Redish, Scherr, and Tuminaro 1 7/5/2005

Reverse engineering the solutionof a “simple” physics problem:Why learning physics is harder than it looksEdward F. Redish, Rachel E. Scherr, and Jonathan TuminaroUniversity of Maryland, College Park, MD

Problem solving is the heart and soul of mostcollege physics and many high school physicscourses. The “big idea” is that physics tells you moreabout a physical situation than you thought you knew— and you can quantify it if you use fundamentalphysical principles expressed in mathematical form.Often, the results of your problem solving can leadyou to understand and rethink your intuitions aboutthe physical world in new and more productive ways.As a result, physics is a great place (some of uswould claim the best place) to learn how to usemathematics effectively in science.

As physics teachers, we often stress theimportance of problem solving in learning physics.Unfortunately, many of our students appear to findproblem solving very difficult. Sometimes theygenerate ridiculous answers and seem satisfied withthem. Sometimes they can do the calculations but notinterpret the implications of the results. Sometimes,despite apparent success in problem solving, theyseem to have a poor understanding of the physics thatwent into the problems.1 We give them explicitinstructions on how to solve problems (“draw apicture,” “find the right equation,” …) but it doesn’tseem to help.

We might respond that they need to take moremath prerequisite classes, but in the algebra-basedphysics class at the University of Maryland, almostall of the students have taken calculus and earned anA or a B. Many of them have been successful inclasses such as organic chemistry, cellular biology,and genetics. Why do they have so much troublewith the math in an introductory physics class?

As part of a research project to study learning inalgebra-based physics,2 the Physics EducationResearch Group at the University of Marylandvideotaped students working together on physicsproblems. Analyzing these tapes gives us newinsights into the problems they have in using math inthe context of physics. One problem is that they haveinappropriate expectations as to how to solve

problems in physics (some of it learned, perhaps, in mathclasses). This is discussed elsewhere.3 A secondproblem seems to lie with the instructors. As instructors,we may have misconceptions about how people thinkand learn, and this has important implications about howwe interpret what our students are doing.

In this paper, we want to consider one example ofstudents working on a physics problem that showed us ina dramatic fashion that we had failed to understand thework the students needed to do in order to solve anapparently “simple” problem in electrostatics. Ourcritical misunderstanding was failing to realize the levelof complexity that we had built into our own “obvious”knowledge about physics.

“Packing” Knowledge Until You Don’t See ItsParts: Compilation

Modern cognitive psychology and neurosciencehave documented that much of our everyday functionalknowledge is dramatically more complex than we give itcredit for. One component of this is automaticity. Oncewe have learned to do something, like tie our shoes orride a bicycle, it becomes easy and we can do it withoutthinking. But we usually understand and remember thelearning that goes into such tasks and we typically havepatience in teaching them to our children.

But we have other knowledge that has componentparts that are invisible to us. Once our knowledgereaches that stage it is hard to see why someone mightnot find the result obvious. Even an apparently simplething like identifying an object is clearly much morecomplicated than it appears.4 When we pick up a cup ofcoffee, its visual image enters the back of our brainthrough the retina. Tactile data about the feel and heft ofthe cup enters in the midbrain, the aroma in theforebrain, and the episodic memories that give us theknowledge of what to do with the coffee, what it willtaste like, and what its effect will be on us are stored yetelsewhere. Yet the overall result is our sensation of theobject as a single integrated and irreducible “thing.”

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Some cognitive “illusions” dramaticallydemonstrate how much unnoticed processing thebrain is doing for us. A nice example is given by EdAdelson and is shown in figure 1. The squares of thecheckerboard marked A and B are, in fact, exactly thesame color. (If you don’t believe this, make a copyof the page with the figure, cut out the squares, andplace them next to each other, or check out Adelson’swebpage on the topic.5) Your brain knows enough torealize that if two objects appear to be the same colorbut one is in shadow, then the one in shadow must be“really” lighter — and that’s how you see it. Thisparticular example appears to be “wired up” verytightly and at a very young age. You can’t see it anyother way.

Fig. 1: An example of automatic processing that youcannot unpack. (E. Adelson, with permission)

A second example is more obviously learned. Ifyou look at figure 2, you will find it impossible tolook at the words and see any of them (say “cat”) as aseries of lines and shapes. You probably not onlysaw the meaning immediately, you had some visualimage associated with the phrase. You have learnedto interpret the shapes as letters, to see combinationsof letters as words, and to associate the words withparticular meanings — objects and actions. Althoughyou can’t undo this easily (looking at it upside downdoes it for some folks), you know that there was atime when all you could see where lines and shapes.

Fig. 2: An example of learned recognition you cannoteasily unpack.

This same sort of process occurs as we learnthroughout our lives. When professional physicistslook at a graph, it is almost impossible for them notto see the y-intercept, the slope at each point, the

maxima and minima of the curve. For many students inintroductory physics, however, this process is not quickand automatic but takes explicit recall and reasoning.

We refer to the process of binding knowledge tightlyso that its parts are inaccessible to the user ascompilation.6 The metaphor here is computer code.Once a program written in a high level computerlanguage has been debugged and is stable, it isconvenient to convert it into machine language so itdoesn’t need to be translated each time it runs. This“executable” is fast, but if you are only given a machine-language executable, it is immensely difficult to back-interpret it to understand what it is actually doing. Tounderstand what is going on in such a program, acomputer programmer who wants to re-create it mayhave to reverse engineer it.7

Once we learn how to do something, it can bedifficult to empathize with someone who does not knowhow to do that thing. In particular, this lack of empathymay lead physics teachers to forget what it is like toactually learn physics—and, therefore, to not understandhow their students are unable to solve “simple” physicsproblems. Some physics teachers may think, “If it takes astudent an hour to solve a problem to which I can justwrite down the answer, then that student does not knowenough physics—and she is wasting her time spendingthat long on such a ‘simple’ problem.” In this paper wewant to demonstrate that this is not the case. To do this,we reverse engineer what solving a “simple” physicsproblem really entails. We analyze a group of students’solution to this simple problem and show that, while thestudents take much longer than the typical teacher tosolve this problem, their solution involves manyactivities that can be seen as part of compiling theirphysics knowledge and that are appropriate for studentsat their stage of knowledge.

The Setting for this Study.The setting for this study is an algebra-based physics

course at the University of Maryland that was non-traditional in many aspects. The most important non-traditional aspect of this course was its shift in emphasisaway from a focus on answers to the idea ofunderstanding principles, process, and concepts. Thisemphasis was expressed through explicit discussion withthe students and through reforms in lecture, recitation,laboratory and homework.8

Students were given about five homework problemsper week and were told that we expected them to spendabout an hour on each problem, even when workingtogether in groups. Many resisted this at first, some notbelieving it (and trying to do the homework in the fifteenminutes before it was due), some not knowing what workto do for an hour. We gave no exercises and ourproblems often contained qualitative questions,

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estimations, and essays. We required that they giveexplanations in words for each problem in order toreceive full credit on the grading. Each week, oneproblem (not identified for the students beforehand)was graded carefully with written feedback (grade of0-5). The others received only an indication ofwhether it was right or not (grade of 0-2). Detailedsolutions were posted on the class website. We setup a workroom, the Course Center, where studentscould gather work on their physics homework. Thecenter was staffed approximately twenty hours perweek by a teaching assistant (TA) or the courseinstructor. The staff members assisted the studentswith their homework, but did not explicitly solve anyof the students’ homework problems. Often, insteadof answering students” questions, TAs directed themto other students working on the problem in thecenter and they were encouraged to formworkgroups.9 By the end of the first semester, mostof the students were working in groups and spendingsignificant amounts of time on their homework (4-6hours/week).

An Example: the Three-Charge ProblemWe examine a videotaped episode in which a

group of students attempt to solve an electrostaticsproblem that we refer to as the Three-ChargeProblem (figure 3). This episode occurred in thesecond week of the second semester of the two-semester sequence and shows three female students(pseudonyms, Alisa, Bonnie, and Darlene) workingtogether. All of the students had taken thetransformed course in the first semester and werefamiliar with the idea that a single problem mighttake a long time to solve and that qualitative andquantitative considerations might both be needed.

Fig. 3: The “three charge” problem.

How instructors solve this problem:

We have asked this problem of numerous physicsinstructors. Some answer immediately. Others have tothink for a few seconds. Occasionally an instructor willgive a quick answer and get it wrong at first — but itrarely takes anyone more than a minute to work theirway through to a correct answer. A typical instructor’ssolution might be: “Well, charge 3 is twice as far awayfrom charge 1 as it is from charge 2. So if the forcesbalance, charge one has to be –4Q, opposite sign tobalance Q and four times as big because Coulomb’s lawsays the force falls like the square of the distance.”

What instructors want students to do

Although most instructors can do this kind of acalculation in their heads, we, as instructors, expected thestudents to go through a bit more math. The problemstates there is “no net electrostatic force” acting oncharge q3. This implies that the sum of all the forcesacting on q3 is equal to zero, which is written formally insymbols as

€

r F q2→q3

+r F q1→q3

= 0 . (1)

(We will see below that this form of this equation isdeceptively simple and hides a great deal of conceptualinformation.) Using Coulomb’s Law to write the forcesin equation (1) yields:

€

kQq3d2

ˆ i +kq1q32d( )2

ˆ i = 0, (2)

where we have set q2 = Q and written (for simplicity aswe did in the class) k=1/4πε0. Finally, we bring thesecond term to the right side of the equation and cancelsimilar terms, which results in the answer:

€

q1 = −4Q.

What the students did

An interesting aspect about the students’ problemsolving approach is that it takes so long compared to atypical instructor’s solution. The students work fornearly 60 minutes before arriving at a solution— almosttwo orders of magnitude longer than the typical teacher!Is this a problem? In our analysis of our students’approach in solving this and in other problems, twofactors seemed critical in understanding what thestudents were doing.

First, we observe that students tend to solveproblems by working in locally coherent activities inwhich they use only a limited set of the knowledge thatthey could in principle bring to bear on the problem. Werefer to each of these activities as an epistemic orknowledge-building game. Each game has allowedmoves, a starting point, a goal or ending point, and aform or visible result. Most important, while students are

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playing one game, they ignore moves that theyconsider as not pertinent, thereby excluding muchrelevant knowledge. We see an example of this inour episode. (These are discussed in detail in ref. 3.)

Second, we note that much of the knowledge thatstudents are using are not integrated; results thatwould be considered trivially identical by aninstructor are treated as distinct and unrelated. Theyhave not yet compiled these distinct knowledgeelements the way experts have.

An example of a knowledge-building gamemany of us have seen when interacting with studentsis Recursive Plug-and-Chug. The student’s goal is tocalculate a numerical answer. The opening move isto identify the target variable to be calculated. Thenext move is to find an equation containing thatvariable. Then, check if the rest of the variables inthe equation are known. If so, calculate the resultingquantity. If not, find another target variable in theequation and repeat the process. Note the absence ofdeveloping a story about the problem, evaluating therelevance of the equation, or making sense of theanswer. The absence of these sense-making movescan produce strikingly inappropriate and even bizarreresults. The output form of this game is a string ofequations — one that might look identical to thatproduced by a student playing a more productivegame, such as Making Meaning with Mathematics.

E-Game DescriptionPhysicalMechanism

using common sensereasoning; “telling thestory” of the problem

PictorialAnalysis

using the e-form of afree-body diagram to“nail down” the relevantforces

MappingMathematics toMeaning

interpreting current stateof mathematicalknowledge in terms ofthe physical elements ofthe problem

MappingMeaning toMathematics

connecting the physicalresult to using formalknowledge

Table 1: Knowledge games played by the students

Knowledge-building games the students used

In the example we are considering, we canidentify five different knowledge-building games thatthis group of students played to solve the three-charge problem. (See table 1.)

Physical mechanism: Understanding the physicalsituation. The students start this problem by attemptingto understand the physical situation articulated in theproblem statement. Their reasoning is based on intuitiveknowledge about and experience with physicalphenomena rather than on formal physics principles.Darlene: I'm thinking that the charge q1 must have

it's...negative Q.Alisa: We thought [q1] would be twice as much,

because it can't repel q2, because they're fixed.But, it's repelling in such a way that it'skeeping q3 there.

Bonnie: Yeah. It has to—Darlene: Wait, say that.Alisa: Like— q2 is— q2 is pushing this way, or

attracting—whichever. There's a certain forcebetween two Q, or q2 that's attracting.

Darlene: q3.Alisa: But at the same time you have q1 repelling

q3.

Darlene initiates this exchange with a possiblesolution to this problem: the charge on q1 is “negativeQ.” Although this is wrong, it has a good piece ofphysics: the charge on q1 must have the opposite sign tothe charge on q2 if the forces they exert on q3 are tobalance. Rather than simply accept or reject thissuggestion, the students discuss the physical mechanismthat acts to keep q3 from moving: q2 is attracting and q1 isrepelling q3, or vice versa. If the students were onlyattempting to find a solution to this problem, then adiscussion about the physical mechanism seemsunnecessary—they would only need to assess thecorrectness of Darlene’s assertion. That the studentsdiscuss a possible physical mechanism involved in thephysical situation is an indication that the students areattempting to develop a conceptual understanding of thisproblem. Much of the research on quantitative problemsolving in physics discusses the importance ofconceptual understanding.10 Although the instructor’ssolution outlined above does not explicitly contain adescription of the physical mechanism underlying thephysical situation, it is clearly implicit — compiled intothe way the instructor thinks about and approaches theproblem. They not only “know the right formula,” theyknow what the formula means and how to use it.

The next exchange indicates, however, that thestudents’ intuitive ideas about the physical situation arenot always consistent with the physics principles that anexpert would use. The students are still struggling withreconciling the principle of superposition with theireveryday ideas.Darlene: How is [q1] repelling when it's got this

charge in the middle?

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Alisa: Because it's still acting. Like if it's biggerthan q2 it can still, because they're fixed.This isn't going to move to its equilibriumpoint. So, it could be being pushed thisway.

Darlene: Oh, I see what you're saying.Alisa: Or, pulled. You know, it could be being

pulled more, but it's not moving.Darlene: Un-huh.

The arrangement of the charges cues Darlene tothink that the presence of q2 somehow hinders orblocks the effect of q1 on q3, which does not agreewith the superposition principle. Exploring thepossibility that q2 blocks the effect of q1 on q3 is not astep in the instructor’s solution outlined above — it’ssomething the instructor knows and takes for granted.The exploration is not a dead-end but a step inhelping them to develop the intuitive sense ofsuperposition. Alisa’s argument is particularinteresting here. She uses an incorrect qualitativeargument (overcoming — “if it’s bigger…it could bebeing pulled more”) rather than the correctquantitative one (superposition) that she later showsshe knows. Within the context of the PhysicalMechanism game, formal arguments are not “legal”moves.

What we see here (and in many other examples)is that when the students are playing a particularknowledge-building game, they tend not to use otherknowledge that they have. That knowledge seems tobelong to a different game and not be easily accessedhere.

Pictorial analysis: Drawing a picture. The studentsmake progress on this problem by attempting todevelop a conceptual understanding of the physicalsituation in terms of their intuitive ideas; however,this is not stable within the group. After theirapparent initial agreement in developing a conceptualunderstanding of the physical situation, and inparticular on determining that charges q1 and q2 hadto have opposite sign, Darlene decides she is notconvinced.Darlene: I think they all have the same charge.Bonnie: You think they all have the same

charge? Then they don't repel each other.Darlene: Huh?Bonnie: Then they would all repel each other.Darlene: That's what I think is happening.Bonnie: Yeah, but q3 is fixed. If it was being

repelled—Alisa: No, it's not. q3 is free to move.Bonnie: I mean, q3 is not fixed. That's what I

meant.

Darlene: Right.Bonnie: So, like...Darlene: So, the force of q2 is pushing away with is

only equal to d.Bonnie: Yeah, but then...Darlene: These two aren't moving.Bonnie: Wouldn't this push it somewhat?Alisa: Just because they're not moving doesn't

mean they're not exerting forces.Darlene: I know.Alisa: What do you think?

The TA (Tuminaro) notices the students’ failing tocommunicate clearly and lock down their apparent gains,and suggests that they draw a picture of the physicalsituation. The students do not use an algorithmicpictorial analysis technique (e.g. free-body diagrams),but rely on their intuitive ideas about the situation togenerate a picture. The picture helps the studentsorganize their thoughts and agree on the relative sign oneach of the charges.Alisa: So, maybe this is pushing...Darlene: That's [q2] repelling and q1's attracting?Bonnie: Yeah, it's just that whatever q2 is, q1 has to

be the opposite. Right?Alisa: Not necessarily.Darlene: Yeah.Bonnie: OK, like what if they were both positive?Alisa: Well, I guess you're right, they do have to be

different, because if they were both positive...Bonnie: Then, they'd both push the same way.Alisa: And, this were positive it would go zooming

that way.Darlene: They would both push.Alisa: And, if this were negative it would go there.Bonnie: It would go zooming that way.Alisa: And, if they were negative...Darlene: It would still—they'd all go that way.Alisa: It would be the same thing.

The picture enhances the students’ ability to reasonabout this problem, which enables them to agree on aclear intuitive understanding of the physical situation:“whatever [the sign of] q2, q1 has to be the opposite.”

Mapping mathematics to meaning: Identifying therelevant physics. At this point, the students have notmade use of Coulomb’s Law—they have relied solely ontheir intuitive ideas. Yet, their intuitive reasoning helpsthem understand the physical situation—and, ultimately,realize that Coulomb’s Law is essential for this problem.Bonnie: Yeah. Negative two Q, since it's twice as

far away.Alisa: And, this is negative Q.Bonnie: Negative two Q.Darlene: Negative two Q.

Redish, Scherr, and Tuminaro 6 7/5/2005

Alisa: Are we going to go with that?Bonnie: I think it makes sense.Darlene: That makes...Alisa: Well, I don't know, because when you’re

covering a distance you’re using it in thedenominator as the square.

Bonnie: Oh! Is that how it works?Alisa: And [...inaudible...] makes a difference.Bonnie: Yeah, you're right.Tuminaro: So, how do you know that?All: From the Coulomb's Law.Bonnie: So, it should actually be negative four

q? Or what? Since it has…Alisa: Cause we were getting into problems in

the beginning of the problem with [thenon-equilibrium three-charge problem –see figure 4] because I thought that like ifyou move this a little bit to the right thedecrease for this would make up for theincrease for this. But, then we decided itdidn't. So, that's how I know that I don'tthink it would just increase it by a factor oftwo.

Fig. 4: Another “three charge” problem

The students relied on their intuitive ideas togenerate a conceptual understanding of the physicalsituation—two mutually exclusive influences actingon q3 that exactly cancel each other. Yet, Alisa,recalling her experience of working on an earlierproblem (figure 4) realizes that their intuitive ideasare not enough—they need Coulomb’s Law.

Mapping meaning to mathematics: Translatingconceptual understanding into mathematicalformalism After some false starts and nearly sixtyminutes, the students finally solve this problem,integrating their conceptual understanding, whichthey developed in terms of their intuitive common-sense ideas, with formal application of Coulomb’sLaw.

Tuminaro: What did you do there?Alisa: What did I do there?Tuminaro: Yeah, can I ask?Alisa: All right, so because this isn't moving, the

two forces that are acting on it are equal: thepush and the pull.

Alisa reiterates the group’s conceptualunderstanding of the physical situation: two mutuallyexclusive influences exactly canceling yielding no result.

Next, Alisa writes down the form of the two forcesin terms of Coulomb’s Law:Alisa: So, the F—I don't know if this is the right F

symbol—but, the F q2 on q3 is equal to this[see eq. 3]. And, then the F q1 on q3 is equalto this [see eq. 4], because the distance istwice as much, so it would be four d squaredinstead of d squared.

23

32 d

kQqF qq =→

(3)

23

31 4d

kxQqF qq =→

(4)Alisa: And, then I used x Q like or you can even

do—yeah—x Q for the charge on q1, becausewe know in some way it's going to be relatedto Q like the big Q we just got to find thefactor that relates to that…Then, I set themequal to each other…

Alisa uses Coulomb’s Law to write the form of thetwo forces, but she does not formally invoke the otherphysics principle outlined in the ideal solution: Newton’s2nd Law. Rather, Alisa relies on her conceptualunderstanding of the physical situation to write that theforces must be equal—not on a formal application ofNewton’s 2nd Law. This feature of her solution is notinsignificant; it lends additional evidence that Alisa ismaking sense of this problem, rather than simplyfollowing some problem-solving algorithm.

After setting up the equation, Alisa is only left withan algebra problem, which she has little trouble solving:Alisa: … and I crossed out like the q2 and the k and

the d squared and that gave me Q equals x Qover four. And, then x Q equals four Q, so xwould have to be equal to four. That's howyou know it's four Q.

The other students then evaluate the plausibility andvalidity of Alisa’s recited solution—yet anotherindication that these students are making sense of thisproblem.Bonnie: Well, shouldn't it be—well equal and

opposite, but...Alisa: Yeah, you could stick the negative.Bonnie: Yeah.

Redish, Scherr, and Tuminaro 7 7/5/2005

Darlene: I didn't use Coulomb's equation, Ijust—but it was similar to that.

Bonnie: That's a good way of proving it.Darlene: Uh-huh.Bonnie: Good explanation.Alisa: Can I have my A now?

Alisa’s final question is meant in jest (“Can Ihave my A now?”), but shows that she realizes thatshe has understood and solved this problem in anexpert-type manner.

Doing good workThe students’ solution, while it took much longer

than the average teacher would take to generate one,has many components of an expert-like solution.However, they show that much of the knowledgethey call on is not yet compiled in an expert fashion.The students have to recall and construct thebackground knowledge needed to make sense of theproblem a step at a time: a long but likely necessarystep in creating their own knowledge compilations.

Even though it takes them longer than the typicalexpert, there are two clues that show that the studentsare working appropriately — in an expert-likemanner: they rely heavily on their conceptualunderstanding and they themselves choose their ownpath.

First, the students solve this problem by usingtheir intuitive understanding of the physicalsituation—two mutually exclusive influences exactlycanceling each other—and formal application ofCoulomb’s Law. At no point during the entireproblem-solving episode do they use or make explicitreference to Newton’s 2nd Law, even though it is therelevant physics principle for why q3 remains inequilibrium. This is not a negative. It shows that thestudents are using their own understanding of thephysical situation to generate a solution (an expert-like characteristic), rather than doggedly applying aformal physical principle (a novice-likecharacteristic).

Second, they generate their own problem-solvingpath. They do not defer to the TA and let him directwhat they should do next. The language that thestudents use is an indication that they do not defer tothe TA—even though the students follow the TA’ssuggestion to draw a picture. The students usephrases like “we thought” or “I decided,” and neveruse phrases like “the TA said” or “the book says,”which offers linguistic evidence that the students arein control of the solution path—and not the TA.

Finally, from the details of the full transcript,11 itis clear that the students are recalling and workingthrough many of the items of basic knowledge that

are new to them (only learned last semester) and thatthey are still reconciling with their intuitive knowledge.A list of some of the physics knowledge the students callon explicitly is given in table 2.

E-Game Physics Knowledge NeededPhysicalMechanism

• Like charges repel, unlike attract• Attractions and repulsions are forces• Forces can add and cancel (one does

not “win”; one is not “blocked”)• “Equilibrium” corresponds to

balanced, opposing forces (not asingle strong “holding” force)

• Electric force both increases withcharge and decreases with distancefrom charge

• Objects respond to the forces they feel(not those they exert)

• Charges may be of indeterminate signand still exert balancing forces on thetest charge

• “Fixed” objects don’t give visibleindication of forces acting on them;“free” ones do

PictorialAnalysis

• Only forces on the test charge requireanalysis

• Each other charge exerts one force ontest charge

• Each force may be represented by avector

• “Equilibrium” corresponds toopposing vectors

• Vertical and horizontal dimensions areseparable

• One dimension is sufficient foranalysis

MappingMathematicsto Meaning

• Electric force both increases withcharge and decreases with distancefrom charge

• Electric force decreases with thesquare of the distance

MappingMeaning toMathematics

• Charges of indeterminate sign areappropriately represented by symbolsof indeterminate sign

• Coefficients may relate similarquantities

• Balanced forces correspond toalgebraically equal Coulomb’s-Lawexpressions

Table 2: Some of the knowledge required by the studentin each game

Recognition vs. formal manipulationIn physics, especially at the college level, we tend to

focus our attention on formal manipulation. We oftendon’t realize how much of an expert’s success is based

Redish, Scherr, and Tuminaro 8 7/5/2005

on a more fundamental cognitive ability: recognition.There is good evidence from both cognitive andneuroscience that we possess a variety of mentalabilities.12 The ability to handle language and formalreasoning is analogous to serial processing incomputer technology — sequential and reasonablyslow. The ability to recognize faces and places isanalogous to parallel processing — all happening atonce and very quickly. When we identify a coffeecup or recognize a friend we don’t go through aformal checklist of properties, we just recognizethem. Many physicists have had the experience ofgoing through a tedious calculation, makingmathematical manipulation after manipulation, andthen reaching a particular point and saying: “Oh!Now I get it.” Either the rest of the calculation nowbecomes trivial or the previous steps now are obviousrather than formal.

The sort of thing we mean is easily demonstratedin more daily activities. The anagram puzzle“Jumble” is familiar to many and appears in hundredsof newspapers. (See figure 5.) The task is torearrange each of four strings of letters into words.The circled letters in each of the four words arecombined and rearranged to provide the answer to thephrase clued by the drawing. Before going on to thenext paragraph, look at figure 5 and see if any of thewords just spring to mind.

Fig. 5: A puzzle illustrating two kinds of thinking(©Jumble, permission applied for)

Many people immediately recognize either thefirst or the third word.13 With these, the bonus phrase(“false alarm”) is reasonably obvious from thecontext. After getting the fourth word (prettystraightforward since there are only three distinctletters), the second word can be approached. Thebonus phrase tells us that the two circled letters in thesecond word must be “a” and “r”. This produces thetwo options: “a __ __ __ r” or “r __ __ __ a” with theletters “u”, “u”, and “g” to be distributed in the threeinterior spaces. There are only six possibilities:

running through them has one recognizable word:“augur.”

Note the two mental processes we’ve described. Inthe first case (see note 13) most people simply look at thescrambled letters and know the answer — directrecognition. In the second case (“augur”), most peoplehave to go through a formal algorithm — checking outall of a limited set of possibilities. But in the end, eventhe second case relies on our recognizing the word“augur.” It’s a fairly uncommon word and many peopledon’t know it. If the recognition process is missing, theformal manipulations won’t help.

Our handling of a physics problem has much incommon with the way we think about the Jumble puzzle.We do formal manipulation until we reach a point wherewe recognize the result as making sense. But when wehave a lot of compiled knowledge, our intuitiverecognition skills become much stronger. When wefocus on teaching our students formal manipulationskills, we are tacitly assuming that the recognition skillswill grow naturally. But if we are unaware of howcompiled our knowledge is, we may not appreciate thework students need to do to compile their newknowledge and build their intuitive recognition skills.

In this paper, we reverse engineered a simplyphysics problem, comparing the way instructors whoalready have lots of compiled knowledge solve it to theway a group of novice students who are still working oncompiling their knowledge solve it. An immediatelyobvious difference between the two solutions is that thestudents’ solution took much longer to generate—nearlytwo orders of magnitude longer. Our analysis shows twothings. First, even simple physics problems are difficultfor novices. There are many conceptual and technicalsubtleties to physics problems that experts tend to forgetabout because they are so familiar with these subtletiesthat they don’t notice them. Second, what we may at firstjudge to be poor student problem-solving behavior mayactually be very good behavior. After careful analysis ofthe students’ solution in the three-charge problem, wesee that the students’ solution shares many aspects withthe expert-like solution. In addition, the students aredoing work towards the consolidation and reconciliationof new knowledge, which is just what they need to bedoing at their stage of learning. To do a better jobhelping our students, we need to better understand bothwhat they know and the hidden components of our ownknowledge. Only then, can we effectively “reverse-engineer” what we know to figure out what our studentshave to go through to build expert problem-solvingskills.

AcknowledgmentsThis work was supported in part by NSF grant REC-0087519. We are very grateful to the University of

Redish, Scherr, and Tuminaro 9 7/5/2005

Maryland Physics Education Research Group formany useful discussions and suggestions.

References 1 E. Mazur, Peer Instruction (Prentice Hall, 1996).2 Learning How to Learn Science: Physics forBioscience Majors NSF grant REC-008 7519.3 J. Tuminaro and E. Redish, “Students understandingand use of mathematics in physics: A cognitivemodel,” submitted for publication; J. Tuminaro, Acognitive framework for analyzing and describingintroductory students’ use and understanding ofmathematics in physics, PhD dissertation, Universityof Maryland (2004).4 O. Sacks, The Man Who Mistook His Wife for a Hat(Touchstone, 1998).5 http://web.mit.edu/persci/people/adelson/checkershadow_proof.html6 This is often referred to in the cognitive andneuroscience literatures as binding. They often donot, however, distinguish between permanent bindingthat is irreducible for the user and dynamic bindingthat is temporary and situational. We introducecompilation to focus on permanent bindings that looktrivial and obvious to the user.7 “Reverse engineering is the process of takingsomething (a device, an electrical component, asoftware program, etc.) apart and analyzing itsworkings in detail, usually with the intention toconstruct a new device or program that does the samething without actually copying anything from theoriginal.”http://en.wikipedia.org/wiki/Reverse_engineering8 See ref. 3 for more details.9 Students in the Course Center typicallyspontaneously formed workgroups with otherstudents who happened to be there. When a groupworked well, they would sometimes arrange a regularmeeting time for subsequent collaborations.10 J. Larkin, “Understanding and teaching problemsolving in physics,” Eur. J. of Sci. Ed. 1: 2, 191-203(1979); J. Larkin, J. McDermott, D. Simon, H.Simon, “Expert and novice performance in solvingphysics problems,” Science 208, 1335-1342 (1980);M. Nathan, W. Kintsch, and E. Young, “A theory ofalgebra-word-problem comprehension and itsimplications for the design of learningenvironments,” Cog. and Instr., 9:4, 329-389 (1992).11 The full transcript is available online athttp://www.physics.umd.edu/perg/dissertations/Tuminaro/3Qtranscript.pdf.

12 G. Fauconnier and M. Turner, The Way We Think:Conceptual Blending and the Mind’s HiddenComplexities (Basic Books, 2003), Chapter 1.13 “Metal” and “fossil”. Each of the anagrams for thesewords contains a significant block of the letters of afamiliar word in the correct ordering. The third(“appall”) may be a bit harder.