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cOGNITIVE SCIENCE 5, 121-
Categorization and Representation
Physics Problems by Experts
and N0vices*
of
MICHELENE T. H. CHI
PAUL J. FELTOVICH
ROBERT GLASER
University of Pittsburgh
The representation of physics problems in relation to the organization of physics knowledge is investigated in experts and novices. Four experiments examine {a~ the existence of problem categories as a basis for representation; (b) differences in the categories used by experts and novices; (c) differences in the knowledge associated with the categories; and (d) features in the problems that contribute to problem categorization and representation. Resultsfrom sorting tasks and pro- tocols reveal that experts and novices begin their problem representations with specifiably different problem categories, and completion of the representations depends on the knowledge associated with the categories. For, the experts initially abstract physics principles to approach and solve a problem representa- tion, whereas novices base their representation and approaches on the problem's literal features.
CATEGORIZATION AND REPRESENTATION OF PHYSICS PROBLEMS BY EXPERTS AND NOVICES This paper presents studies designed to examine differences in the ways expert and novice problem solvers represent physics problems and to investigate impli-
cations of these differences for problem solution. A problem representation is a
*This research program, conducted at the Learning Research and Development Center, is sopported in part by contract No. N00014-78-C-0375,NR 157-421 of the Office of Naval Research, and in part by the National Institute of Education. Portions of this paper were presented by the f'ust author at the meetingof the AmericanEducationalResearch Association,San Francisco, April 1979, and at the meetingof the PsychonomicSociety, Phoenix, November 1979. The authors are grateful for the help of Andrew Judkis, Ted Rees, and Christopher Roth, for comments, data collection, analysis, and editing. We particularlyappreciate the generosityof the physicsprofessors and graduate students who contributed their time, especially Ned S. VanderVen. Jill Larkin deserves special thanks for her contributionsand insightfulcommentsto Study Four. Reprint requests should be sent to MicheleneChi, LearningResearch and DevelopmentCenter, Universityof Pittsburgh, Pittsburgh, Pa. 15260. 121
122 CHI, FELTOVICH AND GLASER
cognitive structure corresponding to a problem, constructed by a solver on the basis of his domain-related knowledge and its organization. A representation can take a variety of forms. Greeno (1977), for example, has proposed the representa- tion of a problem as a constructed semantic network containing various compo- nents. Some of these correspond closely with the problem as stated, including the initial state (i.e., the "givens"), the. desired goal, and the legal problem-solving operators (Newell & Simon, 1972). In addition, a representation can contain embellishments, inferences, and abstractions (Heller & Greeno, 1979). Since embellishment is one way of judging a solver's "understanding" of a problem (Greeno, 1977), it is possible that with increasing experience in a domain, the representation becomes more enriched. The research described here explores the changes in problem representation that emerge as a result of developing subject-matter expertise. It is well known by now that the quality of a problem representation influences the ease with which a problem can be solved (Hayes & Simon, 1976; Newell & Simon, 1972). In physics, Simon and Simon (1978) have attributed the expert's "physical intuition" to the quality of the problem representation. The current consensus is that the expert's representation is superior because it con- tains a great deal of "qualitative" knowledge. De Kleer (1977), for example, has introduced both "quantitative" and "qualitative" components in the expert's representation of a physics problem where the qualitative component includes nonmathematical semantic descriptions of physical objects and their interactions. Novak's (1977) program ISSAC also suggested some characteristics of qualita- tive representation. In this program, physical objects from a problem statement are represented not literally, but rather, as abstract object categories---canonical object frames--each of which serves an equivalent physics role (e.g., pivot,
lever, or point mass). The canonical objectframe is a knowledge structure that
augments the information about an object stated in a problem with associated information from the knowledge base. In his later work, Novak has proposed the inclusion of problem types in categorization by types (Novak & Araya, 1980). Categorization of a problem as a type would cue associated information in the knowledge base. Similarly, Reif (1979) has proposed a problem-solving model in which an initial step is a representation or "redescription of any problem in terms of concepts provided by the knowledge base" (p. 1). This knowledge base is arranged around "problem schemata," each of which contains information necessary to solve a specific category of problems. The hypothesis guiding the present research is that the representation is constructed in the context of the knowledge available for a particular type of problem. The knowledge useful for a particular problem is indexed when a given physics problem is categorized as a specific type. Thus, expert-novice dif- ferences may be related to poorly formed, qualitatively different, or nonexistent categories in the novice representation. In general, this hypothesis is consistent with the "perceptual chunking" hypothesis for experts (e.g., Chase & Simon,
124 CHI, FELTOVlCHAND GLASER
each chapter, and these.were individually typed on 3 x 5 cards. Instructions were to sort the 24 problems into groups based on similarities of solution. The subjects were not allowed to use pencil and paper and, thus, could not actually solve the problems in order to sort them. As a test of consistency, subjects were asked to re-sort the problems after the first trial. Following this, they were asked to explain the reasons for their groupings. The time taken to sort on each trial was also measured.
Analysis of Gross Quantitative Results
No gross quantitative differences between the sorts produced by the two skill groups were observed. There were no differences in the number of categories produced by each group (8.4 for the experts and 8.6 for the novices), and the four largest categories produced by each subject captured the majority of the problems (80 percent for the experts and 74 percent for the novices). Likewise, experts and novices were equally able to achieve a stable sort within the two trials, that is, their second sort matched their first sort very closely. This suggests that their sorting pattern was not ad hoc, but rather, was based on some meaningful representation. There were, however, some differences in the amount of time it took experts and novices to sort the problems. In fact, experts took longer ( 18 minutes or 45 seconds per problem, on the average) to sort the problems in the first trial than novices (12 minutes or 30 seconds). Both groups were relatively fast at sorting the second trial (4.6 minutes for the experts and 5.5 minutes for the novices). The speed with which the problems were sorted on the second trial (about 12 seconds per problem) suggests that subjects probably did not have to go through the entire process of "understanding" each problem again. Since the problems were all categorized after the first trial, the subjects probably needed only to identify the cues that elicited category membership. In general, these quantitative data suggest that both experts and novices were able to categorize problems into groups in a meaningful way. Other than the difference in the time taken to sort on the first trial, there was little difference between skill groups. The critical question then becomes: what are the bases on which experts and novices categorize these problems?
Qualitative Analysis of the Categories
Analyses of Four Pairs of Problems. A cluster analysis (Diameter
method) was performed on the problems grouped together by the experts and those by the novices. Such an analysis shows the degree to which subjects of each skill group agree that certain problems belong to the same group. One way to interpret the cluster analysis is to examine only those problems that were grouped together with the highest degree of agreement among subjects.
CATEGORIZATION AND REPR$ENTATION^125
Our initial analysis centered on four pairs of problems. Figures 1 and 2 contain the diagrams of pairs of problems that were grouped together by the novices and the experts, respectively. These diagrams can be drawn to depict the physical situations described in the problem statements, and are sometimes given along with a problem statement (although no diagrams were given to the subjects in our studies). All eight novices grouped the top pair (Figure 1) together, and seven of the eight novices grouped the bottom pair. Both pairs of problems (Figure 2) were grouped together by six of the eight experts. Examination of the novice pairs (Figure 1) reveals certain similarities in the surface structures of the problems. By "surface structures," we mean: (a) the objects referred to in the problem (e.g., a spring, an inclined plane); (b) the literal physics terms mentioned in the problem (e.g., friction, center of mass); or (c) the physical configuration described in the problem (i.e., relations among physical objects such as a block on an inclined plr.ne). Each pair of problems in Figure 1 contains the same object components and configurations----circular disks in the upper pair, blocks on an inclined plane in the lower pair. The suggestion that novices categorize by surface structure can be con- firmed by examining subjects' verbal descriptions of their categories. (Samples are given in the figures.) Basically, according to their explanations, the top pair of problems involves "rotational things" and the bottom two problems "blocks on inclined planes." To reiterate, the novices' use of surface features may involve either keywords given in the problem statement or abstracted visual configurations, that is, the presence of identical keywords (such as friction) is one criterion by which novices group problems as similar. Yet, novices were also capable of going beyond the word level to classify by types of physical objects. For example, "merry-go-round'~ and "rotating disk" are classified as the same object, as is the case for the top pair of problems in Figure 1. For experts, surface features do not seem to be the bases for categorization. There is neither great similarity in the keywords used in the problem statements, nor visual similarity apparent in the diagrams depictable from each pair of prob- lems shown in Figure 2. Nor is the superficial appearance of the equations that can be used on these problems the same. Only a physicist can detect the similar- ity underlying the expert's categorization. It appears that the experts classify according to the major physics principle governing the solution of each problem. The top pair of problems in Figure 2 can be solved by applying the Law of Conservation of Energy; the bottom pair is better solved by applying Newton's Second Law (F = MA). The verbal justification of the expert subjects confirms this analysis. If "deep structure" is defined as the underlying physics law applicable to a problem; then, clearly, this deep structure is the basis by which experts group the problems.
Analysis of Categories. Further insight into the ways subjects categorize problems is given by the descriptions subjects gave for the categories they
CATEGORIZATION AND REPRSENTATION^127
Diagrams Depicted from Problems Catergorized by Experts within the Same Groups
Problem 6 (21) K = 200 n t / m .6 m
.15 m I! equilibrium
Problem 7 (35)
l-'xperts" I:'xplanotions fi,r Their Similarity Groupings
Expert 2: "'Conservation of Energy '" Expert 3: "Work-A)lergy Theorem. They are all straight-forward problems." Expert 4: "These can be done from energy considerations. Either you should know the Princip/e of Conservation o1" Energy. o r w o r k is l o s t somewhere."
Problem 5 (39)
Problem 12 (23)
,l mg Mg
T Fp = Kv
m g
Expert 2: "These can be solved by Newlot~'s Second Law" Expert 3: " F = m a ; Newton'S Second Law" Expert 4: "Largely use F = ma; Newttm's Second Law"
Figure 2. Diagrams depicted from pairs of problems categorized by experts as similar and samples of three experts' explanations for their similarity are provided. Problem numbers given represent, chapter, followed by problem number from Hailiday and Resnick (1974).
128 CHI, FELTOVICH AND GLASER
created. Tables 1 and 2 show the category descriptions (Column 1) used by more than one expert or nov'ice. These category labels apply to all problems within each of their sorted piles. ~ Column 2 shows the number of subjects who used the category label. Column 3 shows the average size of the category among subjects who used it. And, Column 4 gives the total number of problems (out of 192, 24 problems for each of 8 subjects) according to category. TABLE 1 Expert Categories
Category Labels
Number of Subjects Average Size Number of Problems Using Category Labels of Category Accounted for (N1 = 8) (N2 = 24) (N1 x N2) Second law 6 6.0 36 Energy principles (Conservation of Energy considerations, Work-Energy Thearem)t 6 5.5 33 *Momentum principles ( Conservation of Momentum, Conservation of Linear Momentum, momentum considerations)t 6 5.0 30 *Angular motion (angular speed, rotational motion, rotational kinematics, rotational dynamics)t 6 3.0 18 Circular motion 5 1.6 8 *Center of mass (center of gravity)t 5 1.4 7 Statics 4 1.0 4 Conservation of Angular Momentum 2 1.5 3 *Work (work and kinetic energy, work and power)t 2 1.5 3 Linear kinematics (kinematics)t 2 1.5 3 Vectors 2 1.0 2 *Springs (spring and potential energy, spring and force)t 2 1.0 2 *Indicates the categories used by both novices and experts. tWhen multiple descriptors across subjects were treated as equivalent, these are given in pa- refltheses.
tFor example, if a subject said of a problem group: "These all involve inclined planes, some with a frictional surface, some frictionless," the label "inclined planes" was counted since it applied to all problems in the set.
130 CHI, FELTOVlCH A N D^ GLASER
for the experts, 43 percent for the novices)2 into four categories, there is a slight difference in the distribution of the problems across Categories, which may suggest greater variability in novices' classification. Three major categories ac- counted for a sizable number (33 on the average) of experts' problems, whereas only one major category accounted for a large number (39) of novices' problems. This again suggests that experts are able to " s e e " the underlying similarities in a great number of problems, whereas the novices " s e e " a variety of problems that they consider to be dissimilar because the surface features are different.
Study Two: Sorting Problems with Surface Similarity
The objective of this study was to test our interpretations of Study One: experts categorize problems by laws of physics, and novices by surface features. A new
No. 11 (Force Problem) A man of mass M 1 lowers himself to the ground from a height X by holding onto a rope passed over a massless frictionless pulley and attached to another block of mass M 2. The mass of the man is greater than the mass of the block. What is the tension on the rope?
X
No. 18 (Energy Problem) A man of mass M 1 lowers himself to the ground from a height X by holding onto a rope passed over a massless frictionless pulley and attached to another block of mass M2. The mass of the man is greater than the mass of the block. With what speed does the man hit the ground?
E
.. - _. _ _ _. _ ~
X m Figure 3. Examples of problem types. ~The percentages here do not correspond to those mentioned on page 5. Those were based on the largest sorting piles given by each subject, regardless of their contents or what they were labeled. Percentages here (Tables 1 and 2) are based on the sizes of specifically labeled categories when they were used by subjects.
CATEGORIZATION AND REPRSENTATION 131
set of 20 problems was constructed in which surface features were roughly crossed with applicable physics laws. Table 3 shows the problem numbers and the dimensions on which these problems were varied .3 The left column indicates the major objects described in a problem. The three right headings are basic laws that can be used to solve problems. Figure 3 shows an example of a pair of problems that contain the same surface structure but different deep structure. In fact, they are identical, except for the question asked. Our prediction was that novices would group together problems that have the same surface structure, regardless of the deep structure, and experts would group together those prob- lems with similar deep structures, regardless of the surface structure. Individuals of intermediate competence should exhibit some characteristics of each.
TABLE 3 Problem Categories Principles Momentum Surface Structure Forces Energy (L/near or Angular) Pulley with hanging blocks 20I" 11 19t 14" 3"1" Spring 7 18 16 1 17+ 9 6+ Inclined plane 14" 3"1~ 5 Rotational 15 2 13 Single hanging block 12 Block on block 8 Collisions (Bullet-"block" or Block-block)
4 6 + 10+ *Problems with more than one salient surface feature. Listed multiply by feature. tProblems that could be solved using either of two principles, energy or force. +Two-step problems, momentum plus energy.
3The problems were chosen from texts or constructed (to satisfy the a priori classification scheme) by AndrewJudkis, an assistant in the project who was a senior electrical engineering major with substantial experience in physics. Clearly, some problems could be solved using approaches based on either of two principles, Force and Energy, and in fact Judkis solved them both ways. In these cases, the problem is listed under the principle he judged to yield the simplestor most elegant solution but is marked with a cross. Also, some problems were two-step problems involving both momentumand energy. These are listed under the principle that seemed most important(in this case, momentumconservation)and are marked with a " +. " These two-step problems are not designated explicitly as involving two principles. Some problems involve more than one potential physical configuration, e.g., "a pulley attached to an incline." These are marked with a single asterisk and listed multiply under alternativefeatures.
CATEGORIZATION AND REPRSENTATION 133
Hence, this subject's categorization can serve as a validation for our prior analysis of problem types (Table 3). Only one problem, 9, is sorted according to a principle different from our choice.
TABLE 6 Problem Categories and Explanations for Expert V. V. Group 1: 2, 13 Group 2: 18 Group 3~ 1, 4 Group 4: 19, 5, 20, 16, 7 Group 5: 12, 15, 9", 11, 8, 3, 14 Group 6." 6, 10, 17
"Conservation of Angular Momentum" "Newton's Third Law" "Conservation of Linear Momentum" "Conservation of Energy" "Application of equations of motion" (F = MA) "Two-step problems: Conservation of Linear Momentum plus an energy calculation of some sort" *Problem discrepant with our prior principles analysis.
What would an individual of intermediate competence do? Table 7 shows the groupings of an advanced novice (a fourth-year undergraduate physics major). His representations of the problems are characterized by the underlying principles in an interesting way. These principles are qualified and constrained by the surface components included in the problems. For example, instead of classifying all the force problems together (Groups 4, 6, and 7), as did the expert, he explicitly separated them according to surface entities of the problems. How- ever, although he did not strictly group problems by physics laws, neither did he uniformly group them according to surface features. For instance, Groups 3 and 6 were separated even though they both involved springs. In addition, his group- ings of principle were substantially discrepant with our prior analysis and that of the physics professor (Expert V.V. Table 6).
TABLE 7 Problem Categories and Explanations for Advanced Novice M. H. Group 1: 14, 20 Group 2~ 1, 4, 6, 10, 12" Group 3: 9, 13", 17, 18" Group 4~ 19, 11
Group 5: 2, 15" Group 6: 7", 16" Group 7: 8, 5", 3
"Pulley" "Conservation of Momentum" (collision) "Conservation of Energy" (springs) "Force problems which involve a massless pulley" (pulley) "Conservation of Angular Momentum" (rotation) "Force problems that involve springs" (spring) "Force problems" (inclined plane) Italic numbers mean that these problems share a similar surface feature, which is indicated in the parentheses, if the feature is nat explicitly stated by the sub~t. *Problems discrepant with our prior principles analysis.
134 CHI, FELTOVICH AND GLASER
To summarize the second study, we were able to replicate the initial find- ing that experts categorize physics problems by the underlying physics princi- ples, a kind of "deep structure," whereas novices categorize problems by the surface structure of the problem. Furthermore, with learning, advanced novices begin to categorize problems by the principles with gradual release from depen- dence on the physical characteristics of the problems, although their groupings are still constrained by surface features.
Discussion of the Nature of the Representation The results of the first two studies clearly indicate that the categories into which experts and novices sort problems are qualitatively different. However, neither
group is classifying solely on the basis of the literal description of the problem
statement. Both are able to read and gain some understanding of the problem, that is, to construct a somewhat enriched internal representation of it. What is the relation between categorization and a subject's representation of problems? There are at least two plausible interpretations. One, that after the reading of a problem statement, a representation is formed, and based on that representation, the problem is categorized. The taxonomy of representations proposed by McDermott and Larkin (1978) offers a plausible interpretation for the present results. These authors have proposed that the problem solver pro- gresses through four stages of representations as s/he solves a problem. The first stage is a literal representation of the problem statement (containing relevant keywords) and the fourth stage is the algebraic representation that results once equations are produced. The middle two are the most important. The second stage ("naive") representation contains the literal objects and their spatial rela- tionships as stated in the problem and is often accompanied by a sketch of the situation (Larkin, 1980)4 Such a representation and the accompanying sketch is "naive" because it can be formed by a person who is relatively ignorant of the domain of physics. The third stage ("scientific") representation contains the idealized objects and physical concepts, such as forces, momenta, and energies, which are necessary to generate the equations of the algebraic representation. This stage is related to the solution method. A plausible interpretation based on this framework is to postulate that novices' categorization is based on the con- struction of "naive" representations, with some limited elements of a "scien- tific" representation. Experts, on the other hand, may have constructed a more "scientific" representation, and based their categorizations on the s~milarities at this third level of representation. Such an interpretation would be"consistent with- the timing data of Study One: it could explain why experts actually took longer initially to classify the problems. They had to process the problems more
4In the McDermott and Larkin (1978) paper, they referred to the second stage of representa- tion as the accompanying or produced diagram and to the third stage as the abstracted free body diagram. We took the liberty of corresponding the "naive" representation as the second stage and the "scientific" representation as the third stage, although Larkin (1980) has developed the ideas of "naive" and "scientific" representations beyond that of the diagram and the free body diagram.
136 CHI, FELTOVlCHAND GLASER
indicates that the novice's representation for "inclined plane" is very well de- veloped. His representation contains numerous variablesthat can be instantiated, including the angle at which the plane is inclined with respect to the horizontal, whether there is a block resting on a plane, and the mass and height of the block. Other variables mentioned by the novice include the surface property of the plane, whether or not it has friction, and if it does, what the coefficients of static and kinetic friction are. The novice also discussed possible forces that may act on the block, such as possibly having a pully attached to it. The novice did not discuss any physics principles until the very end, where he mentioned the perti- nence of Conservation of Energy. However, his mentioning of the Conservation of Energy principle was not elicited as an explicit solution procedure that is applicable to a configuration involving an inclined plane, as will be seen later in the case of the expert.
J InclinePlane ~ , ~
Figure4. Network representation of Novice H. P.'s schemaof an indinedplane.
CATEGORIZATION AND REPRSENTATION 137
The casual reference to the underlying physics principle given by the novice in the previous example is in marked contrast to the expert's protocol in which she immediately mentioned general alternative basic physics principles, Newton's laws of Force and Conservation of Energy, that may come into play for problems containing an inclined plane (Figure 5). The expert not only mentioned the alternative methods, but also the conditions under which they can be applied (dotted enclosures in Figure 5). Therefore, the expert appears to associate her principles with procedural knowledge about their applicability.
I I I
I f^ f... .zno^ Law:~ %11[h^ f^ ",'~ - n~ %1i I \ F = M A /II~" - ' - ~ J l
Incline Plane
Figure .5. Network representation of Expert M. G.'s schema of an inclined plane.
CATEGORIZATION AND REPRSENTATION 139
TABLE 9 Novice Productions Converted from Protocols H. P.
- (IF problem involves inclined plane)* THEN find angle of incline with horizontal.
- IF block resting on plane THEN a) find mass of block b) determine if plane is frictionless or not.
- IF plane has friction THEN determine coefficients of static and kinetic friction.
- IF there are any farces an the block THEN..
- IF the block is at rest THEN...
- IF the block has an initial speed THEN...
- IF the plane is frictionless THEN the problem is simplified.
- IF problem would involve Conservation of Energy and height of block, length of plane, height of plane are known THEN cauld solve for potential and kinetic energies. P. g ,
- (IF problem involves an inclined plane)* THEN a) figure out what type of device is used; b) find what masses are given; c) find outside forces besides force coming from pulley.
- IF pulley involved THEN try to neglect it.
- IF trying to find coefficient of friction THEN slowly increase angle until block on it starts moving.
- IF two frictionless inclined planes face each other and a ball is rolled from a height on one side THEN ball will roll to same height on other side.
- IF something goes down frictionless surface THEN can find acceleration of gravity on the incline using trigonometry.
- IF want to have collision THEN can use incline to accelerate one object. *Statements in parentheses were not said explicitly by the subject but are indicated by the context.
directly into "production rules" (Newell, 1973). This can be done simply by converting all statements that can be interpreted as reflecting if-then or if-when structures in the protocols. This transformation is quite simple and straightfor- ward and covers a majority of the protocol data. Tables 8 and 9 depict the same set of protocols as do Figures 4 and 5, except these also include the data of the other two subjects. Such an analysis captures differences between the expert and novice protocols in a more pronounced way, and other differences also become more apparent. As suggested earlier, the experts' production rules (Table 8) contain ex- plicit solution methods, such as "use F = M A , " "sum all the forces to 0. " These procedures may be considered as calls to action schemata (Greeno, 1980).
140 CHI, FELTOVICHAND GLASER
None of the novices' rules depicted in Table 9 contain any actions that are explicit solution procedures. Their actions can be characterized more as attempts to find specific unknowns, such as "find mass" (see rules H. P. 2 and P. D. 1 in Table 9). In addition, one novice (H. P.), exhibited a number of production rules that have no explicit actions. This suggests that he knew what problem cues are relevant, but did not know what to do with them, that is, if we think of the protocols as reflecting contents of an inclined plane schema, the novice's schema may contain fewer explicit procedures. Finally, our network analyses (Figures 4 and 5i suggested that the mention- ing of Conservation of Energy by Novice H. P. was somehow different from the mentioning of Conservation of Energy by the Expert M. G. This difference can now be further captured by this second mode of analysis. In Table 9, it can be seen that Novice H. P. 's statement of Conservation of Energy (Rule 8) was part of a description of the condition side of a production rule, whereas the statement of this principle by both experts (Table 8, see rules M.S. 2 and M.G. 2) is described on the action side of the production rules--supporting our previous interpretation of a difference in the way "Conservation of Energy" was meant when mentioned in the protocols of Novice H. P. (Figure 4) and Expert M. G. (Figure 5).
Study Four: Feature Identification
We have now claimed: (a) that experts and novices categorize problems dif- ferently; (b) that these categories elicit a knowledge structure (a schema) that functions in the representation of a problem; and (c) that at least for experts this schema includes potential solution methods. In this study, we attempt to deter- mine problem features that subjects use in eliciting their category schema and, hence, their solution methods. Subjects were asked to read problem statements and to think out loud about the "basic approach" that they would take toward solving the problem. Subjects were encouraged to report all thoughts and hunches they had while deciding upon a "basic approach," even if these ideas occurred during the reading of the problem. Following this unconstrained thinking period for each problem, sub- jects were asked to state their "basic approach" explicitly and to state the problem features that led them to their choice. The subjects were two physicists who had frequently taught introductory mechanics and two novices who had completed a basic college course in mechanics with an A grade. The 20 problems used in this task were the same (described in Table 3) used for the sorting replication (Study Two); they have surface configurations crossed with principles.
Analyses of "Basic Approaches"
Table 10 gives the f'mal "basic approaches" for all 20 problems, as stated by the two experts. Two aspects of these results are noteworthy. First, "basic ap-
