During the past fifteen years, asteadily increasing number of physicists have been contributing to thegrowth of a new field for scholarly inquiry: the learning and teachingof physics. We have by now a rich source of documented information in themany published reports of this research. At this point, it seems reasonableto ask whether we have learned anything from this collective experiencethat would be useful in current efforts to bring about innovative reformin the introductory course. Results from research indicate that at alllevels of instruction the difference between what is taught and what islearned is often greater than most instructors realize. This discrepancysuggests the following question: Is there a corresponding mismatch betweenhow we teach and how students learn?

I. TRADITIONAL APPROACH TO INSTRUCTION

Instruction in introductory physicshas traditionally been based on the instructor’s view of the subject andthe instructor’s perception of the student. Most teachers of physics areeager to transmit both their knowledge and enthusiasm. They hope that theirstudents will acquire not only specific information and skills but alsocome to appreciate the beauty and power that the physicist finds in physics.Having obtained a particular insight after hours, days, months, or yearsof intellectual effort, they want to share this knowledge. To save studentsfrom going through the same struggles, instructors often teach from thetop down, from the general to the particular. Generalizations are oftenfully formulated when they are introduced. Students are not actively engagedin the process of abstraction and generalization. Very little inductivethinking is involved; the reasoning is almost entirely deductive. By presentinggeneral principles and showing how to apply them in a few special cases,instructors hope to teach students how to do the same in new situations.

In recalling how they were inspiredby their own experience with introductory physics, many instructors tendto think of students as younger versions of themselves. In actual fact,such a description fits only a very small minority. Typically, in the U.S., no more than one in every 30 university students taking introductoryphysics will major in the subject. The trouble with the traditional approachis that it ignores the possibility that the perception of students maybe very different from that of the instructor. Perhaps most students arenot ready or able to learn physics in the way that the subject is usuallytaught.

II. SOME GENERALIZATIONS ABOUTLEARNING AND TEACHING

The generalizations that appear beloware based on results from research on the learning and teaching of physics.† The evidence presented in support of the generalizationsis taken from the cited articles on research by the Physics Education Groupat the University of Washington. However, the same arguments could be basedon findings by other investigators. Similar conclusions have also beenreached by experienced instructors who have probed student understandingin less formal ways in the classroom.

A. Facility in solving standardquantitative problems is not an adequate criterion for functional understanding.Questions that require qualitative reasoning and verbal explanationare essential.

The criterion most often used inphysics instruction as a measure of mastery of the subject is performanceon standard quantitative problems. As course grades attest, many studentswho complete a typical introductory course can solve such problems satisfactorily.

However, they are often dependenton memorized formulas and do not develop a functional understanding ofphysics, i.e., the ability to do the reasoning needed to apply appropriateconcepts and physical principles in situations not previously encountered.We illustrate this first generalization with examples from dynamics andelectricity.

1. Example from dynamics: impulse-momentumand work-energy theorems

In an investigation conducted severalyears ago, we examined whether students could apply the impulse-momentumand work-energy theorems to a simple motion that they could observe.1,The motion was generated by applying a constant force to two objects ofdifferent mass over the same distance. Students were asked to compare thefinal momenta and kinetic energies of the objects. No calculations wereneeded to predict that the heavier object would have a greater momentumand that both would have the same kinetic energy. It was only necessaryto understand the relationship between impulse and momentum and the relationshipbetween work and kinetic energy. For a response to be considered correct,both the right comparison and the proper reasoning were required.

Data were gathered in individualdemonstration interviews. The 28 students who participated came from twoclasses: an honors section of calculus-based physics and a regular sectionof algebra-based physics. Responses ranged from random formula searchesto conscious attempts to apply the theorems. Only a few honors studentswere able to give satisfactory answers initially. With step-by-step guidance,most of these students eventually succeeded. Even with help, however, virtuallyno one in the algebra-based course was able to apply the concepts of impulseand work to make a correct comparison. There was a similar lack of successwhen written versions of the tasks were presented in a regular sectionof calculus-based physics. Among the many errors was the failure of moststudents to recognize the cause-and-effect relationships inherent in thetheorems. Some seemed to treat the symbol "=" as if it represented onlya mathematical relationship in which the variables may take on any values,provided the equality is maintained.

2. Example from electricity:electric circuits

We have been investigating studentunderstanding of electric circuits over a period of several years. Onetask that has proved particularly effective for eliciting common difficultiesis based on three simple circuits consisting of identical bulbs and idealbatteries. One circuit has a single bulb; another has two bulbs in series;the third has two bulbs in parallel. Students are asked to rank the fivebulbs according to relative brightness and to explain their reasoning.This comparison requires no calculations. A simple qualitative model, inwhich bulb brightness is related to current or potential difference, issufficient.

We have administered this task tomore than 500 university students. Almost every possible bulb order hasappeared. Whether before or after instruction, only about 15% of the studentsin a typical calculus-based course give the correct ranking. We have obtainedthe same results from high school physics teachers and from universityfaculty who teach other sciences and mathematics. Many people who are unableto rank the bulbs properly can use Ohm’s law and Kirchhoff’s rules to solvemore complicated problems. Evidently, success on standard problems is nota reliable indicator of functional understanding.

B. A coherent conceptual frameworkis not typically an outcome of traditional instruction. Studentsneed to participate in the process of constructing qualitative models thatcan help them understand relationships and differences among concepts.

Perhaps the most serious difficultythat we have identified is failure to integrate related concepts into acoherent framework. Rote use of formulas is common. To solve standard problems,mathematical manipulation may suffice. To be able to apply a concept ina variety of contexts, however, students must not only be able to definethat concept but also relate it to others. They also need to differentiatethat concept from related concepts.

The question on ranking the bulbswas first administered several years ago on a course examination in a standardcalculus-based course. Lacking a conceptual model on which to base predictions,most students relied on intuition or formulas. About 40% used algebra tofind the equivalent resistances of the series and parallel circuits, substitutedthe values into the formula for the power dissipated in a resistor, andassociated the results with the brightness of individual bulbs in the seriesand parallel networks. Such errors revealed a failure to differentiatebetween two related concepts: the resistance of an element and the equivalentresistance of a network containing that element.

A general instructional strategythat we have found useful for helping students relate electrical conceptsand distinguish one from another is to engage them actively in the intellectualprocess of constructing a qualitative model for an electric circuit. Developmentof the model is based on observations of the behavior of batteries andbulbs, preferably through experiments that the students themselves perform.

Experience has shown that emphasison concept development and model-building does not detract from performanceon quantitative problems. Many students need explicit instruction on problem-solvingprocedures to develop the requisite skills. However, once equations areintroduced, students often avoid thinking of the physics involved. Postponinguse of algebraic formalism until after a qualitative understanding hasbeen developed has proved to be an effective approach. Although less timeis spent on numerical problem-solving, examination results indicate thatstudents who have learned in this way often do better than others on quantitativeproblems and much better on qualitative questions.

C. Certain conceptual difficultiesare not overcome by traditional instruction.Persistent conceptual difficulties must be explicitly addressed by multiplechallenges in different contexts.

Some student difficulties disappearduring the normal course of instruction. Others seem to be highly resistantto change. If sufficiently serious, they may preclude meaningful learning,even though performance on quantitative problems may be unaffected. Anexample of a common difficulty that research has shown to be especiallypersistent is the apparently intuitive belief that current is "used up"in a circuit.

Deep-seated difficulties cannot beovercome through assertion by the instructor. Active learning is essentialfor a significant conceptual change to occur. An instructional strategythat we have found effective for obtaining the necessary intellectual commitmentfrom students is to generate a conceptual conflict and to require themto resolve it. A useful first step is to elicit a suspected difficultyby contriving a situation in which students are likely to make a relatederror. Once the difficulty has been exposed and recognized, the instructormust insist that students confront and resolve the issue.Unlike physicists, students may be willing to tolerate inconsistency.

A single encounter is rarely sufficientto overcome a serious difficulty. Students do not make the same mistakesunder all circumstances; the context may be critical. Unless challengedwith a variety of situations capable of evoking a given difficulty, studentsmay simply memorize the answer for a particular case. To be able to integratecounter-intuitive ideas into a coherent framework, they need time to applythe same concepts and reasoning in different contexts, to reflectupon these experiences and to generalize from them.

D. Growth in reasoning abilitydoes not usually result from traditional instruction. Scientificreasoning skills must be expressly cultivated.

An important factor in the difficultiesthat students have with certain concepts is an inability to do the qualitativereasoning that may be necessary for applying the concept. It is often impossibleto separate difficulties with concepts from difficulties with reasoning.An error may be a symptom of an underlying conceptual or reasoning difficulty,or a combination of both.

A failure to think holistically indealing with compound systems is one kind of reasoning difficulty thatmay be hard to disentangle from conceptual confusion. For example, in predictingbulb brightness, students often considered only the order of a bulb inan array. Many claimed that the first bulb in a series network was thebrightest. This error is consistent with the misconception that currentis "used up" and also with improper use of local sequential reasoning.For interacting systems, such as elements in an electric circuit, it isimpossible to predict the behavior of one without taking into account theeffect of the others. However, instead of considering the circuit as awhole, many students focused on only one bulb at a time. The conservationof current was an abstraction for which they might be able to write anequation but which they could not apply to a qualitative problem.

Predicting the effect of a changein a circuit requires a more sophisticated level of holistic reasoning.In one task, students were shown a circuit diagram in which a network containingtwo branches in parallel was connected in series with other bulbs. Thestudents were asked to predict what would happen to the brightness of abulb in one branch of the parallel network when the other branch was removed.A common response was that the brightness would not change. Often the explanationgiven was that the bulb was part of a parallel combination. In treatingthe parallel branches as independent, the students were not recognizingthe difference between parallel branches connected across a battery andparallel branches connected elsewhere. Instead of using qualitative reasoningto check that their predictions were consistent with what they knew aboutcurrent and potential difference, the students relied on a rule that theyhad incorrectly memorized.

Traditional instruction does notchallenge but tends to reinforce a perception of physics as a collectionof facts and formulas. Students often do not recognize the critical roleof reasoning in physics, nor do they understand what constitutes an explanation.They need practice in solving qualitative problems and in explaining theirreasoning. However, they are unlikely to persevere at developing facilityin scientific reasoning unless the course structure, including the examinations,emphasizes the importance of this ability.

E. Connections among concepts,formal representations, and the real world are often lacking after traditionalinstruction. Students need repeated practice in interpreting physicsformalism and relating it to the real world.

Students are often unable to relatethe concepts and formal representations of physics to one another and tothe real world. An inability to interpret equations, diagrams and graphsunderlies many conceptual and reasoning difficulties.

1. Difficulty with algebraicrepresentations: example from dynamics

Performance on the impulse-momentumand work-energy comparison tasks illustrated the difficulty students frequentlyhave in relating algebraic formalism to physical concepts and to the realworld. The demonstration creates a simple physical situation in which therelevant theorems can be applied. Nevertheless, few students have beenable to connect the mathematical statement of the theorems to the motionof the pucks.

2. Difficulty with diagrammaticrepresentations: example from optics

In another investigation, studentswho had studied geometrical optics participated in interviews in whichthey were shown a demonstration that consisted of an object, a thin converginglens and an inverted real image on a screen. One of the tasks was to predictthe effect of covering half of the lens. Most students claimed that halfof the image would disappear. The ray diagrams that they drew sometimesreinforced this mistaken intuition. Two of the special rays were oftenshown as blocked. In interpreting their diagrams, the students indicatedthat these rays were necessary for forming the image, rather than merelyconvenient for locating its position. 

3. Difficulty with graphicalrepresentations: example from kinematics

Student understanding of the graphicalrepresentation of motion has been a long-term research interest of ourgroup., In one task from this ongoing study, studentsare shown a ball rolling along a track and given a diagram of the motionwith a description similar to the following: The ball moves with steadyspeed on the level segment of the track, speeds up as it moves down an incline, and then continues at a higher constant speed on the last segment.The students are told that position is measured along the track and areasked to represent the motion in graphs of position, velocity and accelerationversus time. The task has been presented to several hundred students whohave studied kinematics. Few students in the standard calculus-based coursehave produced correct graphs.

We have also examined student difficultieswith the reverse process: visualization of a real motion from its graphicalrepresentation. The ability to relate actual motions and their graphicalrepresentations does not automatically develop with acquisition of simplegraphing skills, such as plotting points, reading coordinates and findingslopes. Students need practice in translating both ways: from motion tographs and from graphs to motion.

F. Teaching by telling is an ineffectivemode of instruction for most students. Students must be intellectuallyactive to develop a functional understanding.

All the examples of student difficultiesdiscussed above share a common feature: the subject matter involved isnot difficult. Many instructors expect university students who have studiedthe relevant material to be able to answer the types of questions thathave been illustrated. Yet, in each instance, we found that a large percentageof students could not do the basic reasoning necessary. On certain typesof tasks, the outcome did not vary much from one traditionally taught classto another, nor did it matter when in the course the problems were posed.Enrollment in the associated laboratory course also did not appear to affectthe quality of student performance. Moreover, there was no correlationbetween the success of students and the reputation of the course instructoras a lecturer.

The difficulties that students havein physics are not usually due to failure of the instructor to presentthe material correctly and clearly. No matter how lucid the lecture, norhow accomplished the lecturer, meaningful learning will not take placeunless students are intellectually active. Those who learn successfullyfrom lectures, textbooks and problem-solving do so because they constantlyquestion their own comprehension, confront their difficulties and persistin trying to resolve them. Most students taking introductory physics donot bring this degree of intellectual independence to their study of thesubject.

Although the traditional lectureand laboratory format has disadvantages, it may be the only mode possiblewhen the number of students is large. Such instruction, however, need notbe a passive learning experience. There are several techniques that instructorsof large classes can use to promote active participation by students inthe learning process.

III. IMPROVING THE MATCH BETWEENTEACHING AND LEARNING

The generalizations about learningand teaching presented above have been derived from investigations of studentunderstanding in the context of classical physics. We believe, however,that they have broad applicability and should be taken into account incurrent efforts to introduce new topics and new technology into the introductorycourse.

Physicists generally assume thatstudents will find contemporary topics inspiring. It has been our experience,however, that few students are motivated by exposure to material that theydo not understand. Instead, the outcome may only confirm a belief thatphysics is too difficult for most people. There is also great enthusiasmabout the potential of the computer for enhancing student learning in physics,especially modern physics. Although there is reason for optimism, our experiencesuggests a need for caution. Success on a computer task does not necessarilyindicate development of a skill that can be transferred to other environments.Even a highly interactive program does not insure that students will makethe mental commitment necessary for significant concept development tooccur.

Perhaps the most significant contributionthat research in physics education can make to the improvement of instructionis to underscore the importance of focusing greater attention on the student.The successful incorporation of contemporary topics or advanced technologyinto the introductory course is likely to depend as much on how thematerial is taught as on what is taught. To insure that the curriculumthat is developed will be well-matched to the students for whom it is intended,there is a need for research on the learning and teaching of both classicaland modern topics, with and without the computer.8

Meaningful learning, which connotesthe ability to interpret and use knowledge in situations different fromthose in which it was initially acquired, requires that students be intellectuallyactive. Development of a functional understanding cannot take place unlessstudents themselves go through the reasoning involved in the developmentand application of concepts. Moreover, to be able to transfer a reasoningskill learned in one context to another, students need multiple opportunitiesto use that same skill in different contexts. The entire process requirestime. Inevitably, this constraint places a limit on both the breadth ofmaterial that can be covered and the pace at which instruction can progress.New topics cannot be added without omitting others. Choices must be made.Unless we design instruction to meet the needs and abilities of students,efforts to update the teaching of introductory physics will produce littleof either intellectual or motivational value.

Lillian C. McDermott
Department of Physics
Box 351560
University of Washington
Seattle, WA 98195-1560
 

Endnotes
† Researchconducted by the Physics Education Group since this paper was publishedin the American Journal of Physics has yielded additional evidencein support of the generalizations. For each of the examples given, we havedata from more than 1,000 students. The percentages of correct responseshave remained essentially the same. An up-to-date list of papers can befound at http://www.phys.washington.edu/groups/peg/pubs.html. Some of theseare listed below:
 

O’Brien Pride, T., S. Vokos, andL.C. McDermott, "The challenge of matching learning assessments to teachinggoals: An example from the work-energy and impulse-momentum theorems,"to be published in Am. J. Phys. (1998)

Steinberg, R., G. Oberem, and L.C.McDermott, "Development of a computer-based tutorial on the photoelectriceffect," Am. J. Phys. 64 (11) 1370 (1996).

Grayson, D.J. and L.C. McDermott,"Use of the computer for research on student thinking in physics," Am.J. Phys. 64 (5) 557 (1996).

McDermott, L.C., P.S. Shaffer andM. Somers, "Research as a guide for teaching introductory mechanics: Anillustration in the context of the Atwood’s machine." Am. J. Phys.62 (1) 46 (1994).

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