Knowledge structure and problem solving in physics

Educational Psychologist

ISSN: 0046-1520 (Print) 1532-6985 (Online) Journal homepage: http://www.tandfonline.com/loi/hedp20

Knowledge structure and problem solving in physics F. Reif & Joan I. Heller To cite this article: F. Reif & Joan I. Heller (1982) Knowledge structure and problem solving in physics, Educational Psychologist, 17:2, 102-127, DOI: 10.1080/00461528209529248 To link to this article: http://dx.doi.org/10.1080/00461528209529248

Published online: 01 Oct 1982.

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Educational Psychologist 1982, Vol. 17, No. 2, 102—127

Knowledge Structure and Problem Solving in Physics F. Reif and Joan I. Heller

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University of California, Berkeley This article presents a prescriptive analysis of the kinds of knowledge and procedures leading to effective human problem solving in a quantitative science such as physics. The knowledge about such a science, explicated in the case of mechanics, specifies special descriptive concepts and relations described at various levels of abstractness, is organized hierarchically, and is accompanied by explicit guidelines specifying when and how this knowledge is to be applied. General problem-solving procedures, to be used in conjunction with such domain-specific knowledge, specify how initially to describe and analyze any problem so as to facilitate its subsequent solution; how to search for a solution by methods of constraint satisfaction used together with heuristic methods for decomposing problems and exploring decisions; and how to assess whether the resulting solution is correct and reasonably optimal. The preceding model of effective human problem solving is compared with some relevant observations and with special experiments designed to test such a prescriptive model. It also suggests methods for teaching students improved scientific problem-solving skills. also revealed that the knowledge structures and problem-solving procedures of experts and novices differ in significant ways. Although studies of the thought processes of experts and novices have yielded valuable insights about problem solving, they have significant limitations. For example, it is unwise to assume that experts necessarily perform optimally. Furthermore, educational efforts must do more than merely teach students to perform as experts do. Instead, students must often be taught to use explicit procedures to accomplish tasks which experts perform almost automatically because they recognize familiar patterns as a result of years of experience. The work described in this article has, therefore, aimed to study human problem solving from a more general point of view which transcends the investigation of naturally occuring intellectual functioning. In particular, we have sought to answer the following general question: What kinds of knowledge and procedures enable human subjects to achieve good problem-solving performance in a realistic scientific domain? This question is "prescriptive" and more general • than a descriptive concern with naturalistic phenomena. Thus it may be answered without necessarily trying to simulate what actual experts do and without assuming that experts always perform optimally. Of course, a prescriptive theoretical model of

Recent years have witnessed increasing interest in understanding human cognitive processes in realistically complex domains. In particular, while earlier studies of human problem solving dealt largely with puzzles and games (Newell & Simon, 1972), more recent work has focused attention on problem solving in complex domains of practical educational or scientific interest. For example, such work has been pursued in mathematics (Greeno, 1978a, 1978b), engineering thermodynamics (Bhaskar & Simon, 1977), electronics (Brown & Burton, 1975), and physics (Chi, Feltovich, & Glaser, 1981; Chi, Glaser, & Rees, 1981; Larkin, McDermott, Simon, & Simon, 1980; Larkin & Reif, 1979; Simon & Simon, 1978). Furthermore, work in artificial intelligence has also been extended to scientific domains such as physics (Bundy, 1978; DeKleer, 1977; Novak, 1977), chemistry (Feigenbaum, 1977), and others. Such work has shown that effective problem solving in a realistic domain depends crucially on the content and structure of the knowledge about the particular domain. The studies have This work was supported in part by the National Science Foundation under grant #SED79-20592. We wish to thank Allan Collins for several useful discussions and for playing so well the role of an obliging naive "robot" solving unfamiliar physics problems under external control. The address of F. Reif is Physics Department and Group in Science and Mathematics Education; University of California; Berkeley, California 94720.

Copyright 1982 by Division 15 of the American Psychological Association, Inc.

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good performance may be suggested by naturalistic observations of expert behavior; however, it may also be suggested by purely theoretical task analyses. Correspondingly, the sole criterion of validity of such a prescriptive model is that it lead to predictably effective performance when implemented by a human subject, even if it does not mimic what experts do. . To clarify the distinction between prescriptive and descriptive points of view, imagine that a hypothetical cognitive scientist, working in the year 100 AD, had been trying to formulate a model of good performance in arithmetical problem solving. If the model had suggested the use of the modern place-value representation of numbers, it would have led to very good arithmetical performance and would thus have been an excellent prescriptive model. However, it would have been an unsatisfactory descriptive model of the behavior of experts, all of whom used Roman numerals at that time. A prescriptive perspective is prevalent in artificial intelligence where the aim is to create programs that can perform intellectual tasks effectively, without any necessary relation to human cognition. But, as pointed out elsewhere (Reif, 1979), a prescriptive approach can also be of major interest in work on human cognitive processes. Thus, it can be very useful for identifying essential knowledge required for good performance and can thereby help to make explicit expert knowledge which is often largely tacit. Furthermore, such an approach is of essential importance in any applied work aiming to improve human performance, to design educationally effective instruction, or to exploit the potentialities of systems involving person-computer interaction. In trying to specify the knowledge and procedures leading to good human performance in a scientific domain, we have focused our attention on problem-solving performance in basic college-level physics, specifically in the field of mechanics. This scientific domain is realistically complex, requires flexible problem solving based on a considerable amount of special knowledge, is often difficult for many students, and is representative of other quantitative sciences or engineering fields. On the other hand, this domain is sufficiently simple to permit an analysis of underlying cognitive processes. Furthermore, such an analysis can draw upon insights derived from

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previous observations of experts and novices in this domain (e.g., Larkin & Reif, 1979). Our work has presupposed that the human subjects, who engage in scientific problemsolving tasks, possess well-developed human capabilities, such as natural-language understanding, basic mathematical and algebraic skills, and the ability to draw and interpret diagrams. On the other hand, we do not assume that these subjects possess more sophisticated or strategic forms of knowledge required for scientific problem solving. Indeed, our main aim has been precisely to specify explicitly these more sophisticated forms of knowledge leading to good problemsolving performance. The problem-solving model discussed in the following sections encompasses some general problem-solving procedures to be used in conjunction with a knowledge base about a particular scientific domain. The procedures include procedures for initially describing and analyzing any problem so as to bring it into a form facilitating its subsequent solution, procedures for constructing the solution by methods and decision processes facilitating search, and finally procedures for assessing whether the resulting solution is correct and reasonably optimal. The content and organization of the domain-specific knowledge base are specifically designed to facilitate the implementation of these general problem-solving procedures. In the following pages we shall first discuss the knowledge base, exhibited in the specific domain of mechanics, since this will allow us to illustrate subsequent general remarks with concrete examples. Then we shall turn our attention to the general problem-solving procedures and illustrate how they are used in conjunction with the knowledge base. Finally, we shall mention briefly some experimental tests of this model and point out some of its practical educational implications. We shall strive to provide sufficient details and examples to make clear the nature of the knowledge and cognitive processes contributing to effective human problem solving. On the other hand, we shall try to avoid excessive details that might be distracting or of predominant interest to practicing physicists. Thus the discussion is intended to be understandable to non-technical readers even if they cannot fully interpret specific equations. •


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KNOWLEDGE BASE

Top Level of the Knowledge Base

Problem solving in a complex scientific domain is impossible without a knowledge base containing substantial amounts of specialized knowledge about this domain. It is important to specify not only the content of this knowledge base, but also how this knowledge is organized and symbolically represented to facilitate efficient information retrieval and problem solving. Accordingly, we shall devote this section of the article to propose a particular structure of the knowledge base for the chosen prototype domain of mechanics. This structure is not found in any conventional textbook, but is specifically designed to aid effective problem solving and resembles somewhat the knowledge possessed by experts. The knowledge base discussed in the following pages is organized so as to facilitate information retrieval by progressively narrowing the domain of search, starting from gross information and proceeding to successively more detailed information with relatively few decisions at each step. Thus the entire knowledge is subdivided into knowledge blocks, of a size conveniently small for human processing, with a few ideas in each knowledge block elaborated in subordinate knowledge blocks. The knowledge structure obtained by such successive elaborations, starting from some toplevel knowledge block, is then predominantly hierarchical. (However, it has ultimately the characteristics of a network because overlapping knowledge and cross-references connect knowledge blocks in ways other than superordinate-subordinate relation.) In particular, the top-level knowledge block describes the knowledge about the entire domain at the grossest level of description. The lower-level knowledge blocks, obtained by successive elaboration, then provide progressively more detailed descriptions of selected aspects of knowledge described at higher levels. The kinds of descriptive concepts and the symbolic representations match the level of description. Thus the higher-level knowledge blocks are described in terms of abstract or vague concepts, expressed in the form of words or pic- ' tures, useful for unifying and subsuming more detailed information. On the other hand, the lower-level knowledge blocks are described in terms of more precisely defined concepts and are predominantly expressed in the form of algebraic symbols and mathematical formalism.

The knowledge base for any domain must specify the particular entities of interest in this domain, the particular concepts ("descriptors") useful for describing these entities, the relations existing between such descriptors, and procedures for transforming such relations. In a quantitative science like physics, most important descriptors are "quantities" (i.e., descriptors characterized by numerical values) and relations between such descriptors are often expressed by mathematical equations. These essential features can be concretely exemplified in the specific domain of mechanics (i.e., the science of morion). The top level of the knowledge base for this domain describes, at a gross level, the entire knowledge about mechanics and is summarized in Figure 1. The particular entities of interest in mechanics are "particles" (i.e., objects small or simple enough to be specified by the positions of single points) and "systems" consisting of one or more particles (e.g., the planets, a string, etc.). The aim of the science of mechanics is then to describe and predict how such systems move in the course of time. This aim is furthered by introducing two different kinds of descriptors, "individual descriptors" and "interaction descriptors". An "individual descriptor" is a concept used to describe a system without attention to the interactions of its particles with each other or with other particles. Thus an individual descriptor of a particle involves only specification of this particle alone. (For example, individual descriptors of a particle i include intrinsic descriptors such as its ' 'mass' ' m\ or motion descriptors such as its "velocity" v¡.) By contrast, an "interaction descriptor" is a concept used to describe how particles or systems interact with each other. Thus an interaction descriptor of a particle requires the specification of this particle and of the other particles with which it interacts. (For example, the interaction of some particle / with some other particle/may be described by the "force" F¡¡ exerted on i by/.) There are also composite interaction descriptors (e.g., "energy") which involve combinations of simpler interaction descriptors and individual descriptors. The essential structure of the entire science of mechanics, as summarized in Figure 1, consists then of the following four types of knowledge: (a) One type of knowledge concerns the


OVERVIEW

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OF KNOWLEDGE ABOUT MECHANICS

(entities: particles and systems thereof)

to < the ball > at < the time t2> with and with < components chosen along the downward direction>." Another important constraint is generated by, "Applying to between with ." Removing unwanted parameters from some constraints is a subproblem which can be easily solved by using one or both of the following methods: (a) One can use standard algebraic techniques to combine existing constraints to generate new constraints which involve none of the unwanted parameters. As many as N unwanted parameters can thus be removed if there are (N +1) available independent constraints involving these parameters, (b) If all of the unwanted parameters cannot be removed in this way because there are too few available constraints, one can generate new constraints involving those parameters that cannot be removed. The generation of these constraints, by the methods discussed previously, paves the way for the subsequent removal of these parameters by combining constraints. Removing an unwanted form of a constraint is a subproblem which can be solved by simple algebraic techniques for modifying the form of a constraint. Such techniques (e.g., performing the same operation on both sides of an equation) are too routine to merit further discussion in the present context. It is apparent that additional decisions must be made to decide which particular subproblem to consider, which particular method to choose, and what choices to make in applying this method (particularly the complex choices needed in knowledge instantiation). Such decisions are facilitated by procedures discussed in the next paragraphs. Exploration of Decisions A decision among a few available alternatives can be made judiciously by "exploring the decision", i.e., by a procedure whereby choices are made on the basis of an assessment of anticipated consequences. Such a


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procedure requires more processing, but fewer demands on memory, than one based on stored rules prescribing choices to be made under many possible conditions. It also allows human problem solvers to make flexible choices in diverse and unfamiliar situations. To explore a decision one must be able to characterize any alternative course of action by its "utility", i.e., by a rough measure of its benefits compared to its costs. The "benefits" characterize the state resulting from the action. They are considered positive or advantageous if the resulting state is closer to the desired problem goal or leads to a simplification of the problem; they are considered negative if the opposite is the case. By contrast, the "costs" characterize the process of carrying out the action and are considered large if this process requires much effort or time. The utility, which is a rough measure of the ratio of such benefits compared to such costs, needs only be accurate enough to permit an approximate ranking of alternatives according to their estimated utility, i.e., estimated on the basis of currently available information. The estimated utility of a particular alternative may thus change as more information becomes available during the problem-solving process. (Because the utility involves a comparison of benefits and costs, a course of action offering large benefits may nevertheless have zero utility if it cannot be implemented by the problem solver. Conversely, an alternative offering only minor benefits may have a large utility if it can be implemented very easily.) On the basis of the preceding comments, the procedure for exploring a decision may be summarized by the following major steps: 1. Identification of promising alternatives: Identify, on the basis of available knowledge, one or more alternatives of comparatively large estimated utility. 2. Selection by exploration: Repeat the following until the estimated benefits of finding a more useful alternative outweigh the costs of further exploration: (a) Explore an identified alternative by predicting its major consequences and then assessing its estimated utility on the basis of this information, (b) Select, among the alternatives explored, that of largest estimated utility. 3. Revision of unsatisfactory choices: (a) Implement the selected alternative, (b) If its actual utility is unsatisfactorily small, backtrack and reconsider the decision process on the basis of the new information now available.

Note that this decision process involves progressively more refined choices made on the basis of increasingly more reliable information. Thus the first step makes preliminary choices on the basis of general information stored in the knowledge base (e.g., on the basis of the previously discussed "application guidelines" included in the knowledge base); the second step makes improved choices on the basis of information about anticipated consequences; and the third step may revise such choices on the basis of information about the actual consequences of an implemented action. Note also that the second step of the decision procedure may consider only the most obvious alternative alone if it seems sufficiently useful. Thus the decision procedure does not aim to make an optimum choice, but merely one which is good enough (i.e., which "satisfices"). Despite its seeming complexity, the decision procedure just outlined explicates to some extent processes which persons perform quite commonly in daily life. To provide an example of the exploration of decisions in physics problems, consider a typical kind of decision encountered in an attempt to generate constraints by "knowledge instantiation". In the specific case of the pendulum problem described in Figures 6 and 7, the application guidelines in the knowledge base suggest that it would be useful to apply the energy principle to the ball. However, a further choice must still be made about the times between which this principle should be applied. One alternative, suggested by the problem description in Figure 7, is to apply this principle twice — once between the times tç, and t\ (when the ball is at its lowest point) and then again between the times t\ and ¿2- Another alternative is to apply this principle during the entire time interval between the times t0 and ¿2- The first alternative would lead to the generation of an extra intermediate result (an additional constraint involving the unwanted velocity at the time ti) and would thus involve more processing to obtain a solution. Hence the second alternative has larger estimated utility and would be chosen preferentially.

Multiple Levels of Description A relation expressing a constraint can be described at various levels of detail. The most detailed description is in the form of a mathematical equation which specifies precisely what parameters are related and how they



are related. [For example, Newton's motion principle ma. = F involves three parameters (the mass wz of a particle, its acceleration a, and the force F on it by all other particles) and specifies precisely how these parameters are related.] A less detailed description might merely specify what parameters are related without specifying how they are related [e.g., Newton's principle might merely be described by: rel(*zz,a,F), i.e., as some relation, of unspecified form, between the indicated parameters.] An even less detailed description might omit mention of some of the parameters and might describe other parameters collectively in terms of more abstract or qualitative concepts. [For example, Newton's principle might thus be simply summarized as: rel(motion, interaction), i.e., as a relation between some motion parameter and some interaction parameter.] The flexibility of using multiple levels of description can be effectively exploited to facilitate decisions in problem solving. (Accordingly, it is also helpful if the knowledge base about a domain is desaibed at various levels of detail.) For example, gross qualitative descriptions are very useful when exploring decisions, in the manner discussed in the preceding section, to describe approximately the major anticipated results of contemplated actions and to estimate their utility without becoming enmeshed in complicating details. Multiple levels of description are particularly useful for planning solutions. Indeed, a problem can be solved at any level of description by using the search methods discussed in the preceding sections. Thus one can first construct a schematic problem solution at a gross level of description where decisions can be made without attention to complicating details. This solution can then be used as a plan guiding the construction of a solution at a more detailed level of description. This procedure may be repeated successively until one obtains the completely detailed solution. Planning by successive refinements can greatly facilitate search and provides advantages exploited in many fields (e.g., Hughes & Michton, 1977; Sacerdoti, 1977). The essential advantage is that any decision made at a gross ievel of description need deal with only limited information, unencumbered by excessive details, but narrows greatly the range of alternatives to be considered subsequently in decisions at a more detailed level of description. In this way the domain of search can be rapidly

narrowed down, although only few alternatives are considered at each step. Note that these essential advantages are preserved even if the early gross descriptions of the problem are vague or ambiguous. For example, the following is a grossly described solution of the pendulum problem illustrated in Figures 6 and 7: (a) Apply a motion principle to the ball at the time t2 when its interaction is known to be such that the string is still taut. The result should be a relation specifying a constraint on the motion of the ball at the time t2. (b) Apply motion principles to the ball between the times t0 and t2. The result should be a relation between the motion of the ball at the time t2 and its specified motion at prior times, (c) Combine the preceding information. [Although this schematic solution is vague, it suffices to provide a plan decomposing the solution into its two major steps (a) and (b).]

Efficacy of Methods Used Jointly with Knowledge Base The methods discussed in the preceding sections are designed to help make judicious decisions facilitating the solution of physics problems. As the following comments indicate, these methods are quite powerful if they are used jointly and in conjunction with a wellstructured knowledge base. Decisions about how to transform constraints can be readily made by using the previously discussed heuristic procedures for decomposing problems. The most difficult decisions arc those needed to generate constraints (i.e., decisions about what principle to apply, to what system, at what time, and with what description). But these complex decisions are substantially facilitated because the wellstructured knowledge base and prior problem description greatly restrict the range of alternatives to be considered. The following remarks illustrate this statement. As discussed previously, the knowledge base is hierarchically organized into knowledge categories and subcategories which are accompanied by explicit application guidelines. In particular, these guidelines suggest that the solution of a problem start by application of one of the motion principles. There arc only four such principles in the knowledge base illustrated in Table 4. Furthermore, the application guidelines accompanying these principles


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provide advice which ordinarily limits even further the,number of principles useful in a given situation. Decisions between these very few principles can then easily be made after a little exploration. Once a selected motion principle has been applied, the application guidelines in the knowledge base provide guidelines about a whole set of other decisions needed to elaborate this motion principle. Indeed, as already mentioned, they suggest that motion descriptors in the principle should be related to simpler motion descriptors (by using the knowledge about motion descriptors in Table 1). They also suggest that interaction descriptors should be related to simpler interaction descriptors (by using the knowledge about interaction descriptors in Table 2) and that these should then, in turn, be related to individual descriptors (by using the knowledge about interaction laws in Table 3). (For example, as discussed more fully in the Appendix, one step in the solution of the pendulum problem involves application of Newton's motion principle ma = f to the ball at the time t2. The acceleration a is then elaborated by relating it to the velocity of the ball at this time. Similarly the total force F on the ball is elaborated by relating it to the gravitational force on the ball by the earth and the force on the ball by the string; each of these forces is then, in turn, related to the individual properties of the ball and of the string.) The initial description of a problem helps to identify what particular systems and times should be considered in applying principles from thç knowledge base. (For example, the descriptions of the pendulum problem in Figures 6 and 7 identify the ball as the major system of interest and identify the particularly interesting times t0, tlt and t2 during its motion). Furthermore, the initial theoretical problem description, by deliberately «describing a situation in terms of knowledge provided by the knowledge base, explicates much of the knowledge subsequently needed to apply principles from the knowledge base and also helps to make decisions about what descriptions to use when these principles are applied. (For example, the description of the pendulum problem in Figure 7 shows that the acceleration of the ball, and all forces on it, arc directed downward it the time t2. It is then obviously best to apply the motion principle /wa = F to the ball by describing the principle in terms of components along the downward direction.)

The preceding comments, which indicate how solution methods and the knowledge base interact to facilitate decisions in problem solving, are exemplified further in the Appendix which outlines the detailed solution of the pendulum problem described in Figures 6 and 7. Assessment of Solution After a solution has been constructed, it is important to assess how satisfactory it actually is. We merely mention briefly the following increasingly stringent criteria which must be satisfied to ensure that a solution is free from obvious defects. Any constructed solution is tested against these criteria. Failure to satisfy any one of these criteria requires that the solution be revised until all these criteria are satisfied. Clear interpretation: The answers to a problem should only involve clearly defined parameters and should specify all parameters needed to avoid ambiguity (e.g., units of measurement, magnitudes and directions of vectors, reference frame used to specify motion descriptors, etc.). Completeness: (a) All questions should be answered, (b) Every answer should be expressed entirely in terms of known quantities, (c) Answers should be given for all types of solutions possible for different values of the parameters in the problem. Internal consistency: (a) All steps in the argument should be free of logical errors, (b) Answers should be internally consistent (e.g., the units of quantities in an answer should be consistent). External consistency: (a) Answers should agree with qualitatively expected functional relationships between parameters in the problem, (b) Answers should agree with those obtained for extreme values and other special values of the parameters in the problem. Optimality: (a) The solution of the problem should be as simple as possible, (b) The answers should be expressed in the simplest and/or most easily interpretable way. DISCUSSION AND IMPLICATIONS Scope and Limitations In the preceding pages we sought to specify important cognitive mechanisms facilitating effective human problem solving in the



domain of physics. In particular, we first discussed the content, organization, and description of the knowledge base facilitating such scientific problem solving. Then we discussed procedures used to describe and analyze a problem initially, to search for its solution, and to assess this solution. Finally, we pointed out that effective problem solving depends crucially on the joint application of such a knowledge base used in conjunction with such procedures. Since our main interest has been in human scientific problem solving, our discussion has explicitly presupposed basic informationprocessing capabilities characterizing human subjects. However, some of the ideas discussed are probably also germane to scientific problem solving by computers programmed by the methods of artificial intelligence. Although our discussion has dealt predominantly with the domain of basic mechanics, it could readily be extended to other fields of physics and even to other quantitative or engineering fields. The particular content of the knowledge base would then, of course, be different. But its general characteristics, (e.g., its hierarchical organization or its inclusion of explicit application guidelines) would be similar. Furthermore, the problem-solving procedures discussed by us are quite generally applicable, although they need to be used in conjunction with the particular knowledge base for the scientific domain of interest. Space limitations did not permit us to elaborate or exemplify several issues in greater detail. Moreover, we omitted entirely the examination of some important topics. One such topic concerns the knowledge of individual scientific definitions and principles, knowledge which we explicitly assumed to preexist in the knowledge base. Such knowledge is clearly a necessary, although not sufficient, prerequisite for scientific problem solving. Furthermore, the underlying knowledge required to interpret and use any particular definition or principle is considerable and merits detailed analysis (Reif, Note 1). Deficiencies in such knowledge lead also to many of the commonly observed errors and misconceptions exhibited by students. A second omitted topic concerns humanly useful 'bookkeeping procedures" for managing effectively complex information of the kind encountered during problem solving. Although such procedures are fairly routine, they are

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often not in the repertoire of novice students and their absence can undermine even the most sophisticated problem-solving strategies. Relevant Observations and Experiments The primary focus of the preceding pages has been theoretical, concerned with outlining a prescriptive model of effective human problem solving in a scientific domain such as physics. In the following paragraphs we mention briefly how this model is related to past observations and descriptive studies of experts and novices. Then we point out how such a prescriptive model can be validated by more specific experiments. In some respects the model outlined in the preceding pages simulates the problem-solving behavior of expert physicists. For instance, observations by Larkin and Reif (1979) and ourselves indicate that experts rapidly redescribe problems presented to them, often use qualitative arguments to plan solutions before elaborating them in greater mathematical detail, and make many decisions by first exploring their consequences. Furthermore, the underlying knowledge of such experts appears to be tightly structured in hierarchical fashion. By contrast, novice students commonly encounter difficulties because they fail to describe problems adequately. They usually do little prior planning or qualitative description. Instead of proceeding by successive refinements, they try to assemble solutions by stringing together miscellaneous mathematical formulas from their repertoire. Furthermore, their underlying knowledge consists often largely of a loosely connected collection of such formulas. However, as pointed out at the beginning of this article, our theoretical considerations have been intended to be prescriptive rather than descriptive and have thus not primarily aimed at simulating the problem-solving behavior of actual experts. Accordingly, we have specified explicit rules and procedures to accomplish tasks which experts may actually carry out by using tacit knowledge or pattern recognition derived from long experience. For example, while we have tried to explicate detailed procedures for generating a useful initial description of a problem, experts ordinarily redescribe a problem in theoretical terms very rapidly and without conscious effort. While we have


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formulated explicit application guidelines for important principles, experts often choose suitable principles automatically without explicit awareness of the tacit knowledge guiding their choice. Observations of experts' behavior are thus only peripherally relevant to a prescriptive model of effective human problem solving. Direct tests of the validity of such a model can be provided by experiments which deliberately induce human subjects to act in accordance with the model and which then observe whether the resulting performance has the predicted charateristics and is effective. For example, one can let an individual subject work problems under carefully controlled conditions where the subject implements "on-line" external control directions which specify step-bystep how to describe a problem in accordance with the description procedures outlined in this article. Indeed, we have recently performed such experiments (Reif & Heller, Note 2; Heller and Reif, Note 3) which showed that subjects, guided in this way according to the model, generate excellent descriptions which lead subsequently to good problem solutions (while subjects, working without benefit of the model, generate descriptions which are often incomplete or wrong — and thus fail to solve problems correctly.) Similar experiments can be performed to induce a person to search for problem solutions in accordance with the methods outlined in this article. It is also possible to deliberately induce a person to utilize a knowledge base structured in a particular way. Indeed, Eylon (1979) and Eylon and Reif (Note 4) have performed such experiments to study forms of knowledge organization facilitating human task performance in scientific domains. In all such experiments, where a person is induced to act in accordance with a prescriptive model of good performance, the person's resulting performance can be studied in detail, and protocol data can be collected on the underlying thought processes verbalized by the person. In this way one can assess whether various selected features of the model are sufficient to lead to good performance. One can also ascertain whether such features are necessary by comparing the model with alternative models where these features have been modified or omitted. Such experiments, designed to test a prescriptive model of good problem-solving performance, should be carefully distinguished

from experiments designed to test a model of effective problem-solving instruction. A model of effective instruction must certainly be based on a model of good performance which specifies how the desired ultimate student performance is to be achieved. But it must also incorporate a model describing the student's initial functioning and a model specifying the instructional transformation process. Furthermore, experiments testing a model of effective instruction must ensure that students internalize control directions and other knowledge which can be explicitly externalized in studies designed to test models of good performance.

Educational Implications The most common method of teaching scientific problem solving is to provide students with sufficient examples and practice. This method is neither very efficient nor effective. Indeed, as the previously mentioned observations of novices indicate, most students' problem-solving skills are quite primitive and improve only slowly. Furthermore, many students find the problem solving required in college-level physics courses difficult or even unmanageable. These observations are not too surprising. As our discussion has shown, the cognitive mechanisms needed for effective scientific problem solving are complex and thus not easily learned from mere examples and practice. A potentially more effective instructional method would teach problem-solving skills explicitly on the basis of insights derived from a model of effective problem solving. Our discussion in the preceding pages is highly relevant to such an instructional approach because it identifies essential components contributing to effective problem solving. In particular, it suggests that each of these separate components should be taught explicitly and that all these components should then be integrated to achieve effective problem solving. For example, our discussion suggests that one teach separately how to generate good basic and theoretical descriptions of a problem; how to analyze a problem qualitatively before its actual solution; how to search for a problem solution by decomposing the problem systematically and exploring relevant decisions; and how to assess the merits of the resulting solution. Similarly, our discussion suggests, in keeping with suggestions made by others



(Karplus, 1969, 1981), that considerable attention be paid to the organization of the knowledge acquired by students; that students be taught to structure their knowledge in hierarchical form; and that such knowledge be accompanied by explicit application guidelines. Our discussion also indicates that some common teaching practices may be dysfunctional and hinder the development of students' problem-solving skills. Two examples may suffice: (a) In the quantitative sciences, a desire for precision often impels instructors and textbooks to overemphasize mathematical formalism at the expense of more qualitative modes of description. As a result, students may even come to regard such qualitative descriptions as scientifically illegitimate. Nothing could be further from the truth! As we pointed out, seemingly vague verbal and pictorial descriptions help greatly in the search and planning of solutions; they are also commonly used by experts. Hence one needs to teach both qualitative and quantitative descriptions, and how to use them jointly in procedures of progressive refinement, (b) Too little attention is commonly paid to the organization of the knowledge acquired by students. For example, knowledge is usually presented sequentially, locally organized by topics; but students are given little help to integrate their accumulating knowledge into a coherent structure facilitating flexible use. Furthermore, scientific arguments or illustrative problem solutions axe commonly presented in purely sequential form, rather than in hierarchically structured ways better suited to facilitate the flexible use of such knowledge (Eylon & Reif, Note 4).

application guidelines in Table 4 suggest that the simplest motion principle, Newton's principle mz = F, is the only one of obvious utility for relating motion and interaction at this time. The theoretical description of Figure 7 also indicates that all forces on the ball at the time t2 are directed downward; hence one chooses to describe the principle mz = F in terms of its vector components along the downward direction. Thus one generates the constraint ma

2 = Fdown 0-) where m is the mass of the ball, a2 is the magnitude of its acceleration at the time t2, and F¿ovia is the downward component of the total force on the ball at this time. Here the interaction descriptor Fdown is an unwanted quantity which may be removed by generating another constraint about it. To achieve this aim, one uses the application guideline which suggests that any interaction descriptor be elaborated by first using the knowledge about interaction descriptors (Table 2) and then about interaction laws (Table 3). In the present case the knowledge of Table 2 (the "superposition principle") leads to Fdown = Fg + F s . The knowledge of Table 3 has already been explicated in the theoretical problem description of Figure 7. It implies that the gravitational force has a magnitude F g = mg (where g is the magnitude of the gravitational acceleration near the earth) and that the force exerted by the string has a magnitude F s > 0 (since the string is taut). Hence Fdown > mg so that the elaborated constraint (1) becomes g-

Appendix Solution of the Pendulum Problem We outline here the solution of the pendulum problem, previously described in Figures 6 and 7, to illustrate more fully the major decision processes involved in its solution. The application guidelines in the knowledge base suggest that the generation of constraints should ordinarily start with application of a motion principle. Furthermore, this particular problem specifies a special condition, the tautness of the string, at the time t2. Hence one can usefully begin by applying a motion principle to the ball at this time. The

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(2)

Here the motion descriptor a2 is again an unwanted quantity which may be removed by generating another constraint about it. To achieve this aim, one uses the application guideline which suggests that any motion descriptor be elaborated by using the knowledge about motion descriptors (Table 1). In the present case of circular motion, this table contains the relation ac = v2lr. Applied to the ball at the time t2, this relation yields the constraint a2 = v2V (L.D)

(3)

where v2 is the magnitude of the velocity of the ball at the time t2. Here v2 is again an unwanted quantity which may be removed by generating another


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constraint about it. None of the information about the ball's rhotion before the time t2 has yet been used. In accordance with the previously discussed application guidelines, one again tries to generate such a constraint by applying a motion principle to the ball before t2. One merely wants to relate the unwanted speed v2 to other speeds and positions, without being explicitly interested in any elapsed time. Thus the output information in Table 4 suggests that the energy principle A E = WKS is the motion principle likely to be most useful. An exploration of alternatives, already discussed as an example in the text, leads to the decision that this principle is most usefully applied to the entire time interval between t0 and t2 (rather than between t0 and t\, and then between t\ and t2). Following again the application guidelines, the motion and interaction descriptors in this energy principle are now systematically elaborated by using the knowledge in Tables 1 through 3. The knowledge about interaction descriptors (Table 2) implies that A E = A AT + U.U. The knowledge about motion descriptors (Table 1) implies that A / C = Vim v22. The knowledge of interaction laws (Table 3) implies that the change in the ball's gravitational potential energy A U = mg (L-2D) and that the residual work done by the string T^tcs — 0. Thus the elaborated energy principle A £ = WKS yields the constraint Vi mv¿

+ mg(L-2D) = 0.

(4)

The two unwanted quantities a2 and v2 can now be easily removed by using simple algebra to combine the contraints (2), (3), and (4). The result is Z?>(3/5) L. In other words, the answer to the problem is that the minimum distance between the hook and the peg must be (3/5) L. Reference Notes 1. Reif, F. Making scientific concepts and principles effectively usable: Requisite knowledge and teaching implications. Unpublished manuscript, 1982. Available from F. Reif, University of California, Berkely. 2. Reif, F., & Heller, J. I. Cognitive mechanisms facilitating human problem solving in physics: Formutation and assessment of a prescriptive model. Paper presented at the meeting of the American Educational Research Association, New York, March 1982. 3. Heller, J. I., & Reif, F. Cognitive mechanisms facilitating human problem solving in physics: Empirical validation of a prescriptive model. Paper presented at the meeting of the American Educational Research Association, New York, March 1982.

4. Eylon, B., & Reif, F. Effects of internal knowledge organization on task performance. Paper presented at the meeting of the American Educational Research Association, San Francisco, April 1979.

References Bhaskar, R., & Simon, H. A. Problem solving in semantically rich domains: An example from engineering thermodynamics. Cognitive Science, 1977, 1, 193-215. Brown, J. S., & Burton, R. Multiple representations of knowledge for tutorial reasoning. In D. G. Bobrow & A. Collins (Eds.), Representation and understanding, New York: Academic Press, 1975. Bundy, A. Will it reach the top? Predictions in the mechanics world. Artificial Intelligence, 1978, 10, 129-146. Chi, M. T. H., Feltovich, P. J., & Glaser, R. Categorization and representation of physics problems by experts and novices. Cognitive Science, 1981, 5, 121-152. Chi, M. T. H., Glaser, R., & Rees, E. Expertise in problem solving. In R. Sternberg (Ed.), Advances in the psychology of human intelligence. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1981. DcKleer, J. Multiple representations of knowledge in a mechanics problem solver. Proceedings of the fifth International Joint Conference on Artificial Intelligence. Cambridge, Mass.: MIT Press, 1977. Eylon, B. Effects of knowledge organization on task performance. Unpublished doctoral dissertation, University of California, Berkeley, 1979. Feigenbaum, E. A. The art of artificial intelligence: Themes and case studies of knowledge engineering. Proceedings of the Fifth International Conference on Artificial Intelligence, Cambridge, Mass.: MIT Press, 1977, pp. 1014-1029. Greeno, J. G. A study of problem solving. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 1). Hillsdale, N.J.: Lawrence Erlbaum Associates, 1978.

(a) Greeno, J. G. Understanding and procedural knowledge in mathematics instruction. Educational Psychologist, 1978, 12, 262-283. (b) Hughes, J. K., & Michton.J. I. A structured approach to programming. Englewood Cliffs, N.J.: Prentice-Hall, 1977. Karplus, R. Introductory physics: A model approach. New York: Benjamin, 1969. Karplus, R. Educational aspects of the structure of physics. American Journal of Physics, 1981, 49, 238-241. Larkin, J. H., & Reif, F. Understanding and teaching problem solving in physics. European Journal of Science Education, 1979, 1, 191-203. Larkin, J., McDermott, J., Simon, D. P., & Simon, H. A. Expert and novice performance in solving physics problems. Science, 1980, 208, 1335-1342. Newell, A., & Simon, H. A. Human problem solving. Englewood Cliffs, N.J.: Prentice-Hall, 1972. Novak, G. Representation of knowledge in a program for solving physics problems. Proceedings of the Fifth International Joint Conference on Artificial Intelligence. Cambridge, Mass.: MIT Press, 1977. Reif, F. Theoretical and educational concerns with problem solving: Bridging the gaps with human cognitive engineering. In D. T. Tuma and F. Reif (Eds.), Problem solving and education: Issues in teaching and research. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1979.



Sacerdoti, E. A structure for plans and behavior. New York: Elsevier North-Hollard, 1977. Simon, D. P., & Simon, H. A. Individual differences in solving physics problems. In R. D. Siegler (Ed.), Children's thinking: What develops? Hillsdale, N.J.: Lawrence Erlbaum Associates, 1978.

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