Case of two electrostatics problems: Can providing a diagram adversely impact introductory physics students' problem solving performance?
Case of two electrostatics problems: Can providing a diagram adverselyimpact introductory physics students ’ problem solving performance? Alexandru Maries and Chandralekha Singh Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA (Received 1 October 2017; published 21 March 2018)Drawing appropriate diagrams is a useful problem solving heuristic that can transform a problem into arepresentation that is easier to exploit for solving it. One major focus while helping introductory physicsstudents learn effective problem solving is to help them understand that drawing diagrams can facilitateproblem solution. We conducted an investigation in which two different interventions were implementedduring recitation quizzes in a large enrollment algebra-based introductory physics course. Students wereeither (i) asked to solve problems in which the diagrams were drawn for them or (ii) explicitly told to draw adiagram. A comparison group was not given any instruction regarding diagrams. We developed rubrics toscore the problem solving performance of students in different intervention groups and investigated tenproblems. We found that students who were provided diagrams never performed better and actuallyperformed worse than the other students on three problems, one involving standing sound waves in a tube(discussed elsewhere) and two problems in electricity which we focus on here. These two problems werethe only problems in electricity that involved considerations of initial and final conditions, which maypartly account for why students provided with diagrams performed significantly worse than students whowere not provided with diagrams. In order to explore potential reasons for this finding, we conductedinterviews with students and found that some students provided with diagrams may have spent less time onthe conceptual analysis and planning stage of the problem solving process. In particular, those providedwith the diagram were more likely to jump into the implementation stage of problem solving early withoutfully analyzing and understanding the problem, which can increase the likelihood of mistakes in solutions.
DOI: 10.1103/PhysRevPhysEducRes.14.010114
I. INTRODUCTION
Physics is a challenging subject to learn and it isespecially difficult for introductory students to associatethe abstract concepts they study in physics with moreconcrete representations that facilitate understandingwithout an explicit instructional strategy aimed to aidthem in this regard. Here, by “ representation, ” we meanany of the diverse forms in which scientific knowledge, or,physical concepts to be more exact, are understood andcommunicated [1]. This very broad definition encom-passes nearly anything scientists have used to describe theworld, but to be a bit more precise, in this article, wespecifically refer to verbal, diagrammatic, mathematical,and graphical representations (for more information onhow these are defined see Ref. [1]). Without guidance,introductory students often employ formula oriented problem solving strategies instead of developing a solidgrasp of physical principles and concepts [2 – – –
18] partly becausethe process of constructing an effective representation of aproblem makes it easier to generate appropriate decisionsabout the solution process. Also, getting students torepresent a problem in different ways helps shifttheir focus from merely manipulating equations toward
Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article ’ s title, journal citation,and DOI. PHYSICAL REVIEW PHYSICS EDUCATION RESEARCH = = = – “ span ” the conceptual spaceassociated with an idea. Since traditional courses whichdo not emphasize multiple representations lead to lowgains on the Force Concept Inventory [24,25] and on otherassessments in the domain of electricity and magnetism[26 – ’ understanding ofphysics concepts, many researchers have developedinstructional strategies that place explicit [8,14,29 – –
40] emphasis on multiple representa-tions. Van Heuvelen ’ s approach, for example [14], startsby ensuring that students explore the qualitative nature ofconcepts by using a variety of representations of a conceptin a familiar setting before adding the complexities ofmathematics. Many other researchers have emphasizedthe importance of students becoming facile in translatingbetween different representations of knowledge [29,41 – – ’ s active learning problem sheets (ALPS)[14] adapted from Reif follow a very similar underlyingapproach. Other researchers who have emphasized, amongother things, the importance of diagrams in their approachto teaching students problem solving skills have foundsignificant improvements in students ’ problem solvingmethods [9,13]. In mathematics, Schoenfeld [57,58] advo-cates drawing a diagram (if possible) as the first step.Previous research shows that students who draw dia-grams even if they are not rewarded for them are moresuccessful problem solvers [13]. In addition, students whotake courses which emphasize effective problem solvingheuristics, which include drawing a diagram, are morelikely to draw diagrams even on multiple-choice exams[13]. Furthermore, courses which are rich in use ofrepresentations can have significant positive impact onstudent skills [59]. It is therefore possible that explicitlyasking students to draw diagrams when solving problemsmay result in improved performance. An investigation intohow spontaneous drawing of free body diagrams (FBDs)affects problem solving [60,61] shows that only drawingcorrect FBDs improves a student ’ s score and that studentswho draw incorrect FBDs do not perform better thanstudents who draw no diagrams. Heckler [62] investigatedthe effects of prompting students to draw FBDs inintroductory mechanics by including as the first subpartof each problem an instruction to draw clearly labeledFBDs. He found that students who were prompted to drawFBDs were more likely to follow formally taught problemsolving methods rather than intuitive methods, whichresulted in deteriorated performance.Here, we extend the previous research on the impact ofdrawing diagrams on algebra-based introductory physics(mainly taken by bioscience majors and premeds) studentperformance on electricity problems through two studies.In study 1, we investigated how student performance wasaffected when students were provided with a diagraminstead of being asked to draw one and compared theirperformance to that of students who were asked to draw adiagram (without being any more specific than that) and tothe performance of a comparison group which was neitherasked to draw a diagram nor provided a diagram. Weanalyzed performance on ten problems throughout thesemester given as quizzes; all problems were at theapplication level of Bloom ’ s taxonomy and they werealways related to the topic that was part of the previousweek ’ s lecture and the homework that was due (althoughthe quiz problems were not identical to those in theMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
40] emphasis on multiple representa-tions. Van Heuvelen ’ s approach, for example [14], startsby ensuring that students explore the qualitative nature ofconcepts by using a variety of representations of a conceptin a familiar setting before adding the complexities ofmathematics. Many other researchers have emphasizedthe importance of students becoming facile in translatingbetween different representations of knowledge [29,41 – – ’ s active learning problem sheets (ALPS)[14] adapted from Reif follow a very similar underlyingapproach. Other researchers who have emphasized, amongother things, the importance of diagrams in their approachto teaching students problem solving skills have foundsignificant improvements in students ’ problem solvingmethods [9,13]. In mathematics, Schoenfeld [57,58] advo-cates drawing a diagram (if possible) as the first step.Previous research shows that students who draw dia-grams even if they are not rewarded for them are moresuccessful problem solvers [13]. In addition, students whotake courses which emphasize effective problem solvingheuristics, which include drawing a diagram, are morelikely to draw diagrams even on multiple-choice exams[13]. Furthermore, courses which are rich in use ofrepresentations can have significant positive impact onstudent skills [59]. It is therefore possible that explicitlyasking students to draw diagrams when solving problemsmay result in improved performance. An investigation intohow spontaneous drawing of free body diagrams (FBDs)affects problem solving [60,61] shows that only drawingcorrect FBDs improves a student ’ s score and that studentswho draw incorrect FBDs do not perform better thanstudents who draw no diagrams. Heckler [62] investigatedthe effects of prompting students to draw FBDs inintroductory mechanics by including as the first subpartof each problem an instruction to draw clearly labeledFBDs. He found that students who were prompted to drawFBDs were more likely to follow formally taught problemsolving methods rather than intuitive methods, whichresulted in deteriorated performance.Here, we extend the previous research on the impact ofdrawing diagrams on algebra-based introductory physics(mainly taken by bioscience majors and premeds) studentperformance on electricity problems through two studies.In study 1, we investigated how student performance wasaffected when students were provided with a diagraminstead of being asked to draw one and compared theirperformance to that of students who were asked to draw adiagram (without being any more specific than that) and tothe performance of a comparison group which was neitherasked to draw a diagram nor provided a diagram. Weanalyzed performance on ten problems throughout thesemester given as quizzes; all problems were at theapplication level of Bloom ’ s taxonomy and they werealways related to the topic that was part of the previousweek ’ s lecture and the homework that was due (althoughthe quiz problems were not identical to those in theMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14, II. STUDY 1A. Methodology for study 1
A traditionally taught second-semester class of 111algebra-based introductory physics students was brokenup into three different recitations. The three recitationsformed the comparison group and two intervention groupsfor this investigation. All recitations were taught in atraditional manner in which the TA worked out problemssimilar to the homework problems and then gave a 15 minquiz at the end of class. Students in all recitations attendedthe same lectures, were assigned the same homework, andtook the same exams and quizzes. While the instructoralways used effective problem solving strategies, e.g.,drawing a diagram, listing knowns or unknowns, makinga plan, etc., students were not assessed on whether or notthey followed these strategies during quizzes and exams(e.g., no points taken off if students did not draw adiagram). In the recitation quizzes throughout the semester,all students were given the same problems but with thefollowing interventions:(i) Prompt only group (PO). — in each quiz problem,students were given explicit instructions to draw adiagram with the problem statement.(ii) Diagram only group (DO). — in each quiz problem,students were provided a diagram drawn by theinstructor that was meant to aid in solving theproblem. (iii) No support group (NS). — this group was the com-parison group and was not given any diagram orexplicit instruction to draw a diagram with theproblem statement.The sizes of the different recitation groups varied from22 to 55 students because the TA was the same for all threerecitations and some students asked the TA if they could goto a different recitation than the one they signed up for. TheTA generally allowed students to do this, and the vastmajority of students, after choosing the most convenientrecitation for their schedule, went to that same recitationevery week (e.g., the Tuesday evening recitation). It is alsoimportant to note that each intervention was not matched toa particular recitation. For example, in one week, studentsin the Tuesday evening recitation comprised the compari-son group, while during another week the comparisongroup was a different recitation section. This implies thatindividual students underwent different interventions fromweek to week and we therefore do not expect cumulativeeffects due to the same group of students always being partof the same intervention for the entire semester.In study 1, we investigated the extent to which askingstudents to draw a diagram or providing them with onedrawn by an expert impacts their problem solving perfor-mance. This investigation was carried out for all the quizproblems in a second semester introductory algebra-basedphysics course. We found that the performance of studentsprovided with a diagram was significantly worse than theperformance of students in other groups in two problemsfrom electricity which we discuss below and one problemrelated to standing sound waves in a tube discussedelsewhere [18].The two electricity problems are the following (thediagrams provided to students in DO are shown inFigs. 1 and 2): Problem 1 “ Two identical point charges are initially fixed todiagonally opposite corners of a square that is 1 m on aside. Each of the two charges q is 3 C. How much work isdone by the electric force if one of the charges is movedfrom its initial position to an empty corner of the square? ” FIG. 1. Diagram for problem 1 provided to students in the DOgroup.
CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, roblem 2 “ A particle with mass − kg and a positive charge q of3 C is released from rest from point A in a uniform electricfield. When the particle arrives at point B , its electricalpotential is 25 V lower than the potential at A . Assumingthe only force acting on the particle is the electrostaticforce, find the speed of the particle when it arrives atpoint B . ” These diagrams (shown in Figs. 1 and 2) were drawn bythe instructor and they are very similar to what most physicsexperts would initially draw in order to solve the problems(However, the diagrams may be augmented further in theproblem solving process as needed). Furthermore, thesecond diagram also includes an important piece ofinformation from the problem statement that would nor-mally be included in a known quantities or target quantitiessection of a solution. Neither diagram was meant to trickthe students, but rather they were provided as scaffoldingsupport for them. In the interviews conducted in study 2,students were asked to comment on the diagrams and all ofthem indicated that the diagrams were clear.In order to ensure homogeneity of scoring, we developedrubrics for each problem analyzed and made sure that therewas at least 90% interrater reliability between two inde-pendent raters on at least 10% of the data. The development of the rubric for each problem went through an iterativeprocess. During the development of the rubric, the tworaters also discussed a student ’ s score separately from theone obtained using the rubric and adjusted the rubric if itwas agreed that the version of the rubric was too stringentor too generous. After each adjustment, all students werescored again on the improved rubric. In Table I, we providethe summary of the final version of the rubric used to gradeproblem 1. The rubric for problem 2 is similar and isincluded in the Appendix in Table IV.Problem 1 could be solved by employing two analogousapproaches. The first approach (method 1 in Table I) is touse W ¼ − q Δ V in which q is the charge of the particle and Δ V is the change in electric potential between the initialand final positions of the charge. The second approach(method 2 in Table I) is to use W ¼ − Δ U in which Δ U isthe change in the electric potential energy of the configu-ration of charges between the initial and the final situation.The two approaches are analogous because in both casesone must consider the initial and final situations (charges atopposite corner of the square, charges at adjacent cornersof the square) and determine a change in a physical quantity(it is also evident that the two approaches are analogous ifone uses the connection between electric potential andelectric potential energy, namely, V ¼ U / q ).Table I shows that for each of the two analogousmethods, there are two parts to the rubric: correct andincorrect ideas. Table I also shows that in the correct ideaspart, the problem was divided into different sections andpoints were assigned to each section (10 maximum points).Each student starts out with 10 points and in the IncorrectIdeas part we list the common mistakes students made andhow many points were deducted for each of those mistakes.Using the electrostatic force approach to solve this problemis not an effective strategy for students in an algebra-based TABLE I. Summary of the rubric for problem 1 ( W , V , U , q refer to work done by the electric force, electricpotential, electric potential energy, and charge, respectively).Correct ideasMethod 1 Method 2Section 1 1. W ¼ − q Δ V W ¼ − Δ U V f , V i and find Δ V ¼ V f − V i
2. Solve for U f , U i and find Δ U ¼ U f − U i − p), if no units ( − p)Method 1 Method 2Section 1 1. Used incorrect equation 1. Used incorrect equation − pSection 2 2.1 Solved for V f or V i incorrectly 2.1 Solved for U f or U i incorrectly − p2.1 Solved for V f and V i incorrectly 2.1 Solved for U f and U i incorrectly − p2.2 Did not subtract ( − p), and/orother mistake ( − p) 2.2 Did not subtract ( − p),and/or other mistake ( − p) − / − − pSection 3 3. Incorrect or no units 3. Incorrect or no units − pFIG. 2. Diagram for problem 2 provided to students in the DOgroup. MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
2. Solve for U f , U i and find Δ U ¼ U f − U i − p), if no units ( − p)Method 1 Method 2Section 1 1. Used incorrect equation 1. Used incorrect equation − pSection 2 2.1 Solved for V f or V i incorrectly 2.1 Solved for U f or U i incorrectly − p2.1 Solved for V f and V i incorrectly 2.1 Solved for U f and U i incorrectly − p2.2 Did not subtract ( − p), and/orother mistake ( − p) 2.2 Did not subtract ( − p),and/or other mistake ( − p) − / − − pSection 3 3. Incorrect or no units 3. Incorrect or no units − pFIG. 2. Diagram for problem 2 provided to students in the DOgroup. MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
B. Results for study 1
Before discussing the findings for the two problemsoutlined, we note that the two problems analyzed were partof the same three problem recitation quiz. In the thirdproblem of that quiz, we did not find any statisticallysignificant differences in the performance of the differentgroups (PO, DO, and NS). Furthermore, students in differ-ent groups exhibited almost identical performance onmidterm and final examinations.Table II shows that the average performance of stu-dents provided with diagrams was lower by roughly 20%compared to student performance in the other interven-tion groups. ANOVA [64] indicates that the groups arenot comparable ( p < . ) and post hoc comparisons between individual groups were conducted to investigateperformance differences between groups (we report p values obtained with the Scheffe algorithm for post hoc comparisons). We also calculated effect sizes (Cohen ’ s d [64]) for the performance comparisons between differentgroups. The p values and effect sizes are shown inTable III.Table III shows that students who were provided diagrams(DO group) performed significantly worse than students inthe other two groups. The effect sizes for comparing theperformance of students in the DO group with the othergroups are quite large, especially for problem 2 where theperformance of students in the DO group was on average onestandard deviation lower than the performance of students inthe other groups. Table III also shows that the performancesof students in the PO and NS groups are comparable on bothproblems. We note that, for problem 1, all students drew adiagram even if they were not specifically asked to do so.However, for problem 2, only 57% of the students in the NSgroup drew a diagram. But within the NS group, there are nostatistical differences between the performance of the stu-dents who drew a diagram and those who did not draw adiagram. We performed a t test to compare the performanceof students in the NS group who did not draw a diagram andall students in the DO group. We found that students in theDO group performed significantly worse than students in theNS group who did not draw a diagram ( p value ¼ . ,effect size ¼ . ). Thus, on problem 2, even students whodid not draw a diagram performed better than those who wereprovided a diagram (drawn by the instructor) with theproblem statement. Possible reasons for this counterintuitiveresult were explored and will be discussed in study 2. TABLE II. Group sizes, averages, and standard deviations for the scores of students in the different groups, out of10 points.Problem 1 Group size Average Standard deviationPO (students prompted to draw a diagram) 26 8.5 1.9DO (students provided with a diagram) 34 6.9 2.8NS (students provided with no support) 51 9.0 1.4Problem 2 Group size Average Standard deviationPO (students prompted to draw a diagram) 26 9.0 1.4DO (students provided with a diagram) 34 6.4 3.1NS (students provided with no support) 51 8.6 1.3TABLE III. p values (obtained using the Scheffe algorithm) and effect sizes for comparisons between the differentgroups. DO-PO DO-NS PO-NS p value Effect size p value Effect size p value Effect sizeProblem 1 0.024 0.634 < . < . < . CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
Before discussing the findings for the two problemsoutlined, we note that the two problems analyzed were partof the same three problem recitation quiz. In the thirdproblem of that quiz, we did not find any statisticallysignificant differences in the performance of the differentgroups (PO, DO, and NS). Furthermore, students in differ-ent groups exhibited almost identical performance onmidterm and final examinations.Table II shows that the average performance of stu-dents provided with diagrams was lower by roughly 20%compared to student performance in the other interven-tion groups. ANOVA [64] indicates that the groups arenot comparable ( p < . ) and post hoc comparisons between individual groups were conducted to investigateperformance differences between groups (we report p values obtained with the Scheffe algorithm for post hoc comparisons). We also calculated effect sizes (Cohen ’ s d [64]) for the performance comparisons between differentgroups. The p values and effect sizes are shown inTable III.Table III shows that students who were provided diagrams(DO group) performed significantly worse than students inthe other two groups. The effect sizes for comparing theperformance of students in the DO group with the othergroups are quite large, especially for problem 2 where theperformance of students in the DO group was on average onestandard deviation lower than the performance of students inthe other groups. Table III also shows that the performancesof students in the PO and NS groups are comparable on bothproblems. We note that, for problem 1, all students drew adiagram even if they were not specifically asked to do so.However, for problem 2, only 57% of the students in the NSgroup drew a diagram. But within the NS group, there are nostatistical differences between the performance of the stu-dents who drew a diagram and those who did not draw adiagram. We performed a t test to compare the performanceof students in the NS group who did not draw a diagram andall students in the DO group. We found that students in theDO group performed significantly worse than students in theNS group who did not draw a diagram ( p value ¼ . ,effect size ¼ . ). Thus, on problem 2, even students whodid not draw a diagram performed better than those who wereprovided a diagram (drawn by the instructor) with theproblem statement. Possible reasons for this counterintuitiveresult were explored and will be discussed in study 2. TABLE II. Group sizes, averages, and standard deviations for the scores of students in the different groups, out of10 points.Problem 1 Group size Average Standard deviationPO (students prompted to draw a diagram) 26 8.5 1.9DO (students provided with a diagram) 34 6.9 2.8NS (students provided with no support) 51 9.0 1.4Problem 2 Group size Average Standard deviationPO (students prompted to draw a diagram) 26 9.0 1.4DO (students provided with a diagram) 34 6.4 3.1NS (students provided with no support) 51 8.6 1.3TABLE III. p values (obtained using the Scheffe algorithm) and effect sizes for comparisons between the differentgroups. DO-PO DO-NS PO-NS p value Effect size p value Effect size p value Effect sizeProblem 1 0.024 0.634 < . < . < . CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, p values for comparison of the percentage of poorlyperforming students between DO-PO and DO-NS viaFisher ’ s exact test [64,65] are 0.033 and 0.001, respec-tively, for problem 1 and 0.002 and < . forproblem 2) but the percentages of students with anintermediate score are comparable (all p values identifiedwith Fisher ’ s exact test are larger than 0.5). III. STUDY 2A. Methodology for study 2
In order to investigate possible reasons for the findings ofstudy 1, interviews were conducted with twenty-three paidstudent volunteers who were at the time enrolled in anequivalent second semester algebra-based introductoryphysics course. At the time of the interviews, all studentshad taken an exam in their course which covered electro-statics and their exam scores varied from below average(e.g., a score of 60 when the class average was 70) to wellabove average (e.g., a score of 90 when the average is 70).It was not clear a priori how the interview protocol wouldaffect students ’ reasoning and problem solving approaches.For example, it is possible that the think-aloud protocolwould alter how students engage in problem solvingcompared to the case when they are not thinking aloudwhile solving problems. Therefore, the researchers usedone type of interview protocol for some of the students andanother type of protocol for another set of students. Inparticular, six of these interviews were conducted using athink-aloud protocol [63], while in the other seventeeninterviews, the students solved the problems while being observed by one of the researchers who took detailed notesof what the students were writing down and at what times(we refer to these latter interviews as the “ observational ” interviews and the first six interviews as the “ think-aloud ” interviews). All these interviews took place after studentslearned and were tested in their course on the relevantconcepts required for successfully solving these problems.At the end of all of the interviews, the interviewed studentswere explicitly asked to comment on the diagrams and howtheir problem solving processes were impacted by beingprovided the diagrams in two of the problems.The goal of the interviews was to investigate the extent towhich providing a diagram versus not providing a diagraminfluences how students engage in problem solving. Thus,in the interviews, students were asked to solve an additionalproblem, which required the use of the same concepts(conservation of energy or work, electric potential, electricpotential energy, etc.) as the two problems discussed in thispaper. However, in this additional problem, a diagram wasnot provided. Additional problem “ A particle of mass − kg and charge q ¼ μ C isshot at a speed of m/s directly towards another particlewith charge q ¼ μ C that is held fixed. If the initialdistance between the two particles is 1 m, how close doesthe particle with charge q get to q ? ” It is important to note that since problems 1 and 2 bothinvolved considerations of initial and final situations(and were the only two problems from the ones weanalyzed which fit this description), the additional prob-lem also involves considerations of initial and finalsituations. We also made sure that the additional problemwas of comparable difficulty to the other problems. Wegave the additional problem and problem 2 from study 1as a quiz to a class of 43 algebra-based introductorystudents and developed a rubric to score the students ’ problem solving performance on the additional problem(the rubric was similar to the one shown in Table I and isincluded in the Appendix in Table V). This rubric was FIG. 3. Percentages of students from each intervention group (PO, DO, and NS) who earned a score of 4 or less, earned a score of 5, 6,or 7, or earned a score above 8.
MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14, p value for comparingstudents ’ performance on these two problems was0.626 and the effect size was 0.105, indicating similarstudent performance on these two problems). Thus,the additional problem is of comparable difficulty toproblem 2, and in study 1, we found that problem 1 is ofcomparable difficulty to problem 2 (overall averages were8.04 and 8.27 on problem 1 and problem 2, respectively).Finally, the additional problem was designed such that astudent could potentially solve it without drawing adiagram. Out of the 43 students who solved the additionalproblem, 35% of them did not draw a diagram (and hadcomplete solutions, some correct, some incorrect) indi-cating that a significant fraction of algebra-based intro-ductory students did not think that drawing a diagram forthis problem is necessary in order to solve it.The goal of the interviews was to compare students ’ problem solving approaches in the additional problemwhich did not provide a diagram with the two problemsfrom this study in which the diagrams shown in Figs. 1and 2 were provided. Since it is unclear if the interviewresults would be altered if students solve the additionalproblem first or last, the order in which students wereasked to solve the problems was varied: in the first roundof interviews (six think-aloud interviews and elevenobservational interviews) students solved the additionalproblem first followed by problems 1 and 2 from study 1,and in the second round of interviews (six observationalinterviews), students solved problems 1 and 2 from study1 first followed by the additional problem. In all inter-views, students were provided diagrams for problems 1and 2, but were not provided a diagram for the additionalproblem. In addition, the interviews were designed tomimic the quiz situation as closely as possible andtherefore, students were provided with an equationsheet which was photocopied from the textbook ’ s [66]end of chapter summary (chapter 19, which discusseselectrostatic potential and electrostatic potential energy).Students were provided with this equation sheet becausein quizzes throughout the semester, the teaching assistantprovided students with relevant equations. B. Results for study 2
None of the twenty-three interviewed students men-tioned anything negative about the diagrams and in generalthey thought that the diagrams were helpful when provided.A few students noted that they did not necessarily gainanything from being provided the diagrams because if theyhad not been provided diagrams, they would have drawnsomething similar. Additionally, the interviews suggested that students were interpreting the diagrams provided inproblems 1 and 2 in the intended manner (i.e., they did notfind them confusing).In the six think-aloud interviews, students appeared toapproach problems similarly whether or not they wereprovided a diagram. Three students did not draw adiagram for the additional problem, but they also didnot pay much attention to the provided diagrams inproblems 1 and 2. The other three students drew diagramsfor the additional problem, and in problems 1 and 2, theyappeared to pay attention to the diagrams, or drew theirown diagram even though one was provided. Also, whilereading problems 1 and 2, these students paused to look atthe diagram, then read some more, again looked at thediagram, etc. Regarding the approaches to solving theproblem, we estimated how much time they spent con-ceptually analyzing the problem before moving on to theimplementation stage. This time was estimated by timingstudents from when they first started reading the problemstatement until they wrote down an equation from theequation sheet provided and started performing algebraicsteps. (Note that sometimes students looked at theequation sheet provided and wrote down a formula afterwhich they returned to the equation sheet, or thoughtabout the problem more without performing algebraicsteps or writing other formulas down. In cases such asthese, the interviewer waited until the student actuallystarted performing algebraic steps to estimate the con-ceptual planning time.) We found that students spentabout the same time conceptually analyzing each of thethree problems and also spent about the same timesolving each problem. These interviews suggested thatpartly due to being asked to verbalize their thoughtprocess, each student approached the three problems ina very similar manner and was not influenced by beingprovided diagrams in two of the problems: students oftenexplicitly justified their problem solving approach andtried to provide reasoning. The researchers analyzed thethink aloud interviews and concluded that a think-aloudsetting did not reproduce quiz conditions very well. Inparticular, while solving a problem in a quiz, moststudents do not engage in this type of explicit thinkaloud reasoning and justification. The researchers realizedthat asking students to verbalize their thought process infront of a researcher helped motivate students to come upwith arguments to support their solutions, and incentiv-ized them to understand the problems they were asked tosolve. This conclusion is supported by prior studies aswell. For example, in DeVore et al. ’ s study on thechallenges in engaging students with self-paced learningtools [67], students were likely to engage much moredeeply with a learning tutorial when they were thinkingout loud in front of a researcher compared to whenworking on the tutorial on their own. Students thereforelearned significantly more from the tutorials whenCASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
None of the twenty-three interviewed students men-tioned anything negative about the diagrams and in generalthey thought that the diagrams were helpful when provided.A few students noted that they did not necessarily gainanything from being provided the diagrams because if theyhad not been provided diagrams, they would have drawnsomething similar. Additionally, the interviews suggested that students were interpreting the diagrams provided inproblems 1 and 2 in the intended manner (i.e., they did notfind them confusing).In the six think-aloud interviews, students appeared toapproach problems similarly whether or not they wereprovided a diagram. Three students did not draw adiagram for the additional problem, but they also didnot pay much attention to the provided diagrams inproblems 1 and 2. The other three students drew diagramsfor the additional problem, and in problems 1 and 2, theyappeared to pay attention to the diagrams, or drew theirown diagram even though one was provided. Also, whilereading problems 1 and 2, these students paused to look atthe diagram, then read some more, again looked at thediagram, etc. Regarding the approaches to solving theproblem, we estimated how much time they spent con-ceptually analyzing the problem before moving on to theimplementation stage. This time was estimated by timingstudents from when they first started reading the problemstatement until they wrote down an equation from theequation sheet provided and started performing algebraicsteps. (Note that sometimes students looked at theequation sheet provided and wrote down a formula afterwhich they returned to the equation sheet, or thoughtabout the problem more without performing algebraicsteps or writing other formulas down. In cases such asthese, the interviewer waited until the student actuallystarted performing algebraic steps to estimate the con-ceptual planning time.) We found that students spentabout the same time conceptually analyzing each of thethree problems and also spent about the same timesolving each problem. These interviews suggested thatpartly due to being asked to verbalize their thoughtprocess, each student approached the three problems ina very similar manner and was not influenced by beingprovided diagrams in two of the problems: students oftenexplicitly justified their problem solving approach andtried to provide reasoning. The researchers analyzed thethink aloud interviews and concluded that a think-aloudsetting did not reproduce quiz conditions very well. Inparticular, while solving a problem in a quiz, moststudents do not engage in this type of explicit thinkaloud reasoning and justification. The researchers realizedthat asking students to verbalize their thought process infront of a researcher helped motivate students to come upwith arguments to support their solutions, and incentiv-ized them to understand the problems they were asked tosolve. This conclusion is supported by prior studies aswell. For example, in DeVore et al. ’ s study on thechallenges in engaging students with self-paced learningtools [67], students were likely to engage much moredeeply with a learning tutorial when they were thinkingout loud in front of a researcher compared to whenworking on the tutorial on their own. Students thereforelearned significantly more from the tutorials whenCASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, ’ s study on self-explanations [68] found that students who elicited moreexplanations (remarks related to physics content) whenstudying worked out examples while thinking aloudshowed significantly better performance in solving sub-sequent problems related to the same content thanstudents who elicited fewer self-explanations. If instead,students are not asked to think aloud while solving aproblem or studying worked out examples, they are lesslikely to self-explain, and some may not do it at all.While the think-aloud setting did not reproduce the quizsetting, these six think-aloud interviews provided valuableinformation because they offered further evidence that theadditional problem was well chosen. In particular, studentsspent about the same amount of time conceptually analyz-ing this additional problem as they did conceptuallyanalyzing the other two problems and they spent aboutthe same amount of time solving this additional problem asthey did solving the other two problems. This similarityindicated that the additional problem required a similaramount of time for students to conceptually analyze andcomplete as the other two problems. Also, the students whohad more formula centered approaches to solving problemsdid not draw a diagram for the additional problem,providing further evidence that some students did notconsider that drawing a diagram was necessary or helpfulto solve the additional problem.In the first eleven observational interviews, studentssolved the additional problem (which did not provide adiagram) followed by problems 1 and 2 (which provideddiagrams) from study 1 and in the last six observationalinterviews conducted, the order of the problems wasswitched and students first solved problems 1 and 2 fromstudy 1 (diagrams provided) and then solved the additionalproblem (diagram not provided).It is important to stress that this design ensures that eachstudent acts as their own control. In other words, thedetermination of whether a student spends more or lesstime conceptually analyzing a problem which has a dia-gram provided compared to another problem which doesnot provide a diagram was done for each student. This isimportant because different students may have differentproblem solving strategies and may end up spendingdifferent amounts of time conceptually analyzing a prob-lem. But if each student acts as their own control, we candraw conclusions about the extent to which a provideddiagram may influence the amount of time a student spendsconceptually analyzing a problem.During these seventeen observational interviews, roughlyhalf the students (nine students) started the implementationstage of the problem solving process while solving problems1 and 2, in which the diagrams were provided, noticeablyearlier than when solving the additional problem in which adiagram was not provided. This was found both for students who solved the additional problem first (six out of eleveninterviewed students) and for students who solved theadditional problem last (three out of six interviewed stu-dents). Including all seventeen students, we found that onaverage, students spent 71% more time conceptually plan-ning the additional problem (which did not provide adiagram) than the other two problems. In a few of thosecases, in one problem or the other, this quicker focus onmanipulation of equations appeared to negatively impacttheir performance.Suzana (an interviewed student) had received an 83%on the electricity exam, and an A in the first semesterphysics class. She solved the problems in the order(i) additional problem (no diagram provided), (ii) problem1 (diagram provided), (iii) problem 2 (diagram provided).Her work for the additional problem and problem 2 areshown in Fig. 4 (in her work, EPE refers to electricpotential energy, or U ). While solving the additionalproblem (given first) in which a diagram was notprovided, it appeared that Suzana was aware that electricpotential and electric potential energy are differentbecause she used the equation which relates these twoquantities, EPE ¼ q V and explicitly solved for theelectric potential energy of two charges to obtain EPE ¼ kqq / r . She also correctly used electric potential energyto solve the additional problem and it was apparentfrom the interview that she used the resources for electricpotential energy and electric potential appropriately. Onthe other hand, while solving problem 2, Suzana immedi-ately wrote down two electric potential energies: EPE A ¼ V and EPE B ¼ V, even though the diagram pro-vided contained an equation relating electric potentials( V A − V A ¼ V), not electric potential energies. Anddespite the fact that she used the resources of electricpotential and electric potential energy appropriately in aprevious problem (which did not provide a diagram), sheappeared to have difficulty distinguishing between themin this problem (which did provide a diagram).Another student, Calvin, solved the problems in the order(i) problem 1 (diagram provided), (ii) problem 2 (diagramprovided), (iii) additional problem (no diagram provided). Inthe first two problems, Calvin treated the electric potential aselectric potential energy. For example, in problem 2 (workshown in Fig. 5), after a false start with electric potential, heattempted to use conservation of energy and wrote KE final ¼ V even though 25 V is given to be the change in electricpotential (the diagram also explicitly shows the informationthat V A − V B ¼ V). It appeared that Calvin was attempt-ing to use information provided in the problem beforeanalyzing the problem qualitatively and ensuring he under-stood what the information really meant. In the additionalproblem, on the other hand (work shown in Fig. 6), Calvinspent significantly more time and, while he also usedconservation of energy in that problem, he used the correctMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
None of the twenty-three interviewed students men-tioned anything negative about the diagrams and in generalthey thought that the diagrams were helpful when provided.A few students noted that they did not necessarily gainanything from being provided the diagrams because if theyhad not been provided diagrams, they would have drawnsomething similar. Additionally, the interviews suggested that students were interpreting the diagrams provided inproblems 1 and 2 in the intended manner (i.e., they did notfind them confusing).In the six think-aloud interviews, students appeared toapproach problems similarly whether or not they wereprovided a diagram. Three students did not draw adiagram for the additional problem, but they also didnot pay much attention to the provided diagrams inproblems 1 and 2. The other three students drew diagramsfor the additional problem, and in problems 1 and 2, theyappeared to pay attention to the diagrams, or drew theirown diagram even though one was provided. Also, whilereading problems 1 and 2, these students paused to look atthe diagram, then read some more, again looked at thediagram, etc. Regarding the approaches to solving theproblem, we estimated how much time they spent con-ceptually analyzing the problem before moving on to theimplementation stage. This time was estimated by timingstudents from when they first started reading the problemstatement until they wrote down an equation from theequation sheet provided and started performing algebraicsteps. (Note that sometimes students looked at theequation sheet provided and wrote down a formula afterwhich they returned to the equation sheet, or thoughtabout the problem more without performing algebraicsteps or writing other formulas down. In cases such asthese, the interviewer waited until the student actuallystarted performing algebraic steps to estimate the con-ceptual planning time.) We found that students spentabout the same time conceptually analyzing each of thethree problems and also spent about the same timesolving each problem. These interviews suggested thatpartly due to being asked to verbalize their thoughtprocess, each student approached the three problems ina very similar manner and was not influenced by beingprovided diagrams in two of the problems: students oftenexplicitly justified their problem solving approach andtried to provide reasoning. The researchers analyzed thethink aloud interviews and concluded that a think-aloudsetting did not reproduce quiz conditions very well. Inparticular, while solving a problem in a quiz, moststudents do not engage in this type of explicit thinkaloud reasoning and justification. The researchers realizedthat asking students to verbalize their thought process infront of a researcher helped motivate students to come upwith arguments to support their solutions, and incentiv-ized them to understand the problems they were asked tosolve. This conclusion is supported by prior studies aswell. For example, in DeVore et al. ’ s study on thechallenges in engaging students with self-paced learningtools [67], students were likely to engage much moredeeply with a learning tutorial when they were thinkingout loud in front of a researcher compared to whenworking on the tutorial on their own. Students thereforelearned significantly more from the tutorials whenCASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, ’ s study on self-explanations [68] found that students who elicited moreexplanations (remarks related to physics content) whenstudying worked out examples while thinking aloudshowed significantly better performance in solving sub-sequent problems related to the same content thanstudents who elicited fewer self-explanations. If instead,students are not asked to think aloud while solving aproblem or studying worked out examples, they are lesslikely to self-explain, and some may not do it at all.While the think-aloud setting did not reproduce the quizsetting, these six think-aloud interviews provided valuableinformation because they offered further evidence that theadditional problem was well chosen. In particular, studentsspent about the same amount of time conceptually analyz-ing this additional problem as they did conceptuallyanalyzing the other two problems and they spent aboutthe same amount of time solving this additional problem asthey did solving the other two problems. This similarityindicated that the additional problem required a similaramount of time for students to conceptually analyze andcomplete as the other two problems. Also, the students whohad more formula centered approaches to solving problemsdid not draw a diagram for the additional problem,providing further evidence that some students did notconsider that drawing a diagram was necessary or helpfulto solve the additional problem.In the first eleven observational interviews, studentssolved the additional problem (which did not provide adiagram) followed by problems 1 and 2 (which provideddiagrams) from study 1 and in the last six observationalinterviews conducted, the order of the problems wasswitched and students first solved problems 1 and 2 fromstudy 1 (diagrams provided) and then solved the additionalproblem (diagram not provided).It is important to stress that this design ensures that eachstudent acts as their own control. In other words, thedetermination of whether a student spends more or lesstime conceptually analyzing a problem which has a dia-gram provided compared to another problem which doesnot provide a diagram was done for each student. This isimportant because different students may have differentproblem solving strategies and may end up spendingdifferent amounts of time conceptually analyzing a prob-lem. But if each student acts as their own control, we candraw conclusions about the extent to which a provideddiagram may influence the amount of time a student spendsconceptually analyzing a problem.During these seventeen observational interviews, roughlyhalf the students (nine students) started the implementationstage of the problem solving process while solving problems1 and 2, in which the diagrams were provided, noticeablyearlier than when solving the additional problem in which adiagram was not provided. This was found both for students who solved the additional problem first (six out of eleveninterviewed students) and for students who solved theadditional problem last (three out of six interviewed stu-dents). Including all seventeen students, we found that onaverage, students spent 71% more time conceptually plan-ning the additional problem (which did not provide adiagram) than the other two problems. In a few of thosecases, in one problem or the other, this quicker focus onmanipulation of equations appeared to negatively impacttheir performance.Suzana (an interviewed student) had received an 83%on the electricity exam, and an A in the first semesterphysics class. She solved the problems in the order(i) additional problem (no diagram provided), (ii) problem1 (diagram provided), (iii) problem 2 (diagram provided).Her work for the additional problem and problem 2 areshown in Fig. 4 (in her work, EPE refers to electricpotential energy, or U ). While solving the additionalproblem (given first) in which a diagram was notprovided, it appeared that Suzana was aware that electricpotential and electric potential energy are differentbecause she used the equation which relates these twoquantities, EPE ¼ q V and explicitly solved for theelectric potential energy of two charges to obtain EPE ¼ kqq / r . She also correctly used electric potential energyto solve the additional problem and it was apparentfrom the interview that she used the resources for electricpotential energy and electric potential appropriately. Onthe other hand, while solving problem 2, Suzana immedi-ately wrote down two electric potential energies: EPE A ¼ V and EPE B ¼ V, even though the diagram pro-vided contained an equation relating electric potentials( V A − V A ¼ V), not electric potential energies. Anddespite the fact that she used the resources of electricpotential and electric potential energy appropriately in aprevious problem (which did not provide a diagram), sheappeared to have difficulty distinguishing between themin this problem (which did provide a diagram).Another student, Calvin, solved the problems in the order(i) problem 1 (diagram provided), (ii) problem 2 (diagramprovided), (iii) additional problem (no diagram provided). Inthe first two problems, Calvin treated the electric potential aselectric potential energy. For example, in problem 2 (workshown in Fig. 5), after a false start with electric potential, heattempted to use conservation of energy and wrote KE final ¼ V even though 25 V is given to be the change in electricpotential (the diagram also explicitly shows the informationthat V A − V B ¼ V). It appeared that Calvin was attempt-ing to use information provided in the problem beforeanalyzing the problem qualitatively and ensuring he under-stood what the information really meant. In the additionalproblem, on the other hand (work shown in Fig. 6), Calvinspent significantly more time and, while he also usedconservation of energy in that problem, he used the correctMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14, r : U ¼ kq q / r .Interviews suggested that these students proceeded tomanipulate equations earlier in the problems which pro-vided a diagram and they did not spend sufficient timeconceptually analyzing these problems. In Suzana ’ s case,despite the fact that she had previously realized whilesolving the first problem that electric potential energy andelectric potential are different, in problem 2 she did notappear to distinguish between them which resulted in anincorrect solution. In Calvin ’ s case, he also realized that theelectric potential and electric potential energy are differentonly while solving the problem which did not provide adiagram.In fact, similar to these two students, seven otherinterviewed students, almost immediately after reading thisproblem, which included a diagram, started looking at theequation sheet. Then, they often copied a formula on theirpaper and proceeded to solve the problems using theformula, which sometimes negatively impacted their per-formance on these problems. The observational interviews suggest that roughly half ofthe students were spending less time conceptually analyz-ing the problems in which diagrams were provided. Thesestudents appeared to jump to the implementation stage ofthe problem solving process immediately, which some-times had a negative impact on their performance. IV. DISCUSSION AND SUMMARY
Prior research suggests that students in classeswhich promote conceptual understanding through active-learning methods outperform those from traditionally
FIG. 4. Suzana ’ s solution to the additional problem (left) on which she worked first and problem 2 (right), on which she worked last.FIG. 5. Calvin ’ s work to problem 2 (second problem solved ininterview). FIG. 6. Calvin ’ s work to the additional problem (last problemsolved in interview). CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
FIG. 4. Suzana ’ s solution to the additional problem (left) on which she worked first and problem 2 (right), on which she worked last.FIG. 5. Calvin ’ s work to problem 2 (second problem solved ininterview). FIG. 6. Calvin ’ s work to the additional problem (last problemsolved in interview). CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, – — something many introductory studentsskip without explicit guidance and support, and when theydo skip this important stage, it can lead to deterioratedperformance.It is also important to note that students who were askedto draw diagrams were almost always statistically morelikely to draw productive diagrams (as defined from anexpert ’ s point of view) than students in the other inter-vention groups. Within a cognitive apprenticeship model[76], asking students to draw diagrams is a type ofscaffolding support, and this investigation indicates thatstudents asked to draw a diagram did not perform worsethan those provided with no support on typical quizproblems (i.e., problems that are not very difficult orcomplex). Therefore, an implication of the study reportedhere based on the cognitive apprenticeship model is that inorder to help students learn the usefulness of drawingdiagrams in problem solving, students can be asked to drawdiagrams in various assignments and quizzes throughoutthe semester. This support can be reduced as students begindrawing more diagrams and recognize their usefulness ontheir own. Moreover, since assessment drives learning [77],it is likely that rewarding students for drawing appropriatediagrams will have a beneficial effect. This can be onehelpful step in getting students accustomed to usingproductive problem solving heuristics, and over timemaking them better at performing initial conceptual analy-sis and planning of the problem solution on their own.Finally, prompted by the results of study 1, we haddiscussions with ten instructors who regularly teach intro-ductory physics and they nearly always conjectured thatproviding students with diagrams would likely lead toimproved performance. Our study suggests that providinga diagram was never helpful for students, and, in certainMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
FIG. 4. Suzana ’ s solution to the additional problem (left) on which she worked first and problem 2 (right), on which she worked last.FIG. 5. Calvin ’ s work to problem 2 (second problem solved ininterview). FIG. 6. Calvin ’ s work to the additional problem (last problemsolved in interview). CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14, – — something many introductory studentsskip without explicit guidance and support, and when theydo skip this important stage, it can lead to deterioratedperformance.It is also important to note that students who were askedto draw diagrams were almost always statistically morelikely to draw productive diagrams (as defined from anexpert ’ s point of view) than students in the other inter-vention groups. Within a cognitive apprenticeship model[76], asking students to draw diagrams is a type ofscaffolding support, and this investigation indicates thatstudents asked to draw a diagram did not perform worsethan those provided with no support on typical quizproblems (i.e., problems that are not very difficult orcomplex). Therefore, an implication of the study reportedhere based on the cognitive apprenticeship model is that inorder to help students learn the usefulness of drawingdiagrams in problem solving, students can be asked to drawdiagrams in various assignments and quizzes throughoutthe semester. This support can be reduced as students begindrawing more diagrams and recognize their usefulness ontheir own. Moreover, since assessment drives learning [77],it is likely that rewarding students for drawing appropriatediagrams will have a beneficial effect. This can be onehelpful step in getting students accustomed to usingproductive problem solving heuristics, and over timemaking them better at performing initial conceptual analy-sis and planning of the problem solution on their own.Finally, prompted by the results of study 1, we haddiscussions with ten instructors who regularly teach intro-ductory physics and they nearly always conjectured thatproviding students with diagrams would likely lead toimproved performance. Our study suggests that providinga diagram was never helpful for students, and, in certainMARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14, ’ performance, is quite complex and not at all intuitive. ACKNOWLEDGMENTS
We would like to thank the National Science Foundationfor Grant No. 1524575 and the members of the physicseducation research group at the University of Pittsburgh(E. Marshman, S. DeVore) as well as R. P. Devaty for usefuldiscussions and feedback on the manuscript.
APPENDIX: WORKED OUT SOLUTIONS TO PROBLEMS 1 AND 2 AND RUBRICS USED FORPROBLEM 2 AND THE ADDITIONAL PROBLEM
Figure 7 shows an instructor worked out solutions to problems 1 and 2 as well as how the points were allocated to variousparts of the solutions.
FIG. 7. Worked out solutions for problems 1 (left) and 2 (right) along with how many points were assigned to each part of the solution.Both problems can be solved using two equivalent methods and both methods are shown.TABLE IV. Rubric used to score students ’ problem solving performance on problem 2.Correct ideas Correct ideasMethod 1: work-energy theorem Method 2: conservation of energySection 1 W ¼ − q Δ V U i þ KE i ¼ U f þ KE f W correctly 2 p qV A − qV B ¼ Δ KE W ¼ Δ KE U ¼ qV KE ¼ mv KE ¼ mv − p Section 1 Used incorrect equation − pCalculated W incorrectly a − p Calculated Δ KE incorrectly − pObtained the wrong sign for W − p Obtained the wrong sign for Δ KE − pSection 2 Used incorrect equation connectingthe two parts − p Did not use U ¼ qV , or used incorrectequation − pUsed incorrect equation for KE − p Used incorrect equation for KE − pSection 3 Incorrect or no units − p Section 3 Incorrect or no units − p a If a student uses an incorrect equation, but uses it correctly, these two points are not taken off.
CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
CASE OF TWO ELECTROSTATICS PROBLEMS: … PHYS. REV. PHYS. EDUC. RES.14,
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MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
MARIES and SINGH PHYS. REV. PHYS. EDUC. RES.14,
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