Challenges In Teaching and Learning Introductory Physics Challengesin Teaching and Learning Introductory PhysicsRogerA. FreedmanDepartment of Physics and Collegeof Creative StudiesUniversity of California, Santa BarbaraSanta Barbara, California 93106-9530This article first appearedin a Festschrift for Professor William A. Little of Stanford University:From High Temperature Superconductivity to Microminiature Refrigeration,B. Cabrera, H. Gutfreund, and V. Kresin, eds. (Plenum, New York, 1996),pp. 313-322. ©1996 Plenum Press, New York.INTRODUCTIONIn a volume intended to celebrate BillLittle's many contributions to fundamental and applied research in physics,an article about teaching the most elementary aspects of physics may seemsomewhat out of place. But a substantial part of Bill's well-deserved reputationis based on his excellence as a teacher, especially for the introductorycourses for engineers and biological science majors. This article is offeredin homage to his tradition of excellence in education.During his distinguished career, Billhas served as a mentor to a great number of graduate teaching assistants.In large part, I owe what success I have had as a physics educator to whatI learned from Bill during my own teaching assistant days at Stanford.My goal in this article is to attempt, however feebly, to carry on histradition.In particular, I will discuss some aspectsof the introductory physics course that will be of interest (and perhapssurprising) to new college physics teachers and new teaching assistants.I hope that this article will also be of use to the "old hands"who, like me, discover something new about teaching every time they gointo a classroom or talk with a student.The challenges involved in teaching anintroductory course in physics are legion, so my goal in this brief articleis not to be comprehensive --- merely provocative! References 1and 2 include a variety of othertips and suggestions for physics faculty and teaching assistants.WHY INTRODUCTORY PHYSICS EDUCATIONIS IMPORTANTThe relative importance of teaching inthe physics enterprise has increased dramatically in recent years. As federalfunding for research decreases, the golden era in which newly-minted physicsPh. D.s were guaranteed at least a postdoctoral research position is becomingan ever more distant memory. A larger fraction of the available academicpositions emphasize teaching, especially at four-year and two-year colleges.Even at research universities, teaching is now playing a larger role inpromotions and tenure decisions. In this brave new world, a physics graduatestudent who aspires to an academic career dare not neglect the teachingside of her or his graduate training.While teaching physics at all postsecondarylevels --- beginning undergraduate, advanced undergraduate, and graduate--- is important, the greatest importance attaches to the introductorycourses taken by students in their first two years of college. For thebudding physics major, these courses are the ones in which the studentgets his or her first taste of the subject and decides whether or not topursue the bachelor's degree in physics. The basic understanding achievedin these courses is the foundation for all subsequent study in physics.The real importance of the introductorycourses, however, lies in those students who are not physics majors. Indeed,the vast majority of students in introductory courses are likely to beengineers (in a calculus-based course), premedical students (in an algebra-basedcourse), or humanities majors (in a "conceptual physics" course).These students constitute the educated electorate of the future, and theirintroductory physics courses are the only chance that we physicists haveto plead our case with them.The dominant public perception of physicsis that it is tedious, abstract, and fundamentally irrelevant; the challengein an introductory course is to convince the audience that physics is rewarding,fun, useful, and most of all a worthwhile endeavor. If we fail inthis, and the public perception of physics does not change, there is littlechance that future physics research will be funded at anything more thana token level. In this sense, introductory physics teaching is the foundationnot only of a physics education, but of the physics enterprise as a whole.We neglect the teaching of these courses at our own grave peril.WHY JOHNNY CAN'T ANALYZE CIRCUITSCollege students begin most of their coursesin a state of nearly perfect tabula rasa. Before they take theirfirst course in world history, in economics, or in psychology, they knowlittle or nothing about those subjects. The instructor can then help thestudents to implant fresh knowledge upon the palimpsests of their minds.The situation in an introductory physicscourse is quite different. Although they would be shocked to hear you sayit, students arrive in their first physics course with a set of physicaltheories that they have tested and refined over years of repeated experimentation.How can this be? The reason is that students have spent some eighteen yearsexploring mechanical phenomena by walking, running, throwing baseballs,catching footballs, and riding in accelerating vehicles. They have alsosome more limited experience with electrical phenomena, garnered from usingelectric circuits in the home, and about the behavior of light, lenses,and mirrors. Based on their observations, students have pieced togethera set of "common sense" ideas about how the physical universeworks.Unfortunately, research carried out byphysicists has shown these "common sense" ideas are in the mainincompatible with correct physics. Worse still, these erroneousideas are robust and difficult to dislodge from students' minds, in largemeasure because these ideas are not addressed by conventional physics instruction.As an example, Fig. 1 illustrates somerepresentative "common sense" ideas about electric circuits.In part (a) of the figure, a battery is connected to two identical lightbulbs A and B in series. In part (b), the battery is connectedto a single bulb C which is identical to bulbs A and B. McDermottand Shaffer [3] asked studentsin introductory physics courses to compare the brightnesses of bulbs Aand B in circuit (a) and to compare these with the brightness ofbulb C in circuit (b). Figure 1.(a) A battery connected to two identical light bulbs A and Bin series. (b) The battery is now connected to a single bulb C whichis identical to bulbs A and B. When asked to compare thebrightnesses of the bulbs in these circuits, only an embarrassingly smallnumber of students gave the correct answer even after instruction in circuittheory.The results of this investigation wereincredibly disappointing. The correct answer, that bulbs A and Bin circuit (a) are equally bright and that bulb C in circuit (b)is brighter still, was given by only about 10% of the students in algebra-basedcourses and by only about 15% of the students in calculus-based courses.The most remarkable result of McDermott'sand Shaffer's study is that the types of student errors made on this questionare unrelated to, and unaffected by, conventional instruction.One common student error is the belief that in circuit (a), bulb Awill be brighter than bulb B because bulb A "'uses up'the current first." Another common error is that the brightness ofeach bulb will be the same in either circuit because the battery providesa constant current in all cases. Neither of these incorrect ideasare learned from an introductory course, but neither are they discreditedin a standard introductory course. Indeed, McDermott and Shaffer foundthat student performance on this question was nearly independent of whetherthe question was posed before or after instruction on electric circuits.Similarly disquieting results have been found regarding "common sense"ideas in mechanics [4] [5][6] and in optics [7].Investigations of this sort show thatit is not enough to merely teach students the right way to thinkabout physics. Rather, the challenges to the instructor are to identifypossible student misconceptions, to confront these misconceptions head-on,and to help students to unlearn these misconceptions at the sametime that they are learning correct physics. Failure to do this will invariablyleave students with their erroneous "common sense" ideas intact.In order to rise to these challenges,an essential tool is an introductory physics textbook that addresses "commonsense" ideas explicitly. Sadly, most contemporary textbooks are severelydeficient in this respect. But some very recent textbooks make extensiveuse of research into student misconceptions [8][9], and these should be givenconsideration by instructors who are serious about helping students overcometheir "common sense" ideas about physics."I UNDERSTAND THE CONCEPTS, IJUST CAN'T DO THE PROBLEMS"Every physics instructor has heard thiscomplaint from students at one time or another. All too often, however,what the student really means is the converse:"I can do (some of) the problems,I just don't understand the concepts."Students can usually handle problems thatare akin to the worked examples in their textbook, especially if thereare "special equations" that they can use. Problems that requireusing fundamental concepts, along the lines of how we might expect a physicistto think, are another matter altogether.The proof of this statement is the differencebetween student performance on "standard" physics problems thatrequire computation and calculation and their performance on purely conceptual,qualitative problems. As an example, McDermott and Shaffer [3]found that even students who performed well on standard numerical problemsin circuit analysis, and even students with near-perfect scores on suchproblems, performed poorly on the conceptual question depicted in Fig.1.Part of the difficulty that students havewith conceptual questions stems from the kind of problems that studentsare most often assigned. Instructors commonly assign homework and examproblems that involve computation or calculation, in the belief that theseare "real" physics problems. A corollary to this belief is theassumption that a student's ability to successfully solve such problemsis evidence of complete understanding. Alas, research shows that such isnot the case. One example is an investigation of student understandingof the Newtonian concept of force carried out by Hestenes, Wells, and Swackhamer.[6] By comparing student performance on a set of conceptualquestions posed both before and after a first course on mechanics, theyfound that conventional instruction (including the assignment of conventionalhomework problems) produces only marginal gains in conceptual understanding.If we truly want students to learn aboutthe ideas of physics, we must require them to use these ideasin their homework and then hold them accountable for these ideas in examinations.Most introductory textbooks include a wealth of conceptual questions, andquestions of this sort these should be assigned regularly. My own studentsregularly comment that they find conceptual questions to be much more difficultthan the "ordinary" problems; such comments convince me thatconceptual questions are very useful tools for teaching and learning physics.WHAT DOES THAT EQUATION MEAN?A related issue is the question of howstudents deal with formal, mathematical expressions of physicalconcepts. Two examples are Newton's second law and the work-energy theorem: It is very common for students to interpretEq. (1) to mean that the product of a body's mass and its accelerationis itself a force. In other words, they fail to realize that a mathematicalequality between two quantities does not imply that the two quantitiesare conceptually distinct. As a result, they do not appreciate that accelerationis the consequence of the presence of a net force. Thus studentsfrequently make reference to such chimera as "the force due to acceleration"or "the force due to momentum."A similar confusion arises concerningthe work-energy theorem, Eq. (2). When students are asked to explain whatkinetic energy means, the most common response is that it is "one-halfthe mass times the speed squared." By fixating on the mathematicaldefinition, they fail to grasp the essence of the work-energy theorem:that the kinetic energy of a particle is equal to the total work that wasdone to accelerate it from rest to its present speed, and equal to thetotal work that the particle can do in the process of being brought torest.This tendency to focus on a mathematicaldefinition rather than physical meaning was shown convincingly by Lawsonand McDermott. [10] They presentedstudents with a simple question concerning the work-energy theorem. Asdepicted in Fig. 2, an object of mass m and another object of mass2m are initially at rest on a frictionless horizontal surface. Thesame constant force of magnitude F is then applied to each object.The question to be answered is "Which object crosses the finish linewith greater kinetic energy?" Figure 2.A top view of a "race" between two objects on a frictionlesshorizontal surface. The two objects are of different mass but are subjectedto the same net force. When asked which object crosses the finish linewith greater kinetic energy, only a few students were able to give thecorrect answer.Using the work-energy theorem, and keepingin mind the physical meaning of kinetic energy, it can easily beseen that each object has the same kinetic energy upon reaching the finishline. Yet in interviews with 28 students taken from two classes at theUniversity of Washington, an honors section of calculus-based physics anda regular section of algebra-based physics, Lawson and McDermott foundthat only a few honors students were able to supply the correct answerand the correct reasoning without coaching. While most of the remaininghonors students were able to eventually achieve success with guidance fromthe interviewer, almost none of the students from the algebra-based coursewere able to do so. No less disappointing results were obtained with awritten version of the question presented to a regular section of calculus-basedphysics. I have had similar experiences with my own students: Their performanceon conventional homework-type problems shows that they can compute quantitiessuch as work and kinetic energy, but their performance on conceptual questionsshows that they have much more difficulty explaining or interpretingtheir results.This example shows again that emphasison numerical problem-solving can obscure major conceptual deficienciesin students. It underscores the importance of requiring students to applythe fundamental concepts of physics in a variety of different situations,as well as requiring them to explain the logic that they use in solvingphysics problems of all kinds.RETHINKING THE LECTURE AND DISCUSSIONSECTIONA point that I have stated repeatedlyin this article is that conventional physics instruction tends to be ineffectivein helping students to develop a real understanding of physics. How, then,should the nature of physics instruction be changed? A number of differentapproaches have been suggested and explored; I will summarize the approachesthat I believe to be the most promising.The Misuses of the LectureThe lecture is one of the most ancientof teaching methods. In the teaching of physics, it is typically used todemonstrate physical phenomena, to present derivations; and to show examplesof how to solve problems. The first of these uses of the lecture is animportant one, and is often neglected by instructors who feel compelledto "cover more material" or who regard the demonstrations asa distraction. My own experience is that good lecture demonstrations areabsolutely indispensable as tools for helping students to relate physicalconcepts to the real world. Good lecture demonstrations also have the strengthof being memorable. I have had students come to me a decade after takingone of my classes and tell me how they still remember a certain demonstrationand the physics that they learned from it. (By contrast, I have yet tohave a former student tell me how vividly they remember my derivation ofthe thin-lens formula.) The title "Lecture Demonstrator" is stillin use at certain British universities to denote a science lecturer; thetitle alone speaks volumes about the importance of lecture demonstrations.By contrast, the use of lecture time topresent derivations is typically ineffective. A derivation presented onthe blackboard is less useful to the student than the same derivation presentedin the textbook, where it can be traced through repeatedly at the student'sleisure. My suspicion is that instructors tend to present derivations inlecture because they doubt that their students read the book. While thisis indeed a valid concern, it would seem that using the lecture to reiteratethe contents of the book is ultimately counterproductive; it merely helpsto ensure that the students won't read the book.Far and away, however, the least effectiveuse of lecture time is for presenting the solutions to physics problems.The essential difficulty here is that physics problem-solving is a skillthat has to be learned by repeated practice. In learning a skill, itcan be useful to first watch an expert exercise that skill, but that isby no means the most important part of the learning process. If it were,the millions who watch professional sports would themselves naturally developinto top-notch players; avid movie-goers would inexorably turn into accomplishedactors (who really want to direct); and the poor souls who watch televisedcourt proceedings would slowly but surely mutate into highly paid defenseattorneys. Of course, none of these evolutions really take place. In thesame way, students who watch their instructor (an expert problem-solver)work out a solution on the board may be impressed by the instructor's prowess,but they will augment their own problem-solving skills only marginally.The disappointing problem-solving performance of students who have hadsuch conventional instruction, referred to earlier, is testimony to this.A Lecture Model with "Active Learning"Numerous instructors, myself included,have found that lectures become more useful when students are forced tobecome active participants in the lecture. [11]In my own classes, I speak briefly about each new topic (proceeding underthe assumption that students have read the required material from the textbookbefore class), and do a lecture demonstration or two as appropriate.I then give the students an exercise to work out. They then spend severalminutes working out this exercise, which is chosen to be specific to thetopic at hand: it may involve tasks such as drawing free-body diagrams,writing down (but not necessarily solving) the key equations for a groupof related but distinct situations, or making graphs of different typesof motion. While this is going on, I roam around the classroom inspectingthe students' work. I then instruct the students to confer with their neighborto compare their responses and to resolve any discrepancies. Remarkably,this works very well even in a large lecture hall; the sound level fromthe discussions among 300 students can be quite impressive! Finally, Idiscuss with the students the correct way to tackle the exercise, beingcareful to point out common errors to the students. I typically do twoor three sequences of instructor description --- student work --- instructordiscussion during a typical lecture.This technique has several merits. First,the students have something constructive to do during the lecture; it isa sure-fire cure for the torpor that grips students midway through a conventionallecture. Second, students are forced to discuss physics with their peersand to defend their ideas. Third, students get immediate feedback as towhether or not they understand a concept that has been presented in class,and any points of confusion can be corrected at an early stage in the students'apprehension of the concept. Last, but by no means least, the instructorcan learn a great deal about her or his students' understanding of thematerial. This last point was brought home to me vividly during a lecturewhen I asked students to draw the free-body diagram for a car roundinga banked curve; many of the diagrams I saw while walking around the lecturehall included a number of creative and wholly imaginary forces that I hadnever dreamed existed!When conducting the lecture in this way,it is best if the students have a printed sheet with the exercise on it.These can be time-consuming to develop and to prepare in printed form,however. I have relied heavily on Alan Van Heuvelen's ALPS Kit (anacronym for Active Learning Problem Sheets), and Randy Knight'sStudent Workbook for Physics: A Contemporary Perspective, whichare workbooks containing several hundred exercises and activities expresslydesigned for student use during lecture. [12]Students purchase these inexpensive workbooks at the campus bookstore,and are required to bring them to lecture; happily, I find that almostall of them do so religiously.Some will no doubt complain that thistechnique of "active learning" forces the lecturer to cover lessmaterial. It is indeed true that the lecturer talks about less materialwith this approach; the challenge to the lecturer is to choose betweenthe material that is worthy of discussion during the lecture and the easiermaterial that the students can learn adequately on their own from the textbook.Thus this technique does not require that any material be deleted fromthe course syllabus.Employing "active learning"in the lecture keeps students engaged in the lecture. More importantly,it yields substantially better student performance on exams than does conventionalinstruction. [11] [12]Discussion Sections,Teaching Problem-Solving,and "Cooperative Learning"Most introductory physics courses haveboth a lecture component and a discussion section (or "recitationsection") component. The discussion section, typically led by a teachingassistant (TA), is intended principally to be a forum in which studentsgain insight into problem-solving technique by observing the discussionleader, by practicing solving problems, and by discussions with other students.Unfortunately, physics discussion sectionsvery often fail to live up to this intent. Too many students come to discussionsections with the intent that they will get their weekly homework "donefor them" by the TA. As a result, despite the earnest efforts of hard-workingand talented TAs, it is difficult to cajole students into actually doingproblem-solving work in a discussion section. Furthermore, it is next toimpossible to initiate and sustain any real student discussion within theunstructured format of a typical discussion section. The upshot is thatfew students are able to move beyond the "formulaic" approachto problem-solving, which consists of hunting through the textbook fora likely-looking equation or set of equations into which they can plugthe values stated in the problem. This sad state of affairs is especiallyfrustrating for TAs who, after working diligently with a group of studentsfor an entire term, must grade those students' disappointing work on exams.A very promising effort to rectify theseshortcomings of the discussion section has been described by Heller, Keith,Anderson, and Hollabaugh. [13][14] They reorganized the discussionsections in two rather different physics courses, the first quarter ofa large algebra-based introductory course at the University of Minnesotaand a sophomore modern physics course with a dozen students at NormandaleCommunity College. In both courses, students were taught a general problem-solvingstrategy based on the methods used by expert problem-solvers, and wererequired to write up their problem solutions in a way that explicitly reflectsthe use of that strategy. (In addition to shaping the students' approachto problem-solving, this technique helps to clarify for the grader whatconceptual ideas the students are using.) To discourage "formulaic"problem-solving, students were assigned so-called "context-rich"problems. Such problems do not always explicitly identify the unknown variable,may include extraneous information, and may require reasonable assumptions(e. g., the acceleration is constant) or estimation (e. g., the mass ofa typical cat). In other words, they are less like standard textbook problemsand more like the problems encountered by real scientists and engineers.The folowing is an example of such a "context-rich" problem,taken from Ref. 14:While visitinga friend in San Francisco, you decide to drive around the city. You turna corner and find yourself going up a steep hill. Suddenly a small boyruns out on the street chasing a ball. You slam on the brakes and skidto a stop, leaving a skid mark 50 ft long on the street. The boy calmlywalks away, but a policeman watching from the sidewalk comes over and givesyou a ticket for speeding. You are still shaking from the experience whenhe points out that the speed limit on this street is 25 mph. After you recover your wits, you examinethe situation more closely. You determine that the street makes an angleof 20° with the horizontal and that the coefficient of static frictionbetween your tires and the street is 0.80. You also find that the coefficientof kinetic friction between your tires and the street is 0.60. Your car'sinformation book tells you that the mass of your car is 1570 kg. You weigh130 lb, and a witness tells you that the boy had a weight of about 60 lbsand took 3.0 s to cross the 15-ft wide street. Will you fight the ticketin court?Such "context-rich" problemsare of the kind that we would like our students to be able to solve, butwhich are usually thought too difficult and challenging for students tosolve on their own. Remarkably, students in both of the test groups describedin Refs. 13 and 14 were ableto solve such problems when each discussion section was organized intocooperative groups of three students. The students in each groupwere required to work together to produce a group solution to the assignedproblem, using the problem-solving strategy that they had been taught.All students in the group received the same grade for their group assignment.The students in each group were assigned the roles of Manager (who keepsthe group on task and manages the sequence of steps), Skeptic (who helpsthe group to avoid overly quick agreement and asks questions like "Arethere other possibilities?"), and Checker/Recorder (who checks forconsensus among the group and who writes up and hands in the group solution).These roles were rotated among the students each week. The use of suchdefinite roles, and the challenging nature of the assigned "context-rich"problems, kept the students from simply working independently.To reinforce the use of the problem-solvingstrategy and of the skills used in the cooperative groups, each courseexam included a "context-rich" problem that had to be solvedby the students in their cooperative group during the discussion section.(More conventional individual exams were given during lecture.)Heller et al. found that over two quartersof using these methods, the problem-solving technique of students of allability levels improved. [13] It may not be surprisingthat this proved to be the case for students in the lowest third and middlethird of the class. The structured problem-solving strategy and the requirementto discuss ideas with other students seems well-suited to helping studentswhose understanding of problem-solving was initially only fair or poor.What is remarkable is that participation in cooperative groups also helpedthe best students in the class to improve their problem-solvingskills, and that these students improved at about the same rate as thestudents in the lowest and middle thirds. For example, the percentage ofstudents in the lowest third of the class whose individual solutions followeda logical mathematical progression improved from 20% to 50% over two quarters;this percentage for students in the upper third improved from 60% to 90%.Furthermore, this improvement of all students was found in both group problem-solvingand individual problem-solving.The use of cooperative groups and "context-rich"problems can have a very beneficial effect on student problem-solving skills.We have just begun to implement these innovations in the introductory calculus-basedphysics course at UC Santa Barbara, and the preliminary results look encouraging.This method is not a panacea, however; Heller at al. found that their innovationsdid not have much beneficial effect on students' understanding of the conceptualaspects of physics. [13] This suggests that theseaspects are best addressed in the lecture using the "active learning"technique described previously.CONCLUSIONTeaching and learning introductory physicsare both challenging tasks. While traditional methods have led to frequentlydisappointing results, I have tried to show that there is hope. As instructors,we should heed the lessons about our students' thought processes learnedfrom research into "common sense" ideas about physics and intostudents' difficulty with formal mathematical reasoning. We must see toit that students truly learn how to use the concepts of physics,in order that they may learn how to think like a scientist or engineer.And we should be willing to consider new forms and new approaches for thetime-honored lecture and discussion section. Whatever gains we can makein improving student understanding and appreciation of physics cannot helpbut improve the public perception of physics as a useful, interesting,and above all comprehensible human activity.ACKNOWLEDGEMENTSI thank my many teachers at Stanford forhelping to cultivate my interest in the field of physics education. Inparticular, I thank Alan Schwettman, Dirk Walecka, and especially BillLittle for his service as a role model to generations of Stanford graduatestudent TAs.REFERENCES1. A. B. Arons, TeachingPhysics, Wiley, New York (1997). [First appearancein article]2. M. J. Bozack, Tipsfor TA's: The role of the physics teaching assistant, Phys. Teach.21, 21 (1983). [First appearance in article]3. L. C. McDermottand P. S. Shaffer, Research as a guide for curriculum development: an examplefrom introductory electricity, Part I: Investigation of student understanding,Am. J. Phys. 60, 994 (1992). [Firstappearance in article]4. I. Halloun and D.Hestenes, The initial knowledge state of college physics students, Am.J. Phys. 53, 1043 (1985). [First appearancein article]5. I. Halloun and D.Hestenes, Common-sense concepts about motion, Am. J. Phys. 53,1056 (1985). [First appearance in article]6. D. Hestenes, M.Wells, and G. Swackhamer, Force concept inventory, Phys. Teach.30, 141 (1992). [First appearance in article]7. F. M. Goldberg andL. C. McDermott, An investigation of student understanding of the realimage formed by a converging lens or concave mirror, Am. J. Phys.55, 503 (1987). [First appearance in article]8. L. C. McDermott,Physicsby Inquiry, Wiley, New York (1996). [Firstappearance in article]9. H. D. Young andR. A. Freedman, UniversityPhysics, 9th edition, Addison-Wesley, Reading, MA (1996). [Firstappearance in article]10. R. A. Lawson andL. C. McDermott, Student understanding of the work-energy and impulse-momentumtheorems, Am. J. Phys. 55, 811 (1987). [Firstappearance in article]11. A. Van Heuvelen,Learning to think like a physicist: A review of research-based instructionalstrategies, Am. J. Phys. 59, 891 (1991). [Firstappearance in article]12. A. Van Heuvelen,Overview, Case Study Physics, Am. J. Phys. 59, 898 (1991);R. D. Knight, StudentWorkbook for Physics: A Contemporary Perspective (Addison-Wesley,Reading, MA, 1998). For information on how to obtain the ALPS Kit, contactProfessorAlan Van Heuvelen, Department of Physics, Ohio State University, ColumbusOH 43210-1106. [First appearance in article]13. P. Heller, R.Keith, and S. Anderson, Teaching problem solving through cooperative grouping,Part 1: Group versus individual problem solving, Am. J. Phys. 60,627 (1992). [First appearance in article]14. P. Heller andM. Hollabaugh, Teaching problem solving through cooperative grouping, Part2: Designing problems and structuring groups, Am. J. Phys. 60,637 (1992). [First appearance in article]Go to beginningof articleRogerFreedman's main page |
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