We describe a modeling approach to help students learn expert problem solving. Models are used to present and hierarchically organize the syllabus content and apply it to problem solving, but students do not develop and validate their own Models through guided discovery. Instead, students classify problems under the appropriate instructor‐generated Model by selecting a system to consider and describing the interactions that are relevant to that system. We believe that this explicit System, Interactions and Model (S.I.M.) problem modeling strategy represents a key simplification and clarification of the widely disseminated modeling approach originated by Hestenes and collaborators. Our narrower focus allows modeling physics to be integrated into (as opposed to replacing) a typical introductory college mechanics course, while preserving the emphasis on understanding systems and interactions that is the essence of modeling. We have employed the approach in a three‐week review course for MIT freshmen who received a D in the fall mechanics course with very encouraging results.National Science Foundation (U.S.

Analia Barrantes

Andrew Pawl

Chandralekha Singh

Charles Henderson

David E. Pritchard

Mel Sabella

Pawl, Andrew

Barrantes, Analia

Pritchard, David E.

DSpace@MIT

Modeling Applied to Problem SolvingAndrew Pawl, Analia Barrantes and David E. PritchardPhysics Department, MIT, Cambridge, MA 02139Abstract. We describe a modeling approach to help students learn expert problem solving. Models are used to presentand hierarchically organize the syllabus content and apply it to problem solving, but students do not develop and validatetheir own Models through guided discovery. Instead, students classify problems under the appropriate instructor-generatedModel by selecting a system to consider and describing the interactions that are relevant to that system. We believe that thisexplicit System, Interactions and Model (S.I.M.) problem modeling strategy represents a key simplification and clarificationof the widely disseminated modeling approach originated by Hestenes and collaborators. Our narrower focus allows modelingphysics to be integrated into (as opposed to replacing) a typical introductory college mechanics course, while preserving theemphasis on understanding systems and interactions that is the essence of modeling. We have employed the approach in athree-week review course for MIT freshmen who received a D in the fall mechanics course with very encouraging results.Keywords: physics education research, course reform, introductory physics, modeling instructionPACS: 01.40.Fk, 01.40.gbINTRODUCTIONStandard mechanics teaching materials emphasizedeclarative and procedural knowledge, but not thestrategic knowledge which is essential to the applicationof these facts and procedures in problem solving [1].From this perspective the common student complaint “Iunderstand the material, but I can’t start the problems” isa cry for help in gaining strategic knowledge. This paperpresents a pedagogy for teaching Newtonian mechanicsthat is designed specifically to help students developthe strategic knowledge to apply course material toproblem solving in an expert-like manner. We achievethis objective by helping students to organize the coursematerial into a hierarchy, and to use this organization asa key component of the problem solving strategy.Our approach is a simplification of the modelingphysics approach [2, 3, 4] that is focused on helpingstudents learn expert problem solving. We call our ped-agogy Modeling Applied to Problem Solving (MAPS)to emphasize that its primary aim is enabling studentsto organize and activate their knowledge to solve prob-lems. Restricting modeling to two goals – the organiza-tion of core course material and its application to prob-lem solving – has several advantages. The narrowed fo-cus implies a commensurately reduced set of new con-cepts and terminology, allowing MAPS to be more eas-ily learned. The hierarchical organization helps studentscope with the large amount of material covered in intro-ductory mechanics. The explicit and universal problemmodeling strategy places the emphasis on interactions asthe key to physics models and their application in prob-lem solving. Furthermore, MAPS has deliberately beenstructured in a manner that allows its incorporation intoexisting mechanics courses without significant alterationof the syllabus or specialized training for the instructor.We have implemented the ideas presented here in aone-month review course (hereafter the ReView course)involving 30 MIT students who received a grade of D inthe fall 2008 semester introductory mechanics course.OVERVIEW OF MAPS PEDAGOGYIn MAPS, the scientifically established declarative andprocedural knowledge of Newtonian Mechanics is orga-nized into Models. These Models are process models [5],and are referred to as “basic models” in [3, 4]. We orga-nize these Models into hierarchies under four physicalcategories (motion, momentum, energy, and angular mo-mentum) to facilitate application to problem solving.Strategic application of the Models is encouragedthrough the use of a problem modeling strategy. Thisstrategy helps students represent a given physical situ-ation in a manner that suggests the applicability of a par-ticular basic Model. As in the problem-specific models of[3], it involves the description of the system under con-sideration and the interactions present. In our curriculum,we reserve the term “Model” for the instructor-generatedbasic models that are arranged in our hierarchy. The pro-cess of solution is called “modeling”.TABLE 1. Contrasting the basic Models of the hier-archy with the problem-specific modeling process.Models modelingTheory. Application.General principle. Problem-specific.Organize syllabus. Apply content.Exact, ideal. Approximating, idealizing.Existing knowledge. Student-constructed.SPECIFICATION OF MODELSMAPS pedagogy uses general physical Models topresent the basic physical principles of Newtonian me-chanics. These models function as cognitive “chunks”[6] that link the many disparate facts and formulaerelated to a particular concept. They are specifiedaccording to the template:Compatible Systems: Systems which can be describedby the Model. Some Models can only describe a sin-gle point particle, others can describe the behaviorof a collection of rigid bodies.Relevant Interactions: The class of interaction whichserves as the agent of change in the Model. Mod-els require the separation of interactions into rel-evant and irrelevant types based upon classifica-tions like conservative vs. non-conservative, in-ternal vs. external, and torque-producing vs. non-torque-producing.Law of Change: The heart of each Model is the equa-tion or equations which describe the evolution ofthe state of the system if the Model holds [3].A sample Model specification for Momentum and Ex-ternal Force is shown below.MOMENTUM AND EXTERNAL FORCECompatible Systems: The system may containany number of point particles. Rigid bodiesmay be treated as point particles with positionsspecified by their centers of mass.Relevant Interactions: Only external forces needto be considered. Internal forces do not changethe system’s momentum.Laws of Change:Integral Differential∑sys~p f = ∑sys~pi +∫∑ext~F dt ∑sysd~pdt = ∑ext ~FHIERARCHY OF MODELSTo help the student build an overview of the entire do-main, we categorize each Model under a basic physicalcategory. The typical calculus-track course is concernedwith four such categories:• Motion, Acceleration and Net Force• Momentum and External Force• Mechanical Energy and Non-Conservative Work• Angular Momentum and External TorqueThese categories explicitly indicate the relevant type ofinteraction for all Models that fall into the category, em-phasizing to the students that the type of interaction istheir guide to choosing a Model when solving a prob-lem, and reminding them that only certain types of inter-actions function as agents of change for a given Model.This association of a class of relevant interactions witheach physics principle is the most important vehicle forthe recognition and application of the principles whensolving problems.Within these four categories, a hierarchy of detailedModels is constructed that connect the broad physicalprinciples with more specialized recurring patterns ofnature. One possible detailed hierarchy for the Motion,Acceleration and Net Force category is shown below.Models that are specializations of another Model areindented beneath the more general Model.Motion, Acceleration and Net Force (3-D)General 2-D MotionRotational MotionUniform Circular MotionGeneral 1-D Motion1-D Motion with Constant AccelerationSimple Harmonic MotionPROBLEM MODELING STRATEGYIn the MAPS pedagogy, all problems are approachedusing the S.I.M. problem modeling strategy: System,Interactions, and Model(s). The strategy is central tothe main objective of MAPS: helping students becomebetter problem solvers. Its goals can be summarized asstrategic, universal, and simple:• Give students an expert-like strategic approach forfinding the relevant physical model and associatedequations.• Keep the steps as universal as possible for bettertransfer to other domains.• Make it easy to learn, implement, and incorporateinto the pedagogy of an existing course.Experts possess the insight to sort problems into cate-gories based upon their interactions. They recognize thatif no external forces are applied to a system then momen-tum will be conserved, if non-conservative forces are ab-sent or do no work then mechanical energy is conserved,and so on. Novice students also look for conserved quan-tities, but do so by using superficial features and equa-tion hunting rather than attempting to qualitatively un-derstand the nature of the interactions [6, 7, 8]. For thisreason, when novices are presented with a problem thatthey do not recognize based on pattern matching to previ-ous problems or equations, they are helpless to proceed.The S.I.M. Problem Modeling StrategyDescription of the System: Students should listall the objects that they consider to be part ofthe system and classify them as point particles,rigid bodies, or massless objects.Description of Interactions: Every problem re-quires a careful description of the interactions.Students should classify every relevant forceas internal/external and conservative/non-conservative, since this is the key to choosingthe model which will most efficiently solve theproblem.Choice of Model: A selection from the hierarchy.The consistency of the three parts of the strat-egy shows whether the student has made a rea-soned choice without requiring a detailed ex-planation.The S.I.M. strategy promotes an expert approach tomechanics problems because it fosters a strong under-standing of the interplay between the description of thesystem, the nature of the interactions, and the choice ofthe appropriate basic physical Model. The most impor-tant interdependencies are:• The description of the system dictates which forceswill be internal versus external versus excluded.• The type of forces acting (conservative, internal,torque-producing, etc.) influences which physicalvariables are good descriptors, and hence whichModel(s) will apply.• The choice of Model dictates which interactionsare relevant to, and what system descriptions arecompatible with, the Law of Change governing thesystem’s evolution.With these interdependencies in mind, it is impossibleto view the specification of the system, the description ofthe interactions and the choice of the Model as a linearprocess (e.g. first system, then interactions then Model).These interdependencies are at the heart of strategicproblem solving. By making an insightful choice whenspecifying the system or choosing the model, the rele-vant interactions can sometimes be greatly simplified.The S.I.M. strategy is simplified relative to problemsolving in a typical modeling physics curriculum withoutsacrificing the strategic core of the approach. Becauseit is universally applicable to any mechanics problem,it has the advantage of setting up a clear procedure forthe students to follow when they are initially stuck. TheS.I.M. strategy is succinct enough to be habitually usedby an instructor on all example problems worked duringclass, and to constitute a minimal additional workload inwritten solutions for the students.OUTCOMESImproved Student AttitudesIn January 2009 we taught a three-week ReView for30 students who received a D in the fall 2008 MITmechanics course using the MAPS pedagogy. Studentstaking the ReView and achieving a sufficiently improvedgrade on an equivalent final exam given at its conclusionhad their grade raised to C, giving credit for the course.In contrast with most introductory physics courses (in-cluding the MIT course) which leave students with morenovice attitudes about physics and problem solving thanthey had when they entered [9, 10], our ReView gener-ated favorable shifts in every category of the ColoradoLearning Attitudes about Science Survey (C-LASS) [10](Fig. 1). These shifts were significant at the 2σ level inall five categories related to problem solving and concep-tual understanding.000102030  OverallPersonal InterestReal-world ConnectionProb. Solv. (General)Prob. Solv. Sophist.Conceptual Underst.Applied Conceptual UnderstandingShift in % Favorable ResponsesFIGURE 1. C-LASS shifts by category for the ReView(black circles), MIT 8.01 (squares) [105 voluntary respondents]and a typical university course (gray band) [10].The student responses to the course evaluationsstrongly suggest that this attitude shift was largelybrought about by the problem solving approach em-ployed. Student free-responses to the questions “Whatone or two things did you like about this course?” and“Do you feel you are ready to move on to [the secondsemester course]?” yielded comments including:“The S.I.M. rubric was extremely helpful. It should betaught in [mechanics] the first time around.”“I liked...looking at the basic solving steps that arenormally assumed...”“The systems, interactions, models stuff makes youthink before jumping in.”Overall, 15 of the 33 respondents offered explicit en-dorsements of MAPS similar to those reported above.−3 −2 −1 0 1 2 30123456789Standard Deviations from Course MeanNumber of ReView Students  RetestFall CourseFinal ExamRetest FitFall CourseFitFIGURE 2. Original and retest final exam histograms for theReView. The lines represent Gaussian fits to the data.Improved Student PerformanceOne measure of performance shift for our ReViewcourse is a comparison of the students’ exam scores fromthe fall term with their performance on the final examretest administered upon completion of the ReView. Toallow for the most valid comparison possible, we useda final exam given to a previous freshman class as ourretest. This gave us a set of 571 scores from which togenerate a curve for the test. Since a comparison ofthe absolute score on the retest with the absolute scoreon the 2008 test is not valid, we made the assumptionthat the abilities of the 2008 freshmen were identical indistribution to the abilities of the freshmen from the yearthe retest was used as the final exam, allowing us to makea valid comparison of z-scores. The results, shown in Fig.2, indicate a substantially improved performance for theReView students as a group. The mean z-score of the 29students on the fall 2008 exam was −1.11± 0.08 whiletheir mean z-score on the retest was−0.13±0.15 (withinerror of the overall average for the full freshman class).Another measure is student performance on the Me-chanics Baseline Test [11]. The three-week ReViewachieved a normalized gain of 0.20± 0.07, more thanhalf the normalized gain achieved by the same studentsduring the one-semester fall course (0.38± 0.06). Thegain achieved was sufficient to bring the mean gain ofthe ReView students for the fall course and ReView com-bined (0.52±0.04) into correspondence with the overallmean gain for the fall course (0.48±0.01).CONCLUSIONSThe MAPS pedagogy outlined in this paper proved suc-cessful in significantly improving student attitudes asmeasured by the C-LASS and performance as measuredby an MIT final exam and the Mechanics Baseline Testamong a group of 30 D students taking a three-week Re-View course offered after the regular mechanics course.These results are extremely encouraging, but it must benoted that the test group was strongly motivated to altertheir problem solving approach and already knew muchof the course content, permitting them to concentrate onorganizing the material rather than learning it. In fall2009 we will employ the approach in one section of theregular freshman course at MIT to investigate whethersimilar results can be achieved in a group learning theMAPS approach concurrently with the course content.ACKNOWLEDGMENTSWe thank Daniel Guetta and Evangelos Sfakianakis fortheir vital contributions to the success of the ReViewcourse and their enthusiastic participation in the devel-opment of MAPS, and the NSF for financial support.REFERENCES1. W. J. Gerace, “Problem Solving and ConceptualUnderstanding,” in Proceedings of the 2001 PhysicsEducation Research Conference, edited by S. Franklin, J.Marx and K. Cummings, PERC Publishing, New York,2001, p. 33.2. D. Hestenes, “Toward a modeling theory of physicsinstruction,” Am. J. Phys. 55, 440-454 (1987).3. M. Wells, D. Hestenes and G. Swackhamer, “A ModelingMethod for high school physics instruction,” Am. J. Phys.63, 606-619 (1995).4. D. Hestenes, “Modeling methodology for physics teachers,”in The changing role of physics departments in modernuniversities: International Conference on UndergraduatePhysics, edited by E. F. Redish and J. S. Rigden, CollegePark, Maryland, 1996, pp. 935-958.5. E. Etkina, A. Warren, M. Gentile, “The Role of Models inPhysics Instruction,” Phys. Teach. 44, 34-39 (2006).6. M. T. H. Chi, P. Feltovich and R. Glaser, “Categorizationand representation of physics problems by experts andnovices,” Cogn. Sci. 5, 121 (1981).7. J. H. Larkin and F. Reif, “Prescribing Effective HumanProblem-Solving Processes: Problem Description inPhysics”, Cog. Instr. 1, 177 (1984).8. P. T. Hardiman, R. Dufresne and J. P. Mestre, “The relationbetween problem categorization and problem solvingamong experts and novices,” Mem. Cognit. 17, 627-638(1989).9. E. Redish, J. M. Saul and R. N. Steinberg, “Studentexpectations in introductory physics,” Am. J. Phys. 66, 212(1998).10. W. K. Adams, K. K. Perkins, N. S. Podolefsky, M. Dubson,N. D. Finkelstein and C. E. Wieman, “New instrumentfor measuring student beliefs about physics and learningphysics: The Colorado Learning Attitudes about ScienceSurvey,” Phys. Rev. ST-PER 2, 010101 (2006).11. D. Hestenes and M. Wells, “A Mechanics Baseline Test,”Phys. Teach. 30, 159-165 (1992).

Modeling applied to problem solving

Abstract. We describe a modeling approach to help students learn expert problem solving. Models are used to present and hierarchically organize the syllabus content and apply it to problem solving, but students do not develop and validate their own Models through guided discovery. Instead, students classify problems under the appropriate instructor-generated Model by selecting a system to consider and describing the interactions that are relevant to that system. We believe that this explicit System, Interactions and Model (S.I.M.) problem modeling strategy represents a key simplification and clarification of the widely disseminated modeling approach originated by Hestenes and collaborators. Our narrower focus allows modeling physics to be integrated into (as opposed to replacing) a typical introductory college mechanics course, while preserving the emphasis on understanding systems and interactions that is the essence of modeling. We have employed the approach in a three-week review course for MIT freshmen who received a D in the fall mechanics course with very encouraging results

Modeling applied to problem solving

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