chapter 10 visuospatial reasoning - silc 10 visuospatial reasoning ... all these abstract...

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C H A P T E R 1 0

Visuospatial Reasoning

Barbara Tversky

Visuospatial reasoning is not simply a mat-ter of running to retrieve a fly ball or wend-ing a way through a crowd or plotting apath to a destination or stacking suitcasesin a car trunk. It is a matter of deter-mining whether gears will mesh (Schwartz& Black, 1996a), understanding how a carbrake works (Heiser & Tversky, 2002), dis-covering how to destroy a tumor without de-stroying healthy tissue (Duncker, 1945 ; Gick& Holyoak, 1980, 1983), and designing amuseum (Suwa & Tversky, 1997). Perhapsmore surprising, it is also a matter of decid-ing whether a giraffe is more intelligent thana tiger (Banks & Flora, 1977; Paivio, 1978),whether one event is later than another(Boroditsky, 2000), and whether a conclu-sion follows logically from its’ premises (Bar-wise & Etchemendy, 1995 ; Johnson-Laird,1983). All these abstract inferences, andmore, appear to be based on spatial reason-ing. Why is that? People begin to acquireknowledge about space and the things in itprobably before they enter the world. In-deed, spatial knowledge is critical to survivaland spatial inference critical to effective sur-vival. Perhaps because of the (literal) ubiq-

uity of spatial reasoning, perhaps becauseof the naturalness of mapping abstract el-ements and relations to spatial ones, spatialreasoning serves as a basis for abstract knowl-edge and inference. The prevalence of spa-tial figures of speech in everyday talk atteststo that: We feel close to some people andremote from others; we try to keep our spir-its up, to perform at the peak of our pow-ers, to avoid falling into depressions, pits,or quagmires; we enter fields that are wideopen, struggling to stay on top of things andnot get out of depth. Right now, in this sec-tion, we establish fuzzy boundaries for thecurrent field of inquiry.

Reasoning

Before the research, a few words about thewords. The core of reasoning seems to be, asBruner put it years ago, going beyond the in-formation given (Bruner, 1973). Of course,nearly every human activity requires goingbeyond the information given. The simplestrecognition or generalization task, as well asthe simplest action, require going beyond

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2 1 0 the cambridge handbook of thinking and reasoning

the information given, as according to a farmore ancient saying, you never step into thesame river twice. Yet many of these tasksand actions do not feel cognitive, do not feellike reasoning. However, the border betweenperceptual and cognitive processes may beharder to establish than the borders betweencountries in conflict. Fortunately, psychol-ogy is typically free of territorial politics, soestablishing boundaries between perceptionand cognition is not essential. There seems tobe a tacit understanding as to what counts asperceptual and what as cognitive, althoughfor these categories just as for simpler ones,such as chairs and cups, the centers of thecategory enjoy more consensus than the bor-ders. Invoking principles or requirements forthe boundaries between perception and cog-nition – consciousness, for example – seemsto entail more controversy than the separa-tion into territories.

How do we go beyond the informationgiven? Going beyond the information givendoes not necessarily mean adding informa-tion. One way to go beyond the informationgiven is to transform the information given.This is the concern of the earlier part of themanuscript. Going beyond the informationgiven can also mean transforming the giveninformation, sometimes according rules, asin deductive reasoning. Another way to gobeyond the information given is to make in-ferences or judgments from it. Inference andjudgment are the concerns of the later partof the manuscript. Now some more distinc-tions regarding the visuospatial portion ofthe title.

Representations and Transformations

Truths are hard to come by in science, butuseful fictions, approximate truths, abound.One of these is the distinction between rep-resentations and transformations, betweeninformation and processes, between data andthe operations performed on data. Repre-sentations place limits on transformationsas they select and structure the informa-tion captured from the world or the mind.Distinguishing representations and transfor-mations, even under direct observation of

the brain, is another distinction fraughtwith complexity and controversy. Evidencebrought to bear for one can frequently bereinterpreted as evidence for the other (e.g.,Anderson, 1978). Both representations andtransformations themselves can each be de-composed into representations and transfor-mations. Despite these complications, thedistinction has been a productive way tothink about psychological processes. In fact,it is a distinction that runs deep in hu-man cognition, captured in language assubject and predicate and in behavior asagent/object and action. The distinction willprove useful here, more than as a way oforganizing the literature (for related discus-sion, see Doumas & Hummel, Chap. 4).

It has been argued that the very estab-lishment of representations entails inferen-tial operations. A significant example arethe Gestalt principles of perceptual orga-nization – grouping by similarity, proxim-ity, common fate, and good continuity –that contribute to scene segmentation andrepresentation. These are surely a form ofvisuospatial inference. Representations areinternal translations of external stimuli (orinternal data); as such, they not only elim-inate information from the external world,they also add to it and distort it in the ser-vice of interpretation or behavior. Thus, ifinference is to be understood in terms ofoperating on or manipulating informationto draw new conclusions, then it begins inthe periphery of the sensory systems withleveling and sharpening and feature detec-tion and organization. Nevertheless, the fieldhas accepted a level of description of repre-sentations and transformations, one higherthan the levels of sensory and perceptualprocessing; that level is reflected here.

Visuospatial

What makes visuospatial representationsvisuospatial? Visuospatial transformationsvisuospatial? First and foremost, visuospatialrepresentations capture visuospatial proper-ties of the world. They do this in a way


visuospatial reasoning 2 1 1

that preserves, at least in part, the spatialstructural relations of that information (seeJohnson-Laird, 1983 ; Pierce in Houser &Kloesel, 1992). This means that visuospa-tial properties that are close or above or be-low in the world preserve those relationsin the representations. Visual includes staticproperties of objects, such as shape, texture,and color, or between objects and referenceframes, such as distance and direction. It alsoincludes dynamic properties of objects suchas direction, path, and manner of movement.By this account, visuospatial transformationsare those that change or use visuospatial in-formation. Many of these properties of staticand dynamic objects and of spatial relationsbetween objects are available from modal-ities other than vision. This may explainwhy well-adapted visually impaired individ-uals are not disadvantaged at many spatialtasks (e.g., Klatzky, Golledge, Cicinelli, &Pellegrino, 1995). Visuospatial representa-tions are regarded as contrasting with otherforms of representation, notably linguistic.The similarities (e.g., Talmy, 1983 , 2001 )and differences between visuospatial andlinguistic representations provide insightsinto both.

Demonstrating properties of internal rep-resentations and transformations is tricky foranother reason; representations are manysteps from either (controlled) input or(observed) output. For these reasons, thestudy of internal representations and pro-cesses was eschewed not only by behavior-ists, but also by experimentalists. It was oneof the first areas to flourish after the so-called Cognitive Revolution of the 1960s,with a flurry of innovative techniques todemonstrate form and content of internalrepresentations and the transformations per-formed on them. It is that research that wenow turn.

Representations and Transformations

Visuospatial reasoning can be approachedbottom-up by studying the elementary rep-resentations and processes that presumablyform the building blocks for more com-plex reasoning. It can also be approached

top-down by studying complex reasoningthat has a visuospatial basis. Both ap-proaches have been productive. We beginwith elements.

Imagery as Internalized Perception

The major research tradition studying visu-ospatial reasoning from a bottom-up per-spective has been the imagery program, pi-oneered by Shepard (see Finke & Shepard,1986; Shepard & Cooper, 1982 ; Shepard &Podgorny, 1978, for overviews) and Kosslyn(1980, 1994b), which has aimed to demon-strate parallels between visual perceptionand visual imagery. There are two basictenets of the approach, one regarding rep-resentations and the other regarding opera-tions on representations: that mental imagesresemble percepts, and that mental trans-formations on images resemble observablechanges in things in the world, as in men-tal rotation, or perceptual processes per-formed on things in the world, as in men-tal scanning. Kosslyn (1994b) has persistedin these aims, more recently demonstrat-ing that many of the same neural structuresare used for both. Not the demonstrationsper se, but the interpretations of them havemet with controversy (e.g., Pylyshyn, 1978,1981 ). In attempting to demonstrate the sim-ilarities between imagery and perception,the imagery program has focused both onproperties of objects and on characteristicsof transformations on objects – the former,representations, and the latter, operations ortransformations. The thrust of the researchprograms has been to demonstrate that im-ages are like internalized perceptions andtransformations of images like transforma-tions of things in the world.

representations

In the service of demonstrating that im-ages preserve characteristics of perceptions,Shepard and his colleagues brought evi-dence from similarity judgments as sup-port. They demonstrated “second-orderisomorphisms,” similarity spaces for per-ceived and imagined stimuli that have thesame structure, that is, are fit by the



same underlying multidimensional space(Shepard & Chipman, 1970). For example,similarity judgments of shapes of cutoutsof states conform to the same multidimen-sional space as similarity judgments of imag-ined shapes of states. The same logic wasused to show that color is preserved inimages, as well as configurations of faces(see Gordon & Hayward, 1973 ; Shepard,1975). Similar reasoning was used to demon-strate qualitative differences between pic-torial and verbal representations in a taskrequiring sequential same–different judg-ments on pairs of schematic faces and names(Tversky, 1969). The pictorial and verbalsimilarity of the set of faces was orthogonalso the “different” responses were a clue tothe underlying representation; times to re-spond “different” are faster when more fea-tures between the pairs differ. These timesindicated that when participants expectedthe target (second) stimulus would be a pic-ture, they encoded the first stimulus pictori-ally, whether it had been a picture of a face orits name. The converse also held: When thetarget stimulus was expected to be a name,participants coded the first stimulus verballyirrespective of its presented modality.

To demonstrate that mental images pre-serve properties of percepts, Kosslyn and hiscolleagues presented evidence from studiesof reaction times to detect features of imag-ined objects. One aim is to show that prop-erties that take longer to verify in perceptstake longer to identify in images. For ex-ample, when participants were instructed toconstruct images of named animals in orderto judge whether the animal had a partic-ular part, they verified large parts of ani-mals, such as the back of a rabbit, fasterthan small but highly associated ones, suchas the whiskers of a rat. When participantswere not instructed to use imagery to makejudgments, they verified small associatedparts faster than large ones. When not in-structed to use imagery, participants usedtheir general world knowledge to make judg-ments (Kosslyn, 1976). Importantly, whenthe participants explicitly used imagery, theytook longer to verify parts, large or small,than when they relied on world knowledge.

Additional support for the claim that imagespreserve properties of percepts comes fromtasks requiring construction of images. Con-structing images takes longer when there aremore parts to the image, even when thesame figure can be constructed from moreor fewer parts (Kosslyn, 1980).

The imagery-as-internalized-perceptionhas proved to be too narrow a view of thevariety of visuospatial representations. In ac-counting for syllogistic reasoning, Johnson-Laird (1983) proposed that people formmental models of the situations describedby the propositions (see Johnson-Laird,Chap. 9). Mental models contrast with clas-sic images in that they are more schematicthan classical images. Entities are repre-sented as tokens, not as likenesses, andspatial relations are approximate, almostqualitative. A similar view was developedto account for understanding text and dis-course, that listeners and readers constructschematic models of the situations described(e.g., Kintsch & van Dijk, 1983 ; Zwaan &Radvansky, 1998). As is seen, visuospatialmental representations of environments, de-vices, and processes are often schematic,even distorted, rather than detailed and ac-curate internalized perceptions.

transformations

Here, the logic is the same for most researchprograms, and in the spirit of Shepard’snotion of second-order isomorphisms: todemonstrate that the times to make par-ticular visuospatial judgments in memoryincrease with the times to observe or per-form the transformations in the world. Thedramatic first demonstration was mental ro-tation (Shepard & Metzler, 1971 ): time tojudge whether two figures in different ori-entations (Figure 10.1 ) are the same ormirror images correlates linearly with theangular distance between the orientationsof the figures. The linearity of the relation-ship – 1 2 points on a straight line – suggestssmooth continuous mental transformation.Although linear functions have been ob-tained for the original stimuli, strings of10 cubes with two bends, monotonic, but



Figure 1 0.1 . Mental rotation task of Shepardand Metzler (1971 ). Participants determinewhether members of each pair can be rotatedinto congruence.

not linear functions are obtained for otherstimuli, such as letters (Shepard & Cooper,1982). There are myriad possible mentaltransformations, only a few of which havebeen studied in detail. They may be classifiedinto mental transformations on other objectsand individuals, and mental transformationson one’s self. In both cases, the transforma-tions may be global, wholistic, of the entireentity, or the transformations may be opera-tions on parts of entities.

Mental Transformations on Objects. Ro-tation is not the only transformation thatobjects in the world undergo. They canundergo changes of size, shape, color, in-ternal features, position, combination, andmore. Mental performance of some of thesetransformations has been examined. Thetime to mentally compare the shapes oftwo rectangles differing in size increases asthe actual size difference between them in-

creases (Bundesen, Larsen, & Farrell, 1981 ;Moyer, 1973). New objects can be con-structed in imagery, a skill presumably re-lated to design and creativity (e.g., Finke,1990, 1993). In a well-known example,Finke, Pinker, and Farah (1989) asked stu-dents to imagine a capital letter J centeredunder an upside-down grapefruit half. Stu-dents reported “seeing” an umbrella. Evenwithout instructions to image, certain tasksspontaneously encourage formation of vi-sual images. For example, when participantsare asked whether a described spatial array,such as star above plus, matches a depictedone, response times indicate that they trans-form the description into a depiction whengiven sufficient time to mentally constructthe situation (Glushko & Cooper, 1978;Tversky, 1975).

In the cases of mental rotation, mentalmovement, and mental size transformations,objects or object parts undergo imaginedtransformations. There is also evidence thatobjects can be mentally scanned in a contin-uous manner. In a popular task introducedby Kosslyn and his colleagues, participantsmemorize a map of an island with severallandmarks, such as a well and a cave. Partic-ipants are then asked to conjure an imageof the map and to imagine looking first atthe well, and then mentally scanning fromthe well to the cave. The general finding isthat mental scanning between two imaginedlandmarks increases linearly as the distancebetween them increases (Denis & Kosslyn,1999; Kosslyn, Ball, & Rieser, 1978; Fig-ure 10.2). The phenomenon holds for spa-tial arrays established by description ratherthan depiction, again, under instructions toform and use images (Denis, 1996). Mentalscanning occurs for arrays in depth and forflat perspectives on 3D arrays (Pinker, 1980).In the previous studies, participants weretrained to mentally scan, and directed to doso, leaving open the question of whetherit occurs spontaneously. It seems to be ina task requiring direction judgments on re-membered arrays. Participants first saw anarray of dots. After the dots disappeared,an arrow appeared on the screen. The taskwas to say whether the arrow pointed to



the previous location of a dot. Reactiontimes increased with distance of the arrowto the likely dot, suggesting that participantsmentally scan from the arrow to answerthe question (Finke & Pinker, 1982 , 1983).Mental scanning may be part of catchingor hitting the ball in baseball, tennis, andother sports.

Applying Several Mental Transformations.Other mental transformations on objects arepossible, for example, altering the internalconfiguration of an object. To solve someproblems, such as geometric analogies, peo-ple need to apply more than one mentaltransformation to a figure to obtain the an-swer. In most cases, the order of applying thetransformations is optional; that is, first ro-tating and then moving a figure yields thesame answer as first moving and then rotat-ing. Nevertheless, people have a preferredorder for performing a sequence of mentaltransformations, and when this order is vi-olated, both errors and performance timeincrease (Novick & Tversky, 1987). Whataccounts for the preferred order? Althoughthe mental transformations are performedin working memory, the determinants of or-der do not seem to be related to workingmemory demands. Move is one of the leastdemanding transformations, and it is typi-cally performed first, whereas rotate is oneof the most difficult transformations and isperformed second. Then transformations ofintermediate difficulty are performed. Whatcorrelated with the order of applying succes-sive mental transformations is the order ofdrawing. Move determines where the pencilis to be put on the paper, the first act of draw-ing. Rotate determines the direction in whichthe first stroke should be taken, and it is thenext transformation. The next transforma-tions to be applied are those that determinethe size of the figure and its internal details(remove, add part, change size, change shading,add part). Although the mental transforma-tions have been tied to perceptual processes,the ordering of performing them appearsto be tied to a motor process, the act ofdrawing or constructing a figure. This findingpresaged later work showing that complex

visuospatial reasoning has not only percep-tual, but also motor, foundations.

Mental Transformations of Self. That men-tal imagery is both perceptual and motorfollows from broadening the basic tenets ofthe classical account for imagery. Accordingto that account, mental processes are inter-nalizations of external or externally drivenprocesses, perceptual ones according to theclassic view (e.g., in the chapter title ofShepard & Podgorny, 1978, “Cognitive pro-cesses that resemble perceptual processes”).The acts of drawing a figure or construct-ing an object entail both perceptual andmotor processes working in concert, as domany other activities performed in both realand virtual worlds, from shaking hands towayfinding.

Evidence for mental transformations ofself, or motor imagery, rather than or in addi-tion to visual imagery has come from a vari-ety of tasks. The time taken to judge whethera depicted hand is right or left correlateswith the time taken to move the hand intothe depicted orientation, as if participantswere mentally moving their hands in or-der to make the right/left decision (Parsons,1987b; Sekiyama, 1982). Mental reorienta-tion of one’s body has been used to ac-count for reaction times to judge whethera left or right arm is extended in picturesof bodies in varying orientations from up-right (Parsons, 1987a). In those studies, re-action times depend on the angle of rotationand the degree of rotation. For some orienta-tions, notably the picture plane, the degreeof rotation from upright has no effect. Thisallows dissociating mental transformationsof other, in this case, mental rotation, frommental transformations of self, in this case,perspective transformations, as the latter doyield increases in reaction times with de-gree of rotation from upright (Zacks, Mires,Tversky, & Hazeltine, 2000; Zacks & Tver-sky, in press). Imagining one’s self interactingwith a familiar object such as a ball or a ra-zor, selectively activates left inferior parietaland sensorymotor cortex, whereas imagin-ing another interacting with the same objectsselectively activates right inferior parietal,



Figure 1 0.2 . Mental scanning. Participantsmemorize map and report time to mentally scanfrom one feature to another (after Kosslyn, Ball,& Rieser, 1978).

precuneus, posterior cingulated, and fron-topolar cortex (Ruby & Decety, 2001 ).

There have been claims that visual andmotor imagery, or as we have put it, mentaltransformations of object and of self, sharethe same underlying mechanisms (Wexler,Kosslyn, & Berthoz, 1998; Wolschlager &Wolschlager, 1998). For example, perform-ing clockwise physical rotations facilitatesperforming clockwise mental rotations, butinterferes with performing counterclock-wise mental rotations. However, this maybe because planning, performing, and mon-itoring the physical rotation requires bothperceptual and motor imagery. The workof Zacks and collaborators (Zacks et al.,2000; Zacks & Tversky, in press) and Rubyand Decety (2001 ) suggests that these twoclasses of mental transformations are dis-sociable. Other studies directly comparingthe two systems supports their dissociability:The consequences of using one can be differ-ent from the consequences of using the other(Schwartz, 1999; Schwartz & Black, 1999;Schwartz & Holton, 2000). When peopleimagine wide and narrow glasses filled to the

same level, and are asked which would spillfirst when tilted, they are typically incorrectfrom visual imagery. However, if they closetheir eyes and imagine tilting each glass un-til it spills, they correctly tilt a wide glassless than a narrow one (Schwartz & Black,1999). Think of turning a car versus turn-ing a boat. To imagine making a car turnright, you must imagine rotating the steer-ing wheel to the right; however, to imaginemaking a boat turn right, you must imaginemoving the rudder lever left. In mental ro-tation of left and right hands, the shortestmotor path accounts for the reaction timesbetter than the shortest visual path (Parsons,1987b). Mental enactment also facilitatesmemory, even for actions described ver-bally (Englekamp, 1998). Imagined motortransformations presumably underlie men-tal practice of athletic and musical routines,techniques known to benefit performance(e.g., Richardson, 1967).

The reasonable conclusion, then, is thatboth internalized perceptual transforma-tions and internalized motor transformationscan serve as bases for transformations inmental imagery. Perceptual and motor im-agery can work in concert in imagery, justas perceptual and motor processes work inconcert in conducting the activities of life.

elementary transformations

The imagery-as-internalized-perception ap-proach has provided evidence for myriadmental transformations. We have reviewedevidence for a number of mental per-ceptual transformations: scanning, changeorientation, location, size, shape, color, con-struct from parts, and rearrange parts. Thenwe have motor transformations: motions ofbodies, wholes, or parts. This approach hasthe potential to provide a catalog of elemen-tary mental transformations that are simpleinferences and that can combine to enablecomplex inferences.

The work on inference, judgment, andproblem solving will suggest transformationsthat have yet to be explored in detail. Here,we propose a partial catalog of candidatesfor elementary properties of representations



and transformations, expanding from the re-search reviewed:

� Determining static properties of entities:figure/ground, symmetry, shape, internalconfiguration, size, color, texture, andmore

� Determining relations between staticentities:◦ With respect to a frame of reference:

location, direction, distance, and more◦ With respect to other entities, com-

paring size, color, shape, texture, loca-tion, orientation, similarity, and otherattributes

� Determining relations of dynamic and staticentities:◦ With respect to other entities or

to a reference frame: direction,speed, acceleration, manner, intersec-tion/collision

� Performing transformations on entities:change location (scanning), change per-spective, orientation, size, shape; mov-ing wholes, reconfiguring parts, zooming,enacting

� Performing transformations on self: changeof perspective, change of location,change of size, shape, reconfiguring parts,enacting

individual differences

Yes, people vary in spatial ability. However,spatial ability does not contrast with ver-bal ability; in other words, someone can begood or poor at both, as well as good in oneand poor in the other. In addition, spatialability (like verbal ability) is not a single,unitary ability. Some of the separate spa-tial abilities differ qualitatively; that is, theymap well onto the kinds of mental transfor-mations they require. A meta-analysis of anumber of factor analyses of spatial abili-ties yielded three recurring factors (Linn &Peterson, 1986): spatial perception, spatialvisualization, and mental rotation. Rod-and-frame and water-level tasks load high on spa-tial perception; this factor seems to reflectchoice of frame of reference, within an ob-ject or extrinsic. Performance on embedded

figures, finding simple figures in more com-plex ones, loads high on spatial visualization,and performance on mental rotation tasksnaturally loads high on the mental rotationfactor. As frequently as they are found, thesethree abilities do not span the range of spa-tial competencies. Yet another partially in-dependent visuospatial ability is visuospatialmemory, remembering the layout of display(e.g., Betrancourt & Tversky, in press). Thenumber of distinct spatial abilities as well astheir distinctness remain controversial (e.g.,Carroll, 1993 ; Hegarty & Waller, in press).

More recent work explores the relationsof spatial abilities to the kinds of men-tal transformations that have been distin-guished, for example, imagining an objectrotate versus imagining changing one’s ownorientation. The mental transformations, inturn, are often associated with differentbrain regions (e.g., Zacks, Mires, Tversky, &Hazeltine, 2000; Zacks, Ollinger, Sheridan,& Tversky, 2002 ; Zacks & Tversky, inpress). Kozhevniikov, Kosslyn, and Shepard(in press) proposed that spatial visualiza-tion and mental rotation correspond respec-tively to the two major visual pathways inthe brain – the ventral “what” pathway un-derlying object recognition and the dorsal“where” pathway underlying spatial loca-tion. Interestingly, scientists and engineersscore relatively high on mental rotation andartists score relatively high on spatial visu-alization. Similarly, architects and design-ers score higher than average on embed-ded figure tasks but not on mental rota-tion (Suwa & Tversky, 2003). Associatingspatial ability measures to mental transfor-mations and brain regions are promisingdirections toward a systematic account ofspatial abilities.

Inferences

Inferences from Observing Motionin Space

To ensure effective survival, in addition toperceiving the world as it is we need toalso anticipate the world that will be. This



entails inference, inferences from visuospa-tial information. Some common inferences,such as determining where to intersect aflying object, in particular, a fly ball (e.g.,McBeath, Shaffer, & Kaiser, 1995), or whatmoving parts belong to the same object (e.g.,Spelke, Vishton, & von Hofsten, 1995) arebeyond the scope of the chapter. From sim-ple, abstract motions of geometric figures,people, even babies, infer causal impact andbasic ontological categories, notably, inani-mate and animate. A striking demonstrationof perception of causality comes from thework of Michotte (1946/1963 ; see Buehner& Cheng, Chap. 7). Participants watch filmsof a moving object, A, coming into contactwith a stationary object, B. When object Bmoves immediately, continuing the direc-tion of motion suggested by object A, peopleperceive A as launching B, A as causing B tomove. When A stops so both A and B arestationary before B begins to move, the per-ception of a causal connection between A’smotion and B’s is lost; their movements areseen as independent events. This is a forcefuldemonstration of immediate perception ofcausality from highly abstract actions, as wellas of the conditions for perception of causal-ity. What seems to underlie the perceptionof causality is the perception that object Aacts on object B. Actions on objects turn outto be the basis for segmenting events intoparts (Zacks, Tversky, & Iyer, 2001 ).

In Michotte’s (1946/1963) demonstra-tions, the timing of the contact between theinitially moving object and the stationary ob-ject that begins to move later is critical. IfA stops moving considerably before B be-gins to move, then B’s motion is perceivedto be independent of A’s. B’s movementin this case is seen as self-propelled. Self-propelled movement is possible only for ani-mate agents, or, more recently in the historyof humanity, for machines. Possible pathsand trajectories of animate motion differfrom those for inanimate motion. Preschoolchildren can infer which motion paths areappropriate for animate and inanimate mo-tion, and even for abstract stimuli; they alsooffer sensible explanations for their infer-ences (Gelman, Durgin, & Kaufman, 1995).

From abstract motion paths, adults canmake further inferences about what gen-erated the motion. In point-light films,the only thing visible is the movementof lights placed at motion junctures of,for example, at the joints of people walk-ing or along branches of bushes swaying.From point-light films, people can determinewhether the motion is walking, running, ordancing, of men or of women, of friends(Cutting & Kozlowski, 1977; Johannson,1973 ; Kozlowski & Cutting, 1977), of bushesor trees (Cutting, 1986). Surprisingly, frompoint-light displays of action, people are bet-ter at recognizing their own movements thanthose of friends, suggesting that motor ex-perience contributes to perception of mo-tion (Prasad, Loula, & Shiffrar, 2003). Evenabstract films of movements of geometricfigures in sparse environments can be inter-preted as complex social interactions, suchas chasing and bullying, when they are espe-cially designed for that (Heider & Simmel,1944 ; Martin & Tversky, 2003 ; Oatley &Yuill, 1985) or playing hide-and-seek, but in-terpreting these as intentional actions is notimmediate; rather, it requires repeated ex-posure and possibly instructions to interpretthe actions (Martin & Tversky, 2003).

Altogether, simply from abstract mo-tion paths or animated point-light displays,people can infer several basic ontologicalcategories: causal action, animate versusinanimate motion, human motion, motionof males or females and familiar individuals,and social interactions.

Mental Spatial Inferences

inferences in real environments

Every kid who has figured out a short-cut,and who has not, has performed a spatialinference (for a more recent overview ofkids, see Newcombe & Huttenlocher, 2000).Some of these inferences turn out to beeasier than others, often surprisingly. Forexample, in real environments, inferencesabout where objects will be in relationshipto one’s self after imagined movement in theenvironment turn out to be relatively ac-curate when the imagined movement is a



translation, that is, movement forward orbackward aligned with the body. How-ever, if the imagined movement is rota-tional, a change in orientation, updating isfar less accurate (e.g., Presson & Montello,1994 ; Reiser, 1989). When asked to imaginewalking forward a certain distance, turning,walking forward another distance, and thenpointing back to the starting point, partic-ipants invariably err by not taking into ac-count the turn in their pointing (Klatzky,Loomis, Beall, Chance, & Golledge, 1998). Ifthey actually move forward, turn, and con-tinue forward, but blindfolded, they pointcorrectly. Spatial updating in real environ-ments is more accurate after translation thanafter rotation, and updating after rotationis selectively facilitated by physical rotation.This suggests a deep point about spatial in-ferences and possibly other inferences, thatin inference, mental acts interact with phys-ical acts.

gesture

Interaction of mind and body in inference isalso revealed in gesture. When people de-scribe space but are asked to sit on theirhands to prevent gesturing, their speech fal-ters (Rauscher, Krauss, & Chen, 1996), sug-gesting that the acts of gesturing promotespatial reasoning. Even blind children ges-ture as they describe spatial layouts (Iverson& Goldin-Meadow, 1997).

The nature of spontaneous gestures sug-gests how this happens. When describingcontinuous processes, people make smooth,continuous gestures; when describing dis-crete ones, people make jagged, discontin-uous ones (Alibali, Bassok, Solomon, Syc,& Goldin-Meadow, 1999). For space, peo-ple tend to describe environments as if theywere traveling through them or as if theywere viewing them from above. The planeof their gestures differs in each case, in cor-respondence with the linguistic perspectivethey adopt (Emmorey, Tversky, & Taylor,2000). Earlier, mental transformations thatappear to be internalized physical transfor-mations, such as those underlying handed-ness judgments, were described. Here, we

also see that actual motor actions affect andreflect the character of mental ones.

inferences in mental environments

The section on inference opened with spatialinferences made in real environments. Of-ten, people make inferences about environ-ments they are not currently in, for exam-ple, when they tell a friend where to howto get to their house and where to find thekey when they arrive. For familiar environ-ments, people are quite competent at thesesorts of spatial inferences. The mental repre-sentations and processes underlying these in-ferences have been studied for several kindsof environments, notably the immediatelysurrounding visible or tangible environmentand the environment too large to be seenat a glance. These two situations, the spacearound the body, and the space the bodynavigates, seem to function differently in ourlives, and consequently, to be conceptualizeddifferently (Tversky, 1998).

Spatial updating for the space around thebody was first studied using language aloneto establish the environments (Franklin &Tversky, 1990). It is significant that lan-guage alone, with no specific instructionsto form images, was sufficient to establishmental environments that people could up-date easily and without error. In the proto-typical spatial framework task, participantsread a narrative that describes themselvesin a 3D spatial scene, such as a museumor hotel lobby (Franklin & Tversky, 1990;Figure 10.3). The narrative locates and de-scribes objects appropriate to the scene be-yond the observer’s head, feet, front, back,left, and right (locations chosen randomly).After participants have learned the scenesdescribed by the narratives, they turn to acomputer that describes them as turning inthe environment so they are now facing a dif-ferent object. The computer then cues themwith direction terms, front, back, head, and soon, to which the participants respond withthe name of the object now in that direc-tion. Of interest are the times to respond,depending on the direction from the body.The classical imagery account would predictthat participants will imagine themselves in



Figure 1 0.3 . Spatial framework situation.Participants read a narrative describing objectsaround an observer (after Bryant, Tversky, &Franklin, 1992).

the environment facing the selected object,and then imagine themselves turning to faceeach cued object in order to retrieve the ob-ject in the cued direction. The imagery ac-count predicts that reaction times should befastest to the object in front, then to the ob-jects 90 degrees away from front, that is, left,right, head, and feet, and slowest to objects1 80 degrees from front, that is, objects to theback. Data from dozens of experiments failto support that account.

Instead, the data conform to the spatialframework theory according to which partic-ipants construct a mental spatial frameworkfrom extensions of three axes of the body,head/feet, front/back, and left/right. Timesto access objects depend on the asymmetriesof the body axes, as well as the asymmetriesof the axes of the world. The front/back andhead/feet axes have important perceptualand behavioral asymmetries that are lack-ing in the left/right axis. The world also hasthree axes, only one of which is asymmetric,the axis conferred by gravity. For the uprightobserver, the head/feet axis coincides withthe axis of gravity, so responses to head andfeet should be fastest, and they are. Accord-ing to the spatial framework account, timesshould be next fastest to the front/back axis

and slowest to the left/right axis, the pat-tern obtained for the prototypical situation.When narratives describe observers as reclin-ing in the scenes, turning from back to sideto front, then no axis of the body is corre-lated with gravity so times depend on theasymmetries of the body, and the patternchanges. Times to retrieve objects in frontand back are then fastest because the per-ceptual and behavioral asymmetries of thefront/back axis are most important. This isthe axis that separates the world that can beseen and manipulated from the world thatcannot be seen or manipulated.

By now, dozens of experiments have ex-amined patterns of response times to system-atic changes in the described spatial envi-ronment (e.g., Bryant, Tversky, & Franklin,1992 ; Franklin, Tversky, & Coon, 1992). Inone variant, narratives described participantsat an oblique angle outside the environ-ment looking onto a character (or two!) in-side the environment; in that case, noneof the axes of the observer’s body is cor-related with axes of the characters in thenarrative, and the reaction times to all di-rections are equal (Franklin et al., 1992).In another variant, narratives described thescene, a special space house constructed byNASA, as rotating around the observer in-stead of the observer turning in the scene(Tversky, Kim, & Cohen, 1999). That con-dition proved difficult for participants. Theytook twice as long to update the environ-ment when the environment moved thanwhen the observer moved, a case prob-lematic for pure propositional accounts ofmental spatial transformations. Once theyhad updated the environment, retrievaltimes corresponded to the spatial frame-work pattern.

Yet other experiments have varied theway the environment was conveyed, com-paring description, diagram, 3D model, andlife (Bryant & Tversky, 1999; Bryant, Tver-sky, & Lanca, 2001 ). When the scene is con-veyed by narrative, life, or a 3D model, thestandard spatial framework pattern obtains.However, when the scene is conveyed bya diagram, participants spontaneously adoptan external perspective on the environment.



Their response times are consonant withperforming a mental rotation of the entireenvironment rather than performing a men-tal change of their own perspective with re-spect to a surrounding environment (Bryant& Tversky, 1999). Which viewpoint partici-pants adopt, and consequently which mentaltransformation they perform, can be alteredby instructions. When instructed to do so,participants will adopt the internal perspec-tive embedded in the environment in whichthe observer turns from a diagram or the ex-ternal perspective from a model in which theentire environment is rotated, with thepredicted changes in patterns of retrievaltimes. Similar findings have been reportedby Huttenlocher and Presson (1979), Wraga,Creem, and Proffitt (2000), and Zacks et al.(in press).

route and survey perspectives

When people are asked to describe envi-ronments that are too large to be seen at aglance, they do so from one of two perspec-tives (Taylor & Tversky, 1992a, 1996). In aroute perspective, people address the listeneras “you,” and take “you” on a tour of the en-vironment, describing landmarks relative toyour current position in terms of your front,back, left, and right. In a survey perspective,people take a bird’s eye view of the envi-ronment and describe locations of landmarksrelative to one another in terms of north,south, east, and west. Speakers (and writers)often mix perspectives, contrary to linguistswho argue that a consistent perspective isneeded both for coherent construction ofa message and for coherent comprehen-sion (Taylor & Tversky, 1992 , 1996; Tversky,Lee, & Mainwaring, 1999). In fact, con-struction of a mental model is faster whenperspective is consistent, but the effect issmall and disappears quickly during retrievalfrom memory (Lee & Tversky, in press).In memory for locations and directions oflandmarks, route and survey statements areverified equally quickly and accurately irre-spective of the perspective of learning, pro-vided the statements are not taken verbatimfrom the text (Taylor & Tversky, 1992b). For

route perspectives, the mental transforma-tion needed to understand the location in-formation is a transformation of self, an ego-centric transformation of one’s viewpointin an environment. For survey perspectives,the mental transformation needed to under-stand the location information is a transfor-mation of other, a kind of mental scanningof an object.

The prevalence of these two perspectivesin imagery, the external perspective viewingan object or something that can be repre-sented as an object and the internal perspec-tive viewing an environment from within,is undoubtedly associated with their preva-lence in the experience of living. In life, weobserve changes in the orientation, size, andconfiguration of objects in the world, andscan them for those changes. In life, we movearound in environments, updating our po-sition relative to the locations of other ob-jects in the environment. We are adept atperforming the mental equivalents of theseactual transformations. There is a naturalcorrespondence between the internal andexternal perspectives and the mental trans-formations of self and other, but the humanmind is flexible enough to apply either trans-formation to either perspective. Althoughwe are biased to take an external perspec-tive on objects and mentally transform themand biased to take an internal perspective onenvironments and mentally transform ourbodies with respect to them, we can takeinternal perspectives on objects and ex-ternal perspectives on events. The mentalworld allows perspectives and transforma-tions, whereas the physical world does not.Indeed, conceptualizing a 3D environmentthat surrounds us and is too large to be seenat once as a small flat object before the eyes,something people, even children, have donefor eons whenever they produce a map, isa remarkable feat of the human mind (cf.Tversky, 2000a).

effects of language on spatial thinking

Speakers of Dutch and other Western lan-guages use both route and survey perspec-tives. Put differently, they can use either a



relative spatial reference system or an ab-solute (extrinsic) spatial reference systemto describe locations of objects in space.Relative systems use the spatial relations“left,” “right,” “front,” and “back” to locateobjects; absolute or extrinsic systems useterms equivalent to “north,” “south,” “east,”and “west.” A smattering of languages dis-persed around the world do not describelocations using “left” and “right” (Levinson,2003). Instead, they rely on an absolute sys-tem, so a speaker of those languages wouldrefer to your coffee cup as the “north” cuprather than the one on “your right.” Talkapparently affects thought. Years of talkingabout space using an absolute spatial refer-ence system has fascinating consequences forthinking about space. For example, speak-ers of absolute languages reconstruct a shuf-fled array of objects relative to extrinsic di-rections in contrast to speakers of Dutch,who reconstruct the array relative to theirown bodies. What’s more, when speakers oflanguages with only extrinsic reference sys-tems are asked to point home after beingdriven hither and thither, they point withimpressive accuracy, in contrast to Dutchspeakers, who point at random. The viewthat the way people talk affects how theythink has naturally aroused controversy (seeGleitman & Papafragou, Chap. 26), but is re-ceiving increasing support from a variety oftasks and languages (e.g., Boroditsky, 2001 ;Boroditsky, Ham, & Ramscar, 2002). Tak-ing a broader perspective, the finding thatlanguage affects thought is not as startling.Language is a tool, such as measuring in-struments or arithmetic or writing; learn-ing to use these tools also has consequencesfor thinking.

Judgments

Complex visuospatial thinking is fundamen-tal to a broad range of human activity, fromproviding directions to the post office andunderstanding how to operate the latestelectronic device to predicting the conse-quences of chemical bonding or designing a

shopping center. Indeed, visuospatial think-ing is fundamental to the reasoning processesdescribed in other chapters in this handbook,as discussed in the chapters on similarity (seeGoldstone & Son, Chap. 2), categorization(see Medin & Rips, Chap. 3), induction (seeSloman & Lagnado, Chap. 5), analogical rea-soning (see Holyoak, Chap. 6), causality (seeBuehner & Cheng, Chap. 7), deductive rea-soning (see Evans, Chap. 8), mental models(see Johnson-Laird, Chap. 9), and problemsolving (see Novick & Bassok, Chap. 1 4). For-tunately for both reader and author, there isno need to repeat those discussions here.

Distortions as Clues to Reasoning

Another approach to revealing visuospa-tial reasoning has been to demonstrate theways that visuospatial representations differsystematically from situations in the world.This approach, which can be called the dis-tortions program, contrasts with the classi-cal imagery approach. The aim of the distor-tions approach is to elucidate the processesinvolved in constructing and using men-tal representations by showing their conse-quences. The distortions approach has fo-cused more on relations between objectsand relations between objects and refer-ence frames, as these visuospatial propertiesseem to require more constructive processesthan those for establishing representationsof objects. Some systematic distortions havealso been demonstrated in representationsof objects.

representations

Early on, the Gestalt psychologists at-tempted to demonstrate that memory forfigures got distorted in the direction of goodfigures (see Riley, 1962). This claim was con-tested and countered by increasingly sophis-ticated empirical demonstrations. The dis-pute faded in a resolution: visual stimuliare interpreted, sometimes as good figures;memory tends toward the interpretations.So if o – o is interpreted as “eyeglasses,” par-ticipants later draw the connection curved,whereas if it is interpreted as “barbells,”they do not (Carmichael, Hogan, & Walter,



1932). Little noticed is that the effect doesnot appear in recognition memory (Prentice,1954). Since then, and relying on the sophis-ticated methods developed, there has beenmore evidence for shape distortion in repre-sentations. Shapes that are nearly symmet-ric are remembered or judged as more sym-metric than they actually are, as if peoplecode nearly symmetric objects as symmetric(Freyd & Tversky, 1984 ; McBeath, Schiano,& Tversky, 1997; Tversky & Schiano, 1989).Given that many of the objects and be-ings that we encounter are symmetric, butare typically viewed at an oblique angle,symmetry may be a reasonable assump-tion, although one that is wrong on occa-sion. Size is compressed in memory (Kerst& Howard, 1978). When portions of ob-jects are truncated by picture frames, theobjects are remembered as more completethan they actually were (Intraub, Bender, &Mangels, 1992).

representations and transformations: spatial

configurations and cognitive maps

The Gestalt psychologists also producedstriking demonstrations that people organizethe visual world in principled ways, evenwhen that world is a meaningless array (seeHochberg, 1978). Entities in space, espe-cially ones devoid of meaning, are difficultto understand in isolation, easier to grasp incontext. People group elements in an arrayby proximity or similarity or good continua-tion. One inevitable consequence of percep-tual organizing principles is distorted repre-sentations.

Many of the distortions reviewed herehave been instantiated in memory for per-ceptual arrays that do not stand for anything.They have also been illustrated in memoryfor cognitive maps and for environments. Assuch, they have implications for how peoplereason in navigating the world, a visuospa-tial reasoning task that people of all ages andparts of the world need to solve. Even moreintriguing, many of these phenomena haveanalogs in abstract thought.

For the myriad spatial distortions de-scribed here (and analyzed more fully in

Tversky, 1992 , 2000b, 2000c), it is diffi-cult to clearly attribute error to either rep-resentations or processes. Rather the errorsseem to be consequences of both, of schema-tized, hence distorted, representations con-structed ad hoc in order to enable specificjudgments, such as the direction or distancebetween pairs of cities. When answeringsuch questions, it is unlikely that people con-sult a library of “cognitive maps.” Rather, itseems that they draw on whatever informa-tion they have that seems relevant, organiz-ing it for the question at hand. The reliabilityof the errors under varying judgments makesit reasonable to assume erroneous represen-tations are reliably constructed. Some of theorganizing principles that yield systematicerrors are reviewed in the next section.

Hierarchical Organization. Dots that aregrouped together by good continuation, forexample, parts of the same square out-lined in dots, are judged to be closer thandots that are actually closer but parts ofseparate groups (Coren & Girgus, 1980).An analogous phenomenon occurs in judg-ments of distance between buildings (Hirtle& Jonides, 1985): Residents of Ann Arborthink that pairs of university (or town) build-ings are closer than actually closer pairs ofbuildings that belong to different groups, oneto the university and the other to the town.Hierarchical organization of essentially flatspatial information also affects accuracy andtime to make judgments of direction. Peopleincorrectly report that San Diego is west ofReno. Presumably this error occurs becausepeople know the states to which the citiesbelong and use the overall directions of thestates to infer the directions between cities inthe states (Stevens & Coupe, 1978). Peopleare faster to judge whether one city is east ornorth of another when the cities belong toseparate geographic entities than when theyare actually farther, but part of the same ge-ographic entity (Maki, 1981 ; Wilton, 1979).

A variant of hierarchical organizationoccurs in locating entities belonging to abounded region. When asked to rememberthe location of a dot in a quadrant, people



place it closer to the center of the quadrant,as if they were using general informationabout the area to locate entity contained init (Huttenlocher, Hedges, & Duncan, 1991 ;Newcombe & Huttenlocher, 2000).

Amount of Information. That representa-tions are constructed on the fly in the ser-vice of particular judgments seems to be thecase for other distance estimates. Distancesbetween A and B, say two locations within atown, are greater when there are more crossstreets or more buildings or more obstaclesor more turns on the route (Newcombe &Liben, 1982 ; Sadalla & Magel, 1980; Sadalla& Staplin, 1980a, 1980b; Thorndyke, 1981 ),as if people mentally construct a represen-tation of a path from A to B from that in-formation and use the amount of informa-tion as a surrogate for the missing exactdistance information. There is an analogousvisual illusion: A line appears longer if bi-sected, and longer still with more tick marks(at some point of clutter, the illusion ceasesor reverses).

Perspective. Steinberg regaled generationsof readers of the New Yorker and denizens ofdormitory rooms with his maps of views ofthe world. In the each view, the immediatesurroundings are stretched and the rest ofthe world shrunk. The psychological realityof this genre of visual joke was demonstratedby Holyoak and Mah (1982). They asked stu-dents in Ann Arbor to imagine themselveson either coast and to estimate the distancesbetween pairs of cities distributed more orless equally on an east-west axis across thestates. Regardless of imagined perspective,students overestimated the near distancesrelative to the far ones.

Landmarks. Distance judgments are alsodistorted by landmarks. People judge the dis-tance of an undistinguished place to be closerto a landmark than vice versa (McNamara& Diwadkar, 1997; Sadalla, Burroughs, &Staplin, 1980). Landmark asymmetries vi-olate elementary metric assumptions, as-sumptions that are more or less realized inreal space.

Figure 1 0.4. Alignment. A significant majorityof participants think the incorrect lower map iscorrect. The map has been altered so the UnitedStates and Europe and South American andAfrica are more aligned (after Tversky, 1981 ).

Alignment. Hierarchical, perspective, andlandmark effects can all be regarded as con-sequences of the Gestalt principle of group-ing. Even groups of two equivalent entitiescan yield distortion. When people are askedto judge which of two maps is correct, a mapof North and South America in which SouthAmerica has been moved westward to over-lap more with North America, or the ac-tual map, in which the two continents barelyoverlap, the majority of respondents pre-fer the former (Tversky, 1981 ; Figure 10.4).A majority of observers also prefer an in-correct map of the Americas and Europe/Africa/Asia in which the Americas aremoved northward so the United States andEurope and South America and Africa aremore directly east-west. This phenomenonhas been called alignment; it occurs whenpeople group two spatial entities and thenremember them more in correspondencethan they actually are. It appears not onlyin judgments of maps of the world, but alsoin judgments of directions between cities, in



memory for artificial maps, and in memoryfor visual blobs.

Spatial entities cannot be localized inisolation; they can be localized with re-spect to other entities or to frames of ref-erence. When they are coded with respectto another entity, alignment errors are likely.When entities are coded with respect to aframe of reference, rotation errors, describedin the next section, are likely.

Rotation. When people are asked to placea cutout of South American in a north-southeast-west frame, they upright it. A large spa-tial object, such as South America, inducesits own coordinates along an axis of elon-gation and an axis parallel to that one. Theactual axis of elongation of South America istilted with respect to north-south, and peo-ple upright it in memory. Similarly, peopleincorrectly report that Berkeley is east ofStanford, when it is actually slightly west.Presumably this occurs because they up-right the Bay Area, which actually runsat an angle with respect to north-south.This error has been called rotation; it oc-curs when people code a spatial entity withrespect to a frame of reference (Tversky,1981 ; Figure 10.5). As for rotation, it ap-pears in memory for artificial maps and un-interpreted blobs, as well as in memory forreal environments. Others have replicatedthis error in remembered directions and innavigation (e.g., Glicksohn, 1994 ; Lloyd &Heivly, 1987; Montello, 1991 ; Presson &Montello, 1994).

Are Spatial Representations Incoherent?.This brief review has brought evidence fordistortions in memory and judgment forshapes of objects, configurations of objects,and distances and directions between objectsthat are a consequence of the organizationof the visuospatial information. These arenot errors of lack of knowledge; even ex-perienced taxi drivers make them (Chase& Chi, 1981 ). Moreover, many of these bi-ases have parallels in abstract domains, suchas judgments about members of one’s ownsocial or political groups relative to judg-ments about members of other groups (e.g.,Quattrone, 1986).

What might a representation that cap-tures all these distortions look like? Nothingthat can be sketched on a sheet of paper, thatis, coherent in two dimensions. Landmarkasymmetries alone disallow that. It does notseem likely that people make these judg-ments by retrieving a coherent prestoredmental representation, a “cognitive map,”and reading the direction or distance from it.Rather, it seems that people construct repre-sentations on the fly, incorporating only theinformation needed for that judgment, therelevant region, the specific entities withinit. Some of the information may be visuospa-tial, from experience or from maps, somemay be linguistic. For these reasons, “cog-nitive collage” seems a more apt metaphorthan “cognitive map” for whatever represen-tations underlie spatial judgment and mem-ory (Tversky, 1993). Such representationsare schematic, they leave out much infor-mation and simplify others. Schematizationoccurs for at least two reasons. More exactinformation may not be known and there-fore cannot be represented. More exact in-formation may not even be needed as thesituation on the ground may fill it in. Moreinformation may overload working mem-ory, which is notoriously limited. Not onlymust the representation be constructed inworking memory, but also a judgment madeon the representation. Schematization mayhide incoherence, or it may not be noticed.Schematization necessarily entails system-atic error.

Why do Errors Persist?. It is reasonable towonder why so many systematic errors per-sist. Some reasons for the persistence of er-ror have already been discussed, that theremay be correctives on the ground, that someerrors are a consequence of the schematiza-tion processes that are an inherent part ofmemory and information processing. Yet an-other reason is that the correctives are spe-cific – now I know that Rome is north ofPhiladelphia – and do not affect or evenmake contact with the general informationorganizing principle that generated the error,and that serves us well in many situations(e.g., Tversky, 2003a).



Figure 1 0.5 . Rotation. When asked to place a cutout of South American in aNSEW framework, most participants upright it, as in the left example (afterTversky, 1981 ).

From Spatial to Abstract Reasoning

Visuospatial reasoning does not only entailvisuospatial transformations on visuospatialinformation. Visuospatial reasoning also in-cludes making inferences from visuospatialinformation, whether that information is inthe mind or in the world. An early demon-stration was the symbolic distance effect (e.g.,Banks & Flora, 1977; Moyer, 1973 ; Paivio,1978). The time to judge which of two ani-mals is more intelligent or pleasant is fasterwhen the entities are farther on the dimen-sion than when they are closer, as if peoplewere imagining the entities arrayed on a linecorresponding to the abstract dimension. Itis easier, hence faster, to discriminate largerdistances than smaller ones. Note that a sub-jective experience of creating and using animage does not necessarily accompany mak-ing these and other spatial and abstract judg-ments. Spatial thinking can occur regardlessof whether thinkers have the sensation ofusing an image. So many abstract conceptshave spatial analogs (for related discussion,see Holyoak, Chap. 6).

Indeed, spatial reasoning is often studiedin the context of graphics, maps, diagrams,graphs, and charts. External representationsbear similarities to internal representations,if only because they are creations of the hu-man mind, cognitive tools to increase thepower of the human mind. They also bearformal similarities, in that both internal andexternal representations are mappings be-tween elements and relations. External rep-resentations are constrained by a mediumand unconstrained by working memory; forthis reason, inconsistencies, ambiguities, andincompleteness may be reduced in externalrepresentations.

Graphics: Elements

The readiness with which people map ab-stract information onto spatial informationis part of the reason for the widespread useof diagrams to represent and convey ab-stract information, from the sublime, theharmonies of the spheres, rampant in re-ligions spanning the globe, to the mun-dane corporate charts and statistical graphs.



Graphics, such as these, consist of elementsand spatial relations among the elements. Incontrast to written (alphabetic) languages,both elements and use of space in graph-ics can convey meaning rather directly (e.g.,Bertin, 1967/1983 ; Pinker, 1994 ; Tversky,1995 , 2001 ; Winn, 1989). Elements mayconsist of likenesses, such as road signs de-picting picnic tables, falling rocks, or deer. El-ements may also be figures of depiction, sim-ilar to figures of speech: synecdoche, wherea part represents a whole, common in ideo-graphic writing, for example, using a ram’shorns to represent a ram; or metonomy,where an association represents an entity oraction, common in computer menus, suchas scissors to denote cut text or a trashcan toallow deletion of files.

Graphics: Relations

Relations among entities preserve differentlevels of information. The information pre-served is reflected in the mapping to space. Insome cases, the information preserved is sim-ply categorical; space is used to separate en-tities belonging to different categories. Thespaces between words, for example, indi-cate that one set of letters belongs to onemeaning and another set to another mean-ing. Space can also be used to represent ordi-nal information, for example, listing historicevents in their order of occurrence, groceriesby the order of encountering them in thesupermarket, and companies by their prof-its. Space can be used to represent intervalor ratio information, as in many statisticalgraphs, where the spatial distances amongentities reflect their distances on someother dimension.

spontaneous use of space to represent

abstract relations

Even preschool children spontaneously usediagrammatic space to represent abstract in-formation (e.g., diSessa, Hammer, Sherin, &Kolpakowski, 1991 ; Tversky, Kugelmass, &Winter, 1991 ). In one set of studies (Tver-sky et al., 1991 ), children from three lan-guage communities were asked to placestickers on paper to represent spatial, tem-

poral, quantitative, and preference informa-tion, for example, to place stickers for TVshows they loved, liked, or disliked. Almostall the preschoolers put the stickers on a line,preserving ordinal information. Children inthe middle school years were able to repre-sent interval information, but representingmore than ordinal information was unusualfor younger children, despite strong manipu-lations to encourage them. Not only did chil-dren (and adults) spontaneously use spatialrelations to represent abstract relations, butchildren also showed preferences for the di-rection of increases in abstract dimensions.Increases were represented from right to leftor left to right (irrespective of direction ofwriting for quantity and preference) or downto up. Representing increasing time or quan-tity from up to down was avoided. Rep-resenting increases as upward is especiallyrobust; it affects people’s ability to makeinferences about second-order phenomenasuch as rate, which is spontaneously mappedto slope, from graphs (Gattis, 2002 ; Gattis &Holyoak, 1996). The correspondence of up-ward to more, better, and stronger appearsin language – on top of the world, rising tohigher levels of platitude – and in gesture –thumbs up, high five – as well as in graph-ics. These spontaneous and widespread cor-respondences between spatial and abstractrelations suggest they are cognitively natural(e.g., Tversky, 1995a, 2001 ).

The demonstrations of spontaneous useof spatial language and diagrammatic spaceto represent abstract relations suggests thatspatial reasoning forms a foundation formore abstract reasoning. In fact, childrenused diagrammatic space to represent ab-stract relations earlier for temporal relationsthan for quantitative ones, and earlier forquantitative relations than for preference re-lations (Tversky et al., 1991 ). Corrobora-tive evidence comes from simple spatial andtemporal reasoning tasks, judging whetherone object or person is before another. Inmany languages, words for spatial and tem-poral relations, such as before, after, andin between, are shared. That spatial termsare the foundation for the temporal comesfrom research showing priming of temporal



perspective from spatial perspective, but notvice versa (Boroditsky, 2000). More supportfor the primacy of spatial thinking for ab-stract thought comes from studies of prob-lem solving (Carroll, Thomas, & Mulhotra,1980). One group of participants was askedto solve a spatial problem under constraints,arranging offices to facilitate communica-tion among key people. Another group wasasked to solve a temporal analog, arrangingprocesses to facilitate production. The solu-tions to the spatial analog were superior tothose to the temporal analog. When exper-imenters suggested using a diagram to yetanother group solving the temporal analog,their success equaled that of the spatial ana-log group.

diagrams facilitate reasoning

Demonstrating that using a spatial diagramfacilitates temporal problem solving also il-lustrates the efficacy of diagrams in thinking,a finding amply supported, even for infer-ences entailing complex logic, such as doubledisjunctions, although to succeed, diagramshave to be designed with attention to theways that space and spatial entities are usedto make inferences (Bauer & Johnson-Laird,1993). Middle school children studying sci-ence were asked to put reminders on pa-per. Those children who sketched diagramslearned the material better than those whodid not (Rode & Stern, in press).

diagrams for communicating

Many maps, charts, diagrams, and graphs aremeant to communicate clearly for travel-ers, students, and scholars, whether they areprofessionals or amateurs. To that end, theyare designed to be clear and easy to com-prehend, and they meet with varying suc-cess. Good design takes account of humanperceptual and cognitive skills, biases, andpropensities. Even ancient Greek vases takeaccount of how they will be seen. Becausethey are curved round structures, creatinga veridical appearance requires artistry. Thevase “Achilles and Ajax playing a game” bythe Kleophrades Painter in the Museum ofMetropolitan Art in New York City (Art.

65 .1 1 .1 2 , ca. 500–480 b.c.) depicts a spearthat appears in one piece from the desiredviewing angle, but in three pieces whenviewed straight on (J. P. Small, personal com-munication, May 27, 2003).

The perceptual and cognitive processesand biases that people bring to graphics in-clude the catalog of mental representationsand transformations that was begun earlier.In that spirit, several researchers have devel-oped models for graph understanding, no-tably Pinker (1990), Kosslyn (1989, 1994a),and Carpenter and Shah (1998) (see Shah2003 /2004 , for an overview). These mod-els take account of the particular perceptualor imaginal processes that need to be ap-plied to particular kinds of graphs to yieldthe right inferences. Others have taken ac-count of perceptual and cognitive processingin the construction of guidelines for design of(e.g., Carswell & Wickens, 1990; Cleveland,1985 ; Kosslyn, 1994a; Tufte, 1983 , 1990,1997; Wainer, 1984 , 1997). In some cases thedesign principles are informed by research,but in most they are informed by the au-thors’ educated sensibilities and/or rules ofthumb from graphic design.

Inferences from Diagrams: Structural andFunctional. The existence of spontaneousmapping of abstract information onto spatialdoes not mean that the meanings of diagramsare transparent and can be automatically andeasily extracted (e.g., Scaife & Rogers, 1995).Diagrams can support many different classesof inferences, notably, structural and func-tional (e.g., Mayer & Gallini, 1990). Struc-tural inferences, or inferences about quali-ties of parts and the relations among them,can be readily made from inspection of a di-agram. Distance, direction, size, and otherspatial qualities and properties can be “readoff” a diagram (Larkin & Simon, 1987), atleast with some degree of accuracy. “Readingoff” entails using the sort of mental trans-formations discussed earlier, mental scan-ning, mental distance, size, shape, or direc-tion judgments or comparisons. Functionalinferences, or inferences about the behav-ior of entities, cannot be readily made frominspection of a diagram in the absence of



additional knowledge or assumptions, oftena consequence of expertise. Spatial infor-mation may provide clues to functional in-formation, but it is not sufficient for con-cepts such as force, mass, and friction.Making functional inferences requires link-ing perceptual information to conceptualinformation; it entails both knowing how to“read” a diagram, that is, what visuospatialfeatures and relations to inspect or trans-form, and knowing how to interpret that vi-suospatial information.

Structural and functional inferences re-spectively correspond to two senses of men-tal model prevalent in the field. In both cases,mental model contrasts with image. In onesense, a mental model contrasts with an im-age in being more skeletal or abstract. This isthe sense used by Johnson-Laird in his book,Mental Models (1983), in his explication ofhow people solve syllogisms (see Johnson-Laird, Chap. 9, and Evans, Chap. 8). Here,a mental model captures the structural re-lations among the parts of a system. In theother sense, a mental model contrast withan image in having moving parts, in being“runnable” to derive functional or causal in-ferences (for related discussion on causal-ity, see Buehner and Cheng, Chap. 7, andon problem solving, see Chi and Ohlsson,Chap. 16). This is the sense used in anotherbook also titled Mental Models (Gentner &Stevens, 1983). One goal of diagrams is toinstill mental models in the minds of theirusers. To that end, diagrams abstract the es-sential elements and relations of the systemthey are meant to convey. As is seen, convey-ing structure is more straightforward thanconveying function.

What does it mean to say that a mentalmodel is “runnable?” One example comesfrom research on pulley systems (Hegarty,1992). Participants were timed to make twokinds of judgments from diagrams of three-pulley systems. For true-false judgments ofstructural questions, such as “The upper leftpulley is attached to the ceiling,” responsetimes did not depend on which pulley inthe system was queried. For judgments offunctional questions, such as “The upper leftpulley goes clockwise,” response times did

depend on the order of that pulley in themechanics of the system. To answer func-tional questions, it is as if participants men-tally animate the pulley system in order togenerate an answer. Mental animation, how-ever, does not seem to be a continuous pro-cess in the same way as physical animation.Rather, mental animation seems to be a se-quence of discrete steps, for example, thefirst pulley goes clockwise, and the rope goesunder the next pulley to the left of it, so itmust go counterclockwise. That continuousevents are comprehended as sequences ofsteps is corroborated by research on segmen-tation and interpretation of everyday events,such as making a bed (Zacks, Tversky, &Iyer, 2001 ).

It has long been known that domain ex-perts are more adept at functional inferencesfrom diagrams than novices. Experts can“see” sequences of organized chess movesin a midgame display (Chase & Simon,1973 ; De Groot, 1965). Similarly, expertsin Go (Reitman, 1976), electricity (Egan& Schwartz, 1979), weather (Lowe, 1989),architecture (Suwa & Tversky, 1997), andmore make functional inferences with easefrom diagrams in their domain. Novicesare no different from experts in structuralinferences.

Inferences from Diagrams of Systems. Thedistinction between structural and func-tional inferences is illustrated by work onproduction and comprehension of diagramsfor mechanical systems, such as a car brake,a bicycle pump, or a pulley system (Heiser& Tversky, 2002 ; Figure 10.6). Participantswere asked to interpret a diagram of one ofthe systems. On the whole, their interpreta-tions were structural, that is, they describedthe relations among the parts of the system.Another set of participants was given thesame diagrams, enriched by arrows indicat-ing the sequence of action in the systems.Those participants gave functional descrip-tions; that is, they described the step-by-stepoperation of the system. Reversing the tasks,other groups of participants read structuralor functional descriptions of the systemsand produced diagrams of them. Those who



Figure 1 0.6. Diagrams of a car brake and a bicycle pump (both after Mayer & Gallini, 1990), and apulley system (after Hegarty, 1992). Diagrams without arrows encouraged structural descriptions anddiagrams with arrows yielded functional descriptions (Heiser and Tversky, in press).

read functional descriptions used arrows intheir diagrams far more than those who readstructural descriptions. Arrows are an ex-trapictorial device that have many mean-ings and functions in diagrams, for exam-ple, pointing, indicating temporal sequence,causal sequence, and path and manner ofmotion (Tversky, 2001 ).

Expertise came into play in a study oflearning rather than interpretation. Partic-ipants learned one of the mechanical sys-tems from a diagram with or without ar-rows or from structural or functional text.They were later tested on both structural andfunctional information. Participants high inexpertise/ability (self-assessed) were able toinfer both structural and functional infor-mation from either diagram. In contrast,participants low in expertise/ability couldderive structural but not functional informa-tion from the diagrams. Those participants

were able to infer functional informationfrom functional text. This finding suggeststhat people with high expertise/ability canform unitary diagrammatic mental modelsof mechanical systems that allow spatial andfunctional inferences with relative ease, butpeople with low expertise/ability have anduse diagrammatic mental models for struc-tural information, but rely on propositionalrepresentations for functional information.

Enriching Diagrams to Facilitate FunctionalInferences. As noted, conveying spatial orstructural information is relatively straight-forward in diagrams. Diagrams can use spaceto represent space in direct ways that arereadily interpreted, as in maps and archi-tectural sketches. Conveying informationthat is not strictly spatial, such as changeover time, forces, and kinematics, is lessstraightforward. Some visual conventions for


2 30 the cambridge handbook of thinking and reasoning

conveying information about dynamics orforces have been developed in comics andin diagrams (e.g., Horn, 1998; Kunzle, 1990;McCloud, 1994), and many of these con-ventions are cognitively compelling. Arrowsare a good example. As lines, arrows in-dicate a relationship, a link. As asymmet-ric lines, they indicate an asymmetric rela-tionship. The arrowhead is compelling as anindicator of the direction of the asymme-try because of its correspondence to arrow-heads common as weapons in the world orits correspondence to Vs created by pathsof downward moving water. A survey ofdiagrams in science and engineering textsshows wide use of extrapictorial diagram-matic devices, such as arrows, lines, brack-ets, and insets, although not always consis-tently (Tversky, Heiser, Lozano, MacKenzie,& Morrison, in press). As a consequence,these devices are not always correctly in-terpreted. Some diagrams of paradigmaticprocesses, such as the nitrogen cycle in bi-ology or the rock cycle in geology, containthe same device, typically an arrow, withmultiple senses, pointing or labeling, indi-cating movement path or manner, suggest-ing forces or sequence, in the same diagram.Of course, there is ambiguity in many wordsthat appear commonly in scientific and otherprose, words that parallel these graphic de-vices, such as line and relationship. Neverthe-less, the confusion caused by multiple sensesof diagrammatic devices in interpreting di-agrams suggests that greater care in designis worthwhile.

An intuitive way to visualize change overtime is by animations. After all, an animationuses change over time to convey change overtime, a cognitively compelling correspon-dence. Despite the intuitive appeal, a sur-vey of dozens of studies that have comparedanimated graphics to informationally com-parable static graphics in teaching a widevariety of concepts, physical, mechanical,and abstract, did not find a single exampleof superior learning by animations (Tversky,Morrison, & Betrancourt, 2002). Animationsmay be superior for purposes other thanlearning, for example, in maintaining per-spective or in calling attention to a solution

in problem solving. For example, a diagramcontaining many arrows moving toward thecenter of a display was superior to a diagramwith static arrows in suggesting the solutionto the Duncker radiation problem, how todestroy a tumor without destroying healthytissue (Pedone, Hummel, & Holyoak, 2001 ;see Holyoak, Chap. 6, Figure 6.4). The fail-ure of animations to improve learning itselfbecomes intuitive on further reflection. Forone thing, animations are often complex, soit is difficult for a viewer to know where tolook and to make sense of the timing of manymoving components. However, even simpleanimations, such as the path of a single mov-ing circle, are not superior to static graphics(Morrison & Tversky, in press). The secondreason for the lack of success of anima-tions is one reviewed earlier. If people thinkof dynamic events as sequences of stepsrather than continuous animations, thenpresenting change over time as sequencesof steps may make the changes easierto comprehend.

Diagrams for Insight

Maps for highways and subways, diagramsfor assembly and biology, graphs for eco-nomics and statistics, and plans for electri-cians and plumbers are designed to be con-cise and unambiguous, although they maynot always succeed. Their inventors want tocommunicate clearly and without error. Incontrast are graphics created to be ambigu-ous, to allow reinterpretation and discovery.Art falls into both those categories. Early de-sign sketches are meant to be ambiguous, tocommit the designer to only those aspectsof the design that are likely not to change,and to leave open other aspects. One reasonfor this is fixation; it is hard to “think outof the box.” Visual displays express, suggest,more than what they display. That expres-sion in fact, came from solution attempts tothe famous nine-dot problem (see Novick& Bassok, Chap. 1 4 , Fig. 1 4 .4). Connect allnine dots in a 3 × 3 array using four straightlines without lifting the pen from the pa-per. The solution that is hard to see is toextend the lines beyond the “box” suggested


visuospatial reasoning 2 31

Figure 1 0.7. A sketch by an architect designing a museum. Uponreinspection, he made an unintentional discovery (Suwa, Tversky, Gero, &Purcell, 2001 ).

by the 3 × 3 array. The Gestalt psychologistsmade us aware of the visual inferences themind makes without reflection, grouping byproximity, similarity, good continuation, andcommon fate.

inferences from sketches

Initial design sketches are meant to be am-biguous for several reasons. In early stages ofdesign, designers often do not want to com-mit to the details of a solution, only the gen-eral outline, leaving open many possibilities;gradually, they will fill in the details. Per-haps more important, skilled designers areable to get new ideas by reexamining theirown sketches, by having a conversation withtheir sketches, bouncing ideas off them (e.g.,Goldschmidt, 1994 ; Schon, 1983 ; Suwa& Tversky, 1997; Suwa, Tversky, Gero, &Purcell, 2001 ). They may construct sketcheswith one set of ideas in mind, but on laterreexamination they see new configurationsand relations that generate new design ideas.The productive cycle between reexaminingand reinterpreting is revealed in the protocolof one expert architect. When he saw a new

configuration in his own design, he was morelikely to invent a new design idea; similarly,when he invented a new design idea, he wasmore likely to see a new configuration in hissketch (Suwa et al., 2001 ; Figure 10.7).

Underlying these unintended discoveriesin sketches is a cognitive skill termed con-structive perception, which consists of twoindependent processes: a perceptual one,mentally reorganizing the sketch, and a con-ceptual one, relating the new organizationto some design purpose (Suwa & Tversky,2003). Participants adept at generating mul-tiple interpretations of ambiguous sketchesexcelled at the perceptual ability of findinghidden figures and at the cognitive ability offinding remote meaningful associations, yetthese two abilities were uncorrelated.

Expertise affects the kinds of inferencesdesigners are able to make from theirsketches. Novice designers are adept at per-ceptual inferences, such as seeing proxim-ity and similarity relations. Expert design-ers are also adept at functional inferences,such as “seeing” the flow of traffic or thechanges in light from sketches (Suwa &Tversky, 1997).



Conclusions and Future Directions

Starting with the elements of visuospatialrepresentations in the mind, we end withvisuospatial representations created by themind. Like language, graphics serve to ex-press and clarify individual spatial and ab-stract concepts. Graphics have an advantageover language in expressiveness (Stenning& Oberlander, 1995); graphics use elementsand relations in graphic space to convey el-ements and relations in real or metaphoricspace. As such, they allow inference basedon the visuospatial processing that peoplehave become expert in as a part of theireveryday interactions with space (Larkin &Simon, 1997). As cognitive tools, graphicsfacilitate reasoning, both by externalizing,thus offloading memory and processing, andby mapping abstract reasoning onto spatialcomparisons and transformations. Graphicsorganize and schematize spatial and abstractinformation to highlight and focus the es-sential information. Like language, graphicsserve to convey spatial and abstract conceptsto others. They make private thoughts pub-lic to a community that can then use andrevise those concepts collaboratively.

Of course, graphics and physical and men-tal transformations on them are not identi-cal to visuospatial representations and rea-soning, they are an expression of it. Talkabout space and actions in it were probablyamong the first uses of language, telling oth-ers how to find their way and what to look forwhen they get there. Cognitive tools to pro-mote visuospatial reasoning were among thefirst to be invented from tokens for propertycounts, believed to be the precursor of writ-ten language (Schmandt-Besserat, 1992), totrail markers to maps in the sand. Spatialthought, spatial language, and spatial graph-ics reflect the importance and prevalenceof visuospatial reasoning in our lives, fromknowing how to get home to knowing howto design a house, from explaining how tofind the freeway to explaining how the judi-cial system works, from understanding basicscience to inventing new conceptions of theorigins of the universe. Where do we go fromhere? Onward and upward!

Acknowledgments

I am grateful to Phil Johnson-Laird andJeff Zacks for insightful suggestions on aprevious draft. Preparation of this chapterand some of the research reported weresupported by Office of Naval Research,Grant Numbers NOOO1 4-PP-1 -O649,N0001401 10717, and N000140210534 toStanford University.

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