The mechanism of restructuring in geometry

Mn, onr

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NThe Mechanism of Restructuring

; -Geometry

Stellan Ohisson

The Learning Research and Development Center.

University of Pittsburgh, Pittsburgh, PA 15260

Technical Report No. KUL-90-04

May, 1990

LEARNING RESEARCH AND DEVELOPMENT CENTER

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University of Pittsburgh

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The Mechanism of Restructuring

in Geometry

Stellan Ohisson

The Learning Research and Development Center,

University of Pittsburgh, Pittsburgh, PA 15260

Technical Report No. KUL-90-04

May, 1990

To appear in the Proceedings of the Twelfth Annual Conference of the Cognitive

Science Socie Massachusetts Institute of Technology, Cambridge, Massachusetts,

July 25-28, 1990.

DTI CSE' ,. -'

JUL 3 0 1990!

Preparation of this manuscript was supported, in part, by ONR grant N00014-89-J-1681. Approved for

public release; distribution unlimited. The opinions expressed do not necessarily reflect the positions

of the sponsoring agency, and no endorsement should be inferred. This report is a substantially

rewritten version of an earlier technical report: Ohlsson, S. (1983). Restructuring revisited. Iil.

Re-describing the problem situation as a heuristic in geometric problem solving (Technical Report No.

353). Uppsala, Sweden: Department of Psychology, University of Uppsala.

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11 TITLE (Include Security Classification)

The Mechanism of Restructuring in Geometry

12. PERSONAL AUTHOR(S)

Stellan Ohlsson13a. TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year,Month Day) " PAGE CO ,NT

Technical FROM TO May 1990Nh SUPPLEMENTARY. NOTATIOis report is a substantially rewritten version of an earlier technical report:Ohlsson, S. (1983). "Restructuring revisited. III. Re-describing the problem situation"

17 COSATI CODES 18 SUBJECT TERMS (Conlnue on reverse if necessary and identify by block numOer)

FIELD GROUP SUB-GROUP Search; Restructuring; Function; Mechanism; Heuristic;05 02 Operators; Path; Initial state; Goal state; Description

space.19 ABSTRACT (Continue on reverse if necessary and identify by block number)

Restructuring consists of a change in the representation of the current search state, a processwhich breaks an impasse during problem solving by opening up new search paths. A corpus of52 think-aloud protocols from the domain of geometry was scanned for evidence ofrestructuring. The data suggest that restructuring is accomplished by re-parsing the geometricdiagram.

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U3UNCLASSIF'EDUNLIMITED 0 SAME AS RPT - DTIC USERS Unclassified22a NAME OF RESPONSIBLE INDIVIDUAL 22J2EL)PH(NE (Inclucje Ae, CcdeI 22( -( '.

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O hisson 2 Restructuring in Geometry

Knowledge and Understanding in Human Learning

Knowledge and Understanding in Human Learning (KUL) is an umbrella term for aloosely connected set of activities lead by Stellan Ohisson at the Learning Researchand Development Center, University of Pittsburgh. The aim of KUL is to clarify therole of world knowledge in human thinking, reasoning, and problem solving. Worldknowledge consists of concepts and principles, and contrasts with facts (episodicknowledge) and with cognitive skills (procedural knowledge). The long term goal is toanswer six questions: How can the concepts and principles of particular domains beidentified? How are concepts and principles acquired? How can the acquisition ofconcepts and principles be assessed? How are concepts and principles encoded in themind? How are concepts and principles utilized in performance and learning? Howcan instruction facilitate the acquisition and utilization of concepts and principles (asopposed to episodic or procedural knowledge)? Different methodologies are used toinvestigate these questions: Psychological experiments, protocol studies, computersimulations, historical studies, semantic, logical, and mathematical analyses,instructional intervention studies, and so on. A list of KUL reports appear at the backof this report.

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May KUL-90-04 1990

Ohisson 3 Restructuring in Geometry

Table of Contents

Abstract 4

Introduction 5

Method 6

Results 7

Case 1: Deliberate restructuring 7

Case 2: Goal-driven restructuring 8

Case 3: Hint-driven restructuring 9

Discussion 10

References 11

List of KUL reports 12

May KUL-90-04 1990


Abstract

Restructuring consists of a change in the representation of the current searchstate, a process which breaks an impasse during problem solving by opening up newsearch paths. A corpus of 52 think-aloud protocols from the domain of geometry wasscanned for evidence of restructuring. The data suggest that restructuring isaccomplished by re-parsing the geometric diagram.

May KUL-90-04 1990

Ohlsson 5 Restructuring in Geometry

Introduction

A wide variety of problem solving processes have been analyzed in terms ofheuristic search (Newell & Simon, 1972). For example, in geometry proofs thegeometric theorems (operators) are applied to the mental representation of thediagram (the knowledge state) until the desired proposition (the goal state) has beenattained (Anderson, 1981). The stepwise character of heuristic search contrasts withthe Gestalt hypothesis that problem solving proceeds through (a) an initial,unsuccessful, attack on the problem, (b) a more or less protracted impasse, and (c) arestructuring of the problem, which is typically, but not necessarily, followed byinsight (Ohlsson, 1984a).

Several attemptz have been made to reconcile the information processing andGestalt hypotheses. Simon (1966) proposed that it helps to sleep on a problem,because goal tree information is forgotten faster than problem information. After apause, a new goal tree is built on the basis of more knowledge about the problem.Langley and Jones (1988) interpret an impasse as a failure to retrieve the relevantproblem solving operator. Insight occurs when some external stimulus causes enoughactivation to spread to that operator to allow its retrieval. A related hypothesisclaims that insight occurs when an appropriate analogy is retrieved (Keane, 1988).Both the differential rate of forgetting hypothesis and the spread of activationhypothesis require that the problem solver moves attention away from the problem,and so cannot explain insight during ongoing problem solving. Greeno and Berger(1987) have proposed that insights occur when a problem solver breaks an impasse byconstructing new functional knowledge, i. e., new problem solving operators. A newoperator is constructed by inferring that an object can fulfill a particular function, e.g., that a screwdriver can be used to complete an electric circuit. This follows from thefact that the screwdriver is made of metal, in conjunction with the general principlethat metallic objects conduct electricity. Several researchers have proposed thatproblem representations can be improved by the construction of macro-operators(Amarel, 1968; Korf, 1985). Koedinger and Anderson (in press) have proposed therelated idea that geometry experts combine geometric theorems into larger inferenceschemas, called diagram configuration schemas, which allow them to find a proofwithout step-by-step search of the proof space. The macro-operator and diagramconfiguration hypotheses explain expert performance, but they do not explaininsights by novices. All of these hypotheses locate restructuring in the processes ofproblem solving.

In contrast, I have proposed that restructuring involves a change in the mentalrepresentation of the current search state (Ohlsson, 1984b). A change in therepresentation implies that objects, relations, and properties which initially are seenas instances of certain concepts are being re-encoded as instances of other concepts.

May KUL-90-04 1990


For example, an object which is initially encoded as a hammer might in the course ofproblem solving become re-encoded as a pendulum weight, a line may be re-encodedas a triangle side, and so on. Re-encoding a search state changes the set of operatorswhich are applicable in that state, and thus breaks an impasse by opening up newsearch paths. A similar theory has been proposed by Kaplan and Simon (in press) toexplain restructuring in the Mutilated Checker Board Problem. The critique byMontgomery (1988) does not touch those aspects of the theory that are of mainconcern in this paper. The purpose of the present paper is to provide evidence forre-encoding from the domain of geometry, and to propose a mechanism forre-encoding in that domain.

Table 1. Geometric theorems acquired by the subjects.

Theorem 1. Supplementary angles are congruent.

Theorem 2. Vertical angles are congruent.

Theorem 3. The supplementary angle of a right angle is a right angle.I

Theorem 4. If two angles and their common side in one triangle are congruent to thecorresponding angles and their common side in another triangle, thenthe two triangles are congruent.

Theorem 5. If two sides in a triangle are congruent, then their opposite angles arecongruent; and vice versa.

Method

Three undergraduate psychology students participated in an experimentalcourse in elementary geometry. The experimenter saw each subject individually insessions that lasted approximately one hour each. The subjects learned basictheorems of plane geometry, the first five of which are shown in Table 1. A typicalsession began with free recall of previously learned theorems, continued with theintroduction of new theorems, and ended with problem solving practice. The subjectshad the theorems available during problem solving, and they were instructed tothink aloud. The data consist of 52 think-aloud protocols, representing a total ofapproximately nine hours of problem solving effort.

May KUL-90-04 1990


Results

The protocols were scanned for the occurrence of restructuring events. Ten suchevents were found. The three most informative events will be analyzed below. Theyillustrate deliberate restructuring, goal driven restructuring, and restructuring inresponse to a hint.

Case 1: Deliberate restructuring. Subject S3 was given the problem in Figure 1after she had studied Theorems 1-5 (see Table 1). She began by proving that trianglesAED and BEC are congruent, and then entered an impasse. In fragments F65-F67

(see Table 2) she deliberately sets out to see the problem from many viewpoints. Theprocess of restructuring proceeds through three steps. First, she mentally cuts thefigure along the diagonal CA, forming the triangles CDA and CBA (F68-F70). Shethen mentally cuts the figure along the other diagonal, forming the triangles DCBand DBA (F71-F74). Finally, she keeps one triangle from each pair, as it were, andsets herself the task of proving them congruent (F75-F77). Figure 2 gives adiagrammatic analysis of the process. The geometric objects perceived by the subjectare drawn in bold lines, while the rest of the diagram is drawn in broken lines.Restructuring was not followed by insight in this case. The subject worked on theproblem for twelve minutes without solving it.

.....................................................................................................................

Table 2. Protocol excerpt from Subject S3.

F65. but perhaps one can see this in some other way alsoF66. one can perhaps see this from many viewpoints hereF67. now we shall seeF68. one can see it asF69. CDA and CBAF70. trianglesF71. one can see it onF72. DCB and DBA insteadF73. yes exactly yesF74. those twoF75. wellF76. now I can see this in another wayF77. CDB and CAD ought to be congruent here in some way

May KUL-90-04 1990


Case 2: Goal-driven restructuring. S1 was given the problem in Figure 1 as hisfirst problem after studying Theorems 1-5 (see Table 1). S1 misunderstood the goal ofthe problem to be to prove that angle ADC is congruent to angle BCD. When theprotocol excerpt in Table 3 begins, he has proved that angles EDA and ECB arecongruent by proving them corresponding parts of the congruent triangles EDA andECB. He then sets himself the goal of proving that the remaining parts, i. e., anglesEDC and ECD, are equal (F43). His plan is to prove that they are equal by provingthat the sides of the triangle EDC are equal (F42-F45).


F42. yes now I am thinking about whether one can prove that these two sides [DE,EC] are equally longF43. because if they are then those two angles [EDC, ECD] which are just theremaining parts of those angles which I want to get [ADE, BCD] must be equally longF44. so then this and that angle [ADE, BCD] must be equally bigF45. and then the problem is solvedF46. so it is now a question of proving that it is isoscelesF47. that triangle [EDC]F48. and that I cannotF49. but perhaps one can do it in some other way(What are you thinking?)F50. well now I am thinkingF51. well it is the same problemF52. but from another angleF53. yes if this oneF54. is those two lines [ED, EC] are equally longF55. I am thinkingF56. yes but they must beF57. since they are parts ofF58. it is congruentF59. these two here are congruent [triangles EDA, ECB]F60. and it is [ED, EC] corresponding sides in the triangles [EDA, ECB]F61. therefore these two sides [ED, EC] are equally long

This goal is reformulated as proving that the triangle EDC is isosceles (F46-F47).This view of the problem leads to an impasse (F48-F49). Prompted by theexperimenter to continue to think-aloud, he states that he is thinking about the same

May KUL-90-04 1990

OhIsson Restructuring in Geometry

D C C

D G

A B A B

Prove angles ECD and CDE congruent. Prove line segments AG and BD congruent.

Figure 1. Problem 1. Figure 4. Problem 2.

C C

>E lop / E<

A B A B

Figure 5. Analysis of S2's re-encoding process. Perceived geometric figuresare drawn in bold lines, the rest of the figures in broken lines.

OhLsson 9 Restructuring in Geometry

problem but from another angle (F50-F52): he has re-encoded ED and EC as lines(F54). The goal is still to prove them congruent (F53-F55). He suddenly realizes thatED and EC are corresponding sides of the two triangles EDA and ECB, which he hasalready proved congruent (F56-F61). Figure 3 shows a diagrammatic analysis of theprocess with perceived geometric objects in bold lines and the rest of the diagram--thebackground--in broken lines. The subject quickly completed the correct solution.

Case 3: Hint-driven restructuring. S2 attempted Problem 2 (see Figure 4) afterhaving learned the five theorems in Table 1, plus four others. She decided to provetriangles AED and BEG congruent and quickly reached an impasse. The protocolexcerpt in Table 4 begins


(What other triangles could be congruent?)F109. what othersFl10 could there be others which are congruentF1ll. huh(That could be. You have now been working the hypothesis that the whole point is toprove that those two triangles [AED, BEG] are congruent.)F112. yes(And just now you reached the conclusion that you cannot do that with theinformation you have. Can you find two other triangles which one can find which onecould believe could be congruent?)F113. congruent exactly alikeF114. no that is impossible there are no othersF115. it cannot beF116. there are only one otherF117. also hypothetically then this line hereF118. then there are two hereF119. and those two here can surely never be congruentF120. these two here can surely never be congruentF121. no I do not understand thatF122. butF123. now I see itF124. I have forgotten this one here [AGB or BDA]

May KUL-90-04 1990

OhIsson Restructuring in Geometry

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O hisson 10 Restructuring in Geometry

when the experimenter gives her the hint that there are other pairs of triangles in thefigure that might be congruent. She first rejects this suggestion (F113-F115). Shethen runs through the triangles in the figure (F113-F121), and concludes that thereare no other congruent triangles in the figure (F121). She then suddenly sees thetriangles AEG and BDA (F123-F124). Figure 5 shows a diagrammatic analysis of theprocess with perceived geometric objects drawn in bold lines and the rest of thediagram drawn in broken lines. In spite of this restructuring, the subject failed tosolve the problem.

Discussion

The restructuring process revealed in these three protocol excerpts consists inre-encoding the given figure. The diagram--the set of lines on the paper--containswithin it a large number of different geometric objects (angles, sides, triangles, etc.).Only some of those geometric objects are perceived at any one time. The others recedeinto the background. In particular, if a line configuration is perceived in one way,then alternative encodings of that same line configuration recede into thebackground. Restructuring consists of switching to one of the alternative encodings.How does the switching mechanism work? The data suggest that re-encoding is doneby re-parsing the diagram. During initial problem perception complex objects (e. g.,triangles) are constructed out of simpler objects (e. g., lines). This process is a searchthrough a description space (Ohlsson, 1984b). Alternative interpretations of theperceptual information are possible, so some choices are made, resulting in aparticular encoding of the given diagram. When an impasse forces the problem solverto re-encode the problem, he/she backs up in the description space, dismantles his/herprevious encoding, and traverses another path through the description space. Thisprocess breaks an impasse by allowing other operators (geometric theorems) to applyto the current state. Restructuring is a rare event: There was approximately onerestructuring event per hour of problem solving effort in the present study.Restructuring does not necessarily lead to insight: In two of the three excerptspresented above, the subject failed to solve the problem. This study supports the ideathat diagram parsing is central in geometry (Koedinger & Anderson, in press), butthe validity of the re-parsing mechanism for other domains than geometry remainsan open question. For example, a different mechanism seems to be responsible forre-encoding of the Mutilated Checker Board Problem (Kaplan & Simon, in press).

May KUL-90-04 1990


References

Amarel, S. (1968). On representations of problems of reasoning about actions. In D.Michie (Ed.), Machine intelligence. Vol 3. (pp. 131-171). Edinburgh, UK:Edinburgh University Press.

Anderson, J. R. (1981). Tuning of search of the problem space for geometry proofs.Proceedings of the Seventh International Joint Conference on ArtificialIntelligence (pp. 165-170). Vancouver, Canada: University of British Columbia.

Greeno, J. G., & Berger, D. (1987). A model of functional knowledge and insight(Technical Report No. GK-1). Berkeley, CA: University of California.

Kaplan, C. A. & Simon, H. A. (in press). In search of insight. Cognitive Psychology.

Keane, M. (1988). Analogical problem solving. New York, NY: Wiley.

Koedinger, K. R. & Anderson, J. R. (in press). Abstract planning and perceptual

chunks: Elements of expertise in geometry. Cognitive Science.

Korf, R. E. (1985). Learning to solve problems by searching for macro-operators.Marshfield, MA: Pittman.

Langley, P. & Jones, R. (1988). A computational model of scientific insight. In R. J.Sternberg (Ed.), The nature of creativity. Contemporary psychological

perspectives (pp. 177-201). Cambridge, MA: Cambridge University Press.

Montgomery, H. (1988). Mental models and problem solving: Three challenges to a

theory of restructuring and insight. Scandinavian Journal of Psychology, 29,85-94.

Newell, A. & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ:Prentice-Hall.

Ohlsson, S. (1984a). Restructuring revisisted. I. Summary and critique of the Gestalttheory of problem solving. Scandinavian Journal of Psychology, 25, 65-78.

Ohlsson, S. (1984b). Restructuring revisited. II. An information processing theory ofrestructuring and insight. Scandinavian Journal of Psychology, 25, 117-129.

Simon, H. A. (1966). Scientific discovery and the psychology of problem solving. InMind and cosmos: Essays in contemporary science and philosophy. Vol. III. (pp.22-40). Pittsburgh, PA: University of Pittsburgh.

May KUL-90-04 1990


KUL Reports

1985

Ohlsson, S., & Langley, P. (April, 1985). Psychological evaluation of path hypothesesin cognitive diagnosis (Technical Report No. 1985/2). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

1986

Ohlsson, S. (January, 1986). Some principles of intelligent tutoring (Technical ReportNo. 1986/2). Pittsburgh, PA: Learning Research and Development Center,University of Pittsburgh.

Ohlsson, S. (June, 1986). Computer simulation and its impact on educational researchand practice (Technical Report No. 1986/14). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

Ohlsson, S. (October, 1986). Sense and reference in the design of interactiveillustrations for rational numbers (Technical Report No. 1986/18). Pittsburgh,PA: Learning Research and Development Center, University of Pittsburgh.

1987

Ohlsson, S. (April, 1987). A semantics for fraction concepts (Technical Report No.KUL-87-01). Pittsburgh, PA: Learning Research and Development Center,University of Pittsburgh.

Ohlsson, S. (September, 1987). Trace analysis and spatial reasoning: An example ofintensive cognitive diagnosis and its implications for testing (Technical ReportNo. KUL-87-02). Pittsburgh, PA: Learning Research and DevelopmentCenter, University of Pittsburgh.

Ohisson, S., Nicholas, S., & Bee, N. (December, 1987). Interactive illustrations forfractions: A progress report (Technical Report No. KUL-87-03). Pittsburgh,PA: Learning Research and Development Center, University of Pittsburgh.

Ohlsson, S., & Rees, E. (December, 1987). Rational learning: Deriving arithmeticprocedures from state constraints (Technical Report No. KUL-87-04).Pittsburgh, PA: Learning Research and Development Center, University ofPittsburgh.

May KUL-90-04 1990


1988

Ohlsson, S. (February, 1988). Mathematical meaning and applicational meaning inthe semantics for fractions and related concepts (Technical Report No.KUL-88-01). Pittsburgh, PA: Learning Research and Development Center,University of Pittsburgh.

Ohlsson, S. (March, 1988). Principled understanding of subtraction with regrouping:A mathematical analysis (Technical Report No. KUL-88-02). Pittsburgh, PA:Learning Research and Development Center, University of Pittsburgh.

Ohlsson, S., & Rees, E. (August, 1988). An information processing analysis ofconceptual understanding in the learning of arithmetic procedures (TechnicalReport No. KUL-88-03). Pittsburgh, PA: Learning Research and DevelopmentCenter, University of Pittsburgh.

Ohlsson, S. (December, 1988). Towards intelligent tutoring systems that teachknowledge rather than skills: Five research questions (Technical Report No.KUL-88-04). Pittsburgh, PA: Learning Research and Development Center,University of Pittsburgh.

1989

Ohlsson, S. (January, 1989). Knowledge requirements for teaching: The case offractions (Technical Report No. KUL-89-01). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

Ohlsson, S. (April, 1989). Cognitive science and instruction: Why the revolution is nothere (yet) (Technical Report No. KUL-89-02). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

Robin, S., & Ohlsson, S. (August, 1989). Impetus than and now: A detailedcomparison between Jean Buridan and a single contemporary subject(Technical Report No. KUL-89-03). Pittsburgh, PA: Learning Research andDevelopment Center, University of Pittsburgh.

Ohlsson, S. (Ed.), (October, 1988). Aspects of cognitive conflict and cognitive change(Technical Report No. KUL-89-04). Pittsburgh, PA: Learning Research andDevelopment Center, University of Pittsburgh.

Leinhardt, G., & Ohlsson, S. (November, 1989). Tutorials on the structure of tutoringfrom teachers (Technical Report No. KUL-89-05). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

May KUL-90-04 1990

OhIsson 14 Restructuring in Geometry

Ernst, A., & OhIsson, S. (December, 1989). The cognitive complexity of the regroupingand augmenting algorithms for subtraction: A theoretical analysis (TechnicalReport No. KUL-89-06). Pittsburgh, PA: Learning Research and DevelopmentCenter, University of Pittsburgh.

1990

Ohisson, S., & Rees, E. (January, 1990). Adaptive search through constraintviolations. (Technical Report No. KUL-90-01). Pittsburgh, PA: LearningResearch and Development Center, University of Pittsburgh.

Ohlsson, S., & Hall, N. (February, 1990). The cognitive function of embodiments inmathematics instruction (Technical Report No. KUL-90-02). Pittsburgh, PA:Learning Research and Development Center, University of Pittsburgh.

Ohlsson, S., & Rees, E. (March, 1990). Comparative evaluation of knowledge-basedsimulation models of procedural learning (Technical Report No. KUL-90-03).Pittsburgh, PA: Learning Research and Development Center, University ofPittsburgh.

OhIsson, S. (May, 1990). The mechanism of restructuring in geometry (TechnicalReport No. KUL-90-04). Pittsburgh, PA: Learning Research and DevelopmentCenter, University of Pittsburgh.

May KUL-90-04 1990

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