COGNITIVE

PSYCHOLOGY

11,

The Coding

375-394

(1979)

and Transformation Information

of Spatial

JANELLENHUTTENLOCHER University

of Chicago

AND CLARK Indiana

C. PRESSON University

The present paper examines the mental processes involved in inferring perspective changes that result either from the rotation of a spatial array or from the rotation of the viewer of that array. Piaget has shown that viewer-rotation problems are difftcuh when children must choose among pictures or models of an array from differing perspectives. We showed earlier that, with parallel tasks, array-rotation problems are much easier than viewer-rotation problems. We proposed that in solving these problems, subjects interpret the instructions literally, recoding the position of the viewer vis-a-vis the array for viewer-rotation problems and recoding the array with respect to its spatial framework for arrayrotation problems. At that time, we proposed a second principle to explain why Piagetian perspective problems are so difficult; namely, that children have special difftculty in recoding viewer position (egocentrism). The present experiments show that, when subjects are asked a different sort of question on such tasks, viewer-rotation problems become easy and array-rotation problems become difficult. The results show that the difficulty of the Piagetian perspective task is not due to egocentrism; i.e., to difftculty recoding viewer position. The results of all these rotational-transformation tasks can be explained if we add a different second principle to the principle of literalness of problem interpretation. This new second principle posits that the array is fixed vis-a-vis the spatial context rather than that the viewer is fixed vis-a-vis the array.

Many years ago Piaget and Inhelder (1967) demonstrated a striking limitation in children’s ability to transform spatial information. They presented “perspective” problems in which children were shown a model of three mountains and were asked to indicate how it would look to an observer who viewed it from some different position. Until 9 or 10 years of age, children tended to make “egocentric errors”: When shown representations of the array which depicted a variety of viewing positions, the children selected the one that represented their own vantage point rather than the one that represented the other observer’s vantage point. This The preparation of this paper was supported in part by Research Grant HD 03215 from the National Institutes of Health to the first author. The authors thank Richard Aslin, Christian Gruber, Linda Smith, and especially Deborah Burke, for their helpful comments on the manuscript. Requests for reprints should be sent to Janellen Huttenlocher, University of Chicago, 5835 S. Kimbark Avenue, Chicago, IL 60637. 375 OOlO-0285/79/030375-20$05.00/O Copyright @ 1979 by Academic Press, Inc. AU rights of reproduction in any form reserved.

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result has been widely replicated (e.g., Flavell, 1968; Laurendeau & Pinard, 1970). In an earlier paper (Huttenlocher & Presson, 1973), we attempted to gain a broader understanding of this phenomenon by contrasting perspective problems (henceforth viewer-rotation problems) with problems where subjects are asked how an array of objects would look if it were rotated about its own axis (henceforth array-rotation problems). The two problems are equivalent in the sense that, for both, subjects initially view an array from one vantage point and must anticipate its appearance given a changed relation between viewer and array. The problems differ only in the actual physical operation involved: a movement of the viewer versus a movement of the array. Nevertheless, array-rotation problems were much easier for children than viewer-rotation problems and “egocentric” errors did not appear. These results showed that children approach these two types of tasks differently and that the difftculty with viewer rotation does not involve a uniform inability to carry out rotational transformations. That is, the child is not egocentric in the sense of being completely unable to recode the relation between viewer and array. It seems most plausible to suppose that the reason children approach these two tasks differently is that they interpret the instructions literally, as requiring in the one case rotation of the array and, in the other, rotation of the self. In this context, array-rotation tasks might be easier than viewerrotation tasks because the child can more easily imagine a rotation of an array than a rotation of the self. We tested this hypothesis using a modified viewer-rotation task in which the child actually moved to the new viewing position. First, he was shown an array. Then the array was covered and he moved to a new position and indicated how the array would appear if it were uncovered. Here, as in the regular viewer-rotation task, the child was given initial information about the array and had to anticipate the appearance of that array given a movement of the viewer. Nevertheless, this modified task was easy and there was little tendency toward egocentricity. This is just the result one would expect if children have special difficulty in imagining movements of the self. Therefore, in our earlier paper we tentatively concluded that the child is egocentric in the sense of being unable to imagine a change in his own viewing position. There is reason to seek more evidence before accepting the conclusion that the child cannot imagine changes in his viewing position. Humans continually orient themselves while moving about in an environment of fixed objects. As Howard and Templeton (1966) point out, “the typical environment of man consists of solid objects in more or less consistent spatial relationships to one another together with objects which change their positions. . . . The very static nature of this backdrop provides a stable geographical environment within which man may orient himself or

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navigate” (pp. 9- 10). Insofar as problems which people frequently encounter become easy to represent in thought, it should be easy to imagine oneself moving among fixed objects, contrary to what Piaget’s and our findings seem to indicate. Further investigation of rotational transformation problems also seems warranted because of a special feature of the Piagetian task, namely, that it involves spatial information obtained from a single vantage point (SVP). For such SVP information, the outcomes of rotational transformations can be anticipated. As Attneave (1967) says, “one may build into a perceiving machine perfectly determinate mathematical principles” to execute rotational transformations based solely on what is seen in the initial perspective on an array. However, much spatial information about objects and arrays in the world is obtained from multiple vantage points (MVP) rather than from a single vantage point; for example, the information about different faces of a solid object. Since it is not possible to “anticipate” or deduce what such objects or arrays look like from the other side, their very representation must include information about how they look from various viewing positions. Thus it is possible that children might find transformation problems easier with such MVP information than with SVP information. Still another reason for further investigation is that in the studies done thus far, the differences between the difficulty of viewer-rotation and array-rotation problems could be due to differences in the trajectory of the moving element. With SVP information, the viewer’s trajectory involves a circular path around the array whereas the array simply rotates on a point. Thus if it were easier to imagine rotation on a point than around a circular path, that could explain the relative ease of array-rotation problems. In Experiment 1 we examine the difficulty of viewer-rotation and array-rotation tasks using MVP information. We also evaluate trajectory on a point versus circular path independently of type of rotation (viewer versus array) by using two types of MVP array: a cube with four distinctive sides versus a room with four distinctive sides. In the case of the cube, the viewer is outside the array and would rotate in a circular path around it, and the array, when rotated, would turn on its axis; in the case of the room, the viewer is fixed in the center of the array and would rotate on a point, while the array, when rotated, would make a circular path around him. To anticipate our findings, the results of the first experiment were in sharp contrast with our previous results. Viewer rotation was easier than array rotation, and this was true for both types of MVP arrays. A second experiment was thus conducted with SVP arrays. The results showed that the reversal from our earlier study was not due to the use of MVP informa-

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tion, per se, but rather to the use of a different type of question for MVP arrays (which was necessary because a viewer cannot see the various sides of the array simultaneously for MVP arrays). The finding that viewer rotation was easier than array rotation for both SVP and MVP arrays with a different sort of question discontirmed our earlier hypothesis. Children are not egocentric in the sense of being generally unable to imagine movements of the self. More importantly, the results show an inadequacy in our initial theoretical framework, since we had posited just this sort of egocentrism to explain why viewer rotation was more difficult than array rotation in our earlier study. The present studies, then, led us to reexamine our assumptions about how spatial arrays are coded. We had assumed that the arrays were coded as units in terms of the relations of the items to one another (e.g., as in the diagram shown on the left in Fig. 1). This unit in turn would then be oriented relative to an outside point. We conceived of that outside point as the viewer, whose perspective constituted the starting point of the rotational transformation. To assume that the arrays are units and that the transformations occur within an isolated viewer-array system in this way implies that the viewer-rotation and array-rotation tasks are equivalentexcept with respect to whether it is the viewer or the array which is imagined to move. Such an account cannot explain the present results. Below, we report the experiments and then we present an alternative hypothesis about how spatial arrays are represented. We argue that to account for the present results we must reject both the assumption that arrays are coded as units and the assumption that arrays are coded independently of the more general spatial context in which they are presented. We propose, instead, that the items in an array are treated as having a fixed relation to some element(s) external to the target array, other than ego (e.g., as in the diagram shown on the right of Fig. 1). Consideration of the external spatial context in which the target array appears is critical to providing an overall account of the results reported below as well as previous findings.

FIG. 1. Schematic examples of coding in which array is oriented as a unit (left diagram) or the array is oriented item by item (right diagram).

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1: ROOM AND CUBE

Experiment 1 was designed to contrast the relative difftculty of arrayrotation and viewer-rotation problems with MVP information. It has been reported that viewer-rotation tasks are rather easy with MVP information. Fishbein, Lewis, and Keiffer (1972) found that even preschoolers could choose among four photographs to show which of the different sides of a soldier (saluting with one hand and carrying a candy cane in the other) could be seen by a viewer in another position. Masangkay, McCluskey, McIntyre, Sims-Knight, Vaughn, and Flavell (1974) obtained similar results with several tasks. Using a sheet of paper with a drawing on each side, which the child and experimenter faced from opposite sides, the question of what the adult saw was easy even for 3 year olds. However, these tasks only test the child’s knowledge of what part of the array would be nearest the other viewer. They do not test whether the child also knows how the rest of the array would relate to the viewer; what would be to the adult’s left, across from him, etc. In the present experiment, we evaluate the difficulty of viewer-rotation and array-rotation problems with MVP information by determining if the subject knows the altered positions of all portions of the target array. Since the entire array is not visible at once, each trial will concern only one array element given a particular transformation. Our MVP arrays consisted of four figures, either on the walls of a specially constructed “room,” or on the sides of a cube. These two different MVP arrays allowed us to test the effect of trajectory of the moving element. For the cube, the viewer would circle around the array (viewer rotation) or the array would rotate in place about its own axis (array rotation). For the room (which was on coasters), the viewer would rotate in place or the array would circle around the viewer. Method Subjects. Subjects were 128 third-grade suburb, 64 boys and 64 girls.

children from a public school in a New York

Apparatus. The experimental room was 3.5 ft square and 5 ft high, consisting of four plywood panels, mounted on coasters, with one hinged open for access. The subject sat on a swivel chair located in the center of the room and could not see the larger room outside. In the cube task, a 2-ft cube was mounted on a raised turntable. The same set of four figures (square, circle, triangle, and star) was used in both tasks. Each figure was on a 9-in. square of posterboard, centered at about eye level on a side of the cube or wall of the room. Pieces of black cloth were hung over each square on the wall, and were lifted to display the figures. Colored stickers on the subjects’ hands identified their right and left sides as their “red” and “green” sides. A 4-in. cardboard arrow was used to indicate the extent of rotation, as explained below. A 6-in. square answer board displayed the four figures, and subjects responded by pointing to the appropriate figure. Two answer boards with the figures in different arrangements were presented alternately.

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Design and procedure. There were four experimental groups which had arrayrotation and viewer-rotation tasks with the room and with the cube. Within these groups, order of array presentations (see below) was completely counterbalanced. Subjects predicted the outcome of a clockwise rotation (0,90, 180, or 270”) of themselves (viewer rotation) or of the array (array rotation). On a trial, the question concerned which figure would be in one of four given positions relative to subjects after the rotation: in front of, to the right (“red side”), in the back side, or to the left (“green side”). Subjects had 16 trials, constructed in blocks of 4. Each block included one each of the 4 extents of rotation, and of the 4 questions of relative position. Thus, there were 4 nontransformation trials (0” rotation), and 12 transformation trials (rotations of 90, 180, and 270”). Subjects sat in front of the cube or within the room. Prior to testing, subjects saw the set of four figures twice, once while they moved relative to the array, and once while it moved relative to them. The order of these two presentations was randomized. After both presentations of the array, the colored stickers were put on the subjects’ hands to differentiate their right and left sides. Subjects were asked to point to the figures on the response board which were in each of the four positions relative to them (e.g., “Which figure is to the green side?“). For all tasks, the amount of rotation was indicated to subjects by the use of an arrow. For viewer rotation, the arrow was on a cardboard strip mounted across the arms of the subject’s chair, pointing ahead of him. The subject was asked what figure would be in one of the positions relative to him if he moved until the arrow was in a particular position. For array rotation, the arrow was mounted on the top of the array in front of the subject. The subject was asked what figure would be in one of the positions relative to him if the array moved until the arrow was in a particular position. Results and Discussion

In sharp contrast to our earlier study with SVP arrays, viewer rotation was easier overall than array rotation for both the room (p < .OZ)and the cube (p < .OS).l These overall comparisons are of subjects’ scores that include both transformation and nontransformation scores. For transformation trials alone (see Table l), this task difference was significant with the room (p < .02), but not for the cube. Array rotation was also more difficult than viewer rotation on nontransformation trials, significantly so for the cube (p < .Ol). This suggests that array-rotation transformations impair the retention of initial array positions. During the presentation of the initial array information, we observed a phenomenon which is consistent with this finding. While all subjects saw the arrays under two conditions, the two presentations did not seem to be of equal difficulty. When the array itself rotated, subjects seemed unable to remember the locations of the figures on the four walls. Only when subjects moved relative to the array did they appear to learn where the various figures were. The difference between array-rotation and viewer-rotation tasks on nontransformation trials for the cube makes the transformation data dif* Significance tests between groups throughout this paper involve the Mann-Whitney U test.

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TABLE 1 PERCENTAGE OF RESPONSESINCORRECT AND EGOCENTRIC ERROR PATTERNS FOR THE MVP ARRAYS Transformation trialsb

Nontransformation trials” Task

Percentage incorrect

Percentage incorrect

Percentage errors egocentr&

The room Viewer rotation Array rotation

5 9

Viewer rotation Array rotation

13 25

20 34

44 40

33 44

64 36

The cube

Note. n = 32 per group. (1Four trials per subject. * Twelve trials per subject. c Percentage of transformation

errors of the egocentric

type.

ficult to interpret. That is, differences on transformation trials may simply have resulted from subjects forgetting the original array positions. However, when we examined each subject’s first transformation trial, there were twice as many errors for array rotation than for viewer rotation (44% vs 22%), although this difference was not quite statistically significant with a x2 test. “Egocentric” errors occurred when subjects indicated the figure which would be in the given position without any transformation. There are four possible responses on each trial, three of which are errors. One of these is the egocentric error, so if a subject made errors randomly, one-third of them would be egocentric. To test for systematic egocentricity, we used a sign test to classify each subject according to whether he made more than one-third egocentric errors. For viewer rotation with the cube, more subjects than expected by chance made such an error pattern (p < .05). However, egocentricity varied with degree of rotation. A significant number of subjects were egocentric on 180”trials (p < .OOl), but nor for 90” (p > .40) or 270” (p > .lO) trials. This pattern, then, may simply be because of right-left reversals on 180” trials, rather than egocentrism per se. For the room, subjects did not tend to be egocentric for viewer rotation (p > .40), and subjects were not egocentric for array rotation with either array. The major finding is that the relative difftculty of viewer rotation and array rotation reverses for these MVP arrays compared to our earlier findings with SVP arrays. Thus, it is not always harder to imagine move-

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ments of self than movements of an array. Further, the trajectory of the rotating element was not critical since the pattern of results is the same for the room as for the cube. One possible reason for these results is that SVP and MVP spatial information are coded in fundamentally different ways. Alternatively, however, the reversal in relative task difficulty might be due to the procedural difference in the type of question used with these MVP arrays. The standard questions with SVP arrays concern alternative appearances of the entire array, and subjects select from a set of drawings or models of the array. We will refer to such questions as appearance questions. With MVP arrays the entire array is not visible from any single vantage point, so we tested subjects’ knowledge of the entire array by questioning them about items one at a time. We will refer to the type of questions we used in Experiment 1 as item questions. The critical factor in Experiment 1 may not be MVP versus SVP information, but the specific kind of question the child is asked: one about the total array as a unit or one about the component items of the array. EXPERIMENT

2: EFFECTS OF QUESTION INFORMATION

TYPE FOR SVP

Experiment 2 was designed to directly investigate the effects of question type on array-rotation and viewer-rotation tasks. We used an SVP array, since onfy for these arrays can one pose both appearance and item questions. The two types of questions were used for both types of rotation task. Method Subjects. Subjects were 80 third-grade children from a public school in a New York suburb, 40 boys and 40 girls. Apparatus. The experimental array consisted of four 15in. objects: a ball, a drum, a table, and a house. They formed a diamond shape on a I-ft square plywood board, with one object near each of the four edges at the midpoint. A cardboard lid covered the array. For the item question procedure, the response alternatives were four half-scale objects mounted 2 in. apart on an II-in. posterboard strip. Two answer strips had the four objects in different orders and were presented alternately. Red and green stickers were used as in the previous study to avoid confusion between right and left. For the appearance question procedure, the response alternatives were four half-scale model arrays of the objects on 6-in. squares of posterboard, presented in the four possible orientations side-by-side on a 2-ft long board. Design and procedure. There were four experimental groups which had viewerrotation and array-rotation tasks with item questions and appearance questions. As in the previous study, subjects imagined the result if either they (viewer rotation) or the array (array rotation) were rotated. For the item question procedure, subjects pointed to the item that would be in a particular position after the rotation; in front, in back, to the red side (right), or to the green side (left) of the viewer. For the appearance question procedure, subjects pointed to the model that showed how the whole array would look after the rotation. As in Experiment 1, there were four degrees of rotation. For item questions, questions about the four positions were coun-

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terbalanced across all four rotations. For appearance questions, each degree of rotation was tested four times. Thus all groups had 16 trials constructed in four counterbalanced blocks. Subjects sat at a small table with the covered array in front of them. The array was presented and then covered again. Subjects in the item question condition had the colored patches put on the right and left hands, and the four relative positions of the array were explained. For viewer-rotation groups, subjects were told to imagine themselves standing on a different side of the table looking at the array, and to indicate how the objects would look from that place. For array-rotation groups, a black dot was put on the front of the array cover. Subjects were told to imagine that the array was turned so that the dot was on a different side, and to indicate how the objects would then look. For all groups, after the extent of transformation was specified, the four response alternatives were presented, either the four models (appearance questions) or the four objects (item questions).

Results and Discussion

There was a striking difference in relative task difficulty for the two question procedures (see Table 2). For appearance questions, the results parallel our earlier findings (1973) using this type of question. That is, viewer rotation was much more difficult than array rotation on transformation trials (p < .Ol). In contrast, the results for item questions parallel those in Experiment 1 with MVP arrays. Viewer rotation was significantly easier than array rotation (p < .05). Again, we used a sign test to assess the number of subjects for whom egocentric errors were greater than one-third of their total errors. For appearance questions, more subjects made systematic egocentric errors in viewer rotation than expected by chance; both overall (p < .025) and for each degree of rotation (p < .04 in each case). For item questions, the egocentric error rate in the viewer-rotation condition was significant TABLE 2 INCORRECTANDEGOCENTRIC FOR THE SVP ARRAYS

PERCENTAGEOFRESPONSES ERROR PATTERNS

Nontransformation trials0 Task

Percentage incorrect

Viewer rotation Array rotation

Appearance 4 6

Viewer rotation Array rotation

8 9

Transformation trial@ Percentage incorrect questions 56 15

Percentage errors egocentric’ 80 25

Item questions

Note. n = 20 per group. 0 Four trials per subject. b Twelve trials per subject. p Percentage of transformation

20 32

errors of the egocentric

49 24

type.

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overall (p < .OS),but, as with the cube in Experiment 1, the effect varied markedly with degree of rotation. Again, the egocentricity was greater than chance only on 180” trials (p < .04), which may be attributable to right-left confusions rather than to a general egocentrism. For arrayrotation tasks, there was no tendency to make egocentric errors for either item or appearance questions. Thus, there is something special about appearance questions in viewer-rotation tasks which leads to egocentrism. Recall that at the start of these experiments, we had been assuming that the two types of rotational transformations (viewer and array) were equivalent except for differences in the difficulty of imagining changes in the position of the self as opposed to changes in the position of the array. Experiment 1 showed that it was not difficult to imagine changes in the position of the self for MVP information. The present experiment shows that it is not difficult to imagine such changes for SVP information either. We present one final experiment before discussing the alternative coding hypothesis which explains the differences in difficulty among these tasks. Experiment 3 was designed to determine whether we could find an array which would be coded as a unit, i.e., in terms of the relations among its parts. With such unit coding, which we originally assumed to be the general case, viewer rotation and array rotation should be of equal difficulty. We reasoned that such unit coding should be used for familiar unitary figures which do not have fixed geographic orientation. EXPERIMENT

3: THE TELEPHONE

In Experiment 3 we contrasted viewer-rotation and array-rotation tasks using a telephone as an array. Telephones are moveable objects seen from various vantage points. Such an item should be coded internally in an orientation-free manner. If a telephone is coded as a unit, and there is nothing particularly special about imagining rotations of self, then viewerand array-rotation tasks should be equivalent for this array. Method Subjects.

Subjects were 40 first-grade children from a public school in a New York suburb, 20 boys and 20 girls. Apparatus. The array was a standard black dial telephone with its wall wire trimmed. In the array-rotation condition it rested on a 12-in. turntable. A 12-in. cardboard lid was used to cover the telephone. The response alternatives were four photographs, each taken from a different side at an angle of about 30” from horizontal. The photographs were made against a white background. Design and procedure. There were four experimental groups with array-rotation and viewer-rotation tasks, with the telephone either covered or uncovered after presentation. Again, subjects imagined the appearance of the telephone if either they or it were rotated clockwise 0, 90, 180, or 270”. They chose from among the four photographs. There were 16 trials constructed in blocks of four. Each block had the telephone presented once in all four orientations, and had one question about each degree of rotation.

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Subjects sat at a small table with the telephone in front of them. The experimenter pointed out that the sides of the telephone were different, and presented the photographs showing the different sides. The experimenter explained that he would put the telephone down in a different way each time. Subjects were asked what it would look like either after they moved to a different side of the table (viewer rotation) or after the turntable and telephone turned so that an arrow, on the front of the turntable, would be on a different side (array rotation).

Results and Discussion

With the telephone uncovered, viewer rotation and array rotation were of equal difficulty (see Tabie 3). We had predicted that this result should be obtained if arrays were coded internally as units. However, there were contrasting error patterns for the two types of task which suggests that the tasks are not, in fact, equivalent. In viewer rotation, all subjects who made errors made more egocentric errors than expected by chance (p = .002, sign test). For array rotation, a disproportionate number of errors may be accounted for by positing that subjects based an accurate transformation on a forward facing phone. We have termed this a “prototypic memory” error, since the telephone is treated as if it were in its characteristic orientation with its dial facing the viewer. Most subjects who did make errors on this task made mainly prototypic memory errors (six out of eight subjects), although due to the small numbers this was not statistically significant (p < .15, sign test). With the telephone covered, array rotation was more difficult than viewer rotation, and this difference was significant for transformation TABLE 3 PERCENTAGE

OF REWQNSES INCORRECT AND PATTERNS ERRORFOR THE TELEPHONE

Nontransformation trials”

Task

Percentage incorrect

Viewer rotation Array rotation

2 2

Viewer rotation Array rotation

2 10

OF

Transformation trialsb Percentage incorrect

Percentage of errors’ Egocentric

Prototypic memory

Uncovered array 26 25

90 10

17 75

Covered array 40 61

48 22

28 91

Note. n = 10 per group. o Four trials per subject. b Twelve trials per subject. r Percentage of transformation errors of each type. Prototypic memory error only possible on nine trials.

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trials (p < .05). Looking at the error patterns, some subjects made many egocentric errors in viewer rotation, but overall the number of subjects who made more than one-third egocentric errors was not significant. For array rotation, errors were systematically of the “prototypic memory” type. Very few errors occurred on trials in which the dial did face front (only 14% vs 61% overall). For trials where prototypic memory errors were possible, all subjects made more errors of that type than expected by chance (p < .OOl, sign test). This experiment contrasted array-rotation and viewer-rotation tasks for a familiar movable object which seemed likely to be coded as a unit in an orientation-free manner. However, telephones, like many movable objects, do have an habitual orientation relative to the viewer. The evidence indicates that the habitual orientation of the telephone, with the dial in front, is coded into its memory representation. While subjects used the actual position of the telephone when it was fixed in viewer-rotation tasks, they apparently relied on its prototypic “egocentric” coding in long-term memory rather than its actual starting position when they had to imagine rotating it. DISCUSSION

In this series of experiments, we compared the difficulty of anticipating the outcomes of rotational transformations brought about by viewer rotation or by array rotation. In each case, subjects were familiarized with an array; then they were presented with that array from one particular perspective and had to specify what the relation between the viewer and the array would be after a rotational transformation. We varied the nature of the array, the trajectory of the moving element, and the nature of the questions asked. Subjects treated array rotation and viewer rotation differently in each case, and the major variable affecting relative task difficulty was question type. While there are various ways of coding arrays and of executing rotational transformations, these results rule out a number of hypotheses about how people solve these problems. Below we give one account of the coding and transformation of spatial information which provides an explanation of the observed pattern of results. In our account, we continue to assume that subjects interpret transformation instructions literally-that they attempt to recode the array relative to its framework in array-rotation tasks, and to recode their own positions relative to the fixed array in viewer-rotation tasks. However, we add a new hypothesis concerning coding. Our earlier hypothesis, that subjects code arrays as independent units which can be manipulated freely, offers no way to explain the profound effect of question type. The alternative hypothesis is that subjects code the items in arrays as parts of a larger spatial framework. These arrays cannot then be transformed as units relative to those frameworks. We will now examine more closely the

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hypothesized coding of arrays according to the larger spatial framework and then we will consider the transformation processes within such a coding system. Coding and Spatial Information Orientation-specific codings.

There are a variety of forms of spatial coding which would preclude mental rotation of an array as a unit vis-avis its outside framework. One possibility is that each of the items within the array is separately fixed in relation to the outside framework as shown in the right-hand diagram in Fig. 1. Here the internal relations of the items are determined without being specified directly. For example, the diamond shape of the array is only derivable from the positions of the four elements relative to the outside frame. Alternatively, certain relations among the items as points might be directly specified, but only relative to the outside framework. For example, one item might be coded as being “in front” or “to the left” of another. While such a coding specifies the internal relations between a pair of items, it is based on the two items having a fixed orientation relative to some point external to the array itself. If an array is coded internally as a unit, one item can be next to or across from another but cannot be to the left or right of another except relative to a third point. It is not possible on the basis of the present experiments to specify which of these forms of orientation-specific coding is used. The present position, however, is that some form of orientationspecific coding is necessary to explain the total pattern of children’s performance. Our claim is that children do not represent the test arrays as units that are independent of their external surroundings. Evidence that subjects use outside frameworks in coding of arrays derives from the study with the room and cube. The children found it very difficult to establish the coding of these arrays when the initial presentation of that array involved movement of the cube or room relative to them, although coding was easily established when they moved relative to a stationary array. The fact that ease of coding depends on the array having a fixed position in its surroundings suggests that the coding is in relation to these surroundings. The nature of spatial frameworks. The spatial frameworks within which people operate include stationary landmarks. If landmarks are simply undifferentiated points, three are needed to specify the location of an object (or array) within a horizontal plane. However, only one landmark is necessary if it is differentiated (e.g., an inherent front, left side, etc.). Such landmarks can directly orient objects (or arrays), or less directly, the landmarks can be used to establish a system of orthogonal axes or polar coordinates. The most general such spatial framework, of course, is geographic north, south, east, and west. If the viewer himself is the external landmark, coding may be appropri-

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ately called “egocentric.” Such an egocentric coding is reasonable if the viewer remains stationary (e.g., when building toy block structures) or when the viewer and array move in constant relation to one another (e.g., the case of the telephone). However, egocentric codings cannot accommodate movements of the viewer relative to a stationary array because in that case, items which were in front of others would appear behind them, etc. The subject himself may also be used to establish coding within a spatial framework. That is, he may use his knowledge of his own orientation within a larger spatial framework to code the items in an array in relation to that framework. Indeed, in the absence of observable landmarks, a viewer must use stored knowledge of his own orientation in a spatial framework to establish a framework for coding the array. In that case, the coding is not itself egocentric; the viewer is serving as a compass rather than a landmark. Coding may no doubt involve several coexisting frameworks. First, egocentric coding may be used whenever it will suffice. With respect to spatial frameworks, in an area such as a room, local landmarks may suffice to encode the positions of objects. Such local frameworks may themselves be oriented within more general frameworks. A northsouth-east-west framework can provide organization for other frameworks. For example, the location of a book may be coded as being on a desk in a certain place within an office. The office in turn is located in a building, the building in a neighborhood, and so on. Such a nested system of frameworks would provide an efficient organization for an individual to move about without a continual recoding of the information. He need only maintain the orientation of an object or array to a framework at the most particular level. Information about its orientation in other frameworks would be deducible from the coding network. The role of the viewer’s position in spatial codings. Even when people do not rely on egocentric coding of arrays, their viewing positions may play a central role in coding arrays in terms of spatial frameworks. Indeed, there is evidence that the locations of items in relation to an individual’s viewing position affects the configuration of the elements in the mental representation of that array. First the memory representation of the telephone was affected by the fact that the dial typically faces the viewer. Subjects in Experiment 3 had a strong tendency to treat the telephone as starting with its dial in front in array-rotation tasks. Second, room tasks were easier than cube tasks across all transformations. The most reasonable interpretation for this is that the distinctiveness of the four items relative to the viewer is greater for the room than for the cube. The diagrams in Fig. 2 show the array configurations of the cube and room relative to the viewer’s position. For the room, the four figures were positioned to his front, back, left, and right sides. For the cube, on the

SPATIAL INFORMATION

A. The Cube

389

B. The Room

& + FIG. 2. Array configurations of the cube and room relative to ego.

other hand, the four figures were all positioned in front of him, with one to the left and one to the right of midline. The less distinctive arrangement for the cube might also lead to greater right-left confusion on 180” trials, which in turn would account for the difference in “egocentric” errors for the two arrays. Transforming

Spatial Information

If subjects fix the array by coding items into a larger framework and literally interpret instructions to rotate the array or to rotate the viewer, as we propose, certain questions should be easy and others difficult to answer for each task. Array rotation. For array-rotation tasks, subjects are told to rotate the array. If the array were coded independently of its spatial framework in terms of the relations of the items to one another as we originally assumed, rotation of any one focal item would suffice-the locations of the other items relative to the first could be regenerated. In this case, subjects could deal equally well with appearance questions and item questions. However, if the items in the array were fixed into the spatial framework, rotation of a focal item would not transform the entire array. If there were a processing limit on the number of elements which could be rotated at once, rotation of an entire array might be very difftcult. For item questions, the child may be asked what item is in any of the four possible locations, so that to be consistently correct, he must rotate all the items in the array. We have found that these problems are indeed difficult. For appearance questions, it is important to note, it is not necessary to transform the entire array. The child must choose among pictures containing all the array items in their new positions. The problem, therefore, can be solved by rotating just one element. Thus we argue that array-rotation tasks are only easy with appearance questions because for these, subjects need not rotate the entire array. This analysis leads to the prediction that array-rotation tasks with appearance questions could be made difficult if the alternatives included foils in which the internal rela-

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tions among items were not preserved. Pilot work indicates that arrayrotation tasks are indeed difficult with such appearance questions. Viewer rotation. For viewer-rotation tasks, subjects are told to change the viewer’s position while the array remains fixed within its spatial frame. To answer item questions in this task, the child must establish a new external position for the viewer, and determine the relation of that viewer to the items in the array. This task does not require the subject to recode the array in relation to its spatial framework. That is, when the subject transforms the position of the viewer, he recodes that viewer’s relation to the framework including the array items which are coded into that framework. Apparently it is easy to imagine moving the viewer to a new position in relation to the framework, including the array items, since there were few errors in viewer rotation with item questions. For appearance questions, children must choose among alternative pictures of the array. Three of the four pictures, of course, show the array reoriented with respect to the experimental room by 90, 180, or 270”. When the subject follows the viewer-rotation instructions by recoding the viewer’s position relative to the actual array and room, this alone does not permit him to select the proper picture. That is, while the subject has reoriented the viewer vis-a-vis the array and room, picture selection requires reorienting the array vis-a-vis the room. This requires an extra mental operation; that is, after transforming the viewer, and thus knowing what item(s) would be in a particular position(s) vis-a-vis that imagined viewer, he must then determine which picture has the same relation to himself that the actual array has to that imagined viewer. As we argued in our earlier paper, this second step of making the imagined viewer coincident with the subject involves the same operation required in an arrayrotation task, namely, recoding of the array vis-a-vis the outside framework. Failure to carry out this additional step would lead to the selection of the “egocentric” alternative which depicts the unchanged coding of the array. Consistent with this, a systematic egocentricity in viewer-rotation tasks was found only for appearance questions, not for item questions. Recall that in our earlier study (1973) we gave a “modified” viewerrotation task where subjects actually moved relative to a covered array and were given appearance questions. This modified task was easy with appearance questions. While we argued then that this task was easy because the change in viewer position did not have to be imagined, it now seems clear that the reason it was easy was because the correct alternative picture maintained its initial spatial coding vis-a-vis the framework; i.e., when the child actually moves, the correct picture is the one which has the same orientation as the actual array in the experimental room. To complete our discussion we must consider the effect of still another

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question type, used by Hardwick, McIntyre, and Pick (1976) on the difticulty of viewer-rotation and array-rotation tasks. This is a question concerning the position of a particular item after a transformation (e.g., “Suppose you moved [the array were rotated] 90”; show me where the picture on the wall would now be”). This question is posed in terms of the viewer’s present position and thus to answer it, the target item must be rotated relative to the external framework. Therefore, viewer rotation should be difficult, since establishing a new position of the viewer as instructed does not lead directly to the answer, and therefore the extra step of rotating the target item relative to the framework is required. Array rotation should be easy, since only one focal element must be rotated to answer the question. Hardwick et al. (1976) indeed found array rotation to be easier than viewer rotation. Developmental Issues in the Coding and Transformation of Spatial Information Let us consider the nature of the developmental changes which lead to age-related improvement in the ability to solve the Piagetian perspective problem. Changes in the form of spatial coding do not seem to be the critical factor. Very young children as well as adults code items in terms of outside frameworks. Acredolo, Pick, and Olsen (1975) and Acredolo (1976) have shown that young preschoolers code objects in relation to landmarks. More recently, Acredolo (1978) trained l&month-old babies to expect an interesting event on one side of the room. After the subjects were turned 180”, they looked to that same side of the room even though it involved turning their heads the opposite way. In contrast, younger infants made the same head movements after being turned. This could indicate egocentric coding in young infants. However, the infants may have failed to encode the occurrence of the turn at all, or may simply have been learning that a particular motor response of head turning is rewarded by an interesting event. Moreover, adults also tend to code arrays relative to outside frameworks. Witkin, Lewis, Hertzman, Machover, Meissner, and Wapner (1954) made a distinction between “field dependence” and “field independence,” which concerns individual differences in the extent to which people rely on particular orientation-specific codings. They studied individual differences in tasks where subjects made judgments of the upright in the presence of a visual frame which was itself tilted. Some adults, as well as children, coded the figure in relation to the immediate frame, thus failing to establish its relation to ground. In contrast, “fieldindependent” subjects could “free” the object from the immediate spatial frame and establish upright with respect to the more general gravitational frame of reference. These studies show that with age there is an increas-

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ing, but not universal, ability to treat arrays independently of the most salient immediate spatial framework, but they do not show any change in using orientation-free codings. Indeed, it is reasonable to suppose that young children rely on current perceptible information about position of items relative to self and its most immediate local spatial framework. The use of less immediate spatial frames, especially those which are not currently visible, may increase developmentally. However, such a change would not directly affect the solution processes of rotational transformation problems, as long as arrays are spatially coded relative to some framework. There is no reason to believe that a change in the understanding of array-rotation or viewer-rotation instructions underlies improvements in the ability to solve Piagetian perspective problems either. Even young children seem to understand both instructions. Further, adults as well as children tend to interpret transformation instructions literally. Array rotation and viewer rotation are psychologically different problems according to the psychometric literature. For example, Michael, Guilford, Fruckter, and Zimmerman (1957) reported the existence of two distinct spatio-visual factors in psychometric tasks. In considering the characteristics of the tasks which clustered statistically, they named these “orientation” and “visualization.” In the first, they say, the subject’s actual position relative to the figures is critical, whereas in the latter, the “individual is removed from the stimulus pattern,” operating on it in a detached manner. Thus the transformations of viewer rotation and array rotation seem to be treated as separate sorts of operations in adults as well as in children. We also have evidence that adults interpret instructions literally on our tasks. In an unpublished study, we tested adults on the tasks used in Experiment 2, including reaction times as well as errors. The same pattern of relative task difficulty emerged in adults as in children. The fact that the same pattern of difficulty holds in adults as in children suggests that there are not developmental changes in the processes involved in solving these problems. Insofar as adults as well as children code arrays in terms of their larger spatial framework and interpret instructions literally-in array rotation by recoding the array vis-a-vis its spatial frame and in viewer rotation by recoding the viewer’s positionthe Piagetian task would seem to involve extra mental operations for adults as well as children. The simplest explanation for age-related improvements in the ability to solve Piagetian perspective problems is that there are increases in capacity of various processing components (e.g., working memory). As the child gets older he may become better able to retain the original coding of the array and viewer’s position, while at the same time rotating the array to determine which transformed version on the answer board represents the proper perspective. In addition, there

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could be developmental improvement in problem-solving ability such that older subjects understand better that they must first treat appearance questions as equivalent to item questions, and then rotate the array to select the correct alternative picture. Studies of the Piagetian perspective problem which had emphasized rotation of the array by having subjects turn a model show fewer errors and less egocentricity even in children (Fishbein et al., 1972; Borke, 1975). SUMMARY

AND CONCLUSIONS

We have examined the difficulty of various rotational transformation problems. Two factors affected task difficulty. The first factor was whether the problem was described as a rotation of the array or of the viewer. The second was the type of question. The effect of these two factors interacted: with appearance questions, array-rotation tasks were easy and viewer-rotation tasks were difficult; with item questions, viewer-rotation tasks were easy and array-rotation tasks were difficult. We proposed two principles as to how children treat these problems. First, we proposed that arrays are coded item by item in relation to an outside framework. Second, we proposed that transformation instructions are interpreted literally as involving movement of the viewer or array. For array rotation this entails recoding the array with respect to the framework; for viewer rotation it entails recoding the viewer’s position with respect to both the array and its framework. The cojoint operation of these principles explains the observed interaction of problem type and question type. For array rotation, item questions are difficult since all items must be recoded. Appearance questions are easier since only one item, rather than all the items in the array, need be recoded. For viewer rotation, in contrast, item questions are easy. They require knowledge of the original locations of all the items, but that is just how the items are coded. Appearance questions are very difficult for viewer rotation. The instruction is to maintain the original spatial coding of the array relative to the framework, but the correct response shows the array transformed relative to that frame. Therefore, after transforming the viewer vis-Cvis the fixed array and room, it is necessary to recode the array vis-a-vis the framework to select the proper picture. While our studies involve children, there is reason to believe that these principles operate in the solution of rotational transformation problems in adults as well. REFERENCES Acredolo, L. Frames of reference used by children for orientation in unfamiliar spaces. In G. Moore & R. Golledge (Eds.), Environmenral knowing. Stroudsburg, PA: Dowden, Hutchison & Ross, 1976.

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Acredolo, L. The development of spatial orientation in infancy. Developmental Psychology, 1978, 14, 224-234. Acredolo, L., Pick, H., & Olsen, M. Environmental differentiation and familiarity as determinants of children’s spatial memory for location. Developmental Psychology. 1975, 11,495-501. Attneave, F. Criteria for a tenable theory of form perception. In W. Wathen-Dunn (Ed.), Models for the perception of speech and visual form. Cambridge, MA: MIT Press, 1967. Borke, H. Piaget’s mountains revisited: Changes in the egocentric landscape. Develupmental Psychology, 1975, 11, 240-243. Fishbein, H.. Lewis, K., & Keiffer, S. Children’s understanding of spatial relations: Coordination of perspectives. Developmental Psychology, 1972, 7, 21-33. Flavell, .I. The development of role-taking and communication skills in children. New York: Wiley, 1968. Hardwick, D., McIntyre, C., & Pick, H. The content and manipulation of cognitive maps in children and adults. Monographs of the Society for Research in Child Development, 1976, 41(3, Serial No. 166). Howard, I. P., & Templeton, W. B. Human spatial orientation. New York: Wiley, 1966. Huttenlocher, J., & Presson, C. Mental rotation and the perspective problem. Cognitive 1973, 4, 279-299, Psychology, Laurendeau, M., & Pinard, A. The development of the concept of space in the child. New York: Intern. Univ. Press, 1970. Masangkay, Z., McCluskey, K., McIntyre, C., Sims-Knight, J., Vaughn, B., & Flavell, J. The early development of inferences about the visual percepts of others. Child Development,

1974, 45, 357-366.

Michael, W. B., Guilford, J. P., Fruckter, B., & Zimmerman, W. S. The description of spatial-visualization abilities. Educational and Psychological Measuremenf, 1957, 17, 185- 199.

Piaget, J., & Inhelder, B. The child’s conception of space (F. J. Langdon, & J. L. Lunger, trans.). New York: Norton, 1967 (Originally published, 1948.) Witkin, H. A., Lewis, H. B., Hertzman, M., Machover, K., Meissner, P. B., & Wapner, S. Personality through perception. New York: Harper, 1954. (Accepted March 2, 1979)