Multiple representations of work–energy processesazmodmech2017.weebly.com/uploads/4/7/7/4/47742735/... · problems since, as visual aids, they automatically enhance human perceptual

Multiple representations of work–energy processesAlan Van Heuvelen and Xueli Zou Citation: Am. J. Phys. 69, 184 (2001); doi: 10.1119/1.1286662 View online: http://dx.doi.org/10.1119/1.1286662 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v69/i2 Published by the American Association of Physics Teachers Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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Multiple representations of work–energy processesAlan Van Heuvelen and Xueli ZouDepartment of Physics, The Ohio State University, Columbus, Ohio 43210

~Received 13 September 1999; accepted 25 April 2000!

An energy process can be represented by verbal, pictorial, bar chart, and mathematicalrepresentations. This multiple-representation method for work–energy processes has beenintroduced and used in the work–energy part of introductory college physics courses. Assessmentindicates that the method, especially the qualitative work–energy bar charts, serves as a usefulvisual tool to help students understand work–energy concepts and to solve related problems. Thispaper reports how the method has been used to teach work–energy concepts, student attitudestoward this approach, and their performance on work–energy problems. ©2001 American Association

of Physics Teachers.

@DOI: 10.1119/1.1286662#

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I. INTRODUCTION

Experiments in cognitive science and physics educaindicate that experts often apply qualitative representatisuch as pictures, graphs, and diagrams to help themseunderstand problems before they use equations to solvequantitatively. In contrast, novices use formula-centemethods to solve problems.1 Studies in physics educatioalso have found that student problem-solving achievemimproves when greater emphasis is placed on qualitativeresentations of physical processes.2–10 In this method, a typi-cal physics problem is considered as a physical process.process is first described in words—the verbal representaof the process. Next, a sketch or a picture, called a pictorepresentation, is used to represent the process. This islowed by a physical representation that involves mphysics-like quantities and descriptions such as free-bdiagrams and graphs. Finally, the process is represemathematically by using basic physics principles to descthe process. The pictorial and physical representationsoften called qualitative representations, in contrast toquantitative mathematical representation. In this paper,use of verbal, pictorial, physical, and mathematical represtations is called multiple-representation problem solving.example of multiple representations for a kinematics procis shown in Fig. 1.

In terms of multiple representations, the goal of solviphysics problems is to represent physical processes in dient ways—words, sketches, diagrams, graphs, and equatThe abstract verbal description is linked to the abstract mematical representation by the more intuitive pictorial adiagrammatic physical representations. First, these quative representations foster students’ understanding ofproblems since, as visual aids, they automatically enhahuman perceptual reasoning.12 Second, qualitative representations, especially physical representations, build a bribetween the verbal and the mathematical representatThey help students move in smaller and easier steps fwords to equations. Third, qualitative representations hstudents develop images that give the mathematical symmeaning—much as the picture of an apple gives meaninthe word ‘‘apple.’’ After representing the process, studecan obtain a quantitative answer to the problem usingmathematical representation. However, the main goal isrepresent a process in multiple ways rather than solvingsome unknown quantity~see the example in Fig. 1!.

If students understand the meaning of the symbols in

184 Am. J. Phys.69 ~2!, February 2001 http://ojps.aip.org

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equations and the meaning of the diagram, they can wbackward to invent a process in the form of a sketch orwords that is consistent with the equations or with the dgram. For example, they should be able to construct dgrammatic, pictorial, and verbal representations of thenamics process that is described mathematically in Fig. 2Howard Gardner says, ‘‘Genuine understanding is mlikely to emerge...if people possess a number of waysrepresenting knowledge of a concept or skill and can mreadily back and forth among these forms of knowing.’’13

Thus we might say that an important goal of physics edution is to help students learn to construct verbal, pictorphysical, and mathematical representations of physical pcesses, and to learn to move in any direction between threpresentations.

The examples shown in Figs. 1 and 2 involve kinematand dynamics. As we know, most physics professorsteachers solving dynamics problems rely on diagrammforce representations—a free-body diagram or a forcegram. There is, however, no similar representation for soing work–energy problems. This paper describes qualitawork–energy bar charts that serve the same role for anaing work–energy processes as motion diagrams and fodiagrams serve when analyzing kinematics and dynamproblems. We find that use of these bar charts helps studthink more about the physics of a work–energy procrather than relying on formula-centered techniques that lqualitative understanding.

In the remainder of the paper, a multiple-representatstrategy for helping students analyze work–energy proceis described along with several examples of its use. Welook at strategies that are important for a successful analof a work–energy process. We then introduce the eneequivalent of a force diagram—a qualitative work–enerbar chart. Finally, we analyze student attitudes towardmethod, their problem performance, and their actual usethis strategy.

II. WORK –ENERGY PROCESSES ANDWORK –ENERGY BAR CHARTS

There is considerable diversity in the way that physfaculty solve work–energy problems. The procedure usethis paper has its roots in a book by P. W. Bridgman.14 Muchattention is paid to a system, to its changing character, anits interaction with its environment.

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To use this approach with the concepts of work andergy, students must learn certain conceptual ideas and sincluding:

d choosing a system—the object or objects of interest forprocess being considered;

d characterizing the initial state and the final state ofprocess;

d identifying the types of energy that change as the sysmoves from its initial state to its final state and the signsthe initial and final energies of each type;

d deciding if work is done on the system by one or moobjects outside the system as the system changes sta

d developing the idea that the initial energy of the systplus the work done on the system leads to the final eneof the system—the energy of the universe remains cstant;

d constructing an energy bar chart—a qualitative represetion of the work–energy process; and

Fig. 1. The kinematics process described in the problem can be represby qualitative sketches and diagrams that contribute to understandingsketches and diagrams can then be used to help construct with understathe mathematical representation.

185 Am. J. Phys., Vol. 69, No. 2, February 2001


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The qualitative work–energy bar charts illustrated in thand subsequent sections play a significant role in fillinggap between words and equations when using the concof work and energy to solve physics problems. Studentsprovided with a series of problems in which their only taskto convert words and sketches into qualitative work–enebar charts and then into a generalized form of the worenergy equation. As shown in Fig. 3, a bar is placed inchart for each type of energy that is not zero. For this proc~neglecting friction!, the system includes the spring, thblock, and Earth. The initial energy of the system is telastic potential energy of the compressed spring. This

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Fig. 2. The physical process described in the mathematical equations crepresented by diagrams, sketches, and words. The diagrams and skaid in understanding the symbolic notations, and help give meaning toabstract mathematical symbols.~There could be more than one diagram asketch consistent with the mathematical equations.!

185A. Van Heuvelen and X. Zou


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converted into the final energy of the system, that is,kinetic energy of the block and the gravitational potentenergy due to the separation of the block and Earth. Sincexternal forces affect the process, no external work is don the system. Hence no bar is drawn for the work part ofchart. To conceptually distinguish between work and ene~e.g., work is a process quantity, but energy is a state qutity!, we shade the work part in the bar chart.~In teachingthis work–energy bar chart method, we could either colorwork part differently or show it as a separate region to incate that work is conceptually different from energy.!

The relative magnitudes of the different types of energythe bar chart are initially unknown, just as the magnitudesforces in a free-body diagram are often initially unknowThe chart does, however, allow us to qualitatively conseenergy—the sum of the bars on the left equals the sum ofbars on the right.

We can also reason qualitatively about physical procesusing the charts. For example, for the process representeFig. 3, we might ask students what changes occur inprocess and in the chart if the final position of the block isa higher or lower elevation. What changes occur in the pcess and in the chart if friction is added? What happens toprocess and the chart if the block’s mass is increased wkeeping the other quantities constant?

Fig. 3. The work–energy problem is originally described in the detasketch. Students are asked to convert the sketch into a qualitativechart—a bar is placed in the chart for each type of energy that is not zand the sum of the bars on the left is the same as that of the bars on theThen the generalized mathematical work–energy equation without any nbers is set up with one energy expression for each bar on the chart.~Noticethat the work part in the bar chart is shaded so as to distinguish conceptbetween work and energy, that is, work is a process quantity, but enera state quantity.!



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III. SYSTEMS AND WORK

How do we decide for a given problem whether to calclate the work done by a force~for example, the work done bythe gravitational force that Earth’s mass exerts on an obj!or to calculate a change in energy~for example, a change ingravitational potential energy!? A system approach providea reason for one choice or the other. The book by Bridgmprovides a nice introduction to the effect of system choicethe description of the work–energy process.15 Burkhardt,16

Sherwood,17 Sherwood and Bernard,18 Arons,19 and Chabayand Sherwood20 also discuss the importance of systechoice. A system includes an object or objects of interesa region defined by an imaginary surface that separatefrom its surroundings. Choosing a system is the key tociding what energy changes occur and what work is doWork is done only if an object outside the system exertforce on an object in the system and consequently does won the system as the internal object moves.

This idea is illustrated in Fig. 4 where the same proc~neglecting friction! is analyzed by using three different sytems~an idea provided by Bob Sledz at Garfield High Schoin Cleveland!. Notice that in Fig. 4~a! the cart and the springare in the system, as is Earth. The process is not affecteobjects outside the system. Thus no external work is donethe system. Potential energy change occurs when objectssystem change their shape or their position relative to oobjects in the system. For example, the change in separaof a mass relative to Earth’s mass in the system causgravitational potential energy change. The change in seption of two electric charges in a system causes a changelectrical potential energy. The change in the shape ospring when it compresses or stretches causes a changeelastic potential energy. In Fig. 4~a!, the system’s initial en-ergy, the elastic potential energy of the compressed sprinconverted to the system’s final energy, the kinetic energythe cart and the gravitational potential energy due toseparation of the cart and Earth.

In Fig. 4~b!, the system has been chosen with Earth oside the system. Thus, the Earth’s mass exerts an extegravitational force on the cart and consequently does won the cart as it moves to higher elevation. In this case,do not count the system’s gravitational potential energychanging because Earth is not in the system. As we knthe negative work done by Earth’s gravitational force onleft-hand side of the chart in Fig. 4~b! has the same effect athe positive final gravitational potential energy on the righand side of the chart in Fig. 4~a!.

For the system in Fig. 4~c!, the cart alone is in the systemThus we now include the effect of the spring by analyzithe work done by the external force of the spring on the cand the work done by the external Earth’s gravitationforce. The sum of these two work terms causes the systekinetic energy to increase. There is no elastic potentialergy change in the system since the spring is now outsidesystem.

In the above example, the system chosen in Fig. 4~a! isprobably easiest to use in physics instruction. Gravitatiopotential energy is usually emphasized in high school acollege introductory physics courses and is an easier confor students to understand. It may, in practice, be easierstudents to solve problems if they choose systems thatclude Earth. Similarly, it is more difficult for students t

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Fig. 4. The different systems are chosen for the same physical process.~a! The cart, the spring, and Earth are in the system.~b! The cart and the spring arein the system, but not Earth.~c! The system includes only the cart. For each chosen system there is one work–energy bar chart and the corregeneralized work–energy equation. In practice, it would be easy for students to use a system that includes Earth and the spring, although the cthesystem does not affect the physical results.

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calculate the work done by a spring than to calculateelastic potential energy12 kx2 at the beginning and end ofprocess.

What about friction? Bridgman,21 Sherwood,17 Sherwoodand Bernard,18 Arons,22 and Chabay and Sherwood20 havediscussed examples such as those shown in Fig. 5—oneject sliding along a surface. Let us discuss a nonrealsimple situation first. In Fig. 5~a!, we imagine the block is apoint particle, and this point-particle block is moving untilstops on the frictional floor. The system includes only tpoint-particle block. So the floor exerts on the point-partiblock an external frictional force which points in the oppsite direction of the motion. This frictional force does a negtive amount of work. This negative work in stopping thpoint-particle block has the same magnitude as the bloinitial kinetic energy. But in a real and complex situatiosuch as the car skidding to stop on a rough road showFig. 5~b!, the car is a real object. If the car is the only objein the system, the road that touches the car causes an extfrictional force. The work done by this frictional force ivery difficult to calculate, if we look carefully at what reallhappens at the boundary between the car tires and theAs Arons says~see Ref. 19, p. 151!:

What happens at the interface is a very complicamess: We have abrasion, bending of ‘‘aspirates,’’ weing and unwelding of regions of ‘‘contact,’’ as well ashear stresses and strains in both the block and the fl



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In this situation, it is very difficult to deal with the formadefinition of work done by the frictional force. Ruth Chabaand Bruce Sherwood in their new textbook give a nice adetailed explanation about how to calculate the work donethe frictional force in such a situation—seeMatter and In-teractions~see Ref. 20, pp. 236–241!. According to Chabayand Sherwood’s arguments, the work done by this frictioforce is still negative, but the magnitude of this work is lethan the car’s initial kinetic energy~and not equal to2F frictiond either!; the remaining amount of the car’s initiakinetic energy is converted into internal energy of the c~see the following for detailed discussions about the interenergy!.

Rather than dealing with this difficult work calculationBridgman14 and Arons19 recommend that the two touchinsurfaces, such as the car tires and the road, be included isystem as in Fig. 5~c!. Since friction is no longer an externaforce, we look for energy changes in the system. Frictcauses objects rubbing against each other to becwarmer—the random kinetic energy of the atoms and mecules in the touching surfaces increases. Friction cancause atomic and molecular bonds to break. This causespotential energy holding atoms and molecules togethechange. This happens, for example, when snow meltsperson’s skis rub against the snow or when a meteorite bdue to air friction. A large number of bonds between rubbmolecules are broken when a car skids to stop. Thus, if



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Fig. 5. The physical processes involve friction.~a! A point-particle block slides to a stop on a floor with friction. The system includes only the point-pablock. So the floor exerts an external frictional force on the point-particle block, and this frictional force does a negative amount of work, which hasthe samemagnitude as the block’s initial kinetic energy.~b! A real car skids to a stop on a rough road. The car is the only object in the system. Thus the roatouches the car causes an external frictional force and a difficult work calculation. Chabay and Sherwood argue that for such a real system that is thecar, the amount of~negative! work done on the car by road friction is less than the initial kinetic energy of the car. The remaining amount of the car’senergy is converted into internal energy of the car~see Ref. 20, pp. 236–241 for detailed discussions about this advanced topic!. ~c! One sees a recommendesystem choice that includes the objects and the frictional interfaces between the objects in the system. In this way we can readily include the systes internalenergy change due to friction rather than dealing with a complex work calculation.

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faces with friction are in the system, the system’s interenergy increases due to friction. The internal energy ofsystem is expressed by the last term,DU int~friction!, in thequalitative work–energy bar chart. Helping students visuize this form of energy leads naturally to an introductioninternal energy in thermodynamics.

IV. QUALITATIVE REPRESENTATIONS OFWORK –ENERGY PROCESSES

How do we use the bar charts with students? Figuresare examples of worksheets used by students to qualitatianalyze work–energy processes. Usually, the proceshown in the worksheets are demonstrated with real objin the lecture or laboratory. In Fig. 6~a!, a work–energy pro-cess is described in words and in a sketch. Students are ato construct a detailed sketch that identifies the systemits initial and final states, includes a coordinate system,indicates the values of relevant quantities in the systeinitial and final states. Construction of a sketch such asin Fig. 6~b!, a pictorial representation of the process, is phaps the most difficult task for students.

Having completed a pictorial representation of the pcess, students next construct a qualitative work–energychart for the process~as in Fig. 7!. The student looks at theinitial situation and decides whether the system has kinenergy,K0 . If so, the student places a short bar aboveinitial kinetic energy slot. If there is no initial kinetic energno bar is drawn. The bar for initial gravitational potentienergy,Ug0 , depends on the initial location of an objectthe system relative to the origin of the vertical coordinasystem. The student draws a positive bar if the object ishigher position than the origin of they axis, no bar if at thesame elevation as the origin, or a negative bar if lower th



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the origin. Similarly, the bar for the initial elastic potentienergy,Us0 , depends on whether compressed or stretcelastic objects are initially in the system. The workW bardepends on the presence of objects outside the system

Fig. 6. The work–energy process is described in words and in a skeStudents are asked to construct the pictorial representation, including atem choice and a coordinate system, and indicating the values of the qtities in the system’s initial and final states.



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exert forces on objects inside the system as the sysmoves from its initial state to its final state. The sign of twork depends on the direction of the external force relatto the direction of the displacement of the object in the stem. The final energy of the system is analyzed in the saway as the initial energy.

Having identified nonzero energy terms by placing shbars in the chart, we can now emphasize the conservatioenergy principle by making the sum of the lengths of barsthe left equal to the sum of the bars on the right. The relamagnitudes of the bars on one side are usually unknown,as the relative magnitudes of force arrows in a force diagare often unknown.

Having constructed the bar chart, it is now a simple taskconstruct the generalized work–energy equation to desc

Fig. 7. For the given work–energy process, students are asked to conthe detailed sketch, and then convert it to an energy bar chart. Finally,use the bar chart to apply the generalized work–energy equation.



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this process. There is one term in the equation for each tin the bar chart. When detailed expressions for the typeenergy are developed, students can include these expresin the equations, as in the bottom mathematical represetion in Fig. 7. For the problem in Fig. 7, students couldasked to think about how the chart and the actual procwould change if the coefficient of kinetic friction wadoubled.

In Fig. 8, students start with a complete bar chart andasked to invent a verbal and pictorial description of a phycal process that would lead to that bar chart, and to constthe mathematical representation of the physical proc~There are many processes that could lead to a particchart.! In Fig. 9, students start with the mathematical repsentation of a process and are asked to construct a barthat is consistent with the equation, and to invent a procthat would produce the equation and the chart—a so-caJeopardy problem.23 Students should now have a good quatative understanding of work–energy processes—at lemuch better than when the processes are introduced firsing a formal mathematical approach.

V. QUANTITATIVE WORK –ENERGY PROBLEMSOLVING

Students next solve quantitative problems using tmultiple-representation strategy. They go from words,sketches and symbols, qualitative bar graphs, a generawork–energy equation, a solution, and finally an evaluatto see if their solution is reasonable. An example is shownFig. 10. A set of Active Learning Problem Sheets24 ~the

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Fig. 8. A work–energy process is described by a work–energy bar chStudents start with the bar chart and invent a sketch, a real-world situatiowords, and a generalized work–energy equation that is consistent withbar chart.



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ALPS Kits! has a Work–Energy Kit that includes 38 quatative questions@such as illustrated in Figs. 6~a! and 8# and16 quantitative problems~such as illustrated in Fig. 10!. Stu-dents buy the ALPS Kits at the beginning of the semeste

Fig. 9. The so-called Jeopardy problem starts with the mathematical etion for a work–energy process. Students are asked to construct a barthat is consistent with the equation, to draw a sketch, and to invent a prothat would produce the equation and the chart.



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quarter. They solve some of these problems during lectuduring recitations, and for homework. Problems from ttext are assigned with the proviso that the same formshould be used on these problems. In addition, a set oqualitative and quantitative work–energy problems~includedas part of the ActivPhysics1 CD and workbook!25 is used asactive learning activities during lectures and laboratoriThe problems are computer simulations, which includedynamic work–energy bar charts to help students visuaenergy transformations and conservation during physprocesses.

VI. STUDENT ACHIEVEMENT

Do the energy bar charts and the problem-solving metinvolving the multiple representations of a work–energy pcess help students understand energy concepts and solvlated problems? Do students have better performance onergy problems using this method than using othapproaches? Do students use these multiple-representstrategies when rushing through an hour-long exam or a fiexam? In the following, we try to answer these questiofrom a study done during the Fall 1997 and Winter 19quarters of the calculus-based introductory physics courseThe Ohio State University~OSU!.

A. Student attitudes toward the energy bar charts andthe multiple representations of work–energyprocesses

Students responded very positively on a three-quesfree-response survey concerning the energy bar chartsthe method of the multiple representations. The survey wadministered in the last week of the fall quarter 1997 tohonors engineering freshmen who learned this approachcalculus-based introductory physics class.26 This ten-weekclass covered the regular concepts of mechanics: kinemaNewton’s dynamics, momentum, work and energy, and so

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Fig. 10. One of the quantitative problems includedthe Active Learning Problem Sheets. Students sothese problems using the multiple-representation stegy after having developed skills to construct qualittive representations. These multiple-representatproblems help students develop qualitative understaing about the physical processes and develop problesolving expertise, instead of using only an equatiocentered method.



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rotational dynamics. The students worked with the work aenergy concepts for about six lecture periods, three recitaperiods, one lab period, and in their homework assignmeStudents’ answers on each of the three free-response suquestions are summarized in Tables I, II, and III. Fromresults in Tables I and II~the results in Table III will bediscussed in Sec. VI C!, we can see that energy bar chartsa visual representation play a crucial role in helping studeunderstand energy concepts and solve related probleMost students thought the work–energy multiple represetion technique was helpful for learning physics. They afelt it helped them develop problem-solving expertise,stead of using a formula-centered method.

Table I. Students’ responses for question 1 on the survey; Did usingenergy bar charts help you learn energy concepts and solve work–enproblems? Explain why they were useful or not useful.

92% Useful (N567)64% 64% of the students thought that the energy bar chart

helped them visualize what is happening to different typesof energy in energy problems, and set up the right equationto solve the problems easily.

Two examples of the students’ responses:• Greatly helped, because they provided visual

representation of what’s going on and made figuringout the equations easier.

• The energy bar charts were extremely useful becausthey provided direction for each problem. You wereable tosee what types of energy were being used adifferent stages of the problems and what individualenergy equations to apply to the problems.

15% 15% of the students thought that the energy bar charthelped them understand the abstract energy concepts athe energy conservation better.

Two examples of the students’ responses:• Yes, they were very helpful. They allowed me to see

how and in what forms energy was conserved.• These wereextremely helpful. Doing this made me

change my way of think@sic# from thinking in termsof equation to concepts. Once I had the conceptsdown, then I can choose the right equations.

10% 10% of the students thought that the energy bar charts wehelpful in learning the energy concepts at the beginningand after a while they could create the bar charts mentallrather write them out.

One example of the students’ responses:• They were useful since they gave a visual way to go

about the problems. After a while I got to the pointthat I automatically did them in my head did not needto really write them out.

3% 3% of the students thought that the energy bar charts werhelpful, but they did not explain their reasons.

8% Not useful (N567)8% 8% of the students preferred using equations directly rathe

the energy bar charts.One example of the students’ responses:• No. I had trouble finding relationships between the

energies involved using the energy bar charts. I wasbetter off using the equations. The energy bar chartonly showed before, after, which increased, ordecreased, and which energy was more. Personally,needed exact values, which working equationsprovided. I did not gain a better understanding ofenergy using the charts.



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B. Student performance on a tutorial-type problem

Although the energy bar charts and the multiprepresentation approach are developed as a problem-sostrategy to help students acquire problem-solving expertthey emphasize the development of qualitative understanand reasoning about work–energy concepts and proceDoes performing this type of qualitative analysis improstudent scores on conceptual work–energy questions? Todress this question, we examined our students usintutorial-type work–energy problem~shown in Fig. 11! devel-oped by the Physics Education Group at The UniversityWashington~UW!.27,28

The problem in Fig. 11 asks students to compare the fikinetic energy of two pucks having different masses pusby the same force across the same distance. The OSU hoengineering freshmen were given this problem on a surtest at the end of winter quarter 1998—one quarter after thad studied the introduction of the work–energy methodfall quarter 1997. Out of the 56 students, about 60% of thgave correct answers with correct reasoning in words orusing equations. The same problem was given on the fiexam to a regular OSU engineering calculus-based phyclass after standard instruction in which the bar chartsthe multiple representation strategy had not been used. A20% of 147 students in this class provided correct explations for the problem.

ergyTable II. Students’ responses for question 2 on the survey; Did represethe work–energy processes in multiple ways—words, sketches, bar cand equations—help you learn energy concepts and solve energy problExplain why they were useful or not useful.

84% Useful (N567)84% 84% of the students thought that the multiple-representatio

strategy was helpful to understand the concepts better, anto set up the problems correctly, and was a good teachinmethod to help a variety of students learn physics.

Two examples of the students’ responses:• Doing these problems in many different ways with

different descriptions helped to visualize the deeperbasic concepts underlying them. Seeing a problemonly one way, or learning a concept only one way,causes your knowledge of that concept to be veryone-dimensional. Representing multiple ways helpsme to see all sides of a concept.

• They did help me by letting me see the problemdifferent ways. Also I would like to note that this is agood teaching method to help a variety of studentslearn physics. Often times a student doesn’tunderstand one method but sees another.

9% Useful and not useful (N567)9% 9% of the students thought that the multiple representation

were helpful to understand the energy concepts, but wersometimes a waste of time for quantitative problem-solving

One example of the students’ responses:• Sometimes they were useful but doing each type of

description for each problem was overkill and wasteda lot of time.

7% Not useful (N567)7% 7% of the students thought that the multiple representation

were not helpful except equations.One example of the students’ responses:• No. Only the equations were of any use to me. The

rest just seemed to get in the way.



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The paper by O’Brien Pride, Vokos, and McDermott~seeRef. 28! reports that after standard lecturing for 985 reguand honors calculus-based physics students from UW, 1got correct answers with correct reasoning on this questIn addition, for 74 UW tutorial instructors~most of themgraduate teaching associates! before teaching the Washington work–energy tutorial sections, 65% answered correwith correct explanations. Furthermore, it is interestingsee from the paper that before the tutorials, 65% of 1physics faculty from the national tutorial workshops succefully produced correct answers to the question shown in F11. The percentages of these different groups success

Table III. Students’ responses for question 3 on the survey: Did you~or howdid you! use the energy bar charts and the multiple representations of wenergy processes to solve energy problems while doing homework, gproblems in recitation, problems in lab, and/or on exam problems? Diduse them when first becoming familiar with the concepts and then lesyou become more expert at solving work–energy problems?

95% Use in problem solving(N567)46% 46% of the students used the energy bar charts and th

multiple representations to solve problems at the beginningbut less and less as they became more familiar with thproblems.

Two examples of the students’ responses:• I used them less as I became more of an expert.

began to go straight to writing out the formulas~foreach type of energy! and the equations. But, I kept thebar chart ideas in the back of my mind.

• I did use them. I first used the bar chart just to seewhat parts I wanted in the equations, such as workfrom friction or potential spring energy. Then I wouldwrite the equation down and solve. Now I’ve becomemore able to solve problems. I still use the barswhenever I get confused. They really help to limitmistakes in writing the initial equation.

33% 33% of the students used the energy bar charts and thmultiple representations most of the time when solvingenergy problems.

Two examples of the students’ responses:• I used the charts and representations throughout th

process to be sure I was setting up the problem righwithout overlooking important details. Convertingone representation to another was just one more waI double checked my work.

• I used the charts every time, although I did’t alwayswrite it down on paper. It’s a safe guard@sic# that youare not neglecting any form of energy.

16% 16% of the students used the energy bar charts and thmultiple representations in other ways.

Two examples of the students’ responses:• I personally only used the bar charts if I felt the

situation was difficult enough to require a furtheranalysis. However, at the beginning I used the bacharts for every problem and they really helped megrasp the concepts.

• I used them very little in the very beginning, but thenI realized their uses. Once I became familiar withthem, I was able to be more comfortable solvingproblems. Now I have worked with many differentproblems, I use them to check my work.

5% Never use in problem solving~N567 in total!5% 5% of the students never used this method, and they onl

used equations or got help from the textbook.



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answering this question are summarized in the bar grapFig. 12. The OSU calculus-based honors engineering fremen learning the multiple-representation strategy and usthe energy bar charts performed much better on this probthan regular and honors calculus-based physics studentsfrom OSU and from UW, and they did almost as wellphysics faculty and physics graduate students.

C. Student use of the multiple-representation strategy inproblem solving

How do students use the multiple-representation metto solve energy problems? In the survey with the three frresponse questions administered to the 67 honors engining freshmen, the last question asked how the studentsthe energy bar charts and the multiple representationssolving problems. A summary of the student responsereported in Table III. We find that most of the students usthis method to solve problems. Although about half of tstudents used it less and less when becoming more famwith energy problems, many of them still kept the energy bcharts in their mind as a useful problem-solving toolevaluate their solutions.

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Fig. 11. The problem originally developed by the Physics Education Grat UW was administered to the OSU honors engineering freshmen one qter after they had learned the work–energy method. The students werethat their scores on the problem did not affect their class grades. This plem was also given to 147 OSU regular calculus-based introductory phystudents after standard instruction in which the bar charts and the multrepresentation strategy had not been used.

Fig. 12. The graph shows the percentage of each group that correctlyswered the question shown in Fig. 11. Bars 1 and 2 indicate that no mthan 20% of over 1000 calculus-based physics students from UW and Oanswered and explained the problem correctly after standard instrucBars 3 and 4 indicate that 65% of about 200 physics graduate studentfaculty correctly provided the answers and reasoning for the problempretests before the UW tutorials. Bar 5 indicates that 60% of more thanOSU honors engineering freshmen successfully answered the questioncorrect reasoning.



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But did the students actually apply the multiplrepresentation method when rushing through their exaOn the final exam problem given in this calculus-based hors engineering freshmen class, 66% of the studentsstructed a pictorial representation of the process. 57% ofstudents solved the problem using the work–energy methbut 43% of them applied Newton’s second law. Among t57% of students who used the work–energy approach, a16% of them drew a qualitative work–energy bar chart, a79% of them correctly constructed the generalized woenergy equation.

From the above results, we see that many students mapictorial sketch to help them understand the problem, butmany actually constructed a work–energy bar chart onfinal exam problem. There could be several reasons forWith the time pressure needed to complete an exam, studskip steps that they feel are not needed to complete thelution. Students’ responses on the survey question in TaIII indicate that many students had become more famiwith work–energy problems and had learned to draw thecharts mentally—they no longer needed to draw them expitly. The similar case can be found for the students usingmotion diagrams in solving kinematics problems. The sdents explicitly drew the motion diagrams less and less wthey could mentally use the motion diagrams. This finexam problem may not have been difficult enough for mof the students to need to use the work–energy bar charcomplete the solution. Finally, it might be that some studefelt the bar charts were not useful in problem solving, andthere was no reason to use them.

For a future study, we plan a more detailed experimeninvestigate how the energy bar charts and the multiprepresentation strategy help introductory physics studenunderstand qualitatively work–energy processes and tovelop problem-solving expertise compared to similar sdents learning other work–energy approaches.

VII. SUMMARY

It is well known that students attempt to solve probleby matching quantities listed in the problem statementspecial equations that have been used to solve similar plems. Students move between words and equations, ware very abstract representations of the world, with notempt to connect either representation to more qualitarepresentations that improve understanding and intuition

In an alternative strategy proposed by others and uhere, we view physics problems as descriptions of physprocesses. We ask students to represent these procesvarious ways. Our preliminary study indicates several advtages for this method. For example, the linkage betweenword description of the process and its pictorial and bar chrepresentations helps students produce mental images fodifferent energy quantities. The bar charts help studentssualize the conservation of energy principle—the studecan ‘‘see’’ the conservation of energy. A sketch withsystem choice combined with the qualitative bar chart hestudents reason about physical processes without using mematics and helps them predict conceptually how changevarious factors will affect the process. A bar chart functiolike a bridge, helping students move from the abstract reof words, or from real-life sketches with surface featuresthe abstract realm of scientific and mathematical notatio



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Additionally, this strategy aids students in overcomiformula-centered naı¨ve methods and in developing expertiin problem solving.

The actual classroom strategies are very important. Sdents learn to learn better if they understand the reasonsvarious pedagogical strategies. We find, for example, tstudents in some classes regard the construction of sketand bar charts as activities that are independent and unreto applying the conservation of energy principle to a prolem. We have observed the same response when studenasked to construct free-body diagrams and to apply the cponent form of Newton’s second law—these are consideas unrelated independent activities. For an experienphysicist, a qualitative representation is important for solva challenging problem. For instance, a Feynman diagprovides a more visual representation of a scattering procThese more qualitative representations are used to help athe more abstract mathematical representations correctlyexample, anS matrix for a scattering process. Students munderstand why they use the qualitative representations.

However, an expert may subconsciously use a qualitarepresentation for an easier problem—for example, consta mental free-body diagram. Students may avoid using qutative representations early in their study because they dounderstand what is being represented. It makes no sensestudent to draw a free-body diagram if the student doesunderstand the concept of force or the nature of differtypes of forces or how the free-body diagram is to be usehelp solve a problem. Similarly, a bar chart is not helpfula student does not understand the different types of energthe conservation of energy principle or how the bar chacan be used to solve problems. Students must underswhy they are learning to represent processes in the mqualitative ways and how these qualitative representatican be used to increase their success in quantitative probsolving.

We feel that students accept qualitative representatmore easily, understand them better, and use them morfectively for qualitative reasoning and problem solving if thqualitative representations are introduced before the cosponding mathematical equations are introduced. Therless ‘‘noise’’ in the mind.

Many students have experienced only formula-centedidactic instruction. For these students, it may be difficultapply this new multiple-representation method in their prolem solving. Some students like only equations and thinwastes time or is a redundant task to represent a probledifferent ways. For a novice with little conceptual undestanding, this is not true. However, as students acquirederstanding, some qualitative representations become mschema and constructing paper versions is less necessa

Finally, when students have learned all of the represetion types, their understanding improves if they learnmove among representations in any direction. For examthey might be given an equation that is the application ofconservation of energy principle for some process. Their tis to construct other representations of that process—plaEquation Jeopardy.23 They learn to ‘‘read’’ the symbolicmathematics language of physics with understanding.

ACKNOWLEDGMENTS

We gratefully thank the OSU Physics Education ReseaGroup for their help and feedback in the preparation of t



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paper. We especially acknowledge the help of Leith Allfor her constructive suggestions on the paper and GorAubrecht for his valuable review. In addition, our heartfthanks go to Bruce Sherwood for his critical and insightsuggestions on the manuscript. Moreover, support fromNational Science Foundation~Grant Nos. DUE-9653145DUE-9751719, and GER-9553460! is deeply appreciated. Ihas enabled us to develop new instructional methods, athem to classrooms, and evaluate their effectiveness in hing student learning.

1R. Plotzner,The Integrative Use of Qualitative and Quantitative Knowedge in Physics Problem Solving~Peter Lang, Frankfurt am Main, 1994!,pp. 33–53. In the book, he summarizes research findings on differencproblem solving between novices and experts.

2F. Reif and J. I. Heller, ‘‘Knowledge structures and problem solvingphysics,’’ Ed. Psych.17, 102–127~1982!.

3J. I. Heller and F. Reif, ‘‘Prescribing effective human problem-solviprocesses: Problem description in physics,’’ Cogn. Instrum.1, 177–216~1984!.

4J. Larkin, ‘‘Understanding, problem representations, and skill in physicin Thinking and Learning Skills, edited by S. F. Chipman, J. W. Segal, anR. Glaser~Lawrence Erlbaum Associates, Hillsdale, NJ, 1985!, Vol. 2, pp.141–159.

5J. Larkin, ‘‘The role of problem representation in physics,’’ inMentalModels, edited by D. Gentner and A. L. Stevens~Lawrence Erlbaum As-sociates, Hillsdale, NJ, 1983!, pp. 75–98.

6D. Hestenes, ‘‘Toward a modeling theory of physics instruction,’’ Am.Phys.55, 440–454~1987!.

7D. Hestenes and M. Wells, ‘‘A mechanics baseline test,’’ Phys. Teach.30,159–166~1992!.

8A. Van Heuvelen, ‘‘Learning to think like a physicist: A review oresearch-based instructional strategies,’’ Am. J. Phys.59, 891–897~1991!.

9A. Van Heuvelen, ‘‘Overview, case study physics,’’ Am. J. Phys.59,898–907~1991!.

10P. Heller, R. Keith, and S. Anderson, ‘‘Teaching problem solving throucooperative grouping. 1. Group versus individual problem solving,’’ AJ. Phys.60, 627–636~1992!.

11R. Hake, ‘‘Interactive-engagement versus traditional methods: Athousand-student survey of mechanics test data for introductory phcourses,’’ Am. J. Phys.66, 64–74~1998!.



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12J. Larkin and H. Simon, ‘‘Why a diagram is~sometimes! worth ten thou-sand words,’’ Cogn. Sci.11, 65–99~1987!.

13H. Gardner,The Unschooled Mind~BasicBooks, New York, 1991!, p. 13.14P. W. Bridgman,The Nature of Thermodynamics~Harvard U.P., Cam-

bridge, MA, 1941!.15See Ref. 14, pp. 23–47.16H. Burkhardt, ‘‘System physics: A uniform approach to the branches

classical physics,’’ Am. J. Phys.55, 344–350~1987!.17B. A. Sherwood, ‘‘Pseudowork and real work,’’ Am. J. Phys.51, 597–602

~1983!.18B. A. Sherwood and W. H. Bernard, ‘‘Work and heat transfer in t

presence of sliding friction,’’ Am. J. Phys.52, 1001–1007~1984!.19A. B. Arons,Teaching Introductory Physics~Wiley, New York, 1997!.20R. W. Chabay and B. A. Sherwood,Matter & Interactions~Wiley, New

York, 1999!, preliminary edition.21See Ref. 14, pp. 47–56.22See Ref. 19, pp. 150–153.23A. Van Heuvelen and D. Maloney, ‘‘Playing physics Jeopardy,’’ Am.

Phys.67, 252–256~1999!.24A. Van Heuvelen,Mechanics Active Learning Problem Sheets (the AL

Kits) ~Hayden-McNeil Publishing, Plymouth, MI, 1996!. This set of theALPS Kits is developed for college or high school students to learn mchanics in algebra-based and calculus-based introductory physics couAnother set of the Electricity and Magnetism ALPS Kits can be usedstudents in the electricity and magnetism part of introductory physcourses.

25A. Van Heuvelen,ActivPhysics1~Addison–Wesley Longman, Palo AltoCA, 1997!.

26The course was taught by one of the authors. In this class, studentsactively participated in activities using the ALPS kits and the compusimulations with the workbook from the ActivPhysics1 CD in ‘‘lectures~called Large Room Meetings!. In ‘‘recitations’’ ~called Small RoomMeetings! students cooperatively worked in groups and solved compproblems such as context-rich problems and multipart problems.

27R. A. Lawson and L. C. McDermott, ‘‘Student understanding of the woenergy and impulse-momentum theorems,’’ Am. J. Phys.55, 811–817~1987!.

28T. O’Brien Pride, S. Vokos, and L. C. McDermott, ‘‘The challengematching learning assessments to teaching goals: An example fromwork-energy and impulse-momentum theorems,’’ Am. J. Phys.66, 147–157 ~1998!.

TURNING TO PHILOSOPHY

It is not uncommon for distinguished scientists in the twilight of their careers to turn their handto philosophy. Unfortunately, the failures among such endeavors are generally acknowledged tooutnumber the successes, and. . . ’s contribution to the genre must on the whole be consigned tothe majority.

John Dupre´, in a book review in Science280, 1395~1996!.



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Multiple representations of work–energy processesazmodmech2017.weebly.com/uploads/4/7/7/4/47742735/... · problems since, as visual aids, they automatically enhance human perceptual