two roads diverged: young children's ability to judge distance

Two Roads Diverged: Young Children's Ability to Judge DistanceAuthor(s): William V. Fabricius and Henry M. WellmanSource: Child Development, Vol. 64, No. 2 (Apr., 1993), pp. 399-414Published by: Wiley on behalf of the Society for Research in Child DevelopmentStable URL: http://www.jstor.org/stable/1131258 .

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Two Roads Diverged: Young Children's Ability to Judge Distance

William V. Fabricius

University of Georgia

Henry M. Wellman

University of Michigan

FABRICIUS, WILLIAM V., and WELLMAN, HENRY M. Two Roads Diverged: Young Children's Ability to Judge Distance. CHILD DEVELOPMENT, 1993, 64, 399-414. In the present studies, we investi- gated 4- and 5- to 6-year-olds' ability to compare the distances covered by a direct route to a location and an indirect route to the same location. The distances ranged between 16 and 22 feet. The routes were visible from a single vantage point, and objects serving as landmarks were sometimes located along the routes. We found clear demonstrations of the two classic Piagetian distance errors-the direct-indirect error, in which children judge that a direct route and an indirect route cover the same distance, and the interposed object error, in which children judge that a route is shorter when it is segmented by an object located somewhere along the route. The interposed object error occurred because children focused on only one segment of the route, which was consistent with Piaget's explanation of the error. However, in contrast to Piaget, we found that about 40% of 4-year-olds could successfully avoid the direct-indirect error, and in addition, when the routes were visually displayed, they could also avoid the interposed object error. It is important that they also gave correct explanations for why the indirect route was longer, by referring to the fact that it was not straight. For these children at least, the interposed object error was due to difficulty they had representing routes, rather than to a misconception of distance. We suggest that future research should examine whether that may also be true for younger children.

Children's understanding of distance is a classic developmental topic (Piaget, In- helder, & Szeminska, 1960), and one of re- newed research interest (Bartsch & Well- man, 1988; Miller & Baillargeon, 1990; Schiff, 1983). Piaget (Piaget et al., 1960; see also Shantz & Smock, 1966) pointed out that the concept of distance is fundamental to mature notions of space. The distance between two points is an interval of space between them, and therefore a conception of distance "provides the necessary conditions for the emergence of space as a common medium" (Piaget et al., 1960, p. 70). Research- ers have therefore studied children's understanding of space as revealed in their conceptions of distance.

According to Piaget, young children do

not conceive of distances as fixed units of space. As a result, they cannot conceive of space as a three dimensional medium, or a framework of fixed extent that remains unchanged as objects change positions within it. Instead, according to Piaget, "the perceptual and intuitive universe of young children is subject to constant deformation and is thus much closer to the elastic and contractile space of topology than it is to the invariants of Euclidean space" (Piaget et al., 1960, p. 90). For Piaget, young children's lack of a conception of distance is the common theme underlying a variety of manifestations of their spatial shortcomings, including prob- lems in conserving length, comparing routes, constructing maps, measuring objects, judging speed, and representing paths of movement.

Support was received from a Faculty Research Grant and from the Institute for Behavioral Research, University of Georgia, to the first author. We would like to thank Lynn Cavalier, Melissa Fincher, Carolyn Kruse, and Seung-Ho Park for their assistance, and the McPhaul Child and Family Development Center for their cooperation. Helpful comments were received from Karen Bartsch, Kevin Miller, Nora Newcomb, and Clark Presson. Reprint requests should be addressed to William Fabricius, who is now at the Department of Psychology, Arizona State University, Tempe, AZ 85287-1104.

[Child Development, 1993, 64, 399-414. ? 1993 by the Society for Research in Child Development, Inc. All rights reserved. 0009-3920/93/6402-0018$01.00]



400 Child Development

Specifically, according to Piaget, young children make two classic distance errors. The first, which we will call the direct- indirect error, occurs when children compare a direct route between two locations to an indirect route between those same locations (such as in the Rope or Lake Arrays in Fig. 1). Piaget questioned children about straight and circuitous routes made from match sticks, string, etc. forming small, table-top arrays. Piaget et al. (1960) reported that 84% of children under 41/2 years of age mistakenly judged that a direct route and an indirect route between the same points covered the same distance. According to Piaget children judge that the routes are the same distance because they end at the same point. Without a concept of distance, which speci- fies that movement in any direction uses up an interval of empty space, "detours made enroute matter little" (Piaget, 1946, p. 69).

The second error is that young children judge that the distance between objects A and B will decrease if a third object, S, is placed between them. We call this the interposed object error. When object S is interposed, Piaget held that young children initially are unable to conceive of the overall distance (ASB). "The moment such an object S is interposed between A and B there is no distance relation between them. The only distance relations are AS and BS which cannot be composed because the object S is in the way" (Piaget et al., 1960, p. 74). Piaget held that this error stemmed from a deeper logical deficiency on the part of young children; namely, their inability to recompose mentally a whole once it had been subdi- vided into parts. Moreover, Piaget et al. (1960) held that, even when children begin to be able to conceive of the overall distance (ASB), they will continue to make the inter-

Arrays Used in Experiment 1

Landmark Arrays

Rope Array 1 2 3 4

Child Child Child Child Child

Arrays Used in Experiments 2 and 3

Lake Array Landmark Arrays (Used only in 1 2 3 4 5

Experiment 2)

Child Child Child Child Child Child

FIG. 1.-The arrays used in Experiments 1, 2, and 3



Fabricius and Wellman 401

posed object error for a time because they subtract the space filled by S from their estimate of the distance between A and B.'

Recently, evidence has been accumulat- ing that modifies Piaget's conclusions about young children's inability to understand distance. In a study directly relevant to the direct-indirect error, Bartsch and Wellman (1988) asked children to compare direct versus indirect routes. Children first saw two straight, parallel, elastic "sidewalks," each about 10 inches in length and tacked down on each end. The experimenter then screened both sidewalks from view while she held one in the middle and stretched it to form an angled route. She told the children what she was doing and briefly showed them a small bit of the resulting angle. Four- year-olds were above chance in judging that the angled route was longer. Because children could not see the routes when they made their judgments, this study showed that young children were able to reason correctly, in the absence of perceptual confirmation, about direct versus indirect routes.

In a study directly relevant to the interposed object error, Miller and Baillargeon (1990) found that 3-6-year-olds made the error in the standard questioning condition when asked whether two objects were still just as "near" or "far" apart after a third object was interposed. However, at all ages they were much less likely to make the error when asked to anticipate the length of a stick needed to bridge the distance between the two objects, which also suggests that they are more knowledgeable about distance than Piaget believed.

However, studies such as these that show that children are able to avoid one distance error in isolation are less than defini- tive. For example, it is possible that at an early age different children are able to avoid each error. Or, individuals might be able to avoid both errors in separate tasks but fail when they have to avoid both errors simultaneously. A stronger case could be made that young children understand distance if it were shown that individual children can avoid both errors simultaneously. Conse- quently, we designed a task that incorporated both errors and that allowed us to

identify several alternative approaches that individuals might use to solve the task.

In contrast to previous studies, we used large arrays (the direct route was 16 feet), which remained visible to the child. When distances are small, as in Piaget's studies, they all may seem "short" or equivalent to children. When arrays are screened (Bartsch & Wellman, 1988) it eliminates the possibility of assessing the effects of interposed objects on children's judgments. To gain a complete understanding of children's conception of distance, we need to know how children reason about large, visible distances, of the kind they encounter everyday. One prior study (Fabricius, 1988) suggests that young children may be able to reason about large, visible distances. In that study, 4-year-olds were often able to select shorter routes between objects several yards apart on the basis of whether the routes did or did not involve backtracking, a distance-related consideration. However, that study focused primarily on children's search behavior and not directly on their understanding and judgment of distance. Thus, in the present research, we studied the classic distance errors in the context of large, visible distances.

Experiment 1 METHOD

Overall Design of Stimulus Arrays The arrays that we used are shown in

Figure 1. The only differences in the arrays used in Experiment 1 versus Experiments 2 and 3 were in the number of arrays and in how the routes were displayed. In Experi- ments 2 and 3, we added Landmark Array 5, and the space between the routes in each array was filled in with a piece of blue material (a "lake"). In Experiment 1, the routes were either not displayed (as indicated by the dotted lines in the figure), or were out- lined by ropes. Here we only wish to explain the geometric properties of the arrays, which did not differ among the studies. In all cases the child stood at one end of the array, and there were two routes-one direct and the other indirect-to an "ice cream store" located at the other end. The child was asked, "Which is the long way (or 'short way') to

1 The present study was not designed to test this version of the interposed object error very well, because the objects did not take up very much space in relation to the length of the routes. We will not discuss this version of the error any further for two reasons. We found no evidence for it in our data, and Miller and Baillargeon (1990) specifically looked for it using interposed objects of varying sizes and also found no evidence for it.




the ice cream store?" The child responded by choosing one route or the other.

We designed the Rope and Lake Arrays to test the direct-indirect error. Children who make the error should respond randomly to those arrays. We designed Land- mark Array 1 to incorporate the interposed object error as well. The interposed object was either a "tree" or a "rock" placed at the angle of the indirect route. As shown in Fig- ure 1, the distance from the child to the tree (AS) equalled the distance from the tree to the store (SB), and each was shorter than the direct route. Evidence for the interposed object error would be a decrease in correct responses from the Rope or Lake Array to Landmark Array 1. That is, if, as Piaget suggested, children judged only one part of the distance segmented by the interposed object (either AS or SB), then the indirect route in Landmark Array 1 should appear shorter to them than the direct route.

Finally, we designed Landmark Arrays 2, 3, 4, and 5 to examine the interposed object error in more depth. Piaget repeatedly emphasized the importance children attach to the endpoint, or point of arrival, in making distance-related judgments. These included judging the static distance covered by a path, the length of an object (Piaget et al., 1960), and the distance covered by a moving object (Piaget, 1946). Consequently, we wondered whether children who made the interposed object error would focus on the segment of the route from the interposed object to the endpoint (SB), which we will refer to as the "endpoint segment," rather than on the segment from the startpoint to the object (AS), which we will refer to as the "startpoint segment." To determine which they used, we systematically varied the two segments.

As shown in Figure 1, in Landmark Array 2 the startpoint segments are equal, and the longer endpoint segment is on the indirect route. In Landmark Array 3, the endpoint segments are equal, and the longer startpoint segment is on the indirect route.

In Landmark Array 4, the longer endpoint segment is on the indirect route, but the longer startpoint segment is on the direct route, while the opposite is true in Land- mark Array 5. Thus, use of either type of segment would produce a distinctive pattern of performance across Landmark Arrays 2-5.

Our main analysis strategy was to examine individuals' patterns of performance across arrays, in order to characterize the overall approaches, or rules, that children used. We derived several candidate rules from Piaget's position that children would make both errors. We also considered several types of more advanced rules. Two sets of rules ultimately fit the data from the three experiments. In one set of rules (Rules IE, IIE, and IIIE), interposed object errors resulted from children focusing on the distances of the endpoint segments. In another set (IS, IIS, and IIIS), those errors resulted from focusing on startpoint segments. In both sets, the higher rules incorporated increasing tendencies for children to avoid both errors, as explained below.2 Both sets also culminated in a highest level rule (Rule IV), in which children consistently judged all arrays correctly.

Figure 2 shows the percentages of correct judgments across arrays that are predicted by the above rules. Both types of Rule I (E and S) reflect Piaget's position that children should make both errors. Children using either rule would make the direct- indirect error in the Rope/Lake Arrays and be no better than chance (.50) at judging which route was longer. In Landmark Array 1, they would make the interposed object error and judge the indirect route to be shorter than the direct route. Children using Rule IE would do so by focusing on the endpoint segment of the indirect route, and those using Rule IS would focus on the startpoint segment. In Landmark Arrays 2-5, a different pattern results from each rule. For Rule IE, when the endpoint segments are equal (Landmark Array 3), children

2We also considered alternate sets of rules based on the assumption that, although children focused on one type of segment in the lower rules, they increasingly considered the other type of segment as well in the higher rules. For example, in one higher rule, if the endpoint segments were equal, then children would go on to compare the startpoint segments. This meant that in the higher rules children would still not know that the indirect route was necessarily longer but would instead compare corresponding segments of the two routes, much as an adult might do for two indirect routes that happened to be segmented. In other words, in contrast to the two sets of rules that did fit our data, these rules incorporated an increasing tendency for children to avoid the interposed object error but not the direct-indirect error. We found no evidence for these alternate sets of rules and so will not discuss them further.




Rules

Both No D-I Error Both No D-I Error Neither Errors Partial 1-0 Error Errors Partial I-0 Error Error

Arrays IE IIE IIIE IS IIS IllS IV

V Rope/Lake Array .5 1.0 1.0 .5 1.0 1.0 1.0

Landmark Array 1 .0 .0 .5 .0 .0

.5 1.0

Landmark Array 2 1.0 1.0 1.0 .5 1.0 1.0 1.0

Landmark Array .5 1.0 1.0 1.0 1.0 1.0 1.0 Landmark Array .5 1.0 1.0 1.0 1.0 1.0 1.0

Landmark Array 4 1.0 1.0 1.0 .0 .0 .5 1.0

Landmark Array 5 .0 .0 .5 1.0 1.0 1.0 1.0

FIG. 2.-Predictions of proportions of correct judgments for children using various rules. D-I Error = direct-indirect error; I-O Error = interposed object error.

would respond randomly, either because they judged the routes to be equal or because they at times thought one or the other route was longer as a result of their inability to compare the distances of the segments exactly. However, when one route has a noticeably longer endpoint segment, they would judge that route to be longer. That would lead them to be correct in Landmark Arrays 2 and 4 but incorrect in Landmark Array 5. In contrast for Rule IS, children would respond randomly on Landmark Array 2, correctly on Landmark Arrays 3 and 5, and incorrectly on Landmark Array 4.

Both types of Rules II and III reflect ability to avoid the direct-indirect error, but only partial ability to avoid the interposed

object error. These children would avoid the direct-indirect error in the Rope/Lake Arrays. Rule IIE and IIS children would avoid the interposed object error only when the initial segments are equal (i.e., in Land- mark Array 3 for Rule IIE, and Landmark Array 2 for Rule IIS). In these cases they would consider the whole route, not just one segment, and judge correctly. In all other Landmark Arrays, they would consider only one segment of each route and would resem- ble Rule IE and Rule IS children.

Rule IIIE and IIIS children would always consider the whole routes in all arrays but would experience an unresolvable con- flict and respond randomly when the shorter relevant segment appears on the indirect




route. Thus, on Landmark Array 1, both Rules IIIE and IIIS predict random responses, in contrast to the incorrect performance predicted by their Rule II counterparts. Similar improvement from incorrect to random performance would occur on Land- mark Array 5 for Rule IIIE, and on Land- mark Array 4 for Rule IIIS. In the other arrays, Rules IIIE and IIIS make the same predictions as their Rule II counterparts.

Finally, Rule IV children make neither error. Bartsch and Wellman (1988) argued that young children understand distance much as adults do and that children's understanding includes a direct-indirect distance principle (i.e., that the shortest distance between two points is a straight line). Accord- ingly, we wanted to determine whether Rule IV children would not only consistently judge correctly but also appeal to such a direct-indirect principle in their explanations.

In the present task, however, children might judge correctly by a process of visually scanning along the two routes in each array and mentally comparing the scan times. In that case children could perceptu- ally determine that the indirect route was longer without becoming aware of the direct-indirect principle. If that is what Rule IV children do, then they would fail to appeal to the direct-indirect principle in their explanations. If other children used scanning, we should find evidence for it in a comparison of Landmark Array 1 to the other arrays, for the following reason. It has been demonstrated (Thorndike, 1981) that landmarks along a map route increase the judged distance of the route for adults. This "clut- ter" effect seems due to increases in scanning time as a result of short fixations at each landmark. Thus, if children used scanning they should be most likely to judge Land- mark Array 1 correctly, because in that array only the indirect route has a landmark on it. In contrast, Rules IE-IIIE and IS-IIIS all predict that Landmark Array 1 will be one of the most difficult arrays. To summarize, evidence consistent with scanning would be either an inability on the part of Rule IV children to verbalize the direct-indirect principle, or superior performance on Land- mark Array 1 among children not classified as using any rule.

Subjects Subjects were 12 3'/2-4'/2-year-olds (M

= 4-2; range = 3-8 to 4-6), including four boys, and 12 5-6-year-olds (M = 5-6; range = 5-2 to 6-0), including five boys. They were

recruited from a pool of families who had volunteered to bring their children to the university to participate in developmental research. Information about race and SES was not systematically collected for any of the three studies.

Arrays The arrays used in Experiment 1 were

the Rope Array and Landmark Arrays 1-4 shown in Figure 1. In all arrays, the routes began at the child and ended at a small card- board "ice cream store." The Rope Array was constructed by laying a brown rope on the floor for the direct route, and a white rope for the indirect route. The Landmark Arrays were constructed by removing the ropes and placing an object (either a tree or rock, each cut from an 8.5 x 11-inch piece of posterboard) at a designated point along each route. On the direct route, the object was placed slightly off to the side, as indicated in Figure 1, so as to preserve a straight path to the store. We removed the ropes because in the standard Piagetian assessment of the interposed object error, the path between the two points (AB) is not displayed. In Landmark Array 1, there was no interposed object on the direct route; instead, a small green arrow at the child's feet pointed toward the store to indicate the direct route.

The direct route in the Rope Array was 16 feet long, and each segment of the indirect route was 11.5 feet long. These dimensions were preserved in Landmark Arrays 1-3. In each of those arrays a landmark was put at the corner of the indirect route. In Landmark Array 2 the landmark on the direct route was 11.5 feet away from the child. This meant that the first segments of the two routes were the same length (11.5 feet). In Landmark Array 3 the landmark on the direct route was 4.5 feet from the child, which meant that the last segments of the two routes were the same length (11.5 feet). In Landmark Array 4 the first segment of the indirect route was 4.5 feet and the first segment of the direct route was 11.5 feet. The last segments of the routes were 13 and 4.5 feet, respectively.

Procedure Judgments.-Each child was tested in

one session lasting about 20 min. During half of the session, the child judged which route was the "short way to the ice cream store." In the other half, the child judged which was the "long way to the ice cream store." Before each half, the child received a short pretest on the appropriate term




("short" or "long"). Each pretest included two arrays. In one, two stores were placed about 5 feet apart, and a doll was placed between them but closer to one. In the other, one store was placed about 3 feet away from the doll and the other was placed about 6 feet away in the same direction, forming a narrow and lopsided V between store, doll, and store. In each case a different colored arrow pointed from the doll to each store, and the child was asked, once for each pretest array, which was the short (or long) way to go to the store. The child responded by saying, "the green way" or "the blue way." Four younger children initially gave an incorrect answer to one of the pretest questions. They were corrected, the question was repeated, and they answered all other pretest questions correctly.

After each pretest, the child was seated in a chair that always faced between the direct route and the indirect route in each array. The experimenter sat behind the child. On each array the child was asked to point to the ice cream store and any landmarks. Then the experimenter pointed out and named each route. For Landmark Arrays 2-4, the experimenter said, while pointing, "The tree way goes to the tree and then to the ice cream store, and the rock way goes to the rock and then to the ice cream store." On Landmark Array 1, the experimenter described the direct route (marked by the green arrow) as, "The green way goes this way to the ice cream store." In the Rope Array, the experimenter described the two routes by the color of the ropes. (The ropes were removed for the Landmark Arrays.) The experimenter was careful to make the pointing and description of the two routes in each array take the same amount of time. Then the experimenter asked for the child's judgment. For example, she might say in Landmark Array 1, "Which way is the short [or long] way to the ice cream store, the green way or the tree way [or vice versa de- pending on which route the experimenter had pointed out first]?" The child responded verbally. An assistant set up the next array while the child stood up and turned around to talk briefly with the experimenter.

Each child saw each of the arrays a total of four times, including two trials with the "long" question and two trials with the "short" question. Different children were given different array orders, but within children the order of arrays was kept the same for the four repetitions. Each time the child saw an array it was presented exactly as it

had been the first time, in terms of whether the direct route was on the child's left or right, which landmark was on which route, and which route the experimenter pointed out first. Each child's order of array presenta- tion was constrained so that the correct route alternated from the child's left to right as the five arrays were presented. The first route (direct or indirect) that the experimenter pointed out in each array usually alternated from left to right. The following were all counterbalanced across children: Whether the direct route in each array was on the child's left or right, whether the indirect route in each array had the rock or tree landmark, whether the experimenter pointed out the direct or indirect route first in each array, and whether the child received the "short" or "long" question first.

Explanations.-At the end, we asked children to judge Landmark Array 1 again, and we asked them to explain why they thought one route was longer and one was shorter. We asked children for explanations in all three studies, and we report overall interrater reliability here. Across studies, children gave four types of correct explanations; that is, explanations that referred to the shapes of the routes. Two types of explanations referred to the routes being straight or curved. There were explicit references to straight versus curved. These children said that the short route was "straight" and/or that the long route "curved," "turned," or had a "corner." There were also implicit references to straight versus curved. These children said, for example, that the long route "goes that way and then that way," and that the short route "only goes that way." Two other types of explanations referred more generally to the shapes of the routes. A few children referred to the number of sides. These children said the long route had "two sides" or "two paths" and the short route had "one side" or "one path." We are confident that references to the number of sides were references to the shape of the routes and not simply to the number of landmark-defined segments, because all three children who gave these explanations were in Experiments 2 or 3 where they happened to be asked to explain arrays that had landmarks on both segments. Finally, one child referred to the shape by pointing. This child said the long route was long "because it goes like this" and pointed along the shape of the detour route.

Interrater reliability between two scorers working from transcripts of children's ex-




planations made by the experimenter was calculated by considering as agreements only those cases where both scorers judged an explanation to be one of the above four types. Reliability was 89.9%. Disagreements were resolved by discussion. Across studies, of the 31 children who gave correct explanations, 64.5% made explicit references to straight versus curved, 22.5% made implicit references, 10% referred to the number of sides, and 3% relied on pointing.

RESULTs

Preliminary ANOVAs on percent correct performance for each of the three studies consistently failed to find any sex differences. A 2 (age) x 2 (question: "long way" vs. "short way") x 5 (array) x 2 (trial) ANOVA on proportion correct with the last three factors within-subjects revealed effects due only to age, F(1, 22) = 19.69, p < .001, and array, F(4, 88) = 8.81, p < .001. The age x array means are presented in Table 1. Older children outperformed younger children, and Landmark Array 1 was more difficult than each of the other four arrays (p's < .05, Fisher's Least Significant Difference Test, here and below), which did not differ from each other. The fact that performance was worst on Landmark Array 1 suggested that children were not scanning along the routes and comparing the scan times to estimate the distances of the routes. We also examined performance on the Rope Array, because that was a large-scale version of the direct-indirect tasks used by Piaget et al. (1960) and by Bartsch and Wellman (1988). Consistent with Bartsch and Wellman, the 31/2-41/2-year-olds (M = .71) were significantly above the chance value of 50% correct, t(11) = 2.59, p < .05.

Next we examined individuals' patterns of responses for evidence of rule use. We calculated the binomial p for the fit between an individual's performance across arrays and the predictions of each of the rules shown in Figure 2. In calculating the fit, we ignored performance on arrays for which the rule predicted random performance. For ex-

ample, Rule IE predicted the following pattern of correct and incorrect responses: four incorrect responses to Landmark Array 1, and four correct responses to each of Land- mark Arrays 2 and 4. We assigned a child to the rule that had the smallest statistically significant binomial p. The assigned rules had fits ranging from p = .01 to .000001. (The minimum criterion across studies for assigning a rule was p < .05.) The numbers of children using various rules are shown in Table 2.3

Table 2 shows that all of the 5-6-year- olds, and 50% (6/12) of the 31/2-41/2-year-olds used rules. Half of the 5-6-year-olds used Rule IV, while none of the 31/2-41/2-year-olds did so (Fisher exact p < .05). All the 5-6- year-olds who used Rule IV gave correct explanations for why the detour was longer (five made explicit references to the routes being straight vs. curved, and one made an implicit reference, as defined above). One additional 5-6-year-old, who used Rule IIIE, also gave a correct (explicit) explanation. None of the 31/2-41/2-year-olds explained correctly why the detour was longer.

Of the 12 children who used rules other than Rule IV, all but one used Rules IE, IIE, or IIIE. In those rules, interposed object errors resulted from children's focusing on the endpoint segments of the routes. This was in accord with Piaget's emphasis on the importance of the endpoint or point of arrival in children's reasoning about distance. Fi- nally, 50% (6/12) of the 31/2-41/2-year-olds did not fit any rule. For this subset of children, as for the sample as a whole, performance was lowest on Landmark Array 1, suggesting that they did not use scanning.

DISCUSSION

In the Rope Array 31/2-41/2-year-olds tended to judge correctly that the detour was longer. This finding is consistent with Bartsch and Wellman (1988), who found that, when small displays of direct versus indirect routes were covered and children were simply told that one of the routes de-

3 As an alternate method of assigning rules we used Siegler's (1981) approach with a criterion of 17/20 responses in accord with predictions of the rule, and additional safeguards that a child assigned to one rule was not more likely to have used a different rule. For example, because Landmark Array 1 differentiated Rules IIE, IIIE, and IV, a child had to have no correct responses on that array to be classified Rule IIE, and all correct to be classified Rule IV, and to be classified as Rule IIE or IIIE vs. Rule I, a child would have to have at least 75% correct on both the Rope/ Lake Array and Landmark Array 3. Ninety-two percent of children in Experiment 1 were classified the same using both approaches. We adopted the approach described in the text for all three studies.




TABLE 1

PROPORTIONS OF CORRECT RESPONSES IN

EXPERIMENTS 1, 2, AND 3

ROPEa LANDMARK ARRAYS

LAKEb ARRAY 1 2 3 4 5

Experiment 1: 31/2-41/2-year-olds ...... .71 .31 .71 .56 .63 ... 5-6-year-olds ......... .98 .65 .96 .88 .96

Experiment 2: 3'/2-41/2-year-olds ...... .68 .55 .59 .82 .61 .68

Experiment 3: 312-41/2-year-olds ...... ... .48 .58 .68 .50 .70 41/2-5-year-olds ......... ... .50 .73 .75 .58 .60

" Used in Experiment 1. b Used in Experiment 2.

toured, 4-year-olds were above chance. Both findings indicate that 4-year-old children have some distance knowledge that allows them at times to avoid the direct-indirect error.

In the present task, we examined children's distance knowledge in the context of both classic errors-the direct-indirect error and the interposed object error. The results showed that developmental changes in distance knowledge could be largely charac- terized by a set of rules, which accounted for all of the 5-6-year-olds and half of the 3'12-4'12-year-olds.

The rule analysis showed that half of the 5-6-year-olds (the Rule IV users) reasoned about distance in a principled fashion. They consistently avoided both the direct-indirect and interposed object errors. Furthermore,

in their explanations, they appealed to the principle that the shortest distance between two points is a straight line. In contrast, none of the 31/2-41/2-year-olds was consistently correct, and none expressed the direct-indirect principle.

The rule analysis also showed that prior to acquiring principled distance knowledge, children did not use the estimation procedure of scanning along the routes and comparing the scan times. All of the 5-6-year- olds who had not yet acquired Rule IV, and half of the 31/2-41/2-year-olds used one or the other version (E or S) of Rules I-III. Those rules predict worst performance on Land- mark Array 1, in contrast to scanning which would give best performance on Landmark Array 1. Likewise, the younger children who did not use rules also did not use scanning;

TABLE 2

NUMBERS OF CHILDREN USING VARIOUS RULES OR No RULE IN EXPERIMENTS 1, 2, AND 3

RULES

No RULE IE IIE IIIE IS IIS IIIS IV

Experiment 1: 31/2-4?/2-year-olds ..... 6 2 0 3 1 0 0 0 5-6-year-olds ........... 0 3 3(1) 0 0 0 6(6)

Experiment 2: 31/2-4'/2-year-olds ..... 14 0 1 0 2 0 0 11(9)a

Experiment 3: 312-4'/2-year-olds ..... 14(3) 1(1) 0 0 1 1 0 3(3) 41/2-5-year-olds ........ 9(3) 1 2 0 1 2(1) 0 5(4)

NOTE.-Numbers in parentheses represent the number of children in that cell who explained correctly why the indirect route was longer. a Only 10 of the 11 Rule IV users were asked to explain why the indirect route was longer.




that is, for those children performance was also lowest on Landmark Array 1. It is possible, however, that our procedure of having the experimenter point out the routes for the children may have discouraged children from using scanning. Therefore, in Experi- ments 2 and 3 we let children themselves point out the routes.

Finally, the rule analysis tentatively suggested that children made interposed object errors because they focused on endpoint segments. We sought confirmation of this in Experiments 2 and 3 by attempting to prompt children to focus on startpoint segments.

Experiment 2 We especially wanted to investigate

4-year-olds' distance knowledge further in Experiment 2. Their highest level of performance in Experiment 1 was Rule IIIE, achieved by 25% (3/12) of the younger children. They knew that indirect routes were longer when there were no interposed objects (Rope Array) and when endpoint segments were equal (Landmark Array 3), but they were apparently confused when the endpoint segment on the indirect route was shorter than the direct route (Landmark Array 1). Thus these younger children had some distance knowledge, but the presence of interposed objects disrupted their performance. In addition, we found two other 31/2-4'/2-year-olds (the one using Rule IS and one using no rule) who, like the Rule IIIE users, were consistently correct on the Rope Array and were disrupted in one way or another by interposed objects. That raised to 42% (5/12) the percentage of 31/2-41/2-year- olds who knew that indirect routes were longer but who were still affected by the interposed object error.

We wanted to understand why interposed objects disrupted children who other- wise seemed to have some distance knowledge. In Experiment 1, the Landmark Arrays differed from the Rope Array not only in having interposed objects along the routes but also in not having ropes laid out along the routes. It may have been the absence of a visual display of the routes (i.e., the ropes), more than the presence of interposed objects, that led to interposed object errors. That would suggest that children's errors stemmed from difficulty they had representing routes that were not displayed rather than from an inability to conceive of the overall distance of a segmented route. If so,

the various rules we observed would have stemmed from varying levels of difficulty children had in representing undisplayed routes. For example, the Rule IIIE children would be those who were able to encode the whole routes in all the Landmark Arrays but who did so inconsistently, sometimes encoding whole routes and other times encoding only endpoint segments. That would give the predicted random performance on Land- mark Array 1 and correct performance on Landmark Arrays 2 and 4. On Landmark Array 3, the fact that endpoint segments are equal could have given them pause to recon- sider and then represent the whole routes, which would mean that they would be correct, as predicted, regardless of whether they initially encoded segments or whole routes. The Rule IIE children would be those who were able to encode undisplayed routes only in Landmark Array 3 (where the endpoint segments are equal), and the Rule IE children would be those who were never able to encode whole routes in the Land- mark Arrays. We reasoned that if 4-year-olds could be helped to encode and keep in mind the entire routes in the landmark arrays, then many of them might be able to avoid both errors and be classified as Rule IV. Therefore, in Experiment 2 we displayed the routes in the landmark arrays. For reasons given below, we did this by constructing the arrays around a triangular "lake" (a piece of blue material; see Fig. 1).

We also examined three other issues in Experiment 2. First, it was important to as- sess the robustness of children's ability to avoid the direct-indirect error. There might be something peculiar about the Rope Array that made it especially easy for children to determine that the indirect route was longer. We wished to be confident that young children do, in some visible situations, appre- ciate that direct routes are shorter than indirect ones and are not just responding correctly in some artifactual way to a single sort of display. This was the reason we used the lake in Experiment 2 rather than outlin- ing the routes with ropes.

A second issue involved whether there was something special about children's tendency to focus on the endpoint segment of a route rather than on the startpoint segment, as Piaget (1946) implied. If children's interposed object errors are due to failures to represent the whole route, then in principle there should be no reason why children could not also show the error by focusing on




the startpoint segment. In Experiment 1, we named the routes by the landmark on them ("rock way," "tree way") and children made their judgments by naming the route that was longer or shorter. Use of the landmark names might have served to draw attention to the segment from the landmark to endpoint. In contrast, in Experiment 2 we named the routes by the color of a small arrow placed at the beginning of each route ("green way," "blue way"). We expected that this might draw children's attention to the startpoint segments of the routes.

Third, we let children themselves point out the routes prior to judging to see if that would encourage them to use visual scanning to compare the two routes.

METHOD

Subjects Twenty-eight 31/2-41/2-year-olds partici-

pated (M = 4-2; range = 3-5 to 4-8), including 13 boys. They were recruited from the McPhaul Child and Family Development Center on campus.

Arrays Arrays used were the Lake Array and

Landmark Arrays 1-5 shown in Figure 1. The Lake Array was constructed by laying a blue piece of material on the floor. The Landmark Arrays were constructed by placing the rock and/or tree along the edges of the "lake." As in Experiment 1, the object on the direct route was placed slightly off to the side. The object on the indirect route was placed a similar distance away from the corner of the lake. On each array two small arrows, blue and green, were put at the child's feet, each pointing in the beginning direction of one of the routes. The arrays were approximately the same dimensions as the arrays in Experiment 1. The short and long segments of the two routes in Land- mark Array 5 were also approximately the same lengths as those in Landmark Array 4.

Procedure Each child was tested in one session

lasting about 15 min. Throughout the session the child judged which way was the

"long way to the ice cream store," because Experiment 1 did not find any effect of ask- ing children to judge the long way versus the short way. The child received a short pretest on the term "the long way," which was similar to the pretest in Experiment 1. Two children initially gave an incorrect answer to one of the pretest questions. They were corrected, the question was repeated, and they went on to answer all other pretest questions correctly. Two other children were dropped from the study for repeated failure to answer the pretest questions correctly.

The rest of the judgment procedure was the same as in Experiment 1, with the following exceptions: the child's chair faced the ice cream store in each array,4 the child pointed out both routes in each array in response to the experimenter's questions, "Can you point the whole green (blue) way?" the child was always asked which was the long way, and the child responded by saying either "the green way" or "the blue way.

Each child received the six arrays in a different order. The order took the child through the six arrays once and then was repeated once more. The second time the child received an array the long route was switched from the child's left to right or vice versa, and the landmark and arrow that had been on the long route were switched to the short route. Each child's order was constrained so that the long route alternated from the child's left to right across the six arrays. The first route that the child pointed out in each array also alternated from left to right. Thus the following were all counterbalanced within children: Whether the long route was on the child's left or right, whether the long route had the rock or tree landmark, and whether the child pointed out the long or short route first.

The procedure we used to ask children to explain their judgments was the same as in Experiment 1, with one exception. Rather than giving children Landmark Array 1 again after they had finished all their judgments,

4 Having the child face toward the ice cream store rather than between the two routes was an inadvertent difference in procedure between Experiments 1 and 2. In response to this difference, we tested another group (N = 9) of 4-year-olds (M = 3.9; range = 3-6 to 4-8) to see if there was any effect of the direction in which the child faced. We used the same procedures as in Experiment 1, except that the new group faced the ice cream store and received two trials on the Rope Array only. Mean proportion correct on the Rope Array was .72, essentially equivalent to that in Experiment 1 (.71). The change in the child's orientation most likely does not account for differences in performance between the two studies.




we simply asked children to explain their judgment on whatever array they had received last.

Results and Discussion We will postpone a detailed discussion

of the results of Experiment 2 until after Experiment 3. As shown in Table 1, 31/2- 41/2-year-olds' performance on the Lake Ar- ray (M = .68) was almost identical to what it had been on the Rope Array in Experiment 1 (.71) and again was significantly above chance (.50), t(27) = 3.30, p < .05. This suggests that there was nothing unique or ar- tifactually easy about the Rope Array in Experiment 1. Instead, 4-year-olds' understanding that visible direct routes are shorter than indirect ones has some generality.5

A 6 (array) x 2 (trial) ANOVA on proportion correct showed only an effect of array, F(5, 135) = 3.10, p < .05. The array means are presented in Table 1. Landmark Array 1 (along with Landmark Arrays 2 and 4) differed significantly only from Landmark Array 3. No other pairwise comparisons were significant.6 As in Experiment 1, performance on Landmark Array 1 was not better than that on the other arrays, suggesting that children did not estimate distance by scanning along the routes and comparing scan times. This despite the fact that children pointed out the routes themselves prior to judging them.

We examined rule use with the same procedure as in Experiment 1. The assigned rules had fits of p = .02-.0002. The numbers of children using various rules are shown in Table 2. The most important finding is that the changes in the task in Experiment 2 revealed more 31/2-41/2-year-olds using Rule IV. Eleven (39%) now used Rule IV, which was a significant difference from Experi- ment 1 where no 31/2-41/2-year-olds did so (Fisher exact p < .05). One of the Rule IV

users was inadvertently not asked for an explanation for why the indirect route was longer, but of the 10 who were asked, 9 (90%) offered correct explanations (six made explicit references and one made an implicit reference to the routes being straight vs. curved, one referred to the number of sides the routes had, and one used pointing to refer to the shapes of the routes). No other children gave correct explanations.

Findings regarding whether interposed object errors could be based on startpoint as opposed to endpoint segments were limited due to the small number of children (three) who used rules other than Rule IV. Never- theless, two of the three did focus on startpoint segments (Rule IS). In order to obtain a larger number of children using rules other than Rule IV, we used a larger sample in Experiment 3.

Finally, as in Experiment 1, 50% (14/28) of the 3'/2-41/2-year-olds did not fit any rule. And again for this subset of children, as for the sample as a whole, performance on Landmark Array 1 was inconsistent with use of scanning.

Experiment 3 Our first reason for conducting Experi-

ment 3 was to obtain a larger number of children using rules other than Rule IV, in order to examine the relative frequencies of Rules IE-IIIE versus Rules IS-IIIS. We did this by testing 20 31/2-41/2-year-olds in essentially the same age range as in the first two experiments,7 and 20 41/2-5-year-olds in the age range between the two age groups in Experi- ment 1.

The second reason for Experiment 3 was to provide another test of the generality of young children's distance knowledge. In Experiment 2, we constructed the Landmark

5 We also tested 10 4-year-olds (M = 4-0; range = 3-5 to 4-4) on a within-subject comparison between Rope and Lake Arrays, using the same procedures as in Experiment 2. Proportion correct on the Rope Array (.60) did not differ from that on the Lake Array (.70), t(18) = .55, p > .50. Given the small N, only the latter tended to differ from chance, t(9) = 1.73, p < .06 (one-tailed).

6 Performance on Landmark Arrays 2 and 4 tended to decrease from Experiment 1 to the next two studies, while Array 3 increased. This is consistent with the observed switch from endpoint rules to startpoint rules.

7 We could not make the upper end of the age range for 31/2-412-year-olds exact across the three studies. It ranged from 4-6 to 4-8 to 4-4, respectively. This does not appear to be a problem for two reasons. First, Experiment 3 showed no age differences between 31/2-4112-year-olds, and 41/2-5-year-olds. Second, results were unchanged when we eliminated four children from Experi- ment 2, and switched three from the older group to the younger group in Experiment 3 in order to adjust the cutoff to be 4-6 across studies. For example, in Experiment 2, the new mean for the Rope Array was .65, which is still above chance, t(23) = 2.34, p < .05, and the percentage of children using Rule IV was 33%, which is still significantly more than in Experiment 1.




Arrays around the lake. We now wondered whether showing children the Lake Array, which had no interposed objects on it, may have inadvertently helped them to avoid the interposed object error in the Landmark Arrays. Consequently, we eliminated the Lake Array in Experiment 3.

METHOD

Subjects Twenty 31/2-41/2-year-olds (M = 4-0;

range = 3-6 to 4-4), including 12 boys, and 20 41/2-5-year-olds (M = 4-10; range = 4-5 to 5-2), including seven boys, were recruited from the McPhaul Child and Family Devel- opment Center.

Arrays Arrays used were Landmark Arrays 1-5

shown in Figure 1. They were constructed from the same materials as had been used in Experiment 2.

Procedure Each child was tested in one session

lasting about 15 min. The procedure was the same as used in Experiment 2, except that children were not tested on the Lake Array. Six younger children and three older children initially gave an incorrect answer to one of the pretest questions on the term "the long way." They were corrected, the question was repeated, and they went on to answer all other pretest questions correctly.

RESULTS AND DISCUSSION

Means for the two ages are presented in Table 1. A 2 (age) x 5 (array) x 2 (trial) ANOVA showed there was no overall effect of age, F < 1. The 31/2-4x/2-year-olds aver- aged 59% correct, and the 41/2-5-year-olds, 63%. As before, the analysis showed an effect of array, F(4, 152) = 2.97, p < .05.

However, in this experiment, there was also an effect of trial, F(1, 38) = 7.21, p < .05, qualified by an age x array x trial interaction, F(4, 152) = 3.17, p < .05. It was clear first of all from the pattern of performance across arrays that there was no evidence of use of scanning to estimate distance. On each trial at both ages, performance tended to be lowest on Landmark Array 1, or, consistent with use of Rule IS, on both Landmark Arrays 1 and 4. The same was true for the subset of children who did not use rules. We explored the three-way interaction with a se- ries of two-way analyses, each conducted at one level of the third factor. There were no significant age effects. The interaction ap- peared to stem from children's differential

ability to improve over trials. Separate array x trial analyses for each age group showed a main effect of trial for 41/2-5-year-olds, F(1, 19) = 11.47, p < .01, but a trial x array interaction for 31/2-41/2-year-olds, F(4, 76) = 2.77, p < .05, which was due to their improvement over trials being confined to Array 4.

We examined rule use with the same procedure as before. The assigned rules had fits of p < .05-.001. The numbers of children using various rules are shown in Table 2. Of the nine children across age groups who used rules other than Rule IV, five used Rules IS or IIS. Comparing Experiment 1 to Experiments 2 and 3 combined, there was a significant shift toward use of startpoint segments, Fisher exact p < .05.

Finally, the absence of the Lake Array in this study appears to have resulted in somewhat fewer children using Rule IV than in Experiment 2, though not significantly so. Across both age groups, 20% (8/40) used Rule IV, as opposed to 39% previously, Fisher exact p, N.S. Of those using Rule IV, 88% (7/8) gave correct explanations for why the detour was longer. However, eight other children (four younger and four older) did not use Rule IV but nevertheless gave correct explanations by the end of the task for why the detour was longer. Of the 15 total correct explanations, eight were explicit references and five were implicit references to the routes being straight versus curved, and two were references to the number of sides the routes had.

In summary, in Experiment 3 children's explanations were somewhat in advance of their use of Rule IV. Of the 15 children who gave correct explanations, eight did not use Rule IV, whereas in Experiment 2 all who explained used Rule IV, Fisher exact p < .01. Children in Experiment 3 may have needed some practice on the early trials representing the whole routes, because they never saw an array (the Lake Array) without interposed objects. That is consistent with the fact that trial effects were found only in this study. Apparently then, the absence of the Lake Array in Experiment 3 had only a minor effect in delaying for a few trials some children's ability to determine that the indirect route was longer in arrays that included interposed objects. This seems especially clear for the eight children who correctly explained but did not use Rule IV. They showed strong improvement from Trial 1 (.48) to Trial 2 (.83). On the last two arrays they received, they were correct 94% of the time. If we use a relaxed criterion of either




Rule IV use or correct explanation, then 40% (16/40) of 4-year-olds in Experiment 3 displayed correct distance knowledge, which was essentially identical to Experiment 2 (39%). This is at least partial evidence for the generality of children's distance knowledge; namely, they were still able to give some evidence of it even without seeing an array without interposed objects.

Analyses of Non-Rule-Users across the Three Studies

In order to get some sense of what may have influenced non-rule-users' performance, we conducted a group level analysis of the 31 non-rule-users from Experiments 2 and 3 who did not give correct explanations. We correlated mean performance on the 12 trials (only Experiment 2 children received the two trials on the Lake Array) with predictions from Rules IE and IS. Rule IE correlated -.13, and IS correlated .59, p < .05, with performance, suggesting that non-rule- users in Experiments 2 and 3 tended to base their judgments on startpoint segments.

General Discussion

We will focus on three issues in the following discussion: our findings on children's ability to judge distance, the implica- tions for what develops, and the relation of the current studies to others that have also found earlier reasoning abilities than Piaget predicted. The present findings tell us several things about how children make distance judgments. First, interposed object errors were due to children basing their judgment of distance on only one segment of the route, and task factors affected whether children used startpoint or endpoint segments. In Experiment 1, almost all children who used rules that incorporated interposed object errors relied on endpoint segments, while in Experiments 2 and 3, there was a significant shift toward startpoint segments. The non-rule-users in Experiments 2 and 3 also tended to use startpoint segments.

Second, there was no evidence that children simply scanned along the routes and mentally compared scan times to determine which was longer. Landmark Array 1 was never the easiest array, as it should have been had children used scanning. For Rule IV users, about 90% or more of them correctly explained that the indirect route was longer because of its shape. We might have expected children to use scanning here, because in contrast to most previous studies

the large distances we used would have led to noticeable differences in scan times.

Finally, substantial numbers of 4-year- olds were able to avoid both distance errors, as revealed either in their performance or in their explanations. This confirms and ex- tends previous studies (Bartsch & Wellman, 1988; Miller & Baillargeon, 1990) that showed earlier distance knowledge than Piaget predicted. In Experiment 2, about 40% of 31/2-41/2-year-olds used Rule IV. In Experiment 3, there was no difference between younger and older 4-year-olds, and across groups 40% either used Rule IV or by the end of the task were at least able to explain correctly why the indirect route was longer in an array that had interposed objects. The present studies also showed that children's ability to avoid the distance errors had some generality. Displaying routes with ropes versus a "lake" did not affect performance, and not showing children an array without interposed objects only seemed to delay for a few trials some children's ability to determine that the indirect route was longer in arrays with interposed objects.

Piaget et al. (1960) argued that there were, however, two other limitations to young children's distance knowledge. First, Piaget acknowledged that young children sometimes correctly infer that a route is longer because it is indirect. But he argued that they do so only "when they think in terms of the speed or time taken in rounding a bend but it has no bearing on the geometri- cal properties of the path; that is, its length as such" (p. 109). Several of our children who gave incorrect explanations did refer to speed or time by saying that one would have to stop at or go around a landmark, or walk fast. However, all of our 4-year-olds who gave correct explanations referred to the shapes of the paths, and none referred to the speed or time it would take to round the bend.

Second, Piaget argued that young children "resist the idea that a line can be longer because of angles and curves unless these detours are exaggerated. Smaller detours are therefore insufficient to create an impression of greater length" (p. 109). We were also able to examine this possibility, because in the Lake Array and Landmark Arrays 1, 2, and 3 the indirect route was 7 feet longer than the direct route, while in Landmark Arrays 4 and 5 it was only 1.5 feet longer. If Piaget is right, the claims we have made about 4-year-olds' understanding of distance




would have to be modified. Accordingly, we examined the 19 4-year-olds in Experiments 2 and 3 who used Rule IV.8 The Rule IV users did perform better on the first four arrays (M = .98) than on the last two (.91), F(1, 72) = 6.46, p < .05. But the fact that they were correct over 90% of the time on the latter arrays suggests that, although the error due to degree of detour can be observed, it is rare and does not undermine the claim that many 4-year-olds understand distance correctly.

The present results also provide information on the question of what develops that allows children to demonstrate an understanding of distance. Piaget's answer was that it was the child's conceptual understanding of space. Below we argue that the classic distance errors that children make in the age range Piaget discussed are not the result of an immature conception of space but, instead, are the result of limitations in their ability to represent routes. Further- more, we believe that it is an open question how early in development children acquire the distance knowledge that would allow them to avoid the distance errors.

When the routes were visually displayed in the Rope Array in Experiment 1, 42% (5/12) of the 4-year-olds were consistently able to avoid the direct-indirect error. When we added interposed objects and removed the visual displays in the Landmark Arrays, none of those children was able to avoid consistently the interposed object error. However, in Experiments 2 and 3 we added back the visual displays (the lakes) to the Landmark Arrays and found that the same percentage (40%) of 4-year-olds who had been able to avoid the direct-indirect error in Experiment 1 were now able to avoid both errors, as indicated either by their use of Rule IV or by their correct explanations at the end of the experiment for why the indirect route was longer.

These findings suggest that the interposed object error was not indicative of a fundamentally incorrect conception of distance and space, at least for this advanced group of 4-year-olds. When the routes were not displayed, these children often failed to encode the whole routes and compared segments of the routes instead. But when the

routes were visually displayed they were able to represent and compare the whole routes. In addition to displaying the routes in Experiments 2 and 3, we also allowed children to point out the routes for themselves. This may also have helped them represent the whole routes, though it had no effect on their tendency to use scanning to compare the routes.

Of the remaining approximately 60% of 4-year-olds, the great majority either used one of the versions of Rule I or used no rule. In other words, there was essentially no evidence that these 4-year-olds could avoid either error. One possibility is that Piaget was right about this less advanced group of 4- year-olds (and also about younger children). In that case our results would lower the age at which we would grant children a correct understanding of distance but would not challenge Piaget's more important claim that at some point in development children have a fundamentally incorrect conception of distance. The alternate possibility is that the classic distance errors in younger children may result from the same type of difficulty encoding and representing distances that was the basis of the interposed object error in more advanced 4-year-olds.

There is some evidence regarding the interposed object error to suggest the latter possibility. In Miller and Baillargeon's (1990) bridge condition, having to think about a stick spanning the distance may have functioned like the lake did in our landmark arrays. Both could encourage children to encode the whole route. Miller and Baillar- geon suggested that the bridge condition was easy because children did not have to use terms such as "near" and "far," as they did in the standard condition. A linguistic interpretation cannot explain our results, however, because the linguistic require- ments of our task did not vary with the presence (in Experiments 2 and 3) and absence (in Experiment 1) of the lake.

Regarding the direct-indirect error, our less advanced 4-year-olds may have had difficulty simultaneously representing one route as direct and the other as indirect, even though they were visually displayed in the Rope and Lake Arrays. The goal of future studies would be to try to reduce the repre-

8 We tried but could not draw any conclusions about the eight 4-year-olds in Experiment 3 who correctly explained but did not use Rule IV. Their performance across arrays did not show any systematic pattern, most likely because, as noted above, they showed such a strong tendency to improve over trials.




sentational demands and thereby reduce the direct-indirect error. One possibility is that it may be easier for children to represent direct and indirect routes in smaller arrays. Bartsch and Wellman (1988) used small arrays, but the arrays were screened from view. Piaget's (1946) arrays were small and visible, but the indirect routes they con- tained were complex, sometimes containing "irregular zigzags with variable angles and unequal segments" (p. 64).

Finally, our studies share some important similarities and differences with other studies that have found early logical compe- tence. The similarity is that helping children to construct a representation of the problem often allows them to operate on it successfully. For example, Markman (1983) found that using collection nouns (e.g., "crowd," "forest") rather than class nouns ("people," "trees") leads to improvement relative to standard Piagetian class inclusion tasks. She pointed out that collections are easier to imagine, which helps one hold in mind and compare the whole to one of its parts. Relat- edly, Trabasso (1977) found that, when children were helped to construct a mental im- age of an array of five sticks ordered from smallest to largest, they could answer transitive inference questions correctly by mentally scanning the array and "reading off" the answer. Similarly, when we helped children represent routes by providing visual displays, many 4-year-olds could avoid the interposed object error.

However, in the case of class inclusion and transitive reasoning, the task modifica- tions and training have resulted in children's using processes different from those that Piaget was interested in. Scanning an array is different than using two or more pairwise comparisons to make a transitive inference. And making part-whole comparisons with collections, where the wholes are defined by external, perceptual relations, is not the same as doing so with classes, where the wholes are defined by internal, conceptual relations. This might have happened in our studies. Displaying the routes might have resulted in children simply scanning to estimate distance. However, children demonstrated a conceptual understanding of distance, and this suggests that it might be fruitful in class inclusion and transitive reasoning tasks to find ways to help children construct representations that would allow them to demonstrate the kinds of conceptual understanding that Piaget targeted.

In summary, the present results have

shown that when young children make the interposed object error it is because they tend to focus on only one segment of a route. Which segment that would be was affected by task factors that directed attention to either the beginning or end of the route. There was no evidence that children judged distance by visually scanning along the routes. We found that a substantial proportion of 4-year-olds could avoid the direct- indirect error and also avoid the interposed object error when we displayed the routes (and also perhaps because we allowed them to point out the routes). For these children, at least, the interposed object error was due to difficulty they had representing routes rather than to a misconception of distance. Future research should examine whether the same is true of younger children.

References

Bartsch, K., & Wellman, H. M. (1988). Young children's conception of distance. Developmental Psychology, 24, 532-541.

Fabricius, W. V. (1988). The development of for- ward search planning in preschoolers. Child Development, 59, 1473-1488.

Markman, E. M. (1983). Two different kinds of hierarchical organization. In E. K. Scholnick (Ed.), New trends in conceptual representation: Challenges to Piaget's theory? (pp. 165- 184). Hillsdale, NJ: Erlbaum.

Miller, K. F., & Baillargeon, R. (1990). Length and distance: Do preschoolers think that occlu- sion brings things together? Developmental Psychology, 26, 103-114.

Piaget, J. (1946). The child's conception of movement and speed. New York: Ballantine.

Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child's conception of geometry. London: Routledge & Kegan Paul.

Schiff, W. (1983). Conservation of length redux: A preceptual-linguistic phenomenon. Child De- velopment, 54, 1497-1506.

Shantz, C. U., & Smock, C. D. (1966). Develop- ment of distance conservation and the spatial coordinate system. Child Development, 37, 943-948.

Siegler, R. S. (1981). Developmental sequences within and between concepts. Monographs of the Society for Research in Child Develop- ment, 46(2, Serial No. 189).

Thorndike, P. W. (1981). Distance estimation from cognitive maps. Cognitive Psychology, 13, 526-550.

Trabasso, T. (1977). The role of memory as a system in making transitive inferences. In R. V. Kail & J. W. Hagen (Eds.), Perspectives on the development of memory and cognition (pp. 333-366). Hillsdale, NJ: Erlbaum.



two roads diverged: young children's ability to judge distance

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