A Multiple Case Study Of Novice And Expert Problem Solving .

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A multiple case study of novice and expert problem solving in kinematics withimplications for physics teacher preparation.Carl J. Wenning, CoordinatorPhysics Teacher Education ProgramIllinois State UniversityNormal, IL 61790-4560wenning@phy.ilstu.eduIn recent years, physics education researchers and cognitive psychologists have turned their attention to thequestion of how individuals solve basic physics problems. The author summarizes the surprising results of amultiple case study in which three experts and three novices were observed as they solved kinematicsproblems using a "think aloud" protocol. Follow-up interviews and content analysis led the researcher toconclude that expert problem solvers do not always follow the most efficient routines, nor do they alwaysuse the most effective methods for teaching basic problem-solving skills to students. These circumstanceshave important implications for physics teacher education.The class began on time at 10:25 AM with the instructorasking the students if they had any questions about thekinematics homework problems that they were supposed tohave attempted the night before. A discussion dealing withthree homework problems (and one example problem)ensued for the next 45 minutes. While the instructor wassolving these problem, students observed intently. Themajority of the students listened while the instructor talkedand worked on the board, yet about one third of the studentsceaselessly recorded in notebooks everything that theinstructor wrote.In each case of problem solving, the treatment by theinstructor was consistent and methodical. The instructorbegan with a statement of the problem. Next, he drew apicture. Thirdly, he stated what was known or given as partof the problem. Fourth, he identified a principle by which theproblem could be solved. Fifth, he stated the relevantequation that related the knowns and unknowns. Sixth herestated the knowns and unknowns. He then solved theequation for the required unknown, inserted the knowns, andcarried out the arithmetic calculation. The instructor thenmade reference to checking the answer for reasonableness.The instructor’s approach to problem seemed clear and, yet,something seemed to be missing. During the problemsolving session there were 19 questions asked by students.The questions, interestingly enough, were more frequentlymetacognitive questions (“How do you know when to.?”and “What do you do if.?” and “How do you go about.?”)than any other variety.Beginning at 11:10 AM, the instructor moved on to a 20minute lecture about Newton's first and second laws. He didnot provide many significant real-life examples of the firstlaw, and the second law was treated entirely at a theoreticallevel. During this time, all students appeared to be diligentlytaking notes. At the outset of the lecture portion of the class,the instructor dealt momentarily with the alternativeconception that moving things need a constant force to keepthem in motion.At the end of this session, and near the end of the class,the instructor worked another example problem. He assignedJ. Phys. Tchr. Educ. Online 1(3), December 2002Page 716 exercises for homework at the end of the hour. Eight ofthe exercises were questions, six were “standard” problems,and two were “challenge” problems. The students diligentlyrecorded the list of required homework problems andpromptly left the classroom at the end of the period. Anothertypical introductory physics class had come and gone.What do we leave students with at the end of a series ofsuch introductory physics lessons? Are students better able tosolve physics problems now that they have seen a fewexamples? Do they have a metacognitive understanding ofthis simple problem-solving process that is so frequentlytendered with almost every lecture-based recitation class inwhich problem solving is addressed? Do courses that have astheir greatest emphasis the solution of textbook problemsleave the students with the perception that the scientificprocess is little more than searching for the right equation?How important are concrete examples to true studentunderstanding of physical phenomena? These are only a fewof the questions that might arise from intently watching andseriously reflecting on what happens in many introductoryphysics classes. To focus on all these questions would be toogreat a task in the limited space available for this article and,so, a more narrow view will be centered on the difficultiesassociated with teaching the general problem-solvingparadigm so frequently taught in didactic introductory-levelphysics courses -- find the knowns and unknowns, state therelationship between them, and solve for the unknown.Problem Solving in PhysicsIn recent years physics education researchers andcognitive psychologists have turned their attention to thequestion of how individuals solve physics problems. Recentresearch has focused on two areas as they pertain to physicsproblem solving: (a) the overall plan of attack used to solveproblems, and (b) the identification and use of heuristics inproblem solving. The researchers generally approach a studyof the first focus area by comparing and contrasting theperformance of novices (generally defined to be students inintroductory physics classes) with that of experts (generallydefined to be physics teachers). Studies in the area of 2002 Illinois State University Physics Dept.

problem solving frequently utilize qualitative approaches andinvolve a relatively small number of subjects. "Think-aloud"protocols are normally used in these efforts. Computermodels are generally associated with the heuristic aspect ofproblem solving and will not be dealt with in this article.A clear and concise definition of problem solving mustbe given if the problem statement is to be meaningful. Areview of secondary sources shows that there are a number ofdefinitions of the word "problem," but the definition that ismost apropos to this project is a characterization -- workassociated with those tasks found at the end of chapters ofintroductory physics text books. Typically, these tasksinvolve a statement of information and/or circumstances, andan additional variable or variables are determined on thebasis of the information provided. These tasks tend to bevery specific and the work and goal well defined. Problemsolving then is the process of attaining the goal of anyspecified problem.ContextStudies of novice and expert physics problem solvershave suggest that there are two distinct and contrastingpatterns of problem solving among experts and novices.These variations have led to the formulation of two majormodels for problem solving. According to Larkin et al.(1980), expert problem solving is typified by the KD model,the so-called knowledge-development approach. Noviceproblem solving is typified by the ME model, the so-calledmeans-end approach. In the ME model the student typicallyworks "backward" from the unknown to the giveninformation. Under this scenario the novice problem solver(NPS) essentially writes an equation and then associates eachterm in the equation with a value from the problem. If thereare additional unknowns, the problem solver moves on to thenext equation. In the KD model the expert proceeds in theopposite direction, working forward from the giveninformation. Under this second scenario, the expert problemsolver (EPS) associates each of the knowns with each term ofthe equation as the equation is set up. That is, novices movefrom equations to variables, while the experts move from thevariables to the equation.The research in the area of physics problem solvingaccelerated rapidly in the early 1980s and is now the focus ofattention in the research literature. There are a number ofquestions left unresolved, including those given by Maloney(1994), "What knowledge do novices typically use whenfaced with physics problems?" and "How is the knowledgethat a novice possesses organized in memory?" and "How doalternative conceptions affect novices' representations?"However important these questions, the basis of this researchstill depends upon the answer to the question, “How doproblem-solving approaches differ between novice andexperts?”MethodIn case studies, the researcher is the primary researchinstrument. When this is the case, validity and reliabilityconcerns can arise. The human investigator may misinterpretor hear only certain comments. Guba and Lincoln (1981), asJ. Phys. Tchr. Educ. Online 1(3), December 2002Page 8well as Merriam (1991), concede that this is a problem withcase study work. Yin (1994, p. 56) lists six attributes that aninvestigator must possess to minimize problems with validityand reliability associated with the use of the human researchinstrument. A person should be able to ask good questions -- and tointerpret the answers. A person should be a good “listener” and not be trapped byhis or her own ideologies or preconceptions. A person should be adaptive and flexible, so that newlyencountered situations can be seen as opportunities, notthreats. A person must have a firm grasp of the issues beingstudied, whether this is a theoretical or policy orientation,even if in an exploratory mode. Such a grasp focuses therelevant events and information to be sought tomanageable proportions. A person should be unbiased by preconceived notions,including those derived from theory. Thus a person shouldbe sensitive and responsive to contradictory evidence.The researcher believes that he exhibited these personalcharacteristics, though “no devices exist for assessing casestudy skills.” (Yin, 1994, p. 56)Five kinematics physics problems were written for thisproject. The five questions ranged from simple one-stepproblems with a single output variable, to more complextwo-step problems where more than one output variable wasrequested. These problems used in this study can be found inAppendix A.Three faculty members and four students were then selfselected to participate in this study. All faculty memberswere male; one of four physics students was female. Thoughthis may at first appear to be too large a sample for a casestudy, “any finding or conclusion in a case study is likely tobe much more convincing and accurate if it is based onseveral different sources of information.” (Yin, 1994, p. 92)The problem-solving skills of these individuals wereexamined through observation, interview, and contentanalysis. Such use of multiple data sources also enhancesvalidity and reliability via triangulation.All volunteer faculty members participating in this studyhad experience teaching introductory physics courses fornon-majors. All students were volunteers who were currentlyenrolled in an introductory, algebra-based physics course fornon-majors at a middle-sized Midwestern university.Students were informed that a wide range of problem-solvingabilities were needed, and that excellence in problem solvingwas not a prerequisite for participating in the study. (Thefemale student was subsequently dropped from the study dueto an apparent lack of ability to solve even rudimentaryalgebraic equations.)Three data collection strategies were used in this project.Participants first solved the five physics problems using a"think aloud" protocol. The researcher listened to theproblem solvers, recording pertinent details dealing with thesolution of the problems. He later coded these comments foranalysis. Following problem solving, the researcher collected 2002 Illinois State University Physics Dept.

the written work which would be used in content analysis,and then commenced a semi-structured interview to achievea greater understanding of the problem-solving process. Infollow-up interviews, faculty members were asked threequestions common to all study participants, and twoadditional questions reserved to expert problem solvers.Students were asked the same three common questions andthree additional student-specific questions. The questions canbe found in Appendix B.Findings from ObservationsAppendix C shows the coding plan for problem solverstatements made while working on the problems using aModel12345EPS #112345EPS #223145think aloud protocol. The coding plan consists of steps in atheoretical scheme of problem solving enunciated by Heller,Keith, and Anderson (1992), and modified and extendedslightly for this study. Each step of the problem-solvingprocess is operationally defined with descriptors. Forinstance, a problem solver can be said to be visualizing theproblem if he or she draws a sketch, identifies the knownvariables and constraints, restates the question, or identifiesthe general approach to solving the problem. While problemsolvers were working problem number one (and allsubsequent problems), the researcher recorded statements forlater coding. The results of the coding can be found in Table1.EPS #321345NPS #11234NPS #221345NPS #3134657537345Table 1. Logical approaches used by expert and novice problem solvers to solve problem one.This table shows the logical approaches used by expertand novice problem solvers. If a problem solver uses what istheoretically the most efficient scheme for solving theproblem, then his solution should consist of five sequentialsteps: 1, 2, 3, 4, and 5. If expert problems solvers (EPS’s)depart substantially and consistently from this model, itmight lead the researcher to conclude one of two things:either these particular EPS’s are inefficient, or the modelproposed by Heller et al. is simply wrong.The data tabulated in Table 1 shows that EPS’s do notgenerally follow the same paths to a solution as thetheoretical model. In all three cases, the EPS’s chosedifferent routes to solve the problem. These paths were 123,231, and 213. Novice problem solvers (NPS’s) #1 and #2took similar mixed routes, while NPS #3 departed from thegeneral problem solving model when he failed to includestep two. Among the six problem solvers, this was the onlyperson to neglect this step, leading possibly to the long,convoluted solution to the problem as indicted by the twelvesteps. Interestingly enough, five of the six problems solversmade the effort to mentally check their answers for apparentcorrectness.The overall impression gained by the researcher whileobserving the problem solvers was that the problem-solvingprocedures utilized by novice problem solvers are veryJ. Phys. Tchr. Educ. Online 1(3), December 2002Page 9unstructured and inefficient. Problems are not systematicallyapproached, knowns are rarely written down in equation2form (for instance, a 1 m/s ), starting equations are rarelywritten down, equations are not solved for unknownvariables before inserting the knowns, work is done withoutunits, solving algebraic equations appears to be a problem formost, etc. Students, in many cases, quite randomly chooseequations to solve for the unknown. They, not infrequently,expected a calculator to “solve” the problem for them. Onestudent in particular regularly multiplied and dividednumbers in a random fashion looking for solutions that“looked right.” This procedure might work on a multiplechoice test -- something that is normally used at theintroductory level -- but not in this research project wherestudents had to derive precise answers of their own. Ingeneral, the time required for EPS’s to solve problems wasone third that required by NPSs.Findings from InterviewsIt is clear from the interview process that in the area ofkinematics, students tend to follow the same generalprocedures as the experts when it comes to problem solving:search for knowns and unknowns, establishing or finding arelationship between the knowns and unknowns, and thensolve for the unknown. The general procedure for problem 2002 Illinois State University Physics Dept.

solving is shown in Figure 1. In some cases the studentswould check their answers to see if they made sense; this wasnormally the case with experts. Checking the answergenerally took the form of looking at the magnitude and signof the solved variable. The students interviewed seemed to beclear on the overall process. When they did have trouble, itwas in selecting the appropriate equation to relate the knownand unknown variables through the most direct route. In thisprocedure two faculty members were very efficient;however, one expert problem solver almost invariably startedthe problem-solving process with the same kinematicsequation, no matter what the original given quantities were.Figure 1. Problem solving flowchart. The general problem-solving procedure appears to consist of identifying the known andunknown variables, finding a mathematical relationship between the variables, and then solving for the unknown.Unfortunately, some students do not appear to have a clear understanding of the thought processes that take place in the blackbox entitled "Establish Relationship."Two students were unable to explain clearly the "blackbox "procedure for selecting the appropriate kinematicsequation to relate the variables (see Figure 1). For instance,"I look to fit all the information into a model" and "I seewhat formula gives me the information I need." The result ofthis uncertainty was clearly evident as these two studentsrandomly selected one equation after another in an effort to"plug and chug" their way through the problem set. Onestudent was clear about the procedure, "The equation I wouldselect would be that which has one unknown variable --theone you are looking for. Alternatively, using a formula withtwo unknowns where one of the unknowns can be obtainedwith the use of another formula." All problem solvers,novices and experts alike, appeared to use the means-endsapproach to solve the five physics programs provided.The physics teachers were asked to explain how theytaught kinematics problem solving in their introductorycourses. In all cases teachers indicated that they made use ofexamples almost exclusively. In one case, an instructor notedthat from time to time he would attempt to clarify the processby explaining the process in words; in another case aninstructor indicated that he would never use a metacognitiveapproach. In his words, ".I do not discuss generalstrategies. I'm not sure some students at this level canconceptualize general strategies. Strategies are drawn byexample." Another instructor noted, "I don't think that thereis any particular procedure that you can describe to thestudents for them to become more expert. In special areas Ipoint out what they have to do to recognize the unknown, thedata, and what sort of formula for them to use. Students oftenrandomly search for formulas. I warn them against this." InJ. Phys. Tchr. Educ. Online 1(3), December 2002no case was any attempt made to explain explicitly what wasgoing on in the mind of the instructor to explain the equationselection process.The students interviewed mentioned that they did makeuse of examples to learn how to do kinematics problemsolving. In all three cases the students reported reading overthe example, and sometimes working the example, in aneffort to comprehend the general procedure. They did notindicate using examples as templates for solving problemsexcept in one instance. This student reportedly resorts tousing examples like templates to find one variable in a twostep problem in which the desired variable is notimmediately obtainable directly from an equation.When queried, student expressed the opinion that theyhad learned general problem-solving strategies prior totaking the physics class mentioned in this study. One studentattributed his physics problem-solving skill to a high schoolclassmate; another to life experiences; and yet another torelated coursework in business classes. Students generallyfelt that their problem-solving skills were enhanced by takingthe physics course, and this helped them to gain a broaderperspective on the problem-solving process. There was ageneral consensus that the instructors did very little to helpstudents learn the fundamental intellectual processes ofmathematical problem solving in physics.Findings from Content AnalysisSubsequent to the follow-up interviews, the written workof problem solving was collected for content analysis. Theprocedures used by problem solvers were coded on the basisPage 10 2002 Illinois State University Physics Dept.

of equations used to find intermediate or final unknownsfollowing the work of Simon and Simon (1978). Theequations referred to are those appearing on the problemsheet shown in Appendix A. The first equation is labeled 1,the second 4, the third 5, the fourth 7, and the fifth 8. Thisnumbering sequence was chosen to remain consistent withprevious research on kinematics problem solving. Thecoding procedure is "shorthand" that indicates how problemsolvers approached problems. Forinstance, if a problem solver found the average velocity, v,using equation 5, then the approach was coded (v5). If theinstantaneous velocity, v, was found from equation 5, then#123!4EPS #1v 5 -a8 t 4a4 -x7t4a8!5v8 -t4EPS #2v 5 -a7 -t4a4 -x7t7at 4 -at 4!/t7t7 -v4the approach was coded (v5 ).Table 2 shows the results of coding the mathematicalsteps used by EPS’s and NPS’s. The designations runninghorizontally along the top numerically distinguish EPS’s andNPS’s. The numbers running vertically along the left side ofthe table indicate problem number. Each cell contains theequation-based problem solving approach. False starts havenot been included in this table, nor have unsuccessfulattempts to solve problems. If a cell in the table is blank, it isan indication that the problems solver was unable to find thecorrect solution.EPS #3v 5 -a8 t 4a 4 -x7t4a8!NPS #1v 5 -t1v8 -t4t4 *NPS #2v 5 -t1t4a8!!v8 - v 5 -t1NPS #3v 5 -t1x8 -t7a8v8 -t4* Did not solve for v.!Table 2. Mathematical approaches used by expert and novice problem solvers.From an inspection of the approaches outlined in thistable, it is clear that not all expert problem solvers determineunknowns in the same fashion or with the same efficiency(efficiency being defined as working toward the answer bytaking the most direct route --using the fewest number ofsteps and equations to solve for an unknown). Admittedly,there are several ways to solve each of these problems, withsome routes being different but equally efficient. This can beseen in the solution of problem 5 by expert problem solvers.Differences in problem-solving efficiencies were notableamong EPS’s attacking problem 4. For example, compare theprocedure of EPS #2 with those used by EPS #1 and EPS #3.EPS #2 used a solution procedure that was less efficient thanthat used by other EPS’s. EPS #2 solved for the product of aand t from equation 4, and then divided this product by t7while the other EPS’s solved equation 4 directly. Thisappears to have do with EPS #2's propensity for beginningmost problems with a statement of equation 7, and thensearching for variables to insert into the equation -- notalways the most efficient procedure.Interestingly, some NPS’s exhibited what appears to begreater insight in solving some problems than EPS’s. Forinstance, note how all NPS’s solved problem 1 in a muchmore direct fashion than any EPS, not solving foracceleration (a) in order to find t. Though the table does notshow it, NPSs took a significant number of dead-endapproaches to solving the problems.DiscussionThe findings of this research project do not lend supportto the claim that expert problem solvers tend to use a KEapproach and novice problems solvers an ME approach -- atJ. Phys. Tchr. Educ. Online 1(3), December 2002least in the area of kinematics. Both NPS’s and EPS’s usedthe same technique of searching for an equation among agroup of equations that contains the end variable. They thenworked from this end using any means necessary. One mightargue that their is no alternative to the solution of kinematicsproblems, but the contrasting solution of problem 1 by EPS’sand NPS’s would seem to indicate that the studentsinterviewed have used a more “insightful” KE approach thandid the EPS’s.It appears that the general procedure for solvingkinematics problems (find the knowns and unknowns, statethe relationship between them, and solve for the unknown)are clear to the students studied. It is also clear that thesestudents have not learned detailed problem-solvingprocedures by watching instructors solve example problems.They seem to have done so on their own – in other courses orthrough friends. What students are not consistently clearabout is how to select the appropriate kinematics equation orequations to relate and solve for the problems’ unknown.Evidently some students have been unable to figure out byobservation the relatively sophisticated black box mentalprocess the instructor goes through to select the appropriatekinematics equation.What was not self-evident to the physics instructors isthat students would appear, in some cases, not have a goodunderstanding of the equation-selecting process that goes onquickly in instructors’ minds. Though instructors argue thatstudents appear to learn from example, one of the mostimportant examples that is lacking is that which illustratesthe thinking process that the course instructor goes throughto select the appropriate equation among those available inkinematics. In one case a NPS had a clearer view of this than,Page 11 2002 Illinois State University Physics Dept.

perhaps, an EPS. This same EPS noted that he didn’t thinkthere was a general problem-solving process that studentscould comprehend. Perhaps this is so because that EPS neverestablished a clear procedure for himself as is evidenced bythe rigid, lock-step procedure of attempting to solve thekinematics problems by starting with equation 7 each time.It is clear from subsequent discussions with each of thefaculty members participating in this project that they maywell generally lack a clear understanding of students’problem-solving difficulties. They tend to see a host of astudent problem-solving difficulties such as: (a) failing to usea systematic process to solve problems, (b) failure to identifyvariables with known quantities, (c) adding dissimilar knowstogether such as velocity and acceleration, (d) trying to solveequations without writing them down, (e) using calculators tosolve the problems rather than the equation for the unknown,(f) randomly selecting equations to be solved for theunknown variable, (g) makingalgebraic errors, (h) confusing v with v-bar, (i) failing torecognize simplifying conditions (v 0 at top of flight pathfor a projectile, for instance), and that (j) novices are muchless systematic than experts in both thinking and writingdown their work. The instructors studied do not seem to beaware, however, of the difficulties students face whenattempting to figure out what is going on in the black box ofestablishing relationships between variables. Howwidespread this evident unawareness on behalf of instructorsis not known.Because the faculty members interviewed possibly havenever taken the time to analyze student problem-solvingdifficulties, and then triangulated those observations to lendcredibility to their findings, they seem not to be aware of thecentral issue of problem solving by NPS’s. Additionally, ifthe instructors studied were to more closely examine thenature of the questions that so many students ask duringclass, they might be more aware of the need for students tohave a metacognitive understanding of the problem-solvingprocess being used, and particularly those occurring in thedark recesses of the black box known as “establishrelationship.”Two questions that arose in the mind of the intervieweras he talked with students and faculty members alike were,"Why don't faculty members take the time to take ametacognitive approach to problem solving?” and “Whydon't faculty members talk about the entire problem-solvingrather than expecting students merely to learn by example?"If instructors were to clarify for themselves the most efficientapproaches for solving problems, this might enhance theirteaching and student problem solving as well. As a result,emphasis in the preparation of physics teacher candidatesshould be placed on the metacognitive processes involved inproblem solving. It also bodes well for a structured problemsolving process. A more systematic analysis of and approachto problem-solving difficulties in all areas of physicsteaching promises to pay dividends.References:Chase, W. G. & Simon, H. (1973). Perception in chess.Cognitive Psychology, 4, 55-81.Chi, M. T. H., Feltovitch, P.J. & Glaser, R. (1981).Categorization are representation of physics problems bexperts and novices. Cognitive Science, 5, 121-152.Guba, E. G., & Lincoln, Y. S. (1981) Effective Evaluation:Improving the Usefulness of Evaluation Results throughResponsive and Naturalistic Approaches. San Francisco,Jossey-Bass.Heller, P., Keith, R., and Anderson, S. (1992). Teachingproblem solving through cooperative grouping. Part 1:Group versus Individual problem solving. AmericanJournal of Physics, 60(7): 627-636.Langley, P., Simon, H, Bradshaw, G. & Zytkow, J. (1987).Scientific Discovery: Computational Explorations of theCreative Processes. Cambridge, MA: MIT Press.Larkin, J. H., McDermott, J., Simon, D. P., & Simon, H. A.(1980). Models of competence in solving physicsproblems. Cognitive Science, 4, 317-345.Lawson, A. E. (1995). Science Teaching and theDevelopment of Thinking. Belmont, CA: WadsworthPublishing Co.Maloney, D. (1994). Research on alternative conceptions inscience, In Handbook of Research on Science Teachingand Learning (Dorothy L. Gabel, Ed.) Washington, DC:National Science Teachers Association.Merriam, S. B. (1991). Case Study Research in Education:A Quantitative Approach. San Francisco: Jossey-Bass.Ostlund, K.L. (1992). Science Process Skills: AssessingHands-On Student Performances. Menlo Park, CA:Addison-Wesley Publishing Company, Inc.Rezba, R. J. Sprague, C. & Fiel, C. (2002) Learning andAssessing Science Process Skills. Dubuque, IA:Kendall

Problem solving then is the process of attaining the goal of any specified problem. Context Studies of novice and expert physics problem solvers have suggest that there are two distinct and contrasting patterns of problem solving among experts and novices. These variations have led to the formulation of two major models for problem solving.

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