young children's use of spatial coordinates

Young Children's Use of Spatial CoordinatesAuthor(s): Susan C. Somerville and P. E. BryantSource: Child Development, Vol. 56, No. 3 (Jun., 1985), pp. 604-613Published by: Wiley on behalf of the Society for Research in Child DevelopmentStable URL: http://www.jstor.org/stable/1129750 .

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Young Children's Use of Spatial Coordinates

Susan C. Somerville Arizona State University

P. E. Bryant University of Oxford

SOMERVILLE, SUSAN C., and BRYANT, P. E. Young Children's Use of Spatial Coordinates. CHILD DEVELOPMENT, 1985, 56, 604-613. 4-6-year-old children were given problems in which they had to decide which 1 of an array of points was in line with 2 coordinate markers. The simplest problems had 4 points to choose between and markers perpendicular to the horizontal and vertical axes. Children of all ages were able to extrapolate lines from both coordinates to solve these problems. The older children were also given more complex problems. In some of these, 1 marker was at 450 to an axis, the other perpendicular: in others the array was increased to 16 points and presented sometimes in a regular, sometimes in an irregular pattern. There were developmental im- provements in performance, and the complex problems were more difficult than the simpler ones. However, 5- and 6-year-olds did extremely well even on the complex problems. The results estab- lish that young children's grasp of Euclidean spatial relationships is more adequate than has often been suggested.

Spatial coordinates play a crucial role in many of the things that children have to learn at school. Map reading and the interpretation of graphs depend on an understanding of coordinates. Maps and graphs typically employ vertical and horizontal axes. In both cases points in space are determined by extrapolating lines from positions on each axis and working out where they intersect.

Suppose that children are given the task of producing a graph to represent the relation- ship between the age of a growing plant and its height, and that in this graph the horizontal axis represents age and the vertical axis height. To plot each point they will have to imagine lines extending vertically from the horizontal axis and horizontally from the vertical axis, and to work out where they intersect. Finding a given location on a map in- volves coordinates in just the same way. Letters are usually used to represent the vertical position and numbers the horizontal position in city maps, and letter-number combina- tions specify where particular streets can be found.

There is surprisingly little research on children's use of two coordinates to find a po-

sition in space. This contrasts with the plenti- ful research on children's use of frames of ref- erence to determine the orientation of one or more lines in tasks involving judgments about water levels and plumb lines (e.g., De Lisi, 1983; Liben, 1975; Liben & Golbeck, 1980; Piaget & Inhelder, 1956; Thomas & Jamison, 1975) or about parallelism (e.g., Abravanel, 1977). Work on map reading has dealt with children's ability to determine position with the use of landmarks (Bluestein & Acredolo, 1979; Presson, 1982), but this does not tell us whether they can also use the intersection of lines drawn from coordinate axes. Piaget, In- helder, and Szeminska (1960) suggested that children younger than approximately 8 years cannot locate a position using two coordinates. They showed the child one rectangle with a point marked within it and a blank rectangle identical in size but not orientation to the first. The child had to mark the place in the blank rectangle corresponding to the point in the other one. Children younger than 8 or 9 years did not use the coordinates of the marked point on both the horizontal and vertical sides of the rectangle. However, to do this they would have had to create their own coordinates. The study does not tell us whether

This research was supported in part by NIH grant R01HD13317-05 to the first author and was carried out while this author was a sabbatical visitor in the Department of Experimental Psychology at the University of Oxford. We are grateful to the teachers and children of Orchard Meadow First School, Botley Primary School, and Elms Road Nursery School for their participation in this project. Requests for reprints should be sent either to Dr. S. C. Somerville, Department of Psychology, Arizona State University, Tempe, AZ 85287, or to Professor P. E. Bryant, Department of Experi- mental Psychology, University of Oxford, South Parks Road, Oxford OX13UD, England.

[Child Development, 1985, 56, 604-613. ? 1985 by the Society for Research in Child Development, Inc. All rights reserved. 0009-3920/85/5603-0021$01.00]



Somerville and Bryant 605

young children can use coordinates when these are provided for them.

This issue is important in its own right and also because of a further question about children's understanding of space. Maps and graphical displays typically employ rectangular coordinate systems, where lines must be extrapolated at right angles to the vertical and horizontal axes. Children are more skilled at reproducing lines perpendicular to a baseline than obliquely inclined lines (Freeman, 1980; Ibbotson & Bryant, 1976; Olson, 1970). This may mean that rectangular coordinate systems suit their capacities more than systems using oblique lines.

We conducted two studies in which children had to extrapolate lines from two marked coordinates and choose the point where the lines would meet. One study involved only perpendicular lines; the other both perpendicular and oblique lines. We also varied the absolute and relative distances (from the axes) of the points to which lines were extrapolated. Extrapolations across greater absolute distances and/or past a nearer point also in line with the coordinate might be expected to be more difficult.

Experiment 1 METHOD

Subjects Eight boys and eight girls from the same

school were tested in each of three age groups; 5'/2-, 6-, and 6/2z-year-olds. The mean ages of these groups, respectively, were 65.9 months (range 63-69, SD = 2.1), 72.8 months (70-76, SD = 2.0), and 81.6 months (77- 86, SD = 2.3).

Design and Procedure In two equivalent tasks, children had to

decide which of four positions satisfied a pair of coordinates marked on rectangular axes (see Fig. 1).

Sticks Task.-We showed slides (on a back-projection screen) of a square white board placed above a yellow and a blue rod. One rod protruded from either the left or right side of the square (the vertical axis), the other from its top or bottom side (the horizontal axis). The child could not see what happened to the sticks under the board. Over a series of 16 trials the children had to judge where the sticks met. Four positions were marked by counters on top of the board. Two were in line with the horizontal, two with the vertical stick, and one of these (the correct one) with both sticks. The children had to touch the cor-

rect counter in each picture. They could pick the correct position consistently only by extrapolating lines from both axes and deciding where the lines met.

The projected image of the square was its actual size (39 cm square). It was surrounded by a 13-cm-wide background, and each of the sticks protruded across this background to one edge of the slide. The counters (1.5 cm diameter) were separated by 6.5 cm (center to center) on each side of the square array.

People Task.-The 16 slides showed a 39-cm-square white board and a red and a yellow Lego space character on adjacent sides of the board, facing inward. Four cylindrical blue blocks (1.5 cm diameter) were arranged in the same manner as the counters in the Sticks Task. We told the child that the characters had to meet at one of the blocks and that they could only walk in a straight line, jump- ing over a block if necessary. The child had to touch the correct block in each picture.

Procedure in both tasks.-Six preliminary trials using the actual materials preceded the experimental trials in each task. Half of the children in each age group had the two tasks in one order, the rest in the reverse order. The problems were displayed with coordinates marked on the left (vertical axis) and bottom (horizontal axis) sides of the square (the 00 condition) for half of the children in each order group: for the other half the problems were rotated 1800, with the coordinates on the right and top sides of the square (the 1800 condition).

In each task the 16 problems were generated by varying both the relative and the absolute distances of the correct point from the axes (see Fig. 1). The point was either the nearer or the further (with respect to an axis) of the two in line with a given coordinate. This made four types of problem (variations in relative distances), in which the correct position was (a) the nearer one to both coordinate markers (NN), (b) nearer to the horizontal but further from the vertical marker (NF), (c) further from the horizontal but nearer to the vertical marker (FN), and (d) further from both markers (FF). Second, within each type of problem described above (NN, NF, etc.) there were four different problems (variations in absolute distances) in which the distances of the correct position along the horizontal and vertical axes, respectively, were (1,1), (1,2), (2,1), (2,2), where 1 and 2 represent 13 cm and 26 cm, respectively. Half of the children were given the problems in one random order, the rest in the reverse order.



606 Child Development

Relative Distances NN

Absolute Distances 1, 2

Display Condition 0

Relative Distances NF

Absolute Distances 1.2

Display Condition 180

Relative Distances FN

Absolute Distances 2,1

Display Condition O

Relative Distances FF

Absolute Distances : 2.1

Display Condition : 180"

FIG. 1.-Rectangular problems in Experiments 1 and 2. Relative and absolute distances locate the correct point with respect to the horizontal and vertical markers, in that order, e.g., NF = nearer to the horizontal but further from the vertical marker; 2,1 = distance 2 from the horizontal and distance 1 from the vertical marker. 00 problems have markers on the left and bottom sides of the square, 1800 problems on the right and top.

RESULTS

Can children use two coordinates and their intersection to find a point in space? A child who is simply guessing will have a probability of .25 of choosing the correct one of the four points in any of the arrays. All scores were above this level; no child scored as low as four out of 16 in either task. This on its own does not demonstrate the use of both

coordinates. Other strategies could have produced scores higher than four. Someone using just one coordinate consistently, or sometimes one and sometimes the other but never both simultaneously, would be correct half of the time on average. So we compared the dis- tributions of numbers of correct choices to a binomial distribution based on a probability of .50 of correct responding.




TABLE 1

MEAN CORRECT RESPONSES IN EXPERIMENT 1

RELATIVE DISTANCES

TASK NNa NF FN FF TOTAL

Sticks ........ 3.401' 3.00 3.17 3.13 12.69C (.87) (1.32) (1.06) (1.28) (3.33)

People ....... 3.65 3.50 3.46 3.25 13.85 (.81) (1.15) (1.03) (1.19) (3.18)

Both tasks .... 3.53 3.25 3.31 3.19 13.27

ABSOLUTE DISTANCES

1,1d 1,2 2,1 2,2 TOTAL

Sticks ........ 3.441) 3.00 3.17 3.08 12.69u (.85) (1.19) (.97) (.99) (3.33)

People ....... 3.56 3.44 3.44 3.42 13.85 (.80) (.97) (.87) (.94) (3.18)

Both tasks .... 3.50 3.22 3.30 3.25 13.27

NOTE.-N = 48 in each cell; standard deviations are in parentheses. a NN = correct point the nearer to both markers; NF = nearer to horizontal, further from

vertical marker, etc.

I, Maximum possible score in each cell is 4.

SMaximum possible total score is 16. , 1,1 = correct point 13 cm from both markers; 1,2 = 13 cm from horizontal, 26 cm from

vertical marker, etc.

Given this distribution, the probability that an individual child would score 12 or more out of 16 on either task was .0384. In a group of 16 children, the probability that three or more would score at least 12 was less than .05 (.0218); the probability that five or more would do so was less than .001 (.0003). In the Sticks Task there were nine 51/2-, 11 6-, and 13 61/2-year-olds who scored 12 or more, and in the People Task the corresponding numbers were 11, 12, and 15, respectively. Thus some children in each age group consistently extrapolated lines from both axes to find the intersection.

We looked at the factors affecting performance. Preliminary analyses showed that sex, the display condition (0' vs. 180'), and the order of trials within a task had no effect. We conducted two overall analyses. In one the scores were summed for each child to give a total out of four for each combination of relative distances (NN, etc.), in the other for each combination of absolute distances (1,1 etc.; see Table 1).

The first was a 3 (age) x 2 (order of tasks) x 2 (task) x 2 (relative horizontal distance: N vs. F) x 2 (relative vertical distance: N vs. F) analysis of variance, with repeated measures on the last three factors. There were significant main effects of age, F(2,42) = 3.67, p < .05; of task, F(1,42) = 12.14, p < .01, perfor-

mance being higher on the People (M = 3.46 out of four) than on the Sticks (M = 3.17) Task; of order, F(1,42) = 6.82, p < .05, performance being higher for those who had the People Task first (M = 3.57) than for those who had it second (M = 3.06); and a significant task x order of tasks interaction, F(1,42) = 13.94, p < .001. Post-hoc comparisons (Newman-Keuls, p < .05) showed that the mean score for 61/?-year-olds (3.64) was significantly greater than that for 51/2-year-olds (2.99), but not significantly different from that for 6-year-olds (3.32). An analysis of the interaction showed that the scores on the Peo- ple (M = 3.56) and Sticks (M = 3.58) Tasks did not differ significantly for children who had the People Task first, t(42) = 0.14, but that scores were significantly higher on the People Task (M = 3.36) than on the Sticks Task (M = 2.76) for those who had the Sticks Task first, t(42) = 4.17, p < .001.

The second analysis was the same except that the last two factors involved absolute rather than relative distances (i.e., these factors were distance 1 vs. distance 2 from the horizontal and vertical axes, respectively). We report only effects involving the last two factors. There was a significant main effect of the vertical distance of the correct point, F(1,42) = 8.37, p < .01; qualified by a significant interaction of vertical with horizontal distance,




F(1,42) = 4.73, p < .05. In post-hoc analyses of the interaction we tested the effects of horizontal distance for each level of vertical distance. For problems in which the correct position was at distance 1 from the vertical marker, scores were significantly higher when it was also at distance 1 from the horizontal marker than when it was at distance 2, t(84) = 3.47, p < .001. For problems in which the correct position was at distance 2 from the vertical marker, there was no significant difference between scores when it was at distance 1 and scores when it was at distance 2 from the horizontal marker, t(84) = 0.55 (see Table 1).

Thus children's overall performance im- proved with age, the People Task was easier than the Sticks Task when each was encoun- tered first, and children who had the People Task first were better overall than those who had it second. Variations in the relative distances of the correct point from the axes had no significant effect on children's performance. This may be due in part to the relatively large variances in scores summed over relative distances (see Table 1). The large variances perhaps may be explained by some children making consistent errors on one type of problem, others on another (e.g., on problems involving far horizontal judgments). Variations in absolute distances affected performance at all ages in the same way; problems with the correct point close to both axes (1,1 problems) were easier than all others.

Experiment 2 METHOD

Subjects The three age groups, each with equal

numbers of girls and boys, were 20 4/2-year- olds (M = 55.6 months, range 49-61, SD = 3.8), 16 51/2-year-olds (M = 67.4 months, range 62-73, SD = 3.8), and 16 61'/-year-olds (M = 80.1 months, range 77-85, SD = 3.3). The youngest group attended a nursery school close to the school attended by the older groups.

Design and Procedure We gave children, individually, a pencil

and paper version of our People Task. It contained two tasks, a Rectangular and an Oblique Task. Children had drawings of the space characters and the board, and of the four positions, and had to mark the place where the characters could meet. We did not allow them to draw the lines that they had to extrapolate or to mark them in any way (e.g., with a finger). The drawings were centered on 15 x 21-cm sheets of paper with the board

represented by a square of sides 9 cm and the four positions within it at the corners of a square of sides 1.5 cm.

Rectangular Task.-This task, which was given to all age groups, had the same 16 problems as in Experiment 1. Half of the problems (two of each of the NN, NF, FN, and FF types; and, similarly, two of each of the four absolute distance types) were presented in the 0' condition and half in the 1800 condition, to each child. The problems in each condition and their order of presentation were counterbalanced across children in each age group.

Oblique Task.-The two older groups were also given a task where the orientation of one marker was at 450 to the axis (see Fig. 2). The obliquely oriented marker was on the horizontal axis for half of the problems and on the vertical axis for the rest. The other marker was perpendicular to its axis. So the extrapolated lines met at an angle of 450. There were four positions: two in line with the oblique marker, two with the nonoblique marker, and one (the correct one) in line with both.

The four types of problem were ones where the correct position was (a) nearer to both markers (NN), (b) nearer to the nonoblique but further from the oblique marker (NF), (c) further from the nonoblique but nearer to the oblique marker (FN), and (d) further from both (FF). It was not possible to manipulate the absolute distances of the correct position from the axes while maintaining the same overall dimensions for the problems as in the Rectangular Task. Instead, within each type of problem (NN, etc.), four different problems were created by placing the oblique marker (a) on the horizontal and (b) on the vertical axis and displaying the resulting problems once in the 0' and once in the 180' condition. Two sets of problems were created, so that across all problems of a given type (NN, etc.) four different positions within the square were correct equally often. In one (Set A), the 00 condition had markers on the left and bottom sides of the square; in the other (Set B) on the right and bottom sides (i.e., one set's problems were mirror images of the other's). Half of the children in each age group had Set A, the other half Set B. The order of problems within sets was counterbalanced across children in each age group.

RESULTS

The numbers of children who scored 12 or more out of 16 (significantly above chance) were eight out of20 4/2-, 11 out of 16 52-, and




Relative Distances N N

Set and Type : Set A . H/O

Display Condition : 0*

Relative Distances : NF

Set and Type : Set A. O/V

Display Condition : 180

Relative Distances FN

Set and Type : Set 8B O/V

Display Condition 0

Relative Distances FF

Set and Type : Set B HIO


FIG. 2.-Oblique problems in Experiment 2. Relative distances locate the correct position with respect to the nonoblique and oblique markers, in that order, e.g., NF = nearer to the nonoblique but further from the oblique marker. Set A has markers on the left and bottom sides of the square (0O condition) or on the right and top (180* condition); Set B problems are mirror images of Set A problems. H/O problems have a perpendicular marker on the horizontal axis and an oblique marker on the vertical axis; O/V problems have an oblique marker on the horizontal axis and a perpendicular marker on the vertical axis.

15 out of 16 61/?-year-olds on the Rectangular Task; and 11 out of 16 51/2- and 13 out of 16 61/2-year-olds on the Oblique Task. These numbers were significantly above chance (p < .001).

The 51/2- and 6'/2-year-olds were the same ages as two groups in Experiment 1, and their mean total scores in the Rectangular Task

(13.19 and 15.19, respectively) were compara- ble to those of their counterparts in the Peo- ple Task (12.88 and 14.94, respectively). A 2 (age) x 2 (experiment: 1 vs. 2) x 2 (relative horizontal distance: N vs. F) x 2 (relative vertical distance: N vs. F) analysis of variance, with repeated measures on the last two factors, produced no significant effects involving experiment (and neither did an analysis using




absolute distances). Thus the pictorial and paper methods of presenting the rectangular problems produced similar results.

We looked next at the factors affecting performance in Experiment 2. Preliminary analyses showed that sex, problem order, and display condition had no effects. Our first analysis compared performance in the two tasks by the two older groups. Since the rectangular and oblique problems could not be divided into types on the same basis, this analysis was conducted on the total scores (out of 16) and was a 2 (age) x 2 (task: rectangular vs. oblique) x 2 (order of tasks) analysis of variance, with repeated measures on the second factor. There was a significant main effect of task, F(1,28) = 15.00, p < .001, scores being higher in the Rectangular (M = 14.19) than in the Oblique (M = 12.72) Task; and a significant interaction of task with order, F(1,28) = 5.71, p < .05. The main effect of age was marginally significant, F(1,28) = 3.40, p < .08, the means for 51/2- and 61?-year- olds being 13.22 and 14.19, respectively.

To examine the interaction, we compared the two order groups' performance in the two tasks. Children who had the Rectangular Task first did not find the Rectangular Task (M = 13.69) significantly easier than the Oblique Task (M = 13.13), t(28) = 1.05; but those who had the Oblique Task first did significantly better on the Rectangular (M = 14.69) than on the Oblique (M = 12.31) Task, t(28) = 4.43, p < .001. We concluded that oblique coordinate markers raise more problems than rectangular ones, but that doing a rectangular task helps children with a subsequent oblique task.

Rectangular Task: Effects of age and type of problem.-This task was given to all three age groups in order to test for early developmental changes. We conducted two analyses, with the scores in one summed over conditions defined by relative distances, in the other over conditions defined by absolute distances. The first was a 3 (age) x 2 (relative horizontal distance: N vs. F) x 2 (relative vertical distance: N vs. F) analysis of variance with repeated measures on the last two factors. There was a significant main effect of age, F(2,49) = 8.27, p < .001; and a significant main effect of relative distance from the vertical marker, F(1,49) = 8.66, p < .01, the mean score being higher when the correct point was the nearer (3.48 out of four) than when it was the further (3.11) of two from this marker. The mean scores for 4/2-, 5?/-, and 61?-year- olds, respectively, were 2.79, 3.30, and 3.80; and post-hoc comparisons (Newman-Keuls, p < .01) indicated that the 6/2-year-olds were

significantly better than the 51/2-year-olds, who were significantly better than the 41/2- year-olds.

The second analysis was the same as the first, except that the second and third factors involved absolute distances. The only significant effect was the main effect of age, re- ported above. The absolute distances were all much smaller in the pencil and paper problems than in the slides in Experiment 1, and this may explain why we found effects of absolute distance in the first but not in the second experiment.

Oblique Task: Effects of age and type of problem.-We conducted a similar analysis of the Oblique Task. The scores were summed across conditions defined by relative distances from the nonoblique and oblique markers. In this 2 (age) x 2 (oblique set: A vs. B) x 2 (relative distance from the nonoblique marker: N vs. F) x 2 (relative distance from the oblique marker: N vs. F) analysis of variance, with repeated measures on the last two factors, the only significant effect was the interaction of relative distances from the two markers, F(1,28) = 6.30, p < .05. When the correct point was the nearer of two to the nonoblique marker, its relative distance from the oblique marker did not affect the scores (the means for NN and NF problems were identical, 3.38), but when the correct point was further from the nonoblique marker, the problems in which it was also further from the oblique marker were significantly more difficult (M = 2.69) than the problems in which it was not (M = 3.28), t(28) = 3.10, p < .01. So the problems in which the correct point was further from both markers (FF problems) were more difficult than all others.

Experiment 3 The next question was whether children

could use coordinates successfully in more complex displays. We conducted one further experiment with eight girls and eight boys who had participated in Experiment 2 (mean age in Experiment 3 = 71.9 months, range 66-81, SD = 4.71). The number of positions between which a child had to choose was 16, and we also varied the arrangement of these positions. In some problems they formed a grid, in others an irregular arrangement.

Design and Procedure There were two 16-problem tasks. In

each the child was given line drawings of two space characters and the board and of 16 positions. The tasks were explained in four practice trials with the same Lego materials as be-




Grid : Correct Position (2 . 3)

Display Condition 0

Irregular : Correct Position (4 2 )


Irregular :Correct Position (2 .3)

Display Condition 0

Irregular :Correct Position (4. )


FIG. 3.-Grid and irregular problems in Experiment 3. Correct positions are given in 1.5-cm steps from the horizontal and vertical markers, in that order, e.g. (2,3) = two steps from the horizontal and three from the vertical marker. 00 problems have markers on the left and bottom sides of the square, 1800 problems on the right and top.

fore. In the experimental trials the child had to mark the position where the characters could meet.

Both tasks had 16 problems, each with a 7.5-cm square containing 16 marked positions drawn in the center of an 11.5 x 15-cm sheet of paper. In the Grid Task, the positions formed a 4 x 4 array in which adjacent positions were separated by 1.5 cm, the same sep- aration as in the 2 x 2 arrays in Experiment 2. Thus the discriminations that children had to

make between the correct point and the near- est incorrect ones were the same as in Experi- ment 2, although the square array was slightly smaller. The two coordinate markers were at right angles to the axes, as shown in Figure 3. Each position was correct in one problem.

The Irregular Task had the same square frame, and the same 16 positions within it were each correct once. However, the positions of the 15 incorrect dots were altered to produce irregular arrays, some of which are




shown in Figure 3, in the following way. We started from the original grid and imposed the restriction that dots could not be moved more than 0.5 cm horizontally and/or vertically or moved closer to the frame than in the grid problems. Each incorrect position that was in line with a marker was left in line with it and either moved 0.5 cm along this line or not moved (with .5 probability of a move). If a dot was moved, there was a probability of .5 of moving it in each of the possible directions (e.g., left or right). There were thus four dots in line with each coordinate marker in every problem (to maintain comparability with the grid problems). Dots not in line with either marker in the original grid were equally likely to be (a) moved .5 cm along one dimen- sion, (b) moved .5 cm along both the horizontal and vertical dimensions, or (c) not moved. If they were moved, it was decided randomly what the direction(s) of the movement(s) would be (e.g., up or down). Finally, there were never more than two dots aligned in any row or column of an irregular array apart from the row and column that contained the correct position.

In each task half of the problems were presented to each child in the 00 condition and half in the 1800 condition, with counter- balancing across children as in Experiment 2.

RESULTS

The children did extremely well with these more complex displays. The mean number of correct answers was 15.63 out of 16 (SD = 0.89) for the Grid Task and 14.88 (SD = 2.25) for the Irregular Task. Someone choosing a position in line with just one coordinate would have a probability of .25 of being correct, so we compared the numbers of correct responses to those expected for a binomial distribution based on this probability. The probability of scoring 8 or more out of 16 on a task was .0271, and the probability of scoring 11 or more was .0003. In the Grid Task, 13 children scored 16 and the other three 15, 14, and 13. In the Irregular Task, 10 children scored 16, three scored 15, and one each scored 14, 11, and 8. All of these scores were significantly above chance.

The children's sex and the order and display condition (O0 or 1800) in which problems within each task were presented had no effect on performance. We conducted a 2 (task: grid vs. irregular) x 2 (order of tasks: grid first vs. irregular first) analysis of variance, with repeated measures on the first f`actor. There was a significant main effect of task, F(1,14) = 5.84, p < .05, and a significant task x order

interaction, F(1,14) = 6.17, p < .05. To examine the interaction, we compared scores on the two tasks for the two order groups. The scores on the Grid Task (M = 15.63) and the Irregular Task (M = 15.00) were not significantly different for the group who had the Grid Task first, t(14) = 2.03, but the group who had the Grid Task second scored significantly higher on the Grid Task (M = 15.63) than on the Irregular Task (M = 14.75), t(14) = 2.84, p < .05. This interaction seems to show an improvement, due to practice, in the Irregular but not in the Grid Task; but, alter- natively, there may have been ceiling effects in the Grid Task.

Discussion We have found that young children can

use two spatial coordinates to find a position in space and that they do so with increasing frequency between the ages of 4 and 6 years. Six-year-olds succeed even when the displays are complex ones.

Several implications follow from these results. One is that young children master a skill that is central to the understanding of Euclidean space quite early in life and a great deal earlier than Piaget et al.'s (1960) analysis of Euclidean understanding suggests. In their task, the children first had to realize that the problem they were set could be solved most accurately with the use of coordinates. Our experiments suggest that it was this realiza- tion that eluded them rather than the ability to use coordinates per se.

Young children may fail to use coordinates to solve Piaget et al.'s (1960) task for at least three distinct reasons. The first is that they may not know how to derive information about relative position from one small space (the marked rectangle) and apply it to another (the unmarked rectangle). Problems in which the two spaces differ in size or orientation (the latter in the case of Piaget et al.'s procedure) demand the same symbolic spatial skills as reading a map to find a position not on the map but in the space that it represents. Young children may perhaps lack these symbolic skills. Second, they may not know how to solve the practical problem of drawing or constructing (e.g., with the use of string) two straight lines to arrive at their point of intersection. A third possibility is that young children have both the symbolic and practical skills necessary to use coordinates to solve the task but do not realize that using coordinates will result in a more accurate solution than using simple perceptual estimates. Each of these possibilities can be tested. Our experi-




ments show that children can use coordinates to find a position when the symbolic and practical demands are removed.

The developmental improvement in children's ability to use coordinates should not detract from our discovery that a significant number of 4 V2-year-old children managed our Rectangular Task well above chance level. We cannot say whether the developmental change was due to a genuine increase in the ability to extrapolate two lines to a meeting point or, on the other hand, to some of the youngest children not understanding our task. But the ability of the 6-year-olds to find positions even in irregular displays with 16 choices is truly impressive. They should have no difficulty with the use of coordinates in finding a position on a map or a simple graph. An obvious next step in this research is to examine children's ability to read and interpret maps and graphs, using coordinate information.

Maps and graphs typically require the extrapolation of lines at right angles to the two axes. Our data suggest that the rectangular system is the most congenial to young children. The difference in difficulty between the Rectangular and Oblique Tasks is not at all surprising, because it is more difficult for children to copy nonperpendicular than perpendicular lines (Freeman, 1980; Ibbotson & Bryant, 1976; Olson, 1970). Moreover, people often believe that features of their environ- ments are more rectangular than they are (e.g., Chase, 1983; Lynch, 1960). In maps, coordinates impose a rectangular system on streets and other features of the landscape that themselves are not usually arranged rec- tangularly. The use of rectangular coordinates may take advantage of a natural tendency to process spatial information in terms of rectangular relationships.

In the first experiment, the hardest problems were those that involved extrapolation to a point at the greater of two absolute distances from both markers. In the second experiment, the rectangular problems were harder when they involved extrapolation past the first point in line with the vertical marker (far judgments with respect to the horizontal axis), and the oblique problems were harder when they involved extrapolation beyond the first point in line with both of the markers (far judgments with respect to both axes). This suggests that some of the errors were due to difficulties in extrapolating lines per se rather than in deter- mining the intersection. Some of the developmental differences and the greater difficulty of oblique problems may also be due to problems in extrapolating one or both lines. This

can be investigated with tasks that test children's ability to extrapolate a single straight line. The extrapolation of lines in our tasks may have been relatively easy for young children because the points between which they had to choose provided anchors at the far end of each line. One can investigate this by giv- ing children tasks in which only one end of the line to be extrapolated is marked (e.g., just the coordinate).

Whatever difficulties children have in extrapolating some types of lines, our experiments have shown that they usually can understand the information provided by spatial coordinates. They put together the information derived from two different axes to find a position in space. Their understanding of small spaces seems a great deal more ordered than is often suggested.

References

Abravanel, E. (1977). The figural simplicity of par- allel lines. Child Development, 48, 707-710.

Bluestein, N. L., & Acredolo, L. P. (1979). Map- reading ability in young children. Child Devel- opment, 50, 691-697.

Chase, W. G. (1983). Spatial representations of taxi drivers. In D. R. Rogers & J. A. Sloboda (Eds.), The acquisition of symbolic skills (pp. 391- 405). New York: Plenum.

De Lisi, R. (1983). Developmental and individual differences in children's representation of the horizontal coordinate. Merrill-Palmer Quar- terly, 29, 179-196.

Freeman, N. H. (1980). Strategies of representation in young children. New York: Academic Press.

Ibbotson, A., & Bryant, P. E. (1976). The perpendicular error and the vertical effect. Perception, 5, 319-326.

Liben, L. S. (1975). Long-term memory for pictures related to seriation, horizontality, and verti- cality concepts. Developmental Psychology, 11, 795-806.

Liben, L. S., & Golbeck, S. L. (1980). Sex differences in performance on Piagetian spatial tasks: Differences in competence or performance? Child Development, 51, 594-597.

Lynch, K. (1960). The image of the city. Cambridge, MA: M.I.T. Press.

Olson, D. R. (1970). Cognitive development. New York: Academic Press.

Piaget, J., & Inhelder, B. (1956). The child's conception of space. New York: Norton.

Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child's conception ofgeometry. New York: Norton.

Presson, C. C. (1982). The development of map- reading skills. Child Development, 53, 196-199.

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