier for a student to be given this (xdx) instead of given this (rdm) and say figure it out.” I ask, “What makes it hard for some student engineering students, whoever, r-vector dm notation?” The student responds “Well I think that the difficulty is this is this logical jump…” adding the arrow seen in Figure 6-15.

Figure 6-15 The student adds an arrow to point out what they see as “difficult” about the process of coordinatizing dm into dx.

…like to to see dm here and think like ok, how am I going to integrate over all mass. The student then points back the equation shown in Figure 6-16 that he has written earlier while solving this problem.

Figure 6-16 This equation was written earlier in the interview but is referred to now as a “… sort of hard conceptual thing to think about.”

This is sort of hard conceptual thing to to think about. Whereas this (points back at the dm/dx in Figure 6-14) this you you have to you have to be able to make this connection I guess is the simple one sentence way to say that. Because if you can’t make this connection then you can’t, you can’t do this problem and that’s that’s a hard one to do I say. That’s, that’s, That’s the leap that you get you from here to there. The “from here to there” the student is referencing the xdm seen in Figure 6-16 and a x(x)dx that he has also written on the board. I feel that that’s a hard thing. Like I had to take a moment to think ok, how am I going to turn dm into dx? This student sees a difference between using an equation that is given to you and an equation that you must interpret by one’s own understanding of the linear mass density and its

relationship to the infinitesimal. The student’s discussion of two ways in which this problem can be presented to some extent align with two of the composite symbolic forms of the integral discussed by Jones: the small pieces form and the function matching form. Additionally, Student 4 points out the dm=dx conversion as a specific difficulty. Later on, during the electric field question, Student 4 spends time discussing how dq interacts with different components of both their drawing and their equation work. So, you draw a picture so you can get the geometry right. So here’s your rod. You are looking at some small section with a some dq in it um at this r (drawing a line that connect the point to the dq seen in Figure 6-17) Now, Let’s see here… you have, this is your origin, the r the unprimed is this point so, so like the r is just d and y-hat. If call this x and this y.

Figure 6-17 Student 4 discusses of dq interacts with different features of their drawing and their equation.

Remembering that the goal of this chapter is to explore the correlation seen between including an infinitesimal in a sketch and the correct equation manipulation of the infinitesimal, Student 4 has articulated why this effect might be seen in their work with the statement “you draw a picture so you can get the geometry right”. For this student, getting the right answer for the problem was a motivating factor for including the dq is a sketch. Placing the dq in the sketch allows the student to coordinatize r and helps the student explain the origins of dq = dx in a manner very similar to Student 1.

6.3.5 Student 5 – dq is a small charge and it has stuff Student 5 is the only student to complete Question 2 with a fully correct solution. After completing Question 2, this student also sketches out an alternative method of finding the electric field for the bar of charge by first using integration to find the scalar electric potential of the bar and then taking the gradient of the potential to find the electric field. Student 5 includes labeled infinitesimals in their sketches and correctly manipulates infinitesimals in their equations. In the student’s dialog the student discusses that an

infinitesimal has several characteristics and quantities associated with it. For example, in Question 2, dq: • is “a small charge” • is associated with a dx in the drawing -- “this is dx, a small amount of

charge will be odx = dq” • has a “distance… between any particular dx and P” • has an “electric field due to that, which we would probably need then angle between that point, you know that point on the rod at the point in space and the x axis” • and that has a “dE (that) is the electric field at this point due some small amount of charge”. In Figure 6-18 dq is prominently displayed along with the correct manipulation of the infinitesimal dq=dx. The dq seems to be inside the drawn dx. In the lower sketch the dq is a distance r away from the point P. Although a dE is not included in the sketch, dE is expressed in vector-consistent equations that show the relationship between dq and dE. Combining the description of their actions and their work on the white-board, Student 5 demonstrates a view of a vector integral that is consistent both in their equations and their sketches.

Like Students 1 and 4, this student clearly describes their conceptual understanding of dq and the additional terms in the accompanying mathematical equations they use such as dq = dx

k d q and dE = . Additionally, the drawing assists the student in communicating their x xy +22  understanding and perhaps aiding their manipulations of variables used in the integration. A goal of this section is to look at the correlation between the inclusion of infinitesimals in sketches drawn by students and their equation manipulations of those infinitesimals. Without prompting, Student 5 very clearly drew infinitesimals in their sketches and correctly related the coordinate dependent and independent infinitesimals. It is interesting to note that the student drew two sketches with infinitesimals. The first sketch has the infinitesimal drawn as a block of a notable thickness labeled dx. This thick dx is connected by an arrow to a charge dq that is set equal to dx. In contrast, the second sketch relates the position of the dq with respect to the point at which the E-field is to be found. In this sketch the infinitesimal is represented as a dot. Student 4 had said “you draw a picture so you can get the geometry right”. Here, we see Student 5 drawing an infinitesimal with a thickness when that thickness is relevant to the equation being worked on at that point in the problem- solving process. Then, we see an infinitesimal drawn as a dot when the equation being worked upon includes a point to point distance in which the thickness of the infinitesimal is irrelevant. For this student, the method in which they perform the act of infinitesimal-in-sketch is connected to the conceptual schema in which they are currently engaged.

6.3.6 Student 6 – q is neglected both in the sketch and the equation. I will go into detail of how Student 6 incorporated acts from each theme of coordinatization categories in the next chapter. Here, I briefly mention that this student correctly utilized dx in the mass problem while indicating an infinitesimal, or at least the position of an infinitesimal, in a sketch. The sketch in Figure 6-19 was drawn while the student was saying “I think I can do that because my position vector is just going to go along this axis and I'm going to go from zero to L.” This statement was made in reference to the x in the center of mass integral and after the student had discussed the small piece nature of dm in an integral.

Figure 6-19 The two arrows in this diagram are drawn to the location of various dm’s.

The situation is more complicated for the electric field problem. Here, the student early on indicates an infinitesimal in a sketch as a series of dots. Although these dots are not labeled as dq the student does verbally describe the dots as “little bit(s) of charge in the rod“ at the times the dots were drawn as seen in Figure 6-20.

Figure 6-20 The dots are “little bit(s) of charge in the rod”.

Along with not labeling these little bits of charge dq, the student also does not include a q 1 in the starting-point-equation for the electric field writing instead Ek= . The lack of a charge r 2 in this equation leads to about a half-hour of misdirection in solving the problem. Once the lack of charge is pointed out, the new sketch seen in Figure 6-21, with dq and dl clearly included is produced at the same time the equation dq = dx is written.

Figure 6-21 The student reworks the problem once the lack of “q” in their first attempt is pointed out. In the process, a new picture is drawn.

In the Coordinatization Framework acts should behave like resources. For Student 6, particularly in Question 2, even though the student initially states one that they recognize the

dots they drew as tiny charges, the lack of labeling the dots as dq is matched with the lack of q in their work with equations. When the lack of the symbol q in their work is pointed out, q shows up not just in equations, but also in the sketch. In this part of this problem, once the cognitive resource of charge is activated, the student is readily able to express the relationship between charge and space both in the sketch and a correct equation.

6.3.7 Analyzing 6 students sketch and equation manipulation of the infinitesimal as a group The goal of this chapter was to investigate the correlation that was seen in the data for students who drew a sketch in diagram and the increased likelihood that they correctly came up with the equation that would relate the coordinate independent infinitesimal with the coordinate dependent infinitesimal. The 6 video-interviewed students presented a diverse set of problem- solving strategies and methods of communicating an infinitesimal in their sketches and equations allowing for an understanding of how the correlation being looked at to be developed. Students 1, 3, 4, 5 and 6 drew infinitesimals in their sketches without prompting. Student 2 produced infinitesimals in their sketches but only in response to probing questions. In this section I am going to first define what I am going to call “the physical infinitesimal” and then I am going to propose two symbolic forms for the physical infinitesimal held by these students that might explain at least a portion of the correlation results seen. Table 6-6 lists each student’s infinitesimal-in-sketch and equation-manipulation-of- infinitesimal pair. If one of the two columns in a row is blank, then that sketch or manipulation event happened without an accompanying partner. Table 6-6

Each Student’s Acts of Infinitesimal In Sketch and Equation Manipulation of Infinitesimal Equation Description of relationship between Infinitesimal In Manipulation of Infinitesimal In Sketch and Equation Sketch Infinitesimal Manipulation of Infinitesimal “This is some continuous function that tells you the mass density as a function of x and then if you have some mass Student density at some at some value x then you 1 multiply it by a little sliver of x and you get by definition dm so just by definition dx is dm…some little sliver of mass…”

Table 6-6 (Continued) Equation Description of relationship between Infinitesimal In Manipulation of Infinitesimal In Sketch and Equation Student Sketch Infinitesimal Manipulation of Infinitesimal so we know that the E-field is going to Student be kqq/r2. Um… So what I am going to 1 2 do is say that dE is instead kq1dq/r

The infinitesimal labeled ‘l’ in the sketch was added several minutes after Student the sketch was drawn and just after the 2 dl is written in the integral. The student describes the infinitesimal in the sketch

“a small… small change in l” The student does not write this Student relationship until given the equation

2 rdm r =  cm M tot The student describes the dot in the slower right-hand corner of this cube as dm. When asked what dm means, they respond “that is a really good Student2 question… it means, I guess that I usually that I usually think more in terms of volume… which is a little bit easier for me to quantify. Student Student 2’s original equation. 2

Student 2’s modified equation. I have Q I have a Q, and I have an L. But I can integrate across that length. These are going to be the pieces“ (makes a tap, tap, tap, tap with hand Student across board) “but I don’t want that I 2 want to cut q up into these little itty bitty pieces because q is what determines E the electric field not L so… um… I made some corrections to my original formula because that wasn’t just wasn’t true, dq is this infinitesimal point, so I guess, I am not sure if by convention dr would be Student this distance between one point and this

2 distance between another point. Or if

you would define it as the distance between the individual two points.

Table 6-6 (Continued) Equation Description of relationship between Infinitesimal In Manipulation of Infinitesimal In Sketch and Equation Student Sketch Infinitesimal Manipulation of Infinitesimal Student 3 was consistent in how they Student labeled their infinitesimal in their sketch 3 and what they used in their integral. Student 3 was consistent in how they Student labeled their infinitesimal in their sketch 3 and what they used in their integral. But we need to create this… a dQ Student because it is just a little sliver of charge 3 at point x” The dm was added to the dot in the Student sketch at the same time the student said 4 “you are going to have to find dm in terms of dx” “Some small section with a dq in it… at Student this r.” “The trick for me is to get all 4 the pieces. A dx with a thickness is drawn in the Student sketch. In the equation, the relationship 5 between the dm and dx is shown. This student used dq=dx to describe Student dEx and dEy. dq is drawn with a finite 5 thickness in this sketch. In the next sketch the infinitesimal is a point. The dot in the diagram is used to help Student calculate r. Although the dot is not 5 labeled, it is still considered an infinitesimal in sketch. I need something to give me little pieces Student of mass in terms of length… so I can talk 6 about how much mass there is based upon where I am. Although Student 6 refers to the dots as Student “little bit(s) of charge in the rod”, 6 neither dq nor dx are written. Q is also not written in the equation for E. Once Student 6 is reminded that charge Student is missing in prior work, a new sketch 6 including a dl and dq is drawn and a

correction dq=dx is produced.

The various symbols used by these students for the infinitesimal are: dx, dm, dl, dr, d, dq, and dE. What makes these infinitesimals physical infinitesimals is that for each of these students most infinitesimals they wrote down had some sort of physical meaning for the student at least at some point in the interview. Physical meanings demonstrated included that the infinitesimal: is in a position in space, it creates an electric field, and it is a ‘thing’ such as a piece of charge or mass. All students at least at one point in each interview attributed units to an infinitesimal. Students 1, 4 and 5 drew an infinitesimal in a picture and correctly converted a dm or dq to a dx from the beginning of their work. These three students all demonstrated how the physical infinitesimal interfaced with their problem solving both in their conceptual understanding and in the equations that that built and/or used. Student 1, for example, used the concept of an infinitesimal of charge creating the infinitesimal of field that needed to be integrated: “so we know that the E-field is going to be 2 2 kqq/r … what I am going to do is say that dE is instead kq1dq/r ”. Making this statement while in the process of drawing the infinitesimal and directly connecting to the infinitesimal in the drawing quantities such as x, r and a vector representing dE indicates that the student had a conceptual understanding of each of the variables used in their drawing and how each was associated with dq. Student 4 labeled their infinitesimal in their sketch while saying “you are going to have to find dm in terms of dx” and then articulated their need to draw the infinitesimal claiming “you draw a picture so you can get the geometry right “. For this student the act of drawing was associated with considering the conceptual meaning of the symbols in their equations including the infinitesimal. Student 5 drew two different representations of infinitesimals where the form of each infinitesimal matched the function at that moment – drawing the infinitesimal with a thickness when discussing how much dq is contained inside a thickness of dx and then drawing the infinitesimal as a point when trying to find the distance between the infinitesimal and a point in space. For Students 1, 4 and 5, the sketch of infinitesimal allowed for a visualization of various physical properties of the dm or dq that assisted the students in their calculations. For these students the infinitesimal has “stuff” associated with it.

Student 3 spent some time attempting to determine if it was d or dl that belonged in the integral. This student often, but not always, showed consistency between the label of the infinitesimal in their sketch and the infinitesimal in their equations associated with that sketch. However, unlike students 1, 4 and 5, this student rarely connected other quantities involved in the calculations being done to the infinitesimal indicated in their sketches. For example, in Figure 6-22 the student did draw an infinitesimal in their sketch – the unlabeled nature of the infinitesimal in the sketch coincides with uncertainty in the infinitesimal that should belong in the equation they attempt to use. Additional uncertainty in the meaning held by the infinitesimal included in the sketch might be reflected in an r that is drawn disconnected to the infinitesimal.

Dimensional analysis eventually directed the student’s choice of using dl in the integral instead of d in the first problem. In the second problem, Student 3 initially defined dQ as Q/L. Dimensional analysis later allowed the student to declare dQ=(Q/L)dx being the correct relation between dQ and dx. For this student, the infinitesimal had units and unit consistency was the main method of determining the relationship between different infinitesimals and which infinitesimal should be used. Student 2 was the only student of the six who did not put an infinitesimal in a sketch until probing questions were asked. The equation dm=dr did not appear until after the equation for center of mass was provided and, like Student 3, it was dimensional analysis that drove the production of that manipulation of the infinitesimal. When asked what the dm in the equation means the student responds “…that is a really good question… it means, I guess that I usually… think more in terms of volume… which is a little bit easier for me to quantify.“ The student then drew a volume element with a dot in on corner representing the dm. This student knew that a coordinate dependent infinitesimal had to be used. They used dimensional analysis to obtain the dm to dr relationship expressing difficulty in explaining the physical meaning of dm.

In the electric field question, Student 2 originally had qdr in the integral but that was later converted into the correct dq=dr referencing the dimensional analysis process the student had done for linear mass density in Question 1. In this case, however the student did go a step further and addressed a conceptual reasoning saying “but I don’t want that, I want to cut q up into these little itty bitty pieces because q is what determines E the electric field not L so… um… I made some corrections to my original formula because that wasn’t just wasn’t true”. Thus, even if Student 2’s correct manipulation of the infinitesimal was initially based upon a “units must match” form, the student did shift to a “d has stuff associated with it” form as the interview progressed.

6.3.8 Conclusions from analyzing student acts of including an infinitesimal in a sketch and manipulation of infinitesimals The goal of this chapter is to investigate the correlation seen in Chapter 5. In the written data set, students who use the physically correct expression dx in their algebra are also more likely to include an infinitesimal in a sketch. In the analysis of the video interview data, focusing on the student acts of including an infinitesimal in a sketch and manipulation of infinitesimals, I identified three symbolic forms that are used by the students in this data set. Two of these three symbolic forms are productive for these students in producing the dx expression and one of them is not productive. One of the productive forms is more likely to benefit from an infinitesimal in a sketch than the other. The first symbolic form involves relating the infinitesimal to other elements of the integral. Analysis of the actions of Student 1, 4 and 5 indicate that the students built their integrals based on how the physical nature of the quantity represented in the infinitesimal helped them understand values in other parts of the problem. Students related the type of physical quantity in the infinitesimal to the quantities in the integrand, to the limits of integration and to the final result of the integral. This analysis leads me to conclude that these students held what I call the “d has stuff associated with it” composite symbolic form of an integral. The symbol template that I have created for this form is:

The second symbolic form I observed involves relating two infinitesimals by considering their physical dimensions. Students 2 and 3 reason that the physical units in an equation must be the same on either side of an equal sign. These students invoke a “units must match” symbolic form for infinitesimals and with this both correctly determined that dq=dx in their work. This form is described by the symbol template d = d and is a differential example of a “same amount” symbolic form in Sherin’s work (Sherin, 2001, p. 535). The two students who use this form have a less solid conception of the physical meaning of an infinitesimal. Student 2 in particular only draws an infinitesimal when prompted to do so. Likewise, Student 2 also initially uses a different, and unproductive, symbolic form for the integral. For this student the lack of unprompted sketches of an infinitesimal coincides with a less firm grasp of a conceptual understanding of an infinitesimal which also coincides with the initial lack of a dq=dx in their work. A third symbolic form involves inappropriately sandwiching an equation by an integral sign and a spatial infinitesimal. Student 2 initially produces an integral where the starting-point- equation of the electric field of a point charge was sandwiched between an integral sign and a spatial infinitesimal. The units do not match on either side equal sign in these cases. An integral of this form is also produced by 10 of the 39 Ph 320 and ten of the 32 Ph 431 students who took part in the written diagnostic. From the interview it is not clear what conceptual schema was driving Student 2’s use of a =∫(푠푡푎푟푡푖푛푔푝표푖푛푡푒푞푢푎푡푖표푛)푑 symbol template. The prevalence of the use of this symbol template that leads to incorrect solutions suggests that investigation into the schema or schemas held by students who use the template might be productive in understanding conceptions held by many students both before and after their junior year as physics majors.

– Following one student through their acts of coordinatization

In this chapter I am going to follow one student through all the acts of coordinatization they pursued. The goal of this chapter is to examine how the Coordinatization Framework can be used to understand a student’s work on solving problems that require vector integration.

7.1 Why this student and how I will present their work The student interview that I am going to investigate in this chapter is chosen from the six video interviews for three main reasons: • this student is much more vocally metacognitive while problem- solving than average allowing for a richer source of data from which to work; • this student initially mainly treats the electric field in the 2nd problem as a scalar quantity as did most students both in the video and written interviews – I want to investigate this phenomenon as it was so prevalent; • and, the student has a somewhat unique problem dealing with the coordinatization of the dq in the electric field problem at which I also want to look more closely. In this chapter I am going to present the student’s solution to both problems mostly in chronological order within the interview with very few gaps in the transcript. I am presenting the interview in this manner to: • fully contextualize the acts of coordinatization as this student performs them; • allow the reader to appreciate the foreshadowing of some comments early in the interview that help expose some conceptions held by the student that are more fully revealed by the end of the interview.

7.2 Examining this student’s work from an instructor’s point of view As an instructor who has graded thousands of written exam questions at all levels of introductory physics, I can convey aspects of this student’s work that might stand out to a physics instructor. Figure 7-1 shows all of Student 6’s work on Question 1. The student

mentions that it has been a while since they have encountered a center-of-mass problem and this might contribute to the student carefully showing and discussing all work. In the end, the student correctly solves the problem while describing their problem-solving process both in algebraic equations and descriptive graphs. If I were grading the work as if this were a written exam question, I would have given them full credit even though they neglect to substitute M/L for  in the last step. The student uses a graph to help correctly produce an equation for the linear mass density of the bar. The student correctly states in their written work a relationship between dm and dx. A correct expression for the bar’s center of mass is found and the student analyzed the units of their final answer as a method of testing the reasonableness of their answer.

Figure 7-1 A reproduction of Student 6's solution to the center of mass problem. Pictures of the original work can be found in Appendix B.

The 2nd problem presents a bit more trouble for the student. Figure 7-2 that shows the first several steps the student completed. One aspect of the solution that stands out is that the

student neglects both to put charge or charge density into the equation for electric field and they also do not include charge in their diagram. This would have been a bit unsettling for me as an instructor grading an exam question as it is electric charge that is the physical basis for the presence of electric field and almost nowhere for the entire initial problem solving process is electric charge mentioned by the student. Despite the lack of inclusion of charge, the student does set up the rest of their problem, both in equations and in their sketch, seemingly correctly – as an instructor, I might have just consigned this mistake as a “flopperoo” that would be easily corrected by the student if it were pointed out to them.

Figure 7-2 Reproduction of Student 6's initial work on bar of charge problem. Pictures of the original work can be found in Appendix B

Indeed, when the lack of charge is pointed out to the student the student produces the work that can be seen in Figure 7-3. As an instructor I would have been happy to see what would appear to be promising correct first steps in setting up integrals to solve for the vector components of the electric field for the continuous bar of charge. But, again as an instructor, I would have been confounded as to why the student did not pursue completing the integrals shown in the lower left-hand corner of Figure 7-3 after appearing to set them up so well. I will show in the analysis of this student’s work that the composite symbolic form that the equation

퐸푦푦̂ = ∫ 푑퐸푦푦̂ represents for the student leads to a reasoning that is not productive for the student. In other words, although a person grading such work might see the equation 퐸푦푦̂ =

∫ 푑퐸푦푦̂ as a correct intermediate step in the problem-solving – for the student this equation represents a point where their reasoning leads them to a dead end.

7.3 Examining this student’s work from the coordinatization framework I use the Coordinatization Framework to guide my analysis of this student’s work. This process reveals a student that holds consistent graphical and symbolic forms of the phenomena they are trying to investigate. The template and conceptual understanding of these forms will be described below during the analysis of the interview in sections 7.4 and 7.5. A summary of the graphical forms used and discovered will be presented in section 7.6. In section 7.4 I will present the student’s work on the center of mass problem. Some highlights of the analysis include: • In section 7.4.3, the coordinatization act of drawing-axes leads to the student articulating the combined use of a graphical form and a symbolic form regarding the linear mass density of the bar in the problem.

• In section 7.4.5 acts of manipulating-the-infinitesimal reveals the symbolic form used by the student to move from the coordinate independent dm to the coordinate dependent dx. Section 7.5 will focus on the student’s work on the electric-field problem where the following main observations are made: • In sections 7.5.1 and 7.5.7 the student’s use of infinitesimal-in-sketch and correct-manipulation-of-the-infinitesimal are correlated with suppression and activation of the role of charge in the problem. • In section 7.5.8 I will introduce a not-before-documented composite symbolic and graphical form of the vector integral that is held by the student for the entirety of the electric-field part of the interview. Again, the student communicates logically consistent graphical and composite symbolic forms for a vector integral – the true nature of these graphical forms are hinted at early in the interview in sections 7.5.3 and 7.5.6, but to me, the observer, the conceptual understanding behind the forms are not full revealed until section 7.5.8. Although these forms held by the student are consistent and strongly held throughout the interview, the nature of the forms do not bring the student to a productive conclusion to the bar of charge problem. At the end of this chapter, I will summarize and recap the main findings and claims made from the data collected in the interview of this student.

7.4 Center of mass problem

7.4.1 Drew axes In the first problem presented, the student is asked to find the center of mass of the bar with non-constant mass density. The student’s first item drawn on the board in the interview is an act of coordinatization. The unlabeled axes with an origin placed near the center of a coordinate system shown in Figure 7-4 are drawn at the same time the student is speaking their first words during the problem-solving process. Ok, I think I understand the question…. We have…

Figure 7-4 Student’s first written action and first act of coordinatization is to draw unlabeled axes.

The student continues on labeling the ends of the bar as seen in Figure 7-5. …a bar and there's on the left side is a lambda naught and this is two lambda and there's some kind of a function.

As the coordinate system was the very first thing this student does while solving this problem suggests that defining a coordinate system is important to this student. Throughout this interview this student does pay close attention to the setting up and maintaining a coordinate system – although sometimes, as seen in Figure 7-5, the labels of the axes are missing so that it is now always known what the coordinates of the system are.

7.4.2 Coordinate independent infinitesimal equation, but not in sketch The student then with very little pause says while looking at the sheet that contains the question they trying to solve: What is the total mass of the bar? Okay, so obviously we need some integration. What is the integral of all of the little dm’s?”

At this point, they write the integral expression seen in Figure 7-6.

Figure 7-6 The student’s second act of coordinatization. While writing this the student describes small piece thinking of the integral.

The student’s second act of coordinatization is an equation-manipulation-of-the- infinitesimal by including the coordinate independent infinitesimal dm in their equation. Additionally, the student introduces the infinitesimal on the whiteboard with what seems the most concise explanation possible of Jones’s adding-up-pieces symbolic form by blending the writing down of M d= m with the words “…the integral of all the little dm’s…?”.

7.4.3 Graphical forms revealed while going back to drew axes Immediately after the last exchange above Student 6 changes the figure seen in Figure 7- 4 to what is shown in Figure 7-7 while saying: And we know that at – I'm going to change my coordinate system because I like first coordinate problems.

Figure 7-7 The student erases the vertical axis that they had previously centered in the coordinate system and now draws a new axis on the left side of the coordinate system.

Not only was Student 6’s first act that of drawing the coordinate axes, but those axes continued to be significant in their active thinking and work produced:

So at zero, we have lambda and at L we have two lambda. Goes like this. So, see, I'm going to say that increases steadily means that it's a linear function. And so, we have y equals mx plus b. And my slope is rise over run. Rise is lambda. Run is L.

Sherin’s symbolic forms seek to combine conceptual schemas with algebraic templates. At this point in the interview Student 6 is also using a graphical form to assist their reasoning.

The student implicitly displays experience and confidence with how coordinate systems play into mathematical equations by saying “I'm going to change my coordinate system because I like first coordinate problems”. Indeed, placing the coordinate system on the left side of the bar made use of the well-known equation of a line y = mx + b easier to employ than it would have been if the axes were still at the center of the bar. The student continues on describing how they use their sketch in interpreting the bar of mass creating the graph and equations seen in Figure 7-8. And then we start at lambda, so that should be my function for mass. This is mass per length, that's –no, lambda – what is lambda? Linear mass density lambda. So, it's mass per length. Yes, it is mass per length. And we're dividing by length, so we get mass – yeah having problems. Mass per length over length is mass. x is presumably length. Mass per length. What we'd like to have is mass. So we have a problem.

During this monologue, Student 6 is describing how they are using dimensional analysis to check if the equation they have come up with is dimensionally correct using a  =  +  symbolic form. (Kuo et al., 2013, pg. 31).

Student 6 then ties this back into the  vs x graph:

So this – as I have it set up, this is actually a mass per length axis and this is a length axis. When ideally, we might just want a mass axis.

This indicates that the graphical, algebraic, and conceptual understanding are all cognitive elements being activated in the student’s thinking of this problem. In adhering to Sherin’s assigning of symbolic templates to a symbolic form, the graphical template for the student’s work during this part of the interview might look like what is seen in Figure 7-9.

Figure 7-9 A proposed “Line End Points” graphical template.

In this case the vertical axis of the graph in Figure 7-8 contains the known values of o and 2o which were laid horizontally in the picture of the bar of charge shown in Figure 7-7; meanwhile the horizontal axis of the graph represents the picture’s horizontal position elements. Later, while sense-making the physics units of the situation, Student 6 focuses upon the slope of the line that had been created.

Figure 7-10 A proposed “Slope Gives Value” graphical symbolic form.

Upon stumbling upon what seems to be a graphical extension of Sherin’s Symbolic forms while writing this section, I immediately went to the literature to see if this topic had been investigated before. It turns out that “Graphical Forms” appear to be a new line of investigation first and currently being pursued in the Chemistry Education Research community (Rodriguez, Bain, Towns, et al., 2019) (Rodriguez et al., 2018). Three graphical forms have been identified (Rodriguez, Bain, & Towns, 2019) by this group: steepness as rate; straight means constant and curve means change.

The idea of graphical forms is new and exciting and deserves more discussion that what I have done in this section. As we progress through Student 6’s interview we will see at least one more novel example of a graphical form in section 6.2. In the summary of this chapter I will compare the graphical forms found in this interview with the limited prior work done on the topic. Now, I am going to continue on with Student 6’s interview and continue to track their acts of coordinatization.

7.4.4 Limits of integration In this next section, Student 6 adds limits to the integral in their equation. Like the majority of the students’ work analyzed in Chapter 5, Student 6 chooses limits of integration that are consistent with the infinitesimal in their integral. Student 6 next constructs a function for (x). This process involves developing an equation from the information given and then using dimensional analysis and verbal reasoning to confirm that the equation was correct. Once the student convinces themself that they could describe the density of the bar as a mathematical function they very fluidly write the below integral while stating: “Okay, so now this is good because I can't sum over mass, but I can sum over x. So, this is this good. And we go from zero to L of x, so M equals – this is total mass. We have the lambda over L x plus lambda. I suppose if we're being consistent, we should make sure we do naughts. On the other hand, they're all naughts, so I should probably just drop it, but, you know, here we are. And then this is over dx.“

Figure 7-11 The student’s first coordinate dependent infinitesimal in an equation with matching limits of integration.

This part of the interview again demonstrates that the student holds a very strong symbolic form of the integral that includes not only the concept of the integral being a sum of small parts but also holding beginning and ending points of 0 and L.

Immediately after the last statement I ask the student “All right, so how did you get that?” The student expounds: Because I was worried – I sort of lost track of what I was trying to do, so what I was – I can't sum over mass. But I can sum over distance, so I had to figure out some way of getting mass to distance. And what I knew or what I sort of inferred because I didn't know how to do it if it wasn't, the mass density increases steadily. I'm going to say that linearly. Here, the student is articulating that they cannot sum, or integrate, over the coordinate independent mass but they can sum over a coordinate dependent distance. Student 6 is consistent, both in the center of mass question and later in the electric field equation, in how they linked the use of limits with the infinitesimal in the integral. For example, M d= m was written without limits and statement were made such as “I can't sum over mass…” and then “… but I can sum over x.” leads to

L  m=+() x dx . 0 L o This pattern is repeated with expressions later in the interview such as EdE= net  that later turns into an integral in terms of dx with limits denoted. This student is consistent throughout their work that coordinate dependent infinitesimals have limits on the integrals and coordinate independent infinitesimals do not have limits on the integral. Consistency between the limits of integration and the infinitesimal in the integral is common not just for this student but with most of the 97 students who took part in written interviews. Out of these students only 6 students wrote a non-blank limit that was inconsistent with the infinitesimal in the integral. The consistency between the limits of integration and the infinitesimal is so strong that integration conceptions such as seen in Fig. 7-12 can be discerned. In Fig. 7-12 Student 37 is identifying the dr in the coordinate independent equation as representing the distance between one side or the other of the bar and the point at which the electric field was to be found.

Figure 7-12 An example of limits of integration consistency seen in the written interviews that demonstrates how closely linked this act is connected to the manipulating the infinitesimal act.

When considering Jones’s symbolic forms of the integral, it perhaps makes sense that consistency between the limits of integration and the infinitesimal are so common. With Jones’s integral forms, it does not matter if a student has an adding up the pieces, using the perimeter and area form or the function matching form as in all cases the matching of infinitesimal to the limits makes conceptual sense. For Student 6, who seems to be consistently using the adding up pieces form throughout the interview, limits of integration from 0 to L are displayed for every integral that possesses the coordinate dependent infinitesimal dl and never are limits of integration displayed when the integral contains a coordinate independent infinitesimal such as dE or dq.

7.4.5 Equation manipulation of infinitesimal Continuing immediately from above, Student 6 says:

“So I started trying to build myself a function to give myself the function in terms of length of mass and what I got sidetracked off is I psyched myself off on the units because I thought I wanted units of mass when I clearly didn't 'cause the whole point was trying to not get units of mass.” As Student 6 was saying “the whole point was trying to not get units of mass” they draw a line under the dm in the Mdm=  they had written several minutes earlier. Here the student indicates that the act of coordinatization of equation-manipulation-of-the-infinitesimal has been a significant motivation of the work described in the previous section that yielded the equation

L  mxdx=+() .  o 0 L

“So this, what got me out of it was I realized that actually lambda has units of kilogram meters or whatever they are, masses per length. And I have total consistency 'cause that's length, that's

length, so those cancel out and everything's kilogram-meters. And then if I sum over meters, then I should get kilograms out at the end. And so I can stick my kilogram-meters in here and sum over meters and I should get kilograms.” When asked to clarify how this equation came to be the student elaborates further: “So, I needed something to give me little pieces of mass in terms of length and so this is dm, but it's a broken apart dm so that I can talk about how much mass there is based on where I am. So it is dm, but dm and dx are proportional to each other and that's how they're proportional.”

While saying “so this is dm” they draw the dm seen under the integrand in Figure 7-13 with the dx inside what has been defined as dm. The student directly points to the dm and the dx while saying “dm and dx are proportional” and then points at the integrand while saying “that’s how they’re proportional”.

Figure 7-13 The student draws the bracket and the dm while explaining how little pieces of mass are proportional to dx based upon where In they are at.

The student, with words, gestures and a bracket, articulates what can be construed as a symbolic form for an infinitesimal with a symbol template that looks like this:

Infinitesimal proportionality: dd= .

7.4.6 Dealing with the vector nature of r

rdm Next, this student confronted the numerator in r =  com  dm The first thing the student writes is the equation seen in Figure 7-14. The equation as written is a coordinatization act of vector-inconsistency.

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Figure 7-14 Vector inconsistency displayed in the iconic equation for center of mass.

Vector-inconsistency is an act that was prevalent in the written data that was analyzed. In this center-of-mass problem the student mentions verbally that the r inside the integral is a position vector but does not take notice of the missing vector on rcm in the equation they have written. While writing the above equation followed by the image in Figure 7-15 the student states: So the location of the center of mass is the integral of that stuff over M total. So this is – I'm trying to remember exactly what it is. So, it's a weighted and weights things. The location of center of mass is where there's equal mass on both sides. So where the mass is and how much of it there is all over how much total. So we've got a bar. But I have dropped my naughts because there's no more math.

Figure 7-15 This picture was drawn as the student was trying to describe their physics conception of the meaning of center of mass.

These statements and drawing suggest that the student is explicitly tying the mathematical equation that was given to an understanding of the physics underlaying the problem. The student then rewrites the algebraic equation for dm that was derived earlier and begins to work on the coordinate independent r in the equation saying: “and r is the position vector on where it is (student looking back at the iconic equation)– so before, we just have this integral. Now, we have an r in there. r is just along the x axis. So one-dimensional problem so I think we should replace r with x –

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Figure 7-16 The vectored r in Figure 7-14 has become a scalar x.

The coordinate ambiguous 푟⃗ in the starting-point equation in Figure 7-14 has been replaced with the more coordinate-dependent scalar x. The equation as it now stands in Figure 7- 13 is now Vector-Consistent in the coding system I developed. In the electric field problem, vector inconsistency will occur again and not be so readily resolved as it was here.

7.4.7 Infinitesimal (maybe) now in Sketch During the past minute of interview described immediately above, the student was speaking with big pauses between words and sentences indicating that there was a lot of thought happening. The slow speech suddenly picks up speed; with the below statement and the original picture in Figure 7-15 being modified as shown in Figure 7-17. The student explains: “I think I can do that because my position vector is just going to go along this axis and I'm going to go from zero to L.”

Figure 7-17 The student modifies the already existing picture to draw two position vectors which can count as an act of showing an infinitesimal in a sketch.

This is where the student has indicated an infinitesimal in their sketch. The student multiple times in the interview has clearly indicated in words that they have a small piece thinking of dm. At this point in the interview, while saying “…my position vector is just going to go along this axis…” the student draws the two arrows in Figure 7-17. The student additionally says that “I’m going to go from zero to L” but does not draw the second arrow all the way to L – this can be interpreted as realization that there are infinitely many position vector of which two randomly chosen position vectors have been drawn. If this interpretation is correct, this strengthens my earlier defining of multiple vectors in a picture representing an infinitesimal in a

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sketch. In the electric field problem, we’ll see even more of this thinking and different methods of representing infinitesimals in a sketch.

7.4.8 Wrapping up the center of mass problem At this point the student continues at high speed producing spoken and written work. “And this has all of my mass in terms of zero and L, so when I put it in x, into this expression, I should get where we are and how much mass there is. So I think this is good. So this is then the integral and total mass.”

The student then replaces the r in the original starting-point equation for center of mass with an x that has been replaced in the equation and then multiplied through the expression for dm that was derived earlier and comes up with this for the position of the center of mass of the bar.

Figure 7-18 The coordinate independent “r” has been replaced with an “x” right before the integral is solved.

While solving the center of mass problem this student has: • drawn an unlabeled coordinate system, • correctly manipulated the infinitesimal in their work from the coordinate independent dm to the coordinate dependent dx, • used graphical and symbolic forms to clarify the how their coordinate system and the physical situation of the problem worked together, • used limits of integration in all coordinate dependent integral and no limits of integration in coordinate independent integrals, • written down a vector-inconsistent equation that did, in this case, eventually reach a correct equation for describing the center of mass of the bar on what they called the x-axis, • included in a sketch two arrows that, when put in context of what was said while they were drawn, can be considered an act of infinitesimal in sketch. The student while solving this problem performed at least one act from each category of coordinatization listed in Table 5-1.

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The vector inconsistency in the students’ written initial iconic equation is never addressed in this problem, and, as I stated before this does not prevent the student from getting to a correct answer. In the next section, the coordinatization act of vector inconsistency appears again accompanied with a symbolic form of the vector integral that the student finds problematic.

7.5 Bar of charge problem Once again, I will present the student’s work on this problem in the order in which the work was completed. At several points I will compare the work the student did in the bar of charge problem with the what they did in the center of mass problem. This provides an additional piece of triangulation in my interpretation of the acts of coordinatization observed.

7.5.1 Drew axes and infinitesimals in sketch Student 6 begins the Electric Field problem just as they did the center of mass problem by drawing a picture with a coordinate system that when finished looks like this:

Figure 7-19 The student begins with both the Drew Acts and Infinitesimal in Sketch acts.

While drawing the dots along the rod Student 6 says: “So this is point E and every little bit of charge in the rod which is evenly distributed”. The dots represent coordinatization acts of infinitesimals-drawn-in-sketch as the student recognizes that the dots represent tiny bits of charge. This is going to be the first and only time Student 6 mention charge until the lack of charge in the problem is pointed out to the student much later in the interview. The lack of mentioning charge permeates the words, equations and drawings used by the student until the resource of charge is activated.

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7.5.2 Vector inconsistency begins Student 6 continues describing the size of each dE while writing the equation shown in Figure 7-20: “…Is going to give this uhhh some amount of a vector and they are going to be bigger… bigger the closer they get. So that by the time we are over here we are going to have pretty big vectors so that the net vector is going to look something like that. This is a vector problem so we can break this into x and y components I am changing my mind. So my net E vector is going to be the summation of all my little E’s

The coordinatization act vector-inconsistency occurs in the first equation written by the student in this problem. Student 6 clearly discusses the vector nature of dE while setting up the problem. After a few more sentences of work, Student 6 eventually adds the vector on the dE in the equation shown in Figure 7-20. The student then writes down part of the starting-point- equation for the magnitude of the electric field explaining: “Well, I'm trying to get… ultimately I'd like some sort of a dE and I know E in terms of r, but I'd like to get r in terms of x and this cosine theta is not working out for me.”

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Notice that the student is missing both the charge and the rˆ in what has been written down for E in this equation. The lack of the electric charge in the equations being manipulated is going to persist through the rest of this interview until the lack of charge is pointed out by me. Meanwhile, the directionality of electric field will be addressed by the student several times throughout the interview. Shortly after this drawing, the student pauses for several seconds and then puts a vector on top of the dE as shown in Figure 7-22. When asked why the student responded: “Because the other side is a vector and it was not technically correct because it was a vector...”

Figure 7-22 The student without prompting alters their symbolic template to be vector consistent.

At this point the student has articulated that vector consistency is important. What happens next shows that although the notational inconsistency was quickly remedied, an underlaying conceptual issue would be much more difficult to resolve.

7.5.3 The first glimpse of a newly cataloged symbolic form of a vector integral While attempting to convert the r in the denominator of the equation they have written for E the student says: “And I'm trying to figure out how to break up the problem into vector components and change it… we want to do an x and a y piece of this. The y piece is always constant. So really, the only thing that's changing is x.” “The y piece is always constant” is an important statement in the cognitive resources this student is drawing upon when constructing a vector integral to solve this problem. The full nature of the symbolic form being utilized by the student will not become clear until later in the interview.

7.5.4 But first, a different problem between r and x Eventually, the student comes up with the question shown in Figure 7-23 for r.

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Figure 7-23 The student continues to attempt to put r in terms of x in a form that can be integrated.

The student is dissatisfied with this equation explaining: “…I do technically know everything… if I really want an answer I am going to have to integrate an arctan” Student 6 then writes the following:

Figure 7-24 dr is then (incorrectly) found.

Explaining: The thought process is similar to what I did before. Umm.... Ok, uhhh so what I want to know is how a little piece of r corresponds to a little piece of x. And what I was trying to decide before I did this step was that I had an x down here, but this theta is not a dtheta it is just a theta so I don’t need a dx I just need an x, and d is a constant. So this dr is proportion to dx thru this relationship at each as dx moves the theta changes this number changes and my dr changes. So as theta gets bigger this gets smaller…”

Student 6 correctly realizes that r is dependent upon x but has not developed the correct dependence of dr upon dx. At this point, the student spends over 2 minutes doing a reasonableness test looking at various angles of theta to see if the corresponding value for x makes sense. This might be similar to what happened in the first question after which they said “I sort of lost track of what I was trying to do”.

7.5.5 dq does not make an appearance Student 6 next spends the next four minutes of work focused on the r to x issue. At one point during this discussion, Student 6 points at one of the dots in the sketch when discussing dx.

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Six minutes later, after the student has cycled through sin, cos and tan without getting to where they want to be, I give the student the hint that “the Pythagorean theorem exists”. Student 6 immediately goes “I don’t know why I didn’t see that.” and then quickly produces algebra to (incorrectly) calculate dE. The method to find dE might be considered an act of “zapping with d” that consists of something akin to taking the derivate dE/dx and the moving the dx over to the other side of the equation.

At the beginning of this problem the student did describe each dot of the 5 unlabeled dots sketched into her drawing of the bar of charge were a “little bit of charge in the rod”. However, while this process of coordinatizing dE might be meeting the student’s need to put E in terms of a coordinate dependent variable over which they can integrate, it is ignoring a relationship between charge and electric field.

7.5.6 Pointing out the lack of direction At the end of the whiteboard work in the previous figure, the student once again states

[11:53] “we have Enet equals the integral of dE’s” while writing the final statement on the board just before they move to the next board. Note that the vectors have disappeared from both the

Enet and the dE. This would be vectorially consistent, although no longer communicating the vector nature of the problem. The student spends a couple of minutes substituting in the value they found for dE into the their integral of dE expression and musing whether they will need to perform a u-sub or an integration by parts to evaluate the integral.

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At this point I ask, “So way back in the beginning you mentioned that E was a vector, how are you going to take into consideration the vector nature of this one?” Neglecting the vector nature of the electric field was pervasive both in the written interviews as well as the video interviews. In the video interviews, five of the six students had neglected to consider the vector nature of E. When this was pointed out to the five students each student seemed to immediately activate a “vectors have components” resource and give a run-down of how to deal with the vector nature of E. The one student who dealt with the vector nature of E from the beginning also, after coming up with a solution for 퐸⃗⃗ then outlines how to solve the problem by first using scalar integration to find the electric potential of the rod at the evaluation point and then take the gradient to find 퐸⃗⃗. Student 6 also quickly activates resources dealing with vectors: “I did sort of forget about that, didn't I? Okay, it is a vector. it is a vector, so at the moment I am finding the magnitude, I think, so because d is constant we don’t need to worry about the y but the vectors will still have x and y components.

Figure 7-26 In the process of considering the vector nature of the electric field, the student draws two “dE’s” for the bar of charge.

They continue: “So I am convinced these are the magnitudes” labeling the smaller vector |dE|f and the larger vector |dE|i. The student explains that the i indicates that that dE is the initial dE from the left side of the bar and the f is the final dE from the right side of the bar. Here again the student recognizes that a dE created from one part of the bar has a different magnitude and direction than the dE created from a different part of the bar.

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7.5.7 A Second glimpse at a newly cataloged symbolic form of a vector integral [18:40ish] I just lost all my directionality. I need to do this in the x components and then in the y components. Student 6 begins to address the vector nature of the electric field by sketching this.

Figure 7-27 Pictorially describing how to deal with the vector nature and the integration required to be considered to solve this problem. and then modifying their equation template:

Figure 7-28 Expanding upon their symbolic template for a vector integral.

As an instructor who has seen thousands of students in calculus-based electricity and magnetism courses I would have been quite happy seeing this written discussion from one of my students. However, it turns out, there was something else going on in the student’s thinking. At least twice before the student had alluded to a conceptual idea they were having difficulties with that I was not quite fully understanding. The y direction doesn’t change so this would not be a hard integral because the y direction wouldn’t change… it is a constant. The x would be, I’d have to do something along these line (pointing at the equation manipulations that had just be done).

Here, Student 6 seems to be suggesting a viable path to solve for the Etotal due to the bar at the point in question. Figure 7-27 and 7-28 describe a productive visualization of many dEx’s and dEy’s adding vectorially up to an Etotal wrapping up this part if the discussion saying [21:30] “and those are just the vector components of my final answer” while drawing this picture:

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Figure 7-29 The student describing how to find Etotal from the vector some of the dEx’s and the dEy’s.

The student clearly distinguishes between the y-direction having an integral that “would not be hard” vs the x-direction being something similar to the equational work that had been done. This is going to come back in an interesting manner indicating a strongly held symbolic form of a vector integral just a little bit later in the interview.

7.5.8 The missing dq is pointed out and resources are activated I point out to the student that charge is missing from all the previous work. The student spends a couple minutes reminding themselves of the difference between E, V and F when dealing with electric charges. With no further prompting from myself, the student writes down scalar equations for E, V and F and then eventually modifies the sketch of the problem to include a dl, dq and .

Figure 7-30 Once reminded of the fact that charge, q, has been missing from all the prior work, the student updates the picture with dq’s, dl’s and the relationship between them.

Then, for the next several minutes, several statements are made that demonstrate that activation of new cognitive resources centered around how dq comes into play. 1) Summing up something about dq’s has to happen: …dq, unfortunately which because we have teensy tiny q’s in our bar, so actually we're going in the wrong direction.

2) dq's and dl’s are related: Okay. So this would be a dq and it's either dl. See how the we have to relate the q’s to the l’s and you can do that through a lambda, which I don't know that there is one, but to charge density, but it's a constant charge density.

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3) Figuring out over what to integrate: …I guess the question I'm trying to figure out is would we integrate over q or integrate over r? We do need the dE... And I think I've convinced myself that because we have a constant charge density, the charge is the same at every dx. Okay. Well since we completely forgot that, I say we since I completely forgot that there's a Q in there. But I was trying to decide was do I need to do dq or do I need to do dr or do I need to do both? This is a nice example that even though the q and dq and even the word charge has not been uttered until I bring it up, the student has a full set of conceptual ideas and even a symbolic form around the infinitesimal. One interesting idea that came up is: “…What I was trying to decide was do I need to do dq or do I need to do dr or do I need to do both?” Indeed, several students in the written interviews do have double integrals with double differentials such as dqdr. This is not the path Student 6 takes, but, the comment here does suggest that perhaps the students who included double infinitesimals such as dqdr in their work realize that infinitesimals such as dq and dr are related to each other in the problem and each is important in the problem-solving, but, while taking the diagnostic, they just were not able to articulate the relationship between the two. In this case, the activation of the idea of charge did relatively quickly allow the student to come up with a correct, although still scalar, expression for E and dE as seen in Figure 7-31.

Figure 7-31 After activation of the cognitive resource of the charge Student 6 relatively quickly produces a correct expression for a scalar dE.

Because... If I didn't have the lambda in there, if I hadn't converted it from a dq to a dx, I could have different amounts of charge and I'd get the same answer. You get a really charged bar and a not charged bar. And I'm just suddenly over the x. The x is never changing. And so it doesn't make sense that I get the same E field if I had different charges. So if I'm going to sum over dx, that number

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has to come along because that tells me how much charge for length I have. That last sentence demonstrates that the “units must match” infinitesimal relationship symbolic form is again being used by Student 6 in helping determine the relationship dq = dx. Thus, here, once the concept of charge was pointed out, very quickly a series of cognitive resources dealing with electric charge and integration of a continuous distribution of charge were activated and the correct symbolic relationship between dq and dx was produced.

7.5.9 The full nature of this student’s symbolic form of a vector integral finally revealed What happens next clarifies the Symbolic Form of a vector integral that Student 6 held during this interview. Prior to now Student 6 had mentioned that dE ”would not be a hard integral because  y the y direction wouldn’t change” and that “The y piece is always constant. So really, the only thing that's changing is x.” Before the part of the interview coming up, I thought that the student was merely noting that that the vertical component of r was a constant – it turns out that there was something deeper and more central to the student’s working conception of the vector integral. Then the r-squared question... Oh and this is where we can actually get this, so... I've already said that this is essentially a dE in the x direction. x-hat by any means because that's it. That's a dx. And so I'm talking about going along this axis.

While saying this, the student writes this perfectly valid equation for dEx.

Figure 7-32 This equation for dEx is written. No matching equation for dEy is ever written to accompany this equation.

No similar equation is written for dEy. Instead the student explains: The dy... I have delta function running around in my head. I don't think that's very helpful. [9:07] When I ask what they meant by delta functions the student responded:

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“Well, because of the dx, you have to go along the dx axis and you have charge. And the dy axis you go along it, you have all the charge... And that reminds me of delta function. But that being said, I don't think that's particularly helpful in the current application.”

Figure 7-33 Student 6’s graphical form of the vector integral that had to be solved.

The student describes their thinking with this set of graphs that describes the charge distribution in two graphs, one for each spatial dimension. There are a few things that are very interesting about this: a) In the x-direction, student 6 has a demonstrated a composite symbolic form that matches a symbolic template, with a clear and correct conceptualization of the physics and that matches an accompanying graphical form. b) In the y-direction, student 6 also has a clear and correct conceptualization of the physics of the charge distribution that is accompanied by a matching graphical form, but is not willing to immediately write down a symbolic template saying “that reminds me of delta function… but that being said, I don't think that's particularly helpful in the current application.” c) No matching dEy equation is produced for the next 5 minutes of contemplation and explanation of thinking, and then, only in frustration saying “I mean we sum y from zero to zero essentially, which then kills everything, which is not true. I clearly have an E component of a y component of E.” [16:24]. The tone they use while making this statement leads me to believe that the dq = dy that they write is a symbolic template that doesn’t match their conceptual understanding and is thus being produced in exasperation of not being able to produce a symbol template that completes a symbolic form. The next 10 minutes is spent attempting to find out how to coordinatize the coordinate independent dE with the coordinate dependent dy. Due to the length of this part of the transcript, rather than giving a continuous moment-by-moment replay of this discussion as I have before, I

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am going to focus on a few statements that highlight the two important concepts Student 6 was attempting to reconcile.

Student 6’s Conceptualization #1 – Enet is a sum of dE’s that have different magnitudes and different directions: • “This is the longest r distance, so it’s going to be the smallest vector. This is closer. So this is going to be bigger and this is closest and so

it's going to be the largest and then to get Enet, we would sum all of those vectors and that's what we're trying to find.”

• “These are all the little dE’s, so it's a sum. Sum your Enet. This is the thing we know to be true is that when you sum all the little tiny ones, you get a big one.”

• Me: “Can you and your picture show me where this dEy is?” Student 6: “Yeah. That is this piece.” (pointing to a y-component of a drawn dE-vector)

• “We'll call it EX and the x-hat equals the sum of the little dx’s in the x- hat direction… and the y, y-hat equals the sum of the dEs in y-hat.”

Student 6’s Conceptualization #2 – Integration to find Ex happens with respect to dx and integration to find Ey happens with respect to dy. • “I’ve broken this down to the x but now I need to have my y piece. And I don't quite see how I'm going to do that because I can't get this relationship down here because it's essentially delta function and so I think it's really easy, which is why I'm not seeing it because I know it's one point.” • “Because I'm not summing it over anything. But I am but I am not. And that's what's confusing me is that I need to generate. I'm convinced it's just a constant that I just add on. But I need to find the constant. This is being a constant is going to do most of, that's good. I like that. But my q is still, each little dE is going to have a dq and in order to have a dq we have to have a lambda. And it's going to be the same lambda but then we end up with this. But we are not, I mean we sum Y from zero to zero essentially, which then kills everything, which is not true.”

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• “Because if we say that's dq to get my lambda in there, which I think I need and that's in terms of the dy but the entire staff of charges is sitting at dy. So it's like there's a point charge. So why have a D?” • “What I want to do, and I don't think is correct, is just call this D.” (pointing to the integral of dy)

The interview concludes after these statements without the student ever reconciling these two views of what was supposed to happen in the vector integration required in the problem. This part of the interview starts over an hour into the overall interview and it is possible that had Student 6 been asked the same question the next day, the correct solution would have very quickly been rolled out. However, on this day at this time Student 6 held very strongly on to a symbolic form Edx= ... . Here I have specifically used Sherin’s notation of an ‘x’ as he used in the x  “dependence” cluster to indicate that in this symbolic form whatever dimension the x in the dx represents, the coordinate in Ex must be the same as the coordinate represented by the x in the ...dx .

Finally, to match the graphical form drawn by the student in Figure 7-30, the student writes an equation that matches the symbol template above. In Student 6’s own writing:

Figure 7-34 The (incorrect) Symbolic Form of a vector integral held by the student.

7.6 Conclusions from one student’s actions of coordinatization In this chapter I identified all acts of coordinatization performed by one student as they solved two problems in a clinical interview. Applying the Coordinatization Framework to this student’s work has led to two results beyond those which are discussed in Chapter 6.

The first result adds to the literature in the educational research community. A Graphical Form is a type of form newly being investigated by the educational research community. It is a

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graphical analog to Sherin’s Symbolic Forms. The student demonstrates three Graphical Forms that describe Student 6’s reasoning in sections 7.4.3 and again in 7.5.9.

The second result demonstrates an unproductive composite symbolic form of a vector integral used by the student throughout the interview. This also once again demonstrates that using the Coordinatization Framework to analyze student work can lead to a better understanding of the cognitive resources employed by students while solving problems of this nature.

7.6.1 Student 6 uses graphical forms while problem solving Using the Coordinatization Framework I am able to identify three previously undocumented graphical forms. In section 7.4.3, I describe the reasoning Student 6 uses during their derivation of an equation for the linear mass density (x) in problem 1. The graphical form described in that section combines a conceptual reasoning with a graph drawn by the student. The graph describes a steadily increasing value from a known starting point to a known ending point with a student goal of describing a value that is “steadily increasing”. I call this a “Line End Points” Graphical Form. Additionally, in section 7.4.3, Student 6 describes how the slope of a graph provides knowledge of a quantity that the student was seeking. In this case the student was wanting to make sure that the slope of the line had units consistent with what they thought that quantity should have. I call this a “Slope Gives Sought Value” Graphical Form. Students in lower- division labs frequently plot data so that the slope of the graph is the desired quantity. Student 6 was using the slope of the graph they produced in a similar manner. Finally, in section 7.5.9 Student 6 is confronted with a charge distribution that is essentially one-dimensional: a bar of charge that has a specified length in the horizontal direction compared to an undefined and assumed very small width in the vertical direction. After describing their conception of this scenario in words several times, the student finally describes their conceptual understanding of a charge distribution that is homogeneous in the x-direction and “that reminds me of [a] delta function” concentrated at one spot in the y-direction. At the student is explain this they sketch the graph in Figure 7-33 that consists of a horizontal line with a vertical spike in the middle representing the charge density at the position on the y-axis where

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the charge is concentrated. The “Dirac-Delta” Graphical Form thus represents a conceptual understanding of something being concentrated in one place. Table 7-1 summarizes these newly cataloged graphical forms. Table 7-1

Graphical Forms Identified During This Interview Graphical Form Graphical Form Template Conceptual Schema Name

Values at the extreme ends of a Line End Points steadily increasing condition make a line.

Finding the slope of a line can give Slope Gives you a result that can be used for Sought Value understanding and further calculations.

Everything is concentrated in one The Dirac-Delta place.

7.6.2 The “integration happens in the component’s direction” symbolic form During the interview, Student 6 uses a composite symbolic form of an integral with an equation template that I describe as:

퐸푥 = ∫ … 푑푥. Here I have specifically used Sherin’s notation of an ‘x’ as he used in the “dependence” cluster. This indicates that whatever component of the electric field one is attempting to evaluate, one integrates along an axis of that coordinate. I call this symbolic form “integration happens in the component’s direction”. This composite symbolic form is generally not one that is productive to correctly evaluate the desired quantity. Additionally, the student uses the graphical template shown in in Table 7-1 that resembles a Dirac-delta function to represent the conceptual understanding that if you are traveling through space in the vertical direction across a thin horizontal bar that a great deal of charge density is going to be concentrated in one small space.

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Student 6 could not reconcile the results of thinking with this composite symbolic form with their conceptual understanding of what the y-component of the electric field of the charged bar should look like.

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— Summary and conclusions I went back to graduate school and worked on this dissertation attempting to answer the general question “Why is physics hard?”. After identifying types of problems that I thought were hardest for my students as a lower-division physics instructor, I was able to narrow that question down to the much more manageable research question “How do junior-level physics majors coordinatize a problem that requires integration?” In this chapter I will summarize what I discovered and present ideas on what could be pursued next based upon what I have learned.

8.1 Acts of coordinatization and the coordinatization framework In answering the question: “How do junior-level physics majors coordinatize a problem that requires integration?” I identified and categorized 32 specific acts of coordinatization that I placed into five main categories. Additionally, I saw that no new acts of coordinatization were employed by the post-junior Ph 431 students that were not seen in the pre-junior Ph 320 students in this sample. The five main coordinatization categories coded are: • Drew Axes • Limits of integration • Manipulation of the infinitesimal • Infinitesimal in the sketches • Dealing with the vector nature of vector quantities

The Coordinatization Framework uses acts of coordinatization as a subset of the cognitive resources a student is using when problem solving. This subset is large enough that tracking acts of coordinatization allow an observer to have insight into cognitive structures such as symbolic and graphical forms students invoke while solving some types of physics problems that require integration.

8.2 Six students and six new symbolic and graphical forms Using the Coordinatization Framework, I was able to identify three composite symbolic forms of integrals and three graphic symbolic forms that describe how students can combine their mathematical and conceptual understanding while solving a physics problem. At least one

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more symbolic form is hinted at via the prevalence of a particular act many students performed that is characterized by an integral of a certain characteristic. Table 8-1 lists the new symbolic forms that can be added to the list of symbolic forms previously described in the literature. Table 8-1

Summary of Symbolic Forms Identified Symbolic/Graphical Symbol or Conceptual Schema Form Graphical Pattern Dimensional analysis is one way to understand “units must match” how a coordinate dependent infinitesimal can infinitesimal d = d be produced from a coordinate independent relationship infinitesimal.

Different parts of the integrand of an integral d has stuff associated can be determined by examining how the with it infinitesimal relates to those parts.

This symbol pattern was observed in many Sandwiching the students’ written work. Currently, conceptual starting point equation =∫(푠. 푝. 푒. )푑 schemas associated with this template have not (s.p.e.) been identified. There may be many. When seeking to evaluate a component of a Integration Happens in vector via integration, the integration the Component’s Edx= ... x  infinitesimal must be the same as the vector Direction component.

A steadily increasing value with known Line End Points endpoints from which a line can be generated..

Slope Gives Sought Finding the slope of a line can provide the value Value of a desired quantity.

Everything is concentrated at one point along a The Dirac-Delta direction.

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8.3 Implications for teaching

An experienced instructor excels at their job by leveraging their past experiences with students to inform the strategies they use on current and future students. By identifying some symbolic forms used by students when solving the problems I presented them, I hope that I can save some instructors some trial-and-error in the classroom. The “d has stuff associated with it” composite symbolic form was the most productive symbolic form used by the group of six clinically interviewed students. Adjusting instruction in a way that requires students to clearly identify the infinitesimal in their sketches and how that infinitesimal relates to other values in the problem might help develop this composite symbolic form. This approach is consistent with the textbook author Randall Knight in his sample problem shown in Figure 3-2. The “units must match” infinitesimal relationship described by the symbol pattern d = d was productive in assisting at least two of the interviewed students in their problem-solving. Along with teaching students about the advantages of using unit analysis for error-checking, physics instructors can point out that infinitesimals have units thus giving students an additional tool for sense-making integrals. The “Integration Happens in the Component’s Direction” composite symbolic form informs how I should approach students whose work on an exam or assignment seems to reach a dead-end. Student 6 had set up their solution process nicely and had written equations that made perfect sense to me as an instructor. Then, the student seemed to get stuck or simply quit the problem before the solution had been found. In the case of Student 6, the student held a large number of useful ideas and methods appropriate for problem-solving along with one unproductive view of how vector integration works.

8.4 Pathways for future research

8.4.1 More clinical interviews The Coordinatization Framework was productive in identifying student thinking while solving the problems in this study. With just six clinical interviews, I was able to identify six

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symbolic and graphical forms held by these students. More clinical interviews are likely to allow for the cataloging of additional forms held by students.

8.4.2 What are students thinking when they sandwich a starting point equation? The “Sandwiching the starting point equation” symbol template appeared many times in the written data set. The one student in the clinical interviews who wrote this template down erased and replaced it shortly thereafter with a more productive equation. This student could explain several things that they thought was wrong with the sandwiched equation such as recognizing that it was charge that made the electric field and thus the infinitesimal used should be dq and not dr. I could not, however, get the student to explain how they came up with the sandwich equation to begin with. As this act of coordinatization was used by many students it would be useful to understand what students are thinking and what resources are being activated when a student performs this act.

8.4.3 What role does cognitive load play in problem solving of this type I identified 32 types of acts of coordinatization that a student can perform while solving physics problems that require vector integration. Each act represents a cognitive resource that can be activated while solving problems. Starting with Miller (1956) prior research suggests that there might be 7±2 spaces in a person’s working memory that can be described as a cognitive system that temporarily, often on the order of seconds, holds onto information available for processing. “Chunking” is a way in which resources are bundled together so that working memory can appear to hold onto a larger number of resources at once. Chase and Simon (Chase & Herbert, Simon A., 1973) observed chunking in expert chess players who could accurately recall, after a 5 second period of looking at a set up board, the position of many chess pieces if the pieces were set up in a manner consistent with what one might observe in the middle of a typical chess game – if the same number of chess pieces were then randomly placed on the chessboard, the experts would be little better than novices at being able to memorize and recall the position of the chess pieces after a 5 second glance.

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The majority of students in both the written and clinical interviews neglected to take the vector nature of the electric field fully into consideration when solving the problem I gave them. When I pointed this out to the students in the clinical interviews, they were very quickly able to incorporate the vector nature of the electric field into their overall solution. Thus, even though the appropriate resources existed in the students long-term memories, these resources did not seem to be automatically activated and placed into working memory. Future experiments could more strongly cue the electric field is a vector quantity resource at the beginning of an interview and see if perhaps other important resources “disappear” during the problem-solving process due to strains on working memory capacity.

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129

APPENDICES

130

Appendix A – Observed acts of coordinatization

Table A-1

Ph 320 Fall 2016 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

Set dq = L dx and says "not sure if I built correctly". This could be conflating L This student did draw an x and y that with lambda as dq = lambda dx. In the is later referenced in equations. In 1 integral, qdL appears and then the dL words he does mention "in the x- changes, without explanation, to dx. A direction" and "in the y-direction". large space for "y-direction" is created but left blank.

Statement: "Need to take into account L b/c there are an [infinite] amount of 2 None evident point charges in the rod." "If [infinite] amount of point charge would need to integrate [from 0 to L]"

A dr appears in an equation with no other apparent motivation. A dtheta is 3 None evident also referenced with respect to an r- prime.

A dE vector is defined early on -- it is immediately stated in terms of dr-scalar. The limits of the integral-dr is from d to 4 None evident sqrt(L^2+d^2). Then, in a third step, the limits of integration turn into 0 --> L with no change in dr.

A clear definition of lambda is given (lambda=Q/L) . dQ=lambda dl is given in 5 None evident then 2nd line. Later is u-sub is done -- without attention given to the u-sub- changed limits.

dq=q/L along with a statement "I'm 6 A 0 and L are apparent in picture. pretty sure I'm wrong. It's been a long summer." dx then appears.

Origin clearly indicated with a (0,0) and axes. And x and y are indicated 7 An expression appears with a dy. in picture and are used in the calculation of r.

131

Table A-1 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

8 None evident None evident

There is a lot going on in this one: dE as part of a sum; Q=the integral of charge density over length; a d of an algebraic 9 An x=0 is clearly defined. expression; a reference to a "differential amount of charge in"; a u-sub.; d = differentiate

There is a scribbled out integral that was 10 None evident lacking a d-anything. Then an integral of Edq=q that happens.

The first appearance if an infinitesimal An x and y axis along with a denoted 11 appears to be a dr-vector which, in the origin seem to be indicated. next step, turns into a dx.

First, the student stars with the E-field of a point charge and then creates a An x and y axis seems to be indicated. double integral presumably to sum all of 12 A zero of x and a zero of y also seem the point charges. The student later to be indicated. seems to re-start and just have a dx x- hat

Two incorrect initial relations causes a non-sensical integral: q = dQ/dL and 13 None evident r^2=dl^2 + d^2. There is an integral of dQ that goes to Q.

E --> integral of dE; dE --> dr; later, an 14 Maybe axes are indicated. unused attempt to go from dr to dtheta

An arrow labeled x is shown that goes from the right side of the bar to dx. I A lot happens here: E=integral of de; suspect that this isn't an axis but dE-vector = dEx-vector + dEy-vector; dE- 15 rather an "integration arrow". In a vector --> dq-scalar; Ex and Ey --> dq; dq different place on the page, a --> dL; and then a dL to a dy or theta to L coordinate system is drawn with the point P at the origin.

132

Table A-1 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

There is possibly a vertical axis 16 indicated placed in the middle of the None evident rod.

A definition of lambda followed by a differential expression of dQ=lambda dl. 17 None evident The, crossed out an attempt at a relationship between dQ and Epoint.

"an infinitesimal charge dq corresponds Wrote out "Then, at a point s along to length ds along te rod. Linear charge 18 the rod from right to left electric field density lambda=Q/L and dq = lambda caused by dq is…" ds…"

There are possibly axes centered upon the point at which we are 19 supposed to find the E-field. These A transition from dq to dL. could also be E-field vectors from the extreme ends of the rod.

No infinitesimal even though an integral has been written. Statement "I never understood how to do this integral…" It 20 None evident is unclear if he means he doesn't know how to set up the integral, or, that he thought he had a reasonable integral set up but didn't know how to solve it.

Compares a ratio L/Q = dl/dQ; used 21 None evident delta q notation to explain his work.

I'm not sure what "this point is x" means. Maybe it is an origin… more 22 likely it is pointing out that this is an dr appears important point in space when trying to solve the problem.

This x is used correctly as a term that lambda and dq are define above the varies while integrating over the 23 picture. E is stated as an integral of length of the bar. Perhaps this can dE's. dE goes immediately to dx. be thought of as an axis.

133

Table A-1 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

After "summation across the rod" is A horizonal axis might be indicated 24 stated, and integral for E in terms of dr with -L/2 and L/2 indicated. is given.

Uses x similarly to Student 23 --

although 23's solution went a bit further and had more depth. Perhaps this can be thought of as an 25 An integral w.r.t. dx is simply stated. axis -- or maybe it is just a placeholder allowing for the calculation of r. Later, the x and d are used to find r.

An integral w.r.t. dL is simply stated 26 None evident (there might also be something that was erased -- need to look at original)

There is no use of d or delta, but there is 27 None evident a sum with earlier words talking about point charges.

This x and y aren't used anywhere 28 A dtheta is attempted to be used. else in the problem.

A separate x axis and y axis are drawn, each with a zero. The 0's x and y all correspond with position of 29 None evident the point and thus might indicate more a distance calculation than axes.

30 None evident dL in denominator

None evident -- treated entire bar as a 31 None evident single point charge.

134

Table A-1 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

dx is introduced in first line of equations; it looks like the integral (and 32 None evident thus the dx) is on the bottom part of a fraction meaning that it is r that integrated and not 1/r [sic]

x = 0 and x = L are clearly denoted indicating perhaps a thought of an x- Individual steps are shown including dE axis. Nothing equivalent is done in to dq and dq to dL. That is all put 33 the y-direction. The x=0 and x=L together into one expression that shows might just correspond with the limits the evolution of dE to dx. of integration.

Two different sets of axes might have been drawn in at different times. One set of axes in a different color. x "dq=dr little piece of charge" is written; and y are rotated 90 deg from equation starts with a dr; attempt to 34 traditional x and y. Neg. and pos. find relationship between dq, dr and directions are denoted. A second dlambda is given drawing is made rotated 90 degrees from the first.

"Floating" axes are drawn in. In An (incorrect) expression for dq is given. 35 another place on the image, an x=0 is dx's and dy's are given for Ex and Ey. denoted.

36 Axes are clearly indicated. A single expression with dx is given.

37 None evident An expression appears with a dr

dE in terms of dQ and dr is given; E as a 38 None evident sum of dl's is given

E in terms of an integral of dx is given

without much equation setup; in words "…So, you need a sum of all the field 39 None evident produced. The only changing variable is the distance, so the other terms can be pulled out in front of the integral." an integral without an infinitesimal is given.

135

Table A-2

Ph 320 Fall 2016 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r-hat” Observed Acts of Coordinatization

Infinitesimal Indicated Infinitesimal Indicated In Methods of dealing with the vector Methods of dealing with the vector nature of E Student In Sketch Sketch nature of E and/or r and/or r

Wrote "for the field in the x direction"… and … "…y direction". Then created two columns: one 1 None evident for the "x direction" and one for the "y direction". Left the "y direction" blank, possibly because he didn't have a dy

2 None evident Nothing evident

The rod appears to be broken up into pieces, each piece seems to be 3 described as having a Nothing evident certain distance from the point where they are asked to find the E-Field.

Vector starts out on both sides of the equation, but disappears on the right side and stays 4 None evident disappeared on the right side for the rest of the problem.

E has probably a vector on it, but the right side does not have a vector on it for a total of 6 or 7

steps. At the very end an r-hat appears (not A dQ is drawn on the 5 scanned) Then a discussion "My answer suggests picture. that [mag]E is constant which doesn't seem correct. I think I messed up somehow with the curly-r-hat vector."

6 None evident Nothing evident

A vector on the right-hand side of the equation 7 no vector on the left side. (This is the first and last equation written for the E-field)

136

Table A-2 (Continued)

Infinitesimal Indicated Infinitesimal Indicated In Methods of dealing with the vector Methods of dealing with the vector nature of E Student In Sketch Sketch nature of E and/or r and/or r

A vector on the left side is drawn with no vector 8 None evident on right. (This is the first and last equation written for the E-field)

All E-field equations have vectors on the left and right side of the equation. The r-hat seems to be 9 treated mostly as a constant as demonstrated in this example.

Iconic equation first re-written without vector symbols. Then an attempt to set Edq = q with a 10 None evident vector on one side of an integral but not on another.

r-vector denoted on picture. R-vector stated in < A single point on the rod is , > format. Recognition that E-vector and F- examined. In another vector are related to each other. A format E- 11 sketch, several points vector = the integral of … r-hat times dr-vector. seem to be examined. The integral transitions into < , > form. A relationship between r-hat, r-vector, and mag(r).

E-vec=… r-hat; mag(E)=scalar double integral w.r.t. dxdy; statement "I am drawing a complete The bar seems to be blank on solving the integral…"; Later: E-vector broken up into pieces -- dot E-vector = mag(E)^2; E-vector-x-component = 12 this seems to primarily … dx x-hat. Last line: the vectorness has serve to show that there disappeared -- this is the first inappropriate are meny different r's. vector=scalar statement made -- also writes "Nope. This won't work."

13 None evident E-vector = double integral … r-hat.

E-scalar=…j-hat; E-scalar = … r-hat; an integral of 14 None evident r-hat; mag(r-r')=dj-hat

A piece of the bar is

labeled dx,dq. Several r's

denoted to different points -- although the r of dE-vector = dEx-vector + dEy-vector; Ex-vector the dx is not indicated. 15 and Ey-vector equal to a scalar; E-scalar=integral One theta between two of dE-vector unlabeled dx's shown. A different diagram shows point P with many dE's at it.

137

Table A-2 (Continued)

Infinitesimal Indicated Infinitesimal Indicated In Methods of dealing with the vector Methods of dealing with the vector nature of E Student In Sketch Sketch nature of E and/or r and/or r

r's do different parts of 16 the rod might indicate E-vector = … r-hat small-piece thinking.

17 None evident E-vector = … r-hat

E-vector = … r-hat; < , > notation in integral that 18 None evident is then explicitly discussed as meaning that there is an Ex and Ey

A piece of the bar looks like it was labeled dy with 19 Nothing evident the y later replaced by an L

There are perhaps two

(unlabeled) E-field vectors on the point - this might Almost nothing except at the very end all the 20 be several dE's at a point. scalar work is equal to E-vector There are also vector-like things pointing away from one side of the rod.

21 A dl is indicated. Nothing evident despite lots of work

States "lots of point charges added up" and

then has 3 dotted lines An Ex-vector = … r-hat with a statement saying "I that might be drawn to 22 wish I remembered what that meant" pointing at different dq's -- 2 of these the h-hat. At the very end E-vector = scalar are labeled r. (r end points used correctly later)

A dq seems to be clearly indicated along with dq's r-vector denoted on picture. R-hat stated in i- relationship to dx. 23 hat, j-hat notation. Consistent vector notation Unlabeled arrows that kept thru solution. might be dE vectors are also shown.

138

Table A-2 (Continued)

Infinitesimal Indicated Infinitesimal Indicated In Methods of dealing with the vector Methods of dealing with the vector nature of E Student In Sketch Sketch nature of E and/or r and/or r

E-vector = scalar; where B is a coeff. in the theta-hat dir.; E-vect=. Words "I believe that the magnitude of the E- "summation across the field is the integral along the length of the line. 24 rod". A single r-vector for From there we need to be able to express the a seemingly random point field in three-dimension." Claims cylindrical in the rod is given. coords should be best and the answer should "be in the form E-vector..."

An unlabeled box on the 25 end of the rod might be a none evident … just as an = scalar integration d'box.

Lines drawn from several points on the bar to the Something I can quite make out--maybe Gauss' 26 point might indicate the Law (look at original); Seems to start out with an bar as a collection of mag(E-vector)=scalar integral points thinking.

Starts with e-vector=…r-hat with "A rod can be A picture was drawn that modeled as a function of point charges… distance didn't include d'anythings, from rod ranges from d to sqrt(L^2+d^2)" No but instead had words integral ever written; E-Vector = ... L-hat with a 27 that described point statement "this is wrong, this gives me the E if pt charge thinking: “A rod charge is located at other end of rod from the can be modelled as a point" also "need to review how to put it into a function of point charges” function with a E=...sum...L-hat"

A dtheta and a First line: integral of E-vector dtheta that then corresponding dL seem to gets turned into a r-hat which is then maybe 28 be shown. Lines showing broken up into vector format (with non-standard r's to various points on bar notation) are shown.

A single dL (that might be 29 labeled only L) is shown none evident with an r to the point.

None evident in sketch, e-vector = scalar or, maybe, it is an e-vector = an

but there is a written incomplete idea . Also a discussion that doesn't discussion. “did an 30 include any vector discussion but starts "I can't integral to find the total exactly remember how we treated charge of a affect of every points of rod…" the rod”

None evident -- treated turning of an E=...r-hat into an Ex=...i-hat 31 entire bar as a single point equation and a Ey=...j-hat equation charge.

139

Table A-2 (Continued)

Infinitesimal Indicated Infinitesimal Indicated In Methods of dealing with the vector Methods of dealing with the vector nature of E Student In Sketch Sketch nature of E and/or r and/or r

The r-hat possibly turns into an x-hat in the first 32 None evident step. Then turns back to an r-hat. In between maybe there is an integral of f(r)dr

Starts with iconic equation. The r-hat turns into <,> notation and stays that way. Notational 33 A dx is shown correctness is maintained. A note is made saying "I don't totally remember how to address the r- hat"

A dq=dl is indicated on The r-hat remains untouched. Vector=vector 34 sketch remains good.

A dq and a dl is shown. E-vector is broken up into Ex-vector and Ey- 35 The dl reaches up to an r vector; notation doesn’t remain vector-vector line for unclear reasons. consistent

Lines drawn from several points on the bar to the Tries defining r-hat as = xx-hat+yy-hat+zz-hat. 36 point might indicate the Writes one line where E-vector=integral of … r- bar as a collection of hat points thinking.

37 None evident The r-hat just disappears

A dl, a dQ and lines are Use of absolute value/magnitude signs to "deal 38 indicated. with" the vectoredness of E and dE

Lines drawn from several points on the bar to the 39 point might indicate the E-vector = scalar in four or five places bar as a collection of points thinking.

140

Table A-3

Ph 320 Fall 2016 “Limits of Integration” Observed Acts of Coordinatization

Student limits of integration limits of integration

0 to unclear for dL; 0 to L for dx where x is long the rod 1 (note that the boxed "x direction" is not part of the upper limit)

This person is never really ever able to set up an integral but uses lots of words to explain what is supposed to 2 happen. The only part of the integral that is written is the integral sign and the limits

0 to L wrt dr r seems to be the hypotenuse; 0 to theta 3 wrt theta theta is at dot

4 0 to sqrt(l^2-d^2) wrt dr; r is the hypotenuse

-L/2 to L/2 wrt dl where dl is most likely along rod; a U- 5 sub happens, limits remain unchanged (look at original for back of page)

6 0 to L wrt x where x is probably along rod

7 0 to L wrt dy where y is along the rod

141

Table A-3 (Continued)

Student limits of integration limits of integration

8 none evident (no integral set up)

no limits wrt to dE; 0 to L wrt dL in order to find Q; 0 to L 9 wrt x to find E along with a wrt dE in a middle step; a zero to U(L) wrt to dU for a U-sub

0 to L wrt dq (dq seems to be scribbled in -- note also 10 there is a comment "Is there an integral involved to find a point charge?)

11 0 to L wrt dr-vector and then dx

double integral 0 to d and 0 to L wrt dxdy; a "reset" that 12 is 0 to L wrt x with a recognition that the limits give an infinity

double integral 0 to d and 0 to L wrt a dQ/dL -- there is a 13 note that says "I think an integral over the length is involved, though I'm not sure if I set it up correctly"

14 integral signs with no limits

9 integral signs wrt dq and dL and dE-vector never with 15 limits until one near the end that is 0 to L wrt dx

142

Table A-3 (Continued)

Student limits of integration limits of integration

16 none evident (no integral set up)

17 none evident (no integral set up)

no limits while wrt dq; limits 0 to L when wrt ds with 18 evidence that s is something in the horizontal

no limits while wrt dq; limits 0 to L when wrt dL where dL 19 seems to be along the rod

no limits no infinitesimal note saying "I never understood 20 how to do this integral…"

0 to L wrt dl; then a trig sub with limits disappearing 21 when dtheta appears; limits come back at appropriate time

22 d to sqrt(d^2+L^2) wrt dr

23 0 to L wrt dx; preceded by nothing wrt dE-vector

143

Table A-3 (Continued)

Student limits of integration limits of integration

24 -L/2 to L/2 wrt dr where r is probably the hypotenuse

25 0 to L wrt dx where x is along bar

nothing wrt unknown (look at original); 0 to Ltot wrt dL 26 with dL probably along L

no integral anywhere; there is a sum sign with intervals 27 wrt L

28 0 to arctan(L/d) wrt theta where theta is at dot

0 to L wrt no infinitesimal followed by no limits wrt no 30 infinitesimal

31 no integral anywhere

144

Table A-3 (Continued)

Student limits of integration limits of integration

0 to L wrt dx; this is also a u-sub like action with no limits 32 wrt f(r)dr

33 nothing wrt dE followed by 0 to L wrt dx

d to L wrt dr and then a d to L wrt dL with it looking dr is 34 the hypotenuse and dl is along the bar (from picture) but, there is also a crossed out 0 to L=r wrt dr

-L to zero wrt x and 0 to d wrt dy along with some 35 clarifying notation

36 0 to L wrt dx

d to sqrt(d^2+L^2) wrt dr (limits disappear during 37 manipulation (no charge of variables), but reappear at end)

0 to L wrt dl (in this problem, there are dE's, dQs, dthetas, and drs in interesting places, but only one 38 integral as shown with a statement "cannot remember how to change "integration for respect to r to integrate with respect to L")

0 to L wrt dx along with a discussion that includes: "… the distance L changes, so you replace that with lambda 39 and integrate 0 --> L.". During this discussion a limitless integral with an infinitesimal is used as a discussion aid

145

Table A-4

Ph 431 Fall 2017 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

Starts out with dQ. A lambda is defined, x,y,z axes was drawn with origin although not explicitly in terms of dx and 101 at left of rod. dq. dQ does get turned into a lambda dx.

Starts out with dq. A lambda is defined, An x was indicated on the bar -- although not explicitly in terms of dx and 102 might be using bar as axis. dq. dQ does get turned into a lambda dl. dl=dx is stated. A u-sub happens.

Starts out with dE-vector --> dq r-hat. Defines dq=lambda dl and states lambda=Q/L. dq then goes to dl. dl 103 Floating x,y axes shown. becomes a dx with no explanation -- probably just for calculation manipulations.

Lambda=Q/L. E-scalar=integral(lambda

dr). Integration happens along r- hypotenuse. "This only deals with the magnitude of the E vector. Forgot to 104 Floating x,y axes shown. deal with its direction" 2nd attempt: E- vector=integral(lambda r-hat dr); a dl in the denominator and a r-vector = (d)y- hat + dlx-hat. "I have forgotten how to deal with the unit vector."

Double integral wrt rdrdtheta. One 105 None evident integral has limits of d to sqrt(d^2 + L^2). Nothing more.

a "drdq" is written alone in a corner. An x was indicated on the bar -- What appears to be a double integral is might be using bar as axis -- 106 written wrt dxdlambda. A triple integral might also just be an x of is written in terms of drdthetadlambda integration. that gets turned in to a dxdthetadlamba

107 None evident Mostly blank sheet

A 0 and L are shown on the rod --could be limits of integration Evector=integral of dE-vector. dE vector 108 or using bar as axis. A dot on is then expressed in terms of dqr-vector. the right side of the bar could dq=(Q/L)dl be an origin.

146

Table A-4 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

When trying to use "Gauss's Law?" an integral of Edotds is given. The ds- 109 None evident vector is then turned in to an rdrdphi. Q=integral of rhodl also given and r-f' is presented that has a dr-hat in it

Starts with a E = integral of Q dr'. 110 None evident There is also a Q --> dq?

111 None evident Starts with E = integral q dl

Enet = integralE(d,l)dl (note to self -- don’t use 'd' in a problem where you 112 None evident want to learn more about student reasoning of the infinitesimal). This the goes to = integral Qdl

Maybe some being used to help First line is E = qdr-vector. dr-vector remember trig (note statement 113 then goes to a dL' where dL' appears to reading "wow, trig is tripping be along the length of the bar me up today.")

x,y axes with origin on left side r-hat = x-hat + y-hat; then E = double 114 of rod integral wrt Qdxdy

There is an x on the rod that 115 might be an x of integration starts with an integral of qdx rather than a coordinate axis

116 None evident starts with E-vector = integral of Qdr

147

Table A-4 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

Starts out with integral qdl. Then lots Something axis-like to help with of words happen that discuss point limits of integration and 117 charges. A definition of lambda relationship between r and d happens and the 2nd equation is now and L integral (Q/L) dl

axes with origin on right side of Starts with a integral of wrt dq which 118 rod and coordinates (0,d), (0,0), then goes to Qdx (no L) (-L,0) clearly shown

Starts out with "(If I could remember how at the moment) I would integrate the length of the rod relating dL to 119 x,y axes centered on dot dQ…"; then dE-vector = integral qdr- vector-r-hat; dr to dx is then articulated and we end of with an integral of qdx

r=-L and r=0 position shown on 120 starts with double integral wrt Qdxdy Rod perhaps using rod as axis

121 None evident starts with qdtheta

122 None evident starts with triple integral wrt qrdzdrdphi

Integral(Q/L) dr' w/ r'=sqrt(d+L'). Later

(Q/L)dr, w/integral 0-->L where r^-2dr goes to (-1/L). Later E-vect equals two Axes-like objects at either end, integrals one wrt dx other wrt dy with 123 both of which have a zero lim going in the x- and y-dir respectively. associated with it. dx-->dL' & dy-->dL'; dy integral disappears. Last step integral w/o infinitesimal. Recog something wrong w/ integral.

starts out with integral of lambda dr'. dr' turns into dx'. dx' remains until the 124 None evident last step which is an unevaluated integral

148

Table A-4 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

starts out with a lambda = Q/L => lambdadx=dq. Ey starts out as a point x,y axes clearly shown with 125 charge, Ex = integral lambda dx x^-2 origin at right side of rod with a recognition that something is wrong with the integral

Recognition that the results of Ex and Ey are integrated wrt dx. E-vector(x,y) = 126 Unlabeled axes centered on dot integral (Q/L)rcos(th)dx x-hat + integral (Q/L)rsin(th)dx y-hat

r1, r2, r3, … rL along rod perhaps indicating rod as axis The point charge equation is first 127 even though the r's are presented. Then Integral wrt qdL hypotenuses.

dr = sqrt(dL^2+d^2) then Q/L = k followed by dq = k dL (this k indeed 128 None evident seems to be a lambda). Finally a dE = integral of dL/dr

x,y axes floating but at same E-vector = sum Ei-vector = integral qdq. 129 elevation as rod (also, an x on The qdq then turns into a (Q/L)dx. the right side of bar)

A lambda is defined as Q/(4piepsilon). 130 just a floating x-axis lambdadl is then used from the first line.

In the first line a dq/dsomething appears a floating x-axis and a floating y- in the integral. That dq/dsomething 131 axis as well as a 0 to L drawn in seems to turn into a lambda. A dl just a different place appears from the beginning and sticks around.

E-vectortot=integral(dE); dE --> dq; x,y axes shown mostly centered lambda=(Q/L)=dq/dl --> dq = lambdadl; 132 on left side of bar; another set dl-->dx; dl = cos(theta)dr and then the of axes shown labeled EFD integral (1/r^2)dr is evaluated from 0-- >L.

149

Table A-5

Ph 431 Fall 2017 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r-hat” Observed Acts of Coordinatization

Infinitesimal Infinitesimal Methods of dealing with the vector nature of Student Methods of dealing with the vector nature of E and/or r Indicated In Sketch Indicated In Sketch E and/or r

E-vector = integral (stuff) r-vector/mag(r^3); then r- vector is defined as a vector matrix (L d 0) - (x 0 0); then 101 none evident an integral containing a x-hat + y-hat happens that uses the vector addition rather than the correct trig functions

E-vector = given iconic equation with an integral 102 none evident plopped in and r-hat intact; the r-hat turns into an (x*x- hat + d*y-hat) still in the integral.

E-vector = given iconic equation with an integral There is a dq = 103 plopped in and r-hat intact; the r-hat turns into an (l*i- lambda dl hat + d*j-hat) still in the integral.

E-scalar (probably) = integral dr, then after some work a three dotted lines note that says "This only deals with the magnitdue of going to three the E-vector. I forgot to deal with its direction. Next 104 different points on line is then E-vector = r-hat inside an integral. r-hat is rod from dot then defined as (d)y-hat + dlx-hat. A note later says "I have forgotten how to deal with the unit vector".

The only equation written Is a (nothing =) integral that 105 none evident is scalar

106 none evident (nothing) = scalars everywhere

107 none evident no related work

E-vector = integral dE-vector; dE-vector = given iconic 108 none evident equation with dr-hat intact; E-vector = integral of iconic equation with r-hat still intact. Work then ends

150

Table A-5 (Continued)

Infinitesimal Infinitesimal Methods of dealing with the vector nature of Student Methods of dealing with the vector nature of E and/or r Indicated In Sketch Indicated In Sketch E and/or r

defines r'-vector = dr-hat + 0phi-hat + Lz-hat scribbled out with note "easier in cylindrical". Integral of E-vector dot ds-vector = scalar. E-vector = slightly modified 109 none evident iconic equation with r-hat intact. Then nothing = scalar integral. Then scalar integral = Q-scalar. The E-hat = scalar. A sketch tries to clarify the relationship between r, r', d and the origin.

110 none evident E-vector = integral r-hat then writes "...hit a wall"

many lines, one dl 111 E-scalar = integral r-hat then work ends at end of bar

112 none evident E-net-vector = integral = (sqrt(d^2+l^2))-vector

a picture that explains r-hat. E-vector = integral r-hat * 5 lines (maybe dr-vector. Explains r*r-hat = y*y-hat + x*x-hat with a 113 vectors) -- one note "but what does r=?". Next a note "wow, this is dotted; an L' tripping me up". Then E-vector = integral (y-hat + h-hat)

r-hat = x-hat + y-hat; E-scalar = integral scalar with r-hat 114 none evident either dropped or turned into dxdy

(nothing) = scalars everywhere with note "I forgot how 115 none evident to deal with direction off the top of my head. I'll come back to it later" (but doesn't)

E-vector = integral of given iconic equation with r-hat 116 none evident intact. r-hat sticks round for the ride.

151

Table A-5 (Continued)

Infinitesimal Infinitesimal Methods of dealing with the vector nature of Student Methods of dealing with the vector nature of E and/or r Indicated In Sketch Indicated In Sketch E and/or r

(nothing) = integral r-hat with a statement "So I think 117 none evident that the integral I would need to solve would look something like this:"

E-vector = (r-vector - r'-vector); defines r-vector as d*y- hat and r'-vector = -x*x-hat; states mag(r-vector-r'- vector) = sqrt(x^2+d^2); then sets up integral E-vector = 118 none evident integral (x*x-hat + d*y-hat) and then splits up into two integrals; final answer is in terms of e-vector = x-hat + y+hat

first equation dE-vector = integral dr-vector * r-hat with slice shown; an (r-vector)^2 in the denominator. r- 119 written discussion vector^2=x^2+d^2 --> 2dr=2dx and then the integral of dL to dQ; becomes E-vector=integral dx-scalar*r-hat

E-vector = double integral r-hat --> r-hat turns into 120 none evident costheta*x-hat + sintheta*y-hat. Also, a statement that r-hat is a direction, but not entirely sure how it works

121 none evident (nothing) = scalar of given iconic equation

three dotted lines going to three E-vector = integral r-hat; scribbled out attempt to define 122 different points on dr-vector in terms of r-hat+phi-hat+k-hat rod from dot

(nothing) = integral scalar; later (nothing) = x-hat; later 123 none evident E-vector = scalar integral wrt dx + scalar integral wrt dy

E-scalar = integral r-hat which then turns into xx-hat + yy-hat + zz-hat… the y-hat and z-hat components 124 none evident disappear. Then the x-hat just disappears within the integral.

152

Table A-5 (Continued)

Infinitesimal Infinitesimal Methods of dealing with the vector nature of Student Methods of dealing with the vector nature of E and/or r Indicated In Sketch Indicated In Sketch E and/or r

Ex and Ey components are evaluated separately. It is 125 Dx then stated Etot = Ex + Ey

E-vector = integral * cos(th)x-hat + integral *sin(th) y- 126 none evident hat

multiple lines (or 127 E-vector = integral r-hat -- work then stops vectors)

slice shown; dl; E-scalar = mostly unaltered original equation with r-hat 128 hypotenuse labeled gone dr

E-vector = integral r-hat where r-hat later turns into x*x- 129 none evident hat + d*y-hat

E-vector = integral r-hat. Some work appears to be 3 lines really close 130 done to figure out what r-hat is where they say r-hat = to each other sqrt(l-hat^2 + d-hat^2). Then work stops.

multiple (wavy) 131 E-vector = integral r-hat(l) where r-hat(l) = l*x-hat + DG lines

slice labeled dq,dl;

note that there is something that E-vector = integral dE-scalar = integral dq-scalar = 132 might be an x-axis; integral dl-r-hat. Later the r-hat disappears again. or, it might be an x- Later, a dl turns into a cos(theta)dr. vector going to thte dq,dl

153

Table A-6

Ph 431 Fall 2017 “Limits of Integration” Observed Acts of Coordinatization

Student Methods of dealing with the vector nature of E and/or r Methods of dealing with the vector nature of E and/or r

0 to L wrt dQ (with d added in perhaps as afterthought); 0 101 to L wrt lambda dx

none wrt dq; -l to 0 wrt lambda dl; -l to 0 wrt du; -l to 0 wrt 102 dx

0 to L wrt dq; 0 to L wrt lambda dl; limits disappear during 103 algebra work with dl's, du's, and dw's

104 none wrt to dr and then d to sqrt(L^2 + d^2) wrt dr

double integral: d to sqrt(d^2 + L2) and nothing wrt 105 rdrdtheta

none wrt dr; double integral: 0 to Q and 0 to L wrt 106 dxdlambda; later the dlambda becomes 0 to 1; later a 0 to 2pi wrt theta comes in

107 None

108 none wrt dE-vector; -L to 0 wrt dl

154

Table A-6 (Continued)

Student Methods of dealing with the vector nature of E and/or r Methods of dealing with the vector nature of E and/or r

none wrt ds-vector; -L/2 to L/2 wrt dl; double integral 0 to 109 L and 0 to 2pi wrt rdrdphi

110 0 to L wrt dr' where r=(r'+d)

none wrt dl and written note that says "need to integrate 111 over dl" … "don't remember the bounds"

112 0 to L wrt dl;

113 0 to L wrt dr-vector; 0 to L wrt dL'

114 double integral: 0 to d and 0 to L wrt dxdy

115 0 to L wrt dx

116 d to sqrt(L^2 + d^2) wrt dr

155

Table A-6 (Continued)

Student Methods of dealing with the vector nature of E and/or r Methods of dealing with the vector nature of E and/or r

117 0 to L wrt dl;

118 none wrt dq; -L to 0 wrt dx

119 none wrt dr-hat; 0 to L wrt dx

120 double integral: 0 to d and 0 to -L wrt dxdy

121 0 to theta wrt dtheta

122 triple integral: 0 to L, 0 to rr, 0 to 2pi wrt rdzdrdphi

0 to -L wrt dr' then 0 to L wrt dr; later (0 to L wrt dx + 0 to d wrt dy) then to 0 to L wrt dL' with a circle around the 123 zero that might indicate that the answer doesn't make sense

124 none to start wrt dr'=dx' but ends with 0 to L wrt dx'

156

Table A-6 (Continued)

Student Methods of dealing with the vector nature of E and/or r Methods of dealing with the vector nature of E and/or r

125 0 to L wrt lambda dx

0 to L wrt dx + 0 to L wrt dx In both the x-hat and y-hat 126 directions

127 0 to L wrt dl;

none wrt dE; none wrt dL then 0 to L wrt dL; theta-naught 128 wrt dtheta

L to 0 wrt dq; L to 0 wrt dx; L to 0 wrt dx in x-hat+y-hat 129 directions

130 0 to L wrt dl

131 0 to L wrt dl

132 none wrt dE; none wrt dq; 0 to L wrt dl; 0 to L wrt dr

157

Table A-7

Ph 322 Spring 2018 “Drew Axes” and “Equation Manipulation of Infinitesimal” Observed Acts of Coordinatization

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

first integral has a qdr in it that then goes to a qdL' at 209 none evident the limits of integration shift from L-->0 to 0-->L

an x that might be a leg of a triangle or an axis. 0 210 (Q/L)dr = dq; dq then turns into a (Q/L)dx to L indicated on ends of bar and sketch

0 to L indicated on ends first integral has a n (?)= integral of dL; later there is 211 of bar an E = integral of qdL

first integral nothing = integral of E-vectordl; this turns into a E-vectoratpoint = qintegraldL with words 212 none evident perhaps specifically pointing at dL saying "need to include r-hat term, not sure how". Later there is a Q = integral of qdL = qL

starts with dr-scalar; somehow turns into double integral of dxdy with dy being along y-axis; both 213 none evident integrals "happen" with nothing happening -- dxdy "change" into integral evaluation symbols

probably not, but there was an r-hat vector at An integral called the "Gaussian Integral" just appears the end of the bar that 214 with drd? With a written discussion that little bits could denote the need to be added up. thought of the placement of the origin

215 none evident starts with dL'

216 none evident starts with dr scalar (and work ends there)

158

Table A-7 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

217 None evident none evident

dr = 2xdx/hypotenuse coming from r = hypotenuse (zap with d); a definition of dE vector in terms of dr- 218 a floating x axis vector; E = integral dE which then gets converted to xdx

floating axes; a 0 and -L dE = …dq; dq=lambda dx with lambda = Q/L clearly 219 along bar and a probably defined y axis on bar

starts with the sum of E vector = sum of … qi; qi is

deemed constant and comes out of the sum; a dr 220 -L to 0 along bar appears when the sum is turned into an integral <-- words indicate that student is unsure of this but is then confident in a dr integral.

what could be a 0 to L An equation with a dr just appears; words then say 221 axis and what looks like "dq is the "small" amount of charge associated w/ an erased -L/2 to L/2 axis that "dr" portion of the rod"

none in a double integral that didn't get anywhere; a 222 none evident dx that appears.

probably not, but there 223 a dr appears next to an E-vector is a vector labeled r

could be floating axes, or a dr appears and turns into a dx; later, a dr-vector could be axes drawn to 224 turns into two integrals added together with one wrt assist or clarify an dx and the other wrt dy operation

159

Table A-7 (Continued)

Equation manipulation of Student Drew Axes Drew Axes Equation manipulation of Infinitesimal Infinitesimal

E-vector is the sum of En (from 1 to infinity); which turns into Evector=sumEndn from 1 to infinity 225 none evident followed by words "change of variable" and then dn turns to dr with a bounds change to h^2 to hypotenuse )

dr just appears; dr gets turned into sqrt(dx^2+dy^2+dz^2); dy and dz seem to disappear as "no zi, no variable y-hat; we have thin rod or a line 226 origin defined of changes q; a dex turns into an rdrdz with "switch to cylindrial (sic) coordinates"; there is also an attempt to do something like a zap with d upon r.

160

Table A-8

Ph 422 Spring 2018 “Infinitesimal in Sketch” and “Methods of Dealing with E-vector and r- hat” Observed Acts of Coordinatization

Methods of dealing with the Methods of dealing with the Student Infinitesimal Indicated In Sketch Infinitesimal Indicated In Sketch vector nature of E and/or r vector nature of E and/or r

Rewrote, with hypotenuse written out to replace r, the original given 201 none evident iconic equation. No further work done.

no vector notation written in any work. The first line is basically in 202 none evident given iconic equation with the r- hat just disappeared.

a dq indicated; a dl indicated in a

different place; lots of "r's"; words "small piece of rod has dQ charge, and each dq piece has variable r 203 e-vector = scalar integra. distance from the point". There is also a discussion of a sum of Ei's, but never a dE not even in an integral

Rewrote original given equation then E-vector = scalar integrals. dotted lines to multiple points An E-vector = double scalar 204 along with the words "sum the integral happens too. Words: small charges dQ of the rod" "see how the direction of the vectors pointing to the point charge from the rod affect it"

E-vectorx and E-vectory = two lines that might indicate a dx Evector*trig; but all integral work 205 thinking (dx shows up later in is scalar with no recognition of equations) components

integral of E-vector dx = integral 206 none evident of original iconic equation * dx; r- vector just pops out of integral

recognition that r-hat is a vector; no integrals, but E-vector = E- scalar in all places with a note that 207 none evident if r-vector = (0,0,h) then mag(r- vector) = h. Evidence perhaps that author is trying to treat r as a point.

E = integral of r-vector (not r-hat). r-vector = An Ex-vector and 208 none evident Ex-vector = scalars with incorrect trig-like activity happening.

161

Table A-8 (Continued)

Methods of dealing with the Methods of dealing with the Student Infinitesimal Indicated In Sketch Infinitesimal Indicated In Sketch vector nature of E and/or r vector nature of E and/or r

E-scalar = r-hat is first thing written. An integral just pops up 209 none evident around iconic equation with r-hat just disappeared.

2 dq's shown along with lines 210 vector nature ignored connecting them to point in space

211 1 dL shown as well as three lines E-vector = scalar integral.

r-hat*r = r-vector is written but not used in calculations. Integral of E-vector(dl) = (blank) is written. 212 none evident E-vector = scalar integral is written along with words saying "need to include r-hat term, not sure how".

First line of calculations for E is all scalar -- along with words "1

dimension, otherwise vectors". lines and words provide evidence 213 Then when E-vector is introduced, of small piece thinking a double integral wrt dxdy is created, but E-vector = doubleintegral-scalar.

a single scalar double integral is 214 none evident written

E-vector=integral r-hat. The r-hat then hangs out with no operative 215 none evident action for the ride through the end

not much work done, but no 216 none evident attention paid to vector

162

Table A-8 (Continued)

Methods of dealing with the Methods of dealing with the Student Infinitesimal Indicated In Sketch Infinitesimal Indicated In Sketch vector nature of E and/or r vector nature of E and/or r

E-scalar = r-hat is first thing written. An integral just pops up 209 none evident around iconic equation with r-hat just disappeared.

2 dq's shown along with lines 210 vector nature ignored connecting them to point in space

211 1 dL shown as well as three lines E-vector = scalar integral.

First line of calculations for E is all

scalar -- along with words "1 dimension, otherwise vectors". lines and words provide evidence 213 Then when E-vector is introduced, of small piece thinking a double integral wrt dxdy is created, but E-vector = doubleintegral-scalar.

a single scalar double integral is 214 none evident written

E-vector=integral r-hat. The r-hat then hangs out with no operative 215 none evident action for the ride through the end

not much work done, but no 216 none evident attention paid to vector

163

Table A-8 (Continued)

Methods of dealing with the Methods of dealing with the Student Infinitesimal Indicated In Sketch Infinitesimal Indicated In Sketch vector nature of E and/or r vector nature of E and/or r

only equation written is e-vector = 217 many lines scalar of slightly modified initial iconic equation

dE-vector = stuff*dr-vector (not r- hat). Vector thoughts disappear 218 none evident and E-scalar=integral dE-scalar = integral dx scalar

E-vector = Ex-vector + Ey-vector; there is some mostly correct work 219 none evident that is not quite notational correct that gets down to Ex and Ey components of E.

vector=vector and mag(vector)=scalar notation is 220 none evident good. But, ultimately, r-hat just disappears.

writes one expression with an dr's draw with unlabeled arrows 221 integral in parenthesis followed coming out of them. by an r-hat

a dx, and maybe a dtheta and no apparent attention paid to 222 maybe a dsomething vector nature

Starts out with sum(E-vector) = integral (E-vector dr). Then an E- 223 none evident vector = integral wrt dr scalar written. No further work done.

Ends up with E-vector = integral slightly labeled arrows that 224 dx + integral dy. Also defines a dr- probably represents dE vectors hat

164

Continued A-8 (Continued)

Methods of dealing with the Methods of dealing with the Student Infinitesimal Indicated In Sketch Infinitesimal Indicated In Sketch vector nature of E and/or r vector nature of E and/or r

slightly labeled arrows that probably represents dE vectors E-vector = scalar almost 225 with an Enet; in a separate sketch everywhere there is a dL and some ri's

E-vector = integral r-hat dr-vector = triple integral r-hat. The r-hat 226 a dr-vector shown sticks around for a while until disappearing in a "switch to cylindrical coordinates"

165

Table A-9

Ph 422 Spring 2018 “Limits of Integration” Observed Acts of Coordinatization

Student limits of integration limits of integration

201 0 --> L wrt dl

202 0 --> l wrt dl; also a usub with no limits

203 none wrt dq

0 --> L wrt dQ; 0 --> L wrt dtheta; double integral 204 both 0-->L wrt dthetadQ

0 --> L wrt dx; none wrt dE; none wrt dQ; also, a bit 205 of an equational "discussion" where 0-->L is shown for the first dx integral, but then not for the rest

first line: none wrt dx; then 0-->L wrt dx for a 206 couple of lines; also a 0-->L wrt dL where this may or may not be a u-sub like activity

207 no integrals

208 0 --> L dl

166

Table A-9 (Continued)

Student limits of integration limits of integration

L --> 0 wrt dr that turns into a 0 --> L wrt dL'. Other than the conversion of r to L' here, there is no 209 description of L'. Note that a minus sign appeared after the L-->0 0-->L switch (the - sticks around in the next line of work).

(something erased) then none wrt dq; then 0-->L 210 wrt dx

211 0 --> L wrt dL

212 0 --> L wrt dL

none wrt dr; then double integral h-->0, 0-->L wrt 213 dxdy (evaluation is interesting)

214 double integral: 0-->L, 0-->h wrt drd?

215 0-->L wrt dL' also a u-sub

216 0-->L wrt dr

167

Table A-9 (Continued)

Student limits of integration limits of integration

217 no integrals

218 none wrt dE; 0-->L wrt dx

blank --> Q wrt dq; then 0-->L wrt dx. Note that there is also a minus sign in the 2nd step that might 219 have something to do with a coordinate system drawn on the bar.

220 -L --> 0 wrt dr

H --> h wrt dr (where H is the hypotenuse of L and 221 h)

incomplete double integral with 0-->L and 0-- 222 >sqrt(L^2+h^2) with nothing after; then a single integral 0-->L wrt dx

223 first step none wrt dr then 0-->L wrt dr

none wrt dr; then none wrt dx; then none wrt dr- vector; finally two integrals, perhaps meant to take 224 the vectoredness of E into account coming up with with 0-->L wrt dx + 0-->h wrt dy

168

Table A-9 (Continued)

Student limits of integration limits of integration

1-->infinity wrt dn where n appears to be the 225 number of charges in the rod; later h^2-->h^2+L^2 wrt dr

first nothing wrt dr-vector; then a triple integral wrt sqrt(dx^2+dy^2+dz^2); later a 0-->R wrt dx; then a 226 "switch to cylindrical" and now a double integral 0-- >L, 0-->r wrt rdrdz

169

Appendix B – Student 6 whiteboard work

Figure B-1 Student 6’s whiteboard work. This is the first board of work on Question 1 which focuses on finding the center of mass of a bar with a linearly increasing mass density.

Figure B-2 Student 6’s second and final board of work for Question 1.

170

Figure B-3 Student 6’s first board of work for Question 2. Question 2 asks for the student to find the electric field above one side of a uniformly charged rod.

Figure B-4 Student 6’s second board of work for Question 2.

Figure B-5 The bottom of Student 6’s second board of work for Question 2.

171

Figure B-6 Student 6’s third board of work for Question 2.

Figure B-7 Student 6’s fourth and last board of work for Question 2.