AN ABSTRACT OF THE THESIS OF. Emily M. Smith for the degree of Master of Science in Physics presented on December 3, 2014.

Transcription

1 AN ABSTRACT OF THE THESIS OF Emily M. Smith for the degree of Master of Science in Physics presented on December 3, Title: Student & Textbook Presentation of Divergence Abstract approved: Corinne A. Manogue This thesis explores student understanding of divergence in physics settings. The various uses of divergence in physics are detailed. Textbook approaches to divergence are categorized as geometric, curvilinear coordinate, discussion-based, and algebraic statement. Textbook approaches to the divergence theorem are categorized as geometric, equivalent integrals, mathematical statement with explanation, and mathematical statement without explanation. A preliminary framework for analyzing student understanding of divergence is developed following the framework of Zandieh. Student homework responses are analyzed using this framework. Directions of future studies using the proposed framework are discussed.

2 c Copyright by Emily M. Smith December 3, 2014 All Rights Reserved

3 Student & Textbook Presentation of Divergence by Emily M. Smith A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science Presented December 3, 2014 Commencement June 2015

4 Master of Science thesis of Emily M. Smith presented on December 3, APPROVED: Major Professor, representing Physics Chair of the Department of Physics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Emily M. Smith, Author

5 ACKNOWLEDGMENTS First, I would like to thank my advisor, Dr. Corinne Manogue, for her outstanding support, guidance, and encouragement. I would also like to recognize the other members of my committee, Dr. David Roundy, Dr. Janet Tate, Dr. Emily van Zee, and Dr. Adel Faridani, for their time, patience, and contribution to my success. Next, I would like to thank Dr. Justyna Zwolak for the discussion, input, and excitement shared with her throughout this project. Lastly, I d like to thank the many other people who have contributed to my education throughout the years.

6 TABLE OF CONTENTS Page 1 Introduction Previous Studies on Student Difficulties with Divergence Thesis Outline Textbook Presentation of Divergence and the Divergence Theorem Introduction & Methodology Approaches to Divergence Geometric Curvilinear Coordinate Discussion-Based Algebraic Statement Texts Approaches to Divergence Approaches to the Divergence Theorem Geometric Equivalent Integrals Mathematical Statement with Explanation Mathematical Statement without Explanation Texts Approaches to the Divergence Theorem Developing a Preliminary Framework for Analyzing Student Understanding of Divergence Analysis Framework Methodology Background on Students and the Prompt Developing the Preliminary Framework The Preliminary Framework Graphical Context

7 TABLE OF CONTENTS (CONTINUED) Page Descriptive Context Symbolic Context Example Context Issues with Modifying the Zandieh Framework for Divergence Applying the Preliminary Framework to Student Responses Summary Charts Student A Student B Student C Student E Collective Summary of Students Student Understanding in the Context of Their Backgrounds Conclusions 45 Bibliography 48 A Curvilinear Coordinate Approach 51 B Equivalent Integral Approach 60 C Student Summary Charts 68

8 LIST OF FIGURES Figure Page Figure 1.1 One of two questions posed to graduate students by Singh & Maries Figure 1.2 A diagram from the CURrENT involving a wire with steady current, J Figure 2.1 The geometric approach defines divergence in terms of flux per unit volume.. 8 Figure 2.2 The textbooks summarized by subject area with the primary and secondary approaches used to discuss divergence Figure 2.3 An example of two small volumes which share an internal surface Figure 2.4 Projection, P, of an arbitrary volume, V, onto the x-y plane Figure 2.5 The texts organized by their intended course use along with the approach for introducing the divergence theorem Figure 3.1 The Zandieh summary chart for the concept of derivative Figure 3.2 The prompt given to students in the Math Methods Capstone course Figure 3.3 Student responses which are representative of the graphical context Figure 3.4 Student responses which are representative of the descriptive context Figure 3.5 Student responses which are representative of the symbolic context Figure 3.6 Student responses which are representative of the example context Figure 4.1 A summary chart used to summarize the various aspects of student responses. 31 Figure 4.2 Student A s response color coded by context Figure 4.3 Summary chart for Student A Figure 4.4 Student B s response color coded by context Figure 4.5 Summary chart for Student B Figure 4.6 Student C s response color coded by context Figure 4.7 Summary chart for Student C Figure 4.8 Student E s response color coded by context

9 LIST OF FIGURES (CONTINUED) Figure Page Figure 4.9 Summary chart for Student E Figure 4.10 Student responses summarized collectively Figure 4.11 Summary chart for the instructor s example of gradient Figure A.1 Position vector, r (u 1, u 2, u 3 ), is given in red Figure A.2 The blue curve represents varying u 1 while holding u 2 and u 3 constant Figure A.3 The r, in green, is the difference in position vectors Figure B.1 An example of a type 1 region as defined by Stewart where where D = {(x, y) a x b, g 1 (x) y g 2 (x)} Figure B.2 An example of a type 1 region, D, as defined by Stewart Figure B.3 An example of a region, D, with domain, R Figure B.4 An arbitrary region, D, can be described by particular bounds Figure B.5 A type 1 region can be described by the bounds given by the ranges of x and y. 64 Figure C.1 Student D s answer to the prompt given in the Mathematical Methods course. 69 Figure C.2 Summary chart for Student D Figure C.3 Student F s answer to the prompt given in the Mathematical Methods course. 70 Figure C.4 Summary chart for Student F Figure C.5 Student G s answer to the prompt given in the Mathematical Methods course. 71 Figure C.6 Summary chart for Student G Figure C.7 Student H s answer to the prompt given in the Mathematical Methods course. 72 Figure C.8 Summary chart for Student H Figure C.9 Student I s answer to the prompt given in the Mathematical Methods course. 73 Figure C.10 Summary chart for Student I Figure C.11 Student J s answer to the prompt given in the Mathematical Methods course. 74 Figure C.12 Summary chart for Student J Figure C.13 Student K s answer to the prompt given in the Mathematical Methods course. 74 Figure C.14 Summary chart for Student K Figure C.15 Student L s answer to the prompt given in the Mathematical Methods course. 75 Figure C.16 Summary chart for Student L

10 Chapter 1 Introduction Divergence is a central mathematical concept with many physical applications. It is perhaps most recognizable in the differential version of Maxwell s equations: E = ρ ɛ 0 and B = 0, where E = ρ ɛ 0 describes the charge density at a particular point in space, and B = 0 requires there are no magnetic monopoles. Additionally, divergence is used in the spatial part of conservation laws including conservation of charge, energy, probability, and mass. Continuity equations, describing these conservation laws, equate the divergence of a vector field with the time derivative of another quantity. For example, for conservation of charge, the divergence of the current density is proportional to the time derivative of charge density at a point, J = ρ t. These various continuity equations are used in many different disciplines including mathematics, physics, chemistry, biology, geology, oceanography, engineering, and atmospheric sciences. With the variety of applications using divergence, it is important that students understand the geometric meaning attributed to the mathematical concept in order to fully understand those fundamental physical relationships. 1.1 Previous Studies on Student Difficulties with Divergence Studies regarding student understanding of divergence in physics have documented that, though able to calculate divergence, students struggle with the conceptual meaning both mathematically and in a physical context. Pepper et al. (2012) discuss student difficulties with understanding of the physical meaning of vector derivatives including divergence. Students in first semester upperdivision electricity and magnetism were asked on the first homework of the semester to identify which vector field, out of four diagrams, have nonzero divergence somewhere and which have 1

11 nonzero curl somewhere. On the same homework, students were asked to calculate divergence of the vector field, F = î(x 2 + yz) + ĵ(y 2 + zx) + ˆk(z 2 + xy), in rectangular coordinates. When calculating divergence, students from two different semesters scored on average 90% ± 2%. However, when asked to identify which fields have nonzero divergence somewhere and which have nonzero curl somewhere, students averaged 77% ± 3%. The authors conclude that some students have difficulty with the physical meaning of divergence but are typically able to perform the calculation in rectangular coordinates. Figure 1.1: One of two questions posed to graduate students by Singh & Maries. At the beginning of the electricity & magnetism core graduate course, 49% of students (N=37) answered this multiple choice question correctly. At the end of the course, 67% of students (N=18) answered the same multiple choice question correctly. The difficulty with understanding the physical meaning of divergence persists with higher level students including graduate students in physics. Singh & Maries (2012) posed a qualitative problem regarding divergence and curl to first year graduate students in physics. Students, in their core electricity and magnetism course, were asked to determine if a vector field has nonzero divergence or nonzero curl at any point in a diagram of the vector field. One of the two questions regarding divergence and curl is given in Figure 1.1. About half of the graduate students were able to correctly identify divergence and curl qualitatively at the beginning of the course (N=37). After completion of the course, qualitative identification of divergence and curl rose to approximately two thirds (N=18). The authors conclude that graduate students in physics have conceptual difficulties with divergence and curl despite their ability to calculate them mathematically as well as use them in 2

12 Figure 1.2: A diagram from the CURrENT involving a wire with steady current, J. Students are asked to determine whether divergence inside the wire with decreasing diameter is zero or non-zero. physical situations. Baily & Astolfi (2014) discuss the ways in which students think about and use divergence in upper-division electricity and magnetism. When students were posed with problems similar to those given in Pepper et al. and Singh & Maries, it was found, as expected, that most students were able to calculate divergence given an expression for the vector field. Student difficulties were observed in comparing diagrams of vector fields. Some of these difficulties stemmed from not being familiar with vector plots, and how they differ from field line diagrams. Additionally, Baily & Astolfi describe student understanding of divergence in the context of the Maxwell equation, E = ρ ɛ 0. Students struggled to determine, for a point charge, divergence of the electric field was zero except at the location of the point charge. Baily & Astolfi inferred this could be an indication that some students assumed divergence to be constant for every point within a field. Finally, Baily & Astolfi discuss responses from students at six universities where the Colorado UppeR-division ElectrodyNamics Test (CURrENT) was administered (Baily et. al 2013). One problem included in the CURrENT is an assessment involving a steady current within a wire with decreasing diameter which asks, Inside this section of wire [diagram included in Figure 1.2], is the divergence of the current density J zero or non-zero?. Baily & Astolfi found only 38% of students answered correctly. Of these correct responses 88% provided correct reasoning. Other universities that administer the CURrENT to upper-division electricity and magnetism students receive similar responses from their students. The exception was a university which used tutorials developed by the University of Colorado at Boulder. These tutorials include directly relevant information to this particular question on the CURrENT. Likely as a result of the tutorial, nearly 80% of those students provided a correct response. Baily & Astolfi determined that student reasoning about the divergence of vector fields, particularly within the context of electromagnetism, was strong in certain physical 3

13 relationships (such as the use of Gauss s law) but in other instances students were unable to access relevant knowledge. These past studies have found that student difficulties with conceptual understanding of divergence in both mathematics and physics contexts persist among students of various levels and experience with divergence. 1.2 Thesis Outline This work began initially by investigating the various approaches textbooks use to introduce divergence and the divergence theorem. In a survey of 22 textbooks, mathematics, physics, and mathematical methods for physicists textbooks were examined for various approaches to divergence and the divergence theorem. These textbooks were selected due to their prevalence in typical physics and mathematics courses. Each presents a unique approach to introducing divergence and the divergence theorem. There are approaches which are similar in the mathematical formalism, level of detail, and type of explanation. This work, as generally summarized in Chapter 2, serves only as a summary of the approaches used in a limited number of textbooks and does not attempt to determine the strongest approach to introducing divergence and the divergence theorem. Rather this summary, serves as motivation to determine how students think about divergence given the numerous approaches and variety of resources. In Chapter 3, a preliminary framework to analyze student understanding of divergence is introduced. The framework is a generalization of the work from Zandieh (2000) which presents a framework for analyzing student understanding of the concept of derivative. The Zandieh framework, consisting of contexts and layers of process-object pairs, is explained in detail. Then, an analogous framework for divergence is established by using student-generated and author-generated responses. Issues with constructing a framework for divergence which parallels the Zandieh framework are discussed. In particular, the parallel version of the Zandieh framework does not allow for an accurate definition of divergence within a single context, has some hollow layers of process-object pairs which lack an associated process, and ignores important aspects of divergence. In Chapter 4, the preliminary framework is applied to student responses to an upper-division homework problem to demonstrate how the framework can be used to summarize student responses. The issues with using the Zandieh framework for divergence, as discussed in Chapter 3, are highlighted through some student responses. A summary of all student responses is provided to demon- 4

14 strate the frequency of particular responses. A discussion of high frequency responses in the context of students previous coursework is provided. In Chapter 5, an overall summary of the findings of this work is provided. Suggestions for future studies about student understanding of divergence in physics contexts are made. 5

15 Chapter 2 Textbook Presentation of Divergence and the Divergence Theorem 2.1 Introduction & Methodology Divergence and the divergence theorem, also known as Gauss s theorem, are fundamental mathematical concepts in vector calculus. Understanding of these mathematical concepts is important to the study of electricity and magnetism. These topics are presented using various approaches by textbooks (and presumably instructors) which include geometric, physical, and mathematical perspectives. Here, we summarize the various introductions to divergence and the divergence theorem presented in textbooks used in typical courses in vector calculus, mathematical methods in physics, and upper-division and lower-division physics. This list of textbooks which address divergence and the divergence theorem is not comprehensive. However, several commonly used representative examples were chosen, and we expect the general approaches used in the majority of textbooks and courses which cover these topics are similar. In this survey, 22 textbooks including those intended for use by mathematics, physics, and mathematical methods for physicists courses are used to examine different approaches to divergence and the divergence theorem. These textbooks were selected due to their prevalence in typical physics and mathematics courses. Many of the texts are standard for the material they address; for ex- 6

16 ample, Griffiths Introduction to Electrodynamics (2008) is used in many upper-division electricity and magnetism courses. Others were selected for their unique approach to a standard course; for example, Chabay & Sherwood s Matter and Interactions (2011) covers introductory physics in a unique order as well as incorporating practical techniques including computational elements in the text. There are three main categories of textbooks: those intended for pure mathematics courses including vector calculus, those intended for physics courses at various levels, and those intended for mathematical methods courses for physics students. The general approaches these textbooks use to introduce divergence and the divergence theorem are described in Section 2.2 and Section 2.4, respectively. Additionally, variations to these approaches which occur in some texts are explained. 2.2 Approaches to Divergence Each textbook introduces divergence in a unique manner, however, many of these approaches share elements with other texts. The notation is generalized and is not reflective of the notation used in any particular text. Instead, the notation in this work is selected for ease of reading and consistency throughout this chapter. In this chosen notation, F is always the vector field with components given by subscripts such as F = F xˆx + F y ŷ + F z ẑ in rectangular coordinates, d A is the differential surface element normal to the surface, dv is the differential volume element, S is a surface, and indicates divergence Geometric An overview of the geometric approach to divergence is as follows: 1. Divergence is defined in terms of the flux per unit volume through the surfaces of a rectangular volume placed within the vector field. 2. The rectangular volume is assumed to be infinitesimally small with dimensions dx, dy, and dz. This gives a differential volume of dv = dxdydz. 3. Two surfaces which are constant in the same coordinate, as given in Figure 2.1, are chosen to determine the net flux in that direction: S 1+S 2 F d A = Fy (x, y+ y, z) dx dz F y (x, y, z) dx dz. 4. This argument is extended to the other dimensions. An expression for total flux through the ( ) F infinitesimal volume is determined as x x + Fy y + Fz z dx dy dz 7

17 Figure 2.1: The geometric approach defines divergence in terms of flux per unit volume. One such volume is the rectangular volume depicted with two surfaces, S1 and S2, constant in the y-direction with surface normal vectors in blue, the width of the volume given as y, and an arbitrary vector field F is shown in red. 5. An expression for divergence is written from the one determined for total flux through the rectangular volume: F ( ) F = x x + Fy y + Fz z Variations of the geometric approach to introduce divergence include different physical contexts throughout the proof where the vector field has physical meaning such as the electric field or flow of water. Some texts use the geometric approach using only rectangular coordinates, as presented here, however, others use generalized orthonormal curvilinear coordinates. Additionally, the amount of detail of each step of the proof varies, but the foundations described here are present Curvilinear Coordinate Refer to Appendix A for one example of a curvilinear coordinate approach to divergence. This particular example follows Riley, Hobson, & Bence (1998) but with greater detail than provided in the textbook. Variations using the approach of curvilinear coordinates to introduce divergence are minimal. Varying levels of detail are provided, and the variables are defined very differently in each text. However, the content of the approach is essentially identical in all texts which introduce divergence using this particular presentation Discussion-Based The discussion-based approach to divergence is as follows: 1. A description of divergence is given based on geometry of vector fields or a physical example. For example, Griffiths uses an example of sprinkling sawdust on the surface of a pond where 8

18 if the material spreads out, then you dropped it at a point of positive divergence; if it collects together, you dropped it at a point of negative divergence. Additionally, he provides diagrams of vector fields with positive and zero divergence as well as discussing the concept of sources and sinks. 2. This description is then used to write a mathematical expression for divergence. Variations of the discussion-based introduction to divergence include various contexts of discussion. Some texts present divergence using a common physical example such as electric field or flowing liquid. Other texts discuss divergence in the context of mathematics by the amount of flux of an arbitrary vector field per unit volume Algebraic Statement The algebraic statement approach to divergence is as follows: 1. An algebraic definition for divergence is given. For example: F = Fx x + Fy y + Fz z 2. Minimal or no explanation for physical or mathematical meaning of divergence is given. Variations of the algebraic statement to introducing divergence include the level of detail in explanations and the equations which are given. The explanations provided range from non-existent to a few sentences of minimal explanation. The equations give either an expression for divergence in rectangular coordinates using partial derivatives or the limit definition of divergence. 2.3 Texts Approaches to Divergence The approaches which are described in Section 2.2 are the general approaches, geometric, curvilinear, discussion-based, and algebraic statement, which texts use to introduce divergence. Figure 2.2 summarizes the primary and secondary approaches to divergence presented in each textbook. These approaches vary in their details, however, all texts use a variation of one or more of the previously summarized approaches. The vector calculus texts discuss divergence to various extents. Some texts, such as Briggs & Cochran (2011) and Stewart (2008), present a simple algebraic statement and quickly transition to other topics in vector calculus. There is no introduction to divergence included within these texts other than formulas which can be used to calculate divergence. Other vector calculus texts do provide 9

19 Text Course Use Primary Approach Secondary Approaches Beatrous & Curjel (2002) [4] Vector calculus Discussion Briggs & Cochran (2011) [6] Vector calculus Algebraic Statement Hughes-Hallett et.al. (2002) [14] Vector calculus Geometric Discussion Stewart (2008) [26] Vector calculus Algebraic Statement Schey (1992) [22] Mathematics & Physics Geometric Arfken (1985) [1] Mathematical methods Geometric Boas (1983) [5] Mathematical methods Geometric Curvilinear Coordinate Butkov (1986) [7] Mathematical methods Geometric Curvilinear Coordinate Riley, Hobson, & Bence (1998) [20] Mathematical methods Curvilinear Coordinate Algebraic Statement Chabay & Sherwood (2011) [8] Introductory physics Discussion-Based Knight (2008) [16] Introductory physics N/A Giancoli (2000) [10] Introductory physics Algebraic Statement Halliday et. al (2002) [12] Introductory physics N/A Young & Freedman (2000) [29] Introductory physics N/A Griffiths (2008) [11] Middle- to Upper-division E&M Discussion-Based Geometric Heald & Marion (1995) [13] Upper-division E&M Algebraic Statement Taylor (2005) [27] Middle- to Upper- division Mechanics Algebraic Statement McIntyre (2012) [17] Upper-division Quantum N/A Shankar (1994) [25] Upper-division Quantum N/A Fetter & Walecka (2003) [9] Graduate Mechanics Geometric Jackson (1998) [15] Graduate E&M N/A Sakurai (1994) [21] Graduate Quantum N/A Figure 2.2: The textbooks summarized by subject area with the primary and secondary approaches used to discuss divergence. detailed explanation of divergence by geometric and discussion-based approaches. Interestingly, the geometric and discussion-based approaches are used in similar ways by these vector calculus texts. The discussion in Beatrous & Curjel (2002) centers on the geometric approach but does not use the detailed mathematics which accompanies the geometric approach. Hughes-Hallett et. al (2002) presents a similar approach but includes thorough mathematics in addition to figures which explain the geometry of divergence. Schey (1992) is a unique text which is a hybrid of vector calculus and middle- to upper-division electricity and magnetism. Schey provides extensive detail in the geometry describing divergence. The discussion of the Maxwell equation, E = ρ ɛ 0, is provided simultaneously. This text is unique in many ways, including the discussion of divergence, and is therefore classified in an independent category. Most of the mathematical methods texts include a geometric approach and incorporate, within the geometric approach, physical examples in different amounts of detail. Additionally, most of the texts include the curvilinear approach but typically as a secondary approach. The exception is Riley, Hobson, & Bence (1998) which uses the curvilinear approach without ever discussing divergence geometrically but does provide formulas to calculate divergence in various coordinate systems. 10

20 Griffiths and Schey share many elements with these mathematical methods texts. Additionally, both texts provide a geometric approach to generalized curvilinear coordinate expressions of divergence. Because these approaches are geometric in the arguments, they are not categorized within the curvilinear coordinate approach which incorporates arguments which are not based in geometry but rather vector manipulations. The introductory physics texts typically do not provide much, if any, explanation of divergence. Knight (2008), Halliday et. al (2002), and Young & Freedman (2000) never introduce divergence while Giancoli (2000) only provides an algebraic statement. Divergence and the differential version of Maxwell s equations are beyond the scope most introductory physics courses, so it is not expected that many introductory texts provide additional discussion of divergence. The exception is Chabay & Sherwood which includes a detailed discussion of divergence in the context of electrostatics though it does not go into detailed mathematics. With the middle-division, upper-division, and graduate physics texts, divergence is a topic which is assumed students have encountered previously. Therefore, most of these texts either list an algebraic statement or do not formally define divergence. Griffiths and Fetter & Walecka (2003) are the exceptions to this. Griffiths, as discussed previously, follows closely with the mathematical methods course. Fetter & Walecka assumes previously knowledge of divergence within the main text, by using divergence frequently in various derivations, but does provide a geometric approach to divergence in the appendix. 2.4 Approaches to the Divergence Theorem The approaches to the derivation or explanation of the divergence theorem are summarized in the following section. The notation is consistent with the previous notation used for divergence Geometric An overview of the geometric approach to the divergence theorem is as follows: 1. A large, seemingly arbitrary volume is divided into small volume elements. 2. The outward flux from each of these small volumes is calculated as F da where the sum is the flux over all the faces of the small volume. 3. Using the geometric definition of divergence, the outward flux in each of these small volumes can be equivalently written: F da = F dv 11

21 Figure 2.3: An example of two small volumes which share an internal surface. The two smaller volumes within a larger volume share the internal surface shown in blue. The outward normal surface vectors are in red and are exact opposites which means this internal surface will not contribute to the total flux of the large volume. 4. This expression can be rewritten for any volume by adding up all the small volumes which compose the arbitrary volume. Internal surfaces do not contribute to total flux, as shown in Figure 2.3: V F dv = S F d A Variations of this approach include differing levels of detail. Some texts delve into each step with a complete explanation using figures and more rigorous mathematics while others use minimal detail when using this particular approach for a simplistic explanation of the divergence theorem Equivalent Integrals The description for the equivalent integral approach described here is general and is not complete. In this description there are several missing steps, however, this outline provides a short explanation of the derivation using equivalent integrals. Appendix B gives a detailed example of the equivalent integral approach which follows Stewart (2008). 1. A volume in space is chosen. The projection of the volume onto a plane in rectangular coordinates is shown, as in Figure The surface integral and the volume integral in the definition of the divergence theorem are each expressed independently of each other: F da = S S x F xˆx d A+ S y F y ŷ da+ S z F z ẑ da and F dv = ( ) F x V V x + Fy y + Fz z dv. 3. Each component of the surface integral is stated as being equivalent to the corresponding volume integral. The claim is made that solving for the validity of one of these such equations will prove the surface integral is equal to the volume integral. For example, the proof of the following equality will prove the divergence theorem: S z F z ẑ d A = V F z z dv 4. Using the projection, P, of the volume onto one of the rectangular coordinate planes as given 12

22 Figure 2.4: Projection, P, of an arbitrary volume, V, onto the x-y plane. in Figure 2.4, the volume integral is rewritten using the Fundamental Theorem of Calculus: F z V z dv = F z P z dz dx dy = P [F z(x, y, g(x, y)) F z (x, y, f(x, y))] dx dy. g(x,y) f(x,y) 5. The surface integral is similarly evaluated by determining outward normals of the volume: = P F z ẑ da = S z S top z F z (x, y, z)ẑ da + F z (x, y, z)ẑ da Sz bottom F z (x, y, z)ẑ ( g xˆx g y ŷ + ẑ) dx dy + F z (x, y, z)ẑ (f xˆx + f y ŷ ẑ) dx dy P = [F z (x, y, g(x, y)) F z (x, y, f(x, y))] dx dy P 6. These two integrals are in an equivalent form which means the other coordinates must follow the same derivation. Therefore, the divergence theorem holds. Variations of this approach include the complexity of the mathematical steps. Stewart requires quite intricate mathematics which relies on many previous proofs including mathematical expressions for the unit normal vector and volumes in rectangular coordinates Mathematical Statement with Explanation The mathematical statement with explanation approach to the divergence theorem is as follows: 1. A simple mathematic expression (or expressions) is given to define the divergence theorem such as: ( F )V i F da and V F dv = F da. S 13

23 2. A physical or mathematical meaning of the expression is given which explains the concept of the divergence theorem in terms of vector fields. The major variation of this approach using an explanation with a mathematical statement is the context in which the approach is explained. Some texts work with an arbitrary vector field while others provide a physical context for the field vector Mathematical Statement without Explanation The mathematical statement without explanation approach to the divergence theorem is as follows: 1. A simple mathematical expression is given to define the divergence theorem. 2. The divergence theorem is used as step in another derivation without explanation as to what the divergence theorem is past a mathematical equivalency. 2.5 Texts Approaches to the Divergence Theorem The approaches which are used by various texts to introduce the divergence theorem are listed in Figure 2.5. Unlike the introduction to divergence, only a single approach is used by each text when addressing the divergence theorem. Therefore, there is no column for secondary approaches as there was for introducing divergence. The math texts took two primary approaches to the divergence theorem which are greatly different. Hughes-Hallett et. al and Schey use geometric approaches where the mathematics of the geometry is highly prevalent and detailed. The detail in the mathematics is present in Stewart and Briggs & Cochran, however, the mathematics varies greatly from the geometric approaches. The math texts, overall, provide more mathematical details in introductions to the divergence theorem than the mathematical methods and physics texts. However, the approaches were far more different among math texts than those used in mathematical methods or physics texts. The mathematical methods texts all use similar introductions to the divergence theorem: a geometric approach without detailed mathematics. Most use some physical context throughout the introduction to divergence. Riley, Hobson, & Bence uses a slightly different approach by stating the divergence theorem and explaining the mathematical meaning of the divergence theorem without physical examples. 14

24 Text Course Use Approach Beatrous & Curjel (2002) [4] Vector calculus Math Statement with Explanation Briggs & Cochran (2011) [6] Vector calculus Equivalent Integrals Hughes-Hallett et.al. (2002) [14] Vector calculus Geometric Stewart (2008) [26] Vector calculus Equivalent Integrals Schey (1992) [22] Math & physics Geometric Arfken (1985) [1] Mathematical methods Geometric Boas (1983) [5] Mathematical methods Geometric Butkov (1968) [7] Mathematical methods Geometric Riley, Hobson, & Bence (1998) [20] Mathematical methods Math Statement with Explanation Chabay & Sherwood (2011) [8] Introductory physics Math Statement without Explanation Knight (2008) [16] Introductory physics N/A Giancoli (2000) [10] Introductory physics N/A Halliday et. al (2002) [12] Introductory physics N/A Young & Freedman (2000) [29] Introductory physics N/A Griffiths (2008) [11] Middle- to Upper-division E&M Math Statement with Explanation Heald & Marion (1995) [13] Upper-division E&M Math Statement without Explanation Taylor (2005) [27] Middle- to Upper-division Mechanics Math Statement without Explanation McIntyre (2012) [17] Upper-division Quantum N/A Shankar (1994) [25] Upper-division Quantum Math Statement without Explanation Fetter & Walecka (2003) [9] Graduate Mechanics Math Statement without Explanation Jackson (1998) [15] Graduate E&M Math Statement without Explanation Sakurai (1994) [21] Graduate Quantum N/A Figure 2.5: The texts organized by their intended course use along with the approach for introducing the divergence theorem. The introductory physics texts do not provide details about the divergence theorem. Chabay & Sherwood is the only text to formally mention the divergence theorem when discussing the differential version of Gauss s law. Giancoli uses the divergence theorem to rewrite an integral when touching on the differential version of Gauss s law but does not provide justification or formal mention of the divergence theorem. The physics texts which go beyond introductory level assume students have previously encountered the divergence theorem. Therefore, the divergence theorem is typically stated as a mathematical expression with little to no explanation. These texts tend to introduce the divergence theorem as a mathematical statement which has physical meaning in a specific physical context such as electric fields. Most physics texts use the divergence theorem as a mathematical tool during a proof. For example, texts use the divergence theorem to go from the integral version of Gauss s law to the differential version. Only Griffiths provides an explanation of why the divergence theorem is valid in a physical system. 15

25 Chapter 3 Developing a Preliminary Framework for Analyzing Student Understanding of Divergence 3.1 Analysis Framework In order to understand how students think about divergence and engage in the material, it is important to establish a way in which to analyze student data. Existing mathematics and physics education literature lacks a framework for addressing divergence. Therefore, the Zandieh framework for the concept of derivative is adapted to develop a framework for analyzing student understanding of divergence (Zandieh 2000). By adapting this particular framework, which addresses various aspects of the meaning of derivative, we have begun to establish various ways in which divergence can be understood. Zandieh presents a framework to describe understanding of the concept of derivative in a structured manner. The Zandieh framework uses the idea of the concept image [which] consists of all the cognitive structure in the individual s mind that is associated with a given concept (Tall & Vinner 1981). Zandieh lays out a description of the entire mathematical community s notion of understanding the concept of derivative at the first-year calculus level. Individual student s concept images of derivative are then categorized according to this description of the mathematical community s understanding of derivative. 16

26 The Zandieh framework serves as the foundations for the preliminary framework developed in this work to describe divergence. The Zandieh framework is developed from the way in which textbooks present the concept of derivative, how various instructors discuss the concept in class, and from interviews with high school seniors enrolled in an Advanced Placement Calculus course. Through these observations, experiences, and collected data, Zandieh develops a theoretical framework to describe derivative. This theoretical framework is composed of two major components: contexts and layers of process-object pairs. The contexts in the Zandieh framework are the various representations used to describe derivative including graphical, verbal, paradigmatic physical, symbolic, and other. The graphical context uses the slope of the tangent line to a curve at a point or as the slope of the line a curve seems to approach under magnification as a definition for derivative. In the verbal context, derivative is the instantaneous rate of change. The paradigmatic physical context uses the interpretation of a derivative in a physical context such as speed or velocity. In the symbolic context, derivative is described as the limit of the difference quotient. The other context encompasses any other contexts which are not described in the presented framework. These contexts can be understood through pseudostructural examples, based on pseudostructural conceptions in Sfard (1991), which students might refer to without necessarily demonstrating understanding of the particular example; often, instructors use a set of particular phrases or examples to describe derivative which students may use to describe a particular context without understanding the intricacies of the example. For example, the graphical context can be described by the pseudostructural example of slope. A student may recite that derivative is the slope of the tangent line without demonstrating understanding that the process of determining slope is the limit of the ratio of rise over run at a point. Each of the contexts are described by pseudostructural examples with rate of change referring to the verbal context, velocity to the paradigmatic physical context, and the difference quotient to the symbolic context. These pseudostructural examples tend to be the common phrases or examples which students give within a particular context to describe derivative. The depth of student understanding of the various contexts and corresponding pseudostructural examples can be explored through layers of process-object pairs which include ratio, limit, and function. Zandieh bases these layers on the work of Sfard. Sfard concludes that mathematics can be conceived in two fundamentally different ways: structurally as objects, and operationally as processes. However, these objects and processes are complementary since the processes involve operations on objects which are previously understood. Zandieh uses the Sfard concept of processes and objects to develop process-object pairs a chain of objects and processes. These layers of process- 17

27 Figure 3.1: The Zandieh summary chart for the concept of derivative. This chart, given in Figure 1 of Zandieh 2000, outlines the framework for the concept of derivative by including contexts in columns and layers in rows. object pairs (hereafter, layers) represent the various ways in which derivative can be described within a particular context. For example, in the graphical context of derivative, the ratio process is the rise divided by the run of the secant line connecting two points on a graph. The resulting slope is the object. As the two points which are connected by the secant line are chosen closer and closer together until the line is at a single point, a tangent line results at this particular point in the graph. The slope of the tangent line at that point is then the object which is determined through the limiting process. As this limiting process is repeated for each point on the graph, a function is developed; this object is the derivative function, another graph. Students may mention a pseudostructural example such as the slope which implies the graphical context. They then may state the derivative is the slope of the tangent line, a statement of the ratio layer, without necessarily demonstrating understanding of the process involved. The Zandieh framework acknowledges both the pseudostructural object and demonstration of understanding the process involved of a particular context. Zandieh uses these contexts and layers to develop summary charts, as given in Figure 3.1, to categorize calculus students individual understanding of derivative. The contexts are listed in columns. The layers are listed in rows. Students who demonstrate knowledge of an object within a specific context and layer have an empty circle indicated on their summary chart where the context and layer intersect. Students who demonstrate understanding of both the process and object of a particular context have a filled circle on the summary chart where the context and layer intersect. Zandieh fills out this chart for individual student interviews. For example, a student who says 18

28 that a derivative is the slope of a tangent line at a particular point demonstrates knowledge of the ratio and limit objects. However, without describing the slope as a ratio of rise over run, the process of the ratio layer is not demonstrated. This results in an empty circle at the intersection of graphical context and ratio layer. Similarly, without describing the limiting process, the process of the limit layer is also not demonstrated, but the object is present by the notion of a particular point, resulting in another empty circle at the intersection of graphical context and limit layer. The Zandieh framework provides a way in which to analyze ordinary derivatives. Vector derivatives, including divergence, are more complicated in terms of the mathematical objects and operations used including vector and scalar fields, partial derivatives, and vector multiplication. Therefore, the preliminary framework presented here is intended to summarize student understanding of the central representations of divergence without considering the small intricacies and operational difficulties associated with divergence. Further studies will explore student understanding of the concepts used while calculating and applying divergence to physical situations. This work intends to only begin development of ways to categorize various representations and depth of student knowledge in a broader sense of the concept of divergence. The coarse-grain approach Zandieh takes to the concept of derivative allows for a broad view of the various representations students use when thinking about derivative. In a similar way, the development of a framework analogous to the Zandieh framework for divergence intends to address student understanding through various representations of divergence. This work extends the Zandieh framework to the concept of divergence, a vector derivative. Contexts and layers, which correspond roughly to those presented by Zandieh for derivative, are developed using student work. This preliminary framework for divergence is then applied to student responses to demonstrate how the framework can show the various aspects of student understanding of divergence. 3.2 Methodology Background on Students and the Prompt Oregon State University has redesigned the undergraduate physics major curriculum into the Paradigms courses in the junior year and Capstone courses in the senior year. The Paradigms courses are structured into three week intensive courses focusing on a particular idea within physics and often incorporate many different subdisciplines within physics. For example, in Static Vector 19

29 Fields students learn simultaneously about electric, magnetic, and gravitational fields and potentials. In the Paradigms courses, students meet in class for seven hours per week. Small group activities, lectures, computer simulations, small whiteboard questions, kinesthetic activities, and other active engagement pedagogies are incorporated throughout each course. Over three ten-week terms, students complete nine Paradigms courses, covering electricity and magnetism, classical mechanics, quantum mechanics, thermodynamics, relativity, and solid state physics. During the senior year of the physics major, students take a more traditional sequence of courses referred to as the Capstone courses. These are courses which cover a specific subdiscipline in physics and include courses in math methods, classical mechanics, electricity and magnetism, quantum mechanics, and thermodynamics and statistical mechanics. They are typically taught using lectures with some active engagement activities incorporated throughout the term. The courses meet three times each week for 50 minutes, and students are enrolled for an entire ten-week term. One of the Capstone courses, Math Methods, is taken by junior physics majors simultaneously with the final three Paradigms courses in the Spring term of the junior year. The homework problem which prompted the development of this preliminary framework was given during the first week of the Math Methods Capstone course. The students enrolled in Math Methods had previously taken a vector calculus course, which covers divergence, either concurrently with or prior to two Paradigms courses, Symmetries & Idealizations and Static Vector Fields, covering vector calculus in the context of electrostatics. Therefore, the analyzed responses come from students who have significant experience with divergence prior to the Math Methods course. The homework problem from which this analysis follows asked students to develop a concept image for divergence and curl. The prompt, given in Figure 3.2, gives an example response for gradient prior to asking students for their responses for divergence and curl. Students were able to collaborate and use resources to answer this prompt. The student responses analyzed come from 12 students enrolled in the Math Methods Capstone during the Spring 2014 term. Two of the students had previously completed the Paradigms courses, and the other 10 students were concurrently enrolled in the Paradigms. All students enrolled in the course either major in physics or in a closely related field such as nuclear engineering and mathematics with, at minimum, a minor in physics. With only one response from a female student included this work, gender was not considered as a factor in the analysis. Therefore, all student work will be referred to with the he pronoun regardless of the gender of the student in order to maintain anonymity of the students involved. Given the detailed example in the prompt, which students were free to emulate, and the open 20

30 A concept image is the set of all the things an individual knows about a particular concept. Listed here is a (mostly complete) bullet list of the items in my concept image of gradient. Notations: f, grad f Most common physics examples: E = V, g = Φ A differential operator. Takes a scalar field (i.e. a function often called the potential function, i.e. a number at every point in the domain) to vector fields (a vector at every point in the domain) The gradient only exists where the scalar field (potential) exists, and where it varies smoothly enough that its partial derivatives exist. At a point in the domain, the magnitude of the vector is given by the maximum rate of change of the scalar field at the point. The direction of the vector is the direction (in the domain) of the maximum rate of change of the scalar field. The gradient is a local quantity, i.e. the value at a point depends only on the values of the scalar field in an infinitesimal area around that point. The vector field always points perpendicular to the level curves or level surfaces of the potential function. Relationship to the Master Formula : df = f d r Formula in rectangular coordinates: f = f f ˆx + ŷ + f ẑ x y z I know where to look up the formula in cylindrical, spherical, and general curvilinear coordinates (or I can figure it out from the Master formula and my knowledge of d r in those coordinate systems.) An iconic diagram of the gradient in two-dimensions: (a) Write a similar concept image for divergence. (b) Write a similar concept image for curl. Figure 3.2: The prompt given to students in the Math Methods Capstone course. This prompt was given to students as part of the first problem set for homework. 21

31 resources available, the responses had more variation than initially expected. After comparison among student responses, it became clear that students did not work closely together on answering this problem. Few responses were similar to each other, however, most responses did follow a format similar to the example given for gradient. This indicates that students were most likely attempting to write parallel statements to those given for gradient rather than collaborating among their peers. It was not clear whether most students were using outside resources for their responses. Few students provided entire mathematically correct statements which may be an indication that few outside resources were used; for example, out of the ten students who included a formula in rectangular coordinates, six students gave an incorrect formula Developing the Preliminary Framework In order to analyze these student response, an analogue of the Zandieh framework for the concept of derivative is developed to describe student understanding of divergence. The Zandieh framework establishes contexts for derivative as graphical, verbal, paradigmatic physical, and symbol with each of these contexts supplemented with pseudostructural examples of slope, rate of change, velocity, and difference quotient respectively. Each of these contexts are then divided into process-object layers: ratio, layer, and function. Divergence, as a specific type of derivative, can be understood using similar contexts and layers. Using student responses to the prompt and textbook approaches to divergence, parallel contexts and layers were drawn to the Zandieh framework for divergence. Student responses generally prompted the contexts of graphical, descriptive, symbolic, and example which are explained in detail in Section 3.3. These are similar to the Zandieh framework s contexts, however, there are clear distinctions between derivative and divergence which require different contexts. The graphical context for divergence involves a diagram with a vector field and a volume which is similar to the slope of a function for derivative. The descriptive context was established to parallel the verbal context within the Zandieh framework. This particular context within the preliminary framework for divergence is not well defined and may be a more central theme to the other contexts of divergence rather than a distinct context itself. A symbolic context is established using formulas to calculate divergence which lack the coherency of the Zandieh symbolic context for derivative using different forms of the difference quotient. However, the analogous framework provides a foundation to begin further work in understanding the various representations students can use. Lastly, the example context is meant to be the rough equivalent of the Zandieh paradigmatic physical but with a larger allowed range of applications including more abstract phys- 22

32 ical interpretations such as sources and sinks. The development of these various contexts serves as a pilot study to distinguish the various representations which exist to contribute to understanding divergence. The Zandieh framework illuminates these representations, however, the framework does not provide the necessary elements and flexibility to accommodate vector derivatives. This will be discussed further in Section 3.4. Within each of these contexts, obvious layers paralleling the Zandieh framework of ratio, limit, and function layers emerge. The limit and function layers remain unchanged in this preliminary framework. The ratio layer does not accurately describe divergence and is instead replaced by the finite layer which describes divergence using finite volumes. These contexts and layers are considered to be part of the preliminary framework since they are not well defined, are not encompassing all contexts and layers, and do not reflect all processes and objects which are part of a vector derivative such as divergence. This pilot study demonstrates the various aspects of divergence as well as the existing limitations in applying a framework intended for ordinary derivatives to a vector derivative. 3.3 The Preliminary Framework The work presented in this section is primarily student work from Spring 2014, though includes responses from student in Fall 2014 following a similar prompt, which is representative of a particular strong response. However, some layers of particular contexts are not strongly represented by any existing student work. Therefore, some of these responses are rewritten from student work or are author-generated to demonstrate the layer and context. This provides a clear distinction between various contexts and layers prior to applying the framework to entire student responses. Studentgenerated and author-generated responses are indicated in the captions of each figure Graphical Context The graphical context of the preliminary framework refers to any drawings or diagrams students made to visually show the meaning of divergence. In the prompt, students were asked to draw an iconic diagram and therefore were explicitly asked to represent divergence in using a graphical representation. In Figure 3.3, the various layers of the graphical context are defined using student-generated and author-generated responses to demonstrate the meaning of each layer. In the finite layer, a student has drawn a volume within a vector field. The student does not make any specification that the volume must be infinitesimal, and therefore, this diagram uses a finite volume in order to determine 23

33 (a) Finite layer (b) Limit layer (c) Function layer Figure 3.3: Student responses which are representative of the graphical context. These particular responses demonstrate an ideal response using the graphical context to convey the concept of divergence. The finite and limit layers are student-generated responses. The function layer is an author-generated response. divergence of the vector field. For the limit layer, the student draws a similar volume within a vector field, however, he specifies that the volume is dτ which is an infinitesimal volume. Additionally, he draws a point labeled P within the volume which presumably is a point within the drawn vector field. These two cues indicate he is using the limit layer of the graphical context. The final diagram within Figure 3.3 is an author-generated response; it is a response which might be given by students through an interview and is included here to explain the function layer of the graphical context. The function layer is similar to the limit layer but includes infinitesimal volumes for several points within a vector field. This conveys that divergence has a value for every point within a vector field Descriptive Context The descriptive context of the preliminary framework is the most vague context, perhaps because it is a central theme to the other contexts rather than a distinct representation. Despite this, the descriptive context, as developed, is used typically when students write statements which pertain to specific aspects of divergence. The expected description in this context is for the student to state that divergence relates to the flux per unit volume in a vector field. Figure 3.4 provides example responses which are included within the descriptive context. In the finite layer, the student states that divergence is the flux over volume of a vector [field] of a determined volume in a position in the [field]. This student does not specify that the volume must be infinitesimal nor does he specify that the position is a point in the vector field. Therefore, despite stating that divergence is flux per volume, he does not convey knowledge that there is a limiting process involved and instead uses the finite layer. In the response using the limit layer, the student clearly conveys that divergence is the flux per unit volume out of a surface that encloses the point, 24

34 (a) Finite layer (b) Limit layer (c) Function layer Figure 3.4: Student responses which are representative of the descriptive context. These particular responses demonstrate an ideal response using the descriptive context to convey the concept of divergence. The finite and limit layers are student-generated responses. The function layer is an author-generated response. as the surface shrinks to the point. This is a description of a limiting process, and therefore falls under the limit layer of the descriptive context. Finally, an author-generated response is provided for the function layer which expands on the limit layer by stating that divergence is defined for all points in the field Symbolic Context The symbolic context of the preliminary framework refers to the equations which students give for calculating divergence. These equations are given in symbolic notation, and therefore were relatively easy to identify when written by students. (a) Finite layer (b) Limit layer (c) Function layer Figure 3.5: Student responses which are representative of the symbolic context. These particular responses demonstrate an ideal response using the symbolic context to convey the concept of divergence. Note that in the limit layer, the notation is not entirely correct because the vector field, F, is not specified as a vector quantity. The responses are all student-generated. Figure 3.5 provides sample responses from student work which illustrate each layer of the symbolic context. It is immediately apparent that the finite layer of this symbolic context is conflated with the finite layer of the descriptive context. Whether there is a distinction between the two is unresolved, but in the current framework they are distinguished by the representation; the symbolic finite layer is an equation while the descriptive finite layer uses words without any form of an equation. In the 25

35 finite layer of Figure 3.5, the student chooses to represent the conceptual idea of divergence as flux per unit volume in a formula. However, in the descriptive context, students represent this same idea through words without a formula. The limit and function layers of the symbolic context are quite straightforward. The limit layer provides the limit definition of divergence; the student response provided in Figure 3.5 states that divergence is equal to the limit of a volume going to a point of a flux integral over a surface enclosing a volume divided by that volume. This particular student does not use entirely correct notation, however, the limit definition of divergence is clearly conveyed. A student uses the function layer of the symbolic context when writing any formula used to evaluate the divergence at any point for a given vector field Example Context The example context is a broad context which encompasses a variety of responses. Some students chose to give physical examples in terms of specific physical variables such as charges. Other students gave more general examples, typically in the language of sources and sinks. Essentially, students were describing the same general idea about divergence whether they chose to use general language or language specific to a particular physical situation. (a) Limit layer (b) Function layer Figure 3.6: Student responses which are representative of the example context. These particular responses demonstrate an ideal response using the example context to convey the concept of divergence. The response for the limit layer is student-generated but rewritten for clarity by the author. The response for the function layer is author-generated. There were no students who provided an example which fit a finite layer nor was there an obvious author-generated response to provide. Therefore, in this preliminary framework, that layer is disregarded. The responses given in Figure 3.6 are both author-generated because student responses using the example context varied greatly and often gave explanations which were difficult to interpret without extensive knowledge of the course context. Therefore, these author-generated 26

36 responses were developed to summarize and explain student responses. The limit layer is described as a statement that divergence can determine whether there is a source or sink at a point in the field. This statement tends to only consider a single point within the vector field which is similar to the limiting process, and therefore is considered within the limit layer. The function layer then describes divergence as a way in which to determine whether there is a source or sink at each point for a given vector field. 3.4 Issues with Modifying the Zandieh Framework for Divergence The Zandieh framework addresses student understanding of ordinary derivatives. Applying this framework to a vector derivative is problematic for many reasons. Although it is possible to superficially select parallel contexts and the corresponding layers, as presented in this chapter, these combinations of contexts and layers do not encompass all aspects of divergence. For example, students who are able to recite formulas to calculate divergence do not necessarily have any understanding of where the formulas come from, the implications of the formula, and the operations associated with the formulas. Zandieh s symbolic context has coherence among layers where the ratio layer is a difference quotient, the limit layer is the limit of that quotient for a specific point, and the function layer is the general limit for every value in the domain of the function. Divergence does not have corresponding layers of process-object pairs because the limit definition of divergence does not allow for distinctions between limit and function layers in a similar way to Zandieh s derivative framework. Therefore, choosing the function layer of the symbolic context to be a formula for calculating divergence would be similar to stating that a derivative is described by df(x) dx. While df(x) dx = lim h 0 f(x+h) f(x) h, the limit definition of derivative for the function layer of the symbolic context describes student understanding of a derivative as a difference quotient. The standard notation allows for this limit to be written as df(x) dx, but the meaning for this notation is embedded within the limit definition of derivative. Similarly with divergence, reciting a formula which can be used to calculate divergence is distinct from understanding that divergence is flux per unit volume when the limit of the volume goes to a point. As presented in this chapter, there is no obvious way in which to distinguish these ideas, and therefore, the symbolic context for divergence is hollow and incomplete; the function layer is a pseudostructural object without an associated process. The function layer, if expanded from this preliminary framework, can encompass an associated 27

37 process. The geometric approach to introducing divergence, as explained in Chapter 2, requires an embedded limit to determine the pseudostructural object of a formula to determine divergence. In Step 3 of Section 2.2.1, the total flux through two surfaces which are constant are determined to be: S 1+S 2 F d A = Fy (x, y + y, z) dx dz F y (x, y, z) dx dz Rewriting this as: F y (x, y + y, z) dx dz F y (x, y, z) dx dz = F y(x, y + y, z) F y (x, y, z) y dx dz y and then taking the limit as y approaches zero requires that this expression becomes: lim y 0 F y (x, y + y, z) F y (x, y, z) y dx dz = F y dx dy dz y y Taking the limit for each surface of the flux through of the infinitesimal box gives an expression for total flux through the infinitesimal volume: ( Fx x + F y y + F ) z dx dy dz z Using the definition of divergence as flux per unit volume, a typical expression for divergence can be written in rectangular coordinates. The development of function layer which isn t hollow for the symbolic context requires a description of this limiting process. This limiting process is different from the limit layer because it results in an object which describes the divergence at any point in the field. The process requires detailed geometric arguments and for three separate limits to be taken when calculating the flux out of various surfaces enclosing an infinitesimal volume. In the student work analyzed, there was no indication that any student understood the limiting process embedded in the geometric approach to divergence. While this may be an interesting aspect of divergence to investigate in the future, in this work, the function layer is considered only as a pseudostructural object without an associated process. Divergence cannot be accurately described using only one context. While the Zandieh framework allows for one context to accurately describe derivative, a single context in this preliminary framework does not provide an accurate definition of divergence. In order to provide a clear description of divergence, several contexts within this framework must be used together. For example, in order 28

38 to use any of the diagrams in the graphical context, the descriptive context must be used simultaneously to demonstrate understanding of divergence. Divergence is a spatial vector derivative which is defined as the flux per unit volume at a point in a vector field. Therefore, representing divergence graphically or symbolically still requires a description and explanation which is identical to the descriptive context. There is not a clear way to distinguish contexts but rather the contexts rely on each other to accurately describe divergence. Lastly, this preliminary framework ignores many aspects of divergence. For instance, divergence is a unique linear combination of partial derivatives of a vector field. Within the framework there is nothing to describe why these partial derivatives are important or what they describe. Additionally, there is no way in which to account for descriptions of divergence which are not directly related to a flux per unit volume. For example, many students stated that divergence takes a vector field to a scalar field. This is an important aspect of divergence and is an aspect of the function layer. However, there is no clear way to include this in the current framework. Another representation issue which is not accounted for within this pilot study is the two frequent and important graphical representations of fields: vector fields and field lines. Here, only vector field representations are considered, however, field lines are frequently used and associated with divergence. Additionally, many students included mathematical and physical equations where divergence is used such as the divergence theorem and Maxwell s equations. These applications of divergence are especially important for physics students to understand, but within the preliminary framework, applications of divergence are generally ignored. The example context provides some allowance for applications, however, mentioning equations which use divergence do not fit into the example context without sufficient explanation. Overall, despite student responses which fit superficially into similar categories as those established by Zandieh for derivative, this particular choice in framework is inappropriate for divergence. However, it is productive in illuminating the various ways in which divergence can be understood and described conceptually. 29

39 Chapter 4 Applying the Preliminary Framework to Student Responses This chapter provides four student responses which were selected to provide examples of various responses and how each response is interpreted within the framework. The responses of these students are categorized within a summary chart describing the various contexts and layers of the preliminary framework developed in Chapter 3. Section 4.1 details how these summary charts are filled, in general, prior to providing the student responses and the corresponding summary charts for each response. These student responses serve as examples for how the framework could be used to understand which aspects of divergence students understand and use in their thinking. In Appendix C, student responses and corresponding summary charts are provided for students who are not discussed within this section. Section 4.6 uses all of the student summary charts and tallies the number of students including particular responses. Here, for brevity, particular solutions were selected which demonstrate a broad range of understanding of divergence. 4.1 Summary Charts Each student s response to the prompt given in Figure 3.2 is summarized in a summary chart as given in Figure 4.1. This summary chart was constructed to imitate the Zandieh summary chart for the concept of derivative. The columns refer to the context and the rows refer to the layer. This way, a student response can be quickly summarized in a single chart to show the various aspects of divergence which the student used to convey the concept of divergence. 30

40 Student Graphical Descriptive Symbolic Example Finite Limit Function Figure 4.1: A summary chart used to summarize the various aspects of student responses. columns refer to the contexts. The rows refer to the layers. The These summary charts were completed using empty circles,, and crossed circles,. These symbols roughly correspond to the empty and filled circles given in the Zandieh framework. Zandieh uses an empty circle to indicate that the student has demonstrated a pseudo-object understanding of the given context and layer. A filled circle is then used when students have demonstrated understanding of the process involved in the context and layer. Written responses do not allow for this particular depth of analysis of student thinking and understanding. Therefore, the empty circle,, is used when a student has attempted to write a response which includes a particular context and layer, however, the response may not be complete or correct. The crossed circle,, is used when a student includes an entire and correct response within a particular context and layer. It is not possible to assess, given these written responses, whether students are demonstrating a pseudo-object understanding, demonstrating an understanding of the process, or copying from an outside source. Therefore, the preliminary framework uses empty and crossed circles which do not directly parallel the empty and filled circles used in the Zandieh framework. 4.2 Student A Student A s response to the prompt is given in Figure 4.2 with color coding corresponding to the various contexts presented in the framework. The response, in its entirity, is categorized in Figure 4.3 by layer and context. Beginning with the graphical context outlined in blue in Figure 4.2, the student draws a vector field with positive divergence, however, the student gives no indication that divergence is specific to a location within the vector field. Therefore, the student attempts to use a graphical representation, however, does not provide an entire graphical representation of divergence. This particular example is categorized with an empty circle under the finite layer of Figure 4.3 because it is not clear that the student is thinking of divergence at a point. There is no indication in his drawing that divergence has a value at a particular point; this makes the response difficult to place. Due to his drawing of a simple vector field which has nonzero divergence, his representation is considered a finite representation. 31

41 Figure 4.2: Student A s response color coded by context. Blue outlines the graphical context. Red outlines the descriptive context. Green outlines the symbolic context. It is likely, if asked to explain his drawing, this student could provide an explanation which would demonstrate understanding of the function layer where he might state that divergence is nonzero at all points in the given vector field. However, this explanation is not provided within his response. Without this explanation, a vector field such as this with no associated explanation is categorized with an empty circle in the graphical context and finite layer. This particular response highlights one limit of the use of a framework addressing divergence which is analogous to the Zandieh framework. The descriptive context, as indicated in Figure 4.2 by red, states that the divergence is a local measure of the outgoingness of the vector field at a point. This response shows that the student understands divergence is an object which measures some property of a vector field at a point. This response, which includes a reference to a particular point, is indicative of a limit. However, the student is not explicit in his description of what the outgoingness of the vector field might be. Although this student may be able to explain what he means by outgoingness, he was not able to convey this information in his response. Therefore, this circle remains empty in Figure 4.3. Lastly, this student uses the symbolic context of divergence by attempting to write a formula for the divergence as a function with a value at any point, outlined in green in Figure 4.2. As previously explained in Chapter 3 Section 3.4, this is a hollow layer within the developed framework because the process to develop the formula for divergence requires detailed understanding of the geometric 32

42 approach to divergence where the limiting process is repeated in each dimension. It is not expected that students would express this process through a written response, therefore a response which includes a correct equation to calculate divergence is considered a sufficient response. However, the equation which the student writes does not reflect his response that divergence takes a vector field to a scalar field and instead uses the components of F as vectors themselves. This equation is both contradictory to other responses and incorrect, so it is not clear whether this student does not understand the difference between vector and scalar fields or whether he simply miswrote the equation. Therefore, in Figure 4.3 an empty circle is given in the intersection of symbolic context and function layer. Student A Graphical Descriptive Symbolic Example Finite Limit Function Figure 4.3: Summary chart for Student A. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 4.3 Student B Student B s response is given, with color coding corresponding to particular contexts, in Figure 4.4. The corresponding summary chart is given in Figure 4.5. Student B draws an iconic diagram for divergence which is highlighted in blue in Figure 4.4. This response establishes his understanding of divergence in a graphical context. He draws a vector field with a small volume placed around a point. Additionally, he labels the small volume with dτ which is interpreted as an infinitesimal volume due to his statements highlighted in red and the use of dτ as an infinitesimal volume in some of his previous courses. He only draws the infinitesimal volume around one point which means he is showing that the divergence is defined at a point in the vector field. He clearly demonstrates provides a response in the limit layer in the graphical context by drawing how he would determine the divergence at a given point in the vector field. However, this student fails to state that this could be done at any point within the vector field and therefore does not provide a response using the function layer. The statements which Student B gives also imply understanding of divergence on the limit layer but within the descriptive context. The first statement, highlighted in red in Figure 4.4, is divergence is defined as the flux per unit volume as the volume goes to zero while the second is 33

43 Figure 4.4: Student B s response color coded by context. Blue outlines the graphical context. Red outlines the descriptive context. Green outlines the symbolic context. divergence depends on the flux divided by the infinitesimally small volume around a point. Both of these statements imply taking a limit of an enclosed volume around a point. Therefore, this student is demonstrating understanding of the limit layer of the descriptive context. Additionally, the student begins to touch on the function layer when he states that divergence is only defined for points in continuously differentiable vector fields. His choice to write points shows that he likely understands divergence can be defined as a function which has a value for each point in a vector field. However, he did not explicitly state that the flux per unit volume can be determined for every point in the vector field which results in an empty circle in the function layer of the descriptive context. 34

44 Student B is an example of a student who gives more than one formula for calculating divergence, as highlighted in green in Figure 4.4. From these equations, two layers of the symbolic context are addressed. The student incorrectly gives a formula in rectangular coordinates where he writes the components of vector, F, as vectors. Although this student clearly attempts a symbolic representation of divergence using the function layer, he does not give the correct formula in rectangular coordinates. He also gives the limit definition of divergence in symbolic form. This equation is missing that the surface that integration occurs over must be closed, however, his other responses such as divergence depends on the flux divided by infinitesimally small volume around a point implies that he understands that this surface must enclose a volume. This is indicative of the limit layer of the symbolic context. Lastly, he gives a formula for divergence as F (ρ) = Φ dτ which is a ratio of the flux, Φ, per unit volume, dτ. This formula could either be finite or a limit depending on how the student is thinking about dτ. If the student is thinking of dτ as a small, but not infinitesimal, volume, then this formula would be in the finite layer. However, from the student s statements about an infinitesimally small volume around a point, this formula is interpreted as one which uses the limit layer of the symbolic context. Student B Graphical Descriptive Symbolic Example Finite Limit Function Figure 4.5: Summary chart for Student B. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 4.4 Student C Student C s response, given in Figure 4.6, provides a more detailed response than Student A and has some similarities to Student B. The corresponding summary chart is provided in Figure 4.7. This student includes statements which are categorized in the example context a context that is not found in the work of Student A or Student B. Beginning with the graphical context of divergence, this student clearly illustrates divergence. He draws examples of vector fields with positive, negative, and zero divergence. Additionally, the student indicates that a small volume is required to calculate the divergence at a point in the vector field; each of the volumes within the vector field are labeled dv, as can be clearly seen in Figure 4.6 outlined in blue. He is clearly considering infinitesimal volumes while drawing these diagrams. 35

45 Figure 4.6: Student C s response color coded by context. Blue outlines the graphical context. Red outlines the descriptive context. Green outlines the symbolic context. Yellow outlines the example context. 36

46 Because of this distinction, in the graphical context, he demonstrates the limit layer of the graphical context. In the descriptive context, the student states that divergence is the rate of outflow per unit volume. This statement falls within the categorization of the finite layer by the stated ratio of outflow per unit volume. The rate of outflow statement, however, shows that the student confuses flux with a rate. Therefore, the summary chart for Student C, given in Figure 4.7, has an empty circle on the finite layer of the descriptive context. Student C additionally states that divergence is a local quantity (depends on an infinitesimal volume around a given point) which touches on divergence as the limit of flux per unit volume as the volume goes to zero. This statement suggests he understands that divergence has a value at a given point and this involves an infinitesimal volume, as is done when taking a limit. However, due to his confusion of flux with a rate the student s response has an empty circle for the limit layer of the descriptive context in the summary chart. Student C uses the symbolic context of divergence by writing the formula for divergence in rectangular coordinates. The formula given in rectangular coordinates is complete and correct, outlined in green in Figure 4.6, demonstrating the function layer of the symbolic context. As mentioned previously in Chapter 3 Section 3.4, simply providing a formula which can be used to calculate divergence does not provide any indication of understanding the limiting processes involved in establishing a formula. Within the limitations of the preliminary framework, the response the student provides in the symbolic context is sufficient for the function layer. Finally, Student C mentions sources and sinks in his response. This response corresponds to the example context of the framework. He states that divergence determines if a point in space is a source or sink of field lines. This statement is categorized as the limit layer because the student mentions a specific point in the vector field rather than the vector field as a whole. Additionally, the student specifies that divergence determines whether the point is a source or sink of field lines. This is a vague statement but can be easily understood in the context of electrostatics where the divergence of the electric field is directly related to the charge density. Electric field is often represented by field lines and where they converge, there is a source or sink; either positive or negative charge is located at that point. The student does not clarify his statement, and therefore, there is an empty circle for this particular response in the limit layer of the example context. 4.5 Student E Student E s work, color coded by context, is given in Figure 4.8, and the corresponding sum- 37

47 Student C Graphical Descriptive Symbolic Example Finite Limit Function Figure 4.7: Summary chart for Student C. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. mary chart is given in Figure 4.9. This student touches on each of the contexts established in the preliminary framework. Student E draws a diagram representing divergence, outlined in blue in Figure 4.8, which uses the finite layer of the graphical context. He does not specify that in the two-dimensional vector field that the area must be infinitesimal around a point and even labels his diagram with divergence is flux out of box divided by box area. Because he recognizes the field he has drawn is in two-dimensions, the use of a box rather than a volume is appropriate in this particular response. This clearly indicates that graphically he is considering divergence of a ratio of flux to a finite area. Therefore, he demonstrates the finite layer in a graphical representation of divergence. This response does highlight one of the deficiencies of the proposed framework. The diagram drawn by the student does not sufficiently describe divergence until he modifies the diagram with the statement outlined in red. The combination of the two, separate contexts, does accurately describe one aspect of divergence, however, neither is sufficient explanation independently of the other. Within Student E s response, he makes several statements which are within the limit layer of the descriptive context. His statement, outlined in red in Figure 4.8, that at a point in the domain, the output is the flux per unit volume out of a surface that encloses the point, as the surface shrinks to the point demonstrates his knowledge of divergence as a limit of a specific finite ratio. Therefore, this statement is complete in that it addresses the aspects which are central to the descriptive context and limit layer. Additionally, his statement which he gives to clarify his diagram, which he labels as a two-dimensional diagram, says that divergence is flux out of box divided by box area. In this case, his box area is a finite area and the statement does not imply any sense of a limit. Therefore, this second statement falls within the finite layer of the descriptive context. Student E provides one formula for calculating divergence: The correct formula for calculating divergence in rectangular coordinates. He has no other formulas given within his responses. Therefore, he is providing a response at the function layer of the symbolic context. Lastly, Student E has two brief statements which are categorized as the example context because he is stating that divergence can determine whether a point in the vector field is a source or sink. He 38

48 Figure 4.8: Student E s response color coded by context. Blue outlines the graphical context. Red outlines the descriptive context. Green outlines the symbolic context. Yellow outlines the example context. 39

49 only states that a point is a source and that a point is a sink ; therefore, this demonstrates the limit layer of the example context. Had he stated that the divergence of a vector field can determine where there are sources and sinks in the vector field, he would have been working on the function layer of the example context. However, he discusses only single points in this context and therefore has only demonstrated the limit layer as indicated in the summary chart of Figure 4.9. With Student E s response, along with the other responses, it is clear that the framework does not include many aspects of divergence. In particular, Student E makes statements such as divergence is a local quantity, a differential operator, and takes a vector field to a scalar field as well as providing notations and common examples. The framework does not account for any of these statements about divergence. Therefore, using this framework to analyze student understanding has deficiencies despite providing some insight into the various ways in which divergence can be represented and understood. Student E Graphical Descriptive Symbolic Example Finite Limit Function Figure 4.9: Summary chart for Student E. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 4.6 Collective Summary of Students In Figure 4.10, the instances of student responses are tallied for each context and the associated layers. Summary charts for each student which contribute to this collective summary chart can be found in Appendix C. By providing a collective summary chart, it is apparent which contexts and layers are typically used by students. The students whose responses are used to develop this preliminary framework have similar backgrounds of coursework where divergence is addressed. Additionally, these students all previously took the same course, Static Vector Fields, from the Math Methods instructor, where divergence is introduced. The prompt, given in Figure 3.2, provides an instructor example using gradient which students seem to follow closely when constructing responses for divergence. Therefore, given the similarity and familiarity of this group of students with divergence and the instructor, it is expected that their responses follow particular trends for how they choose to explain divergence on a homework problem. 40

50 Total Student Graphical Descriptive Symbolic Example Finite N/A N/A Limit Function Figure 4.10: Student responses summarized collectively. The collective summary chart clearly shows that this group of students tends to describe divergence using particular representations, or contexts, more frequently than others. This chart does include particular responses which include multiple layers of a particular context, so tallies do not give the true frequency of a context provided by a student. The graphical, descriptive, and symbolic contexts have most students, 10, 11, and 11 out of 12 respectively, provide responses within the particular context, likely due to the example provided for gradient. However, only half of the students provide responses which included the example context. Since student responses typically follow the example provided by the instructor, it is expected that these responses fall into similar categories. The example for gradient given by the instructor in the prompt, in Figure 3.2, provides students with a sample solution. The instructor s response for gradient can be analyzed in a similar way to divergence. Although a parallel framework has not been developed for gradient, a summary chart in Figure 4.11 roughly corresponds to the instructor response for gradient. The instructor provides an iconic diagram of gradient with a scalar field drawn using contour lines and the gradient drawn perpendicular to the level curves at several points; this representation is similar to the function layer of the graphical context. The instructor provides a formula for gradient in rectangular coordinates similar to the function layer of the symbolic context. The statement of at a point in the domain, the magnitude of the vector is given by the maximum rate of change of the scalar field at a point. The direction of the vector is in the direction (in the domain) of the maximum rate of change of the scalar field provides the necessary information for a parallel descriptive context for gradient. The instructor specifies at a point in that particular statement which would indicate the limit layer, but also notes that gradient takes a scalar field (i.e. a function often called the potential function, i.e. a number at every point in the domain) to vector fields (a vector at every point in the domain) which clearly is describing a function. Therefore, the instructor provides an example parallel to the descriptive context which includes both the limit and function layer. Whether there is a corresponding finite layer or example context is not immediately clear 41

51 Instructor Graphical Descriptive Symbolic Finite Limit Function Figure 4.11: Summary chart for the instructor s example of gradient. Roughly parallel contexts and layers to those presented for divergence are constructed from the prompt given in Figure 3.2. This chart summarizes the equivalent statements made by the instructor when describing gradient. from the instructor s example. Despite the cuing from the given example for gradient within the prompt, within each context there are layers which students choose to represent divergence most often. Within the graphical context, the finite layer is most prevalent. The descriptive context has more variation in addition to more responses which include more than one layer of the context. Most often, students represent divergence using the limit layer of the descriptive context, however, the finite layer is used nearly as often. Students primarily give responses within the symbolic context which are included in the function layer; the layer which lists formulas which can be used to calculate divergence in a chosen coordinate system. This requires little to no conceptual understanding of divergence, unlike the other layers, and can be easily referenced by students. Lastly, within the example context, both the limit and function layer are used by students. With the current number of responses, there is not any significance which can be attributed to students responding using a particular layer in this context. The application of this preliminary framework provides a foundation for analyzing student understanding of divergence. As previously mentioned, the student responses are limited in number and do not necessarily represent what an individual student knows and understands about divergence. Rather, the responses are representative of how students choose to convey the meaning of divergence along the lines of a similar example from the instructor. Given these restrictions, these conclusions elaborate on how to develop a framework from this foundation as well as begin to illuminate how an analysis might proceed from further results. 4.7 Student Understanding in the Context of Their Backgrounds The trends which are identified in the student responses, including most students answering using the graphical, descriptive, and symbolic contexts and the tendency for students to use a particular 42

52 layer within a given context, can be understood by considering the experiences these students have in vector calculus topics. Further analysis on students with varying backgrounds might reveal other emphases. The following provides a brief explanation of some possibly reasons this small group of students may have a tendency to represent divergence in these particular ways. For instance, in the graphical context, the students whose responses include this context typically use the finite layer. In the Fall term during the Static Vector Fields Paradigm, the students are introduced to divergence with an activity, Visualizing Divergence (Paradigms in Physics Group 2012). In this activity, the students are asked to qualitatively identify vector fields with positive, negative, and zero divergence with given vector fields on a Mathematica worksheet. This activity, included within the Paradigms, emphasizes divergence as a flux per unit volume within a vector field using a qualitative graphical context as a finite ratio of flux per volume. It should come as no surprise then that students who have been introduced to divergence through this activity tend to respond using the finite layer in the graphical context. In the descriptive context, the finite and limit layers are most prevalent among students. As stated in the discussion on the graphical context, students in the Paradigms curriculum are asked to identify divergence as the flux per unit volume in the activity, Visualizing Divergence, for given vector fields. Therefore, the descriptions as flux per unit volume or flux per unit volume as the limit of the volume goes to zero are expected from this particular group of students. It is likely that the majority of these students have working knowledge of the function layer of this context, however, it is not immediately evident from their written responses. This context might be unique to Paradigms students because the coursework emphasizes the notion of divergence as a flux per unit volume. Students from other backgrounds may not have this particular emphasis, and therefore, this particular group of students may have unique descriptions of divergence. The tendency for students to respond within the function layer of the symbolic context is expected. Students typically learn divergence as a mathematical operation which can be calculated using a specific formula. These formulas are all part of the function layer of the symbolic context. The finite and limit layers require some conceptual understanding of what divergence is perhaps even some intuition about what divergence measures. Understanding the finite layer requires students to demonstrate understanding of the descriptive context which requires conceptual understanding of divergence, unlike the function layer. Similarly, the limit layer requires taking the limit as the volume goes to zero of an equation with a flux integral divided by a volume. These layers require conceptual understanding of divergence; therefore, it is not surprising that students respond typically in the function layer with formulas to calculate the divergence of a given vector field. This implies that 43

53 the function layer presented in this preliminary framework is a pseudostructural object, and the symbolic context does not have true layers of process-object pairs. 44

54 Chapter 5 Conclusions In Chapter 2, a survey of 22 textbooks is used to establish the various approaches which are used to describe divergence and the divergence theorem in mathematics and physics. The mathematicallydetailed definitions of divergence rely either on geometry or extensive algebraic manipulations: the geometric and curvilinear coordinate approaches, respectively. Textbooks which do not provide mathematically-detailed definitions of divergence either discuss divergence in a conceptual or physical context or simply list formulas which can be used to calculate divergence. The divergence theorem is also described by textbooks in two types of mathematically-detailed derivations: geometric and equivalent integrals. Many texts do not provide a derivation of the divergence theorem but do provide a mathematical statement of the theorem either with or without an explanation. Establishing the general ways in which divergence and the divergence theorem can be presented to students illuminates the many variations in how the concepts can be taught to and understood by students. Although there is no attempt in this work to comment on the strengths and weaknesses of the various approaches, this textbook survey does provide a foundation for such an analysis. Despite the current lack of analysis of the approaches, from completing this textbook survey it is apparent that some textbooks present divergence and the divergence theorem clearly and effectively. However, other textbooks require extensive referencing to previous sections, rely on mathematics which is difficult to follow, and do not provide the necessary background and clarity which students first introduced to the concepts likely require. The textbooks which use the geometric and discussion-based approaches to divergence tend to provide a stronger explanation of divergence than textbooks which use the curvilinear coordinate approach or simply provide an algebraic statement. Similarly, for the approaches to the divergence theorem, the geometric and mathematical statement 45

55 with explanation approaches are the most coherent and straightforward. The equivalent integral derivation of the divergence theorem is particularly difficult to follow and likely does not provide an introduction to the divergence theorem which is easily understood by students. Depending on the students disciplines and levels, students may benefit from a specific approach to divergence or the divergence theorem within their courses. In Chapter 3, an analogue to the Zandieh framework is developed for the concept of divergence. Student homework responses, following an instructor s example for gradient, are used to develop the various representations, or contexts, and layers which are used when describing divergence. Establishing this framework highlights the various ways in which divergence can be understood and represented by students, experts, and textbooks. However, this framework was found to be deficient in providing an appropriate way to analyze student understanding of divergence. It ignores many aspects of divergence which contribute to student understanding of the vector derivative including applications, statements which do not fall within any predefined category, and conceptual interpretation of calculations. Additionally, unlike the Zandieh framework for the concept of derivative, each context does not provide an accurate definition of divergence. Several contexts of this framework must be used simultaneously to define divergence. Attempting to draw contexts and layers parallel to the Zandieh framework resulted in contrived categories which are not representative of all aspects of divergence and do not provide the most appropriate foundations from which student understanding of divergence can be analyzed. Chapter 4 illuminates how the preliminary framework was used as well as demonstrating the deficiencies of the framework by using the entireity of the student homework responses to explain the various categorizations. From the student work, it is not possible to distinguish the processes and objects as done in the Zandieh framework. Instead, there are only pseudostructural objects presented in student work which are categorized here as being either correctly conveyed or not. This makes it particularly difficult to analyze student understanding of divergence because demonstrating understanding of the processes involved is crucial to understanding of the concept. In order to analyze student understanding, different types of responses should be collected and another framework should be used. For example, open-ended questions regarding divergence are important to establishing what an individual student knows and understands. Also, the example provided in the prompt constructed an unintended format for student responses where the students gave answers which were meant to imitate those given by the instructor for gradient. These written responses encourage students to simply list objects which use and describe divergence rather than demonstrate understanding 46

56 of the processes. In order to gain a broader analysis of student understanding, further student explanations are required which touch on these processes and applications involved with divergence. These may include group work students engage in during a class on divergence and individual interviews regarding divergence. 47

57 Bibliography [1] Arfken, G. (1985). Mathematical Methods for Physicists (3rd ed.). Academic Press, Inc., San Diego. [2] Baily, C. & Astolfi, C. (2014). Student reasoning about the divergence of a vector field. ArXiv e-prints (Submitted to PERC 2014) [3] Baily, C., Dubson, M., & Pollock, S.J. (2013). Research-based course materials and assessments for upper-division electrodynamics (E&M II). AIP Conf. Proc. 1513, (2013). [4] Beatrous, F. & Curjel, C. (2002). Multivariable Calculus: A Geometric Approach (1st ed.). Prentice-Hall, Inc., Upper Saddle River. [5] Boas, M.L. (1983). Mathematical Methods in the Physical Sciences (2nd ed.). John Wiley & Sons, New York. [6] Briggs, W.L. & Cochran, L. (2011). Calculus Early Transcendentals (1st ed.), Pearson Education, Inc., San Francisco [7] Butkov, E. (1968). Mathematical Physics (1st ed.). Addison-Wesley Publishing Company, Reading. [8] Chabay, R.W., & Sherwood, B.A. (2011). Matter & Interactions (3rd ed.). John Wiley & Sons, Inc. [9] Fetter, A.L & Walecka, J.D. (2003). Theoretical Mechanics of Particles and Continua (2nd ed.). Dover Publications, Mineola. [10] Giancoli, D.C. (2000). Physics for Scientists & Engineers (3rd ed.). Prentice Hall, Upper Saddle River. [11] Griffiths, D.J. (2008). Introduction to Electrodynamics (3rd ed.). Pearson, San Francisco. 48

58 [12] Halliday, D., Resnick, R., & Krane, K.S. (2002). Physics Volume 2 (5th ed.). John Wiley & Sons, Inc., New York. [13] Heald, M.A, & Marion, J.B. (1995). Classical Electromagnetic Radiation (3rd ed.), Saunders College Publishing, Fort Worth. [14] Hughes-Hallett, McCallum, W.G., D., Gleason, A.M., et.al (2002). Calculus: Single and Multivariable (4th ed.). John Wiley & Sons Inc. [15] Jackson, J.D. (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons, Inc. [16] Knight, R.D. (2008). Physics for Scientists and Engineers: A Strategic Approach (2nd ed.). Pearson Addison-Wesley, San Francisco. [17] McIntyre, D.H. (2012). Quantum Mechanics: A Paradigms Approach (1st ed.). Pearson, Boston. [18] Paradigms in Physics Group, Oregon State University (2012). activities:main. Portfolios Wiki. Retrieved 30 November, 2014 from [19] Pepper, R.E., et. al (2012). Observations on student difficulties with mathematics in upperdivision electricity and magnetism. Phys. Rev. ST Phys. Educ. Res., 8, [20] Riley, K.F., Hobson, M.P., & Bence, S.J. (1998). Mathematical Methods for Physics and Engineering (1st ed.). Cambridge University Press, Cambridge. [21] Sakurai, J.J. (1994). Modern Quantum Mechanics (Rev. ed.). Addison-Wesley Publishing Company, Reading. [22] Schey, H.M. (1992). Div Grad Curl and all That: An Informal Text on Vector Calculus (2nd ed.). W.W. Norton & Company, New York. [23] Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, [24] Singh, C. & Maries, A. (2013) Core graduate courses: A missed learning opportunity?. AIP Conference Proceedings, 1513(1), [25] Shankar, R. (1994). Principles of Quantum Mechanics (2nd Edition). Springer. [26] Stewart, J. (2008). Multivariable Calculus: Early Transcendentals (6th ed.). Thompson Brooks/Cole Publishing Company. 49

59 [27] Taylor, J.R. (2005). Classical Mechanics (1st ed.). University Science Books, Sausalito. [28] Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, [29] Young, H.D. & Freedman, R.A. (2000). University Physics (10th ed.). Addison-Wesley, San Francisco. [30] Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS Issues in Mathematics Education, 8,

60 Appendix A Curvilinear Coordinate Approach This approach is described by following the Riley, Hobson, & Bence (1998) description of divergence. There is more detail included here than in the textbook in order to fully describe the approach. The notation follows the notation used in the textbook. Beginning with general curvilinear coordinates, we can describe a position in space with the coordinates (u 1, u 2, u 3 ). These points can be described as functions of positions in rectangular coordinates. Therefore, we can write each as follows. u 1 = u 1 (x, y, z) u 2 = u 2 (x, y, z) u 3 = u 3 (x, y, z) We can then consider that coordinate curves can be drawn which are similar to x, y, and z axes. We do this by varying one u i while holding the other two constant. Figure A.1 shows how we can define any point in space using a position vector in a general coordinate system. Figure A.2 is drawn by varying u 1 while holding u 2 and u 3 constant. A second position vector, r(u 1 +du 1, u 2, u 3 ) can be drawn. Figure A.3 shows this at a larger scale where the r is the difference in position vectors: r = r(u 1 + du 1, u 2, u 3 ) r(u 1, u 2, u 3 ). As the two points get infinitesimally close to each other, r approaches d r. This means that d r becomes the tangent vector at a point as you move along the u 1 -varying curve. Additionally, we can write that a little change in the position vector, a differential, can be written by definition as: d r = r u 1 du 1 + r u 2 du 2 + r u 3 du 3. In the particular case which is described, we are 51

61 Figure A.1: Position vector, r (u 1, u 2, u 3 ), is given in red. This position vector describes the position of point, (u 1, u 2, u 3 ), in general coordinates. Figure A.2: The blue curve represents varying u 1 while holding u 2 and u 3 constant. The general orthogonal coordinates are u 1, u 2, and u 3. Two position vectors are in red which describe two points along the u 1 curve. 52

62 Figure A.3: The r, in green, is the difference in position vectors. The position vectors are given in red. The difference can be found by r = r(u 1 + du 1, u 2, u 3 ) r(u 1, u 2, u 3 ). moving along u 1 while holding u 2 and u 3 constant. Therefore, d r = r u 1 du 1. This is consistent with taking the limit as u 1 goes to zero: lim u 1 0 r u 1 = d r du 1. We take r u 1 = e 1. Now if we extend this to the other coordinates, we find the same for each which allows us to define a small change in position vector for any direction as d r = r u 1 du 1 + r u 2 du 2 + r u 3 du 3. As previously stated, this is the definition of a differential. We can then define unit vectors of our curvilinear coordinate system as in the direction of increading u i -curves. If we use the original point defined in Figure A.1 and follow the u 1 -curve from that point, the d r is the tangent vector at each point. The unit vector is then defined as ê 1 = 1 h 1 r u 1. This comes from the way in which d r is defined. d r = r u 1 du 1 + r u 2 du 2 + r u 3 du 3 = e 1 du 1 + e 2 du 2 + e 3 du 3 The magnitudes of e 1, e 2, and e 3 are defined as h 1, h 2, and h 3 respectively. This means we can rewrite each vector as e i = hi h i r u 1. ê 1 = e 1 = h 1 r h 2 = 1 r 1 h 1 u 1 h 1 u 1 e 1 = h 1 ê 1 d r = ê 1 h 1 du 1 + ê 2 h 2 du 2 + ê 3 h 3 du 3 We can now define an arbitrary scalar field, Φ. The differential of this scalar field in our curvilinear coordinates is as follows. dφ = Φ u 1 du 1 + Φ u 2 du 2 + Φ u 3 du 3 Knowing dφ = Φ d r, we can equate these two equations. 53

63 Φ du 1 + Φ du 2 + Φ du 3 = Φ u 1 u 2 u d r 3 Φ u 1 du 1 + Φ u 2 du 2 + Φ u 3 du 3 = Φ (ê 1 h 1 du 1 + ê 2 h 2 du 2 + ê 3 h 3 du 3 ) Φ du 1 + Φ du 2 + Φ du 3 = Φ e1 h 1 du 1 + Φ e2 h 2 du 2 + Φ e3 h 3 du 3 u 1 u 2 u 3 [A.1] Collecting like terms we obtain the following. Φ u 1 = Φ e1 h 1 Φ u 2 = Φ e2 h 2 Φ u 3 = Φ e3 h 3 We get that Φ = 1 h 1 If we now let Φ = u i : Φ u 1 ê Φ h 2 u 2 ê Φ h 3 u 3 ê 3. u i = 1 h i u i u i ê i = êi h i [A.2] u i = ( ) 1 êi êi h i h i 2 = ( 1 h 2 i ) 1 2 = 1 Consider an arbitrary vector field defined in terms of our unit vectors. h i a = a 1 ê 1 + a 2 ê 2 + a 3 ê 3 For now, consider just the first component of the vector field. Later, this will be extended to the other components of the vector field. Taking divergence of the first component gives: a = (a 1 ê 1 ). The unit vector, ê 1, can be rewritten as ê 1 = ê 2 ê 3 which can be written this way because we are defining our unit vectors in an orthogonal curvilinear basis. From Equation A.2, we can write u i = êi h i. ê i = h i ui 54

64 ê 1 = ê 2 ê 3 = h 2 u2 h 3 u3 We can now rewrite the divergence of the first component. (a 1 ê 1 ) = ) (a 1 h 2 u2 h 3 u3 = (a 1 h 2 h 3 u2 u ) 3 [A.3] This is the divergence of a cross product, so we should find a vector identity which can be used. The following algebraic manipulations are performed in order to determine an appropriate vector identity to use. First, the scalar and vector fields are renamed as follows. A = a 1 h 2 h 3 B = u 2 C = u 3 We can write these fields in any spatial coordinates of three dimensions. For simplicity, we will write them in rectangular coordinates. 55

65 ( AB C ) ˆx ŷ ẑ = A B x B y B z C x C y C z = [A [ˆx (B y C z B z C y ) + ŷ (B z C x B x C z ) + ẑ (B x C y B y C x )]] = x [AB yc z AB z C y ] + y [AB zc x AB x C z ] + z [AB xc y AB y C x ] = B y C z x A + AC z x B y + AB y x C z B z C y x A AC y x B z AB z x C y +B z C x y A + AC x y B z + AB z y C x B x C z y A AC z y B x AB x y C z +B x C y z A + AC y z B x + AB x z C y B y C x z A AC x z B y AB y z C x = C y x (B za) + C z x (AB y) + AB y x C z AB z x C y +C x y (AB z) C z y (AB x) AB x y C z + AB z y C x C x z (AB y) + C y z (B xa) + AB x z C y AB y z C x [ = C x y (AB z) C x z (AB y) + C y z (B xa) C y x (B za) + C z [ ] A B x y C z B x z C y + B y z C x B y x C z + B z x C y B z y C x [ ( ) ( ) ( )] = C A ˆx + ŷ + ẑ y B z z B y z B x x B z [ ( AB ˆx y C z ) ( z C y + ŷ z C x x C z ˆx ŷ ẑ ˆx ŷ ẑ = C A x y z A B x y z B x B y B z C x C y C z = C ( ) AB AB ( ) C ) + ẑ x (ab y) C z x B y y B x ( x C y )] y C x ] y (AB x) Now we have equivalent expressions: ( AB C ) = C ( ) AB AB ( ) C Next, we want to show that the vector identity, ( ) D E = 0, holds. If D = A B = F and E = C, we have the following: 56

66 ( ) F C = C ( ) F = E F ( ) C ( D ) F ( E ) For any scalar field, G, we now need to show that G = 0. We can write the gradient of the scalar field, G, in terms of rectangular components. G = g xˆx + g y ŷ + g z ẑ In this notation, g x, g y, and g z are functions of x, y, and z and are defined as follows. g x = G x g y = G y g z = G z ˆx ŷ ẑ G = x y z G G G x y z ( 2 ) ( G = ˆx y z 2 G 2 ) ( G + ŷ z y z x 2 G 2 ) G + ẑ x z x y 2 G y z = (0) ˆx + (0) ŷ + (0) ẑ = 0 Therefore, G = 0 which means: ( ) F C = E ( ) D F ( ) E = E (0) F (0) = 0 Now, returning to Equation A.3, we will apply these vector identities in order to determine the general expression for divergence in curvilinear coordinates. 57

67 (a 1 ê 1 ) = (a 1 h 2 h 3 u2 u ) 3 We can rewrite this as follows: (A 1 ê 1 ) = ( (a 1 h 2 h 3 ) u2 u ) ( 3 + a 1 h 2 h 3 u2 u ) 3 We have shown the second term must be zero. (a 1 ê 1 ) = ( (a 1 h 2 h 3 ) u2 u ) 3 Now we replace u 2 and u 3 with their equivalent expressions. u 2 = ê2 h 2 u 3 = ê3 h 3 ( ) (a 1 ê 1 ) = ê2 (a 1 h 2 h 3 ) ê3 h 2 h 3 ( ) = ê1 (a 1 h 2 h 3 ) h 2 h 3 Now there is a dot product with ê 1, so the only term from the gradient which will remain is the ê 1 component. If, from Equation A.1, we let Φ = a 1 h 2 h 3 : (a 1 h 2 h 3 ) = Φ = 1 h 1 Φ u 1 ê h 2 Φ u 2 ê h 3 Φ u 3 ê 3 (a 1 ê 1 ) = 1 h 1 Φ u 1 ê 1 = 1 Φ h 1 h 2 h 3 u 1 ê 1 h 2 h 3 58

68 Now, if this process is repeated for the other components of the vector field, a, the following would be obtained: (a 1 ê 1 ) = 1 Φ ; Φ = a 1 h 2 h 3 h 1 h 2 h 3 u 1 (a 2 ê 2 ) = 1 Φ ; Φ = a 2 h 1 h 3 h 1 h 2 h 3 u 2 (a 3 ê 3 ) = 1 Φ ; Φ = a 3 h 1 h 2 h 1 h 2 h 3 u 3 Combining these, we find the divergence of the vector field, a. 1 a = h 1 h 2 h 3 [ u 1 (a 1 h 2 h 3 ) + u 2 (a 2 h 1 h 3 ) + ] (a 3 h1h 2 ) u 3 Therefore, divergence in general curvilinear coordinates can be expressed as follows: 1 a = h 1 h 2 h 3 [ u 1 (a 1 h 2 h 3 ) + u 2 (a 2 h 1 h 3 ) + ] (a 3 h1h 2 ) u 3 59

69 Appendix B Equivalent Integral Approach The derivation of the divergence theorem presented here follows Stewart (2008). The equations referenced are directly from the textbook and are referenced as Stewart Equation each occurrence. All notation is precisely from Stewart. This derivation deviates from Stewart by including additional explanation and presenting all aspects of the derivation in one continuous derivation. The referenced equations provide some insight to how Stewart relies on previous derivations and explanations presented in other chapters and sections in order to derive the divergence theorem. A vector field, F, can be written in terms of its components, F = P î + Qĵ + Rˆk. Therefore, divergence of this vector field is F = P x + Q y + R z. Integrating the divergence over a volume, E, gives the following: F dv = E = E E ( P x + Q y + R z P x dv + E ) dv Q y dv + E R z dv Using the divergence theorem, S F ˆndS = E F dv, we can write this as follows. 60

70 Figure B.1: An example of a type 1 region as defined by Stewart where where D = {(x, y) a x b, g 1 (x) y g 2 (x)}. S F ds = = = = S S S S F ˆndS ( ) P î + Qĵ + Rˆk ˆndS ( ) P î ˆn + Qĵ ˆn + Rˆk ˆn ds P î ˆndS + Qĵ ˆndS + S S Rˆk ˆndS To prove the divergence theorem, we must prove that the following integrals are equivalent. E E E P x dv = P î ˆndS S Q y dv = Qĵ ˆndS R z dv = S S Rˆk ˆndS To prove the final equation from above, we use the fact that E is a type 1 region. A type 1 region is defined (roughly 150 pages earlier) as a plane region, D, which lies between the graphs of two 61

71 Figure B.2: An example of a type 1 region, D, as defined by Stewart. The region, D, is defined where D = {(x, y) a x b, g 1 (x) y g 2 (x)}. This region is the projection of a volume, E, onto the x-y plane. The volume, E, has a bottom surface, S 1, and a top surface, S 2. The other surface is chosen to be parallel to the z axis. continuous functions of x. Figure B.1 shows an example of a type 1 region where D = {(x, y) a x b, g 1 (x) y g 2 (x)}. This plane region comes from the projection of a graph onto a particular plane. In our case with the volume, E, we can describe the projection of E onto the x-y plane. This gives a type 1 region, D, in the x-y plane. Figure B.2 shows an example of such a projection. Stewart Equation gives E f (x, y, z) dv = [ ] u2(x,y) D u f (x, y, z) dz 1(x,y) da. This is contructed by an argument which leads to Stewart Equation Stewart Equation states that a continuous function f (x, y, z), on a type 1 region, D, where D = {(x, y) a x b, g 1 (x) y g 2 (x)} then D f (x, y) da = b g2(x) a g 1(x) f (x, y) dydx. The argument that leads to this statement is that a functions which is defined for a specific region, such as: f (x, y) : if (x, y) is in D F (x, y) = 0 : if (x, y) is not ind (B.1) Figure B.3 shows a region, D, with domain, R, which modifies the previous equations to the following: 62

72 Figure B.3: An example of a region, D, with domain, R. f (x, y) : if (x, y) is in D F (x, y) = 0 : if (x, y) is not in D but in R (B.2) The double integral of the function, f, over the region, D, is then defined as f (x, y) da = D R F (x, y) da. The left hand side of the equation refers to integrating over all of the region, D, where F (x, y) = f (x, y) in the region, D. The right hand side of the equation refers to integrating over all of the domain, R, where F (x, y) = f (x, y) in the region, D, but outside of D and in R, F (x, y) = 0. These must be true by definition, however, Stewart provides further explanation in Section Additionally, Stewart goes on to define type 1 regions as was done previously in this explanation. For type 1 regions, in order to evaluate the integral, f (x, y) da, a rectangle that contains D region, D, can be chosen. In general, any function can have specific bounds chosen without being a particular type of function. An example of this is given in Figure B.4 which does not fall into a specific categorization provided by Stewart. Stewart uses a particular region which is a type 1 region in order to continue the derivation of the divergence theorem. This rectangle is shown in an example in Figure B.5. The domain, R, can be defined by R = [a, b] [c, d] where [a, b] is the range of x and [c, d] is the range of y. 63

73 Figure B.4: An arbitrary region, D, can be described by particular bounds. These bounds are given by the range of x, [a, b], and the range of y, [c, d]. Figure B.5: A type 1 region can be described by the bounds given by the ranges of x and y. The type 1 region is given as D. The range of x is given by [a, b]. The range of y is given by [c, d]. The lower curve is described by y = g 1 (x), and the upper curve is described by y = g 2 (x). 64

74 f (x, y) da = D = F (x, y) da R b d a c F (x, y) dy dx Similarly, we can describe the region using the curves which bound the top and bottom of the region, D, as shown in Figure B.5. Therefore, we can rewrite d c F (x, y) dy as g 2(x) g 1(x) F (x, y) dy. When y is greater than g 2 (x), then F (x, y) = 0, and similarly, when y is less than g 1 (x), then F (x, y) = 0. Therefore, only f (x, y) can be considered. d c F (x, y) dy = This allows us to write D f (x, y) da = b a g2(x) g 1(x) f (x, ) dy g2(x) g 1(x) f (x, y) dy. Now, by extending this to three dimensions, we can write the equivalent statement. E f (x, y,, z) dxdydz = D ( ) u2(x, y) f (x, y, z) dz dxdy u 1(x, y) With this established relationship, we can use the Fundamental Theorem of Calculus, b f (x) dx = a F (b) F (a) and apply it. u2(x, y) u 1(x, y) f (x, y, z) dz = F (x, y, u 2 (x, y)) F (x, y, u 1 (x, y)) E f (x, y, z) dx = D [F (x, y, u 2 (x, y)) F (x, y, u 1 (x, y))] dxdy Now referencing the original equivalent integrals, such as S Rˆk ˆndS = E write the following expressions: R z dxdydz, we can E R z dxdydz = = = D D D [ u2(x, y) u 1(x, y) [ u2(x, y) u 1(x, y) ] R z dz dxdy ] R z dz da [R (x, y, u 2 (x, y)) R (x, y, u 1 (x, y))] da [B.1] 65

75 The surface which encloses the volume, E, has three pieces: the top, S 2, the bottom, S 1, and the side, S 3. An example of one such surface enclosing a volume, E, is shown in Figure B.2. S 3 might not appear if the volume, E, is a volume such as a sphere. In the choice of volume, E, in Figure B.2, S 3 is vertical therefore, ˆk ˆn = 0 since the unit normal lies parallel to the x-y plane orthogonal to ˆk. Therefore, S 3 Rˆk ˆndS = S 3 0dS = 0. Rˆk ˆndS = S = = Rˆk ˆndS + Rˆk ˆndS + Rˆk ˆndS S 1 S 2 S 3 Rˆk ˆndS + Rˆk ˆndS + 0 S 1 S 2 Rˆk ˆndS + Rˆk ˆndS S 1 S 2 The bottom surface of the volume, E, is S 1 which can be described by z = u 1 (x, y). Similarly the top surface, S 2 is described by z = u 2 (x, y). Therefore, the unit normal vector of S 1 has a component in the ˆk direction while the unit normal vector of S 2 has a component in the +ˆk direction. ˆk ˆndS = S 1 ˆk ˆndS = + S 2 D D R (x, y, u 1 (x, y)) da R (x, y, u 2 (x, y)) da S ˆk ˆndS = ˆk ˆndS + ˆk ˆndS S 1 S 2 = R (x, y, u 1 (x, y)) da + R (x, y, u 2 (x, y)) da D D = [R (x, y, u 2 (x, y)) R (x, y, u 1 (x, y))] da D We showed previously in Equation B.1, this expression is equivalent to E S Rˆk ˆndS = E R z dxdydz R z dxdydz. With similar arguments along the other dimensions, the remaining integrals can be shown to be equivalent. 66

76 true. S S S P î ˆndS = Qĵ ˆndS = Rˆk ˆndS = E E E P x dv Q y dv R z dv This means that the assumption we originally made that the divergence theorem holds must be S F ˆndS = E ( F ) dv 67

77 Appendix C Student Summary Charts The following summary charts correspond to student work used to construct Figure

78 Figure C.1: Student D s answer to the prompt given in the Mathematical Methods course. Student D Graphical Descriptive Symbolic Example Finite Limit Function Figure C.2: Summary chart for Student D. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 69

79 Figure C.3: Student F s answer to the prompt given in the Mathematical Methods course. Student F Graphical Descriptive Symbolic Example Finite Limit Function Figure C.4: Summary chart for Student F. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 70

80 Figure C.5: Student G s answer to the prompt given in the Mathematical Methods course. Student G Graphical Descriptive Symbolic Example Finite Limit Function Figure C.6: Summary chart for Student G. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 71

81 Figure C.7: Student H s answer to the prompt given in the Mathematical Methods course. Student H Graphical Descriptive Symbolic Example Finite Limit Function Figure C.8: Summary chart for Student H. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 72

82 Figure C.9: Student I s answer to the prompt given in the Mathematical Methods course. Student I Graphical Descriptive Symbolic Example Finite Limit Function Figure C.10: Summary chart for Student I. When divergence is conveyed accurately within a particular context and layer by a student, it is indicated in the chart by. An inaccurate or incomplete representation of the concept by the student is indicated by. 73