A Taxonomy of Introductory Physics Problems - comPADRE

Enhancing Cognitive Developmentthrough Physics Problem Solving:A Taxonomy of Introductory Physics ProblemsRaluca Teodorescu, Cornelius Bennhold and Gerald FeldmanDepartment of Physics, The George Washington University, Washington, DC 20052Abstract. As part of an ongoing project to reform the introductory algebra-based physics courses at George WashingtonUniversity, we are developing a taxonomy of introductory physics problems (TIPP) that establishes a connectionbetween the physics problems, the type of physics knowledge they involve and the cognitive processes they develop instudents. This taxonomy will provide, besides an algorithm for classifying physics problems, an organized and relativelyeasy-to-use database of physics problems that contains the majority of already created text-based and research-basedtypes of problems. In addition, this taxonomy will reveal the kinds of physics problems that are still lacking and that arefound to be necessary to enhance students’ cognitive development. For this reason, we expect it to be a valuableteaching resource for physics instructors which will enable them to select the problems used in their curricular materialsbased on the specifics of their students’ cognition and the learning objectives they want to achieve in their class. Thisorganization scheme will also provide a framework for creating physics-related assessments with a cognitive component.Keywords: physics problem solving, taxonomy, physics problems, physics education research.PACS: 01.40.Fk, 01.40.HaINTRODUCTIONDeveloping proficiency in problem solving haslong been recognized as one of the primaryeducational objectives in introductory science courses.Current educational requirements indicate thatintroductory science courses in particular shouldemphasize skill building in quantitative and qualitativeproblem solving along with developing a knowledgebase [1, 2]. Yet research in the cognitive sciences overthe years has revealed the complex and dynamiccharacter of the problem-solving process [3-10]. Fortyyears of research in expert-novice problem-solvingbehavior have elucidated specific characteristics of thethinking processes of the two categories of learners[11]. In parallel, physics problem developers havemade efforts to create physics problems that move thenovices towards a more expert-like status [12-16].Moreover, there are currently successful models thatdescribe different aspects of the thinking process thattakes place during physics problem solving [17, 18].This project proposes to create a taxonomy ofintroductory physics problems (TIPP) that organizesphysics problems according to the manner in whichdisciplinary knowledge is processed within a learner’scognitive system. The desired outcome is expected tobe in good agreement with existing novice-expertresearch as well as modern models of students’thinking.By problem solving we mean “cognitive processingdirected at achieving a goal when no solution methodis obvious to the problem solver” [19, 20]. We adhereto the characterization of the problem-solving processproposed by Mayer and Wittrock [5]: “Problemsolving is cognitive, that is, it occurs internally withinthe problem solver’s cognitive system and can only beinferred indirectly from the problem solver’s behavior.Second, problem solving is a process, that is, itinvolves representing and manipulating the knowledgein the problem solver’s cognitive system. Third,problem solving is directed, that is, the problemsolver’s cognitive processing is guided by the problemsolver’s goals. Fourth, problem solving is personal,that is, the individual knowledge and skills of theproblem solver help determine the difficulty or easewith which obstacles to solutions can be overcome.”METHODOLOGYTIPP is substantially based on the taxonomy ofeducational objectives developed by Marzano [21].Fig. 1 shows schematically the theoretical inputs thatinfluence TIPP. According to Marzano’s model ofbehavior, the mental activity that humans perform isthe result of the interaction of three mental systems:the self-system (which decides to engage or not in acertain task), the meta-cognitive system (which setsgoals and strategies) and the cognitive system (whichprocesses the relevant knowledge). It is well known

that what students believe about physics as a scienceand what they expect from their physics coursesdetermine their attitude and motivation towards theprocess of learning physics. Ultimately, those factorsinfluence their overall achievement in a physicscourse. For the purpose of this project we will restrictourselves to the cognitive system and not include themeta-cognitive aspects in our taxonomy.PhysicsknowledgeCognitivescienceContemporaryeducationalrequirementsExisting collectionsof physics problemsTIPPModern models ofstudent thinkingResearch onnovice-expertphysics problemsolving behaviorFIGURE 1. The theoretical basis of the Taxonomy ofIntroductory Physics Problems (TIPP).While developing TIPP we address the followingquestions: Can physics problems be categorizedaccording to cognitive processes and knowledgedomains? What is the relationship between physicsproblems, knowledge domains and cognitiveprocesses? Are there cognitive processes that are notactivated by the existing physics problems? FollowingMarzano’s taxonomy [21], we have developed analgorithm to organize physics problems according to atwo-dimensional scheme. The first dimension of TIPPrefers to the knowledge domain that is involved in acertain problem while the second deals with thecognitive processes that a solver needs to perform inorder to solve it. These two dimensions arefundamentally different. The knowledge domainsencountered in physics problems are referred to:declarative knowledge or information and proceduralknowledge or mental procedures. The cognitiveprocesses required to solve physics problems are:retrieval, comprehension, and analysis and knowledgeutilization. Fig. 2 provides succinct descriptions ofthese complex cognitive processes, their componentprocesses and how they act on different types ofknowledge. All of the cognitive processes operate inthe same manner on both types of knowledge, with theone exception of retrieval — this acts on informationdifferently than on mental procedures (as shown inFig. 2). These four cognitive processes are hierarchicalrelative to the level of consciousness required tocontrol their execution [21].COGNITIVE PROCESSES4) Knowledge utilization – the solver applies or usesknowledge to create new products and new ideas. Thefocus here is on the products and ideas, not on theknowledge.Component processes: decision making, problem solving,experimenting, investigating.Much more consciousprocessing necessaryfor control3) Analysis – the solver elaborates on the knowledge ascomprehended, examines knowledge in fine detail andgenerates new conclusions. The focus here is on theknowledge.Component processes: matching, classifying, analyzingerrors and generalizing and specifying.More conscious processingnecessary for control2) Comprehension – the solver translates knowledge intoa form appropriate for storage in permanent memory.Component processes: integrating and symbolizing.Some conscious processingnecessary for control1) Retrieval – the solver activates and transfers knowledgefrom permanent memory to working memory. The processof retrieval depends on the type of knowledge that has tobe retrieved. Information is recognized and recalled, whilemental procedures are also executed.Declarative: vocabularyterms, facts. time sequences,generalizations and principlesOPERATE ONKNOWLEDGE DOMAINSProcedural: single rules,algorithms, tactics,macro-proceduresFIGURE 2. The cognitive processing of knowledgeaccording to Marzano [21].

How do we label a physics problem?To characterize an individual physics problem, wehave defined the following parameters: the type of knowledge involved in the problem the highest complex cognitive process that isnecessary to solve it (for both information andmental procedures) the number of intermediate complex cognitiveprocesses required to solve it (with respect toinformation and mental procedures).Example Problem: Rank each case from thehighest to the lowest acceleration based on thedrawings shown in the figure below. Assume allaccelerations are constant and use the coordinatesystem specified in the drawing. Note: zero is greaterthan negative acceleration, and ties are possible. [22]type of knowledge – the problem involves bothdeclarative and procedural knowledge;number of cognitive processes involved‣ for information: retrieval, comprehensionand analysis;‣ for mental procedures: retrieval;highest cognitive process involved‣ for information: analysis;‣ for mental procedures: retrieval.How do we build different classes ofphysics problems?Depending on the purpose of the problemclassification, meaningful groups of physics problemswith similar parameters can be defined. Forinstructional needs, we find that a classification takinginto account the type of knowledge and the highestcomplex cognitive process required to solve theproblem is often enough. However, for research andassessment purposes, a finer-grained characterizationof physics problems is needed, which also includes thecomponent cognitive processes and the type ofdeclarative or procedural knowledge (refer to Fig. 2).TESTINGA student required to solve this problem should knowthe concepts of velocity and acceleration. Thereforewe conclude that the problem involves declarativeknowledge. Also, the student should know thealgorithms of interpreting the diagrams. Hence theproblem involves procedural knowledge as well. Thesetwo types of knowledge are processed in differentways during the problem solving protocol. Concerningdeclarative knowledge, the student has to: recall the concepts of velocity andacceleration; decide what are the key elements that need tobe taken into account (integrate the facts); represent the information (symbolize the facts); rank the time sequences.Regarding procedural knowledge the student needs toexecute the algorithm of extracting the informationfrom diagrams.In conclusion, here is how this problem would beclassified according to TIPP (refer to Fig. 2):The validity of our taxonomy will be established inthe few ways [23]. First, Marzano’s taxonomy will beconfronted with the existing cognitive science andexpert-novice research concerning physics problemsolving. This will be done in order to ensure welldocumentedand complete descriptions of the cognitiveprocesses that take place during physics problemsolving. Second, we will evaluate the extent to whichthe taxonomy is in agreement with modern models ofstudents thinking. In this way we seek to adaptMarzano’s taxonomy to physics problem solving.Third, we will check if certain cognitive processesassigned to a problem (using TIPP) are in fact used bystudents when they solve it. To do this, we will use 5PER articles that publish extensive interviews withstudents solving physics problems.To assess the reliability of our classificationscheme, we are selecting from [11-16, 24] a set ofabout 100 text-based and PER-based physics problemswith a high degree of diversity. These problems willfirst be categorized in terms of TIPP parameters by theproposal team, and then they will be presented toothers who will make an independent categorization.The results for the assigned parameters will becompared to the proposal team results. As a statisticalmeasure, Cohen’s kappa coefficient [25] will becalculated in order to determine the degree of interraterreliability. Moreover, the process will be

epeated at a later time to verify that the results do notvary over time (stability reliability or “test-retest”reliability).CONCLUSIONSWe are developing a taxonomy for introductoryphysics problems that can be used for designingcurricular materials with a cognitive component andfor modeling the physics problem solving process.While doing so, we analyze existing collections ofphysics problems and aim to identify the cognitiveprocess needed to solve them. After an extensivereview of text-books we find that, for the knowledgedomain of information, most standard text-bookproblems engage cognitive processes up to analysis,while PER-based problems reach the cognitive level ofknowledge utilization. With respect to mentalprocedures, we find that text-books contain very fewproblems that require cognitive processes higher thanretrieval, while some PER-based problems go up toanalysis.ACKNOWLEDGMENTSWe thank to Alan van Heuvelen, Gay Stewart,Alan Giambattista, Joe Redish and Pat Heller forfruitful discussions and comments regarding our work.We would also like to acknowledge the UMD PERGmembers for hosting interesting PER seminars that weattended.REFERENCES1. NRC, National Science Education Standards.Washington, DC: National Academy Press, 1996.2. AAAS Benchmarks for Scientific Literacy. New York:Oxford University Press, 1993.3. B. H. Ross, “Cognitive Science: Problem Solving AndLearning For Physics Education” in Physics EducationResearch Conference 2007, Greensboro, NC, 2007 .4. L. D. Conlin, A. Gupta, R. E. Scherr and D. Hammer,“The Dynamics of Students’ Behaviors and Reasoningduring Collaborative Physics Tutorial Sessions” inPhysics Education Research Conference-2007,Greensboro, NC, 2007.5. R. E. Mayer and M. C. Wittrock, “Problem Solving” inHandbook of Educational Psychology, 2 nd Ed., edited byH. Alexander and R. Winnie, (Wiley, NY, 2006), pp.287- 303.6. W. J. Gerace and I. D. Beatty, Teaching vs. Learning:Changing Perspectives on Problem Solving in PhysicsEducation, Invited talk at the 9 th Conference of theCyprus Physics Association and Greek PhysicsAssociation, Nicosia, Cyprus, 2005.7. E. F. Redish, Teaching Physics with the Physics Suite,(John Wiley & Sons, Hoboken, 2003).8. R. J. Dufresne, W. J. Gerace, J. P. Mestre and W. L.Leonard, ASK.IT/A2L, University of MassachusettsTechnical Report UMPERG-2000#09-SEP#1-28pp.9. R. E. Mayer, Instructional Science 26, 49-63 (1998).10. J. Clement, “Use of Physical Intuition and ImagisticSimulation in Expert Problem Solving” in Problemsolving and education: issues in teaching and research,edited by D. T. Tuma and F. Reif, ( Erlbaum Associates,Hillsdale, NJ, 1980), pp. 204 - 244.11. D. P. Maloney, “Research on problem solving”, inHandbook of Research on Science Teaching andLearning, edited by D. L. Gabel, (Simon & SchusterMacMillan, 1994), pp. 327-354; L. Hsu, E. Brewe, T.Foster and K. Harper, Am. J. Phys. 72, 1147-1156(2004).12. V. Shekoyan and E. Etkina, “Introducing Ill-StructuredProblems in Introductory Physics Recitations” inPhysics Education Research Conference-2007, Physics,Greensboro, NC, 2007.13. I. D. Beatty, W. J. Gerace, W. J. Leonard and R. J.Dufresne, Am. J. Phys. 74, 31-39, (2006).14. A. van Heuvelen and E. Etkina, The Physics ActiveLearning Guide, (Addison-Wesley, San Francisco,2006).15. C. J. Hieggelke, D. P. Maloney, S. E. Kanim and T. L.O’Kuma, E&M TIPERs: Electricity & Magnetism Tasksinspired by Physics Education Research, (Prentice-Hall,Upper Saddle River, 2006).16. Interactive Lecture Demonstrations from UMD PERGhttp://www.physics.umd.edu/perg/ILD.htm; M. Sabella,Ph. D. Dissertation, University of Maryland, CollegePark, MD, 1999.17. J. J. Clement, “Thoughts experiments and imagery inexpert protocols”, in Model-based reasoning in scienceand engineering, edited by L. Magnani, (King’s CollegePublications, in press), pp. 1-16.18. J. Tuminaro, Ph.D. Dissertation, University of Maryland,College Park, MD, 2004.19. M. C. Lovett, “Problem Solving” in Stevens’ Handbookof Experimental Psychology, vol.2, Memory andcognitive processes, 3 rd Ed., edited by D. Medlin (Wiley,New York, 2002).20. R. E. Mayer, Thinking, Problem Solving, Cognition, 2 ndEd., (Freeman, New York, 1992).21. R. J. Marzano and J. S. Kendall, The New Taxonomy ofEducational Objectives, 2 nd . Ed. (Corwin Press,Thousands Oaks, 2007).22. T. L. O’Kuma, D. P. Maloney and C. J. Hieggelke,Ranking Task Exercises in Physics, (Prentice-Hall,Upper Saddle River, 2000).23. W. Wiersma and S. G. Jurs, Educational Measurementand Testing (Allyn and Bacon, Boston, 1990).24. R. D. Knight, Physics for Scientists and Engineers: AStrategic Approach, (Addison-Wesley, San Francisco,2004); A. Giambattista, B. McCarthy Richardson, R. C.Richardson, College Physics, (McGraw Hill, New York,2007); P. P. Urone, College Physics (Brooks/ColePublishing, Pacific Grove, 1998).25. J. Cohen, Education and Psychological Measurement 20,37 (1960).