Do students benefit from drawing productive diagrams themselves while solving introductory physics problems? The case of two electrostatic problems
EEuropean Journal of Physics
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Do students benefit from drawing productivediagrams themselves while solving introductoryphysics problems? The case of two electrostaticsproblems
To cite this article: Alexandru Maries and Chandralekha Singh 2018
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This content was downloaded from IP address 130.49.180.35 on 16/12/2017 at 18:16 o students bene fi t from drawingproductive diagrams themselves whilesolving introductory physics problems?The case of two electrostatics problems Alexandru Maries and Chandralekha Singh Department of Physics, University of Cincinnati, Cincinnati, OH 45221, United Statesof America Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA15260, United States of AmericaReceived 26 June 2017, revised 19 September 2017Accepted for publication 2 October 2017Published 13 December 2017
Abstract
An appropriate diagram is a required element of a solution building process inphysics problem solving and it can transform a given problem into a repre-sentation that is easier to exploit for solving the problem. A major focus whilehelping introductory physics students learn problem solving is to help themappreciate that drawing diagrams facilitates problem solving. We conducted aninvestigation in which two different interventions were implemented duringrecitation quizzes throughout the semester in a large enrolment, algebra-basedintroductory physics course. Students were either ( ) asked to solve problemsin which the diagrams were drawn for them or ( ) explicitly told to draw adiagram. A comparison group was not given any instruction regarding dia-grams. We developed a rubric to score the problem solving performance ofstudents in different intervention groups. We investigated two problemsinvolving electric fi eld and electric force and found that students who drewproductive diagrams were more successful problem solvers and that a higherlevel of relevant detail in a student ’ s diagram corresponded to a better score.We also conducted think-aloud interviews with nine students who were at thetime taking an equivalent introductory algebra-based physics course in order togain insight into how drawing diagrams affects the problem solving process.These interviews supported some of the interpretations of the quantitativeresults. We end by discussing instructional implications of the fi ndings. European Journal of PhysicsEur. J. Phys. ( ) ( ) https: // doi.org / / / aa9038 Original content from this work may be used under the terms of the Creative CommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to theauthor ( s ) and the title of the work, journal citation and DOI. / / + eywords: problem solving, visual representation, Physics EducationResearch
1. Introduction
Introductory physics is a challenging subject to learn. It is dif fi cult for introductory students toassociate the abstract concepts they study in physics with more concrete representations thatfacilitate understanding without an explicit instructional strategy aimed to aid them in thisregard. Without guidance, introductory students often employ formula oriented problemsolving strategies instead of developing a solid grasp of physical principles and concepts.There are many reasons to believe that multiple representations of concepts along with theability to construct, interpret and transform between different representations that correspondto the same physical system or process play a positive role in learning physics. First, physicsexperts often use multiple representations as a fi rst step in a problem solving process [ – ] .Second, students who are taught explicit problem solving strategies emphasising use ofdifferent representations of knowledge at various stages of problem solving construct higherquality and more effective representations and perform better than students who learn tra-ditional problem solving strategies [ ] . Third, multiple representations are very useful intranslating the initial, usually verbal description of a problem into a representation moresuitable to further analysis and mathematical manipulation [
5, 6 ] partly because the process ofconstructing an effective representation of a problem makes it easier to generate appropriatedecisions about the solution process. Also, getting students to represent a problem in differentways helps shift their focus from merely manipulating equations toward understandingphysics [ ] . Some researchers have argued that in order to understand a physical conceptthoroughly, one needs to be able to recognise and manipulate the concept in a variety ofrepresentations [
5, 7 ] . As Meltzer puts it [ ] , a range of diverse representations is required to ‘ span ’ the conceptual space associated with an idea. Since traditional courses, which generallydo not emphasise multiple representations, lead to low gains on the Force Concept Inventory [
9, 10 ] and on other assessments in the domain of electricity and magnetism [
11, 12 ] , in orderto improve students ’ understanding of physics concepts, many researchers have developedinstructional strategies that place explicit emphasis on multiple representations [
1, 5, 13, 14 ] while other researchers developed other strategies with implicit focus on multiple repre-sentations [
6, 15 – ] . Van Heuvelen ’ s approach [ ] , for example, starts by ensuring thatstudents explore the qualitative nature of concepts by using a variety of representations of aconcept in a familiar setting before adding the complexities of mathematics.One representation useful in the initial conceptual analysis and planning stages of asolution is a schematic diagram of the physical situation presented in the problem. Dia-grammatic representations have been shown to be superior to exclusively employing verbal ormathematical representations when solving problems [
3, 20 – ] . It is therefore not surprisingthat physics experts automatically employ diagrams in attempting to solve problems [
1, 7, 23 ] . However, introductory physics students need explicit help to ( ) understand thatdrawing a diagram is an important step in organising and simplifying the given informationinto a representation which is more suitable to further analysis [ ] , and ( ) learn how to drawappropriate and useful diagrams. Therefore, many researchers who have developed strategiesfor teaching students effective problem solving skills use scaffolding support designed to helpstudents recognise how important the step of drawing a diagram is in solving physics pro-blems, and guidance to help them draw useful diagrams. In Newtonian mechanics, Reif [ ] Eur. J. Phys. ( )015703 A Maries and C Singh
1, 7, 23 ] . However, introductory physics students need explicit help to ( ) understand thatdrawing a diagram is an important step in organising and simplifying the given informationinto a representation which is more suitable to further analysis [ ] , and ( ) learn how to drawappropriate and useful diagrams. Therefore, many researchers who have developed strategiesfor teaching students effective problem solving skills use scaffolding support designed to helpstudents recognise how important the step of drawing a diagram is in solving physics pro-blems, and guidance to help them draw useful diagrams. In Newtonian mechanics, Reif [ ] Eur. J. Phys. ( )015703 A Maries and C Singh as suggested that several diagrams be drawn: one diagram of the problem situation whichincludes all objects and one diagram for each system that needs to be considered separately.Also, he described in detail concrete steps that students need to take in order to draw thesediagrams as follows: ( a ) describe both motions and interactions, ( b ) identify interacting objects before forces, ( c ) separate long range and contact interactions, and ( d ) label contact points by the magnitude of the action – reaction pair of forces.Van Heuvelen ’ s Active Learning Problem Sheets [ ] adapted from Reif follow a verysimilar underlying approach. Other researchers who have emphasised, among other things,the importance of diagrams in their approach to teaching students problem solving skills havefound signi fi cant improvements in students ’ problem solving methods [
2, 5, 25 ] .Previous research shows that students who draw diagrams even if they are not rewardedfor it are more successful problem solvers [ ] . In addition, students who take courses whichemphasise effective problem solving heuristics which include drawing a diagram are morelikely to draw diagrams even on multiple-choice exams [ ] . An investigation into howspontaneous drawing of free body diagrams ( FBDs ) [ ] affects problem solving [ ] showsthat only drawing correct FBDs improves a student ’ s score and that students who drawincorrect FBDs do not perform better than students who draw no diagrams. Heckler [ ] investigated the effects of prompting students to draw FBDs in introductory mechanics byexplicitly asking students to draw clearly labelled FBDs. He found that students who wereprompted to draw FBDs were more likely to follow formally taught problem solving methodsrather than intuitive methods which sometimes caused deteriorated performance.This study is part of a larger investigation on the impact of using multiple representationsin physics problem solving [ ] , and one of the principal types of representations investigatedwere diagrammatic representations. The broad questions in this larger investigation as per-taining to diagrammatic representations were:1. Is there a correlation between drawing diagrams and problem solving performance evenwhen ( i ) students are not graded on drawing diagrams, and ( ii ) the solution to theproblem involves primarily mathematical manipulation of equations?2. Should students be provided diagrams or asked to draw them while solving introductoryphysics problems?The larger investigation also explored mathematical and graphical representations [ ] ,in particular, the extent to which students ’ mathematical and graphical representations ofelectric fi eld were consistent with one another and the impact of two different scaffoldingsupports designed to help students make the connection between the two representations. Inthis study, we primarily explored question 1. ( i ) , although the insight gained from this studycan be used to inform question 2. We investigated how prompting students to draw diagrams ( without being more speci fi c, e.g. prompting students to draw FBDs as in [ ] ) affects theirperformance in two electrostatics problems and how their performance is impacted whenprovided with a diagrammatic representation of the physical situation described in the pro-blems. There has been much research on students ’ conceptual understanding of electricity andmagnetism [ – ] . However, here, we are primarily interested in how students who drawdiagrams themselves as part of the problem solving process bene fi t ( in their problem solvingperformance ) from their diagrams and we also investigate how diagrams drawn by students,classi fi ed as productive or unproductive based upon certain criteria, affect student perfor-mance. In addition to the quantitative data collected, we conducted think-aloud interviews Eur. J. Phys. ( )015703 A Maries and C Singh
2, 5, 25 ] .Previous research shows that students who draw diagrams even if they are not rewardedfor it are more successful problem solvers [ ] . In addition, students who take courses whichemphasise effective problem solving heuristics which include drawing a diagram are morelikely to draw diagrams even on multiple-choice exams [ ] . An investigation into howspontaneous drawing of free body diagrams ( FBDs ) [ ] affects problem solving [ ] showsthat only drawing correct FBDs improves a student ’ s score and that students who drawincorrect FBDs do not perform better than students who draw no diagrams. Heckler [ ] investigated the effects of prompting students to draw FBDs in introductory mechanics byexplicitly asking students to draw clearly labelled FBDs. He found that students who wereprompted to draw FBDs were more likely to follow formally taught problem solving methodsrather than intuitive methods which sometimes caused deteriorated performance.This study is part of a larger investigation on the impact of using multiple representationsin physics problem solving [ ] , and one of the principal types of representations investigatedwere diagrammatic representations. The broad questions in this larger investigation as per-taining to diagrammatic representations were:1. Is there a correlation between drawing diagrams and problem solving performance evenwhen ( i ) students are not graded on drawing diagrams, and ( ii ) the solution to theproblem involves primarily mathematical manipulation of equations?2. Should students be provided diagrams or asked to draw them while solving introductoryphysics problems?The larger investigation also explored mathematical and graphical representations [ ] ,in particular, the extent to which students ’ mathematical and graphical representations ofelectric fi eld were consistent with one another and the impact of two different scaffoldingsupports designed to help students make the connection between the two representations. Inthis study, we primarily explored question 1. ( i ) , although the insight gained from this studycan be used to inform question 2. We investigated how prompting students to draw diagrams ( without being more speci fi c, e.g. prompting students to draw FBDs as in [ ] ) affects theirperformance in two electrostatics problems and how their performance is impacted whenprovided with a diagrammatic representation of the physical situation described in the pro-blems. There has been much research on students ’ conceptual understanding of electricity andmagnetism [ – ] . However, here, we are primarily interested in how students who drawdiagrams themselves as part of the problem solving process bene fi t ( in their problem solvingperformance ) from their diagrams and we also investigate how diagrams drawn by students,classi fi ed as productive or unproductive based upon certain criteria, affect student perfor-mance. In addition to the quantitative data collected, we conducted think-aloud interviews Eur. J. Phys. ( )015703 A Maries and C Singh ] with nine students who were taking an equivalent introductory algebra-based physicscourse at the time, to gain insight into how drawing ( or not drawing ) diagrams may affecttheir problem solving performance. The interviews provided possible interpretations for someof the quantitative fi ndings.
2. Methodology
A traditionally taught class of 120 algebra-based introductory physics students was enroled inthree different recitation sections. The three recitation sections formed the comparison groupand the two intervention groups for this investigation. All recitations were taught tradition-ally: the TA worked out problems similar to the homework problems and then gave a 15 minquiz at the end of class. Students in all recitations attended the same lectures, were assignedthe same homework, and took the same exams and quizzes. In the recitation quizzesthroughout the semester, students in the three different recitation sections were given the sameproblems but with the following interventions: ( ) Prompt only group ( PO ) : in each quiz problem, students were given explicit instructionsto draw a diagram with the problem statement; ( ) Diagram only group ( DO ) : in each quiz problem, students were provided a diagramdrawn by the instructor intended as scaffolding support to aid in solving the problem; and ( ) No support group ( NS ) : this group, the comparison group, was not given any diagram orexplicit instruction to draw a diagram with the problem statement.We note that students in the DO group were provided with copies of the quiz problemswhich had the diagrams drawn. Some students annotated the provided diagrams by addingrelevant details, other students drew their own diagrams, and yet other students did notannotate the provided diagrams but did not draw their own diagrams either.The sizes of the different recitation groups varied from 22 to 55 students because thestudents were not assigned a particular recitation; they could choose to attend any of the threeeach week. For the same reason, the size of each recitation group also varied from week toweek, although not as drastically because most students ( ≈ ) would stick with a particularrecitation. Furthermore, each intervention was not matched to a particular recitation. Forexample, in one week, students in the Tuesday recitation comprised the comparison group,while in another week the comparison group was a different recitation section. This isimportant because it implies that individual students were subjected to different interventionsfrom week to week so that we do not expect cumulative effects due to the same group ofstudents always being part of the same intervention.In this paper, we analyse two problems: the fi rst problem is one-dimensional and has twoalmost identical parts, one dealing with electric fi eld and the other dealing with electric force.This problem was given both in a quiz ( a week after students learned about these concepts ) and in a midterm exam ( several weeks after learning the concepts ) . Note that the interventionswere only implemented in the quiz and not in the midterm. Also, students received feedbackfrom the TAs about their performance on the quiz ( i.e. the TAs graded student solutions,marked mistakes and returned the quizzes ) . Solutions to the quizzes were also provided to thestudents before the midterm exam. The second problem is a two-dimensional problem onelectric force which was given in a quiz only. The two problems and the diagrams provided tostudents in the DO group ( shown in fi gures 1 and 2 ) are the following: Eur. J. Phys. ( )015703 A Maries and C Singh
A traditionally taught class of 120 algebra-based introductory physics students was enroled inthree different recitation sections. The three recitation sections formed the comparison groupand the two intervention groups for this investigation. All recitations were taught tradition-ally: the TA worked out problems similar to the homework problems and then gave a 15 minquiz at the end of class. Students in all recitations attended the same lectures, were assignedthe same homework, and took the same exams and quizzes. In the recitation quizzesthroughout the semester, students in the three different recitation sections were given the sameproblems but with the following interventions: ( ) Prompt only group ( PO ) : in each quiz problem, students were given explicit instructionsto draw a diagram with the problem statement; ( ) Diagram only group ( DO ) : in each quiz problem, students were provided a diagramdrawn by the instructor intended as scaffolding support to aid in solving the problem; and ( ) No support group ( NS ) : this group, the comparison group, was not given any diagram orexplicit instruction to draw a diagram with the problem statement.We note that students in the DO group were provided with copies of the quiz problemswhich had the diagrams drawn. Some students annotated the provided diagrams by addingrelevant details, other students drew their own diagrams, and yet other students did notannotate the provided diagrams but did not draw their own diagrams either.The sizes of the different recitation groups varied from 22 to 55 students because thestudents were not assigned a particular recitation; they could choose to attend any of the threeeach week. For the same reason, the size of each recitation group also varied from week toweek, although not as drastically because most students ( ≈ ) would stick with a particularrecitation. Furthermore, each intervention was not matched to a particular recitation. Forexample, in one week, students in the Tuesday recitation comprised the comparison group,while in another week the comparison group was a different recitation section. This isimportant because it implies that individual students were subjected to different interventionsfrom week to week so that we do not expect cumulative effects due to the same group ofstudents always being part of the same intervention.In this paper, we analyse two problems: the fi rst problem is one-dimensional and has twoalmost identical parts, one dealing with electric fi eld and the other dealing with electric force.This problem was given both in a quiz ( a week after students learned about these concepts ) and in a midterm exam ( several weeks after learning the concepts ) . Note that the interventionswere only implemented in the quiz and not in the midterm. Also, students received feedbackfrom the TAs about their performance on the quiz ( i.e. the TAs graded student solutions,marked mistakes and returned the quizzes ) . Solutions to the quizzes were also provided to thestudents before the midterm exam. The second problem is a two-dimensional problem onelectric force which was given in a quiz only. The two problems and the diagrams provided tostudents in the DO group ( shown in fi gures 1 and 2 ) are the following: Eur. J. Phys. ( )015703 A Maries and C Singh roblem 1 ‘ Two equal and opposite charges with magnitude 10 − C are held 15 cm apart. ( a ) What are the magnitude and direction of the electric fi eld at the point midway betweenthe charges? ( b ) What are the magnitude and direction of the force that would act on a 10 − C charge if itis placed at that midpoint? ’ Problem 2 ‘ Three charges are located at the vertices of an equilateral triangle that is 1 m on a side.Two of the charges are 2 C each and the third charge is 1 C. Find the magnitude and directionof the net electrostatic force on the 1 C charge ’ .These diagrams were drawn by the instructor. They are very similar to what most physicsexperts would draw in the initial stage of problem solving. Of course, subsequently, physicsexperts would most likely augment these diagrams by drawing arrows to indicate thedirections of electric fi eld / force vectors. Providing students in the DO group with thesediagrams was intended as a scaffolding support based upon the hypothesis that the pictorialrepresentation of a problem situation can aid students in visualising the problem.In order to ensure homogeneity of grading, we developed rubrics for each problemanalysed and ensured that there was at least 90% inter-rater reliability between two differentraters on at least 10% of the data. The development of the rubric for each problem wentthrough an iterative process. During the development of the rubric, the two graders alsodiscussed a student ’ s score separately from the one obtained using the rubric and adjusted therubric if it was agreed that the version of the rubric was too stringent or too generous. Aftereach adjustment, all students were graded again on the revised rubric. Problem 1 is comprisedof two subproblems, part ( a ) which asks for electric fi eld and part ( b ) which asks for electricforce. Therefore, parts ( a ) and ( b ) were scored separately. In table 1, we provide the summaryof the rubric for part ( a ) ( electric fi eld ) of the fi rst problem. The rubric for part ( b ) ( electricforce ) is very similar. Student performance on problem 2 was scored in a similar manner ( used a rubric developed through an iterative process and ensured 90% inter-rater reliabilitybetween two different raters on at least 10% of the data ) . Figure 1.
Diagram for problem 1 given only to students in the DO group.
Figure 2.
Diagram for problem 2 given only to students in the DO group.
Eur. J. Phys. ( )015703 A Maries and C Singh
Eur. J. Phys. ( )015703 A Maries and C Singh able 1 shows that there are two parts to the rubric: correct and incorrect ideas. Table 1also shows that in the correct ideas part, the problem was divided into different sections andpoints were assigned to each section. Each student starts out with 10 points and in theincorrect ideas part we list the common mistakes students made in each section and howmany points were deducted for each of those mistakes. We note that it is not possible todeduct more points than a section is worth ( the mistakes labelled 2.1 and 2.2 are exclusivewith respect to all other mistakes in section 2 and with each other ) . We also left ourselves asmall window ( labelled 2.5 ) if the mistake a student made was not explicitly in the rubric ( only 5% of the cases ) .For example, one common mistake on problem 1 was to use the equation for themagnitude of the electric fi eld of a point charge = ( ) ∣ ∣ E , kQ r plug in the magnitude of onecharge ( − C ) and the distance between them (
15 cm ) to obtain 4 × N C − . Somestudents who did this drew arrows which indicate that the charges are attracted to each other ( i.e. a rightward force on the negative charge and a leftward force on the positive charge ) , anddid not indicate the direction of the net electric fi eld. Here is how the rubric in table 1 wasapplied to this type of student solution: (cid:129) Section 1: 1 point since students used the correct equation (cid:129)
Section 2: 1 point because 2.2 is the mistake students made ( also, as mentioned above,mistake 2.2 was considered to be exclusive with all other mistakes in section 2 ) (cid:129) Section 3: no points because the direction of the net fi eld is not indicated (cid:129) Section 4: 1 point because units are correct (cid:129)
Total: 3 /
10 points.
In addition to the quantitative in-class data collected, individual interviews were conductedusing a think-aloud protocol [ ] with nine students who were at the time enroled in a secondsemester algebra-based introductory physics course. During the interviews, students were Table 1.
Summary of the used rubric for part ( a ) of problem 1.Correct ideasSection 1 Used correct equation for the electric fi eld 1pSection 2 Added the two fi elds due to individual charges correctly 7pSection 3 Indicated correct direction for the net electric fi eld 1pSection 4 Correct units 1pIncorrect ideasSection 1 Used incorrect equation for the electric fi eld − − fi nd electric fi elds due to both charges − ( not relevant here ) or obtained zerofor electric fi eld − r / fi nd the electric fi eld − ( s ) in fi nding the electric fi eld − fi eld − − Eur. J. Phys. ( )015703 A Maries and C Singh
Summary of the used rubric for part ( a ) of problem 1.Correct ideasSection 1 Used correct equation for the electric fi eld 1pSection 2 Added the two fi elds due to individual charges correctly 7pSection 3 Indicated correct direction for the net electric fi eld 1pSection 4 Correct units 1pIncorrect ideasSection 1 Used incorrect equation for the electric fi eld − − fi nd electric fi elds due to both charges − ( not relevant here ) or obtained zerofor electric fi eld − r / fi nd the electric fi eld − ( s ) in fi nding the electric fi eld − fi eld − − Eur. J. Phys. ( )015703 A Maries and C Singh sked to solve the problems while thinking aloud and, after they had fi nished working on theproblems, they were asked follow-up questions related to the physics concepts required forsuccessfully solving the problems. The interviews provided qualitative data which providedan interpretation for some of the quantitative fi ndings.
3. Research questions
Below, we discuss the research questions investigated in this study. The fi rst two are speci fi cto the interventions and the other two are more general and related to the effect of drawing adiagram on problem solving performance. RQ1: How do the different interventions affect the frequency of students drawing pro-ductive diagrams?
Physics experts would most likely augment the diagrams provided by drawing arrowswhich represent the directions of electric fi eld / electric force vectors. Therefore, it was con-sidered that a productive diagram should include at the very least, in addition to the charges,two electric fi eld or electric force vectors ( for example, for problem 1, two vectors whichindicate the direction for electric fi elds / electric forces explicitly drawn at the midpoint,whether or not another charge is placed there. For problem 2, two vectors which indicate thedirections of the two forces which act on the 1 C charge ) . Any diagram which did not includevectors to indicate directions of electric fi elds and / or forces was considered to be unpro-ductive. Productive diagrams can include more relevant detail. For example, in problem 2, inaddition to the two forces that act on the 1 C charge, a student can explicitly draw thecomponents of those forces. It is worthwhile noting that for both problems, students in the DOgroup were provided unproductive diagrams. RQ2: To what extent is student performance in fl uenced, if at all, by the interventions? Since the fi rst step of most physics experts in problem solving is conceptual planning andanalysis, which typically includes drawing one or several diagrams, it is possible thatprompting students to draw diagrams can make it more likely that they engage in thisplanning stage, which may help their problem solving performance. Providing a diagrammight also affect their performance. We investigated how students in the two differentintervention groups performed compared to the students in the comparison group. RQ3: To what extent does drawing a productive diagram affect problem solvingperformance?
In a previous investigation [ ] , we found that students who drew productive diagramsperformed better than students who did not draw a productive diagram for a probleminvolving a standing harmonic of a sound wave in a cylindrical tube. We investigated whetherthis effect also arises in the context of the problems discussed here. RQ4: What are some possible cognitive mechanisms that can explain the effect ofdrawing a productive diagram on student performance?
In order to shed light on possible cognitive mechanisms which could partly explain howstudents ’ problem solving performance is affected ( or not affected ) by drawing a diagram,nine think-aloud interviews were conducted with students enroled in a different, butequivalent algebra-based introductory physics course. At the time of the interviews, studentshad fi nished the study of electrostatics and also had been tested on this material via an in-class exam. Eur. J. Phys. ( )015703 A Maries and C Singh
In order to shed light on possible cognitive mechanisms which could partly explain howstudents ’ problem solving performance is affected ( or not affected ) by drawing a diagram,nine think-aloud interviews were conducted with students enroled in a different, butequivalent algebra-based introductory physics course. At the time of the interviews, studentshad fi nished the study of electrostatics and also had been tested on this material via an in-class exam. Eur. J. Phys. ( )015703 A Maries and C Singh . Results For both problems, all students drew a diagram. However, not all diagrams drawn by studentswere considered to be productive ( for the purposes of solving the problems ) . In problem 1,intervention PO resulted in signi fi cantly increasing the percentage of students who drew aproductive diagram ( p value = [ ] ) whilethe percentage of students in DO who drew a productive diagram is nearly identical to thepercentage of students in NS, as shown in table 2. Note that since students in the DO groupwere provided with an unproductive diagram, only 45% of them added more detail to thosediagrams to obtain a productive diagram. For problem 2, neither intervention affected thepercentage of students who drew productive diagrams signi fi cantly ( data shown in table 3 ) .We note however, that problem 2 is two-dimensional while problem 1 is one-dimensional andthat more students drew productive diagrams for problem 2 than for problem 1 (
77% com-pared to 50% ) . Similar to the percentage of students who drew a productive diagram discussed above, itappears that while the interventions had some effect on student performance for problem 1,they did not have an effect for problem 2. Table 4 lists the average score for each group ( PO,DO, NS ) on the two different parts for problem 1 ( given in a quiz, one week after studentslearned about electric fi eld and electric force ) . ANOVA [ ] indicates no statistically sig-ni fi cant difference between the three groups on the electric fi eld part ( p = ) , but on theelectric force part, the three groups are not all comparable in terms of performance ( p = ) . In order to investigate further, pair-wise t -tests [ ] were carried out for theelectric force part which indicate that students in the PO group performed signi fi cantly better Table 2.
Percentages ( and numbers ) of students in the three intervention groups whodrew a productive diagram for problem 1.% of students who drew a productive diagram ( number of students whodrew a productive diagram ) PO 66% ( ) DO 45% ( ) NS 41% ( ) All students 50% ( ) Table 3.
Percentages ( and numbers ) of students in the three intervention groups whodrew a productive diagram for problem 2.% of students who drew a productive diagram ( number of students whodrew a productive diagram ) PO 82% ( ) DO 79% ( ) NS 66% ( ) All students 77% ( ) Eur. J. Phys. ( )015703 A Maries and C Singh
Percentages ( and numbers ) of students in the three intervention groups whodrew a productive diagram for problem 2.% of students who drew a productive diagram ( number of students whodrew a productive diagram ) PO 82% ( ) DO 79% ( ) NS 66% ( ) All students 77% ( ) Eur. J. Phys. ( )015703 A Maries and C Singh han students in the two other groups ( comparing PO with DO: p value = = p value = = ) . These effect sizescorrespond to medium effects.On problem 2, ANOVA indicated no statistically signi fi cant differences between thedifferent groups ( p = ) , possibly because on problem 2, the percentages of students whodrew a productive diagram in the three different groups were comparable. The averages andstandard deviations of students in the three different groups are shown in table 5 ( the sizes ofthe intervention groups in tables 4 and 5 do not match because the two problems investigatedhere were given in two different quizzes and the interventions were implemented in differentrecitations in different weeks ) .It therefore appears that for problem 1, students who were asked to draw a diagramperformed signi fi cantly better ( in the force part of the problem at least ) , perhaps because theywere more likely to draw productive diagrams, while for problem 2, the interventions did notshow signi fi cantly different trends ( percentage of students drawing a productive diagram orscore ) . Students who draw productive diagrams perform better than students who do notAs mentioned earlier, productive diagrams for both problems include the basic physicalsetups ( i.e. two charges from problem 1 ) and vectors which indicate the directions of electric fi eld or electric force vectors. Table 6 shows the performance of students who drew pro-ductive diagrams and those who did not for both problems regardless of the intervention ( i.e.all students are put together ) . Our results indicate that students who drew a productivediagram signi fi cantly outperformed students who did not on both problems ( both p values areless than 0.001 and effect sizes correspond to large effects ) , which is similar to a resultpreviously encountered in the context of students ’ problem solving performance on a probleminvolving standing sound waves in tubes [ ] . Table 4.
Number of students ( N ) , averages ( Avg. ) and standard deviations ( S.d. ) on thetwo parts of problem 1 for the two intervention groups and the comparison group out of10 points. Field part Force part N Avg. S.d. Avg. S.d.PO 29 7.0 3.25 8.6 2.80DO 40 7.1 2.61 6.6 3.77NS 51 7.9 2.78 6.8 3.59
Table 5.
Number of students ( N ) , averages and standard deviations ( Std. dev. ) onproblem 2 for the two intervention groups and the comparison group out of 10 points. N Average Std. dev.PO 50 5.8 3.1DO 39 6.7 2.5NS 29 5.3 3.3
Eur. J. Phys. ( )015703 A Maries and C Singh
Eur. J. Phys. ( )015703 A Maries and C Singh higher level of relevant detail in a student ’ s diagram corresponds to better performanceFor both problems 1 and 2, students drew productive diagrams which included varyinglevels of relevant detail. For example, for problem 1, some students drew the two charges aswell as two electric fi eld vectors at the midpoint ( relevant detail 1 ) . Other students drew thetwo charges, and also drew two electric fi eld and two electric force vectors at the midpoint ( relevant detail 2 ) . Typically, students who drew the latter type of diagram had two separatediagrams, one for the electric fi eld part and one for the electric force part. And yet otherstudents drew an unproductive diagram which does not include vectors indicating directionsof electric fi eld or force vectors at the midpoint. We note that students may also add details tothe diagrams that are not directly relevant for the problem solving process. For example, somestudents drew vectors going outward from the positive charge and inward towards thenegative charge in all directions. Interviews suggest that some of these students were repli-cating what the electric fi eld looks like around isolated positive and negative charges. Whilethis is related to the physical situation presented in the problem, it is not directly useful forsolving the problem unless the students recognise that they need to consider the midpoint andthink about the direction of the electric fi eld caused by each charge at that point. Thus, theresearchers considered that a productive diagram must have detail that is directly relevant tosolving the problem.For problem 2, productive diagrams included, e.g. the three charges and the two forcesacting on the 1 C charge ( relevant detail 1 ) , or the three charges, the two forces acting on the1 C charge and their x and y components drawn for a particular choice of coordinate system ( relevant detail 2 ) . An unproductive diagram included only the three charges. Similar toproblem 1, some students added details to their diagram that were not directly relevant forsolving the problem. For example, some students drew vectors indicating the direction of theforces acting on the two 2 C charges. Those details may be useful if the problem asks for thenet forces acting on the 2 C charges, but they are not useful for fi nding the net force on the 1 Ccharge. Thus, for both problems 1 and 2, our consideration of what features of a diagrammake it productive relates to visualising relevant information from the problems that is usefulfor solving them ( e.g. directions of electric fi eld or electric force vectors ) .Table 7, which shows the performance of students who drew these different types ofdiagrams for both problems, indicates that a higher level of relevant detail in a student ’ sdiagram corresponds to a higher score. For problem 1 (
1D problem ) , which was given both ina quiz and in a midterm, there is a statistically signi fi cant difference between students who Table 6.
Number of students ( N ) , averages and standard deviations ( Std. dev. ) forstudents who drew productive diagrams and those who did not on problems 1 and 2 outof 10 points, and p values and effect sizes for comparing the performance of studentswho drew a productive diagram with the performance of students who did not draw aproductive diagram. N Average Std.dev. p value Effect sizeProblem 1 — drew prod. diag. 58 8.3 2.2 < — did not drawprod. diag. 62 6.3 2.6Problem 2 — drew prod. diag. 91 6.6 2.9 < — did not drawprod. diag. 27 4.1 2.5 Eur. J. Phys. ( )015703 A Maries and C Singh
Number of students ( N ) , averages and standard deviations ( Std. dev. ) forstudents who drew productive diagrams and those who did not on problems 1 and 2 outof 10 points, and p values and effect sizes for comparing the performance of studentswho drew a productive diagram with the performance of students who did not draw aproductive diagram. N Average Std.dev. p value Effect sizeProblem 1 — drew prod. diag. 58 8.3 2.2 < — did not drawprod. diag. 62 6.3 2.6Problem 2 — drew prod. diag. 91 6.6 2.9 < — did not drawprod. diag. 27 4.1 2.5 Eur. J. Phys. ( )015703 A Maries and C Singh rew unproductive diagrams and students who drew diagrams which included more relevantdetail ( both p values for comparing students who drew relevant detail 1 or 2 diagrams withstudents who drew unproductive diagrams are less than 0.001, and the effect sizes are large ) ,but the difference in performance between students who drew relevant detail 1 diagrams andstudents who drew relevant detail 2 diagrams is not statistically signi fi cant, as shown intable 8. This was found both in the quiz and in the midterm. For problem 2 (
2D problem ) ,students who drew relevant detail 1 diagrams performed signi fi cantly better than students whodrew unproductive diagrams ( p = = ) and students who drew relevantdetail 2 diagrams performed signi fi cantly better than students who drew relevant detail 1diagrams ( p < = ) . The differences between the averages of the groupsare quite noticeable and the effect sizes point to medium to large effects despite the largevariation within each group. The performance of students who drew diagrams with thehighest level of relevant detail is nearly twice that of students who drew unproductivediagrams! As mentioned earlier, individual interviews with nine students who were at the time taking anequivalent second semester of an introductory algebra-based physics course were carried outusing a think-aloud protocol [ ] . These interviews suggested that for problem 2, cognitive Table 7.
Numbers of students ( N ) , averages ( Avg. ) and standard deviations ( Std. dev. ) for groups of students with diagrams including different levels of relevant detail forproblem 1 in the quiz and the midterm, and for problem 2 in the quiz.Problem 1 — Quiz Problem 1 — Midterm N Avg. Std. dev. N Avg. Std. dev.Unproductive diagram 62 6.4 2.6 45 7.0 2.6Relevant detail 1 49 8.3 2.2 51 8.4 2.0Relevant detail 2 9 8.9 1.4 25 9.0 1.4Problem 2 — Quiz N Avg. Std. dev.Unproductive diagram 27 4.1 2.5Relevant detail 1 58 5.7 2.9Relevant detail 2 33 8.0 2.2
Table 8. p values and effect sizes for comparison of the performance of students withdiagrams including different levels of relevant detail ( UD = unproductive diagram,RD1 = relevant detail 1, RD2 = relevant detail 2 ) for problem 1 in the quiz and in themidterm and for problem 2 ( in the quiz ) .UD-RD1 RD1-RD2 p value Effect size p value Effect sizeProblem 1 — Quiz < — Midterm 0.003 0.62 0.133 0.35Problem 2 0.008 0.61 < Eur. J. Phys. ( )015703 A Maries and C Singh
Table 8. p values and effect sizes for comparison of the performance of students withdiagrams including different levels of relevant detail ( UD = unproductive diagram,RD1 = relevant detail 1, RD2 = relevant detail 2 ) for problem 1 in the quiz and in themidterm and for problem 2 ( in the quiz ) .UD-RD1 RD1-RD2 p value Effect size p value Effect sizeProblem 1 — Quiz < — Midterm 0.003 0.62 0.133 0.35Problem 2 0.008 0.61 < Eur. J. Phys. ( )015703 A Maries and C Singh oad theory [ ] may be one possible framework that can explain why students who explicitlydrew the components of the two forces performed better. In particular, two of the studentsinterviewed were almost identical in terms of their majors and grades ( both in the currentphysics course and the previous one ) . Karen and Dan were both Biology majors and in the fi rst semester of physics they obtained similar grades ( B + and A − , respectively ) . In thesecond semester physics class, on the fi rst exam ( class average 75 / ) , they both obtained81 /
100 and on the second exam ( class average 65 / ) they also both obtained 81 / fi rst exam was focused primarily on electrostatics and included questions which askedstudents to calculate the net electric fi eld due to a con fi guration of charges and the net forceacting on a particular charge, but the questions were in other contexts.When solving problem 2, Karen recognised that she needed to fi nd the x and y com-ponents of both forces due to each of the 2 C charges and, before she proceeded to fi nd them,she drew all the components on the diagram provided as shown in fi gure 3. She then fi guredout all the components and combined them correctly to determine both the magnitude of thenet force and its direction ( angle below the x axis ) . While working on this problem, it wasevident that Karen was focusing on only a few things at a time and was being systematicabout the way in which she found the net force. For example, when fi nding the components ofthe oblique ( not horizontal ) force, she redrew a triangle in which this force was the hypo-tenuse and identi fi ed the angles. Karen ’ s only mistake was that she used an angle of 45 ° instead of 60 ° to fi nd these components.Dan also immediately recognised that components should be considered and proceededto fi nd them after redrawing the 1 C charge ( see fi gure 4 ) and the two forces acting on it due to Figure 3.
Forces due to the two individual charges on the 1 C charge and theircomponents as drawn by Karen ( student ) . Figure 4.
Forces acting on the 1 C charge due to the two 2 C charges as drawn by Dan ( student ) . Eur. J. Phys. ( )015703 A Maries and C Singh
Forces acting on the 1 C charge due to the two 2 C charges as drawn by Dan ( student ) . Eur. J. Phys. ( )015703 A Maries and C Singh he two 2 C charges. He worked more slowly than Karen on this problem, but after some time,he correctly determined the x and y components of the oblique force and wrote them down ( trigonometric functions were still included, i.e. he wrote down the y component as 18 × cos 30 ) . However, unlike Karen, he did not draw these components on his diagram; hisdiagram of the forces ( shown in fi gure 4 ) only included the two forces and their magnitudes.When Dan combined the components, he made two mistakes: ( ) his net y componentdid not include the trigonometric function which he had previously written down ( when hefound the y component of the oblique force ) . As he was determining the net y component hesaid: ’ this one [ horizontal force ] doesn ’ t have a y component, so it [ the y component of the netforce ] is just 18 times 10 [ magnitude he found for the oblique force ] ’ and ( ) he subtractedthe two x components instead of adding them ( he subtracted the horizontal force from the x component of the oblique force ) . In particular, he wrote the following on the paper for the net x component: = ´ - ´ x Net 18 10 sin 30 18 10 .
It is possible that part of the reason why he subtracted the components is because he didnot explicitly draw the x component of the oblique force and perhaps, due to the fact that theoblique force is in the fourth quadrant ( which should be dealt with carefully ) , he implicitlyassumed that one of its components must be negative, or that something must be subtracted.He subtracted the horizontal force from the x component of the oblique force even though thepicture he drew clearly indicated that the horizontal force is in the positive x direction. Afterhe fi nished working on all the problems to the best of his ability, in the second phase of theinterview, he was asked for clari fi cations of points he had not made clear earlier and someadditional questions. For example, Dan was asked a simpler question. He was asked to addtwo forces: one in the positive y direction, the other in the fi rst quadrant, making an angle of30 ° with the horizontal. Here too, he did not draw the components explicitly in the diagramand ended up subtracting the y components of the two forces in exactly the same manner inwhich he subtracted the x components in problem 2 ( the triangle problem ) i.e. he subtractedthe vertical force from the y component of the oblique force. When asked why he subtractedthese components he looked at the diagram for a few seconds and said: Actually, you ’ re adding [ K ] sorry, I don ’ t know why [ I did that ] [ K ] , you ’ readding because there ’ s a positive y component here [ vertical force ] and apositive y component here [ of the oblique force ] . The approaches of these two students differed mainly in that Karen explicitly drew allforces and components, whereas Dan only drew the forces. Dan subtracted the x componentswithout providing a reason, and when he was asked to add two forces in a mathematicalcontext ( similar to the two forces in the physics context ) , he made exactly the same mistakefor the two components that were supposed to be added. When questioned about why hesubtracted them, he realised this mistake on his own almost immediately, which suggestedthat when he solved both problems ( problem 2 and the simpler mathematical problem whichhad similar addition of vectors ) he was not focusing on the appropriate information. Once hisattention was drawn to the issue of whether the vectors should be added or subtracted in thesimpler mathematical problem, he clearly knew that the y components must be added. Beforequestioning, he did not draw the components of the oblique force and appeared to be sub-tracting the components automatically, without a clear reason. Also, when asked why hesubtracted the components, he did not start by trying to justify this ( for example by beginninga sentence with ‘ I subtracted them because K ’ ) , which suggested that there was no clearreason for why he subtracted the y components. In other words, it is possible that he did not Eur. J. Phys. ( )015703 A Maries and C Singh
It is possible that part of the reason why he subtracted the components is because he didnot explicitly draw the x component of the oblique force and perhaps, due to the fact that theoblique force is in the fourth quadrant ( which should be dealt with carefully ) , he implicitlyassumed that one of its components must be negative, or that something must be subtracted.He subtracted the horizontal force from the x component of the oblique force even though thepicture he drew clearly indicated that the horizontal force is in the positive x direction. Afterhe fi nished working on all the problems to the best of his ability, in the second phase of theinterview, he was asked for clari fi cations of points he had not made clear earlier and someadditional questions. For example, Dan was asked a simpler question. He was asked to addtwo forces: one in the positive y direction, the other in the fi rst quadrant, making an angle of30 ° with the horizontal. Here too, he did not draw the components explicitly in the diagramand ended up subtracting the y components of the two forces in exactly the same manner inwhich he subtracted the x components in problem 2 ( the triangle problem ) i.e. he subtractedthe vertical force from the y component of the oblique force. When asked why he subtractedthese components he looked at the diagram for a few seconds and said: Actually, you ’ re adding [ K ] sorry, I don ’ t know why [ I did that ] [ K ] , you ’ readding because there ’ s a positive y component here [ vertical force ] and apositive y component here [ of the oblique force ] . The approaches of these two students differed mainly in that Karen explicitly drew allforces and components, whereas Dan only drew the forces. Dan subtracted the x componentswithout providing a reason, and when he was asked to add two forces in a mathematicalcontext ( similar to the two forces in the physics context ) , he made exactly the same mistakefor the two components that were supposed to be added. When questioned about why hesubtracted them, he realised this mistake on his own almost immediately, which suggestedthat when he solved both problems ( problem 2 and the simpler mathematical problem whichhad similar addition of vectors ) he was not focusing on the appropriate information. Once hisattention was drawn to the issue of whether the vectors should be added or subtracted in thesimpler mathematical problem, he clearly knew that the y components must be added. Beforequestioning, he did not draw the components of the oblique force and appeared to be sub-tracting the components automatically, without a clear reason. Also, when asked why hesubtracted the components, he did not start by trying to justify this ( for example by beginninga sentence with ‘ I subtracted them because K ’ ) , which suggested that there was no clearreason for why he subtracted the y components. In other words, it is possible that he did not Eur. J. Phys. ( )015703 A Maries and C Singh ave any cognitive resources free to use for thinking about whether the components should beadded or subtracted due to having to process too much information at one time in his workingmemory ( e.g. forces, trigonometric angles, vector addition, etc ) . When it was time to utilisethis information about the components of the oblique force to fi nd the x component of the netforce, he forgot to correctly account for the x component. On the other hand, Karen had thecomponents explicitly drawn on the paper as opposed to keeping this information in her headand she was able to look back at her components and account for the sign of the x componentof the oblique force correctly. Cognitive load theory [ ] , which incorporates the notion ofdistributed cognition [
44, 45 ] , provides one possible explanation for Dan ’ s unsuccessful andKaren ’ s successful addition of vectors in this context: lack of information about componentson Dan ’ s diagram required him to keep this information in his working memory, while Karendid not need to keep this information in her working memory since she included the com-ponents explicitly in her diagram. As Dan ’ s working memory was processing a variety ofinformation during problem solving, he may have had cognitive overload and the informationabout the components that he planned to use at the opportune time to fi nd the components ofthe net force was not invoked appropriately.Interviews with other students who drew diagrams which included higher levels of detailsuggested that including information on a diagram can help free up cognitive resources forprocessing information about vector addition and about the problems in general which helpedthem perform appropriate calculations and fi nd their mistakes. On the other hand, studentswho drew unproductive diagrams or no diagrams at all sometimes seemed to have cognitiveoverload since, similar to Dan, they made mistakes while solving the problems initially.However, when coming back to the problem after being asked about their solutions, theysometimes identi fi ed their mistakes on their own. This suggested that when they solved theproblems initially they may not have carefully carried out decision making regarding theproblem solution partly because they had reduced cognitive processing capacity. Includinginformation about the problems explicitly, e.g. by using diagrammatic representations canhelp increase the students ’ cognitive processing capacity by distributing their cogni-tion [
44, 45 ] .
5. Discussion and summary
We found that for problem 1, students who were explicitly asked to draw a diagram weremore likely to draw a productive diagram. We also found that students who drew productivediagrams performed better than students who drew unproductive diagrams. Among the stu-dents provided with a diagram ( which was unproductive unless modi fi ed by the student byadding force and / or fi eld vectors at the midpoint ) , less than half added relevant details to thediagram provided in order to use a productive diagram. This is a statistically signi fi cantlylower percentage compared to the percentage of students who used a productive diagram inthe group of students who were prompted to draw one. This fi nding suggests that in anintroductory physics course, prompting students to draw a diagram may provide betterscaffolding for solving problems than providing a diagram and should be incorporated inhelping students learn effective problem solving strategies. Furthermore, we also found thatdiagrams which included more relevant details from the problems ( that are useful for solvingthe problems ) corresponded to better performance. This fi nding suggests that students shouldnot only be incentivized to draw diagrams, but also guided to learn to include as muchrelevant information as is necessary in their diagrams to facilitate problem solution. As notedearlier, one theoretical framework that can provide a possible explanation for why students Eur. J. Phys. ( )015703 A Maries and C Singh
We found that for problem 1, students who were explicitly asked to draw a diagram weremore likely to draw a productive diagram. We also found that students who drew productivediagrams performed better than students who drew unproductive diagrams. Among the stu-dents provided with a diagram ( which was unproductive unless modi fi ed by the student byadding force and / or fi eld vectors at the midpoint ) , less than half added relevant details to thediagram provided in order to use a productive diagram. This is a statistically signi fi cantlylower percentage compared to the percentage of students who used a productive diagram inthe group of students who were prompted to draw one. This fi nding suggests that in anintroductory physics course, prompting students to draw a diagram may provide betterscaffolding for solving problems than providing a diagram and should be incorporated inhelping students learn effective problem solving strategies. Furthermore, we also found thatdiagrams which included more relevant details from the problems ( that are useful for solvingthe problems ) corresponded to better performance. This fi nding suggests that students shouldnot only be incentivized to draw diagrams, but also guided to learn to include as muchrelevant information as is necessary in their diagrams to facilitate problem solution. As notedearlier, one theoretical framework that can provide a possible explanation for why students Eur. J. Phys. ( )015703 A Maries and C Singh ith diagrams with more relevant details performed better is the cognitive load theory [ ] ,which incorporates the notion of distributed cognition [
44, 45 ] . In problem 2, students had toadd forces by using components, so students who did not draw the force vectors or theircomponents they had to add vectorially would have to keep too much information in theworking memory [ – ] while engaged in problem solving ( individual components of thetwo forces, angles required to get those components, what trigonometric function needs to beused for each component, etc ) . This can lead to cognitive overload and deteriorated perfor-mance. Explicitly drawing the forces and their components can reduce the amount ofinformation that must be kept in the working memory while engaged in problem solving andmay therefore make the student better able to go through all the steps necessary withoutmaking mistakes.It is also important to note that these problems were given in the second semester of a oneyear introductory physics course for algebra-based students. These students had done pro-blems for which they had to fi nd the net force in Newtonian mechanics, and still less than30% of the students realised that they should draw the components of the electric force inproblem 2 presented here. Also, only 42% of all students indicated a direction for the netforce. This can partly be an indication of a lack of transfer from one context to another [
49, 50 ] . Students ’ performance also suggests that many algebra-based introductory studentsdo not have a robust knowledge structure of physics nor do they employ good problemsolving heuristics and their familiarity with addition of vectors may also require an explicitreview. Earlier surveys at the start of the course have found that only about 1 / fi cient knowledgeabout vectors to begin the study of Newtonian mechanics [ ] . Here we fi nd that even after asemester of instruction in physics that involves a fair amount of vector addition, the fractionremains about the same and students had great dif fi culty dealing with vector addition incomponent form.This study suggests that students drawing and using productive diagrams can helpimprove their problem solving performance, and suggests multiple avenues for futureresearch. For example, one can conduct a more detailed investigation of the features thatconstitute a productive or an unproductive diagram and how those are correlated with pro-blem solving performance. We should note that our study suggests that the features of aproductive diagram are related to representing relevant information from the problem tofacilitate problem solving, but this could be explored in more detail in future studies. Also,while this study suggests that asking students to draw a diagram may provide useful scaf-folding for students, one can also investigate other possible interventions, for example,providing students with a diagram and explicitly asking them to add details to it, or providingstudents with productive diagrams. Future studies could also explore possible reasons whysome students draw diagrams while others do not, as well as the characteristics of teacher-student interactions that may help students recognise that they should not only draw diagramsbut also ensure that relevant details are included in those diagrams. Our study suggests thatrepresenting relevant information from electrostatics problems on a diagram that studentsdrew helped them in the problem solving process. Instructors may emphasise this in theirteaching, as well as discuss that visual information is much easier to process than verbalinformation and this is partly why physics experts always draw diagrams when solvingproblems. While the extent to which such practices may be effective is beyond the scope ofthis work, future research can explore these issues along with other approaches ( e.g. pro-viding grade incentives for drawing diagrams ) . Eur. J. Phys. ( )015703 A Maries and C Singh
49, 50 ] . Students ’ performance also suggests that many algebra-based introductory studentsdo not have a robust knowledge structure of physics nor do they employ good problemsolving heuristics and their familiarity with addition of vectors may also require an explicitreview. Earlier surveys at the start of the course have found that only about 1 / fi cient knowledgeabout vectors to begin the study of Newtonian mechanics [ ] . Here we fi nd that even after asemester of instruction in physics that involves a fair amount of vector addition, the fractionremains about the same and students had great dif fi culty dealing with vector addition incomponent form.This study suggests that students drawing and using productive diagrams can helpimprove their problem solving performance, and suggests multiple avenues for futureresearch. For example, one can conduct a more detailed investigation of the features thatconstitute a productive or an unproductive diagram and how those are correlated with pro-blem solving performance. We should note that our study suggests that the features of aproductive diagram are related to representing relevant information from the problem tofacilitate problem solving, but this could be explored in more detail in future studies. Also,while this study suggests that asking students to draw a diagram may provide useful scaf-folding for students, one can also investigate other possible interventions, for example,providing students with a diagram and explicitly asking them to add details to it, or providingstudents with productive diagrams. Future studies could also explore possible reasons whysome students draw diagrams while others do not, as well as the characteristics of teacher-student interactions that may help students recognise that they should not only draw diagramsbut also ensure that relevant details are included in those diagrams. Our study suggests thatrepresenting relevant information from electrostatics problems on a diagram that studentsdrew helped them in the problem solving process. Instructors may emphasise this in theirteaching, as well as discuss that visual information is much easier to process than verbalinformation and this is partly why physics experts always draw diagrams when solvingproblems. While the extent to which such practices may be effective is beyond the scope ofthis work, future research can explore these issues along with other approaches ( e.g. pro-viding grade incentives for drawing diagrams ) . Eur. J. Phys. ( )015703 A Maries and C Singh cknowledgments We thank the US National Science Foundation for award 1524575 which made this workpossible. Also, we are extremely grateful to professors F Reif, J Levy and R P Devaty, and allthe members of the Physics Education Research group at University of Pittsburgh for veryhelpful discussions and / or feedback. ORCID iDs
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