What integration cues, and what cues integration in

What integration cues, and what cues integration in intermediate electromagnetism Leanne Doughty, Eilish McLoughlin, and Paul van Kampen Citation: American Journal of Physics 82, 1093 (2014); doi: 10.1119/1.4892613 View online: http://dx.doi.org/10.1119/1.4892613 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/82/11?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Problem roulette: Studying introductory physics in the cloud Am. J. Phys. 83, 76 (2015); 10.1119/1.4894061 Motivating Students to Do Homework Phys. Teach. 52, 295 (2014); 10.1119/1.4872413 Comparison of electromagnetic and gravitational radiation: What we can learn about each from the other Am. J. Phys. 81, 575 (2013); 10.1119/1.4807853 Restructuring the introductory electricity and magnetism course Am. J. Phys. 74, 329 (2006); 10.1119/1.2165249 Students’ reasoning across contexts AIP Conf. Proc. 720, 109 (2004); 10.1063/1.1807266

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PHYSICS EDUCATION RESEARCH SECTION The Physics Education Research Section (PERS) publishes articles describing important results from the field of physics education research. Manuscripts should be submitted using the web-based system that can be accessed via the American Journal of Physics home page, http://ajp.dickinson.edu, and will be forwarded to the PERS editor for consideration.

What integration cues, and what cues integration in intermediate electromagnetism Leanne Doughty, Eilish McLoughlin, and Paul van Kampena) Centre for the Advancement of Science and Mathematics Teaching and Learning & School of Physical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

(Received 7 May 2013; accepted 29 July 2014) We report on semi-quantitative research into students’ difficulties with integration in an intermediate-level electromagnetism course with cohorts of about 50 students. We have found that before they enter the course, students view integration primarily as a process of evaluation, even though viewing integration as a summation process would be more fruitful. We confirm and quantify earlier results that recognizing dependency on a variable is a strong cue that prompts students to integrate and that various technical difficulties with integration prevent almost all students from getting a completely correct answer to a typical electromagnetism problem involving integration. We describe a teaching sequence that we have found useful in helping students address the difficulties we identified. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4892613] I. INTRODUCTION In recent years, a significant number of publications has appeared on the challenges of teaching junior and upper level electromagnetism.1–14 Early work by Dunn and Barbanel1 identified a need to dovetail the teaching of calculus with the teaching of electromagnetism and described an integrated course that achieved this aim. Manogue et al.2 considered problems involving Ampe`re’s law and also noted that there is generally a mismatch between the aims of calculus courses and physics courses. They observed that in moving beyond the introductory level, for the first time students are required to connect various pieces of mathematics and physics knowledge they already have, and develop these into coherent problem-solving strategies. Work carried out at the University of Colorado4–9 has charted the territory in terms of developing a new approach to junior-level electromagnetism. They have identified some persistent conceptual and mathematical difficulties, and developed a course around the three central ideas of developing mathematical sophistication, developing problem-solving expertise, and developing the students as physicists.8 Detailed research on students’ use of integration within an introductory electromagnetism course was carried out by Meredith and Marrongelle3 and Rebello and co-workers.12 Von Korff and Rebello developed a framework for helping students progress through a physics problem from identifying a finite interval Ð Dt as a starting point to writing an integral expression dt.15 These studies used student interviews as their primary research instrument, and focused in the main on two important issues: (i) What cues can lead students to recognize that integration is called for? and (ii) What difficulties do students encounter with setting up and evaluating an integral? 1093

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The research described in this paper extends this body of research. It pertains to an intermediate electromagnetism course taught at Dublin City University, Ireland, since 2006. The aims of the earliest versions of the course were quite broad: we made explicit links to mechanics and other parts of physics, and included a variety of mathematical techniques such as differentiation, integration, solving differential equations, and vector calculus. Classroom observations and analysis of examination papers suggested that many of our students had difficulties with integration within the context of electromagnetism. In the last four years, we have carried out systematic research into the teaching and learning of electromagnetism in our course, with a focus on the students’ use and understanding of integration. In Sec. IV of this paper, we discuss our students’ “concept image” of integration and investigate whether integration in electromagnetism can be viewed as a transfer problem. In Sec. V, we describe various ways in which integration is cued in a physics context, how we designed and implemented tutorials involving integration, what kind of evidence we used to identify the occurrence of the cues, and the prevalence of each cue. In Sec. VI, we describe technical difficulties our students encounter with integration, and we draw our final conclusions in Sec. VII. II. DESIGN OF THE STUDY AND RESEARCH QUESTIONS Integration is perhaps the foremost mathematical technique in intermediate level electromagnetism courses. Integration plays a central role, e.g., in calculating electric fields due to continuous charge distributions, developing the concepts of electrostatic potential and emf, and applying Maxwell’s equations in integral form. In each of these C 2014 American Association of Physics Teachers V

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instances, integration is the process in which infinitely many infinitesimally small parts of a quantity are summed. Students can successfully tackle problems that require integration if they recognize the need for integration and are capable of setting up and evaluating the definite integral. In addition, ideally the students would also know why they are integrating. However, when interpreting a problem, features of that problem may cue students to integrate without them necessarily considering their integral as a summation. Previous studies have shown that being cued to integrate in this way often leads to students setting up the integral incorrectly.3,12 Figure 1 depicts what we consider an ideal solution pathway for students to tackle problems that require the use of integration. For students to follow this pathway they need to identify a varying quantity related to the result being calculated and to recognize that due to the non-constancy of the quantity, small bits of the result must be summed. Students must recognize that integration is the tool that allows you to do this and must be able to set up and evaluate the integral. For this reason, our approach incorporates student recognition of summation as a cue for integration. To investigate how students can be encouraged to adopt our ideal approach, each of the phases and the transitions should be investigated. We have elicited the views of integration our students bring with them from prior instruction in calculus courses and developed an understanding of how they generally approach problems requiring integration. This allows us to identify aspects of their current knowledge and approaches that might be productive in encouraging students to develop and adopt a strategy for tackling problems that require integration. We have focused on the following phases and transitions of our ideal approach: how students transition from interpreting a problem to recognizing that summation is needed; how likely is it that students who recognize the need for summation will integrate; and how students set up and evaluate integrals. The aspects of our study that are addressed in this paper are summarized in four research questions:

1. At the start of the course, what are our students’ views of integration? 2. At the end of the course, what fraction of our students gives evidence of recognizing the need for integration in electromagnetism problems, and how prevalent were the possible cues for integration? 3. To what extent do our students who recognize the need for summation recognize the need for integration? 4. What difficulties do our students experience with setting up and evaluating integrals, given an expression for an infinitesimal quantity? Question 1 builds on work in the mathematics education literature by looking at students’ conceptual understanding of integration through a different lens. Questions 2 and 3 extend and quantify earlier work in the physics education literature. III. SETTING Our intermediate electromagnetism course covers elements from a standard calculus-based introductory electromagnetism course and the first 5 chapters of Griffiths’ textbook on Electromagnetism.16 The course is taken by a diverse group of students: second-year engineering and physics students, as well as fourth-year pre-service high school science and mathematics teachers. Typically, about 50 students enroll in the course annually. The students’ preparation was also quite varied. All students had successfully completed mathematics courses that included a more or less standard treatment of introductory level integration; some students had also completed a vector calculus course, while others took such a course concurrently with the electromagnetism course. Most students had completed an introductory physics course that is calculus-based, but some had taken an algebra-based physics course.17 The results we report in this paper are obtained from two different but equivalent student cohorts.18 The corresponding author is responsible for delivering the electromagnetism course, and introduced a system of providing one 50-min lecture and two 50-min tutorials per week during a twelve-week semester. Most lectures aimed to reacquaint students with materials they had already seen in a different context (for example, within mathematics courses, within non-calculus contexts, etc.), to provide some historical background, and to introduce new vocabulary illustrated by one or two examples. The tutorials formed the backbone of the course. In many cases, they were paired, with a conceptual physics tutorial preceding a conceptual mathematical tutorial. Most conceptual tutorials were adapted from Tutorials in Introductory Physics;19 the corresponding author created a new conceptual tutorial on Ampe`re’s law. Conceptual mathematical tutorials20 were patterned after these and were for the most part developed by the corresponding author. Tutorials took place in a small-group setting with students working on problems and staff facilitating where necessary through semi-Socratic dialogue. One faculty member and up to three postgraduate tutors (“TAs”) facilitated the tutorials, with the staff-student ratio varying from approximately 1:12 to 1:20. IV. OUR STUDENTS’ VIEWS OF INTEGRATION

Fig. 1. Possible pathway by which students can solve problems requiring integration. 1094

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We have investigated our students’ views of integration by means of a written pretest at the very start of the course, Doughty, McLoughlin, and van Kampen

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designed to probe each student’s concept image. This approach, which we describe below, has gained considerable traction in the mathematics education literature. A. Students’ conceptions of integration in mathematics education literature Integration involves many concepts and ideas, for example that of antiderivative, area under the curve, limits, and Riemann sum. Studies carried out by mathematics education researchers have found that many students complete calculus courses with a proficiency in evaluating integrals but only a very limited conceptual understanding of integration. For example, Orton21 conducted interviews with 110 high-school and college students to investigate their understanding of integration-related items. He specifically targeted understanding of limits and integral as a limit of sums, and concluded that lack of understanding of integration as a limit of sums presented a major difficulty. Petterson and Scheja22 asked 20 students to reflect on limits and integrals. After interviewing some of these students, they concluded that their students were aware of many aspects of integration in an unconnected manner. Grundmeier et al.23 surveyed a group of 52 students on completion of an introductory calculus course and asked them to define a definite integral, first using symbols and then using words. They found that very few gave a correct symbolic definition, but about one-third of their students defined the integral as area under the curve between two points while another third mentioned area in some other way; 10% of students mentioned an infinite sum. According to Thompson and Silverman24 and Sealey,25 it is vital that students regard the integral not only as area under the curve, but also as the sum of infinitely small bits. B. Integration concept image The research mentioned above focused on some specific aspects of conceptual understanding of integration. In our research, we set out to obtain a broader picture of the many aspects of integration. Generally speaking, in a situation where a concept needs to be recalled and used, many mental processes may be activated that can affect the meaning and usage of that concept. Tall and Vinner26 define the mental pictures of properties and processes relating to a concept that are built from experience as the concept image. A concept image is unique to each individual. In the case of integration, a person’s concept image may contain generic ideas like antiderivative, area under the curve, and Riemann sum, but also specific instances such as “the indefinite integral of 2x is x2 þ C.” The concept image thus may, and usually does, contrast with the concept definition, which uses words and symbols to specify a concept and is generally accepted by the wider community. A concept image is not static but contextdependent and may contain elements that are contradictory or incorrect. Tall and Vinner note that despite the functional difference between informal statements that describe a mathematical concept and a formal definition that prescribes an instance of that concept, instructors move readily between the two, without necessarily making the distinction clear to students. As a result, students do not always focus on formal definitions nor do they always use them in their reasoning about a concept; 1095


rather, they usually rely on the ideas they form from their experience with a concept.26 Alcock and Simpson27 found that even in situations where students are introduced to the formal mathematical definition of a concept before being given examples, they may still base their learning mainly on the latter. In many calculus courses, including those taken by our students, the concept definition and the idea of a Riemann sum are introduced first, but the conceptual emphasis is on the definite integral as a way of finding the area under a curve. Alcock and Simpson warn that this may lead to students developing a limited concept image, so that they only know how to use the concept in a small number of contexts. The practical emphasis in mathematics courses is often overwhelmingly on mastering a plethora of techniques such as partial integration and substitution to evaluate integrals. If the findings of Alcock and Simpson have general validity, then there is a serious risk that students find the conceptual understanding of integration they bring to an electromagnetism class seemingly inapplicable.

C. Pre-module concept image questionnaire We investigated what integration cued with our students (in the terminology of Tall and Vinner,26 the evoked concept image) by administering a pretest, before students received any instruction in the electromagnetism course, in the form of a short written questionnaire handed out at the start of the first lecture (see Fig. 2). We presented Ð b one cohort Ð b (N ¼ 50) of our students with two integrals, a dx and a nðxÞdx, and asked them to interpret these integrals. The questions concerned generic functions, as we tried to gain insight into students’ concept images of integration as an abstract process. We did not ask for definitions and did not use a sequence of questions to promote a particular conceptual approach. In this way we attempted to get a picture of how likely the many aspects of integration are to be cued. Our approach thus complements the mathematics education research described above. We do not suggest that the view we obtain of students’ concept image isÐ absolute, in the Ð R sense that more specific integrals b such as a kðxÞdx and 0 ar2 dr would necessarily evoke the same responses with the same prevalences. We do think that questions like ours may reveal what the term “integral” cues with students when used without a clearly defined context.28 We used the notation n(x)—as opposed to, say, f(x)—to lessen the likelihood that students would rely on recall in their responses. By asking students to write down everything they think of when they see these integrals,29 it may not be possible to elicit how complete the students’ concept image of integration was, but we feel that it allows us to quantify what students thought of as the most important aspects of integration as a process. For example, both these integrals may evoke definite integral, anti-derivative, and sum of an infinitesimal quantity as aspects of a student’s concept

Fig. 2. Pretest question probing students’ concept image of integration. Doughty, McLoughlin, and van Kampen

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Ðb image. Note that students can evaluate Ðb a dx as b – a, but it is not possible to similarly evaluate a nðxÞdx. To organize our analysis, we have grouped the aspects of integration students mentioned in two broad categories: conceptual and technical.

method used to find the area between certain values For the second integral, seven students mentioned area under the curve. One student mentioned that the second integral would give an area without elaborating on this.

D. Students’ conceptual views of integration Students’ responses to these questions are summarized in Table I. Some students gave more than one interpretation for the integrals and therefore the percentages sum to more than 100%. Despite the differences in prior courses taken, we found no significant difference between the answers given by the physics students, the engineering students, and the pre-service science teachers, as measured by a v2 test. Few students mentioned any aspect of integration we deemed conceptual. In fact, only two conceptual aspects were mentioned: summation30 (by fewer than 10% of our students) and area under the curve (by at Ðmost 20%). b Three students described the integral a dx as a sum, with varying levels of correctness and completeness: Another word for summation sum of x from a to b this integral describes the sum of parts between a and b of infinitesimal length dx Only Ð b one student mentioned summation in the description of a nðxÞdx: sum of parts between a and b of infinitesimal length dx for the function n(x) Five students interpreted the first integral as the area under a curve. Of these, two students simply stated this without further elaboration; two used dx as the function and stated that the integral will give them “the area under the curve dx”; and one stated that there is no curve in this case and concluded that there is nothing to be evaluated. Another five students wrote that the integral will yield an area without using the word “curve” or indeed any further elaboration. Two of the students’ explanations strongly suggest that they meant area under a curve, for example, area from b to a closed by the x-axis It is possible that the other three students also had area under a curve in mind, but since they only use the word “area” it is difficult to tell from their explanations, for example,

Table I. Categorization of students’ interpretations of two integrals. Type of understanding

Conceptual

Technical

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Ðb

Category

dx (N ¼ 50)

Sum Area under a curve Area (not under a curve) Definite integral Anti-derivative Evaluation Verbalized integral Other No answer

6% (3) 10% (5) 10% (5) 16% (8) 6% (3) 76% (38) 50% (25) 6% (3) 2% (1)

a


Ðb

a nðxÞdx (N ¼ 50)

2% (1) 12% (7) 4% (1) 12% (6) 2% (1) 60% (30) 44% (22) 2% (1) 6% (3)

E. Students’ technical views of integration All other responses concerned what we termed technical aspects of integration. Eight students identified the first integral as a definite integral; six students did so for the second integral. Similarly, three students mentioned that the first integral gives the anti-derivative or that integration is the “reverse process of differentiation” without elaboration; one student mentioned that the second integral gives an antiderivative. For both integrals, approximately half the students restated in words what the symbols meant at a very basic level. Such verbalizations are typically not considered to be part of a concept image as they are merely a recapitulation of the definition and reveal no conceptual aspects, but we take their frequent occurrence as an indication of how these students approach such a symbolic expression. Typical answers in this category were: Integral of 1dx from a to b An integral with limits b and a integrated with respect to x Integration of a function ‘n(x)’ with respect to a variable ‘x’ between parameters a and b For five students, such verbalization was their only response. The evaluation category was the largest by far, with over Ðb 75% of students either evaluating a dx or describing the evaluation process. Typical examples are: ½xba ) b a Increase the power of x by 1 and divide by the value of the new power. Insert the limits (upper limit – lower limit) For just over half of all students this was the only part of their answer that Ðwe considered an aspect ofÐ their evoked b b concept image of a dx. Perhaps surprisingly, a nðxÞdx was interpreted similarly. Ð b Some 60% of students explicitly evaluated the integral a nðxÞdx, despite it not being possible to do so. None gave expressions like N(x) or N(b) – N(a), which would be the equivalent of F(x) as the canonical indefinite integral of f(x). Twenty-one of the 30 students who evaluated this integral treated n as a constant and obtained 1 2 2 2 nðb a Þ, for a variety of reasons. Six students explicitly stated that n is a constant. It is hard to know whether they really thought so or simply reshaped the question so that they could do what integration cued: evaluation.31 Four students explicitly stated that n(x) is a function but still ended up with the same evaluation. Three students whose answer Ðb comprised only evaluating Ðand verbalizing the integral a dx b did not attempt to evaluate a nðxÞdx. We infer that these students knew that this second integral cannot be evaluated. These findings confirmed our practitioners’ knowledge built up from teaching the course for many years: despite knowing how to compute integrals and to verbalize symbolic expressions, many students have little conceptual Doughty, McLoughlin, and van Kampen

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understanding of integration. We are reasonably certain that the wording of the question did not play a significant role. Only the cohort of engineering students may not have been asked to interpret an expression before; the physics students would have taken tutorials from Tutorials in Introductory Physics19 in previous courses, and the pre-service science teachers would have taken modules from Physics by Inquiry.32 Moreover, we saw no significant difference in the prevalence of this kind of answer between the three cohorts, which suggests that the clarification given in the question sufficed. F. Implications for integration in an electromagnetism course In answer to Research Question 1, very few of our students spontaneously mentioned summation when asked to interpret the two definite integrals of Fig. 2, and at most 20% linked the integrals to an area under the curve. A significant majority of students evaluated the integral, whether this was possible or not, and verbalized the symbols. Our results on our students’ concept image of integration are broadly consistent with other students’ specific conceptual difficulties described in the literature.26,27 The vast majority of our students appeared not to associate integration with summation. Informal conversations with many students in the classroom confirmed that this finding is not a test effect; rather, the pretest affords us a way to quantify the prevalence of the problem. Hence, it seems likely that students may realize that summation is required to solve a problem without being cued that integration is a suitable tool for the job. In this sense, our conclusions are potentially more far-reaching than those of Manogue et al.:2 not only do students have to piece together bits of knowledge they already had, as these authors plausibly claim; it appears that the crucial piece of knowledge that integration is a process of summation is not readily evoked by most students. Calculus-based introductory texts and texts used at junior level such as Griffiths16 routinely include, e.g., the calculation of the electric field above the mid-point of a uniformly charged rod. Once the techniques have been mastered, it is easy to overlook how many pieces of mathematics knowledge must be called upon: not only integration, but also superposition, vector addition, geometry, understanding what a function is, etc. An additional problem now emerges: the understanding of integration the pretest elicits in our students is not the kind that is likely to be useful in setting up a physics problem. Some researchers have suggested that students perform poorly when required to use mathematics in physics problem solving due to an inability to transfer knowledge from their calculus courses to physics10 or an inability to apply the mathematical skills they have in a physics context.33 For example, all eight students interviewed in the study by Cui et al.10 were confident of their calculus knowledge, and almost all felt that their knowledge would be adequate for use in physics. However, Cui et al. found that these students were unsure when to use calculus in physics problems. They also found that, while seven from a different set of eight students were able to integrate in solving a particular physics problem, only three could provide an adequate explanation for why integration was required. Our results, obtained for a cohort size of 50, suggest that students’ difficulties with integration in electromagnetism are unlikely to be a transfer issue. Stated baldly, it appears 1097


that one crucial bit of knowledge is not there to be transferred. The kind of knowledge of integration our students acquire from their mathematics courses comprises evaluation, which is useful, but lacks the idea of summation. V. CUEING INTEGRATION IN A PHYSICS CONTEXT Our findings concerning our students’ concept images of integration suggest that integration may not cue summation for many students. However, even though our students do not seem to identify integration with summation, the reverse could still occur: when students identify the need for summation, they may automatically recognize that integration is called for. It is also interesting to see in what other ways integration may be cued. In the first instance, we turn to recent literature for some important findings. A. Previous findings Meredith and Marrongelle3 identified three important types of cue that may lead introductory-level students to recognize that integration is called for: 1. Recall cue: Students may remember having used integration in similar contexts, without understanding why it is used. 2. Dependence cue: Recognition that when a quantity varies (for example, with position or with time), integration may be needed. 3. Parts-of-a-whole cue: Recognition that a quantity (the whole) consists of many parts that can be evaluated individually. To illustrate this classification, consider a problem where students are asked to calculate the charge on a rod for a given varying linear charge density. Students may or may not consider (i) that the total charge is a summation of all charges on the rod and (ii) whether or not the linear charge density is constant or dependent on the position along the rod. If they do neither, by pure recall they may decide to multiply linear charge density by length (appropriate only when the linear charge density is constant) or integrate the linear charge density along the rod (which always works but is overelaborate when the linear charge density is constant). In the latter case, we would say that recall cued integration. If students identified a dependence on position without thinking of summation and integrated, we would say that dependence cued integration. If students identified the total charge as a summation of individual charges without thinking of a dependence, we would say that the idea of parts-of-a-whole cued integration. Ideally, students would do both, and indeed Meredith and Marrongelle3 suggested that a parts-of-a-whole cue may even be conditional on identifying a dependence. Thus, the types of cue are listed above in increasing order of usefulness and desirability. The recall and dependence cues in isolation would bypass two phases in the scheme of Fig. 1. Recall might work for solving isomorphic problems, but students relying on the recall cue are unlikely to recognize the need for integration in problems that involve new contexts; and although recognition that some quantity is varying will cue students to use integration through a dependence cue, it may cause students to set up the integral incorrectly in situations where the infinitesimal is not identical to the varying quantity (for example, when calculating Doughty, McLoughlin, and van Kampen

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the electric field due to a uniformly charged rod, the varying quantity is ~ r but the infinitesimal is dx; this point is discussed in more detail in Sec. VI A). A parts-of-a-whole cue is likely to indicate a more complete picture of what is being asked, and is a natural fit with the summation view of integration. In their study, Meredith and Marrongelle3 interviewed students working on four electrostatics problems that required the use of integration. They found that the dependence cue featured most often, with eight of the nine students using it for at least one question. Unsurprisingly, the recall cue featured only if students had repeatedly experienced integration in a number of similar problems. The parts-of-a-whole cue featured with five students at some point, and mostly in conjunction with the dependence cue; it featured less frequently with increasing problem complexity. The authors concluded that an essentially correct interpretation of the physical situation is essential (but not sufficient) for integration to be cued. From this, we may infer that cueing integration is not purely a matter of mathematics. A similar study by Nguyen and Rebello11 found that students did not have difficulty in recognizing the need for an integral, using the recall cue in problems that were familiar to them and the dependence cue in those that were not. The authors did not mention students’ use of the parts-of-a-whole cue. As they cited the Meredith and Marrongelle paper extensively, we infer that this means they did not see the parts-of-a-whole cue used. We investigated whether similar findings held true for our student cohort. B. Treatment of integration in the course In this section, we report on data obtained from a different cohort of 45 students early in the study.18 The first time our students encountered integration in the electromagnetism course, they were reminded in lecture that integration of any function f(x) allows determination of the area under a curve, with an emphasis on the addition of the area of infinitely many rectangles.34 It was then pointed out that the summation idea is often more productive than the area under the curve idea; as an example, the electric field above the midpoint of a uniformly charged rod was derived in this lecture. Throughout the course a physicist’s interpretation was used: the infinitesimal was treated as a very small quantity with a physical meaning. In later lectures, students were shown how integrals are used in a number of more or less standard sample problems; in each case, the idea of summing infinitely many contributions from infinitely many parts was emphasized. In tutorial, students were led to combine the graphical representation of an integral with the idea of a Riemann sum in determining the total charge on a non-uniformly charged rod, and in five more physics problems.35 These tutorials were always structured in the same manner, intended to develop both dependence and parts-of-a-whole cues. First, students were prompted to think about the non-constancy of a quantity involved (e.g., a varying charge density or distance to a point, etc.), to help them see that dependence may lead to integration. They were then asked to verify an expression for an infinitesimal quantity (dE, dUE , dB, etc.). These expressions were complex enough that in essence the students had to derive the expression, and they could use the given expression for verification of their own work. Having completed this, to help students develop the idea that integration applies when a macroscopic quantity is a part-of-a-whole, they were asked to consider why the macroscopic quantity had to be 1098


derived from integrating the infinitesimal expression. Finally, given the indefinite integral, they were asked to derive an expression for the macroscopic quantity (E, UE , B, etc.). An example is given in Fig. 3; it was the second integration problem that the students encountered (after calculating the total charge on a surface), and the first in which a scalar quantity is computed through integration of a dot product.36 The problem is sufficiently complex to make for a rich problem in terms of proportional reasoning, algebra, vector algebra, and multivariable calculus. The granularity and degree of scaffolding37 have been fine-tuned over the years to best suit our students’ needs and to elicit productive in-class conversations. Question A is designed to cue the need for integration. At first, many students think in terms of absolute rather than relative variations (e.g., “it depends on whether a is big or small”), ignore variation in direction, and state that the variation of the magnitude depends on a. When asked whether this is the relevant variation to consider, many realize that the relative variation is germane to this question, andpfigure ffiffiffi out that the magnitude of the field varies by a factor 3 and the direction by 45 . Aspects of the problem we would like students to bring up in conversation about Question B build on this, and we encourage students to relate this problem to the charged rod problem seen in lecture. In this way we encourage students to consider that Coulomb’s law does not work for the entire sheet or rod since a single relevant distance or angle cannot be defined; that splitting the sheet or rod into (infinitely) many small parts is fruitful since Coulomb’s law can be applied to these parts; and that addition of dot products is different from superposition of vectors of different magnitude and direction. Question C provides crucial scaffolding for our students; as argued earlier, solving problems such as this requires many small steps to be recognized and carried out, and what looks like an obvious substitution to an expert may be a valuable stepping stone to our novice students. Question D allows for a discussion of the technical problems students encounter when integrating; they need explain how integration performs addition and why the limits are appropriate. Question E is a valuable check on the answer that allows students to grasp some aspects of the symmetry argument. Thus, the tutorial is much more than a simplified problem from a textbook or a training exercise in how to apply an algorithm. Having worked through this tutorial, students have thought about the need for integration and have been assisted with some technical elements. Students are given the indefinite integrals they need, as well as the final answer, which allows them to progress without having to wait for confirmation from a tutor. Our other integration tutorials are similar in structure. In tutorials early in the course almost all students try to pattern-match; as the semester progresses, more and more students become convinced that success is more likely to come from attempting to answer the question and checking at the end if their answer matches ours. During class, we are unable to check that all students wrote good explanations all the time, but we are able to get a good idea of students’ progress by assessing isomorphic homework questions. C. Post-test Figure 4 shows a post-instruction exam problem in which students have to use integration to obtain an expression for a quantity. (Our experience in tutorials and homework problems Doughty, McLoughlin, and van Kampen

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Fig. 3. Sample tutorial involving integration.

suggests that asking similar questions as a pretest would not yield useful research data as it is beyond most students’ capabilities.) In this question students were required to find the magnetic field due to a current-carrying thin ribbon. Superficially, the question was unlike any integration questions the students had seen in tutorials: they had integrated to find charge, electric field, electric flux, potential, and magnetic force, but not magnetic field, and had not seen linear current density in any form. A recall cue as defined by Meredith and Marrongelle could therefore be eliminated as a likely cue for integration. However, the students had encountered twodimensional current density, integrals involving onedimensional charge density, analogous integrals requiring superposition of electrostatic fields with similar geometric and algebraic manipulations, and integrals involving charge density distributions where sheets had to be cut up into strips. Thus, the setting, though challenging, should be accessible for post-introductory level students, and confusion 1099


about what is being asked is unlikely to be a major factor in students’ responses. This expectation is supported by students’ answers to Question (b), where they were asked to explain that I ¼ KL. Despite having no previous experience with linear current density, only 15% of the students who attempted the problem did not answer Question (b) or gave an incorrect or unclear answer. The problem was intended as a superposition problem, where students would use the known field of an infinitely long wire to determine the field of the ribbon. Thus, in solving the problem students do not encounter any of the difficulties associated with applying Ampe`re’s law.2,5,38 The problem is the magnetostatic equivalent of finding the electric field due to a uniformly charged rod: it concerns vector addition of infinitesimal magnetic fields and the recognition of symmetry in doing so, but students do not need to find clever loops that allow calculation of the magnetic field. Doughty, McLoughlin, and van Kampen

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Fig. 4. Post-test question on calculation of a magnetic field.

Question (a) was designed to check whether students identified the non-constancy of the magnitude and direction of the magnetic field caused by different parts of the currentcarrying ribbon, isomorphic to the tutorial on the charged rod. In doing so, we could investigate whether dependence was recognized. Following the same pattern as in the tutorials, we provided students with the expression for the infinitesimal magnetic field due to the small segment in Question (c), along with an expression for the indefinite integral (not shown in the figure). As in the tutorial question on the flux through a sheet, students were cued but not explicitly told why integration is an appropriate technique. This breakdown of the problem allowed us to see to what extent students recognize the need for integration from this starting point. Stating explicitly that there are segments (parts) giving rise to an infinitesimal magnetic field and that there is a macroscopic field (the whole) due to the entire sheet hints at summation. More superficially, seeing an infinitesimal like dB may cue integration by recall, i.e., without any deep consideration of the physical situation. In different homeworks and exams, we have asked many similar questions for a variety of quantities (electric field, electric potential, and magnetic field) that followed the same pattern. Thus the separation of deriving expressions for infinitesimal and macroscopic quantities also allowed us to check if Meredith and Marrongelle’s assertion holds true, that understanding of the physical situation is an important part of integration. If the physical situation does not matter, then at the very least the prevalence of students attempting to obtain a macroscopic expression from the infinitesimal expression should be independent of context; this holds true whether or not they mechanically follow the question pattern they have now become familiar with. D. Results Forty-five students (N ¼ 45) took the exam; 38 of these attempted the problem. Since we do not know why the remaining seven students did not attempt the problem—they may have run out of time or found the problem entirely intractable—we only consider the 38 students who attempted at least some of the parts. This choice does not substantially affect our conclusions. When asked to find the exact magnetic field at point P due to the entire sheet, for 39% of our 1100


students integration was not cued, with 34% not responding to Question (e) and 5% giving an answer that involved only algebraic manipulations, not integration. To investigate how frequently students who knew to use integration were cued by a dependence, the responses to Question (a) were correlated with the responses to Question (e). This data is shown in Table II. In 15 of the 23 cases where integration was cued, students had identified at least one reason for why the magnetic field due to the sheet could not be expressed using the given expression. Most frequently, this reason was the varying distance from the point to different parts of the sheet. The remaining eight students may have been helped by the breakdown of the question. The converse was also investigated, i.e., how likely it was that students who recognized that the microscopic quantity dB was not constant did integrate. Seven of the 15 students for whom integration was not cued had either not answered Question (a) or had not correctly identified why the given expression was only approximate. For the remaining eight students, neither recognizing dependence nor the structure of the question cued integration. These results show that the dependence cue is important, but suggest that it is neither necessary nor sufficient. This finding does not necessarily invalidate the suggestion by Meredith and Marrongelle3 that it is necessary, because our question allowed for additional cueing. Six students who recognized the non-constancy gave evidence of using a parts-of-a-whole cue for integration, as they included a description of integration as a sum or an explanation that a sum was required in some part of the question. Although none of the following typical answers are completely correct, they reveal an understanding of integration as a process of summation: K divides the sheet with a current I into many small sheets with current DI… by adding all the DI we get the original total current Looking at the sheet as a group of … n wires, each with current I/n, the magnetic field would be a simple sum of each wire ~ ¼ sum of all small segments B As no other students used the parts-of-a-whole cue, our findings support those by Meredith and Marrongelle3 that the parts-of-a-whole cue occurs in conjunction with the dependency cue. Over the years, we have asked many similar post-test questions, always structured in the same manner as the tutorials, albeit with fewer steps. On one or two occasions the questions looked unlike any integration questions the students had seen in tutorials, to eliminate recall as a likely cue for integration, as shown above; but more commonly they Table II. Dependence cue investigation, correlating integration with recognition that the magnetic field due to different parts of the current-carrying ribbon was not constant (N ¼ 38).

Integration cued Integration not cued Total

Non-constancy recognized

Non-constancy not recognized

Total

39% (15) 21% (8) 61% (23)

21% (8) 18% (7) 39% (15)

61% (23) 39% (15) 100% (38)

Doughty, McLoughlin, and van Kampen

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were variations on problems the students had seen before. Each exam comprised two integration questions, one in an electrostatics context and one in a magnetostatics context. The fraction of students that do not attempt to integrate always varies even within the same exam, and students who appear to grasp why integration is required in one problem may not do so in the other. This is evidence that understanding the physical situation influences whether students attempt to integrate, even though from a mathematical point of view the physical context is irrelevant. E. Summary: Cues for integration We asked our students to solve various problems in electromagnetism that require integration. In each case, we asked students to derive a given expression for a microscopic quantity and to obtain from this an expression for the macroscopic quantity. Using this sequence of questions, we find that in general between one-third and two-thirds of our students recognize the need for integration. It may seem unlikely that for so many students the expression for an infinitesimal quantity did not cue integration. However, Kanim39 found similar percentages when he investigated students’ approaches to calculating the net electric field at a given point due to a charged rod. In his case, 40% of the students showed that they knew integration was required, and only 10% correctly calculated the field. For a specific question involving the magnetic field due to a thin strip with uniform current density, we found that twothirds of students who integrated recognized a non-constancy, indicating that a dependence cue was involved. One quarter of the students who integrated gave evidence of recognizing the macroscopic magnetic field as parts-of-a-whole. This parts-of-a-whole cue always occurred in conjunction with a dependence cue, as surmised by Meredith and Marrongelle.3 VI. TECHNICAL DIFFICULTIES WITH INTEGRATION A. Known difficulties The aforementioned studies by Meredith and Marrongelle3 and Nguyen and Rebello11 also discussed the difficulties students have in applying integration (i.e., setting up and evaluating integrals) in physics problems. When setting up an integral, understanding the origin of the infinitesimal was found to be the main difficulty. Although the infinitesimal has its own physical meaning and writing an integral without it is physically meaningless, Nguyen and Rebello11 found that many students either neglected to include it in the set-up of an integral or placed it after the integrand without realizing how it changed the quantity being summed. Meredith and Marrongelle3 also identified the latter difficulty and described it as a failure of the dependence cue when the non-constant quantity is not a density or rate of change. For example, when asked to find the electric field due to a uniformly charged rod, students using the dependence cue correctly identified that the varying quantity was distance, but then erroneously appended dr to the formula for a point charge to obtain kq dr/r2. Our findings for students’ concept image of integration may explain this in part. If students see integration primarily as evaluating a certain function, as suggested by the results of Table I, they may see the dx term essentially as a punctuation mark, and they may only pay attention to the function whose antiderivative must be found. 1101


A failure to pay attention to how infinitesimal terms should be added is another identified difficulty. Approximately, half the students involved in the Nguyen and Rebello11 study did not account for the vector nature of electric field and integrated the whole of dE rather than dEx and dEy separately. The authors suggested that this was due to students’ lack of visualization of the physical situation presented. This study also found that when evaluating the integral, students had difficulties understanding the physical meaning of symbols in the integral, recalling basic mathematical equations needed to write all variables in terms of x if the infinitesimal is dx, determining the limits for integration, and computing the integrals. B. Our results We found that almost all of our students had considerable difficulty with integration. Many did not recognize the need for integration, and the 23 who used integration experienced technical difficulties in setting up the integral for B (even though the expression for the corresponding infinitesimal dB was given), and in evaluating it. Nineteen of these 23 students obtained the given expression for (c). All of these showed that the distance to each infinitesimally thin wire pffiffiffiffiffiffiffiffiffiffiffiffiffiffi was z20 þ x2 , but ten students substituted Kd x for current without explanation. While it is impossible to tell whether they are merely inferring that dI ¼ Kdx from the answer given without understanding what it means, all nine students who explained why they replaced I with Kdx did so correctly. A typical explanation was: K ¼ I/L and dx is a very small length, thus multiplying K by dx should give a small current. The four students who answered (c) incorrectly gave widely varying responses; two appended the infinitesimal to their answer incorrectly, or without explaining its origin. For example, one student appeared to pattern-match: this student started from the expression for the field of a long, straight wire, written somewhat incorrectly as dB ¼

^ l0 I / ; 2pz0

(1)

but multiplied this expression by a factor cosh, which made it possible to obtain an expression that looked correct apart from notationally equating a scalar to a vector: dB ¼

l0 I ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi /: 2p z20 þ x2

(2)

This student then stated: however now we are not assuming I to act at one point so we must replace I with a charge density K. In the final line of this student’s answer, a factor dx appeared in the numerator, accompanied by the text “width sheet.” This example is not unlike some responses quoted by Nguyen and Rebello11 to illustrate that their students often append an infinitesimal factor without appearing to understand its origin; but it also seems that this student has a tenuous grasp of the physical situation. Generally our findings neither confirm nor contradict those of Refs. 3 and 11: our question was not designed to elicit difficulties with the infinitesimal and allowed such difficulties to remain hidden. Doughty, McLoughlin, and van Kampen

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Table III. Categorization of most common errors during integration, for the magnetic field question shown in Fig. 4. Error

(N ¼ 23)

~ treated as scalar) Incorrect summation (dB No limits Incorrect limits Integral written down without attempt at evaluation Mistake in process

86% (19) 23% (5) 45% (10) 23% (5) 18% (4)

Only one student evaluated the integral completely correctly. Table III shows the types of integration errors made by the other 22 students. Some students made more than one of these errors. Nineteen students integrated the expression for dB as if the magnetic field were a scalar. A number of factors could be at play here: they may not have thought of the infinitesimal magnetic field as a vector, they may not have known how to add vectors, they may not see integration as a summation process, or a combination of these reasons. Failure to consider how quantities should be added up has previously been identified as a difficulty.11 Determining the limits for integration was also a problem for students, with five students leaving them out and ten students using incorrect limits. Four students used 0 as the lower limit and L as the upper limit, and this was their only mistake during the integration process. This suggests that students are either careless about limits or struggle to determine the limits when the lower limit is nonzero. Five students who had recognized the need for integration did not attempt to compute the integral. Four more students made different mistakes during the evaluation process. In other problems, students were required to evaluate a double integral, which also caused difficulties. The general picture we obtained is that for different problems students have similar difficulties but with different prevalences. C. Summary: Technical difficulties We have identified that students have difficulties with vector addition, determining limits, manipulating algebraic expressions, and double integration. Any of these difficulties can prevent students from obtaining a correct answer. Our findings support the conclusions by Meredith and Marrongelle3 and Nguyen and Rebello11 that recognizing the need for integration is necessary but not sufficient for a student to come to a correct answer. As pointed out by Manogue et al.,2 students struggle because so many aspects of physics and mathematics knowledge need to be cued and applied correctly to come to an answer. VII. CONCLUSIONS AND IMPLICATIONS FOR TEACHING Over the course of several years, we have investigated how diverse cohorts of about 50 students interpret simple integrals and approach problems in electromagnetism that require integration. We have found that prior to the electromagnetism course between 60% and 75% of our students saw integration as an evaluation, while only 5% identified integration as a process of summation. 1102


This limited view of integration is likely to exacerbate the difficulties students have with solving problems that require integration in the electromagnetism course. If students had internalized that an integral is a summation before starting the course, a teacher could concentrate on helping students learn to identify when and how a continuously varying quantity should be added. In the present situation, a teacher needs to convince students that integration is required because it is a summation—a tautology to an expert, but a source of wonder for a student who sees an integral mostly as an evaluation tool. Early in the study we have found that, depending on the problem, varying fractions of our students recognized the need for integration on post-test problems similar to the problem described in this paper (Fig. 4). Only a small number of students gave evidence of using a parts-of-a-whole cue for integration, while recognizing a dependency was the strongest integration cue. However, 20% of our students who recognized the non-constancy in the magnetic-field problem did not integrate. It is possible that these students did not realize that this meant that a summation was required, or perhaps they did not recognize integration as the required tool for summation. These are issues that need to be investigated further in future work. Many technical difficulties with integration also surfaced, confirming and quantifying findings by other researchers. Both classroom observations and the semi-quantitative research described in this paper suggest to us that for instruction in integration within an electromagnetism course to be successful, students must repeatedly work with the idea that integration is an infinite summation. Far from “wasting time” on re-teaching mathematics within a physics course, such an approach augments and enhances what students have learned in a mathematics context. Our approach in the early stages of the research worked to an extent, and students felt they were making progress. Based on the early results from the magnetic field post-test and the progress made by many students in the tutorials, we adapted the course and emphasized integration as summation more strongly. We replaced the introductory lecture and tutorials on integration that relied on the concept of area under a curve with an interactive lecture that helps students become familiar with the infinite summation concept and with a new tutorial that guides our students through the phases and transitions of our ideal approach (Fig. 1) in which students calculate the total charge on a non-uniformly charged rod. In doing so, the “summation cues integration” phase is now developed in the approach recommended by Von Korff and Rebello.15 We have also introduced an additional tutorial with the same structure as that shown in Fig. 3. On completion of the present course, around 85% of our students recognize the need for integration in examination problems. ACKNOWLEDGMENTS The authors gratefully acknowledge in-depth discussions with Brien Nolan, Eabhnat Nı Fhloinn, Mossy Kelly, Paul Grimes, Scott McDonald, and Mieke De Cock. The authors are indebted to two anonymous referees whose valuable comments and insights strengthened the paper significantly. a)

Electronic mail: [email protected] J. W. Dunn and J. Barbanel, “One model for an integrated math/physics course focusing on electricity and magnetism and related calculus topics,” Am. J. Phys. 68(8), 749–757 (2000). 2 C. A. Manogue, K. Browne, T. Dray, and B. Edwards, “Why is Ampe`re’s law so hard? A look at middle-division physics,” Am. J. Phys. 74(4), 344–350 (2006). 1


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D. C. Meredith and K. A. Marrongelle, “How students use mathematical resources in an electrostatics context,” Am. J. Phys. 76(6), 570–578 (2008). 4 S. J. Pollock, “Longitudinal study of student conceptual understanding in electricity and magnetism,” Phys. Rev. ST Phys. Educ. Res. 5, 020110-117 (2009). 5 C. S. Wallace and S. V. Chasteen, “Upper-division students’ difficulties with Ampe`re’s law,” Phys. Rev. ST Phys. Educ. Res. 6, 020115-1–8 (2010). 6 S. V. Chasteen, S. J. Pollock, R. E. Pepper, and K. K. Perkins, “Transforming the junior level: Outcomes from instruction and research in E&M,” Phys. Rev. ST Phys. Educ. Res. 8, 020107-1–18 (2012). 7 S. V. Chasteen, R. E. Pepper, M. D. Caballero, S. J. Pollock, and K. K. Perkins, “Colorado Upper-Division Electrostatics diagnostic: A conceptual assessment for the junior level,” Phys. Rev. ST Phys. Educ. Res. 8, 020108-1–15 (2012). 8 S. V. Chasteen, S. J. Pollock, R. E. Pepper, and K. K. Perkins, “Thinking like a physicist: A multi-semester case study of junior-level electricity and magnetism,” Am. J. Phys. 80, 923–930 (2012). 9 R. Pepper, S. V. Chasteen, S. J. Pollock, and K. K. Perkins, “Observations on student difficulties with mathematics in upperdivision electricity and magnetism,” Phys. Rev. ST Phys. Educ. Res. 8, 010111-1–15 (2012). 10 L. Cui, N. S. Rebello, P. Fletcher, and A. Bennett, “Transfer of learning from college calculus to physics courses,” in Proceedings of the NARST 2006 Annual Meeting (NARST, Reston, VA, 2006), . 11 D.-H. Nguyen and N. S. Rebello, “Students’ understanding and application of the area under the curve concept in physics problems,” Phys. Rev. ST Phys. Educ. Res. 7, 010112-1–17 (2011). 12 D.-H. Nguyen and N. S. Rebello, “Students’ difficulties with integration in electricity,” Phys. Rev. ST Phys. Educ. Res. 7, 010113 (2011). 13 H. R. Sadaghiani, “Using multimedia learning modules in a hybrid-online course in electricity and magnetism,” Phys. Rev. ST Phys. Educ. Res. 7, 010102-1–7 (2011). 14 E. R. Savelsbergh, T. de Jong, and M. G. M. Ferguson-Hessler, “Choosing the right solution approach: The crucial role of situational knowledge in electricity and magnetism,” Phys. Rev. ST Phys. Educ. Res. 7, 010103-1–12 (2011). 15 J. Von Korff and N. S. Rebello, “Teaching integration with layers and representations: A case study,” Phys. Rev. ST Phys. Educ. Res. 8, 010125 (2012). 16 D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Upper Saddle River, NJ, 1999). 17 The calculus-based textbook was Young & Freedman, University Physics, 12th ed. (Addison-Wesley, Reading, MA, 2007); the algebra-based textbook D. C. Giancoli, Physics, 6th ed. (Pearson, 2010). Both courses cover the relevant material on introductory electricity and magnetism. 18 We deem the two cohorts to be equivalent since in many pretests (not reported in this paper) we obtain very similar responses. 19 L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the University of Washington, Tutorials in Introductory Physics (PrenticeHall, Inc., Upper Saddle River, NJ, 2002). Alterations made to the tutorials were typically small and served to dovetail these existing conceptual physics tutorials with the conceptual mathematical tutorials. 20 The case for this kind of tutorials was made eloquently by B. S. Ambrose, “Investigating student understanding in intermediate mechanics: Identifying the need for a tutorial approach to instruction,” Am. J. Phys. 72(4), 453–459 (2004). 21 A. Orton, “Students’ understanding of integration,” Educ. Stud. Math. 14(1), 1–18 (1983). 22 K. Pettersson and M. Scheja, “Algorithmic contexts and learning potentiality: A case study of students’ understanding of calculus,” Int. J. Math. Educ. Sci. Tech. 39(6), 767–784 (2008).

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T. A. Grundmeier, J. Hansen, and E. Sousa, “An exploration of definition and procedural fluency in integral calculus,” Prob. Res. Iss. Math. Undergrad. Stud. 16(2), 178–191 (2006). 24 P. W. Thompson and J. Silverman, “The concept of accumulation in calculus,” in Making the Connection: Research and Teaching in Undergraduate Mathematics, edited by M. P. Carlson and C. Rasmussen (Mathematical Association of America, Washington, DC, 2008), pp. 43–52. 25 V. Sealey, “Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient,” in Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by S. Alatorre et al. (Universidad Pedagogica Nacional, Merida, Yucatan, Mexico, 2006), Vol. 2, pp. 46–53. 26 D. Tall and S. Vinner, “Concept image and concept definition in mathematics with particular reference to limits and continuity,” Educ. Stud. Math. 12(2), 151–169 (1981). 27 L. Alcock and A. Simpson, Ideas from Mathematics Education: An Introduction for Mathematicians (Higher Education Academy, Birmingham, 2009). 28 A similar approach was adopted by Vinner while researching students’ concept image of functions. See S. Vinner and T. Dreyfus, “Image and definitions for the concept of function,” J. Res. Math. Educ. 20(4), 356–366 (1989). 29 Students appear to have interpreted our elaboration on the term “interpret” as intended: “write” was not meant to exclude diagrammatic representations. Students who mentioned area under the curve especially tended to draw diagrams to illustrate their answers. 30 We use the term “summation” throughout, but note that in the mathematics education literature the term “accumulation” is commonly used. 31 It cannot be denied that there is an ambiguity in the notation: n(x) could mean “multiply n by x,” although no expert would interpret n(x) that way. 32 L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the University of Washington, Physics by Inquiry (Wiley, New York, 1996). 33 J. Tuminaro and E. Redish, “Understanding students poor performance on mathematical problem solving in physics,” AIP Conf. Proc. 720, 113–116 (2004). 34 Of course we are and were aware of the general ineffectiveness of teaching by telling; we were under the mistaken impression that we were merely reminding students of something they had already internalized. 35 These five tutorial problems entailed: finding the total charge on a thin semicircular disk of radius R with varying surface charge density; calculating the electric flux through a flat sheet due to a point charge located above one of the corners of the sheet; determining the potential due to a uniformly charged rod; calculating by integration the circulation along a rectangular loop of the magnetic field due to a straight current-carrying wire; and calculating the magnetic force on a square current-carrying loop due to a straight wire. 36 We realize that at first glance, this may look like a “dumbed down” version of Problem 2.10 of Griffiths’ textbook.16 We are aware that the answer can be obtained in one line from symmetry considerations, but have found it a useful problem to tackle through integration. 37 D. J. Wood, J. S. Bruner, and G. Ross, “The role of tutoring in problem solving,” J. Child Psych. Psychol. 17(2), 89–100 (1976). 38 J. Guisasola, J. M. Almundı, J. Salinas, K. Zuza, and M. Ceberio, “The Gauss and Ampere laws: Different laws but similar difficulties for student learning,” Eur. J. Phys. 29, 1005–1016 (2008). 39 S. Kanim, “An investigation into student difficulties in qualitative and quantitative problem solving: Examples from electric circuits and electrostatics,” Ph.D. dissertation (unpublished), Department of Physics, University of Washington (1999).


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