Theory of constructions and set in problem solving · CONSTRUCTIONS AND SET IN PROBLEM SOLVING 447 B Given: M IS the midpoint of AS and C15: A('''BD Prove LAMe' IBMO Figure 2. Problem

Memory & Cognition1979, Vol. 7 (6), 445461

Theory of constructions and setin problem solving

JAMES G. GREENO and MARIA E. MAGONEUniversity ofPittsburgh, Pittsburgh, Pennsylvania 15260

and

SETH CHAIKLINUniversity of Washington, Seattle, Washington 98195

Hierarchically organized knowledge about actions has been postulated to explain planning inproblem solving. Perdix, a simulation of problem solving in geometry with schematic planningknowledge, is described. Perdix's planning knowledge enables it to augment the problem spaceit is given by constructing auxiliary lines. The planning system also provides a mechanism thatcan result in problem solving set. Results of three experiments involving set and constructionsseem consistent with the kinds of knowledge structures hypothesized in the model. Protocolsgiven during solution of 11 geometry problems showed general agreement with the explanationof constructions and set based on planning knowledge, but they also indicated processes ofhuman problem solving not represented in the model. Finally, the explanation of constructionsis discussed in relation to the general question of ill-structured problems and creativity, andthe explanation of set is discussed in relation to other phenomena in the problem solvingliterature, including functional fixedness.

Several recent analyses of knowledge used in solvingproblems have hypothesized a structure in whichknowledge about actions is hierarchical. Knowledgeabout each action includes knowledge about preconditions that are required for the action to be performed,knowledge about consequences of the action, andknowledge of the component subactions that areperformed in the process of performing the more globalaction. A system based on this principle was developedby Sacerdoti (1977) in the domain of robotics, andsystems designed along similar lines have been proposedfor a number of other problem domains. These includean analysis of geometry theorem proving by Goldstein(Note 1), analyses of solving textbook problems inphysics by Bhaskar and Simon (1977) and by Larkin(Note 2), an analysis of designing electrical circuits bySussman (1977), an analysis of writing prose by Flowerand Hayes (in press), analyses of designing computerprograms by Atwood, Polson, Jeffries, and Ramsey(Note 3), Goldstein and Miller (Note 4), and Long(Note 5), and a general discussion by McDermott(1978). These theories are all consistent in a generalway with earlier suggestions by Berlyne (1965), Duncker(1945), Hull (1930), Miller, Galanter, and Pribram(1960), and others regarding hierarchical structure ofthinking and behavior. The more recent work hasincluded hypotheses about the specific knowledgestructures involved in performing the various kinds oftasks that have been studied.

This research was sponsored by the Personnel and TrainingResearch Programs, Psychological Sciences Division, Office ofNaval Research, under Contract N00014-78-C-D022, ContractAuthority Identification Number NR 157-408.

This paper presents an analysis of hierarchicalknowledge for planning in solving textbook problems inhigh school geometry. A simulation model of problemsolving called Perdix (Greeno, 1978) has been extendedin a way that involves hierarchical planning knowledge.This development has provided two extensions ofprevious theories. First, an explanation is provided forthe occurrence of constructions in problem solving.Second, the knowledge of global actions used inplanning provides an explanation of problem solving set.

The developments presented here permit a conceptualintegration of several general issues in the theory ofproblem solving. The process developed for constructions is relevant to issues in the theory of ill-structuredproblem solving and the theory of creativity. There isa strong connection between constructions and set,because the main cognitive structure that causes set isalso the structure that permits constructions to occur.

One feature of ill-structured problems is that important materials needed for solution are not provided inthe initial problem space. Auxiliary lines needed forsolving some geometry problems are materials thatmust be added to the problem space by the problemsolver. Thus, Perdix's ability to add constructionsprovides some new understanding of an aspect of illstructured problem solving. In the domain of geometry,the problem solving demands are relatively modest incomparison, for example, with musical composition(Reitman, 1965). Nonetheless, the kind of process thatenables Perdix to construct auxiliary lines seemsapplicable in other domains as well.

The process represented in Perdix for augmenting aproblem space involves a form of problem fmding.

Copyright 1979 Psychonomic Society, Inc. 445 0090-502X/79/060445-l7$Ol.95/0

446 GREENO, MAGONE, AND CHAIKLIN

Problem finding has been identified as a major factor increative work on ill-structured problems such as painting(Getzels & Csikszentmihalyi, 1976). Thus, the theoretical development to be presented may provide a steptoward a more rigorous analysis of some aspects ofcreative processes in problem solving.

The explanation of set that is provided by hierarchical knowledge of actions is entirely consistent withconventional explanations of set in experimentalpsychology (e.g., Bourne, Ekstrand, & Dominowski,1971). The representation in Perdix specifies knowledgestructures for solving a class of problems in a way thatshows how set can occur. The point of greatest interestis the close relationship between the cognitive structuresinvolved in set and constructions. The analysis alsosuggests some ways to begin to interpret the considerable experimental literature on problem solving set inrelation to recently developed theoretical concepts ofinformation processing and representation of knowledge.

We will begin by presenting illustrations of thephenomena of constructions and set in geometry problem solving, in order to provide a concrete context forthe theoretical concepts that we will develop. Next, wewill describe the knowledge used by Perdix for planningand the way in which Perdix simulates phenomena ofconstructions and set. Then, we will describe the resultsof three experiments and a set of thinking-aloud protocols that provide further illustrations of the principlesthat are incorporated in the model, and we will alsoindicate some important aspects of human processinginvolving constructions that are beyond the capabilitiesprogrammed in Perdix. Finally, we will discuss generalimplications of the analysis, concerning the relationof constructions to the issues of ill-structured problemsand creativity and the general question of set in problemsolving.

Example of ConstructionAn example of a problem that is solved with a simple

construction is the following: Prove that if two sides ofa triangle are congruent, then the angles opposite thosesides are congruent. (An alternative statement is:Prove that the base angles of an isosceles triangle arecongruent.)

Table 1 shows a protocol of a solution of thisproblem. The student began by drawing a triangle,labelin~ts vertices A, B, and C, marking the congruentsides AC and BC, and marking the angles at A and B,which were to be proved congruent, as shown inFigure 1. At *1 in the protocol, the student drew aline from C to the base of the triangle, labeling thepoint of intersection D. At *2 the student tentativelyspecified that CD would be the bisector of LACB andconfirmed this at *3, when the student realized that,since CD is in both triangles, it provides the neededsecond pair of congruent sides for the side-angle-sidepattern of congruence of triangles.

Table 1Protocol for the Base Angles Theorem

s: Okay, so, I want to prove that angle A is congruent toangle B. Now, let's see. Do you want ... ?

*1 E: Yeah ... why are you drawing that line?S: I don't know yet. Okay, uhm ... okay, then I could ...

if I drew a line ...E: Mm-hmm.

*2 S: That would be the bisector of angle ACB, and that wouldgive me ... those congruent angles ... no. (pause)

*3 Yeah, well that would give me those congruent angles, butI could have the reflexive property, so this would be equalto that. Okay, I got it.

E: Now, before you go ahead and write it all down, whenyou said you were going to draw the line ...

S: Yeah ...E: And I said, why are you doing that, and you said you

didn't know yet, what do you think happened to giveyou the idea of making it the bisector?

*4 S: Okay. Well, I have to try to get this ... I have to try toget triangle ACD congruent to BCD, because if I do that,then angle A is congruent to angle Bbecause correspondingparts of congruent triangles are congruent.

E: So you were drawing the line to give yourself triangles, isthat the idea?

*5 S: No, to ... to get a side that was in both triangles.E: Okay.

*6 S: And to get congruent angles.E: So that's why you drew it as the bisector.S: Yeah.

c

DFigure I. Diagram drawn by student during solution given in

Table l.

CONSTRUCTIONS AND SET IN PROBLEM SOLVING 447

B

Given: M IS the midpoint of AS and C15:

A('''BDProve LAMe' IBMO

Figure 2. Problem of the kind studied by Luchins (1942).

Example of SetConsider the problem shown in Figure 2. The

problem can be solved by proving that trianglesare congruent: I1ACM == I1BDM (side-side-side); thenLAMC == LBMD, since they are corresponding partsof congruent triangles. However, an easier solution ispossible. LAMC and LBMD are vertical angles, so theircongruence can be inferred directly.

Luchins (1942) gave a problem similar to the one inFigure 2 to 10 student subjects who had just solved aseries of four problems in which congruence of angleswas shown by proving that triangles were congruent.Eight of the 10 students solved the problem by provingthat triangles were congruent, rather than by the simplersolution involving vertical angles. Another set of 10students solved four problems involving congruence oftriangles, followed by a problem similar to Figure 1,in which the given information was insufficient to provethe triangles congruent. Six of these 10 students failedto fmd a solution to the problem in the 2.5-min periodthat was allowed. In contrast, the problem was solvedeasily by all 12 students in a group that was given theproblem without preceding problems to induce theattempt to prove triangles congruent.

GENERAL HYPOTHESES

The gist of the explanation of set and constructionsto be proposed here is that both are consequences ofthe way in which planning occurs. The hypothesis tobe proposed about planning is a hierarchical systemthat includes a set of plan-schemata that are associatedwith subgoals. Each plan-schema includes informationabout global features of the situation that are reqUiredfor the plan to be feasible; for example, the plan ofproving triangles congruent requires that two trianglesbe present in the diagram. When a plan has beenadopted, work proceeds in relation to the specificsubgoals that are required for that plan to succeed.

This planning system produces constructions by aprocess of partial pattern matching and pattern comple-

tion. Perdix recognizes that a situation partially matchesa pattern that satisfies the prerequisites for a planschema and then activates productions that completethe required pattern. For example, a pattern thatfrequently enables the plan of proving congruence oftriangles is a pair of triangles with a shared side. Beforeline CD is drawn in Figure 1, the diagram contains some,but not all, of the lines, angles, and points that constitutethat pattern. A system that can detect that partial matchand identify the discrepancy between the situation andthe pattern it has in memory has the knowledge it needsto choose an appropriate construction for this problem.

This idea also provides an explanation of problemsolving set. A plan may be adopted on the basis ofglobal features, and because of the top-down nature ofthe system, the plan will become the basis of work onthe problem until either the plan succeeds or it isdetermined that the plan is impossible. For example,in Figure 2 the prerequisites are present for the plan ofproving that angles are congruent by showing that theyare corresponding parts of congruent triangles. If thisplan is adopted, the system focuses on the subgoal ofproving congruence of the triangles and can easily failto notice that the angles are vertical angles.

PLANNING KNOWLEDGE IN PERDIX

We will give a brief survey of the characteristics ofPerdix as a problem solving system. More detailedinformation about Perdix's geometry knowledge hasbeen given in other articles (Greeno, 1976, 1977, 1978).

Perdix is a system that solves problems by matchingand completing patterns. It is a production system witha general control structure adapted from Anderson's(1976) ACT system. The problem situation is repre·sented as a semantic network. Points, lines, angles, andclosed figures are represented by nodes. Connectionsbetween nodes include part-whole relations (e.g., a pointthat is part of a line segment or a segment that is part ofa triangle) and propositional structures that correspondto relations of congruence and other geometric relations.

The productions that constitute Perdix's problemsolving knowledge are in three categories: inferentialprinciples, feature tests for pattern recognition, andstrategic knowledge. The main problem solving steps areaccomplished by inferential productions. These represent knowledge of propositions such as "Vertical anglesare congruent," or "Angles formed by bisection of anangle are congruent." The actions of these productionsadd relations to the semantic network that representsthe situation. For example, if two angles have previouslybeen identified as vertical angles, the relation ofcongruence can be added between them.

Simple pattern recognition is involved in the conditions that are tested in each production. More complexpatterns are identified by sets of productions organizedin the form of decision networks with terminal nodescorresponding to patterns and relations that can beidentified. Patterns and relations that are identified


during problem solving are added to the representationof the problem situation.

Strategic knowledge consists of productions forsetting goals and adopting plans. Simple goal settinginvolves associations that select productions that arerelevant to the current problem solving goal. Morecomplex goals are set in the actions of productions, anda push-down store memory is used to retain goals thatmust be recalled when their subgoals have been achieved.

The theoretical development to be reported in thispaper is a set of productions that perform planning.Perdix's knowledge for planning consists of some globalactions, or plan-schemata. The characteristics of planschemata are like those used by Sacerdoti (1977) in histheory of planning based on a procedural network.Goldstein (Note I) used a similar organization ofknowledge for geometry problems; his system includesexperts that apply knowledge in specific topic areas,such as triangles or parallel lines. Each plan-schema inPerdix is characterized by three components of information: (1) a set of prerequisite features that make the planpromising, (2) pointers to pattern recognition andinferential productions that are used in executing theplan, and (3) tests to determine whether execution ofthe plan has succeeded. The procedures for planningalso include checks for a few salient features that wouldallow the goal to be achieved directly without goingthrough the usual process of planning and executing aseries of steps. Technically, these shortcuts could beclassified as additional plans.

In Perdix's knowledge structure, plans are associatedwith general goals. Some goals are attempted directlythrough productions for pattern recognition. Other goalsrequire propositional inferences. The latter goals activateplanning productions. When one of these goals is generated during problem solving, a series oftests is performed.Each test is an attempt to fmd the general features thatmake one of the plans for the goal promising. Eachplanning system is thus a pattern recognizer in whicheach identifiable pattern corresponds to a set of featuresneeded to make one of the plans promising.

Two planning systems have been implemented inPerdix. One is associated with the goal of fmding themeasure of an angle, and the other is associated with acategory of goals of finding or proving a quantitativerelationship such as congruence between two angles ortwo segments. One plan for finding the measure of anangle uses the sum of angles in a triangle. The otheruses a quantitative relationship with another angle. Theplanning system is essentially a decision tree thatchooses one of these plans, based on whether theunknown angle is part of a triangle and on the nature ofavailable information about the measures of other anglesin the situation.

We will describe the planning system for inferringquantitative relations in some detail. Perdix has threeplans for inferring quantitative relations between angles

or segments. It can infer congruence between angles orsegments that are corresponding parts of congruenttriangles. This plan is abbreviated TRICONGQREL:TRIangle CONGruence is used to infer a QuantitativeRELation. A second plan uses relations between angleswith a shared vertex: RTNRELQREL means that arelation based on rotation, RTNREL, is used to infer aQuantitative RELation. The third plan uses relationsbetween angles with parallel sides: PRLRELQREL.

The procedure that has been implemented forconstructing auxiliary lines is part of the planningsystem for quantitative relations. If none of the preconditions for Perdix's three main plans is found, Perdix canidentify a partial match to a pattern that provides theprerequisites for one of its plans, the congruence oftriangles. If a sufficient subset of the features of thatpattern is present, Perdix completes the pattern andwill then choose the plan that its own construction hasenabled.

Figure 3 shows the components of the planningprocedure for finding a quantitative relation betweentwo angles or segments. VI and VII denote the anglesor segments in that goal. There are tests at Steps 1,2, and 3 for a few features that can provide quicksolutions. Next, at Step 4, Perdix tests whether thetarget elements are parts of triangles. If they are,triangles are tested at Steps 5, 6, and 7 to see whetherthere are features that would make it feasible to tryto prove that they are congruent. These tests have twofunctions. First, if the triangles are not promising,the system can look for other plans that might beeasier. Second, since many diagrams include more thanone pair of triangles, the tests at Steps 5, 6, and 7 serveto select triangles that contain congruent components. Ifone of these tests is successful, then, at Step 8, the planis adopted to fmd a quantitative relation based oncongruent triangles, and at Step 9, the goal is set toshow that the selected triangles are congruent.

If promising triangles are not found, Perdix testsat Steps 10 and 13 for features of its other generalplans for finding quantitative relations. If the targetsare angles with a single vertex, then at Step 11, theplan RTNRELQREL is adopted to infer a quantitativerelation from a relation based on rotation, and atStep 12, the goal is set to find a relation of rotationbetween the angles. The system will then engage inpattern recognition that can identify vertical angles,linear pairs, or angles formed by perpendicular lines. Ifthe test at Step 13 shows that the angles are formedby parallel lines intersected by a transversal, then atStep 14, the plan PRLRELQREL is adopted to infer aquantitative relation from a relation involving parallelsides. Under the goal PRLREL, set at Step 15, Perdixwill perform pattern recognition that can identifycorresponding angles, alternate interior angles, orinterior angles on the same side of the transversal. Ifthese plans are not feasible, Perdix will try to use


v..

1221set goal:

CHAINCONG

VI, VII

no

construct point

on collinear

sides: va

bind sides:V5. V15. andintersectionpoint; V6

ve.

no

bind inte,section pOint:

vB

bind 1'..V2, V12

191set goal:

TRICONGV2, V12

181set plan:

TRICONGQRELVI, VII, V2, V12

1191construct

segment:V6, V8

Figure 3. Procedure for planning, including construction of new components, when goal is to find a quantitative relationbetween angles or segments.


triangles that seemed unpromising earlier, if they arefound in the test at Step 16.

The process for constructing auxiliary lines is usedif the test at Step 16 is negative. If the prerequisitefeatures of none of the available plans are present,Perdix checks whether the problem situation hasfeatures that provide a partial match for a schema thatincludes the prerequisites for one of the plans. Theschema consists of a pair of triangles with a sharedside. It contains 11 components: two triangles, fivesegments, and four points. The partial match can consistof two segments that intersect, found at Step 17, andeither of two further conditions: two additional intersecting segments at Step 18 or a single segment foundat Step 20 that can be divided to form two neededsegments. If the appropriate components are found,Perdix constructs a segment that completes the patternat Step 19 and returns to the top of the planningprocess.!

If the features needed to perform this constructionare not present, then the system adopts the goal offinding a chain of relations between angles at Step 22.This is a kind of default plan, involving an attempt tofind some third angle that is congruent to one of theangles that can then be related to the other target,perhaps through mediation of further linking angles.

When the planning process is finished, one of threethings will have happened. First, the goal may have beenachieved on the basis of a feature permitting a quicksolution. Second, a plan may have been chosen; thisgenerally involves setting a subgoal that will permit theplan to work. Adoption of a plan may have beenpreceded by a construction to make the plan feasible.Finally, the system's planning knowledge may beinsufficient for the problem at hand, and there mayhave been no successful outcome for planning. (Inthe system shown in Figure 3, this leads to settingthe goal of forming a chain of congruent components.)In the situation in which a plan is chosen, the systemthen executes procedures that are associated withthe plan, including pattern recognition and propositionalinferences.

EXPLANATION OF CONSTRUCTIONS AND SETThe planning process in Perdix is an example of

one kind of knowledge structure that can produceconstructions and set. The relevant general feature is theknowledge of global actions and planning systems thatchoose a general approach to the problem from amongthe available global actions. Figure 3 is an example: Thealternative global actions are called TRICONGQREL,RTNRELQREL, and PRLRELQREL. The planningsystem uses general features of the problem situation,such as the presence of triangles with known relationsbetween their components, to select one of the globalactions as a plan.

The general idea about constructions illustrated inPerdix involves knowledge of prerequisite features forplans and the ability to recognize partial satisfaction of

a set of prerequisites. When Perdix finds that the prerequisites for none of its plans are satisfied, it canrecognize that the situation has some of the componentsof a pattern that permits TRICONGQREL to be used.Perdix has problem solving procedures that permit itto complete this pattern and thus to produce theprerequisites that are required for its planning systemto choose a global action. This illustrates a general kindof process in which knowledge of needed prerequisitespermits a system to identify an auxiliary problem. Thenew problem arises because there is a discrepancybetween this situation and conditions known to beprerequisites for making progress on the main problem.In a hierarchical system, a deficit that arises at the levelof global actions can generate an auxiliary problem thatis solved in a problem space different from the space ofthe main problem. In the geometry example, the operators for the main problem are inferential propositions,such as "Vertical angles are congruent." The operatorsin the auxiliary problem space are a set of rules for adding lines to a diagram, such as "Two points may beconnected by a line segment," or "The bisector of anangle can be constructed." The process of identifying adeficit in the main problem space constitutes a form ofproblem finding and thus provides some insight intocreative problem solving (Getzels & Csikszentmihalyi,1976) and solution of ill-structured problems (Simon,1973).

The occurrence of set is explained in a straightforward way by the top-down nature of the planningsystem. Consider the problem shown in Figure 2. Thetwo alternative solutions correspond to different plans,one involving congruent triangles and the other involvinga relation between angles having the same vertex.Whether Perdix fmds the simple solution involvingvertical angles or proves that triangles are congruentdepends on the order in which the productions testingfeasibility of these plans are included in the productionlist. In the configuration shown in Figure 3, the featuresneeded for the plan of congruent triangles are testedbefore the feature needed for the plan of a relationinvolving rotation. With the productions ordered inthis way, Perdix agrees with the majority of Luchins'(1942) subjects on the way to solve problems similar toFigure 2. The test "VI, VII in triangles?" passes, asdoes the subsequent confirming test, "VI, VII intriangles with sides or angles congruent?" This leads toadoption of the plan to prove the triangles congruent,and the system works on that subproblem and eventually succeeds in solving the problem by proving thetriangles congruent by side-side-side.

The relative strengths of productions are representedin Perdix by the order in which their conditions aretested. For example, the test for "VI, VII angles withshared vertex?" could be performed before "VI, VIIin triangles?" In that case, Perdix would find thesimpler solution based on vertical angles. The mechanisminvolving ordered productions is not intended as aplausible hypothesis about the way in which stronger


productions dominate performance; other systems suchas Anderson's (1976) version of ACT or the OPS system(McDermott & Forgy, 1977) include a variety of mechanisms that have greater psychological plausibility.It would be natural to assume that the strength of aproduction would be increased by the major factorsthat are usually assumed to influence set, such as thefrequency and recency of successful use of a plan inproblems similar to the present one. The present analysisdoes not provide any new ideas about the way in whichset occurs. The contribution of this analysis is inshowing that the knowledge structures responsible forset may be the same as those that are responsible forplanfulness and creativity in problem solving.

EXPERIMENTS ON CONSTRUCTIONS AND SET

The model described above provides a way ofinterpreting the results of three experiments we haveconducted. The first experiment involved a problemrequiring a simple construction. The problem waspresented in two different forms; in one form theconstruction was produced by a greater proportion ofsubjects than in the other. A plausible interpretationof the difference between the two forms can be givenin terms of the planning model that we have presented.

The second experiment used the same problem asExperiment 1. We investigated conditions that could beexpected to make it more likely that subjects wouldproduce the construction needed for solution in theharder version of the problem. One of the conditionsthat we used was successful in making that version ofthe problem easier, and the effect is consistent with theidea of pattern-based planning that is incorporated inPerdix.

The third experiment was a traditional study of set,focused on a problem requiring a construction. We useda problem that has two solutions, one based on parallellines and the other based on congruent triangles. We gavedifferent groups of subjects different series of problemsbefore presenting the ambiguous problem. The problemsin one induction sequence used parallel lines, and theproblems in the other induction sequence used congruent triangles. The majority of the solutions thatsubjects chose for the ambiguous problem agreed withthe plan they had been using, and this result agreeswith the interpretation of problem solving set that wehave given based on the model of schematic planningpresented above.

Experiment 1

The experiment involved two problems, presentedto different groups of subjects. Both problems used thediagram shown in Figure 4. Point 0 is the center ofthe circle, and BC is a diameter. In the problem given toone group of subjects, given information included themeasure of arc AC, 80 deg, and the goal was to find themeasure of the angle LABC. In the other form, the

B

cFigure 4. Diagram for problems in Experiments 1 and 2.

measure of LABC was given as 40 deg, and the goal wasto find the measure of arc AC. To solve either problem,a subject had to give a series of steps justifying eachinference, as is customary in high school geometry.The solution of either problem depends on making theconstruction AO. This creates the central angle LAOC,and, by definition, the measure of a central angle and themeasure of its intercepted arc are the same. Constructionof AO al~creates an isosceles triangle L10AB, with sidesOA and DB that are congruent, being radii of the circle.The angles at A and B, LOAB and LABC, are congruent,since they are the base angles of the isosceles triangle.Using these relationships, a solution for problems withthe arc given is as follows: (l) Measure of arc AC is80 deg (given), (2) construct segment AO, (3) measureof LADe is 80 deg (central angle), (4) measure ofLAOB is 100 deg (linear pairs are supplementary),(5) sum of measures of LOAB and LABC is 80 deg(sum of angles in a triangle is 180 deg), and (6) measureof LABC is 40 deg (LOAB and LABC are equal). Asolution for the problem with the angle given is asfollows: (1) Measure of LABC is 40 deg (given), (2) construct segment AD, (3) measure of LOAB is 40 deg(base angles of isosceles triangle are equal), (4) measureof LAOB is 100 deg (sum of angles in a triangle is180 deg), (5) measure of LAOC is 80 deg (linear pairsare supplementary), and (6) measure of arc AC is 80 deg(are intercepted by central angle).

MethodSubjects were students in introductory psychology at the

University of Pittsburgh who participated to fulfill a courserequirement. Fifty-three subjects were given the problem withthe angle given, and 52 subjects were given the problem with


the arc given. Subjects participated in groups of up to six in size.Experimental sessions began with a review of geometric conceptsneeded to solve either problem, including the congruence ofbase angles of an isosceles triangle, the relation between anexterior angle and its opposite interior angles in a triangle, andthe definitional equality of the measure of a central angle andthe arc it intercepts on a circle. The review of concepts wasgiven in the form of 10 problems, which were worked individually by the subjects in the group. The experimenter monitoredthe subjects' work and explained how to solve any problemsthat any subjects were unable to solve. The experimenter alsocalled attention to features of the various training problems thatwere relevant to the critical problem. The 11 th problem in theseries was Figure 4, with either the measure of arc AC or themeasure of LABC given and the other unknown. No assistancewas given on this problem, and a period of 4 min was allowedfor its solution. At the end of the allotted time, subjects whohad not solved the problem were asked to tum to a second sheetcontaining the same problem, and they were given the hint thata line drawn between Points A and 0 would be helpful in solvingthe problem. An additional 4-min period was allowed to seewhether the subjects could solve the problem with the hintgiven.

ResultsThe problem with the arc given was considerably

easier than the problem with the angle given; proportions of subjects solving the problem in the initial 4 minwere .731 and .471, respectively. The difference isstatistically reliable [X 2 (I) ::: 7.34, p < .0I] .

An important question for our theoretical interpre.tation involves the degree to which the difficulty of theproblem with the angle given was caused by a difficultyin seeing the correct construction. Of the 28 subjectswho had the angle given and failed to solve the problem,7 drew the correct construction during the initial 4 minof work; of the 14 subjects who had the arc given andfailed to solve, 2 drew the correct construction duringthe initial 4 min. If these subjects are counted alongwith the subjects who solved, the difference between theconditions is attenuated and becomes marginally nonsignificant [.769 vs..604; x2 (l)::: 3.32, .05 < P < .10].However, the status of these unsuccessful constructionsis unclear. Only two subjects showed evidence in theirprotocols of relating the construction to appropriaterelational patterns (one of the seven subjects with anglegiven mentioned the isosceles triangle, and one of thetwo subjects with arc given mentioned the central angle).An alternative interpretation is that some subjectswho failed to see the correct pattern simply drew anadditional line in the diagram somewhere, and thesignificance of the constructed line was not evident inmany cases. It may be noted that irrelevant constructions were inserted by nine subjects: A radius perpendicular to the diameter was drawn by seven subjects (sixwith angle given, one with arc given), and Points A and Cwere connected by two subjects (one in each condition).

There was also some marginal evidence that theproblem with the arc given was easier than the problemwith the angle given when the construction was givenas a hint. Thirteen of the 14 subjects who had the arcgiven and failed to solve in the first 4 min did solve afterthe hint was given; 20 of the 28 initial nonsolvers who

had the angle given solved the problem after the hintwas given. The difference was marginally nonsignificant[.929 vs..714; x2 (I)::: 3.18, .05 <p < .10].

DiscussionThe major fmding of the experiment was that the

problem with arc given was easier to solve than theproblem with the angle given. The most reasonableconjecture is that the difficulty had two sources:First, subjects were apparently somewhat less likely togenerate the needed construction when the angle wasgiven; second, the sequence of inferences was apparentlyharder to find with the angle given, even when thesegment AO was present in the problem situation.

An interpretation of the result can be given usingthe idea of schematic planning. A relevant fact is thatstudent subjects usually work forward in geometryproblems. Subjects who were given the measure of thearc probably tried to find features of the problemrelated to the arc, and subjects who were given themeasure of the angle probably tried to find features ofthe problem related to the angle. It seems reasonableto suppose that the pattern of the central angle wasrelatively easy to retrieve with the arc as a cue. Themeasure of the arc and the measure of the central angleare related definitionally, and the central angle is arather simple pattern. On the other hand, it wasprobably less likely that the pattem of the isoscelestriangle formed by radii would be retrieved with theinscribed angle as a cue. There is no direct definitionalconnection between the measure of the inscribed angleandthe triangle formed by radii of the circle, and thatpattern is somewhat more complex than the inscribedangle. Thus, it seems reasonable that the greater difficulty of the problem with the angle given was causedat least partly by a lower probability of retrieving thepattern from memory that was needed to provide abasis for planning the first major step in the solution ofthe problem.

Experiment 2

This study explored two ways of providing experience that might facilitate solution of the harder form ofthe problem of Figure 4. One way to make a solutionmore likely would be to increase the probability ofretrieving the pattern involving an isosceles triangleformed by radii of the circle. A second possibility wouldbe to induce a strategy of planning backward from thegoal of the problem, this could lead subjects to considerproblem components that can be related to the measureof the are, and thus to retrieve the pattern involving thecentral angle.

MethodThree conditions were used. A control condition was identi

cal to the condition used for the problem with the angle given inExperiment 1. For the other two conditions, the sequence ofreview problems was condensed to seven problems, and threeadditional problems were included. For the isosceles pattern


F,nd i ACB

ResultsThe proportions of subjects who solved the problem

in the three conditions were as follows: control, .436;isosceles pattern, .649; backward planning, .527. Thedifference was not quite large enough to give a significant test statistic [X2 (2) = 5.14, .05 < p < .10].However, a chi-square statistic involving just the controland isosceles pattern conditions was significant [X2 (1) =5.10, P < .025], and the results for the two groups ofsubjects were very consistent, providing some additionalconfidence. Proportions of solvers from introductorypsychology were .500, .724, and .586; proportions ofpaid subjects who solved were .370, .571, and .462.It seems reasonable to conclude that solution was mademore probable in the isosceles pattern condition. Aconclusion is not warranted regarding the backwardplanning condition; it may have provided no facilitation,or it may have facilitated problem solving as much asthe isosceles pattern condition.

A substantial number of nonsolvers added the appropriate construction to the diagram, although they failedto solve the problem using it. Of the 31 nonsolvers inthe control group, 15 added AO to the diagram in thefirst 4 min of work. Of the 26 nonsolvers with backwardplanning induction, 9 added AO; 6 of the 20 nonsolverswith the isosceles pattern added AO. The proportions(including solvers) who made the correct constructionwere clearly not significantly different: control, .709;isosceles pattern, .754; backward planning, .691. Again,the status of the correct constructions by nonsolvers isuncertain. Thirteen of the 30 correct constructions madeby nonsolvers were accompanied by additional linesin the diagram that were not useful in the solution,and 21 of 47 subjects who did not make the correctconstruction added some line or lines in their attemptsto solve the problem. It seems likely that relativelyhaphazard modifications of a diagram can be made bysubjects, and some of the constructions in our experiments must have been of that nature. Perhaps the bestinterpretation of the results is that training with theisosceles pattern increased subjects' ability to identifythe pattern and thus to produce a well motivatedconstruction.

The effect of giving the hint to draw line AO wasinconclusive. Proportions of initial nonsolvers whosolved the problem following the hint were .613, .750,and .654. The result is not inconsistent with the ideathat the backward planning and isosceles pattern conditions facilitated problem solving with the completeddiagram as well as facilitating making the construction,but the difference was far too small to be significant[X2 (2) = 1.03, P > .50] .

of SUbjects from these two sources in the three conditions wereas follows: control, 28, 27; isosceles pattern, 29, 28; backwardplanning, 29, 26.

Given LSOE =LASo

AS = DE

LACo = 55"

Find LCAS

D

(e)

Ar- ---.B

c(b)

A

B'--------~'-------_..:t

E

.... -Given: so II EF

so and AF bisect each other.

iCFo=65~

, ABC = 75

(a)Opposite sides of a parallelogram are equal and parallelShow that opposite angles are equal, that is. i. A = i D

condition, the three added problems involved isosceles trianglesthat were formed by radii of circles. For the backward planningcondition, the three added problems involved solution sequencesthat were easier to fmd by working backward from the goalthan by working forward from the given information. In thebackward planning condition, the experimenter specificallyinstructed the subjects to fmd a sequence of steps that startedwith the unknown quantity and progressed toward the giveninformation and coached them so they identified appropriatesequences for the three problems.

The procedure was the same as in Experiment 1, wi th theproblem of Figure 5 presented as the lIth problem in all threeconditions, with the measure of LABC given and the measure ofarc AC the goal. Two groups of subjects participated: 86students from introductory psychology at the University ofPittsburgh, who participated to fulfill a course requirement, and81 students who answered an advertisement in the campusnewspaper, who were paid for their participation. The numbers

Figure 5. Problems used in experiment on constructions and Discussionset. The facilitation of problem solving in the isosceles

Figure 6. Problem involving puallel lines, and constructionsformed by subjects.

4S4 GREENO, MAGONE, AND CHAIKLIN

pattern condition is consistent with the hypothesis ofa planning system that is based on knowledge of patternsthat are identified as conditions for adopting plansand that are used in making constructions that take theform of pattern completion. Our interpretation is thatthe additional problems given with the isosceles patternmade that pattern more salient for subjects, and thusmade it more likely that they would retrieve that patternfrom memory in the problem situation.

Experiment 3

This experiment involved the role of problem solvingset in a problem requiring a construction. Our explana·tion of problem solving set given above is that thepattern-based planning system becomes biased towardchoice of a plan that has been successfully used inrecent problems. Our explanation of constructionsis that a pattern associated with a plan is partiallyidentified, and the system completes that pattern inorder to have the prerequisites for a plan. This suggeststhat if a problem can be solved with alternative construe·tions related to different plans, the construction that ischosen should be influenced by a series of problemsthat are solved using one or the other of the two plans.

The problem used in the experiment is shown inFigure Sa. The problem can be solved by showing thattriangles are congruent. This requires constructing thediagonal line BD; the resulting triangles are congruentby side-side-side. The problem can also be solved usingrelations based on parallel lines. This solution usuallyuses a construction such as extending line BD upward;in this case, LA is congruent to the exterior angle at B(alternate interior angles), and that angle is congruentto LD (corresponding angles).

(a)

(b)

(c)

(d)

r--.-----_B

D"-----L.--------E......Given: AS II DE

Find:mLx

J>----Method

The subjects were 12 University of Michigan students whowere paid for participating. Thinking-aloud protocols weretape-recorded to permit later identification of subjects' solutionstrategies. The experimental sequence began with a series ofreview exercises that reminded the subject of ideas involvedin proving congruence of triangles and in proving theoremsinvolving angles formed by parallel lines and an intersectingtransversal. Then there was a series of 14 induction problems.For six subjects, the induction problems all used congruenttriangles; the other six subjects had 14 problems that usedrelations between angles based on parallel lines. Problem 15 forall subjects was the problem shown in Figure 5a. After thisambiguous problem, each subject received a problem thatrequired use of the plan that had not been used in the inductionsequence that the subject had received. These problems did notrequire constructions, but they gave further opportunity toobserve the biasing effect of problem solving set at the level ofgeneral planning. For subjects who had been induced to usecongruent triangles, Problem 16 was the one shown in Figure 5b.This is an easy problem using parallel lines: AB and CE areparallel (LBDE and LABD are alternate interior angles); hence,LACD and LCAB are supplementary (interior angles on the sameside of the transversal). Problem 16 for subjects who receivedinduction of parallel lines is shown in Figure 5c. The trianglesare congruent by side-angle-side, because of vertical angles;then, LBAC =65 deg (congruent to LCFD), and LACB can befound by subtraction from 180 deg. For the subjects with

induction of congruent triangles, Figure 5c was Problem 14;for the subjects with induction of parallel lines, Figure 5b wasProblem 14. This permitted comparison between subjects forwhom the problem was consistent with the induction seriesand subjects for whom it was not.

ResultsThe manipulation was strong enough to produce

reliable differences, even with only six subjects percondition. In the ambiguous problem, Figure Sa, fiveof the subjects who received induction with congruenttriangles solved the problem by constructing thediagonal and proving that the resulting triangles werecongruent. Only one of the six subjects who wereinduced to use parallel lines found the proof based ontriangles; the other five found proofs using relationsformed by parallel lines. The difference is significant(p < .02) on a sign test for which the random event isagreement between the subject's solution and theinduction sequence.

The problem in Figure 5c must be solved by provingthat triangles are congruent. All six subjects who wereinduced for congruence of triangles adopted a plan


involving triangles initially. In the group induced withparallel lines, only two subjects initially worked with aplan for proving the triangles congruent; the other foursubjects tried to use the relation of parallel lines. This isalso significant by a sign test (p < .02).

The problem in Figure 5b is solved using parallellines. In the group for which induction was with parallellines, four of the six subjects used the plan involvingparallel lines initially; the other two failed to see thatparallel lines could be inferred but mentioned parallellines as a subgoal. In the group for which induction waswith congruent triangles, three of the six subjects beganwith a plan involving congruent triangles, and only oneused parallel lines; the other two subjects initially useda plan involving the sum of the interior angles of aquadrilateral. Counting just those subjects whose initialplan was either congruence or parallel lines, the difference between the conditions was significant (p < .04,by sign test).

DiscussionThis experiment on set provides a supportive illustra

tion of the idea of a strong connection between theprocess of making constructions and the process ofplanning. The idea that a single planning process isresponsible for both problem solving set and the makingof constructions seems a plausible one.

THINKING-ALOUD PROTOCOLS

A collection of thinking-aloud protocols was obtainedfrom six students who were interviewed individuallyapproximately once each week during the year in whichthey took a course in high school geometry. In theinterviews, students solved a wide variety of problems,only some of which are relevant to the issues in thispaper. Solutions of nine of the problems used constructions, and two of the problems provided illustrations ofproblem solving set. Protocols from these problems wereexamined to proVide an informal evaluation of thegeneral hypothesis of schema-based planning processes.A brief summary of the conclusions of this informalanalysis will be given here. A more detailed discussionof the protocols from the nine problems involvingconstructions is available in a technical report (Greeno,Note 6).

Problems Involving ConstructionsPlan-based constructions. Nine problems that were

given required construction of auxiliary lines. Mostof the protocols indicated solution processes thatwere generally consistent with the kind of processprogrammed in Perdix, in which constructions arerelated to schemata that function as plans. Variousschemata were involved in the different problems,including formation of triangles and other geometricfigures, and the central angle of a circle.

An example involving the pattern of parallel linesintersected by a transversal is shown in Figure 6. This

example has special interest because different constructions were used by subjects, based on differentschemata. This problem was given to students earlyin the course, before they had studied congruence oftriangles. They had been taught about relations betweenpairs of angles with parallel sides, such as correspondingangles, alternate interior angles, and so on. Constructionsused in solving the problem are shown in Figures 6b, 6c,and 6d. The construction shown in Figure 6b completestransversals that intersect the parallel lines by extendinglines already in the diagram. The protocol in whichFigure 6b occurred indicated that the subject probablycompleted the pattern and made justified inferencesbased on alternate interior angles and then realized thatthe presence of triangles permitted further inferences,which then led to the solution. The subject who gavethe construction in Figure 6c specified the added lineas the perpendicular going through the vertex andapparently foresaw using the 90-deg angles to permitinference of the remaining angles in the triangles basedon the sum of angles in a triangle being 180 deg. Atthe time this problem was presented, the class had notstudied triangles, so the sum of angles of a triangle wasrecalled from earlier study by the subjects who usedFigures 6b and 6c. The construction in Figure 6dpermitted solution without using the triangle-sumproperty. This construction was not used by anysubjects on the problem in Figure 6a, although onesubject saw that it could be used when a pattern ofthree parallel lines was presented in another problem,and another subject used it in another constructionproblem involving parallel lines.

Working-forward pattern completion. While most ofthe protocols were consistent with Perdix's procedurefor constructing auxiliary lines, several included significant features that could not be simulated by the kindof procedure that has been programmed. One processinvolves pattern completion that does not seem to bedirectly related to the problem goal. An exampleoccurred in a problem of proving that if sides of atriangle are unequal, the angles opposite those sidesalso are unequal. This problem requires a constructionto form an isosceles triangle inside the given triangle.One student made a construction similar to the one inFigure 1, bisecting the angle between the two specifiedsides of the triangle. It seems likely that features of theproblem reminded the subject of the pattern involvingtwo triangles containing the angles in the problem goal.However, that pattern is not a useful one for provinginequality of angles.

Procedures for making forward-working constructionshave not been programmed for Perdix; however, they arequite straightforward. The kind of process needed couldbe represented either as single productions activated byfeatures of the situation with actions that add theauxiliary lines or set goals for adding lines. An examplerelevant to the problem used in Experiments 1 and 2would be a production in which the condition is anarc of a circle with known measure and the action is


construction of the central angle corresponding to thearc.

In order for a construction produced in workingforward to be useful, there must be a procedure forrecognizing the results of the construction that maybe helpful in achieving the goal of the problem. Theplanning process represented in the present version ofPerdix is well suited for that purpose, since the planningprocedure consists of a pattern recognition system foridentifying features that are favorable for the availableplans. Thus, if a forward-working production were toproduce a new feature of the problem that could enablechoice of a plan, the system would only have to invokeits planning system in the usual way for that newfeature to be recognized and used. For example, in theproblem used in Experiment 1, the construction of thecentral angle produces a triangle that has the unknownangle as one of its components. This configuration offeatures is then available for a planning process that hasthe form of a pattern recognition system.

A problem that must be solved in a system thatincludes forward chaining is that many inferences maybe generated that are not useful in the problem. Thiswould become especially critical in a system that permitsuse of auxiliary lines, since this greatly increases thenumber of potential objects about which inferencesmight be made. Nevins (Note 7) and Ullman (Note 8)have given some attention to this difficulty in systemsfor geometry theorem proving. Nevins' system restrictsforward-chaining inferences to those that are relatedto generally useful subgoals that he called paradigms,such as finding congruent triangles. Ullman's systemapplies a general criterion of compactness of inferredinformation, such as the inference that a closed figure isa parallelogram, for setting aside the process of forwardchaining. It seems quite likely that some mechanismsuch as these would keep a process of forward chainingunder control in Perdix, especially if forward chainingwere used only when none of the plans that the systemhas provide a basis for progress.

Stages in specifying constructions. Another featureof some protocols that would require extension ofPerdix occurred when constructions had to be givensome geometric property in addition to the spatialfeatures used in drawing the line. A construction ingeometry is often added in a single step, for example,by connecting two points in the diagram. However,some constructions are more complicated, requiringspecification of some geometric property in additionto simply locating a line in the diagram. An exampleis the construction used in proving that base anglesof isosceles triangles are congruent. In Table I there isa clear indication that the subject drew the line thatcompleted the pattern of two triangles and then chosethe specification of the line as the bisector of the angleat Point C, rather than identifying the construction asthe angle bisector initially.

Geometry constructions illustrate a general feature ofproblems in which material added to the problem space

must satisfy constraints. In geometry, the constraintsare specified in a set of theorems that assert that certainconstructions are permitted. An example of such apermissive theorem is that an angle has exactly onebisector, which is used to justify construction of thebisector of an angle. Students are taught that anauxiliary line added to a diagram should not be eitherunderdetermined or overdetermined. That is, justenough conditions should be specified for the line thatexactly one line can be drawn to meet the conditions.

The process of forrning a construction apparently hastwo major components. One is a process that motivatesthe construction; the other specifies properties of theconstruction that satisfy constraints. The analysis givenin this paper has dealt mainly with the process ofmotivating a construction; protocols seem consistentwith the idea that constructions are motivated byknowledge of patterns in plan-schemata that can becompleted by adding auxiliary lines. The process ofsatisfying constraints is relatively simple in geometry,in which permissive theorems give a complete list of theconditions that may be used. When a problem solverencounters a situation in which an auxiliary line isdesired, the construction theorems are consulted and theconstruction is added if the conditions of a permissiveconstruction theorem are satisfied in the situation.

Three distinguishable patterns of forming constructions of this kind occurred in the protocols availablefor this study. In one pattern, found in two of the fiveprotocols for the base angles theorem, subjects specifiedthe construction completely at the outset. A secondway of forming the construction shown in Table I andby one other subject involved drawing the line, thengiving the geometric specification in relation to a goalof the problem. In this case, the goal is to generatecomponents of triangles that can be proved congruent,and, by specifying the auxiliary line as an angle bisector,there are congruent angles. The third way of formingthe construction involved drawing the line, then givinga geometric specification that was not related to the goalof the problem in any defmite way. The subject whoshowed this pattern drew the line, then said, "I wouldprobably say that this is a median." After completingthe problem, the subject answered a question aboutthe construction by saying, "All I wanted was just aname for a line that was going to give me two triangles."

In its present form, Perdix satisfies multiple constraints on constructions because its productions forconstruction have appropriate specifications as parts oftheir actions. This is inconsistent with most of theconstructions observed in the protocols used in thisstudy. Most of the time when an angle bisector or a lineparallel or perpendicular to a given line was constructed,there was no evidence of special consideration givento the legitimacy of the construction. Of the threeexamples in which permissive theorems seemed to beused, one was consistent with the process used byGelernter's (1953) system, in which the need for theconstruction arises as a subgoal, and there is explicit


checking of the permissive theorems stored as a list tosee whether a justification can be found. In the othertwo cases, a specification was apparently chosen inorder to satisfy subgoals that were anticipated for alater stage of the problem. This kind of anticipatoryplanning has not been implemented in Perdix, but itis not incompatible with the general kind of planningmechanism that Perdix uses. In Sacerdoti's (1977)model, NOAH, unbound variables can be included inhigh-level components of plans, so that further analysiscan be performed before deciding precisely how tospecify the actions that will be performed.

Problems Involving SetTwo problems were given that could be solved either

by ratios of sides of special right triangles or by congruence of triangles. Simpler solutions could be obtainedusing congruence, but the class topic at the time wasproperties of special triangles, and a majority of thesolutions obtained used that approach.

These protocols give further illustrations of the ideathat general approaches to problems can be influencedby recent use. The currency of the properties of specialtriangles in the geometry course apparently provided acontext that was sufficient to induce a problem solvingset for most of these subjects. On the other hand,examination of the protocols provided evidence againstthe specific form of planning that is used in Perdix toexplain problem solving set. Perdix has plans associatedwith goals and checks these plans one at a time untilone of them is selected. Performance by some subjectswas more flexible, involving inference of some propositions in an apparent working-forward process andconsideration of congruence of triangles. One subject,thinking retrospectively about solving a problem, said,"I wondered why you were giving me these things, andI went at the problem saying, 'Well, let me try anduse these things,' and I said, 'Well I give up.' Or, 'Igive up, I see there's an easier way in my mind.' ButI-I was working on it until I saw that things just camemore naturally in another direction."

Performance of some human subjects would besimulated more accurately if Perdix's planning productions could be tested occasionally during problemsolving. Plan-schemata could be represented as structuresthat have some probability of being activated at anytime during problem solving, rather than only at thetime a goal is activated. Effects of set could still besimulated if the probability of a plan-schema's activationdepended on the strengths of relevant productions,and these strengths were influenced by factors such asrecency of successful use of the various plans in thesystem (cf. Anderson, 1976; McDermott & Forgy,1977).

GENERAL DISCUSSION

This final section includes comments on the relation-

ship of constructions to the general question of illstructured problem solving and creativity, and on thegeneral issue of problem solving set.

Constructions in m-Structured ProblemsA property of many ill-structured problems is that

the problem space initially given in the problem isinsufficient for the solution to be achieved. The initialsituation may contain few if any of the componentsthat are eventually generated and brought together ina solution of the problem. Examples that are often usedinclude problems of composition, such as music, poetry,paintings, or scientific articles, in which the problemsolver adopts the goal of composing something andcreates most of the elements of the problem space, aswell as their arrangement, in order to solve the problem.

Geometry problems are not poetry, of course, butproblems with constructions seem to provide a nontrivialbut manageable case in which the initial problem spacemust be enriched in order for the problem to be solved.In the model we have described in this paper, constructions are required when available problem solving planshave prerequisites that are not present in the situation.Then the prerequisites of a plan become a goal thatthe problem solver tries to achieve. Construction ofthe prerequisites is itself a well-structured problem.However, solution of this problem requires operators(drawing lines) that are not in the set of operatorsavailable in the problem space of the main problem.2

In this analysis, constructions occur as solutionsto auxiliary problems that arise in solving the mainproblem. This view puts problems requiring constructions into the class of problems requiring more than oneproblem space in their solution. Simon (1973) discussedgeneral characteristics of problem solving with multipleproblem spaces and pointed out that solutions requireaccess to information that is used to generate newsubgoals, new constraints, and new problem solvingoperators.

For ill-structured problems of this kind, the theoryof well-structured problem solving must be extended byproviding a mechanism for coordinating activity indifferent problem spaces. In Perdix, coordination isachieved by use of general schemata that guide patternrecognition and pattern completion processes of different kinds. A new problem space is generated when apattern required by the planning process in one problemspace is incomplete. Then, problem solving activity istransferred to a problem space that includes operatorsappropriate for the kind of pattern completion that isrequired.

The general nature of this mechanism seems aplausible basis for productive problem solving in manysituations that involve composition. It seems a reasonable conjecture that an important component ofcomposition is a process of pattern completion guidedby the composer's knowledge of general schemata thatrepresent general forms of material that is included in


the work at various levels of structure. Flower andHayes' (in press) analysis of planning processes inwriting seems consistent with this general idea.

This idea also provides an intriguing possibilityregarding an important aspect of creativity. Investigatorswho have studied creative problem solving in artisticcomposition have emphasized the importance ofprocesses that result in the formulation of interestingproblems. Apparently, skill in problem formulation isimportant in distinguishing artists whose work isrecognized in the field from those who are competent,but not influential (Getzels & Csikszentmihalyi, 1976).The analysis presented here suggests that the ability toformulate problems may be based on knowledge in theform of general schemata and procedures for recognizingthe possibility of creating specific instances and combinations of general structures.

The general conclusion of this analysis is that inan ill-structured problem with an insufficient initialproblem space, problem solving takes place in a numberof different problem spaces. The contents of the initialproblem space depend on the problem solver's knowledge about the domain, so problems that are illstructured for some problem solvers may be wellstructured for more experienced problem solvers. Animportant feature of ill-structured problems is that newproblem spaces must be generated in the process ofproblem solving. There are probably many ways inwhich this occurs, but the mechanism that is in thepresent version of Perdix, involving generation ofproblems based on schemata that are also the basis ofplanning, seems to provide an interesting example ofthe process.

It should be remembered that the issue ofill-structuredproblems involves other factors in addition to insufficient initial problem spaces. Another way in which aproblem can have weak structure is to have an indefmitegoal that can be achieved by alternative combinationsof components or features. The idea of an indefmitegoal is represented in Perdix as a pattern recognitionsystem that identifies the various combinations ofcongruent components that are sufficient to prove thattriangles are congruent (Greeno, 1976). Since it canwork both with indefmite goals and incomplete initialproblem spaces, Perdix seems to provide a moderatelysuccessful extension of problem solving theory intothe domain of ill-structured problems. Of course, thereare important aspects of ill-structured problem solvingthat are' not represented in Perdix, such as the development of new problem solving operators and thediscovery of new concepts of strategies in the course ofproblem solving. Recent analyses of learning processesin problem solving systems (Anzai & Simon, 1979;Neches & Hayes, 1978; Sussman, 1975) suggest thatthese generative aspects of creative problem solvingmay be understood best as instances of learning.

Genc(:~l Nature of SetThr. traditional behaviorist explanation of set is that

a higher level response that has high strength dominatesperformance on the problem. The explanation in Perdixis consistent with this idea and adds a specific mechanism by which higher level responses can have theirdominating effect. The explanation gains in interestbecause the cognitive structure that causes set is thesame as the structure that permits constructions. It isof some interest to consider the relationship betweenthe mechanism proposed here and various phenomenaof problem solving set that have been investigatedexperimentally.

There are two general kinds of problem solving set.One kind is typified by Luchins' (1942) water-jugtasks and the examples from geometry discussed inthis paper, in which the problem solver is biased towardthe use of a problem solving plan or method. A secondkind of phenomenon is involved in functional fixedness,studied by Duncker (1945), in which use of an objector idea in one context prevents the problem solverfrom recognizing a potential use of that object or ideain a subsequent problem situation.

Biased top-down planning. The principle of schemabased planning represented in Perdix gives a straightforward explanation of problem solving set that involvesbias toward use of a specific problem solving method.For example, a system such as Perdix involving schematafor solution types in Luchins' (1942) water-jug taskwould show biases of the same kind as it shows for thegeometry problems considered in this paper, in agreement with performance of human subjects.

Lewis (1978) has also developed an analysis of set touse a specific problem solving method. In Lewis' system,a sequence of actions that occurs regularly can becomeintegrated into a single action. This integration caneliminate tests for appropriateness of intermediate stepsin the solution. Without performing these tests, theproblem solver may miss opportunities to solve theproblem in a more efficient way, as in the situationstudied by Luchins (1942). Using a letter-substitutiontask, Lewis obtained evidence that subjects integratedproblem solving steps in a way consistent with hismodel, but they maintained a capability of noticingconditions for applying alternative operations whenthey were instructed to use the alternatives if possible.It seems possible that Lewis' system could provide anexplanation for the development of structures thatcorrespond to plans in the version of Perdix describedin this paper. Plans are integrated action sequences,but, in general, use of a plan is optional, so that withappropriate instructions, a plan's inappropriate use couldbe avoided.

Functional fIXedness. Phenomena involving functional fixedness seem to be of a different character fromthose involving a specific problem solving method.Weisberg and Suls' (1973) analysis of the candle problemand Magone's (1977) analysis of the two-string problemidentified analysis of features of objects in the situationas an important component of the process of solvingthe problem. For example, in the candle problem, in


which a box must be seen as a potential supportingsurface, a condition for solution may be the problemsolver's noticing the flatness of the box. Magone foundthat subjects achieved a greater variety of solutions inthe two-string problem if they were initially promptedto consider features of objects than if they were initiallyprompted to find a solution of a specified kind, such asextension of one of the strings or causing a string toswing back and forth.

There are two alternative mechanisms that could bepostulated in the framework of a production systemfor explaining functional fiXedness, viewed as preventionof recognizing some feature of an object in the situation.One mechanism would cause functional fixedness byengaging in a top-down search for objects with featuresthat could be used in a plan-schema that was previouslyselected. For example, if the system had selected theplan of extending a string in the two-string problem,then the analysis of objects in the situation wouldinvolve their lengths rather than other features thatwould fit into different possible plans for solving theproblem. A second mechanism would involve competingproductions for identifying features of objects. Supposethat an object has two features that can be identifiedby different productions, and one of these is muchstronger than the other because of the way the objecthas already been used in the situation. Then, whenthat object is considered by the problem solver, theproduction that could identify the second feature maybe prevented from executing by the rapid executionof the stronger production.

The mechanism of top-down search is very similarto the mechanism that is represented in Perdix toprovide an explanation for problem solving set basedon top-down planning. Consider Figure 2 and recallthat Perdix solves this problem by proving that trianglesare congruent. The solution involves recognizing thepresence of triangles in the diagram, rather than anotherfeature that is also present, the fact that the two anglesinvolved in the goal of the problem have the samevertex. Perdix fails to notice a relevant feature of theproblem because a general plan-schema that includestests for different features is active and prevents testingfor the feature of a shared vertex. A simulation ofproblem solving in situations such as the candle ortwo-string problems would require a somewhat differentuse of this kind of structure, since there are severaldifferent objects in the situation, and, in some cases,failure to find a feature of an object needed for thecurrent plan apparently leads the problem solver topass by that object rather than to test the object foradditional features. However, it seems reasonable toconjecture that the basic structure of schema-basedplanning that causes Perdix to show problem solvingset in geometry would also produce functional fixednessin situations in which attempts to use a general planlead to the neglect of important features of objects.

Phenomena recently reported by Halstead-Nussloch(1978) can be interpreted by assuming a top-down

search for problem features. Halstead-Nussloch traineddifferent groups of subjects to identify serial patternsusing two different pattern languages taken from analysesby Restle (1970) and Simon (1972). Results indicatedthat subjects trained to represent serial patterns in termsof hierarchies of transformations applied that schemain the analysis of sequences that could be analyzedeither as hierarchies of transformations or as sequencescomposed of two or more interleaved subsequences.Another kind of set phenomenon that can be interpretedas resulting from top-down search is the difficulty ofsimple discrimination learning following a few trials ofrandom feedback, reported by Levine (1971). Levine'sinterpretation was that, during random feedback trials,subjects rejected simple hypotheses about stimulusattributes and removed them from consideration andthus were prevented from recognizing simple relationships between stimulus attributes and feedback onlater trials. Search for solution was apparently guidedby a metahypothesis that the correct rule was a sequential pattern, which permits a very large number ofpossibilities. Plans, as represented in Perdix, establishcategories of hypotheses that guide search for a solutionto a problem, and thus planning processes provide aspecific procedural hypothesis with which to understandLevine's findings.

The second mechanism, in which use of an objectin one way causes a feature of the object to be ignored,probably involves a different functional level of productions from the planning procedures that we haveconsidered in this paper. Some cases in which a featureof an object is ignored probably involve simple perceptual salience. In the candle problem, in which use of abox as a candlestand is inhibited by its use for holdingtacks, solution is facilitated if the word "box" is printedon the box, relative to the situation in which "tacks"is printed on the box (Glucksberg & Weisberg, 1966).This presumably aids recognition of the physical featuresof the box through simple direction of attention. Asimilar phenomenon has recently been reported bySweller and Gee (1978), who found that effects ofset in Luchins-type water-jug problems (Luchins, 1942)and Levine-type induction problems (Levine, 1971)were greatly reduced by perceptual manipulations. Inthe water-jug situation, Sweller and Gee virtuallyeliminated the effect of set by having subjects writetheir solutions numerically, rather than in terms ofletters designating the jars used. In induction problems,the manipulation involved giving feedback that describedthe correct stimulus, rather than stating whether thesubject's response was correct or incorrect. Both ofthese manipulations seem to be interpretable as avoidingthe removal of attention from aspects of the situationthat are important in solving a problem, and theythus support the hypothesis that set occurs because ofrelatively low salience of the important stimulusfeatures.

In other cases, however, suppression of detection ofa feature is apparently related to a more active process.


In Duncker's (1945) gimlet problem, use of the gimletas a hook was inhibited by the subject's use of thegimlet to bore holes for the conventional hooks thatwere available. In a production system, solution of theproblem of boring holes would strengthen productionsfor identifying features of the gimlet used in boringholes and could make less likely the recognition offeatures involved in serving as a hook.

All of the explanations of functional fIxednessproposed here are based on the idea that productionsmust be executed for the relevant features of objectsto be recognized. The required productions can beprevented by the dominance of other recognitionproductions, involving top-down search, ordinaryperceptual salience, or recent use in another context.

REFERENCE NOTES

I. Goldstein, L P. Elementary geometry theorem proving(Artificial Intelligence Memo 280). Cambridge, Mass: M.LT.,1973.

2. Larkin, J. H. Skilled problem solving in physics: A hierarchical planning model. Unpublished manuscript, University of California at Berkeley, September 1977.

3. Atwood, M. E., Polson, P. G., Jeffries, R., & Ramsey. H. R.Planning as a process of synthesis (Tech. Rep. SAI-78-144-DEN).Englewood, Colo: Science Applications Inc., December 1978.

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NOTES

1. The part of the system in which the productions forconstructions interact with the present planning system has notbeen debugged. At its present size, the system is relativelyexpensive to work with, and I believe that only technical insightswould be gained from completing the programming task. Theproductions for construction were debugged in an earlier system,described at meetings of the Midwestern Psychological Association in 1976, in which the decision to construct a line was madewhen conditions for setting a subgoal of congruent triangles weresatisfied.

2. This emphasizes that a problem is well or ill structured

only in relation to a specified problem space. In the specificationassumed here, problem solving operators are the propositions ofgeometry that permit inferences about properties and relations,and operators that permit drawing lines are in another problemspace. Of course, an expanded problem space could be specifiedin which both sets of operators were available. In that case,problems requiring constructions would be well dermed in theusual sense, and the construction of auxiliary lines would beguided by ordinary subgoals.

(Received for publication November 15,1978;revision accepted August 25,1979.)

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