A cross-context look at upper-division student difficulties with integration
aa r X i v : . [ phy s i c s . e d - ph ] A ug A cross-context look at upper-division student difficulties with integration
Bethany R. Wilcox and Giaco Corsiglia
Department of Physics, University of Colorado Boulder, Boulder, CO 80309
We investigate upper-division student difficulties with direct integration in multiple contexts in-volving the calculation of a potential from a continuous distribution (e.g., mass, charge, or current).Integration is a tool that has been historically studied at several different points in the curriculumincluding introductory and upper-division levels. We build off of these prior studies and contributeadditional data around student difficulties with multi-variable integration at two new points in thecurriculum: middle-division classical mechanics, and upper-division magnetostatics. To facilitatecomparisons across prior studies as well as the current work, we utilize an analytical frameworkthat focuses on how students activate, construct, execute, and reflect on mathematical tools duringphysics problem solving (i.e., the ACER framework). Using a mixed-methods approach involvingcoded exam solutions and student problem-solving interviews, we identify and compare students’difficulties in these two different context and relate them to what has been found previously in otherlevels and contexts. We find that some of the observed student difficulties were persistent accrossall three contexts (e.g., identifying integration as the appropriate tool, and expressing the differencevector), while other difficulties seemed to fade as students advanced through the curriculum (e.g.,expressing differential line, area, and volume elements). We also identified new difficulties thatappear in different contexts (e.g., interpreting and expressing the current density).
PACS numbers: 01.40.Fk
I. INTRODUCTION
Physics Education Research (PER) as a field has along history of conducting student difficulties researchfocused on identifying the challenges and problems thatstudents face when dealing with specific physics conceptsor mathematical tools [1, 2]. The findings from thesestudies have historically been used to inform the devel-opment of curricular materials, pedagogical approaches,or classroom practice in order to specifically target thesechallenging areas to directly address and overcome knownstudent difficulties [2, 3]. One clear example of this pro-cess was the creation of the “Tutorials in IntroductoryPhysics” [4], which were developed based on extensiveinvestigations of specific student difficulties with intro-ductory physics content and have been show to improvestudent learning gains [5].Historically, studies of student difficulties have beenperformed in relation to a single concept, topical area,or mathematical tool and within the context of a singlecourse. However, the cyclic nature of the physics cur-riculum means that students often encounter individualconcepts and mathematical tools multiple times, and inmultiple contexts, over the course of their undergraduatecareer. It is reasonable to assume that over this timeperiod, and with multiple exposures, students difficul-ties with these concepts and tools may naturally evolve.For example, some difficulties faced early on may resolvethemselves over time as students see these ideas mul-tiple times in the standard curriculum and gain moreexperience, while other difficulties may be more persis-tent, reappearing despite repeated exposures. Alterna-tively, as the complexity of the physics content associatedwith these topics and tools increases, new difficulties notpresent in earlier contexts may also begin to appear. Understanding the nature of how students difficultieswith common concepts and mathematical tools change asstudents are exposed to these ideas in new contexts canhave important implications for both instructors and re-searchers. By attending to how students’ ideas evolve,we can avoid spending time and effort on developing newmaterials to target difficulties that are already being suf-ficiently addressed in the standard curriculum in favor offocusing on difficulties that persist across multiple con-texts and exposures. Yet, despite the potential advan-tages, investigations of student difficulties with specifictopics or tools across multiple exposures is rare withinthe PER literature. In lieu of comprehensive investiga-tions looking at the evolution of student difficulties overtime, another option for investigating this dynamic wouldbe to make explicit comparisons between distinct studiesinvestigating the same topic or tool at different levels ofthe curriculum. However, such cross-study comparisonswould be difficult without a guiding framework to fa-cilitate making connections across different studies. Oneframework that was created to provide just such a guidingstructure is the ACER framework [6], which focuses onhow students Activate, Construct, Execute, and Reflecton mathematical tools in the context of physics problemsolving [7].One example of student difficulties research that fo-cuses on a mathematical tool that appears multiple timesthroughout the undergraduate curriculum is direct inte-gration. Here, use “direct integration” to refer to theprocess of calculating a physical quantity (e.g., the grav-itational potential) by directly summing the contribu-tions to that quantity from each ‘bit’ of a physical dis-tribution (e.g., a three dimensional mass distribution).We build on the existing research on students difficultieswith direct integration to construct a pseudo-longitudinaland cross-context view of how students’ difficulties evolveover time. To accomplish this, we first review the existingliterature on students difficulties with direct integrationat all levels of the curriculum (Sec. II). We then build onthis work by presenting methods (Sec. III) and findings(Sec. IV) from a pseudo-longitudinal investigation of stu-dent difficulties with direct integration in middle-divisionclassical mechanics (in the context of gravitation) andupper-division electricity and magnetism (in the contextof both the scalar and vector potential). Throughout thispaper, the ACER framework provides a consistent struc-ture that facilitates making comparisons across topicalareas as well as to previous studies.
II. BACKGROUND: STUDENT DIFFICULTIESWITH DIRECT INTEGRATION
Here, we discuss prior work investigating student diffi-culties with direct integration. Throughout this section,the ACER framework will be used facilitate comparisonsof findings across these diverse studies. As such, we beginwith a summary of our own prior work using the ACERframework to investigate students use of direct integra-tion in the context of upper-division electrostatics andthen continue to discuss comparisons with work in othercontexts and at other levels.
A. The ACER framework and an example inupper-division electrostatics
The ACER framework is an analytical framework de-veloped to help structure and organize investigations ofstudents’ difficulties when utilizing sophisticated math-ematical tools in the context of physics problem solv-ing [7]. The framework considers four main componentspresent in this type of mathematically involved problemsolving: activation of the mathematical tool, construc-tion of the mathematical model, execution of the mathe-matics, and reflection on the results. These componentswere identified by studying the general structure of ex-pert problem solving using modified task analysis[8]. Theframework itself is grounded in both a resources [9] andan epistemic framing [10] perspective on the nature ofknowledge and the process of learning. While the broadstructure of the ACER framework was designed to beapplicable across contexts and mathematical tools, theframework was also designed to be operationalized foruse with specific topics and tools. This operationaliza-tion process is important to ensure that the frameworkcaptures the more tool- and context-specific aspects ofstudents’ difficulties. The specific operationalization ofthe ACER framework for the contexts of gravitation andthe vector potential will be discussed in greater detail inSec. III B. For the purposes of cross-study comparisons,we will focus categorizing students’ difficulties within thegeneral components of the framework (i.e., activation, construction, execution, and reflection).One of the investigations which prompted and in-formed the development of the ACER framework wasthat of direct integration in the context of upper-divisionelectrostatics, specifically the calculation of the electricpotential via the integral form of Coulomb’s Law [7].This initial study laid the foundation for the work de-scribed in the remainder of this manuscript and featuresvery similar contexts and methodologies including quan-titative analysis of students’ exam solutions as well asqualitative analysis of interview data. From this analysis,we identified two broad categories of difficulties that thestudy population exhibited with integration in the con-text of Coulomb’s law. The first was difficulty identify-ing direct integration as the appropriate solution method(i.e., difficulty with the activation component). This dif-ficulty was exhibited by roughly a quarter of the studentpopulation [7]. The second was difficulty operationalizingthe Coulomb’s law integral for the specific physical situ-ation given (i.e., difficulty with the construction compo-nent). The two most common difficulties observed werewith correctly expressing the differential charge element, dq , and the difference vector (i.e., Griffith’s “script-r”) ina way consistent with the geometry of the given physicalsituation. These two difficulties were each exhibited byroughly half of the students [7]. Additionally, we foundthat very few students in our population showed explicitand spontaneous attempts to interpret or check their so-lutions (i.e., operating within the reflection component)[7]. B. Interpreting other prior work through theACER framework
In addition to our own prior work investigating studentdifficulties with direct integration in the context of upper-division electrostatics, a number of others have investi-gated student difficulties in introductory courses. Here,we summarize this prior work from the perspective ofthe ACER framework to facilitate cross-study compar-isons. It is important to note that these studies were notdesigned or conducted using the ACER framework, andoften had goals that went beyond pure investigations ofstudents’ difficulties (e.g., theoretical development suchas understanding students’ resource activation or con-ceptual blending); however, for the purposes of this sum-mary, we focus on the aspects of these findings that di-rectly relate to student difficulties with integration as amathematical tool in physics problem solving.Many of the existing studies on student difficulties withintegration were conducted in the context of introductoryphysics courses. For example Nguyen and Rebello [11] ex-amined student difficulties when calculating the electricfield from a straight or curved line segment. They focusedexplicitly on understanding how students recognized theneed for integration and how they set up and then com-puted the integrals [11]. From the point of view of ACER,this framing aligns well with a focus on the activation and construction components of the framework as well as, po-tentially, the execution component. With respect to ac-tivation , Nguyen and Rebello found that students oftenused simple recall of similar problems to activate integra-tion as the correct mathematical tool. In questions wherethey had less experience with similar problems, studentshad more difficulty identifying the need for a integral, andthose who did were most often cued by the presence of anon-constant value in the prompt. With respect to con-struction , the two primary difficulties focused around theinfinitesimal quantity and accounting for the vector na-ture of the electric field. Nguyen and Rebello noted thatmany students struggled to recognize the need for, andphysical meaning of, the infinitesimal quantity as well asto express it in a useful way (e.g., expressing dq as λdx ).Finally, with respect to execution , Nguyen and Rebellodocumented some difficulties related to maintaining anawareness of the physical meaning of the symbols whileperforming the integrals. They also noted some difficul-ties with respect to the actual mechanics of computingthe integrals (e.g., u-substitutions).Meredith and Marrongelle [12] also conducted an in-terviews with students solving several problems requir-ing integration in the context of introductory electrostat-ics, including one asking for the electric field near a barof charge. The focus of this work was on identifyingstudents’ mathematical resources when solving integrals;however, they also documented a number of difficultiesthat their interview students had working through thesetypes of questions. In terms of activation , they foundthat recognizing the need for an integral in the case ofthe bar of charge was a challenge for as many as half oftheir students, with many of these student instead treat-ing the bar as a point charge at the rod’s center. Forstudents who successfully identified the need for integra-tion, Meredith and Marrongelle identified three possiblecues commonly used: recall of prior examples, identifica-tion of a term or variable on which the quantity in ques-tion depended, or the conceptualization of a building up awhole by summing up smaller constituent parts. In termsof construction , Meredith and Marrongelle also observeddifficulties with the infinitesimal, finding that studentssometimes incorrectly used the idea of dependence on aparticular variable to simply assume that would be thevariable of integration without consideration of the phys-ical meaning of the integrand.Both of the two studies discussed above identified dif-ficulties with the infinitesimal/differential term in thelarger context of solving a physics problem involving in-tegration, an element that appears in the construction component of the ACER framework. Others have tar-geted this issue more directly. For example, Hu and Re-bello [13] conducted an interview study with introduc-tory physics students in which they focused specificallyon students applications of differentials when calculatingthe electric field from a charged bar. They identified dif-ferent mathematical resources and conceptual metaphors that students used when considering the need for, andphysical meaning of, the differential term in these inte-grals. Similarly, Amos and Heckler [14] investigated therelationship between introductory students’ understand-ing of differentials (e.g., dx ), differential products (e.g., λdx ), and integrals. They found that understanding ofdifferentials alone did not relate strongly to performanceon integral problems, but rather required an additionalstep of explicitly incorporating ideas about differentialproducts.Others, rather than focusing on the construction com-ponent, focused instead on the activation componentthrough investigations of how students identify integra-tion as the correct mathematical tool for a particularphysics problem. To investigate this, Savelsbergh et. al [15] used problem sorting tasks with introductory physicsstudents which asked them to determine the correct ap-proach to various problems relating to electricity andmagnetism; several of these tasks included problems forwhich direct integration via Coulomb’s law or the Biot-Savart Law were the correct approach. They found that,despite knowing the possible solution types for theseproblems, students had difficulty mapping the problemat hand onto one of the know problem types. In terms ofthe ACER framework, these results would indicate thateven if a student can productively recognize or replicatework in the construction or execution components, theymay still struggle to identify the correct mathematicaltool in the activation component.Taken together, the studies described above tell a rel-atively coherent story with respect to the difficulties stu-dents at the introductory level face with multi-variableintegration in the context of, for example, calculating theelectric field from a charge distribution. These difficultiesare focused on the activation and construction compo-nents of the ACER framework and emphasize difficultiesidentifying integration as necessary and manipulating thedifferential term. Both of these difficulties also appearin our own prior work investigating student difficultieswith these same types of calculations in the context ofupper-division electrostatics, suggesting that these diffi-culties are relatively persistent across several exposuresto the use of multi-variable integration in physics prob-lems. However, as the majority of the studies at theintroductory level are based primarily on student inter-view data, conclusions about potential changes in the fre-quency of these difficulties between the introductory andupper-division population are not possible. Missing fromthe majority of these prior studies is explicit attentionto students’ reflections , which is also consistent with ourfinding that students rarely exhibit reflective behaviorsspontaneously.The current study builds on this prior work by con-tributing data on students difficulties at an additionaltwo points within the undergraduate curriculum: middle-division classical mechanics (in the context of gravita-tion) and upper-division magnetostatics (in the contextof the vector potential). Additionally, we will leverageboth quantitative exam data and qualitative interviewdata, along with comparison to our prior work, to makestatements about how the frequency of these difficultiesshifts (or not) over the course of these multiple exposures. III. METHODSA. Context
Data for this study were collected from two courses atthe University of Colorado Boulder (CU) over the courseof three distinct semesters. Both courses in this studyare the first in a two semester sequence - one target-ing a combination of classical mechanics and math meth-ods (Class. Mech.), and the other targeting electricityand magnetism (E&M). Class. Mech., covers chapters 1-5 in Taylor’s text [16] as well as various chapters in Boas[17], and students are typical sophomores and juniors.E&M, covers the first 6 chapters of Griffths’ text [18](i.e., electrostatics and magnetostatics), and students inthis course are juniors and seniors. For both courses,most students are physics, astrophysics, and engineeringphysics majors. During the semesters of data collection,both courses were taught with varying degrees of inter-activity through the use of research-based teaching prac-tices including peer instruction using clickers [19] andtutorials [20].We collected data from two primary sources forthis investigation: student solutions to instructor-designed questions on traditional midterm exams, andgroup think-aloud, problem-solving interviews. Our ap-proached mirrors that of our earlier investigations of stu-dents’ difficulties with direct integration described in theprior section (Sec. II A). In this approach, exams pro-vided quantitative data identifying common difficultiesand interviews offered deeper insight into the nature ofthose difficulties. At CU, Class. Mech. is a pre-requisitefor E&M, and, thus, students’ responses in these twocourses during the same semester represent a pseudo-longitudinal view of the evolution of students’ reason-ing over the course of several semesters. These data arepseudo-longitudinal, rather than truly longitudinal, be-cause rather than tracking individual students and com-paring their performance at different points in time, weare comparing the aggregate performance of two differentsets of students at different points in the curriculum. Thepseudo-longitudinal nature of our data has implicationsfor the interpretation of this data that will be discussedin more detail in the following section (Sec. IV).Midterm exam data were collected from one semesterin Class. Mech. ( N = 77 students and 1 instructor) andtwo semesters in E&M ( N = 163 distinct students and3 instructors). Of the four total instructors involved inthese courses, two were traditional research faculty andtwo were physics education researchers. One semester ofthe E&M course was co-taught by one PER faculty mem-ber and one of traditional research faculty with shared ex- ams, activities, and homework. The exam questions usedfor the study were developed via collaboration betweenthe traditional research faculty and the PER faculty. Theexam questions were all variations of calculating the po-tential from a short line or strip segment (see Fig. 1). Forexample, in Class. Mech. the students were asked for thegravitational potential energy from a rod of linear massdensity. The E&M students were asked two questions:one asking for the scalar potential from a short chargedrod, and one asking for the vector potential from a shortline carrying constant current current or a short stripcarrying uniform surface current density. The most ef-ficient solution approach in each case is to calculate therequested potential directly using Eqns. 1, 2, or 3. U ( ~r ) = − GM Z V dm | ~r − ~r ′ | (1) V ( ~r ) = 14 πǫ Z V dq | ~r − ~r ′ | (2) ~A ( ~r ) = µ o π Z V ~J | ~r − ~r ′ | dV ′ (3)Here G , ǫ o , and µ o are fundamental constants, M is themass of the of the test object, ~r is the vector from theorigin to the field point, and ~r ′ is the vector from theorigin to the source point.While all exam questions in the study were some vari-ation on calculating the potential from a short rod orstrip, there were several important differences betweenthe prompts used in each case. All questions provideda set of Cartesian axes in the figure; however, the ori-entation of the distribution (i.e., what axis it lay along)varied between prompts. More significantly, one versionof the two vector potential questions provided studentswith the correct mathematical tool to use (i.e., Eqn. 3),whereas in all other exam questions students had to de-cide for themselves which tool to use. From the point ofview of ACER, this means that this specific vector po-tential prompt bypassed the activation component. Ad-ditionally, only the two scalar potential prompts resultedin integral expressions that could actually be calculatedby hand; in the other cases, students were either askedonly to set up the integral or were provided the result ofthe integral. With respect to ACER, this means that thequestions targeting gravitation and the vector potentialall effectively bypassed the execution component.To address the lack of explicit reflection and other lim-itations with the exam data, we also conducted think-aloud interviews with pairs of students. Four interviewswere conducted with students ( N = 8) who had previ-ously completed or were currently enrolled in the Class.Mech. course, and three interviews were conducted withstudents ( N = 6) who had previously completed, or werecurrently enrolled in, the E&M 1 course. In these in-terviews, students were asked to respond to a slightly A thin rod of length L lies along the x-axis, withits center at the origin, as shown in the figure. Thelinear mass density of the rod λ is a constant. Amass m is a vertical distance h above the center ofthe rod.Calculate the gravitational potential energy of themass m due to the mass of the rod. (a) Consider a flat “ribbon” of current of width a andlength b , flowing in the x - z plane as shown. This isa surface current in the y = 0 plane with K = ( K o ˆ z where < x < a where x < or x > a Find an expression for the magnetic vectorpotential, A ( x o , , (b) FIG. 1. Examples of the exam questions used in the study. Variations on (a) included asking instead for the scalar potentialfrom a short rod of charge or for the vector potential from a short line segment of current. more challenging version of the exam questions, whichasked them to calculate the potential along an axis otherthan the central axis from a ring of uniform density (seeFig. 2). The increase in difficulty in the interviews wasto ensure the question was sufficiently challenging for apair of students working together. The paired interviewstructure was adopted to create a more authentic inter-view environment in which interview participants wouldnaturally encourage each other to express their reasoningwithout prompting from the interviewer. In both inter-view sets, students were not told which mathematicaltool to use; however, a correct solution to the problemwill produce an integral that is not easily evaluated byhand. As such, interviewees who reached this integralwere asked to stop at that point. From the perspectiveof the ACER framework, these interview questions tar-geted activation , construction , and potentially reflection ,but did not capture the majority of the execution com-ponent. This design was based on findings from our priorwork suggesting that pure execution errors are rarely theprimary difficulties encountered by students [7].The exams solutions contained little evidence of spon-taneous reflection (see Sec. IV D). In one semester, theexam question asking for the scalar potential include twofollow up questions asking students to discuss the limit-ing behavior of the potential in the large- r limit and toconfirm that their solution exhibited this behavior. Theinterviews also included an additional question designedto examine students’ ability to come up with meaningfulreflections when prompted to do so. Five of the sevenpairs (those with interview time remaining after finish-ing the first question) completed this second question tar- geted directly at reflection . Students were asked to assessthe plausibility of three expressions for either gravita-tional potential energy or vector potential caused by anunspecified, but localized, mass or current distribution.Valid approaches included (but are not limited to) check-ing units or limits, but neither approach was suggestedexplicitly. It can sometimes be useful to model electrons inorbitals around an atom as small rings of current. Inthe figure below, we have provided a diagram of a ringcarrying current I in the counter-clockwise direction asviewed from above. Calculate the vector potentialalong the x-axis from this ring of current.FIG. 2. The vector potential version of the interview ques-tion. For the Class. Mech. students, the distribution was de-scribed as a mass distribution with uniform λ (and the arrowindicating direction of current was removed), and studentswere asked to calculate the gravitational potential energy fora small mass placed on the x -axis. B. The operationalized framework
The process of operationalizing ACER is presented indetail in Ref. [7]. Briefly, in order to operationalize theframework, a content expert utilizes a modified form oftask analysis [7, 8] in which they work through the prob-lems of interest while carefully documenting their stepsand mapping these steps onto the general components ofthe framework. Additional content experts then reviewand refine the resulting outline until consensus is reachedthat the key elements of the problem have been accountedfor. This expert-guided scheme then serves as a prelim-inary coding structure for analysis of student work. Ifnecessary, the operationalization can be further refinedto accommodate aspects of student problem-solving thatwere not captured by the expert task analysis.In prior work, we operationalized the ACER frame-work for the use of multi-variable integration in the con-text of calculating the electric potential from a contin-uous charge distribution [7]. Here, we modify this op-erationalization such that it is appropriate for the useof multi-variable integration to calculate potentials in awider range of contexts including gravitation and magne-tostatics. The elements of the modified operationalizedACER framework are detailed below. Element codes arefor labeling purposes only and are not meant to suggest aparticular order, nor are all elements always necessary forevery problem. In particular, the elements of Construc-tion and Execution are unlikely to occur in the specificorder listed as experts can, and often do, iterate backand forth between setting up and evaluating expressions.
Activation of the tool : The first component of theframework involves the selection of a solution method.The modified task analysis identified four elements thatare involved in the activation of resources identifyingmulti-variable integration as the appropriate tool. A1 : The problem asks for a vector field or the associ-ated potential. A2 : The problem provides an expression for, or de-scription of, the distribution that is the source ofthe field or potential. A3 : The provided distribution does not have appro-priate symmetry to utilize mathematically simplerapproaches (e.g., variations of Gauss’ Law). A4 : Direct calculation of the potential is more efficientthan direct calculation of the associated field.Elements A1–A3 are cues typically present in theproblem statement. Element A4 is specific to problemsasking for the potential and is included to account forthe possibility of solving for potential by first calculatingthe associated field. This method is valid but often moredifficult. Construction of the model : Here, mathematicalresources are used to map the specific physical situationonto the general integral expression. The integral expres-sion produced at the end of the construction componentshould be in a form that could, in principle, be solved with no knowledge of the physics of this specific problem.We identify four key elements that must be completed inthis mapping. C1 : Use the geometry of the distribution to select acoordinate system. C2 : Express the differential element (e.g., dm or ~JdV ′ )in the selected coordinates. C3 : Select integration limits consistent with the dif-ferential charge element and the extent of thephysical system. C4 : Express the difference vector, | ~r − ~r ′ | , in theselected coordinates.Elements C2 and C4 can be accomplished in multipleways, often involving several smaller steps. In order toexpress the differential element, the student must com-bine the charge density and differential to produce anexpression with dimensions consistent with the physicalquantity being summed over (e.g., ~JdV ). Construction ofthe difference vector often includes a diagram that iden-tifies both the source, ~r ′ , and field point, ~r , vectors. Execution of the Mathematics : This componentof the framework deals with the mathematics required tocompute a final expression. In order to produce a formuladescribing the potential or field, it is necessary to: E1 : Maintain an awareness of which variables are beingintegrated over (e.g., r ′ vs. r ). E2 : Execute (potentially multi-variable) integrals inthe selected coordinate system. E3 : Manipulate the resulting algebraic expressions intoa form that can be readily interpreted. Reflection on the result : The final component ofthe framework involves verifying that the expression isconsistent with expectations. While many different tech-niques can be used to reflect on the result, these twochecks are particularly common for problems involvingmulti-variable integration: R1 : Check the units of intermediate and final expres-sions. R2 : Check the limiting behavior to ensure it is con-sistent with the nature and geometry of thedistribution.Element R2 is especially useful when the student al-ready has some intuition for how the potential or fieldshould behave in particular limits. However, if they donot come in with this intuition, reflection on the resultsof this type of problem is a vital part of developing it.In the next section, we will apply this operationaliza-tion of ACER to investigate and compare students’ rea-soning and prevalent difficulties when solving the physicsproblems described in the previous section (Sec. III A)involving integration in the context of gravitation, elec-trostatics, and magnetostatics. IV. RESULTS
Here we present results from our investigations of stu-dents’ use of direct integration. Since our goal is to com-pare changes in students’ reasoning across multiple pointsin the curriculum, we organize our results here by compo-nent of the ACER framework and present data from thecontext of gravitation, electrostatics, and magnetostaticstogether. Throughout this section, we also compare toour prior work investigating students’ use of integrationin the context of electrostatics to provide an additionaldata point to understand differences in students’ reason-ing across multiple contexts. This earlier work was de-scribed in Sec. II A and also involved analysis of students’exam solutions and interview responses to questions sim-ilar to those considered here for the scalar potential, butinvolving calculating the electric potential from a line,surface, and volume charge distributions.
A. Activation of the tool
One of the two vector potential exam question in theE&M course provided students with Eqn. 3, effectivelybypassing the activation component of the problem-solving process. Here, we will focus on data from theremaining courses, representing N = 325 exam solutions.Evidence for the individual elements of activation withinthe exam solutions can be difficult to identify becausestudent often do not articulate their thought process onthe exams. In particular, there was rarely explicit ev-idence that students attended to elements A1 (i.e., theprompt asked for the potential) or A2 (i.e., the promptprovided a distribution). These elements, while a criticalpart of fully justifying direct integration as the appro-priate tool, are often a tacit step in the problem-solvingprocess.Elements A3 (i.e., recognizing the distribution does nothave sufficient symmetry) and A4 (i.e., recognizing di-rect calculation of the potential is most efficient) weremost easily identified when students did not approachthe problem using the appropriate mathematical tool (seeTable I for a summary). For example, students who at-tempt to calculate the potential by first calculating theassociated field or force by treating the distribution ashighly symmetric (e.g., treating it as a point source orusing Gauss’s Law) have demonstrated a difficulty withelement A3. This difficulty was manifested by roughly atenth of the solutions in the contexts of both gravitation(7%, N = 5 of 77) and the scalar potential (14%, N = 23of 160). The increase in the fraction of students exhibit-ing difficulties with element A3 between Class. Mech. andE&M was driven in part by students’ using Gauss’s Lawto calculate the electric field. Difficulty with element A3was observed in none of the N = 50 vector potential solu-tions, possibly because no “point-source” approximationexists for the vector potential due to the lack of mag-netic monopoles. However, we also observed no student attempting to approach the rod of current question usingAmpere’s Law.Another observed difficulty with activation was at-tempting to calculate the potential by first calculatingthe associated field or force via direct integration andthen taking the line integral or, in the case of gravita-tion, using U = mgh . The former approach is valid in thecase of gravitation and the scalar potential, though sig-nificantly more difficult, and represents a difficulty withelement A4. The latter approach represents a difficultywith both A4 and A3, where the student has taken aharder route by opting to calculate the gravitational field ~g , and also over generalized the equation U = mgh , whichis only valid near the surface of the earth. Students withdifficulties with element A4 represented just over 10% ofstudents in the context of both gravitation (12%, N = 9of 77) and the scalar potential (14%, N = 23 of 160).In the case of the vector potential, the approach of firstcalculating the magnetic field is not a valid approach asthe relation between the vector potential and magneticfield is more complex (i.e., ~B = ∇ × ~A ), and only oneof the N = 50 vector potential solutions attempted thisapproach.Overall, roughly three-quarters of students (80%, N =62 of 77 in gravitation; 70%, N = 113 of 160 in electro-statics) correctly activated direct integration via Eqns. 1-2 as the correct approach to the exam questions (Fig. 1).These results are consistent with what was found in ourearlier investigation in electrostatics where just underthree quarters (73% [7]) of students correctly used Eqn. 2to calculate the electric potential in problems involvingvarious charge distributions with the most common al-ternative approach being to attempt the problem by firstcalculating the electric field via either Coulomb’s law orGauss’s law. This suggests that, while Eqn. 2 is at leastthe second time in the curriculum that students have en-countered the direct integration approach, they are notshowing a significant increase in success rates for activa-tion of this tool, and potentially showing some increaseddifficulties. In both of these cases, students typically en-ter the course already familiar with the potential or vec-tor field in question in the context of simple and sym-metric distributions. Class. Mech. students have seenthe gravitational field from a point mass or near the sur-face of the earth (where U = mgh is a valid expression),and E&M students have calculated the electric field fromhighly symmetric charge distributions using Gauss’s Law.Together these results suggest in all contexts, the chal-lenge for students with respect to activation may be elim-inating inappropriate but simpler and/or more familiarmethods as viable solution paths. This difficulty appearsin 20-30% of solutions in both contexts where such sim-pler/more familiar methods exists, though the pseudo-longitudinal nature of our data make it impossible todetermine if this is because the students who struggle inthe context of gravitation continue to struggle later inelectrostatics, or if some students who did not encounterthis difficulty in gravitation later encounter it in electro- TABLE I. Difficulties activating direct integration via Eqns. 1-3 as the appropriate mathematical tool for exam questionsfeaturing a rod or strip distribution (e.g., Fig. 1). Percentages are given with respect to the subset of students with difficultieswith activation (given in parentheses). Codes are not exhaustive or exclusive (so percents may not total to 100%) but representthe most common themes.
Gravitation Electrostatics MagnetostaticsElement Difficulty N % N % N %
A3 Point source or Gauss 5 33% (of N = 15) 23 49% (of N = 47) 0 0% (of N = 1)A4 Calculate of force or field 9 60% (of N = 15) 23 49% (of N = 47) 1 100% (of N = 1) statics and vice versa .The interviews provide additional insight into the is-sue of activation . Neither the gravitation nor the vectorpotential interview questions prompted students to uti-lize Eqns. 1 or 3, and difficulties with elements A3 andA4 were observed in both the sets of interviews. Oneof the four pairs of gravitation interviewees treated thegiven ring (see Fig. 2) as a point mass, representing afailure in A3. One student argued that any rigid bodycould be treated as a point at its center of mass; theother was less confident in this approach, but said thatthey did not know how to account for the details of thisparticular mass distribution. When prompted to considerapproaching the problem via direct integration, both stu-dents expressed concerns about the potential for “horri-ble algebra.”These latter two sentiments also arose in the vector po-tential interviews wherein two of three pairs attemptedto calculate the magnetic field via Ampere’s law, plan-ning to use the result to “solve” for the vector potentialby inverting the relationship ~ ∇ × ~A = ~B (i.e., a difficultywith element A3). This approach is ultimately not pos-sible, and both pairs struggled to find a useful Amperianloop. When directly prompted by the interviewer to con-sider Eqn. 3, both pairs struggled to use it (discussed inSec. IV B), and subsequently returned to the Ampere’slaw strategy despite their earlier trouble. This difficulty,while not observed in the exam solutions, is consistentwith our hypothesis that difficulties with activation arelinked primarily to eliminating simpler or more familiarapproaches. The interviews provide additional detail tothis interpretation by suggesting that both perceived andactual difficulties with the direct integration approachmay impede activation of this tool and make studentsless likely to eliminate what they perceive as easier ap-proaches even if they doubt their applicability.The choice to calculate the magnetic field via Ampere’slaw reflects difficulties with element A3. None of the vec-tor potential interview students attempted to calculatethe magnetic field via direct integration as a step towardfinding the vector potential (i.e., element A4), thoughone pair spontaneously considered it, but preferred notto attempt it. Alternatively, in the gravitation context,difficulties with element A4 were observed independentof difficulty with A3; two of the four pairs of gravitationinterviewees calculated the gravitational force instead of potential energy. This includes the pair who first an-swered with the point mass expression for potential en-ergy, but, when prompted by the interviewer to attemptintegration, switched to calculating the force. However,only one of these two pairs could name the correct rela-tionship between force and potential energy that wouldallow them to answer the question fully. B. Construction
In all three contexts, the largest number of studentdifficulties were observed in the construction componentwhere students are operationalizing the general integralforms in Eqns. 1-3 for the specific physical situation givenin the problem. Overall, N = 72 (out of 77) of the grav-itation solutions, N = 130 (out of 160) of the scalar po-tential solutions, and N = 135 (out of 138) of the vectorpotential solutions included at least one element of con-struction in their solution (i.e., they used direct integra-tion and did more than just write an equation or finalanswer). With respect to element C1 (i.e., selecting anappropriate coordinate system), all exam prompts pro-vided a figure which included a pre-selected origin and la-beled Cartesian axes (effectively bypassing this element),and only 3 students in the entire data set utilized any-thing other than Cartesian variables. Similarly, all butone pair of interviewees appropriately chose polar coor-dinates, with the final pair opting for Cartesian coordi-nates. One E&M interviewee did however express con-cerns about the use of curvilinear coordinates, worriedthat the inseparability of Poisson’s equation for ~A in thissystem may also invalidate Eqn. 3.Element C2 proved more challenging for students withroughly a fifth of the gravitation solutions (17%, N = 12of 72), just over a tenth of the scalar potential solutions(14%, N = 18 of 130), and a third of the vector potentialsolutions (30%, N = 41 of 135) providing responses re-flecting difficulties expressing the differential element (seeTable II). For the questions considered here, these diffi-culties could manifest in two primary ways: difficultiesexpressing the density (e.g., λ , ~J ) or difficulties express-ing the differential line or area element ( dl or dA ).For the students in the context of gravitation, diffi-culty expressing the differential line element was mostcommon (83%, N = 10 of 12) and included express- TABLE II. Difficulties expressing the differential element (i.e., dm , dq , or JdV ′ ). Percentages are given with respect to thesubset of students with difficulties with element C2 (given in parentheses). Codes are not exhaustive or exclusive (so percentsmay not total to 100%) but represent the most common themes. Gravitation Electrostatics MagnetostaticsDifficulty N % N % N %
Incorrect differential line or area element 10 83% (of N = 12) 11 61% (of N = 18) 21 51% (of N = 41)e.g., dl = dx rather than dx ′ or dl = da Incorrect expression for density 5 42% (of N = 12) 8 44% (of N = 18) 27 66% (of N = 41)e.g., λ = M or I = J ing dl as a function of the field variables (i.e., r vs. r ′ , N = 5) and expressing the differential element as an areaor volume element ( N = 2). Similar difficulties with thedifferential line element were observed in the E&M stu-dents’ solutions, though represented a smaller fractionof the difficulties with element C2 (61%, N = 11 of 18in electrostatics; 51%, N = 21 of 41 in magnetostatics).Moreover, focusing only on the exam questions featuringline distributions, the overall fraction of students exibit-ing difficulties with the differential line element decreasesfrom 14% ( N = 10 of 72) to 9% ( N = 20 of 218) fromClass. Mech. to E&M. This downward trend across con-secutive exposures suggests that difficulty with elementC2 may be one that fades over time as students gain moreexperience.None of the interviewed Class. Mech. pairs had trou-ble writing the differential line element in polar coordi-nate. It is possible that some of these students may havehad more difficulty on their own — one, for example,first suggested dl = Rdθdφ — but as pairs, the studentswere able to correct each others mistakes. The E&Minterviewees were equally successful, with the exceptionof the pair working in Cartesian coordinates, who wrote dl = p dx + dy . However, this pair did generate thecorrect form once prompted to switch to polar coordi-nates.Difficulties expressing the linear mass density λ wereless common (42%, N = 5 of 12) for students the contextof gravitation and limited almost exclusively ( N = 4)to incorrectly substituting the total mass of the rod for λ . This difficulty was observed with similar frequency inthe context of the scalar potential (44%, N = 8 of 18).At first glance, this trend would also seem to suggestthat students difficulties around the density stay rela-tively constant over time; however, this pattern does notcontinue in the context of magnetostatics. One versionof the vector potential exam question provided studentswith the expression for the surface current density ~K (seeFig. 1(b)), effectively bypassing the need for students toexpress this quantity. Alternatively, the other version ofthe question required students to express (or translate)the volume current density ~J for the line of current ontheir own. In this case, difficulties around the currentdensity represented the majority of the difficulties with element C2 (90%, N = 27 of 30 solutions exhibiting diffi-culties with C2). This difficulty most often (63%, N = 17of 27) manifested as students simply leaving J in theirfinal expression without transitioning to the current I carried by the wire segment (i.e., incorrectly replacing JdV ′ → Jdz ′ ) and resulting in a final expression withthe wrong units.Consistent with the results of the exam coding, noissues with the linear mass density arose in the Class.Mech. interviews, but all three E&M pairs experiencedsome degree of difficulty with the current density. Twopairs did, after unguided discussion, generate a valid ex-pression for the magnitude of ~J using delta-functions, butboth omitted the direction (although one pair caught thiserror much later on). Both pairs used the units of cur-rent density to guide their choice of delta-functions. Thethird pair also referenced units, but argued that the unitsof ~J were current per volume. They were unable to findan expression for the current density and were eventuallyprovided with the one-dimensional form of Eqn. 3.One interpretation of this sudden increase in the fre-quency of a difficulty that otherwise seemed to be holdingsteady as students progressed may be that it is, in fact,a totally new difficulty. The volume current density, ~J ,is a conceptually and mathematically more challengingquantity than either mass or charge density. Moreover,while the quantity ρdV ′ (where ρ is volume mass den-sity) has a clear physical interpretation as the total masson each differential chunk of the object (i.e., dm ), thequantity ~JdV ′ has no such clear physical interpretation.One exchange between E&M interviewees discussing themeaning of the integral in Eqn. 3 demonstrates this diffi-culty. First, they associated “the current” with the entirering of current, making the choice of a specific ~r ′ whichrepresents the entire circle ambiguous. One student thenhad the insight that ~r ′ points to “a current density ata given location along the loop,” and used the phrase“for each J” to describe this. Their partner objected tothis language as there is only one function ~J , and asks ifthey meant “differential J .” While over the course of thisexchange the pair made progress towards understandingEqn. 3, their interpretation of the differential element wasstill dimensionally incorrect (they had not accounted for0the volume element). Moreover, their difficulties were ex-acerbated by the lack of any common terminology withwhich to refer to the quantity being summed over.After constructing an expression for the differential el-ement, students must select limits of integration that areconsistent with this element and the physical extent ofthe system (element C3). Difficulties with the limits ap-peared in less than a fifth of the gravitation solutions(19%, N = 14 of 72), roughly a quarter of the scalarpotential solutions (28%, N = 36 of 130), and roughlya tenth of the vector potential solutions (12%, N = 16of 135). In all cases, the two most common issues werenot integrating over the actual extent of the rod ( N = 11of 14 in gravitation, N = 26 of 36 in electrostatics, and N = 7 of 16 in magnetostatics), and not including limitsat all ( N = 1 of 14 in gravitation, N = 8 of 36 in electro-statics, and N = 9 of 16 in magnetostatics). Combinedwith findings from our previous work in the context ofelectrostatics showing difficulties with limits in only 14%of solutions [7], these data suggest that difficulties withelement C3 may fluctuate somewhat over students firstexposures to direct integration in these contexts, but staypresent to at least some extent even after multiple expo-sures. In the interviews, none of the six pairs who madeit to this point in the interview struggled to choose limitsof 0 to 2 π for the ring. This includes the pair workingin Cartesian coordinates who still articulated this choice,suggesting that for some the limits may have been an au-tomatic response to the circular geometry rather than aconsidered decision.The fourth element in construction (C4) deals with thedifference vector ( ~r − ~r ′ ) which points from the sourcepoint to the field point. In our prior investigation, nearlyhalf of students encountered difficulties when attemptingto express this vector [7] for a disk-shaped charge distri-bution. The distributions provided in this study resultsin a simpler expression for difference vector, and, unsur-prisingly, students had somewhat less difficulty produc-ing a correct expression with 30-45% of students unableto do so (30%, N = 22 of 72 in gravitation; 45%, N = 45of 130 in electrostatics; 35%, N = 47 of 135 in magneto-statics). However, the fact that the fraction of studentsexhibiting difficulties with the difference vector remainshigh, and perhaps even increases slightly, suggests thatthis difficulty is persistent across multiple exposures.Lending detail to the results from the exams, two typesof difficulties related to the difference vector arose in theinterviews. In two of the four Class. Mech. interviews, thestudents generated incorrect expressions for the magni-tude of the difference vector. When prompted to considermore carefully or to draw the relevant vectors, however,both pairs came to a valid expression. These students ap-peared to understand the meaning of the difference vectorterm in the integrand (i.e., the distance from source totest), but had made geometric errors in calculating itsmagnitude. Alternatively, some of the E&M pairs strug-gled with the meaning of the ~r − ~r ′ term. This difficultyappeared to be related to Griffiths’ “script- r ” notation (script- r = ~r − ~r ′ ) [18]). A total of four pairs of stu-dents used this notation either spontaneously or whenprompted. At least one student in each pair expresseddoubts about the relationship between script- r , ~r , and ~r ′ or the meaning of each vector (e.g., switching the primedand un-primed variables). Two of these pairs had suf-ficient difficulty remembering or generating these ideasthat they never attempted to write an expression for thedifference vector or its magnitude. While the conventionof using script- r , ~r , and ~r ′ permits concise formulas, itrequires students to parse or, in the case of the intervie-wees in this study, simply try to remember the meaningof these symbols before they can proceed to analyzingthe geometry of the problem at hand. This finding isconsistent with our previous findings in the context ofelectrostatics [7]. C. Execution
The Execution component of the framework deals withthe procedural aspects of working with mathematicaltools in physics. Overall, N = 66 of the gravitation solu-tions, N = 128 of the electrostatics solutions, and N = 77of the magnetostatics solutions included at least one ele-ment of execution . On one of the vector potential examquestions, students were only asked to set up the inte-gral and thus did not include execution . In the remainingexam questions, it is not possible to know for certain froman exam solution whether students keep track of whichvariables are being integrated over (i.e., source vs. fieldvariables - element E1); however, we can identify caseswhere students explicitly integrated over the wrong vari-ables (i.e., integrating over the field variables). Roughlya tenth of the students in both courses ( N = 5 of 66 ingravitation, N = 20 of 128 in electrostatics, and N = 7 of77 in magnetostatics) explicitly integrated over variablesin their expression which represented the field point.Only the scalar potential questions resulted in inte-grals that could be reasonably calculated by hand, andjust over a third (37%, N = 47 of 128) of solutions ex-hibited one or more mistakes in doing so. The otherquestions resulted in integrals for which the general solu-tion was provided to the students (effectively bypassingelement E2). However, some students still attempted toperform integrals, typically because a mistake in an ear-lier step of the solution resulted in a solvable integral. Ofthe N = 23 Class. Mech. students who attempted to per-form integrals just under half (43%, N = 10 of 23) madesignificant errors in the process. This fraction droppedto only 17% ( N = 3 of 18) in the context of the vectorpotential students who attempted to perform an integral.Note that these students had, by necessity, already madeone or more errors in the activation and construction components of the framework before reaching execution .Given the high pressure and individual nature of bothexams, we expected that many students would makemathematical errors particularly with element E3 (al-1gebraic manipulation). To account for this, we distin-guish in our analysis between small math errors (e.g.,dropping a factor of 2 or minus sign) and more funda-mental mathematical errors (e.g., dropping the bottomlimit or executing integrals incorrectly). In the contextof the scalar potential, just over half of the students whomade mathematical errors made more fundamental errors(60%, N = 28 of 47); however, for roughly three quartersof these students (71%, N = 20 of 28), the mistake madein the execution stage was preceded by a significant diffi-culty in the activation or construction components. Thistrend is consistent with the findings of our prior workthat suggested difficulty performing integrals was rarelythe primary barrier to students’ success when using inte-grals in the context of physics problems [7].In interviews, students were not asked to evaluate theintegral expression they derived in the interviews, al-though one pair did switch from integrating over thesource variables to over the field variables when consid-ering how they might do so. Otherwise the interviewsprovided little insight into the execution component. D. Reflection
The reflection component deals with the process ofchecking and/or interpreting the final expression. It isoften the case that mistakes in the construction or exe-cution components resulted in an expression for the po-tential that had the wrong units or did not have the cor-rect behavior in particular limits (i.e., elements R1 andR2, respectively). Overall, we found that very few of ourstudents ( N = 20 of the 329 students in any of the threecontexts to complete the question) made explicit, spon-taneous attempts to reflect on their solution using eitherof these checks on exams. This number should be inter-preted as a lower bound on the frequency of spontaneousreflections, as it is possible that more of the exam stu-dents made one of these checks and simply did not writeit down explicitly on their exam solution; however, theinterview results suggest this is less likely. Although allthree E&M interview pairs considered units when writ-ing ~J , and one pair checked extreme cases for their differ-ence vector expression, none of the interviewees sponta-neously reflected on their final integral expression. Someexplained that they used these tools only when partic-ularly worried about a result, and only if they felt theyhad time.Another strategy for understanding reflection involveslooking at the number of solutions where the final expres-sion included an error that would have been detected byone or more of these checks. Table III lists this along withthe number of solutions that explicitly included each re-flective check. Overall, these results suggest that an ex-plicit check of units would likely be the most effective re-flective practice for students in terms of detecting errors,but that our students are rarely executing this (or other)checks spontaneously. This finding is consistent with our TABLE III. Number of exam students who explicitly utilizedeach of the two common reflective checks ( N explicit ) alongwith the number of solutions that included an error thatwould have been detected by this check ( N incorrect ). Datafrom all three contexts have been combined here due to low N explicit . The total number of students who progressed farenough in their solution to utilize one of these checks was N = 329. Reflective check N incorrect N explicit Units (R1) 78 3Limits (R2) 62 10 prior investigation [7] and implies that students do notgrow more likely to spontaneously execute these kinds ofreflective behaviors as they progress further through thecurriculum.We also investigated whether students can performthese reflective checks when asked to do so. In one of thetwo scalar potential exam questions, two followup ques-tions asked students to discuss how the potential shouldbehave as you got far from the charge distribution andto confirm that their expression was consistent with thisexpectation. Most students (72%, N = 47 of 65) cor-rectly articulated that the potential should fall off as 1 /r in the limit of large- r . The most common alternative an-swer was simply that the potential “goes to zero” withno discussion of how it goes to zero (17%, N = 11 of65). When asked to show that their answer matched thisbehavior, nearly half (44%, N = 24) of the 55 studentswho responded executed an appropriate Taylor expan-sion of their prior solution, and a further quarter (25%, N = 14) had made prior mistakes in their solution suchthat their solution already had a pure 1 /r dependence.The remaining quarter of students (24%, N = 13) simplyplugged in r = ∞ into their expression and showed thattheir solution went to zero for large- r .The interviews also provided insight into studentsprompted reflection. Five of the seven interviews in-cluded a second question targeted at reflection . All fivepairs suggested and were able to carry out a units check,although only one pair did so without first expandingthe relevant constant ( G or µ ) in fundamental units.All five also checked that the expressions vanished in thelimit r → ∞ . On the other hand, three pairs expectedthe expressions to blow up in the r → r functional form of this answer, and only whenprompted to do so.Interviewees were also asked to reflect on their finalanswers for the potential from the ring given in the firstinterview question. Their strategies here matched thoseemployed in the second question, but it is worth notingthat two students expressed uncertainty as to the effectof the integral on units or limits. Combined with theexam results, this suggests that students are capable of2checking units and limiting values (e.g., that a functiongoes to zero) but are both less inclined towards, and havemore difficulty, checking limiting forms (e.g., how thepotential goes to zero). V. SUMMARY AND IMPLICATIONS
Here, we build on prior work investigating students’use of direct integration as a mathematical tool in physicsproblem solving. We extend this prior work, which fo-cused on students difficulties in the context of junior-levelelectrostatics, by investigating students use of integra-tion in two additional points in the undergraduate cur-riculum: in the context of gravitation at the sophomorelevel, and magnetostatics at the junior-level. With thegoal of making pseudo-longitudinal comparisons acrossthese three different content areas, we again utilized theACER framework to structure the design and analysis ofour investigations. This has allowed us to directly com-pare the reasoning and difficulties students exhibit at keypoints in the problem-solving process when utilizing di-rect integration, and determine whether and how thesedifficulties evolve or shift as students encounter the samemathematical tool multiple times in different contexts.With respect to activation , we found that across allthree contexts some students (roughly a quarter) showeddifficulties in identifying direct integration as the correctapproach. In nearly all cases, students who did not usedirect integration for the potential instead tried to ap-proach the problem by first calculating the associatedfield by another method. Moreover, in interviews, stu-dents often expressed concern about direct integrationbeing too challenging, instead opting for more familiarapproaches or approaches perceived as being easier. Thistendency to avoid particular approaches because theyare “too hard” may well be something these studentshave learned implicitly their courses. Physicists tend togive students questions that can be solved in a relativelystraight-forward manner and often employ tricks to sim-plify the mathematics of a problem. This tendency mayhave the unintended consequence of making students lesswilling to attempt approaches they consider to be math-ematically complex in the belief that there must be aneasier approach. Moreover, this difficulty does not appearto fade significantly over multiple exposures. This sug-gests that instructors may need to place greater empha-sis on explicit discussions of how to identify the correctapproach to a problem and how to eliminate mathemat-ically simpler approaches when they are not applicable.In terms of construction , we found a greater degreeof variation in students’ difficulties in the different con-texts. In all contexts, one of the primary construction difficulties encountered related to expressing the differen-tial element; however, the nature of this difficulty variedsignificantly. Difficulties expressing the differential lineelement were most prevalent in the context of gravita-tion, and gradually became less frequent in the context of electrostatics and then magnetostatics. This may sug-gest that this difficulty fades somewhat over the courseof the current curriculum. Alternatively, the primarydifficulties in expressing the differential element in thecontext of magnetostatics related to the current density,while difficulties with the mass or charge density were lesscommon in the context of gravitation and electrostaticsand occurred with roughly the same frequency. We arguethat this difficulty, while manifest in the mathematics ofthe problem, actually reflects a conceptual difficulty re-lating to interpreting the current density itself and thephysical meaning of the ~JdV ′ term. This represents anexample of a situation where a new difficulty not observedin early exposures to direct integration arose specificallyas a consequence of a change in the physical context ofthe problem. One other element of construction whichsaw significant difficulties was expressing the differencevector. However, the frequency of this difficulty stayedmore consistent across the context of three contexts, andperhaps even got worse after the introduction of “script-r,” with between a third and a half of students displayingdifficulties in both cases, suggesting that this difficulty isalso persistent over multiple exposures.Based on the findings of the earlier study of students’use of direct integration in electrostatics, which foundthat difficulties around the execution were rarely theprimary barrier to student success, the current studyincluded only a few questions asking students to workthrough the mechanics of actually performing integra-tions. However, consistent with this previous finding,the majority of execution errors observed in this studywere made by students who had already made one ormore significant mistakes in the activation or construc-tion components of the their solution. This result, onceagain, suggests that the procedural mistakes made dur-ing the integration process were not the most significantissues encountered by these students.Finally, with respect to reflection , we found that in allthree contexts spontaneous bids for reflection were veryrare in both the exams and interviews. This suggeststhat multiple exposures to direct integration in physicsproblem solving do not appear to encourage studentsto execute reflective checks on their solutions when notprompted to do so. Exams and interviews further sug-gested when prompted to come up with possible reflectivechecks, students in all three contexts are able to identifyappropriate possibilities (e.g., checking units or limits).However, interviews also suggested that actually execut-ing these checks, particularly in cases where doing so re-quires extra steps such as executing a Taylor expansion,may be more challenging for students, particularly afteronly early exposure to direct integration in the contextof gravitation.Overall, the results of this study, combined with ourearlier investigation, provide insight into students’ use ofdirect integration in multiple contexts and provides ex-amples of cases where observed students difficulties getbetter, change, and remain the same over the course of3these multiple exposures. The results can be used tohelp instructors identify areas where additional effort isneeded to address persistent issues (e.g., identifying in-tegration as the appropriate tool, and expressing the dif-ference vector), versus areas where difficulties fade withexperience (e.g., expressing differential line, area, andvolume elements), as well as to anticipate new difficul-ties that appear in different contexts (e.g., interpretingand expressing the current density).This work represents a novel example of how the ACERframework can serve as a standardized tool to facilitatecross-study comparisons of students’ use of mathematicaltools in physics problem solving. Future work will involvecross-context comparisons of students use of other math-ematical tools in other physics contexts. For example,building on prior work with the Dirac delta function in the context of electrostatics could be extended to includemore mathematically complex uses in quantum mechan-ics or Fourier transforms. Similarly, investigations of stu-dents’ use of separation of variables in electrostatics couldform the basis of understanding changes in students’ dif-ficulties when they encounter this tool again in quantummechanics. ACKNOWLEDGMENTS
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