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Student understanding of energy: Difficulties related to systems

Beth A. Lindsey, Paula R. L. Heron, and Peter S. Shaffer  

Citation: Am. J. Phys. 80, 154 (2012); doi: 10.1119/1.3660661 

View online: http://dx.doi.org/10.1119/1.3660661 

View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v80/i2 

Published by the American Association of Physics Teachers 

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Student understanding of energy: Difficulties related to systems

Beth A. Lindseya)

 Department of Physics, Penn State Greater Allegheny, McKeesport, Pennsylvania 15132

Paula R. L. Heronb) and Peter S. Shafferc)

 Department of Physics, University of Washington, Seattle, Washington 98195-1560

(Received 15 April 2011; accepted 24 October 2011)

Choosing a system of interest and identifying the interactions of the system with its environment are

crucial steps in applying the relation between work and energy. Responses to problems that weadministered in introductory calculus-based physics courses show that many students fail to recog-

nize the implications of a particular choice of system. In some cases, students do not believe that

particular groupings of objects can even be considered to be a system. Some errors are more prevalent

in situations involving gravitational potential energy than elastic potential energy. The difficulties are

manifested in both qualitative and quantitative reasoning.VC  2012 American Association of Physics Teachers.

[DOI: 10.1119/1.3660661]

I. INTRODUCTION

In a previous paper,1 we reported common student diffi-culties with the relation between work and energy in intro-ductory mechanics. We   also described the development of instructional materials2 that can help students to achieve abetter functional understanding of the work-energy principle.

The research described in Ref.   1   involved questions thathad one element in common: The problem statements madeexplicit the particular system that students should use in their analysis. They were not asked to consider more than one sys-tem in a given problem. In most cases, the system was under-going changes in more than one type of energy (for example,a system composed of a block and a spring, in which both ki-netic and potential energy were changing). In most of theresponses, confusion related to the choice of system did notarise explicitly, and we had no direct evidence that studentswere considering a system other than the one we had intended

(for example, the system composed of the block alone). Wefound that even after modified instruction using the tutorials,many common difficulties persisted. We hypothesized thatthese difficulties might be tied to a fundamental confusionabout the implications of the particular choice of system.Thus, we decided to investigate more thoroughly studentunderstanding of the importance of choosing a system of in-terest, and the implications of the choice in the subsequentenergy analysis. To that end, we designed a series of questionsin which students are either asked to consider multiple sys-tems for a given problem or given no explicit prompt to con-sider a particular system. The results led to further insightsinto student thinking about mechanical systems.

The investigation took place in the introductory calculus-

based mechanics course at the University of Washington(UW). This large enrollment course is the primary setting inwhich   we have been developing   Tutorials in Introductory

 Physics,3 instructional materials   intended to supplementinstruction by lecture and textbook.4 Data were also obtainedin a weekly seminar devoted to preparing graduate and under-graduate teaching assistants (TAs) for teaching using the tuto-rials and for other current and future instructional roles.5

II. BACKGROUND

Several recent articles have re-examined the way inwhich energy conservation is presented in introductory

physics.6 – 8 In particular, several authors advocate a unifiedapproach to energy instruction based on the first law of thermodynamics. (The more common approach is to relyon a version of the work-energy theorem derived from

Newton’s second law.) Because the first law of thermody-namics relates the change in energy of a system to thework done on and heat transfer to the system, applicationof this law implicitly depends on the choice of the systemunder consideration.

The example in Fig.   1   illustrates how a description of energy transfers and transformations depends on the choiceof system. A block on a level frictionless surface is connectedby an ideal spring to a wall. The block is released from restwith the spring stretched. Later, it passes   x¼ 0, where thespring is at its equilibrium length. If we consider the systemto be just the block, the spring is the only external agentdoing work on the system during this interval. This work ispositive and causes an increase in the block’s total energy (in

this case, entirely kinetic):   W on block by spring¼D K block. If,instead, we consider a system consisting of the block and thespring, no external agents do work on the system so its totalenergy does not change and  D Etotal¼ 0. The increase in ki-netic energy is balanced by the decrease in potential energy:D K system¼DU system. Although the two analyses are equiva-lent, one choice might lead more readily to the solution of aparticular problem.

In this example, we did not attribute potential energy tothe system consisting of just the block, which undergoes nodeformation; instead, we associated potential energy withthe configuration of the system consisting of the block andspring. As described by Jewett,9 many instructors and in-troductory texts do not emphasize this distinction. For 

example, in discussing the motion of an object near theEarth’s surface, textbooks often refer to “the potentialenergy of the object” rather than of the system consistingof the object and the Earth. The common expressionU grav¼mgh  might reinforce this interpretation, because theheight  h  might appear to be a property of the object rather than of the object-Earth system. This lack of precisionmight encourage students to double-count this interactionby erroneously including a term that represents the workdone on the object by the Earth and a term representing thechange in gravitational potential energy. Several papershave commented on difficulties that   might arise if thechoice of system is not made clear,6 – 8 and some describe

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instructional strategies that might be used to clarify issuesassociated with choice of system.10

Empirical studies examining the learning and teaching of work and energy at various levels have largely focused   onissues other than the choice of system. Loverude   et al.11

investigated the ability of university students to apply the

definition of work, and Lawson and McDermott12 docu-

mented student use of the wor k-energy theorem for point

objects. Singh and Rosengrant13 focused on the development

and validation of a multiple-choice test of energy and mo-

mentum concepts. This test invokes the idea of potential

energy but the paper did not document student difficulties

related to the choice of the system.14 A multiple-choice test

developed by Ding15 addresses the choice of system exten-

sively but has been used mainly to assess the ef fectiveness of 

the   Matter and Interactions   curriculum.16 Driver and

Warrington17 investigated energy reasoning among precol-

lege students and concluded that an increased emphasis on a

systems point of view might   improve student ability to

reason with work and energy.18 Difficulties related to the dis-

tinction between system   and surroundings have been docu-

mented in other contexts.19 We know of no other study that

has documented such problems in the context of energy and

the first law of thermodynamics in a traditional introductory

mechanics course.

III. INVESTIGATION OF STUDENT

UNDERSTANDING

The discussion in Sec.   II   outlines some of the elementsneeded to understand the importance of the choice of systemin applying the concepts of work and energy. We suspectedthat many students do not have a functional understanding of these points. Therefore, we designed problems to probe theextent to which students recognize that certain interactionscan be associated with either potential energy or work,depending on the choice of system, and their ability to decidewhether a system is closed, that is, undergoes no change intotal energy.

Each problem was given online as a pretest for the subse-quent tutorial.20 Students selected an answer from a list of options and then typed their explanation. Each problem wasgiven in more than one course section. In some cases, stu-dents had completed all lecture instruction on work andenergy, a three-hour laboratory experiment on work andchanges in kinetic energy, and a tutorial on work and energythat did not explicitly cover the idea of systems.1,2 In other cases, students had been introduced to work or energy in lec-ture, laboratory, or tutorial, but instruction had not yet con-cluded. We have found that differences in lecturer, textbook,and other aspects of the course typically lead only to smalldifferences in student performance on a given question.Occasionally, these small differences are statistically signifi-cant. Our interest is in characterizing the general nature of student performance in a typical introductory course anddetermining which interventions lead to a large improvementin performance. Thus, we did not distinguish between differ-ences in performance of less than 10% and have grouped stu-dents together, except as noted.21

In this section, we illustrate the kinds of problems admin-istered to students that involved elastic interactions and dis-cuss specific difficulties that arose. We then summarize aseries of analogous problems involving gravitational interac-

tions and an additional difficulty that arose in those prob-lems. The problems we administered are listed in Table I.

A. Representative research problems: Elastic contexts

During this investigation, we designed and administered awide variety of qualitative and quantitative problems. Tworepresentative problems are discussed involving a blockinteracting with a spring on a level, frictionless surface.

We devised the “explicit block-spring” problem to gaugewhether students could successfully isolate an arbitrary sys-tem by correctly identifying the system’s interactions withits environment and the effect of these interactions. Studentswere presented with a physical situation and asked to con-

sider two possible systems composed of different groupings

Fig. 1. The “explicit block-spring” problem. A block attached to a stretched

ideal spring is released from rest and moves to the left on a level frictionless

surface. Students were asked about work done on and changes in energy in a

system consisting of just the block, and a system consisting of the block and

the spring.

Table I. Summary of the problems described in this paper.

Problem

Section where

first described Corresponding figure Context Systems Results

Explicit block-spring III A Fig. 1   Elastic (1) Block (2) Block and spring Tables II and III

Block-spring equations III A Fig. 2   Elastic Block (implicit) Tables IV and V

Explicit pendulum-Earth III C Fig. 3   Gravitationa l (1) Ball (2) Ba ll a nd Ea rth Ta bles VI and VII

Block-Earth equations III C Fig. 4   Gravitational Block (implicit) Table V

Block-Earth dialogue III C Fig. 4   Gravitational Block (implicit) Described in the text

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of the objects. We refer to this problem as “explicit” becausestudents were explicitly presented with the opportunity toconsider one interaction from the viewpoint of multiple sys-tems. In contrast, in several other problems described in thepaper, the choice of system is implicit.

The problem involves a block that is released from rest att i, after the spring connecting it to a wall is extended past itsequilibrium length (see Fig. 1). Students were first instructedto consider a system consisting only of the block and were

asked whether the net work on this system is positive, nega-tive, or zero, and whether the total energy of this systemincreases, decreases, or stays the same for the intervalbetween   t i   and   t f   (when the spring is at its equilibriumlength). In some versions of the question, students were alsoasked whether the kinetic and potential energies of the sys-tem increase, decrease, or remain the same for this interval.They were asked to explain their reasoning for eachresponse. Students were then asked the same questions for the system consisting of the block and the spring. Our expe-rience in the classroom suggested that some students wouldclaim that one or the other of these groupings of objectscould not be considered as a single system. Therefore, weincluded this option as a possible response for each question.

The reasoning required to arrive at the correct answers issummarized in Sec. II.Results for the explicit block-spring problem are summar-

ized in Tables   II   and   III. For the system consisting of theblock alone, 33% of the students stated correctly that its totalenergy increases, even though 69% responded correctly thatthe net work is positive. For the block-spring system, 74%responded correctly that the total energy of the systemremains the same and 53% indicated correctly that the network is zero. About 20% of the students gave a correct answer for the change in the total energy for both systems, while 56%gave the same response for the change in energy of both sys-tems. Specific difficulties with systems that were elicited bythe problem are discussed in Sec. III B. Difficulties in relating

work and energy were discussed in detail in Ref.  1.To answer this problem correctly, students must be able to

interpret the term “system” and to recognize that the two sys-tems under consideration are different. We suspected that

some students might not be able to analyze a problem from thepoint of view of two or more different systems, but they mighthave an understanding that is sufficiently robust to answer 

quantitative questions correctly. To investigate this possibility,we designed problems that probe student ability to relate workand energy mathematically. In these problems, as is commonin most introductory textbook problems, students were not ex-plicitly prompted to consider any particular system.

A representative example, the “block-spring equations”problem, is shown in Fig.  2 in which a block is attached byan ideal spring to a wall. At  t i, the block is at rest on a sur-face of negligible friction and the spring is at its equilibriumlength. Between  t i  and   t f , the block is pushed by a hand sothat the spring extends and the block has nonzero speed at  t f .

In the first part of the problem, students were asked whether each of the following four quantities is positive, negative, or 

Table II. Results from two parts of the explicit block-spring problem (see

Fig.  1) in which students were asked about the change in the total energy

during a specified time interval for two choices of system: the block alone

and the block-spring system. The correct responses are indicated in bold.

The results are the mean across four classes. Uncertainties reflect the 95%

confidence interval based on variances among these four samples. In this

and all other tables, the percentages include both students who did and who

did not provide correct explanations.

Results

 N ¼595

Total

energy

increases

Total

energy

decreases

Total

energy stays

the same

Cannot be

grouped

as a system

Block   33%6 6%   9%65% 54%69% 4%6 1%

Block and spring 12%6 9% 9%63%   74%64%   4%6 3%

Correct for both

systems

20%6 5%

Same answer for 

both systems

56%6 6%

Table III. Results from a version of the explicit block-spring problem (see

Fig. 1) in which students were asked about the change in potential energy

during a specified time interval for the block alone and the block-spring sys-

tem. Uncertainties are not given because the question was administered in

only one section of the course. The correct responses are indicated in bold.

Results

 N ¼148

Potential

energyincreases

Potential

energydecreases

Potential

energy staysthe same

Cannot be

groupedas a system

Block 13% 53%   28%   6%

Block and spring 18%   49%   26% 5%

Correct for both

systems

13%

Same answer for 

both systems

45%

Fig. 2. The “block-spring equations” problem. A block is attached to a

spring, which is attached to a wall. At time  t i  the block is at rest on a level

surface of negligible friction; the block is pushed to the right such that it

passes position xf  at time t f  with speed vf . Students were asked whether each

of the following quantities were positive, negative, or zero: The work on the

block by the hand, the work on the block by the spring, the change in poten-

tial energy of the block, and the change in kinetic energy of the block. They

were also asked to write an equation relating any nonzero terms.

156 Am. J. Phys., Vol. 80, No. 2, February 2012 Lindsey, Heron, and Shaffer 156

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zero between   t i   and   t f : the work done on the block by thehand,  W BH; the work done on the block by the spring,  W BS;the change in kinetic energy of the block, D K ; and the changein potential energy of the block, DU. They were also given theoption of saying that it was not possible to determine the signof any of these quantities. They were then asked to explaintheir reasoning. For this problem we did not probe which sys-tem the students chose, but whether they would deal with allfour quantities in a manner consistent with some implicitchoice. The wording of the questions suggested that a systemconsisting of just the block be considered.

In the second part of the problem, students were asked if was possible to write a single equation that relates all of thequantities that they had indicated were nonzero. They were

then asked to write an equation relating as many of thesenonzero quantities as possible and to explain their reasoning.

Students could use the definition of work to conclude thatW BH is positive and W BS is negative. Similarly, D K  is positive,because the block begins at rest and ends with a nonzero ve-locity.  DU  is zero because the block does not deform (and thespring is outside the system). It is possible to relate the threenonzero quantities with the work-energy theorem, W net¼D K ,in the form   W BHþW BS¼D K , or with   W BHW BS¼D K ,which implicitly relates the absolute values of these quantities.Because there is not a unique correct solution for this problem,we categorized responses according to whether the equationswould lead to correct or incorrect results if the students weregiven numbers. Three equations are “correct” according to

this criterion:  W BHþW BS¼D K  (for the system consisting of the block alone),  W BH¼D K þDU  (for the system consistingof the block and spring) and  W BS¼DU   (which relates thetwo systems).22 The relation  W BHþW BS¼D K þDU   can beconsidered to be correct if DU  or  W BS is identified as zero.

The results from the first part of the problem, in which stu-dents identified the signs of the four quantities, are shown inTable  IV.23 For the first three quantities, between 68% and94% of the responses were correct for a system consisting of the block alone. The most common response for the questionabout the change in potential energy of the block was that itis positive, which is consistent with thinking about a systemconsisting of the block and spring together.

Only 10% of the students gave a set of four responses con-sistent with the choice of the block alone as the system of in-terest. About 59% identified all four quantities as beingnonzero, claiming that both the work done on the block bythe spring and the change in potential energy of the block are

nonzero. These students might have made no attempt to an-swer all four questions with respect to a single system, or they might not have recognized that interactions must onlybe taken into account once. The equations they wrote pro-vide some insight. The most common ones are shown in col-umn I of Table   V. Only 18% of the students answeredcorrectly by writing only correct equations; another 9%wrote at least one cor rect equation but included one or moreincorrect equations.24

B. Summary of specific difficulties identified

We have asked many variations on both of the representa-tive block-spring problems described in Sec. III A. In the fol-

lowing, we interpret the responses in terms of commondifficulties related to the choice of system and discuss someimplications. These difficulties are not mutually exclusive: aparticular response could fit into more than one category.However, we have found these categories useful for describ-ing patterns in student responses.

1. Failure to recognize that an energy analysis dependson the choice of system

Students often fail to match their analysis of work, energy,and energy transfers to the system under consideration. Manyseem to treat different groupings of objects as identical. This

Table IV. Results from the first part of the block-spring equations problem

(see Fig. 2) in which students were asked whether various quantities listed

are positive, negative, or zero over a specified interval. Results from the

block-Earth equations problem were similar to these, except as described in

the text. The responses consistent with the choice of the block alone as the

system of interest are shown in bold. The results are the mean across four 

classes. Uncertainties reflect the 95% confidence interval based on variances

among these four samples.

Results

 N ¼531

Positive Negative Zero

Work on the block by the

hand (W BH)

94%66%   4%65% 1%62%

Work on the block by the

spring (W BS)

23%67%   68%64%   7%62%

Change in kinetic

energy of the block (D K )

74%64%   11%64% 11%62%

Change in potential

energy of the block (DU )

49%68% 29%65%   17%64%

Table V. Results from parts of the block-spring equations problem (see Fig. 2)

and the analogous block-Earth equations problem (see Fig.   4) in which stu-

dents were asked to write equations relating the quantities they had identified

in part 1 as nonzero. A possible interpretation for each of these equations is

included. The first three equations represent the possible correct options. The

results are the mean across four classes for the block-spring equations problem

and three classes for the block-Earth equations problem. Uncertainties reflect

the 95% confidence interval based on variances among these samples.

I II

Block-spring Block-Earth

 N ¼531   N ¼380

W BHþW BS or E¼D K 

(correct for the block system)

16%6 4% 7%6 11%

W BH¼D K þDU 

(correct for the block-spring or 

block-Earth system)

6%6 4% 13%6 3%

W BS or E¼DU 

(relates block system to block-spring or 

block-Earth system)

6%6 5% 6%6 11%

W BHþW BS or E¼D K þDU 

(double-counting)

20%6 5% 24%6 18%

(Ref. 26)

W BH¼W BS or E

(true if the block were to

begin and end at rest)

13%6 4% 10%6 15%

DK¼DU

(true if the energy of the block-spring or 

block-Earth system were constant)

12%6 4% 10%6 11%

W BH¼D K  (possible misapplication of the

work-energy theorem)

5%6 2% 5%6 6%

W BHþW BS or E¼DU  (possible misapplication

of the relation W conservative ¼DU )

5%6 4% 5%6 7%

157 Am. J. Phys., Vol. 80, No. 2, February 2012 Lindsey, Heron, and Shaffer 157

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failure to tailor their answer to the system under considera-tion is reflected in Table II, which shows that more than half of the students gave the same answer for the change in totalenergy of both the block and the block-spring systems in theexplicit block-spring problem. The failure to distinguishbetween systems is closely related to other difficulties whichwe identified.

 2. Tendency to associate potential energy with a single

(point-like) object rather than with a collection of objects

It is common for students to ascribe potential energy tothe system consisting of an object by itself. They oftenreferred explicitly to the potential energy of the block, evenon versions of the problems that do not specifically ask aboutpotential energy. For instance, in the explicit block-springproblem, a student who stated that the energy of the blockremains constant gave the explanation:

[When] “the spring is stretched and the block is in place, allof the block’s energy is in the form of potential energy. As theblock is released, the energy translates from potential energyto kinetic energy but the total amount remains the same.”

The same student also claimed (correctly) that the energy

of the block-spring system remains the same and explainedthat “This is the same situation as in [the block system]. Dueto the law of conservation of energy, the energy translatesamong potential, elastic, and kinetic, but the total remainsthe same.”

As mentioned, one version of the explicit block-springproblem asked students about the change in potential energyof the block. As shown in Table III, 28% indicated correctlythat the potential energy of the block system does notchange, and 53% indicated that it decreased (a correctresponse for the block-spring system). On average, 21% of the students explicitly stated that potential energy is con-verted into kinetic energy for the system consisting of theblock alone. Similar statements were seen among responses

to the block-spring equations problem, on which 78% of thestudents indicated that the change in the elastic potentialenergy of the block was nonzero.

These results are not surprising, because many instructorsand textbooks informally refer to the potential energy of sin-gle objects. However, many students gave no indication thatthey recognize that potential energy should be associatedwith a system consisting of multiple interacting objectsrather than with one object by itself. As shown in Table  III,49% of the students who were asked explicitly about thepotential energy of the block-spring system recognized thatthe potential energy decreases and 26% stated it does notchange. As we will show, confusion over the proper systemwith which to associate potential energy can lead to signifi-

cant difficulties with energy in general.

 3. Assumption that the energy of any system is constant 

In the explicit block-spring problem, 46% of the studentsclaimed that the total energy of both systems remains con-stant. For either system, 18% of the students gave as their rea-son that “energy is conserved” or “conservation of energy”with no further explanation. Others justified this reasoningwith an appeal to the changes in kinetic and potential energy,as in the previous student quote.

Responses to the block-spring equations problem seemto reflect the same assumption. One of the most common

relations given by students was  D K ¼DU , which expressesthe idea that the energy of the system does not change. (Inthis case it is incorrect because external work is done on theblock-spring system.) We have previously  shown that stu-dents frequently over-generalize this relation.1

Although we have reported similar tendencies before,1

these findings are significant because in the explicit block-spring problem, the students were asked to consider the samephysical situation twice, using two different choices of the

system of interest, whereas in Ref.  1 they had been asked toconsider only one. In the present case, the problem remindedthem that more than one choice is possible. Moreover, theymight have expected that different choices of system wouldyield different answers.

 4. Tendency to double-count work and energy terms

The tendency to include a single interaction twice was evi-dent in the block-spring equations problem. As shown inTable   V, the most common relation given by students wasW BHþW BS¼D K þDU . This relation represents the classicexample of double counting because the effect of the springappears  both as a work term (W BS) and as an energy term

(DU ).25

Some students were very explicit about their reasonfor writing this equation: “The only changes that can occur in energy must be in potential and kinetic energy. The sumof the two must be equal to the sum of the work done in theentire system.”

This difficulty can be considered a subset of the previousthree difficulties—it is highly unlikely that a student who isdouble-counting work and energy terms is not also associat-ing a potential energy with a specific object, rather than witha system, and students may believe that both sides of thisequation are separately equal to zero. We include this diffi-culty in a separate category because it seems likely to haveparticular implications for problem solving with work andenergy.

C. Representative research problems: Gravitationalcontexts

For each of the problems, we asked an analogous probleminvolving gravitational interactions. In this context, we alsoasked an additional problem requiring students to make qual-itative comparisons between the work done on a system byvarious external agents.

A gravitational analog of the explicit block-spring prob-lem is the “explicit pendulum-Earth” problem shown inFig.   3. Students were asked to consider an interval thatbegins the instant the pendulum bob is released from rest atpoint  A, and ends at the instant the bob passes point  B, the

lowest point on its trajectory. They were asked about the network done on, and the change in total energy of the systemconsisting of the bob by itself (on which positive work isdone by the Earth, leading to an increase in kinetic energy),and the system consisting of the bob and the Earth (on whichno external work is done; the potential energy of the systembecomes the kinetic energy of the bob). In some versions of the question, students were also asked about the change inkinetic energy and the change in potential energy of eachsystem.

The administration of this problem was identical to theexplicit block-spring problem. (In several sections, it wasgiven to the same students on the second page of the same

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pretest.) The results are summarized in Tables  VI   and   VII.Six percent of the students responded correctly for the totalenergy of both systems and an average of 62% gave thesame responses for the total energy of both systems. We dis-cuss some differences in the responses to this problem andthe analogous block-spring problem in Sec. III D.

The analog for the block-spring equations problem, the“block-Earth equations” problem, is given in Fig.  4. At  t i, ablock is at rest near the surface of the Earth. A hand then liftsthe block so it moves upward with nonzero speed at  t f . Thequestions and correct responses are analogous to those for the block-spring equations problem (see Sec. III A).

On the first part of the block-Earth equations problem inwhich students were asked about the sign of the work doneby the hand, the work done by the Earth, the change in ki-netic energy, and the change in potential energy, the resultswere similar to those reported in Table   IV   for the block-spring equations problem, with one exception. The exceptionwas on the question about the change in potential energy of the block. In the gravitational context, 79% of the studentsindicated that this quantity is positive, compared with only49% in the elastic context. Only 7% of the students on theblock-Earth equations problem indicated that the potentialenergy of the block does not change, compared with 17%for the block-spring equations problem. In both problems, a

similar percentage of students stated that all four quantitiesare nonzero ( 68%).

The results for the second part of the problem, in whichstudents were asked to write equations relating the quantitiesthey believed to be nonzero, are summarized in column II of Table V.22 Only 17% of the students answered correctly bywriting one or more correct equations. Another 8% wrote atleast one correct equation, but included one or more incor-rect equations.

The “block-Earth dialogue” problem was designed toassess the degree to which double-counting influences stu-dents’ qualitative reasoning. No analogous question wasasked in the elastic context. The setup is similar to that of theblock-Earth equations problem shown in Fig.  4 except that inthis case the block begins and ends at rest. Students wereasked to consider the following discussion between two fic-

tional students and to state whether they agreed with student1, student 2, both, or neither. They were also asked toexplain their reasoning.

Student 1: “I exert an upward force on the block, whichmoves upward, so I do positive work on the block. The Earth

Fig. 3. The “explicit pendulum-Earth” problem. A ball is released from rest

at point   A  at time   t 1   and passes point   B   at   t  2.   Students were asked about

work done on and changes in energy in a system consisting of just the ball,

and a system consisting of the ball and the Earth.

Table VI. Results from the explicit pendulum-Earth problem (see Fig. 3) in

which students were asked about the change in total energy for two possible

choices of system over a specified interval. The correct response is indicated

in bold. Values given are the mean score across four classes. Uncertainties

reflect the 95% confidence interval based on variances among the four 

samples.

Results

 N ¼621

Total

energy

increases

Total

energy

decreases

Total

energy stays

the same

Cannot be

grouped

as a system

Ball   14%6 5%   7%6 6% 74%6 6% 3%6 2%

Ball and Earth 9%6 8% 5%6 5%   67%6 8%   16%6 9%

Correct for both

systems

6%6 4%

Same answer for 

both systems

62%6 11%

Table VII. Results from the explicit pendulum-Earth problem in which stu-

dents were asked about the change in potential energy during a specified

time interval for two choices of system. The results are from one class that

had been given both questions. Uncertainties are not given because the ques-

tion was administered in only one section of the course. The correct

responses are indicated in bold.

Results

 N ¼174

Potentialenergy

increases

Potentialenergy

decreases

Potentialenergy

stays the same

Cannot begrouped

as a system

Ball 7% 83%   7%   1%

Ball and Earth 11%   58%   14% 17%

Correct for both

systems

3%

Same answer for 

both systems

56%

Fig. 4. The “block-Earth equations” problem. At time t i  the block is at rest

at height  h i; the block is lifted by a student such that it passes height  hf  at

time   t f   with speed   vf . Students were asked whether each of the following

quantities were positive, negative, or zero: The work on the block by the

hand, the work on the block by the spring, the change in potential energy of 

the block, and the change in kinetic energy of the block. They were also

asked to write an equation relating any nonzero terms. The setup for the

“block-Earth dialogue” problem is similar except that the block begins and

ends at rest.

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exerts a downward force, so it does negative work on theblock. The block begins and ends at rest, so its kinetic energyhasn’t changed. That means that the net work on the block iszero, so the work on the block by my hand and the work onthe block by the Earth must have the same absolute value.”

Student 2: “But the block has moved upward, so its poten-tial energy has increased. That means that the net work onthe block must be positive, so the work on the block by your hand must be greater than the work on the block by the

Earth.”Student 1 gives a correct analysis of the problem, whilethe reasoning used by student 2 incorporates the effects of the Earth twice as both external work on the system, and aninternal change in potential energy of the system.

This problem was administered in four sections of intro-ductory calculus-based physics at UW before any tutorialinstruction on work and energy. Students had completed alllecture and laboratory instruction on work and energy.

In three sections of the course ( N ¼ 420), 17% of studentscorrectly agreed with student 1 and 59% agreed with student2. In the remaining section of the course ( N ¼ 150) 56% of the students agreed with student 1 and 21% agreed with stu-dent 2. In all sections approximately 16% of students agreed

with neither student, and 6% agreed with both. The reasonfor the discrepancy between sections, which is highly un-usual in our experience, may be related to increased empha-sis on the work-energy theorem,   W net¼D K , in one of thelecture sections. Students in that section were more likely tocite the work-energy theorem as their reason for agreeingwith student 1, while students in other sections were morelikely to claim that the forces by the hand and by the Earthcancel. In this question, application of the work-energy theo-rem leads to a correct response for a system consisting of thebook alone.

D. Difficulties that arose in the gravitational context

All of the student difficulties we identified on problemsinvolving elastic interactions with springs also arose in thegravitational analogs. In most cases, they were more preva-lent. The similarities and differences are briefly described inthis section, followed by a discussion of an additional diffi-culty that arose only in a gravitational context.

1. Difficulties that arose in both the gravitational and elastic contexts

Many students appeared not to realize that an energy anal-ysis depends on the choice of system and gave identicalresponses for each system described within a particular con-text. The tendency to associate potential energy with a par-

ticular point-like object rather than with a system, and theassumption that the energy of any system remains constantwere more pronounced in the problems involving gravita-tional interactions than in those involving elastic interac-tions. In the explicit pendulum-Earth problem, for instance,88% of the students said the potential energy of the systemconsisting of the ball alone changes, compared with 66% for the elastic case. About 37% of the students mentioned a con-version of potential energy to kinetic energy for the systemconsisting of the ball alone, compared with 21% in theexplicit block-spring problem. We speculate that in an elasticinteraction, students are able to identify the spring as a physi-cal object with which to associate a potential energy and are

thus less likely to associate potential energy with the block. Ina gravitational interaction, the ball is the physical object theycan see; they do not perceive a different “object” that canstore potential energy. The relative frequency with which stu-dents associate potential energy with a particular object mayalso be related to the common expressions for potentialenergy. Gravitational potential energy is commonly given asU g¼mgh,   in which the height  h   of the ball appears to be aproperty of the ball rather than of the system. The equivalent

expression for elastic potential energy,  U e¼ 1

=2  kx

2

, dependson a property of the spring.The prevalence of double counting was similar in the elas-

tic and gravitational contexts. In the two analogous quantita-tive problems, an average of 20%–25% of the students wroteequations that suggest this error (see Ref.   26). The block-Earth dialogue problem also probed this tendency. On thisproblem, student 2 makes the classic double-counting error.The students’ readiness to agree with student 2, even whenpresented with the correct reasoning expressed by student 1,provides further evidence that many students are unaware of the implications of choosing a particular system and makeerrors that lead to incorrect calculations.

 2. Additional difficulty that arose primarily in gravitational contexts: Failure to recognize that any group of objects can be treated as a system

In responding to the explicit pendulum-Earth problem,16% of the students indicated that the ball and Earth cannotbe grouped as a system. The most common explanation wasthat “The ball and the earth are not in contact.” Students alsofrequently indicated that there must be external forces on asystem:

“If the ball and the Earth were grouped as a system, therewould be no external forces, since gravity is the only actingforce here. It doesn’t seem to make sense that they could begrouped as a system.”

Students also referred explicitly to the relative motion of the objects within a grouping: “All parts of the system arenot moving in a like fashion,” and to differences   in scale:“The ball is negligible compared with the Earth.” 27

Fewer than 5% of the students claimed that a particular setof objects cannot be treated as a system for any other systemin the problems that we describe in this paper. Students whoclaim that particular groupings are not allowed typically givereasoning opposite to that given by students who state thatthe ball and Earth cannot be grouped together: they indicatethat the string must be grouped together with the ball,because it is in contact with the ball, or because it exerts anexternal force on the ball.

Although we mainly observed the failure to recognize that

any group of objects can be treated as a system in the prob-lems involving gravitational interactions, it may also arise inelastic contexts. On versions of the explicit block-springproblem, which are not given here for sake of brevity, weasked students about the changes in energy of systems thatinclude the wall and/or the Earth in addition to the block andthe spring. In these cases, many students indicated that theblock, spring, and wall cannot be grouped into a single sys-tem because the block and wall are not in contact. We con-sistently saw a marked increase in the percentage of studentsindicating that a set of objects could not be grouped together as a system if the Earth was one of the objects. This findingis especially significant because in most courses, after 

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gravitational potential energy is introduced, the Earth is partof the system.

IV. RESPONSES OF PHYSICS TEACHING

ASSISTANTS

The block-Earth equations problem was administered to22 graduate and undergraduate teaching assistants at UWduring a weekly TA preparation seminar. Participants in this

seminar work through the tutorial they will teach to studentsafter taking the accompanying pretest. The pretest adminis-tered to TAs is designed to mirror the pretest administered tothe students. It is given on paper and the questions are posedin an open-ended format rather than as multiple-choice. Atthe time they responded to the questions, the TAs had notworked through any tutorials on work or energy.

On the pretest, 77% of the TAs said that the work done onthe block by the hand is positive, the work done on the blockby the Earth is negative, the change in kinetic energy of theblock is positive, and the change in potential energy of theblock is positive. This combination of responses suggeststhat the TAs did not respond to these questions with one sys-tem in mind. About 59% of the TAs (compared with an aver-

age of 24% of introductory students) wrote that the work onthe block by the hand plus the work on the block by the Earthis equal to the sum of the changes in the kinetic and potentialenergies, in other words.  W BHþW BE¼D K þDU . Althoughmany TAs did not explain their reasoning, at least one fol-lowed this relation by the sentence “net work equals changein total energy.” Although this statement is true, the TAapparently did not recognize the need to apply this relationto a particular system.

These results reveal that even advanced students make thedouble counting error. Although many of these TAs mightperform a correct calculation if asked to solve one of theseproblems numerically, they do not necessarily understandthis material well enough to teach it. The sample is small,

but the results suggest that advanced study in physics doesnot necessarily help students to recognize the need to iden-tify a system and use it consistently.

V. THE EFFECT OF TARGETED INSTRUCTION

The data we have presented are from several sections of the UW introductory calculus-based physics sequence taughtby different lecturers. The lecturers were all experiencedinstructors, but they did not focus on the idea of systems. Wealso collected data from a section at the same institutiontaught by one of the authors of this paper. This instructor emphasized the importance of the choice of system and thefact that only extended systems undergo changes in potential

energy. Several clicker questions addressed this issue.28

When the problems were administered after this targetedinstruction, students performed somewhat better on certainparts of the problems described here. Notably, they weremore likely to indicate that the total energy of an object byitself can change (51% compared with 33% correct in theexplicit block-spring problem and 38% compared with 14%in the explicit pendulum-Earth problem). They were alsoless likely to refer to the potential energy of the block or of the ball. On other parts of the problems, their performancewas almost identical to that of other sections. One issue inthis case may be the lack of consistency between differentelements of the physics course at UW. Although the lecture

and tutorial instruction in this particular section emphasizedthe fact that potential energy must be associated with a sys-tem of multiple objects, the course homework and the treat-ment of energy in the laboratory did not.

We also identified some improvement in student perform-ance in a course in which the instructor did not emphasizethe idea of systems. The anomalous result on the block-Earthdialogue problem described in Sec.   III C  suggests that stu-dent responses may be sensitive to other aspects of lecture

presentation. On this question, it appears that an emphasis onthe work-energy theorem led more students to apply this the-orem for a block being lifted near the surface of the Earth.Nevertheless, the students who responded correctly to theblock-Earth dialogue problem using the work-energy theo-rem to support their reasoning were no more successful thantheir peers in responding to other energy questions on thesame pretest. These results indicate that although lectureinstruction specifically targeting these ideas can improve stu-dent performance on some questions, many difficulties canpersist.

VI. DISCUSSION AND IMPLICATIONS FOR

INSTRUCTION

The results we have described are consistent with our pre-vious work1 and provide new insights into student under-standing of work and energy. In Ref. 1, we demonstrated thatmany students are unable to determine the change in energyof a system correctly even in cases for which they were ableto determine the net work done on the system by its sur-roundings. In particular, many students associated work witheither a change in kinetic energy or potential energy, rather than with the total energy of a particular system. Many stu-dents also claimed that changes in kinetic and potentialenergy cancel. We speculate that these common difficultiesmight be due to an underlying difficulty involving studentunderstanding of systems.

Students’ tendency to assume that the energy of any sys-tem is constant deserves further consideration. In both theexplicit block-spring and the explicit pendulum-Earth prob-lems, there exists a system with constant mechanical energy.The existence of such a system might be a reason why stu-dents tended to indicate that the energy of any system inthese physical situations stays the same. Our previousresearch,1 however, suggests that many students also indicatethat the energy of a system is constant in physical situationsfor which there is no choice of system that has constant me-chanical energy. Careful instruction on systems might helpstudents to recognize that the energy of an individual systemcan change.

Our research has particular implications for instruction on

potential energy. It is clear that many students do not recog-nize that the Earth is implicitly included in many systemsthat undergo a change in gravitational potential energy. Thisresult highlights the importance of consistency in assigning apotential energy only to a system of multiple objects, never to a single object. Failure to assign a potential energy only tosystems of multiple objects might lead some students tomake incorrect conclusions about the change in the totalenergy of a particular system.

An example may help to illustrate this last point. In manypopular textbooks,29 a change in potential energy is definedby  DU ¼W c, where W c is the work done by a conservativeforce. In Sec.   III A, we describe how this relation can be

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used to relate the energy analysis of two different systemswithin the same context (for example, to relate the system of the block alone to the system that includes both the blockand the spring). In the block-spring and block-Earth equa-tions problems, as well as in Ref.  1, we found that studentstend to over-generalize this relation, and associate the network done on a system with the system’s change in potentialenergy rather than its total change in energy. Careful instruc-tion on systems, including the way in which  DU ¼W c  can

be used to convert from one choice of system to another when analyzing the same physical situation, might help stu-dents to recognize the role of this particular relation, whichin turn might promote a better understanding that the network done on a system by its surroundings causes the totalenergy of the system to change.

Many instructional tools have been developed that mighthelp students to recognize the implications of the choice of system. For instance, energy bar charts might be used to helpstudents to visualize the difference in energy analysis   for different systems within the same physical situation.10 InRef. 1, we described the development of a pair of tutorials2

that include exercises designed to help students to recognizehow an energy analysis depends on the choice of system.

These exercises proved insufficient to help students to over-come many common difficulties, particularly in a physicscourse in which other elements of the course do not makeexplicit the importance of choice of system. The   Si x Ideasthat Shaped Physics30 and   Matter and Interactions16 curric-ula are notable for being consistent and explicit in describingthe implications of a particular choice of system. Ding15

found that students who had studied from Ref. 16  were oftenable to identify the correct system with which to associate apotential energy, and many were also able to relate the network done on a system to the change in energy of that sys-tem. We expect these students might also perform better onour problems.

VII. CONCLUSIONS

Many physics educators are aware that the choice of a sys-tem of interest is not always made explicit to students, andthat the informal use of expressions associating potentialenergy with an object rather than with a system is common.It might be thought that in many cases there is little harmdone because the correct solutions to problems can often beobtained despite a lack of precision in discussing systems.However, our results reveal that inconsistencies in consider-ing systems can have serious implications.

We have illustrated some of the difficulties that mightpose barriers to the development of a functional understand-ing of energy conservation. Results from the questions

described in this paper indicate that many students do notrecognize that any combination of objects can be grouped to-gether as a system. Students often place artificial constraintson the set of objects that can be grouped together, for exam-ple, requiring that all objects have the same motion or be incontact. They also often have difficulty describing thechange in total energy of a system, regardless of whether itconsists of a single object or multiple objects. Many alsoidentify changes in potential energy for systems that have nointernal structure, which can lead to a tendency to doublecount interactions. The difficulties we have described occur both in gravitational contexts and in contexts involvingsprings, although some difficulties are more evident in the

former. Differences between student responses in situationsinvolving gravitational potential energy and elastic potentialenergy support the common practice of treating these topicsseparately in instruction, because student reasoning aboutone type of potential energy might not be consistent withtheir reasoning about another type of potential energy.

Results from the TAs reveal that the difficulties we havedescribed are not restricted to physics novices and emphasizethe difficulty of the concepts of work and energy. They also

suggest that advanced physics courses do not necessarily de-velop or strengthen the key idea that one must be careful tochoose and consistently use one particular system in analyz-ing a physical situation. To help students to gain a functionalunderstanding of energy conservation, we must help them torecognize the importance of choice of system.

ACKNOWLEGMENTS

The authors wish to thank the members of the PhysicsEducation Group at the University of Washington, both pastand present, who contributed to this research. Thanks areespecially due to Lillian C. McDermott and MacKenzie

R. Stetzer for their many contributions to this project. BethLindsey would also like to thank the Department of Physicsat Georgetown University for providing her with supportduring portions of this project. The authors gratefullyacknowledge the support of the National Science Foundationthrough Grant Nos. DUE-0096511 and 0618185.

a)Electronic mail: bal23@psu.edub)Electronic mail: pheron@phys.washington.educ)Electronic mail: shaffer@phys.washington.edu1

B. A. Lindsey, P. R. L. Heron, and P. S. Shaffer, “Student ability to apply

the concepts of work and energy to extended systems,”   Am. J. Phys.

77(11), 999–1009 (2009).2The curricular materials consisted of a sequence of two tutorials on work

and energy, “Work and changes in kinetic energy” and “Conservation of 

energy.” They will be included in the 2nd edition of  Tutorials in Introduc-tory Physics (see Ref. 3 for the 1st edition.)

3L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the

University of Washington,   Tutorials in Introductory Physics   (Prentice

Hall, Upper Saddle River, NJ, 2002).4Details about the tutorials and the environment in which they are used can

be found in P. S. Shaffer and L. C. McDermott, “Research as a guide for 

curriculum development: An example from introductory electricity. Part

II: Design of an instructional strategy,”  Am. J. Phys.  60(11), 1003–1013

(1992).5More details about the TA seminar can be found in K. Wosilait, P. R. L.

Heron, P. S. Shaffer, and L. C. McDermott, “Development and assessment

of a research-based tutorial on light and shadow,”  Am. J. Phys.   66(10),

906–913 (1998).6A. B. Arons, “Development of energy concepts in introductory physics

courses,” Am. J. Phys. 67(12), 1063–1067 (1999).7

R. W. Chabay and B. A. Sherwood, “Modern mechanics,”  Am. J. Phys.72(4), 439–445 (2004).

8A series of articles in  The Physics Teacher  provides a summary of many

of these ideas. The first three articles in the series are particularly relevant.

These are J. W. Jewett, “Energy and the confused student I: Work,”  Phys.

Teach.  46, 38–43 (2008); J. W. Jewett, “Energy and the confused student

II: Systems,”  ibid.   46, 81–86 (2008); and J. W. Jewett, “Energy and the

confused student III: Language,” ibid. 46, 149–153 (2008).9Articles 2 and 3 in Ref. 8 provide a particularly accessible summary of 

some of these issues.10A. Van Heuvelen and X. Zou, “Multiple representations of work-energy

processes,”  Am. J. Phys. 69(2), 184–194 (2001).11M. E. Loverude, C. H. Kautz, and P. R. L. Heron, “Student understanding

of the first law of thermodynamics: Relating work to the adiabatic com-

pression of an ideal gas,”  Am. J. Phys. 70(2), 137–148 (2002).

162 Am. J. Phys., Vol. 80, No. 2, February 2012 Lindsey, Heron, and Shaffer 162

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12R. A. Lawson and L. C. McDermott, “Student understanding of the work-

energy and impulse-momentum theorems,”  Am. J. Phys.  55(9), 811–817

(1987).13C. Singh and D. Rosengrant, “Multiple-choice test of energy and momen-

tum concepts,” Am. J. Phys. 71(6), 607–617 (2003).14A more complete listing of articles relating to work and energy in a ther-

modynamic sense, particularly to student understanding of heat and tem-

perature, can be found in L. C. McDermott and E. F. Redish, “Resource

Letter: PER-1: Physics education research,” Am. J. Phys.  67(9), 755–767

(1999).15L. Ding, “Designing an energy assessment to evaluate student understand-

ing of energy topics,” Ph.D. dissertation, Department of Physics,North Carolina State University, 2007.

16R. W. Chabay and B. A. Sherwood,   Matter and Interactions, 3rd ed.

(John Wiley & Sons, Hoboken, NJ, 2010).17

R. Driver and L. Warrington, “Students’ use of the principle of energy

conservation in problem situations,” Phys. Educ. 20(4), 171–176 (1985).18In addition to the articles we have described, there exists a vast literature

on the learning and teaching of energy at the precollege level. Some repre-

sentative articles include N. Papadouris, C. P. Constantinou, and T.

Kyratsi, “Students’ use of the energy model to account for changes in

physical systems,”   J. Res. Sci. Teach.   45(4), 444–469 (2008) and H.

Goldring and J. Osborne, “Students’ difficulties with energy and related

concepts,”  Phys. Educ. 29(1), 26–32 (1994).19

W. M. Christensen, D. E. Meltzer, and C. A. Ogilvie, “Student ideas

regarding entropy and the second law of thermodynamics in an introduc-

tory physics course,” Am. J. Phys. 77(10), 907–917 (2009).20

Students were given credit for completing the pretest, regardless of thecorrectness of their answers. They had 15 min to complete the entire pre-

test and were free to consult textbooks and the web during that time. Based

on the reasoning provided in student responses, we believe that access to

these resources did not affect student responses significantly.21On some of the questions we have described, we have seen variations

larger than is typical. To give a sense of these variations, in the tables we

report values as the mean across several classes, with error bars that repre-

sent the 95% confidence interval based on the score variances.22For the purposes of this analysis, we did not distinguish between student

use of positive or negative quantities. In other words, the relations

W BHþW BS¼D K   and   W BHW BS¼D K   were treated identically regard-

less of whether students had identified the quantities as positive or 

negative.23Students were also given the option of stating that it was not possible to

determine the sign of each of these quantities. These data are not included

in the table because in every case, fewer than 5% of the students chose this

option.

24Although students had been asked only to write one equation, many wrotemore than one. The tables indicate the percentage of students writing each

equation, not the percentage of equations written. Thus the percentages in

the tables sum to more than 100%.25

About 20% of the students wrote this equation. About 15% of all students

had indicated elsewhere that each of the terms in the equation is nonzero.26We observed a larger spread in percentages of students giving the double-

counting response on the block-Earth equations problem than is typical in

our research. (The percentage ranged from 18% to 32% across three sam-

ples.) This result suggests that further research on this topic is needed.27This last response is similar to the results reported in Ref. 19 in which stu-

dents experienced similar scale-related difficulties when asked about the

change in entropy of an object and its (much larger) surroundings.28

Clicker questions are multiple-choice questions to which students respond

in real time using hand-held personal response systems, or clickers.

Clicker questions are used for formative assessment of student understand-

ing as well as to promote interactive engagement in the classroom. For more information on this technique, see E. Mazur,   Peer Instruction: A

User’s Manual (Prentice Hall, Upper Saddle River, NJ, 2007).29See, for example, D. Halliday, R. Resnick, and J. Walker,   Fundamentals

of Physics, 9th ed. (John Wiley & Sons, Hoboken, NJ, 2011).30T. A. Moore,  Six Ideas That Shaped Physics, Unit C: Conservation Laws

Constrain Motion, 2nd ed. (McGraw-Hill, New York, 2003).

163 Am. J. Phys., Vol. 80, No. 2, February 2012 Lindsey, Heron, and Shaffer 163