zhang (1994) representations in distributed cognitive taskswexler.free.fr/library/files/zhang...

Preprint: To appear in Cognitive Science, 18, 87-122, 1994.

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Representations in Distributed Cognitive Tasks

Jiajie ZhangThe Ohio State University

[email protected]

Donald A. NormanApple Computer, [email protected]

ABSTRACT

In this paper we propose a theoretical frame-work of distributed representations and amethodology of representational analysis for thestudy of distributed cognitive tasksÑtasks thatrequire the processing of information distributedacross the internal mind and the external envi-ronment. The basic principle of distributed rep-resentations is that the representational systemof a distributed cognitive task is a set of internaland external representations, which togetherrepresent the abstract structure of the task. Thebasic strategy of representational analysis is todecompose the representation of a hierarchicaltask into its component levels so that the repre-sentational properties at each level can be inde-pendently examined. The theoretical frameworkand the methodology are used to analyze the hi-erarchical structure of the Tower of Hanoi prob-lem. Based on this analysis, four experiments aredesigned to examine the representational proper-ties of the Tower of Hanoi. Finally, the nature ofexternal representations is discussed.

This research was supported by a grant to DonaldNorman and Edwin Hutchins from the Ames ResearchCenter of the National Aeronautics & Space Agency,Grant NCC 2-591 in the Aviation Safety/AutomationProgram, technical monitor, Everett Palmer. Additionalsupport was provided by funds from the AppleComputer Company and the Digital EquipmentCorporation to the Affiliates of Cognitive Science atUCSD.

We are very grateful to Ed Hutchins for many in-spiring discussions throughout this study. We wouldalso like to thank David Kirsh, Jean Mandler, Mark St.John, Hank Strub, and Kazuhiro Takahashi for manygood comments during this study, James Greeno andDaniel Reisberg for thoughtful comments and suggestionson an earlier version of the manuscript, and SandraHong, Lin Riley, and Richard Warren for their assis-tance in the experiments.

Correspondence and requests for reprints should besent to Jiajie Zhang, Department of Psychology, The OhioState University, 1885 Neil Avenue, Columbus, OH43210.

People behave in an information richenvironment filled with natural and ar-tificial objects extended across space andtime. A wide variety of cognitive tasks,whether in everyday cognition, scien-tific practice, or professional life, requirethe processing of information dis-tributed across the internal mind andthe external environment. It is the in-terwoven processing of internal and ex-ternal information that generates muchof a person's intelligent behavior (e.g.,Hutchins, 1990, in preparation;Norman, 1988, 1991, in press).

The traditional approach to cogni-tion, however, often assumes that rep-resentations are exclusively in the mind(e.g., as propositions, schemas, produc-tions, mental images, connectionistnetworks, etc.). External objects, if theyhave anything to do with cognition atall, are at most peripheral aids. For in-stance, written digits are usually consid-ered as mere memory aids for calcula-tion. Thus, because the traditional ap-proach lacks a means of accommodatingexternal representations in its ownright, it sometimes has to postulatecomplex internal representations to ac-count for the complexity of behavior,much of which, however, is merely areflection of the complexity of the envi-ronment (e.g., Kirlik, 1989; Simon, 1981;Suchman, 1987).

This paper addresses the represen-tational issues in distributed cognitivetasksÑtasks that require the processingof information distributed across the in-ternal mind and the external environ-

Distributed Cognitive Tasks2

ment, focusing on three problems: (a)the distributed representation of infor-mation; (b) the interaction between in-ternal and external representations; and(c) the nature of external representa-tions. In the first part of this paper, webegin with an introduction to the repre-sentational effect, then we propose atheoretical framework of distributedrepresentations and a methodology ofrepresentational analysis for the studyof distributed cognitive tasks. In thesecond part, we illustrate the principlesof distributed representations and thestrategies of representational analysisthrough the analysis of the representa-tional structure of the Tower of Hanoiproblem. In the third part, we designfour sets of Tower of Hanoi isomorphsfor empirical investigations. In the lastpart, we summarize our major claimsand discuss the general properties ofdistributed cognitive tasks.

PHENOMENON, THEORY, ANDMETHODOLOGY

The Representational EffectThe representational effect refers to thephenomenon that different isomorphicrepresentations of a common formalstructure can cause dramatically differ-ent cognitive behaviors. One obviousexample is the representation of num-bers (for cognitive analyses, seeNickerson, 1988; Norman, in press;Zhang, 1992; Zhang & Norman, 1993).We are all aware that Arabic numeralsare more efficient than Roman numer-als for multiplication (e.g., 73 ´ 27 iseasier than LXXIII ´ XXVII), eventhough both types of numerals repre-sent the same entitiesÑnumbers. Themost dramatic case is probably theCopernican revolution, where the

change from the geocentric representa-tion of the solar system (Ptolemaic sys-tem) to the heliocentric representation(Copernican system) laid the founda-tion of modern science and fundamen-tally changed people's conception of theuniverse.

Psychological studies of the repre-sentational effect in problem solvingand reasoning have focused on a fewwell-structured problems, includingthe Tower of Hanoi problem (e.g., Hayes& Simon, 1977; Kotovsky & Fallside,1989; Kotovsky, Hayes & Simon, 1985;Simon & Hayes, 1976), the Chinese Ringpuzzle (Kotovsky & Simon, 1990), theHobbits-Orcs problem (e.g., Greeno,1974; Jeffries, Polson, & Razran, 1977;Thomas, 1974), Wason's selection task(e.g., Cheng & Holyoak, 1985; Evans,1983; Margolis, 1987; Wason, 1966;Wason & Johnson-Laird, 1972), andWason's THOG problem (e.g., Griggs &Hewstead, 1982; O'Brien, Noveck,Davidson, Fisch, Lea, & Freitag, 1990;Wason & Johnson-Laird, 1969). The ba-sic finding is that different representa-tions of a problem can have dramaticimpact on problem difficulty even if theformal structures are the same. Onecharacteristic of these problems is thatthey all require the processing of bothinternal and external information.However, most of these studies eitherexclusively focused on internal repre-sentations or, when taking external rep-resentations into account, failed to sepa-rate them from internal representa-tions.

In this paper, we argue that inter-nal and external representations are twoindispensable parts of the representa-tional system of any distributed cogni-tive task. To study a distributed cogni-tive task, it is essential to decomposethe representation of the task into its in-


ternal and external components so thatthe different functions of internal andexternal representations can be identi-fied.

Distributed RepresentationsThe basic principle of distributed repre-sentations is that the representationalsystem of a distributed cognitive taskcan be considered as a set, with somemembers internal and some external.Internal representations are in themind, as propositions, productions,schemas, mental images, connectionistnetworks, or other forms. External rep-resentations are in the world, as physi-cal symbols (e.g., written symbols, beadsof abacuses, etc.) or as external rules,constraints, or relations embedded inphysical configurations (e.g., spatial re-lations of written digits, visual and spa-tial layouts of diagrams, physical con-straints in abacuses, etc.). Generally,there are one or more internal and ex-ternal representations involved in anydistributed cognitive task.

Figure 1 shows the representa-tional system of a task with two internaland two external representations. Eachinternal representation resides in a per-son's mind, and each external represen-tation resides in an external medium.The internal representations form aninternal representational space, and theexternal representations form an exter-nal representational space. These twospaces together form a distributed repre-sentational space, which is the represen-tation of the abstract task space that de-scribes the abstract structures and prop-erties of the task.

We need to clarify our use of theterm "representation". In our presentstudy, the referent (the representedworld) of a representation (the repre-senting world) is an abstract structure.

For simple tasks, a representation andits referent can be perceived by boththeorists and task performers. For ex-ample, both theorists and task perform-ers know that written numerals,whether they are Arabic or Roman, arethe representations of abstract numbers.For complex tasks, however, a represen-tation and its referent are usually onlymeaningful from the point of view oftheorists. To a task performer, a repre-sentation does not represent anything:it is simply the medium (internaland/or external) on which the task per-former performs the task. For example,for the Tower of Hanoi problem that wewill consider later, the three problemsin Figure 12 all represent the same ab-stract structure (i.e., problem space, seeFigure 3) to a theorist. To a task per-former, however, they are simply threedifferent problems and they do not rep-resent anything, though the task per-former might notice some regularitiesacross these problems. Our notion ofrepresentation is essential for our pre-sent studies of the representationalproperties of distributed cognitive tasks.By considering alternative representa-tions of a common abstract structure,we can identify the factors that affect thecognitive behavior in distributed cogni-tive tasks.

Our current approach to dis-tributed cognitive tasks demands: (a)the consideration of the internal and ex-ternal representations of a distributedcognitive task as a representational sys-tem; (b) the explicit decomposition ofthe representational system into its in-ternal and external components; and (c)the identification of the different func-tions of internal and external represen-tations in cognition. The traditionalapproach to cognition is not appropriatefor the study of distributed cognitive


Internal 1

Internal 2

External 1

External 2

Abstract Task Space

Distributed Representational Space

Abstract

Internal

Repres

entatio

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SpaceExternal RepresentationalSpace

Figure 1. The theoretical framework of distributed representations. The internal representa-tions form an internal representational space, and the external representations form an exter-nal representational space. The internal and external representational spaces together form adistributed representational space, which is the representation of the abstract task space.

tasks, because (a) it considers externalrepresentations as mere peripheral aidsto cognition, and (b) it often mixes ex-ternal representations with internalrepresentations. Furthermore, the tra-ditional approach often mistakenlyequates a task's distributed representa-tion that has both internal and externalcomponents to the task's internal repre-sentation. This confusion often leadsone to postulate complex internalmechanisms to explain the complexstructure of the wrongly identified in-ternal representation, much of which ismerely a reflection of the structure ofthe external representation.

Representational AnalysisRepresentational analysis is a method-ology for the study of the representa-tional effect in distributed cognitivetasks. It is based on hierarchical repre-sentations, isomorphic representations,and distributed representations.

Many distributed cognitive taskshave multi-level hierarchical represen-tations (see Zhang, 1992, for a few ex-amples). At each level of a task's hier-archical representation, there is an ab-stract structure that can be implementedby different isomorphic representations.For some levels, the isomorphic repre-sentations can be distributed representa-tions. By decomposing the representa-


tion of a task into its component levels,we can identify the representationalproperties at each level that are respon-sible for a different aspect of the repre-sentational effect. It should be notedthat different tasks usually have differ-ent hierarchical structures and have dif-ferent representational properties attheir component levels. The key of themethodology of representational analy-sis is the strategy of decomposing a taskinto its component levels and studyingthe representational properties at eachlevel.

The methodology of representa-tional analysis is fully illustrated in theanalysis of the representational struc-ture of the Tower of Hanoi problem innext section. It has also been used tostudy several real world problems, in-cluding numeration systems, relationalinformation displays, and cockpit in-strument displays (Zhang, 1992).

THE REPRESENTATIONALSTRUCTURE OF THE TOH

The Tower of Hanoi (henceforth TOH;see Figures 2 and 3) is a well-studiedproblem. Much of the research has fo-cused on its isomorphs and problemrepresentations (e.g., Hayes & Simon,1977; Kotovsky & Fallside, 1989;Kotovsky, Hayes & Simon, 1985; Simon& Hayes, 1976). The basic finding is thatdifferent problem representations canhave dramatic impact on problem diffi-culty even if the formal structures arethe same. Many of these studies eitherexplicitly or implicitly mentioned ex-ternal representations. However, theydid not consider internal and externalrepresentations as two indispensableparts of the representational system ofthe TOH, and they did not separate ex-ternal representations from internalones. Furthermore, the hierarchical

structure of the TOH was not analyzedin these studies. In this section, we usethe principles of distributed representa-tions and the methodology of represen-tational analysis described in last sectionto analyze the representational struc-ture of the TOH in a systematic way.

Figure 2. The standard Tower of Hanoi prob-lem. The task is to move the three disks fromone configuration to another, following tworules: (1) Only one disk can be transferred ata time; (2) A disk can only be transferred to apole on which it will be the largest.

Rule RepresentationsThe standard TOH1 shown in Figure 2has two internal rules: (1) only one diskcan be transferred at a time; (2) a diskcan only be transferred to a pole onwhich it will be the largest. These tworules have to be memorized. The TOHin Figure 2 also has one external rule:(3) only the largest disk on a pole can betransferred to another pole. In the rep-resentation shown in Figure 2, Rule 3need not be stated explicitly because thephysical structure of the disks and polescoupled with Rule 1 guarantee that itwill be followed. But if the disks werenot stacked on poles, explicit statementof Rule 3 would be necessary. In ourstudies we used four rules2 :

1 The disk sizes of the traditional TOH are the reverseof those shown in Figure 2: the largest disk is at thebottom and the smallest is at the top. The disk sizes havebeen reversed to make all experimental designsconsistent. We call the size-reversed TOH with disksand poles in Figure 2 our standard TOH, because allother TOH isomorphs in our present study were derivedfrom this version.2 Experiments 1A and 1B used all four rules.Experiments 2 and 3 only used Rules 1, 2, and 3, whichare the three rules for the standard TOH.


S1 S2

S3

E1

E2

E3

Figure 3. The problem space of the Tower of Hanoi problem. Each rectangle shows one of the 27 pos-sible configurations (states) of the three disks on the three poles. The lines between the rectanglesshow the transformations from one state to another when the rules are followed. S1, S2, and S3 arethree starting states, and E1, E2, and E3 are three ending states. They will be used later in the ex-periments.

Rule 1: Only one disk can be transferredat a time.

Rule 2: A disk can only be transferred toa pole on which it will be thelargest.

Rule 3: Only the largest disk on a polecan be transferred to anotherpole.

Rule 4: The smallest disk and the largestdisk can not be placed on a sin-gle pole unless the mediumsized disk is also on that pole.

Any of these four rules can be ei-ther internal or external. Internal rulesare memorized rules that are explicitlystated as written propositions in the in-structions for experiments. Externalrules are not stated in any form in theinstructions. They are the constraintsthat are embedded in or implied byphysical configurations and can be per-ceived and followed without being ex-plicitly formulated. Some externalrules may sometimes depend on otherinternal rules and/or some background


knowledge. For example, the externalRule 3 in the standard TOH depends onthe internal Rule 1, because Rule 3 is arule about the movement of a single ob-ject3 . In addition, Rules 2 and 3,whether they are implemented as ex-ternal or internal rules, require thatonly one hand can be used to move theobjects and the objects can only beplaced on the three poles (or plates, seeFigures 11 & 12), because otherwise wecan always use the spare hand or an-other place (e.g., table surface) as a tem-porary holder (equivalent to the addi-tion of a fourth pole for the standardTOH) such that Rules 2 and 3 can be by-passed. These two extra requirementsare embedded in the cover stories forthe experiments (see Figure 9), and theyare not counted as internal rules.Furthermore, Rule 2 in the Waitressand Coffee TOH (see Figure 11) is an ex-ternal rule that needs some culturalknowledge: spilling coffee in front of acustomer is not a good behavior for awaitress or a waiter (see Experiment 1Bfor details). Even though some externalrules are not fully independent and nottruly external, we still call them exter-nal rules in the sense that they are notstated and memorized but neverthelessfunctionally equivalent to those that areexplicitly stated and memorized.

In the experiments that follow, wevaried the number of external rules. InCondi t ion I 1 (Figure 11A, inExperiment 1B) and Condition I123(Figure 12A, in Experiment 2), no rule isexternal. In Conditions I1-E3 (Figure11B, in Experiment 1B) and I 1 2 - E 3(Figure 12B, in Experiment 2), Rule 3 is

3 The four rules for the Tower of Hanoi are not fully or-thogonal. Rules 2, 3, and 4 are orthogonal to one an-other and Rule 4 is orthogonal to Rule 1. However,Rules 2 and 3 are not orthogonal to Rule 1, because Rule1 is the prerequisite of Rules 2 and 3.

external. In Condition I1-E23 (Figure11C, in Experiment 1B; and Figure 12C,in Experiment 2), Rules 2 and 3 are ex-ternal. In Condition I1-E234 (Figure 11D,in Experiment 1B), Rules 2, 3, and 4 areexternal. Detailed explanations of theseexternal rules are described inExperiment 1B and Experiment 2.

Problem Space StructuresA problem space of the TOH is com-posed of all possible states and movesconstrained by the rules. Figures 4A-Eshow the problem spaces constrained byRules 1, 1+2, 1+3, 1+2+3, and 1+2+3+4,respectively (see footnote 3) They arederived from the problem space shownin Figure 3. The rectangles (problemstates) in Figure 3 are not shown inFigure 4 for the reason of clarity.Figures 4A-D have the same 27 problemstates as in Figure 3. Figure 4E only has21 problem states (shown by the dots),which are the outer 21 rectangles inFigure 3. Lines with arrows are uni-directional; lines without arrows are bi-directional. One important point is thatthese five spaces can represent internal,external, or mixed problem spaces, de-pending upon how the rules construct-ing them are distributed across internaland external representations. A prob-lem space constructed by external rulesis an external problem space, one con-structed by internal rules is an internalproblem space, and one constructed by amixture of internal and external rules isa mixed problem space. Figure 4B is theinternal problem space of the standardTOH because Rules 1 and 2 are internal.If the physical constraints imposed bythe disks themselves are such that onlyone can be moved at a time (e.g., thedisks are large or heavy), then Figure 4Cis the external problem space of thestandard TOH because under this cir-


cumstance Rules 1 and 3 are both exter-nal. These two spaces form the dis-tributed problem space of the standardTOH (Figure 5), and their conjunction

forms the abstract problem space, whichis equivalent to the problem spaceshown in Figure 4D.

(A) Rule 1 (B) Rules 1+2

(C) Rules 1+3 (D) Rules 1+2+3

(E) Rules 1+2+3+4

¥¥¥¥

¥¥

¥¥

¥ ¥ ¥ ¥ ¥¥¥¥ ¥

¥

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Figure 4. Problem spaces constructed by five different sets of rules. They are derived fromthe problem space shown in Figure 3. Lines with arrows are uni-directional. Lineswithout arrows are bi-directional. The rectangles (problem states) are not shown here forthe reason of clarity. (A)-(D) have the same 27 problem states as in Figure 3. (E) onlyhas 21 problem states (shown by the dots), which are the outer 21 rectangles in Figure 3.


Distributed Problem Space

External Problem Space

Internal Problem Space

Abstract Problem Space

Figure 5. The distributed representation of the TOH. The distributed problem space is composed ofthe internal and the external problem spaces. The abstract problem space is the conjunction of theinternal and the external problem spaces.

Dimensional RepresentationsThe standard TOH (Figure 2) has threedisks, which possess two dimensions ofproperties. The first is the ordinal di-mension represented by the sizes of thethree disks, which has three levels:large > medium > small. The ordinalinformation on this dimension is re-

quired by the rules. The values on thisdimension are constants, that is, thesizes of the disks are fixed. The secondis the nominal dimension representedby the locations of the disks, which alsohas three levels: left, middle, right.This dimension is called nominal di-mension because only the categorical


information of the three locations isneeded for the rules. (For a descriptionof psychological scales, see Stevens,1946, 1951.) The values on this dimen-sion are variables, that is, a disk can beat any of the three locations. The threepoles in the standard TOH shown inFigure 2 are not essential: they are onlyused to construct the external Rule 3.

The ordinal and nominal dimen-sions do not have to be represented bysizes and locations of the disks, as in thestandard TOH. They can be representedby any properties of any objects. For ex-ample, Figure 6 shows two TOH iso-morphs whose ordinal and nominaldimensions are represented by differentproperties. In Figure 6A, the ordinaldimension is represented by the sizes ofthe balls (large > medium > small), andthe nominal dimension is representedby the locations of the balls (left, middle,and right). In Figure 6B, the ordinaldimension is represented by the loca-tions of the triangular cylinders (top >middle > bottom). Each cylinder hasthree different colors (R = red, G =green, Y = yellow) on its three sides,which represent the nominal dimen-sion. The sizes of the balls in Figure6A are mapped to the locations of thecylinders in Figure 6B, and the locationsin Figure 6A are mapped to the colors ofthe cylinder sides in Figure 6B. For ex-ample, moving the largest ball from theright location to the left location inFigure 6A corresponds to rotating thetop cylinder from the yellow side to thered side in Figure 6B. More examples of

the dimensional representations of theTOH are shown in Figure 14.

The ordinal and nominal dimen-sions of the TOH can be represented ei-ther internally or externally. If the or-dinal dimension is represented by phys-ical properties that are on an ordinal (orhigher) scale (e.g., sizes or ordered loca-tions), it can be represented externally.For example, the sizes of the balls inFigure 6A are an external representa-tion of the ordinal dimension becausethe ordinal relation is embedded in thephysical properties (sizes) of the balls. Ifthe ordinal dimension is represented byphysical properties that are on a nomi-nal scale, it must be represented inter-nally. For example, if we use colors torepresent the ordinal dimension, wemust first arbitrarily assign an order tothem (e.g., red > green > yellow) andthen internalize this ordinal relation(see Figures 14G, 14H, 14I for examples).The nominal dimension can be repre-sented externally by physical propertiesthat are either on a nominal or on anordinal scale, because an ordinal scale isalso a nominal scale (but not viceversa).

Furthermore, the ordinal andnominal dimensions of the TOH can berepresented by either visual properties(such as size, color, texture, and shape)or spatial properties (such as location).The separation of spatial propertiesfrom visual properties is significant be-cause many studies have found thatthere are important anatomical and


Ordinal Dimension

(large > medium > small)

Nominal Dimension

(top > middle > bottom)R

Y

G

(left, middle, right)

(red, green, yellow)

(A)

(B)

Figure 6. The mapping between dimensions of two TOH isomorphs. The three triangularcylinders in (B) can be rotated independently around their axis. The three sides of eachcylinder have three different colors (R = red, G = green, Y = yellow). The sizes of theballs (ordinal dimension) in (A) are mapped to the locations of the cylinders (ordinaldimension) in (B), and the locations of the balls (nominal dimension) in (A) are mappedto the colors of cylinder sides (nominal dimension) in (B). For example, moving thelargest ball from the right location to the left location in (A) corresponds to rotating thetop cylinder from the yellow side to the red side.

functional differences between visualand spatial representations. Mishkin,Ungerleider, & Macko (1983) found thatthere are two separate systems for vi-sual and spatial information processing:visual information processing follows aprojection from the occipital to thetemporal cortex, and spatial informa-tion processing follows a projectionfrom the occipital to the parietal cortex.Goldman-Rakic (1992) showed thatamong the prefrontal neurons respon-sible for working memory, some onlyrespond to spatial locations while oth-ers only respond to visual properties.Treisman & Gelade (1980) found thatthe visual and spatial properties of astimulus are processed differently inperceptual tasks and that location is re-quired for focused attention to conjunc-tive stimuli. The differences betweenvisual and spatial representations mayhave important implications for com-plex cognitive tasks such as problemsolving and reasoning. One of the pur-poses of Experiment 3 in the presentstudy is to examine the different func-

tions of visual and spatial dimensionsin the TOH task.

The Abstract Structure of the TOHThe TOH has a goal, a set of rules, twodimensions, and three objects. Figure 7shows the abstract structure of the TOH(see also Figure 8, which will be ex-plained in the following section). Thegoal is defined by an initial and a finalproblem state. The rules can vary innumbers, that is, any subset of the fourrules or all of them can be chosen for agiven version of the TOH. The rules ina given set can be distributed across in-ternal and external representations indifferent ways. The two dimensionscan be represented by different physicalproperties. O1, O2, and O3 are the threelevels of the ordinal dimension, andN 1, N2, and N3 are the three levels ofthe nominal dimension. An object OBJiis described as OBJi = (Oi, Nl), which canbe at three different levels on the nom-inal dimension: (Oi, N1), (Oi, N2), (Oi,N 3). The three objects can be repre-sented by different physical entities.


¥ Two property dimensions.Ordinal dimension. 3 levels: O1 > O2 > O3.Nominal dimension. 3 levels: N1, N2, N3.¥ Object: OBJi = (Oi, Nl). i = 1, 2, 3; l = 1, 2, 3.¥ Problem state: S(l, m, n) = ((O1, Nl), (O2, Nm), (O3, Nn)). l, m, n = 1, 2, 3.¥ Operation: OP(Oi, Nl) = (Oi, Nm). l ¹ m.¥ Rules:1: OP is a unary operator.2: When OBJj = (Oj, Nm), OP(Oi, Nl) = (Oi, Nm) is true if Oi > Oj.3: When OBJi = (Oi, Nl) & OBJj = (Oj, Nl), OP(Oi, Nl) is true if Oi > Oj.4. OBJ1 = (O1, Nm) & OBJ3 = (O3, Nm) is true if OBJ2 = (O2, Nm)¥ Goal: S(lÕ, mÕ, nÕ) ® S(lÓ, mÓ, nÓ).

Figure 7. The abstract structure of the TOH. O1, O2, and O3 are the three levels of the ordinaldimension, and N1, N2, and N3 are the three levels of the nominal dimension. An object OBJi isdescribed as OBJi = (Oi, Nl), which can be at three different levels on the nominal dimension: (Oi,N1), (Oi, N2),(Oi, N3). All four rules or any subset of them can be chosen for a given problem.

The Hierarchical Representation of theTower Of Hanoi

From the abstract structure of the TOH(Figure 7), we can identify four levels ofrepresentations: problem space struc-tures, rule representations, dimensionalrepresentations, and object representa-tions. At each level, there is an abstractstructure that can be implemented bydifferent representations. The differentrepresentations at each level are iso-morphic in the sense that they all sharethe same abstract structure at that par-ticular level. The hierarchical represen-tation of the TOH is summarized inFigure 8, and the details are described asfollows.

The Level of Problem Space StructuresThe problem space of a problem is

composed of all possible states and allmoves constrained by the rules of theproblem. The TOH is not bound to thethree rules of the standard TOH.

Different sets of rules construct differentproblem spaces. Figure 4 shows prob-lem spaces constructed by different setsof rules: Rule 1, Rules 1+2, Rules 1+3,Rules 1+2+3, and Rules 1+2+3+4.Problems that have different problemspaces are isomorphic at this level iftheir goals (initial and final states) arethe same. At this level, the formalproperties of problem space structures(such as the connections between prob-lem states and the total number of prob-lem states) are the major factors that af-fect problem solving behavior.

In Figure 4, when the number ofrules increases, problem space struc-tures become more constrained andproblem solving behavior mightchange accordingly. The effect of thestructural change of a problem space onproblem solving behavior might de-pend on the nature of the rules(whether internal or external).Experiments 1A and 1B were designedto examine these two factors. In


Experiment 1A, the change of the prob-lem space structure was caused by in-ternal rules, while in Experiment 1B, itwas caused by external rules.

The Level of Rule RepresentationsGiven a set of rules, say, Rules 1, 2,

and 3, the abstract problem space con-structed by them is fixed. However, therules can be distributed across internaland external representations in differ-ent ways. For example, Rule 1 may beinternal and Rules 2 and 3 external, orRules 1 and 2 internal and Rule 3 exter-nal, or all three rules internal (seeFigure 12). Problems that have thesame set of rules but different distribu-tions of rules are isomorphic to eachother at this level in the sense that theyall have the same abstract problemspace. Experiment 2 was designed tostudy the effect of the distributed repre-sentation of rules on problem solvingbehavior.

The Level of DimensionalRepresentations

The ordinal and the nominal di-mensions of the TOH do not have to berepresented by sizes and locations as inthe standard TOH. They can be repre-sented by any properties (see Figure 6and Figure 14 for examples). Problemswhose ordinal and nominal dimen-sions are represented by different prop-erties are isomorphic to each other atthis level in the sense that the differentproperties represent the same ordinalinformation on the ordinal dimension

and the same nominal information onthe nominal dimension.

There are two factors at this levelthat might affect the problem solvingbehavior of the TOH. The first factor iswhether the ordinal dimension of theTOH is represented internally (e.g., bycolors) or externally (e.g., by sizes andlocations). The second factor is whetherthe ordinal and nominal dimensions ofthe TOH are represented by visualproperties (e.g., size and color) or spatialproperties (e.g., location). The differentrepresentational properties of internaland external dimensions and those ofvisual and spatial dimensions mayproduce different processing strategiesfor the problem solving of the TOH.Experiment 3 was designed to examinethese two factors.

The Level of Object RepresentationsThe problems that are isomorphic

at all previous three levels can be iso-morphic at still another levelÑthelevel of object representations. At thislevel, different objects can be used. Forexample, when the ordinal dimensionis represented by sizes, we can use dif-ferent sized disks or different sized balls.For the same reason, when the nominaldimension is represented by colors, wecan use different sets of colors to repre-sent the nominal dimension, e.g., red,green, and yellow, or purple, blue, andpink. The representational properties atthis level usually do not have as largeeffects as those at the first three levels.


External Space(Rule 1)

Internal Space(Rules 1, 2, & 3)

External Space(Rules 1 & 3)

Internal Space(Rules 1 & 2)

Size-Location

GR Y

Size-Color

Location-Size Location-Location

Size-LocationSize-Location

Size-Location Size-LocationSize-LocationSize-Location Size-LocationSize-LocationSize-Location

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Figure 8. The hierarchical representation of the TOH. At the level of problem space structures,different sets of rules construct different problem spaces. At the level of rule representations, thesame set of rules can be distributed across internal and external representations in different ways.At the level of dimensional representations, the ordinal and nominal dimensions can be representedby different properties. At the level of object representations, different objects can be used.


EXPERIMENT 1A: PROBLEM SPACESTRUCTURES (INTERNAL)

In Experiments 1A and 1B, we studyisomorphic representations at the levelof problem space structures .Isomorphic representations at this levelhave the same goal but different prob-lem spaces constructed by different setsof rules. Figure 4 shows that the prob-lem spaces become more constrainedwith the increase of the number ofrules. How does the structural changeof the problem space affect problemsolving behavior? There are at leasttwo rival factors. On the one hand, thefewer rules, the more paths there arefrom an initial state to a final state.Hence, fewer rules might make theproblem easier. On the other hand, themore rules, the fewer the choices.Subjects can simply follow where thehighly constrained structure forcesthem to go. So, more rules might makethe problem easier. This implies thatthe problem difficulty might not changemonotonically with the number ofrules. The effect on problem solvingbehavior caused by the structuralchange of the problem space might alsodepend on the nature of the rules(whether internal or external).Experiments 1A and 1B examine theseeffects. In Experiment 1A, all rules areinternal. We change the number of in-ternal rules. In Experiment 1B, all butRule 1 are external. We change thenumber of external rules.

Experiment 1A has four condi-tions, in which all rules were internal.Condition I1 has Rule 1, Condition I13has Rules 1 and 3, Condition I123 hasRules 1, 2, and 3, and Condition I1234has Rules 1, 2, 3, and 4. We made arestaurant story (Waitress and Oranges)for the instructions. The instructions

for Condition I1234 are shown inFigure 9. The instructions forConditions I123, I13, and I1 were thesame as for I1234, except that differentsets of rules were stated in the instruc-tions.

MethodSubjects

The subjects were 24 undergradu-ate students enrolled in introductorypsychology courses at the University ofCalifornia, San Diego, who volunteeredfor the experiment to earn course credit.

MaterialsThree plastic orange balls of differ-

ent sizes (small, medium, and large)and three porcelain plates were used forall four conditions.

DesignEach subject played all four games,

once each. There were twenty-four pos-sible permutations for the four games.The twenty-four subjects were assignedto these permutations randomly. Dueto a limitation in the number of sub-jects available, the first, second, third,and fourth games always started at posi-tions S1, S2, S3, and S1 and ended atpositions E1, E2, E3, and E1, respectively(see Figure 3). This treatment was notexpected to cause significant systematicdeviation because the task structures ofthe four problems each subject solvedwere different from each other, and thegames were randomized.

ProcedureEach subject seated in front of a

table and read the instructions aloudslowly. Then the subject was asked toturn the instruction sheet over and toattempt to repeat all the rules. If thesubject could recite all the rules twice


Waitress and OrangesA strange, exotic restaurant requires everything to be done in a special manner. Here is anexample. Three customers sitting at the counter each ordered an orange. The customer on theleft ordered a large orange. The customer in the middle ordered a medium sized orange.And the customer on the right ordered a small orange. The waitress brought all threeoranges in one plate and placed them all in front of the middle customer (as shown inDiagram 1). Because of the exotic style of this restaurant, the waitress had to move theoranges to the proper customers following a strange ritual. No orange was allowed to touchthe surface of the table. The waitress had to use only one hand to rearrange these threeoranges so that each orange would be placed in the correct plate (as shown in Diagram 2),following these rules:

¥ Only one orange can be transferred at a time. (Rule 1 )¥ An orange can only be transferred to a plate on which it will be the largest. (Rule 2)¥ Only the largest orange in a plate can be transferred to another plate. (Rule 3)¥ The smallest orange and the largest orange can not be placed on a single plate unless

the medium sized orange is also on that plate (Rule 4).How would the waitress do this? That is, you solve the problem and show the movement oforanges the waitress has to do to go from the arrangement shown in Diagram 1 to thearrangement shown in Diagram 2.

Diagram 1

Diagram 2

Figure 9. The instructions for the I1234 condition in Experiment 1A. The instructions for other threeconditions were the same as the one shown here, except that different sets of rules were stated.

without error, the subject was in-structed to start the games. Otherwisethe subject reread the instructions andwas again tested. The cycle continueduntil the subject reached the criterion.The final states were presented to thesubject in diagrams. A subjectÕs handmovements and speech were recordedby a video camera. The solution time,which was from the time the experi-menter said "start" to when the subjectfinished the last move, was recorded bya timer synchronized with the videocamera.

ResultsThe results are shown in Figure 10. Theminimum numbers of steps from thestarting state to the final state are 2, 4, 7,and 8 for Conditions I1, I13, I123, and

I1234, respectively. In order to makemeaningful comparisons, solutiontimes, solution steps, and errors foreach condition were normalized by be-ing divided by the number of mini-mum steps for each condition. The pvalues of the main effects and multiplecomparisons are shown in the upperhalf of Table 1. When solution timesand errors are used as the difficultymeasurements, the difficulty order was,from easiest to hardest: I1 < I13 < I1234 £I123. The difference between I1234 andI123 was not statistically significant.When solution steps are used as the dif-ficulty measurement, the difficulty or-der remained the same (I1 < I13 £ I1234< I123), but in this case the difference be-tween I13 and I1234 was not statisticallysignificant.


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Figure 10. The results of Experiments 1A and 1B. Solution times were measured in seconds. The min-imum numbers of steps to solve the problems are 2, 4, 7, and 8 for Conditions I1, I13, I123, and I1234,respectively. In order to make meaningful comparisons, solution times, steps, and errors for eachcondition were normalized by being divided by the number of minimum steps for each condition.

Table 1. p Values of Experiments 1A and 1BMeasurements

Comparisons Times Steps ErrorsExperiment 1A

Main Effect < .0001 < .0001 < .0001I1 vs. I13 < .05 < .005 < .06I1 vs. I123 < .00001 < .00001 < .001I1 vs. I1234 < .00001 < .01 < .001I13 vs. I123 < .02 < .01 < .005I13 vs. I1234 < .005 > .6 < .01I123 vs. I1234 > .36 < .003 > .75

Experiment 1BMain Effect < .0001 < .0001 ÑI1 vs. I1-E3 < .01 < .1 ÑI1 vs. I1-E23 < .0001 < .0001 ÑI1 vs. I1-E234 < .00001 < .00001 ÑI1-E3 vs. I1-E23 > .18 < .01 ÑI1-E3 vs. I1-E234 < .0001 < .0001 ÑI1-E23 vs. I1-E234 < .03 > .4 ÑNOTE. Fisher PLSD test was used for the multiple comparisons.


EXPERIMENT 1B: PROBLEM SPACESTRUCTURES (EXTERNAL)

Experiment 1B was exactly the same asExperiment 1A, except that Rules 2, 3,and 4 were external rather than inter-nal. There were also four conditions.In Condition I1 (Waitress and Oranges,Figure 11A), Rule 1 was the only inter-nal rule. In Condition I1-E3 (Waitressand Straws, Figure 11B), Rule 1 was in-ternal and Rule 3 was external. Asmaller straw could be dropped into alarger straw and a larger straw could beplaced outside (around) a smaller straw(that is, Rule 2 was not implementedhere). However, the diameters of thethree straws were so small (1 cm, 1.5 cm,and 2 cm, respectively) that a smallerstraw inside a larger one could not bemoved out without the larger straw be-ing moved away first (external Rule 3).In Condition I1-E23 (Waitress andCoffee, Figure 11C), Rule 1 was internaland Rules 2 and 3 were external. Allcups were filled with coffee. A smallercup could not be placed on the top of alarger cup (external Rule 2), as thiswould cause the coffee to spill. All sub-jects understood that spilling coffee wasnot a good behavior because they had toimagine that they were waitresses orwaiters working in a restaurant (see the

cover story in Figure 9). In this sense,Rule 2 was not a truly external rulewith rigid physical constraints, but anexternal rule grounded in culturalknowledge. Rule 3 in this conditionwas external because a cup could not bemoved if there was another cup on itstop. In Condition I1-E234 (Waitress andTea, Figure 11D), Rule 1 was internaland Rules 2, 3, and 4 were external. Allcups were filled with tea. Rules 2 and 3in this condition were external for thesame reason described in Condition I1-E23: a smaller cup could not be placedon the top of a larger one (Rule 2), andonly the largest (topmost) cup could bemoved (Rule 3). Rule 4 were externalbecause the bottom of the largest cupwas smaller than the top of the smallestcup and the bottom of the smallest cupwas smaller than the top of the largestcup, that is, the largest cup and thesmallest cup could not be placed on thetop of each other.

The instructions for these fourconditions were the same as for theI1234 condition in Experiment 1A, ex-cept that different words for differentmaterials were used and only internalRule 1 was stated in the instructions (allother rules were not stated because theywere external).

(A) I1 (B) I1-E3

(C) I1-E23 (D) I1-E234Figure 11. The four conditions of Experiment 1B. See text for explanations. (A) I1: Rule 1 isinternal. (B) I1-E3: Rule 1 is internal and Rule 3 is external. (C) I1-E23: Rule 1 is internal and Rules2 and 3 are external. (D) I1-E234: Rule 1 is internal and Rules 2, 3, and 4 are external.


MethodSubjects

The subjects were 24 undergradu-ate students enrolled in introductorypsychology courses at the University ofCalifornia, San Diego, who volunteeredfor the experiment to earn course credit.

MaterialsMaterials for Condition I1 were

the same as in Experiment 1A. InCondition I1-E3, the straws and tinyplates were made from paperboard. Thediameters and heights of the small,medium, and large straws were approx-imately 1 cm, 1.5 cm, and 2 cm, and 3cm, 6 cm, and 9 cm, respectively. InCondition I1-E23, three plastic cups ofdifferent sizes (small, medium, andlarge) and three paper plates were used.All three cups were filled with coffee.In Condition I1-E234, three cups madefrom metal cans and three paper plateswere used. All three cups were filledwith tea.

Design and ProcedureThe design and procedure were the

same as in Experiment 1A.

ResultsThe results are shown in Figure 10. Thesolution times, solution steps, and er-rors for all four conditions were nor-malized by being divided by the mini-mum number of steps for each condi-tion, as in Experiment 1A. The p valuesof the main effects and multiple com-parisons are shown in the lower half ofTable 1. If solution times are used asthe difficulty measurement, the diffi-culty order was, from easiest to hardest:I1 < I1-E3 £ I1-E23 < I1-E234. The differ-ence between I1-E3 and I1-E23 is not sta-tistically significant. If solution stepsare used as the difficulty measurement,

the difficulty order remained the same(I1 < I1-E3 < I1-E23 £ I1-E234), but thedifference between I1-E23 and I1-E234 isnot statistically significant. SubjectsdidnÕt make any errors in this experi-ment.

DiscussionFrom the results of Experiments 1A and1B, we can see that problem space struc-ture is an important factor of problemdifficulty. For example, the solutiontime difference between the easiestproblem (I1) and the hardest problem(I123) can be as large as a ratio of one toten. We can also see that the effect ofthe structural change of a problem spaceon problem solving behavior dependedon the nature of the rules. When allbut one rule were external (Experiment1B), a problem became more difficultwhen its problem space became moreconstrained (more rules). However,when all rules were internal(Experiment 1A), the hardest problemwas neither the most constrained one(I1234) nor the least constrained one(I1), but the intermediately constrainedproblem (I123).

Two factors, working memory andproblem structure, might contribute tothe different difficulty orders caused bythe nature of rules. In Experiment 1A,all rules were internal. The workingmemory load was high for both I123and I1234 (three and four internalrules, respectively). One possible expla-nation is that subjects could not domuch planning because most of theworking memory was loaded by theprocessing of the internal rules. In thiscase, the structure of the problem spacemight be the dominant factor of prob-lem difficulty. Moves were to a largeextent guided by the structure of theproblem space. I1234 was easier than


I123 problem because it was more con-strained than I123 . In Experiment 1B,all but Rule 1 were external. In thiscase, planning was probably the domi-nant factor of problem difficulty becausethere was little load on working mem-ory. This explains why I1-E23 was easierthan I1-E234, even though I1-E234 wasmore constrained than I1-E23.

All four rules in Experiment 1Awere internal and three of the fourrules in Experiment 1B were external.Comparing the results in these two ex-periments, we found that the condi-tions with external rules in Experiment1B were easier than their correspondingconditions with internal rules inExperiment 1A. In other words, exter-nal rules could make problems easier.This was explicitly tested in Experiment2, which follows.

EXPERIMENT 2: THE DISTRIBUTEDREPRESENTATION OF RULES

In this experiment, we study isomor-phic representations at the level of rulerepresentations. We have shown that aproblem space is constructed by a set ofrules. Given the same set of rules, theabstract problem space is fixed.However, the rules can be distributedacross internal and external representa-tions in different ways. Different distri-butions may have different effects onproblem solving behavior, even if theformal structures are the same. Thisexperiment examines these effects. Ourhypothesis is that the more rules aredistributed in the external representa-tion, the easier the problem.

There were three conditions in thisexperiment, which all had three rules.Though the three rules were the samein their abstract forms, they were dis-

tributed across internal and externalrepresentations in different ways. Inthe I123 (Waitress and Oranges ) condi-tion (Figure 12A), Rules 1, 2, and 3 wereall internal. In the I12-E3 (Waitress andDonuts) condition (Figure 12B), Rules 1and 2 were internal, and Rule 3 was ex-ternal. This is the standard TOH. Rule3 was external because the physical con-straints (coupled with Rule 1) guaran-teed that it was followed (see the previ-ous discussion in the section R u l eRepresentations for more explanations).In the I1-E23 (Waitress and Coffee) con-dition (Figure 12C), Rule 1 was internal,and Rules 2 and 3 were external. Thiswas identical to the I1-E23 condtion inExperiment 2B. The instructions for allthree conditions were the same as forthe I1234 condition in Experiment 1A,except that different words suitable forthe materials used in the current exper-iment were used and that only internalrules were stated in the current instruc-tions (external rules need not be statedexplicitly).

MethodSubjects

The subjects were 18 undergradu-ate students enrolled in introductorypsychology courses at the University ofCalifornia, San Diego who volunteeredfor the experiment to earn course credit.

MaterialsThe materials in the I123 condition

were the same as in the I123 conditionin Experiment 1A. In the I12-E3 condi-tion, three plastic rings of different sizes(small, medium, and large) and threeplastic poles were used. The materialsin the I1-E23 condition were the same asin the I1-E23 condition in Experiment1B.


(A) I123

(B) I12-E3 (C) I1-E23Figure 12. The three conditions of Experiment 2. See text for explanations. (A) I123: All three rulesare internal. (B) I12-E3: Rules 1 and 2 are internal and Rules 3 is external. (C) I1-E23: Rule 1 isinternal and Rules 2 and 3 are external.

DesignEach subject played all three games,

one for each of the three conditions,once in a randomized order (e.g., I1-E23,I123, I12-E3). There were six possiblepermutations for the three games. Eachpermutation was assigned to a subjectrandomly. There were a total of eigh-teen subjects. Due to a limitation in thenumber of subjects available, the start-ing and ending positions were not ran-domized. That is, for each subject, thefirst, the second, and the third gamesalways started at positions S1, S2, and S3and ended at positions E1, E2, and E3,respectively (see Figure 3). The startingand ending positions should not causesignificant systematic deviation becausethe three pairs of starting and endingpositions were exactly symmetric, andthe order of the three games played byeach subject was randomized.

ProcedureThe procedure was the same as in

Experiment 1A.

ResultsThe average solution times, solutionsteps, and errors are shown in Figure 13.The p values for the main effects andmultiple comparisons are shown inTable 2. Problem difficulty measured in

solution times, steps, and errors for thethree problems was consistent. Themore rules were external, the easier theproblem. The order of difficulty was,from hardest to easiest: I123 > I12-E3 ³I1-E23. The difference between I12-E3and I1-E23 was not statistically signifi-cant. All errors made were for internalrules: none were for external rules.

DiscussionTwo of the three conditions in the pre-sent experiment, I123 and I12-E3, weremodifications of the Dish-move andPeg-move problems of Kotovsky, Hayes,and Simon (1985). The results from thepresent study are consistent with theirresults: subjects took more time to solveI123 than I12-E3. Kotovsky, Hayes, andSimon only reported solution times intheir study. The numbers of steps anderrors in the present study are all con-sistent with solution times. In the pre-sent experiment, the more rules exter-nalized, the easier the problem. In addi-tion, external rules seem to be errorproof: subject did not make any errorsfor external rules. This effect might bedue to the fact that the external ruleswere either perceptually available orphysically constrained. The errors forinternal errors might be caused by theload of working memory.


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Figure 13. The results of Experiment 2. Problem difficulty decreased with the increase of thenumber of external rules.

Table 2. The p Values of Experiment 2Measurements

Comparisons Times Steps ErrorsMain Effect < .05 < .05 < .005I123 vs. I12-E3 < .1 < .1 < .03I123 vs. I1-E23 < .01 < .02 < .001I12-E3 vs. I1-E23 > .3 > .4 > .2NOTE: Fisher PLSD test was used for the multiple comparisons.

Why did external rules make prob-lems easier? There might be two fac-tors. The first is the checking of rulesbefore each move. External rules can bechecked by perceptual inspection, whileinternal rules must be checked men-tally. The processing of internal rulesin the mind, which demands more re-sources of working memory, might in-terfere with other processes critical forproblem solving, such as planning. Thesecond factor is the recursive strategyÑthe strategy to reduce a three-objectproblem to a two-object problem, or inother words, move the smallest objectto its destination first. From the prob-lem space of the TOH (Figure 3) we cansee that for any pair of starting and end-ing states, as long as either the small orthe medium object is in its final state,only three more steps are needed to

solve the problem. Among the fifty-four games played by the eighteen sub-jects, the last three steps were solved inthree steps in fifty-two games and infive steps in only two games.Therefore, if either the small or themedium object is in the final state, thegame is virtually solved. Rule 2 mightbe critical for the discovery of the recur-sive strategy. For the I1-E23 condition,Rule 2 was external: a smaller cup ofcoffee could not be placed on the top ofa larger one. This external representa-tion might have prompted the subjectsthat the smallest cup had to be movedto its final state first. Out of the eigh-teen games for the I1-E23 condition, six-teen were solved by moving the small-est cup to its final state first. The othertwo were solved by moving themedium sized cup to its final state first.


For the I123 and the I12-E3 conditions,Rule 2 was internal. Thus, the recur-sive strategy was harder to discover.Out of the eighteen games for the I123condition, only eleven were solved bymoving the smallest object to its finalstate first. The other seven were solvedby moving the medium sized object toits final state first. Similarly, out of theeighteen games for the I12-E3 condition,the numbers for the two cases were tenand eight, respectively. Thus, the easeof discovering the recursive strategy inthe I1-E23 condition might have alsocontributed to the difficulty order.

EXPERIMENT 3: DIMENSIONALREPRESENTATIONS

In this experiment, we study isomor-phic representations at the level of di-mensional representations. The ordi-nal and nominal dimensions of theTOH can be represented by any proper-ties. We chose size, location, and colorto represent these two dimensions,which generated nine isomorphic rep-resentations of the TOH (Figure 14).(See Figure 6 as well as Figure 14 for ex-planations on the dimensional repre-sentations and the mappings betweendimensions).

The nine isomorphs in Figure 14were the nine conditions of this exper-iment, which had the same three inter-nal rules. Among these nine iso-morphs, some might be easier thanothers. We consider two factors of prob-lem difficulty. The first factor is the na-ture (internal or external) of the ordinaldimension. The ordinal dimension isrepresented externally by sizes and

(ordered) locations, because the ordinalrelation is embedded in the physicalproperties of sizes and locations. But itis represented internally by colors, be-cause the order of colors is arbitrary andhas to be internalized. (The nominaldimension is represented externally bysizes, locations, and colors.) The secondfactor is whether the ordinal and nom-inal dimensions are represented by vi-sual properties (sizes and colors) or spa-tial properties (locations). The instruc-tions for the O(size)-N(size) conditionare shown in Figure 15. The instruc-tions for other conditions were thesame as for the O(size)-N(size) condi-tion, except that modifications weremade for different materials used in dif-ferent conditions.

We hypothesize that external di-mensions need less mental processingthan internal dimensions, and that spa-tial dimensions support more efficientperceptual processing than visual di-mensions. Thus, we have the follow-ing predictions. For the ordinal dimen-sion, sizes should be better than colorsbecause the former represents the ordi-nal dimension externally while the lat-ter represents it internally, and loca-tions should better than sizes becausethe former is a spatial dimension whilethe latter is a visual dimension. In ad-dition, locations should be better thancolors because the former is not only anexternal dimension but also a spatialdimension. For the nominal dimen-sion, locations should be better thansizes and colors because the former is aspatial dimension while latter two arevisual dimensions. Sizes and colorsshould not differ from each other be-cause they are both visual dimensions.


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Figure 14. Nine isomorphs at the level of dimensional representations. O = Ordinal Dimension, N= Nominal Dimension. For ordinal dimensions, the colors have a priority: red > green > yellow.All triangular cylinders can be rotated around their axes. (A) Each cylinder has three differentsized circles on its three sides. O = sizes of the cylinders; N = sizes of the circles. (B) O = sizes ofthe balls; N = locations of the balls. (C) Each cylinder has three colors (red, green, yellow) on itsthree sides. O = sizes of the cylinders. N = colors of the sides. (D) Each cylinder has threedifferent sized circles on its three sides. O = locations of the cylinders (top, middle, bottom). N =sizes of the circles. (E) The three disks can be moved horizontally among three locations. O =vertical locations (top, middle, bottom). N = horizontal locations (left, middle, right). (F) Eachcylinder has three colors (red, green, yellow) on its three sides. O = locations of the cylinders (top,middle, bottom). N = colors of the sides. (G) Each cylinder has a color (red, green, or yellow) on allthree sides. On the three sides of a cylinder there are three different sized circles. O = colorpriority. N = sizes of the circles. (H) O = color priority. N = locations of the balls. (I) Eachcylinder has a color (red, green, or yellow) on all three sides. On the three sides of each cylinderthere are three circles of different colors (red, green, yellow). O = the color priority of thecylinders. N = colors of the circles.


Legend has it that in ancient India there was a Buddhist scripture in a lockedcrypt hidden in a cave. Many people tried to acquire it because it possessed the secret oflife. However, none of them succeeded because if the crypt was not opened in a veryshort time, it would disappear. The lock of the crypt was remotely controlled by threemagic triangular cylinders of different sizes (small, medium, and large). On the threesides of each cylinder, there were three different sized circles (also small, medium, andlarge). In order to open the crypt, one had to rotate the three cylinders to a specificconfiguration, strictly following a set of rules. One day, a wise monk entered the cave.Inspired by the wisdom of Buddha, he skillfully opened the crypt within the specifiedtime. How did he do it? You are given a replica of the magic cylinders. Try to solvethis puzzle as fast as you can. The rules are given below. The initial configuration youwill start with and the final configuration that opens the lock will be given to youwhen you have memorized these rules.

Rule 1: Only one cylinder can be rotated at a time.Rule 2: After rotating a cylinder, if the new facing side has a same sized circle

as that of another cylinder, the size of the cylinder you rotated must belarger.

Rule 3: Before rotating a cylinder, if any cylinders have matching sized circlesfacing you, only the largest cylinder can be rotated.

Figure 15. The instructions for the O(size)-N(size) condition. The instructions for other conditionswere the same as the one shown here, except that modifications were made for different objects indifferent conditions.

MethodSubjects

The subjects were 36 undergradu-ate students at the University ofCalifornia, San Diego who were paid forparticipating in this experiment.

MaterialsThe triangular cylinders in Figure

14 were made from paperboard andcould be rotated around their verticalaxes. In Figures 14D and 14F, the threecylinders were stacked on a vertical rod.In Figure 14B, three different sized plas-tic balls were used. In Figure 14E, thethree plastic disks could only be movedhorizontally. In Figure 14H, threesponge balls of different colors (red,green, yellow) were used.

DesignThis was a mixed design. The

within-subject factor was the ordinaldimension. It had three levels: size, lo-cation, and color. The between-subjectfactor was the nominal dimension. Italso had three levels: size, location, andcolor. There were three games acrossthe three within-subject levels at eachbetween-subject level. Each subjectplayed these three games once in a ran-domized order (e.g., O(color)-N(size),O(size)-N(size), and O(location)-N(size)). For the three games at each be-tween-subject level, there were six pos-sible permutations. There were a totalof 18 possible permutations across thethree between-subject levels. Eachpermutation was assigned to two sub-jects randomly. There were a total of 36subjects. Due to a limitation in thenumber of subjects available, the start-ing and ending positions were not ran-domized. That is, for each subject, the


first game always started at position S1and ended at position E1 (see Figure 3),the second at position S2 and at positionE2, and the third at position S3 and atposition E3. The starting and endingpositions were not expected to causemuch systematic deviation because thethree pairs of starting and ending posi-tions were exactly symmetric, and thethree games played by each subject wererandomized.

ProcedureFor each game, the subjects were

asked to read the instructions aloudonce and given two minutes to memo-rize the rules. The subjects were thenasked to recite all the rules. If the sub-jects could recite all the rules withouterrors, they were shown two examplesfor each rule. Otherwise, the subjectsread the instructions again and wereagain tested until they could recite allthe rules without errors. Before thesubjects started the games, they weregiven the initial and final states. Thefinal state for each game was presentedby a set of identical objects. Subjects'hand movements and speech wererecorded by a video camera.

ResultsThe solution times, steps, and errors areshown in Figure 16. The p values areshown in Table 3. For the ordinal di-mension, when solution times and er-rors were used as the difficulty mea-surement, locations were significantlybetter than sizes and colors. Thoughthe solution times and errors for sizeswere smaller than for colors, the differ-ences were not statistically significant.When solution steps were used as thedifficulty measurement, locations weresignificantly better than sizes, but not

statistically better than colors, thoughthe solution steps for locations werefewer than for colors. The difference ofsolution steps between colors and sizeswas not statistically significant.

For the nominal dimension, whensolution times and errors were used asthe difficulty measurement, locationswere significantly better than sizes andcolors. The difference between sizesand colors was not statistically signifi-cant. When solution steps were used asthe difficulty measurement, none of thedifferences between sizes, locations, andcolors were statistically significant,though solution steps for locations werefewer than for sizes and colors.

DiscussionWe identified two factors that might af-fect problem solving behavior at thelevel of dimensional representations.The first factor is whether the ordinaldimensional is represented internallyor externally. The results showed thatalthough solution times and errors forcolors, which represent the ordinal di-mension internally, were larger thanfor sizes, which represent the ordinaldimension externally, the differenceswere not statistically significant. Thismight be because, compared with othermental activities during the problemsolving tasks, internalizing the colorpriority (red>green>yellow) was a triv-ial task. The second factor, whether adimension is represented by visual orspatial properties, played a major role indetermining problem difficulties.Locations were much better than sizesfor the ordinal dimension and thansizes and colors for the nominal dimen-sion. The superiority of locations overcolors for the ordinal dimension was aneffect caused by both factors.


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Figure 16. The results of Experiment 3.

Table 3. The p Values of Experiment 3Measurements

Comparisons Times Steps ErrorsO(size) vs. O(location) < .01 < .05 < .03O(size) vs. O(color) > .17 > .53 > .15O(location) vs. O(color) < .0007 > .2 < .004N(size) vs. N(location) < .003 > .24 < .0001N(size) vs. N(color) > .66 > .62 > .96N(location) vs. N(color) < .016 > .17 < .007NOTE. The comparisons were between dimensions, not between individual conditions. Forexample, O(size) vs. O(location) was the comparison between the ordinal dimensionrepresented by sizes and the ordinal dimension represented by locations. None of theinteractions were statistically significant.


Why were locations so special? Inthe analysis of the dimensional repre-sentations of the TOH, we reviewedsome empirical studies showing theimportant anatomical and functionaldifferences between visual and spatialrepresentations (Goldman-Rakic 1992;Mishkin, Ungerleider, & Macko 1983;Treisman & Gelade, 1980). Though theresults from the present experiment donot allow us to offer a direct explana-tion of why locations (spatial dimen-sion) were better than sizes and colors(visual dimensions), we can compareour results with some perceptual tasks.The ordinal and nominal dimensionsin the present experiment are conjunc-tive dimensions. Treisman & Gelade(1980) proposed that location informa-tion is necessary for the perception ofconjunctive stimuli. Nissen's (1985)study of conjunctive stimuli based onvisual maps is more directly related tothe present study. According to her,color and shape are registered in sepa-rate maps. When a conjunctive stimu-lus was composed of color and location,she showed that selecting an item bycolor and reporting its location was asaccurate as selecting by location and re-porting color. In this case, selection bylocation did not have special advantage,because location itself was part of theconjunctive stimulus. However, whena conjunctive stimulus is composed ofcolor and shape, location is necessaryfor cross-referencing between the sepa-rate shape and color maps. She showedthat when subjects reported the shapeand location of an item cued by its color,the accuracy of shape judgments de-pended on the accuracy of locating thecued color. Comparing Nissen's studywith the present study, we have parallelresults. If neither of the two conjunc-tive dimensions (ordinal and nominal

dimensions) is represented by locations,an extra step of invoking location in-formation is needed to link the twoseparate visual maps of size and color.In this case, the three values on the or-dinal dimension and the three valueson the nominal dimensions, as well asthe three locations, have to be checkedbefore an operation (moving or rotat-ing) can be made. If one of the two con-junctive dimensions is represented bylocations, the extra step of invoking thelocation information is not needed, be-cause the location information itself isin the conjunctive dimensions. In thiscase, only the three values on the di-mension not represented by locationsand the three values on the dimensionof locations have to be checked beforean operation can be made. Thus, if oneof the ordinal and nominal dimensionsor both of them are represented by loca-tions, the problem is easier.

GENERAL DISCUSSION

In this paper, we developed a theoreti-cal framework of distributed representa-tions and a methodology of representa-tional analysis for the study of dis-tributed cognitive tasksÑtasks that re-quire the processing of information dis-tributed across the internal mind andthe external environment, and appliedthem to the empirical studies of theTower of Hanoi problem. Our approachto distributed cognitive tasks demands(a) the consideration of the internal andexternal representations of a task as arepresentational system, (b) the explicitdecomposition of the representationalsystem into its internal and externalcomponents, and (c) the identificationof the different functions of internaland external representations in cogni-


tion. The traditional approach to cogni-tion is not appropriate for the study ofdistributed cognitive tasks, because it of-ten ignores the important functions ofexternal representations in cognition.In addition, it often confuses a task'sdistributed representation that has bothinternal and external components withthe task's internal representation. Thisconfusion often leads to unnecessarilycomplex accounts of cognition.

Formal Structures, Representations ,and Processes

Any distributed cognitive task can beanalyzed into three aspects: its formalstructure, its representation, and its pro-cesses. Our present study focused onrepresentations and their relation toformal structures, for the following rea-sons. First, in order to understand theprocesses involved in a distributed cog-nitive task, we first have to understandwhat information is processed and howthe information to be processed is rep-resented. Second, different processesare activated by different representa-tions, but not vice versa. For example,perceptual processes are activated by ex-ternal representations, while cognitiveprocesses are usually activated by inter-nal representations. Third, from a rep-resentational perspective, tasks thatlook dramatically different may in facthave a common structure. However, ifwe start our analysis with processes, wemay fail to capture this common struc-ture, because different representationalformats of the common structure canactivate completely different processes.

The three aspects of distributedcognitive tasks are closely interrelated:the same formal structure can be im-plemented by different isomorphic rep-resentations, and different isomorphicrepresentations can activate different

processes. Though an representationalanalysis should be the first step for thestudy of distributed cognitive tasks, aprocess model is also essential. In orderto understand the nature of distributedcognitive tasks, we need to study all ofthe three aspects. Two interesting is-sues worth of further studies are the re-lationship between representations andprocesses and the interplay betweenperceptual and cognitive processes.

The Nature of External RepresentationsExternal objects are not just peripheralaids to cognition, they provide a differ-ent form of representationÑan externalrepresentation. By decomposing therepresentational system of a distributedcognitive task into its internal and ex-ternal representations, we can separatethe functions of external representa-tions from those of internal representa-tions. The empirical studies reportedhere on the Tower of Hanoi problemsuggest the following properties of ex-ternal representations.

1. External representations canprovide memory aids. For example, inall of the TOH experiments reportedhere, the goal problem states didn'tneed to be memorized, because theywere placed in front of the subjects ei-ther by diagrams or by physical objects.This is the most acknowledged propertyof external representations. To manypeople, this is the only one.

2. External representations canprovide information that can be directlyperceived and used without being in-terpreted and formulated explicitly. Forexample, in the I1-E23 version of theTOH, Rules 2 and 3 were not told to thesubjects: they were built into the physi-cal constraints. They could be perceivedand followed directly by the subjects.When the subjects were asked to explic-


itly formulate the rules after the games,few could do it. External representa-tions seem to provide affordances(Gibson, 1979).

3. External representations can an-chor and structure cognitive behavior.The physical structures in external rep-resentations constrain the range of pos-sible cognitive behaviors in the sensethat some behaviors are allowed andothers prohibited. For example, in theI1-E23 version of the TOH, externalRules 2 and 3 could not be violated.They constructed an action space inwhich only the actions that did not vio-late these two rules were permitted.This action space is the external prob-lem space of the I1-E23 problem.

4. External representations changethe nature of a task. Norman (1991)proposed that there are two differentviews of cognitive artifacts. From thesystem's view (internal + external rep-resentations), external representationscan make a task easier; from the per-son's view (internal representationsonly), external representations changethe nature of the task. For example, inthe I123 version of the TOH, a subjecthad to process three internal rules,while in the I1-E23 version the subjectonly had to process one internal rule.Though the two versions had the sameabstract structure, their cognitive pro-cesses were different. Nevertheless,when considered as systems, I1-E23 wasmuch easier than I123.

5. External representations are anindispensable part of the representa-tional system of any distributed cogni-tive task. This property is a direct re-flection of the nature of distributed cog-nitive tasks, which require the process-ing of information distributed across in-ternal and external representations.

Related ApproachesAlthough the mainstream approach tocognition has focused on internal repre-sentations, the role of the environmentin cognition has long been acknowl-edged by several alternative approaches.For example, Gibson (1966, 1979) arguedthat perception is the direct pickup ofe n v i r o n m e n t a l i n f o r m a t i o n(invariants) in the extended spatial andtemporal patterns of optic arrays, andthat information in the environment issufficient for perception and action. Thesociohistorical approach to cognition(Leontiev, 1981; Luria, 1976; Vygotsgy,1978, 1986) argues that it is the continu-ous internalization of the informationand structure from the environmentand the externalization of internal rep-resentations into the environment thatproduce high level psychological func-tions.

More recently, the role of the envi-ronment in cognition has become thecentral concern in several fields of cog-nitive science. In the studies of the rela-tionship between images and pictures, ithas been shown that external represen-tations can give us access to knowledgeand skills that are unavailable from in-ternal representations (e.g., Chambers &Reisberg, 1985; Reisberg, 1987). The sit-uated cognition approach argues thatthe activities of individuals are situatedin the social and physical contextsaround them and knowledge can beconsidered as a relation between the in-dividuals and the situation (e.g.,Barwise & Perry, 1983; Greeno, 1989;Lewis, 1991; Suchman, 1987). Studieson diagrammatic reasoning have alsofocused on the functions of externalrepresentations in cognition (e.g.,Chandrasekaran & Narayanan, 1990;Larkin, 1989; Larkin & Simon, 1987).For example, Larkin & Simon (1987) ar-


gue that diagrammatic representationssupport operators that can recognizefeatures easily and make inferences di-rectly.

Although our current approachshares the same interest with others inthe function of the environment incognition, it differs in several aspects.First, our approach focuses on the rep-resentational properties of distributedcognitive tasks: how the informationneeded for a distributed cognitive task isrepresented across the internal mindand the external environment. Second,our approach demands the considera-tion of the internal and external repre-sentations of a distributed cognitive taskas a representational system. We arguethat external representations are an in-dispensable part of any distributed cog-nitive task. Third, our approach de-mands the explicit decomposition of therepresentational system of a distributedcognitive task into its internal and ex-ternal components. With such a de-composition, we can identify the differ-ent functions of internal and externalrepresentations in cognition. Fourth,we suggested that in order to under-stand the nature of a distributed cogni-tive task, we need to study the task'sformal structure, representation, andprocesses and the interrelations amongthem. Finally, the principles of dis-tributed representations and themethodology of representational analy-sis developed in the present study havebeen applied to several real world prob-lems, including numeration systems,relational information displays, andcockpit instrument displays (Zhang,1992).

ConclusionThe distributed representations ap-proach offers a novel perspective for the

study of cognition. It has both theoreti-cal and practical implications.Theoretically, it can shed light on someissues regarding the nature of cognition,such as whether cognition is solely inthe mind or distributed across the mindand the environment and whetherpeople reason on formal structures oron content-specific representations.Practically, it can provide design princi-ples for effective representations.

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