Development in children's representations of transformations and movements

Development in Children’s Representations Transformations and Movements ANNE I-. DEAN. VKKI A. GKOS. AND Stm

This

study

tirst-grade

tested children

the

hypothehi\

to represent

middle. and end state constructed

states. On IS different by the experimenter

in the shape of an arc) and told “go best with” the experimenter‘s to choose Fourth-graders tinuous

the

that

basis upon predominantly

movements

fourth-grade

transformattonc

KUNLN

children

tend

as ordered

serie\

more

than

of beginning.

tasks. children were presented an object (e.g., a ball of clay or a pipe cleaner bent

to construct state(s).

which to relate constructed

or transformation\.

of

one or two other The\c instructions their states

wherea\

ob,ject allowed

states that children

states to the experimenter\’ that were component? of confirst-graders

predominantly

con-

structed slates that related to the experimenters’ on the haTis of figural feature\. In a second phase of the procedure. we re-pre\ented the \tatea made by children and ached them to reconstruct the associated experimenter-constructed state\ from memory. As predicted, memory wab better integrated states in transformation\ or movements states on the basis to similar findings

of perceptual features. The in other cognitive domains.

for

children than for

who children

\pontaneou~ly who related

results were diccu\\ed in relation r ,4X~ Ac:dcm,r ,YC\\. Illi

From the perspective of modern information-processing psychology. the title of Piaget and Inhelder’s ( 197I) book. M~~trrl Itnrrgrty itI the Child, is a misnomer. The experiments contained therein examined the changes that children anticipate in objects undergoing movements or transformations. For example, given a ball of clay, children were asked to anticipate and draw its appearance in beginning, middle. and end states of transformation to a sausage shape. Whether children used imagery. verbal processes. or both to accomplish this task was not an issue. Rather, the issue was the relationship between children’s anticipations This research in part by &ant We acknowledge

was based G-79-0092 the help

on a master‘s thesis from the National of the administration.

by the Institute faculty,

second ‘mthor of Education and students

and wa\ supnor-ted to the fir-s1 author. of the Immaculate

Conception and St, Joseph school\ for their cooperalion in procurrng subjects. and to Susan Lackey for her help in constructing te\t materials. Keque\t\ for r-cprint\ should be sent to Anne L. Dean. Department of Psychology. University of IJew Orlean\. New Orleans. IA 7014X.

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and their understanding of the logic underlying transformations. Preschool and early grade school children, according to Piaget’s theory (Piaget. Grize, Szeminska, & Bang, 1977), use one-way or functional logic of the form .v = f(y’) when reasoning about transformations. They may know that the action (xl of rolling a ball of clay back and forth on the table causes the clay to get longer (y). They may also know that the same action causes the clay to get thinner (v’). However. they may not understand the mutual reciprocal relation between the two outcomes; that is, that y = .fl.v’) and y’ = J-v) (see also Case, 1985, for a similar description). Thus, when drawing objects in beginning, middle. and end states of a transformation, these children might anticipate changes in only one dimension (e.g., length or width), or fail to coordinate anticipated changes in two dimensions. Piaget and Inhelder’s (1971) conclusions have been widely disputed (see reviews by Gelman, 1978, and Mandler, 1983). In particular, critics point to two studies by Marmor (1975, 1977) in which preschool children accurately judged whether two teddy bears (or ice-cream cones) in different orientations were the same or different. Decision times increased linearly as a function of the difference in orientation between the to-be-compared stimuli, thus suggesting that judgments were made after mentally rotating one object into congruence with the other. The requirements of Marmor’s task. however, differ from those of Piaget and Inhelder’s tasks. To succeed on Marmor’s task, children must mentally track the motion of at least a part of an object from initial to end state. There are no requirements, however, to differentiate the movement itself into a series of middle states, to reconstruct the appropriate spatial or logical relations among middle states, or to conserve the spatial relations among the objects’ parts as it rotates. All of these requirements must be met to succeed on Piaget and lnhelder’s tasks. Moreover, as exemplified in three studies by Dean and her associates. the conceptualization of the object in middle states of movement or transformation poses particular difficulty for children in kindergarten and first grade. In one study (Dean, Scherzer, & Chabaud, 1986). kindergartners who generated linear reaction time functions on a replication of Marmor’s teddybear task could not arrange states of the same rotation movement in order from 0” to 180”. In a second study (Dean & Deist. 1980). children judged whether movies accurately depicted how a horizontally oriented stick would look while moving from left to right. Kindergarten children’s judgments were predicated on how the stick looked in the end :state of movement, regardless of whether it expanded and contracted or otherwise transformed during the movement itself, while older children’s judgments took account of the movement and its end state. In a third study (Dean. Duhe, & Green, 1983). kindergartners and first-graders. in contrast to older children, were unable to keep track of the successive positions of

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a pointer undergoing an imagined rotation. as evidenced by flat. nonincrementing functions relating rotation times to anticipated rotation distances. The objective of the present study was to test directly the hypothesis that fourth-graders have more of a predisposition than first-graders to spontaneously conceptualize transformations as series of ordered beginning. middle, and end states. We used a conceptual preference task, in which children were shown an object such as a ball of clay or piece of wire bent like an arc. and then were asked to make something that “goes best with” the object they saw. Children were thus free to choose the basis upon which they would relate their own constructions to the presented object. If children have not yet differentiated middle states in their representations of movements or transformations. as previous studies suggest for first-graders, then they are unlikely to spontaneously conceptualize individual object states as component parts of movements or transformations. On the contrary, if children’s representations are differentiated into component states and integrated into a logical series. as previous studies suggest for fourth-graders. then individual component states might cue connecting states in a transformation series. In a second phase of this study. we represented childrens’ own constructed states. and then asked children to reconstruct the associated experimenter-constructed state(s) from memory. Previous Piagetian memory studies (Liben. 1975a. 1975b; Piaget & Inhelder. 1973) have shown that children who understand the logical relations implicit in a visual configuration remember the contiguration more accurately than children who do not understand the implicit logical relations. The presumed reason for this effect is that children who understand the logical relations encode the configuration transformationally. Indeed. that transformational encodings facilitate memory is one of the most robust findings in the cognitive developmental literature (Brown. Bransford, Ferrara. & Campione, 1983). In the present study. therefore, children’s performances in the memory phase provided a mean\ of independently validating our categorizations of children’s state constructions as transformations 01 nontransformations. METHOD Forty first-graders and 40 fourth-graders participated in the experiment proper. Mean ages were 6.3 years for first-graders and 9.3 years fat fourth-graders. An additional 38 first-graders, 20 fourth-graders, and IO adults participated in various control studies. All children attended parochial schools in middle-class neighborhoods in New Orleans, Louisiana.

MENTAL

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REPRESENTATION

n __

---_ ?

1 Slldlnq

3 Squares I

m ~~__ 1

3

2 Circle

Rc~laflon 3

0 ,/ __-_

___

f/ ,1 2

1 Addhvr

3

Classification 1:

b,

,an L __2

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FIG. purple.

I. Examples of tasks 13 = blue, G = green,

3

Tr,anqtes

2

presented in the one-state R = Red. Y = yellow.

II

condition:

0

= orange.

P L

There were three phases to the experiment: state construction, immediate memory, and long-term memory. In the state-construction phase, children made object states to go with those constructed by the experimenter. A total of 15 different tasks were used, each of which could be presented in three conditions-one state, logical two state. and illogical two state. Each child performed on 5 of the 15 tasks in the one-state condition, 5 in the logical two-state condition, and 5 in the illogical two-state condition. For individual children, tasks were assigned to conditions randomly. with the restriction that each task was performed under each condition by 10 children in both grades. In the memory phases, objects constructed by children in the state-construction phase were re-presented for each of the 15 tasks, and children were asked to reconstruct from memory the associated experimenter-constructed tasks.

All 15 taska are illustrated in Figs. I, 2, dnd * 3. Figure I shows 5 tasks as they appeared in the one-state condition. Fig. 2 shows 5 other tasks as they appeared in the logical two-state condition, and Fig. 3 shows

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Silcki

the remaining five tasks as they appeared in the illogical two-state condition. The illustrations in these figures represent only one-third of the total number of task x condition combinations, since each of the IS tasks could be presented in each of the three conditions. Conditions. In the one-.sttrtc condition. the experimenter showed the child a 55 x 3S-cm poster board with three horizontal lines, one drawn on the left, one in the middle, and one on the right side of the board. The lines were numbered I . 2. and 3. A construction made by the experimenter was located over Line 2. The child’s task wah to construct something on Line 1 and something else on Line 3 that “goes best” with the experimenter’s construction on Line 2. Tasks given in the onestate condition provided the most stringent test of children’h transformational processing, since the experimenter’s object state provided minimal cues for transformations or movements. In the lo~ic~ll ttrw-state condition. two experimenter-constructed states were presented on the poster board, one over Line I and one over Line 2. State 2 for each task was a continuation of a movement or transformation begun in State I. In the ilhgid tnw-.state condition, two experimenter-constructed rtatch were presented over Lines I and 2. as in the logical two-state condition.

MENTAL

Rotating

Squares

a: Bc~ 1

0 1

365

REPRESENTATION

:!

3

K ‘\b 2

3

FKJ. 3. Examples of tasks presented in the illogical two-state condition: B = blue. P = purple. 0 = orange. Y = yellow. G = green. In State I for pegboard task. two peg\ on left are red and two peg\ on right are blue. In State 2. pegs on left are blue and pegs on right are red.

The defining criterion for logical two-state tasks was that State 3 could not be a continuation of any single movement or transformation begun in State 1. Illogical two-state tasks assessed whether first- and fourthgraders differed in their bases for relating or for remembering states that could not be components of a movement or transformation. The one-state and logical two-state versions of a given task had the same State 7. Logical and illogical two-state versions of a given task had the same State 1, but a different State 2. Tmb rind nzatcricrls. The 15 tasks were mostly taken from experiments in Piaget and Inhelder’s (1971) Mrntd imngrvy in the child. The tasks, and the movements or transformations implied in the logical two-state condition, were the following. Arcs, involving the transformation of pipe cleaner from the shape of an arc to a straight line Slidiq square, involving the transposition of one square in relation to a second, stationary square Circle rotation. involving the rotation of a circle with smaller circles of different colors fixed to the perimeter

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Addithxc> c.lLtss~~fic.crtiotz, involving the completion of an additive classification matrix Rotating trimgle. involving the rotation of a triangle around a stationary triangle Fulling stic.l\.s, involving the rotation of a stick from 0” to W, around a stationary base pivot Six progression, involving the addition of strips of increasingly greatel length to the end of a series Rrrtzttirrg [~c)p/e. involving the overtaking and eventual passing of one figure by another C’rrrw~litrg .stttril, involving the movement of a snail around the inner circumference of a circle Figurr cotnplcticttz. involving the completion of an incomplete square Rofaritzg syrr~r, involving the rotation of a square around a stationary square PLI~~I. jbldin~, involving the folding of a piece of paper into halves. fourths, and eighths Rotrrlirzg clrhc. involving the turning of a cube with different colored sides Peg hoard. involving the stretching of ;I rubber band from a squat-c shape to a triagular shape Slzapc elong:rrtion, involving the rolling of a ball of clay into a sausage shape Materials for children’s own states were presented in small plastic drawers. Each drawer contained several replications of one of the components used to construct the experimenter’s state(s). Additional components not used by the experimenter were presented on only two task+--size progression (longer strips were needed) and additive classification (triangles of additional colors were needed). By limiting children’s choices of materials to those used in the experimenter’s states. we hoped to focus their attention on the spatial or logical relations among the component parts of states that could cue the implied movements or transformations. At the same time, the range of possible states that children could make was still quite extensive. given that they were free to use any of the components they wished. and to arrange them in whatever configuration they wished. Procrdurrs. Instructions to children on every task in the one-state condition were “1 (the experimenter) built this (pointing to the state OVCI Line 2) that goes on the middle line. Now you build something here (pointing to Line I) that goes best before the one I built. and one here (pointing to Line 3) that goes best after the one I built.” In the logical and illogical conditions, instructions were “I built this one (pointing to Line I). then I built this one (pointing to Line 71, now you build something

MENTALREPRESENTATION

267

here (pointing to Line 3) that goes best after these two that I built.” The use of the words before and after in these instructions emphasized temporal order in the construction of states, and thus was an additional cue for transformational processing. For each task, children were asked to explain why their own constructed states “went best” with those of the experimenter. Task presentation order was randomized for each child, with the restrictions that (a) the first two tasks received by every child-crawling snail and shape elongation-were two of the easiest for both first- and fourth-graders to transform (as determined by Control Study 3), and (b) each condition immediately followed each condition in the presentation order approximately the same number of times (e.g., one-state tasks immediately followed other one-state. logical two-state. and illogical twostate tasks approximately the same number of times). Scoring. For Piaget (1970). transforming means “intelligible change. which does not transform things beyond recognition at one stroke. and which always preserves invariance in certain respects” (pp. 70-31). For example. when a ball of clay is rolled back and forth, it becomes longer and thinner, while the amount of clay remains unchanged. These changes are intelligible because we know that the act of rolling does not change the amount of a substance, and that if one dimension changes as a result of the action, the other must change in a compensatory fashion. Thus. intelligibility is always relative to a particular action; given that action, changes and invariances across object states are either intelligible or not. In this study, childrens’ construction were scored as either strict transformations. loose transformations, or nontransformations. Mean percentage of agreement among three raters was 96.8%. Strict transformations conformed to Piaget’s notion of intelligible change: that is. an action could be identified that could fully account for the change(s) and invariances implied across states. Strict transformations were possible on one-state and logical two-state tasks, but highly improbable or impossible on illogical two-state tasks. As the examples of illogical tasks in Fig. 3 show. there is no action that could fully account for the changes and invariances between State 1 and State 2. On the rotating squares task, for example, two actions would be needed to change State 1 into State 2: 180” rotation of the whole figure, followed by clockwise rotation of the top square. On paper folding. State 1 could become State 2 by unfolding the paper and then folding up the bottom edge. Changes resulting from the concurrent effects of two independent actions, however, do not fit Piaget’s definition of an intelligible transformation, since features that would remain invariant under the influence of one action could be changed by the other. In such cases, the object in State 2 could appear to be a different object than in State 1. rather than the same object in a different state. Examples of strict transformations for 2 one-state tasks. Running people and Shape

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DEAN.

FIG. 4. transformations 0

=

orange

Transformations of these running

two

GROS.

AND

KUNEN

of two task% in the one-\t~~te tasks by lirstand fourth-grader\.

condition. G

I-igurc green

include\ running

all man:

man.

elongation, are shown in Fig. 4. These two tasks were chosen for illustration because they produced the greatest variety of different transformations. The action that accounts for the changes and invariances across states is indicated to the right of each construction. Like strict transformations. loose transformations showed changes in certain features of the object that could be understood as the product of a particular action. Unlike strict transformations, loose transformations either (a) failed to maintain something that would be maintained. given the implied action. (b) failed to change something that would change concurrently with the depicted change. or (c) failed to extend the logicalmathematical relations implied by the experimenter-constructed state(s). An example of (a) is a construction from the Sliding squares task. As

MENTAL

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269

seen in Fig. I, the experimenter-constructed State 2 in the one-state condition showed two squares with one slightly overlapping the other. One child’s loose transformation showed one, two, and three squares, respectively, on Lines 1, 2, and 3, implying a numerical transformation, but the overlapping relation between squares in State 2 was not maintained in State 3. An example of (b) comes from the circle rotation task. A rotation of the larger circle would necessarily entail a rotation of all of the smaller ones affixed to its perimeter. Some loose transformations implied rotation of some, but not all of the smaller circles. An example of(c) comes from the additive classification task. A loose transformation consisted of filling two, three and four spaces in the matrix in States 1, 2, and 3, respectively, while ignoring the logical basis for the choice of shapes. Since loose transformations were not logical in the sense implied by strict transformations, they were possible on illogical tasks as well as on one-state and logical two-state tasks. For example, states on the illogical version of the Falling stick task shown in Fig. 3 would be scored a loose transformation if the child constructed a State 3 that showed the top square even more rotated in a clockwise direction. This construction would imply a rotation action, while failing to maintain certain invariances. Constructions scored as nontransformations depicted states that could not be placed along a continuum of change; that is, no action could be identified that linked features across all three states. To illustrate. nontransformations of the Running people and Shape elongation tasks in the one-state condition are shown in Fig. 5. Although some constructions in the figure suggest an action that could change State 1 to State 2 (e.g., reversing the direction of movement for the running figures in Construction 3). a continuation of that same action would not produce State 3. Efforts to further categorize nontransformational constructions were unsuccessful due to their highly idiosyncratic nature. Certain relationships or changes between states were evident in some nontransformational constructions. These included state combinations (in which the two presented states were added together in some manner), partial changes (depicting some change between two but not across all three states), symmetries (in which State 1 was reproduced in State 3). and exchanges of one part for another (as when two different colored parts are exchanged for each other). The usefulness of conceptualizing nontransformations in terms of these categories was limited for three reasons: some categories categories were inapplicable on some tasks. some constructions could not be fit into any of the categories, and others could be fit into two or more of the categories. Interrater reliability in categorizing nontransformational constructions was below an acceptable level (90% or better agreement) on all except 3 of the 15 tasks.

270

DEAN.

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B1

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mmG1 -G --3 O-

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FIG. 5. Nontransformations nontransformabx of these

AND

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of two tak\ in the one-stats condition. tasks hy first- and fourth-grader\.

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41

The first memory phase immediately followed the state construction phase. Children were re-presented all 15 tasks in the same order as task5 were presented in the state construction phase. On each task, children were shown their own constructed state(s) and asked to reconstruct the associated experimenter-constructed state(s). The memory phase required about the same amount of time as the state construction phase (approximately 30 min). Thus, the time interval between performing on a task in the state construction phase. and reproducing the experimenter’s states for that same task in the first memory phase, was approximately the same for each of the 15 tasks. The procedures for the second memory phase, which occurred I week later. were identical to those in the first memory phase. Scoring. In the memory phases, constructions were scored as correct or incorrect. To qualify as a correct construction, Memory States I. 2. and 3 had to be correctly ordered states that were part of the same transformation underlying the original States 1, 2. and 3. Memory states did not have to be identical to (i.e., cuclidean replicas of) the original states with respect to exact angles or sizes. Percentage of agreement between two scorers on 50 randomly selected tasks was 100%.

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Studies

In one control study, the I5 illogical tasks were presented to IO firstgraders, 10 fourth-graders, and 10 adults. The objective was to determine whether the illogical tasks were truly illogical-i.e., whether a strict transformation not envisioned by the experimenter was possible on any of the tasks. The same instructions were given as in the state-construction phase of the experiment proper. No subject discovered an action that linked States I and 2 in a strict transformation on any of the I5 tasks. Since State 2 in the logical and illogical conditions differed figurally. an important question was whether or not they differed in the degree to which they could be reconstructed from a mental representation. In a second control study. IO first- and IO fourth-graders were given an immediate memory task, in which State 2 from both the logical and illogical conditions were presented successively for all 15 tasks. For each of the 30 states, the procedure was as follows: (a) children were asked to study the one state in front of them, (b) after 20 s, the state was removed from view, tc) all materials necessary to reconstruct the state were placed within reach, and td) the children were asked to reconstruct the state they had just viewed. The mean number of states reconstructed correctly was 14.6 (max = IS) for both conditions. A third control study was conducted to determine whether both firstand fourth-grade children could mentally represent the actions implied by the logical two-state tasks, if specifically instructed to do so by the experimenter. This information was necessary to control for the possibility that nontransformational performance on the state-construction tasks in the experiment proper might simply reflect unfamiliarity with the specific actions implied in our tasks. The procedure used in the third control study was. in our judgment, equivalent to Marmor’s (1975. 1977) in two ways. First. it required children to mentally represent an action on an object. Second, it did not require children to represent the appearance of the object in successive states of transformation. Thus. since preschoolers were successful in Marmor’s studies. we expected that first-graders would be successful in our control study. The subjects for Control Study 3 were 18 first-graders (mean age = 6.8 years) and IO fourth-graders (mean age = 10.3 years). Each child was presented all 15 tasks in the logical two-state condition in random order. For each, the experimenter constructed State I while saying, “First, I’m going to make this on Line I.” Next. the experimenter constructed State 2, while saying “Now. I’m going to change this (pointing to State 1) to make it look like this (pointing to State 3). Now you make something here on Line 3 that shows how this (pointing to State 2) would look if you changed it even more the way I did. This is how it looked in the beginning (pointing to State I). this is how it looked after 1 changed

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it (pointing to State 2). How would it look after you changed it even more that way (pointing to the space above Line 3)?” Children, therefore. saw an action performed by the experimenter and were simply required to perform that same action on State 2 to produce State 3. Constructions were scored as transformations or nontransformations, according to the same criteria as defined for the experiment proper. In the first grade, 100% of the children constructed transformations on 10 of the 15 tasks. For all except one of these 10 tasks (figure completion), the transformations were those intended when the tasks were designed. The five tasks on which first-graders did not exhibit perfect performances. and the percentages of children constructing transformations, were (a) rotating squares (72%), (b) additive classification (78%), (c) circle rotation (83’%), (d) cube rotation (830/o), and (e) running people (94%). In the fourth grade, 100% of the children constructed transformations on all except the rotating squares (809:) and the additive classification (80%) tasks. Chi-square tests indicated that the frequencies of first- and fourth-grade children who constructed transformations did not differ significantly for any of the 15 tasks. These results indicate that children in both grades can construct the end states of movements and transformations involved in the I5 logical two-state taskx. given specific instructions to do so. RESULTS Pres~ntution

Order

Numbers of strict transformations made by children who reccivcd ;I static, logical, or illogical task first were compared in a one-way ANOVA. The results were nonsignificant for both grades.

Grude und cwldition c:flL~ts. Table I shows the mean number5 of transformations made by children in the one-state. logical. and illogical conditions. In the top half of the table arc means for strict transformations. and in the bottom half are means for strict or loose transformations. A repeated measures ANOVA performed on transformations scored under the strict criteria yielded a significant grade x condition interaction. F(?. 116) = 11.47. p < .Ol. Simple F tests indicated that fourth-graders made significantly more transformations than first-graders in the one-state condition, F( I, 58) = 12.67, p < .Ol, and in the logical two-state condition. F( I, 58) = 28.67. p < .OOl. As intended, very few first- or fourth-graders discovered transformations for illogical tasks. For both groups, NewmanKeuls tests indicated that more children transformed tasks in the logical two-state condition than in the one-state condition, I, -:I .Ol, and more children transformed tasks in the one-state than in the illogical two-state

MENTAL

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REPRESENTATION

TABLE I NL~MBERS OF TRANSFORMATIONS Condition

Grade

One-state

Logical Strict

I 4

1.3(1.7) 2.3 1.3)

I ‘l

1.31.1) 1.7t I .‘I)

Strict

No/r.

Numbers

in parentheses

two

state

transformations 1.9(1.2) 331.1)

or loose

are standard

transformation5 2.31.3) 3.7(l.l)

Illogic~ll

two

state

O.O(O.~l 0. l(O.3) 0.3(0.5) 0.3(0.5)

deviations.

condition, p < .Ol. Comparable analyses performed on transformations scored under the strict or loose criteria yielded an identical pattern of results. Children were also categorized according to whether they predominantly transformed tasks. “Predominant” was defined as transformations on at least three of the five tasks presented in a given condition. Using the strict transformation criterion, frequencies of predominant transformers for the one-state, logical two-state. and illogical two-state conditions, respectively, were 3, 10, and 0 for first-graders and 16, 13. and 0 for fourth-graders. A x2 test indicated that more fourth-graders than firstgraders predominantly transformed tasks in the one-state condition, x? I ) = 13.09, p < ,001, and in the logical two-state condition, x’(l) = 11.45. p < .Ol. Sign tests indicated that more children in the both grades predominantly transformed logical two-state tasks than one-state tasks, p, < .05. Using strict or loose criteria, frequencies of predominant transformers were 6, 13, and 0 for first-graders, and 19, 74, and 0 for fourthgraders, in the one-state, logical two-state. and illogical two-state conditions. respectively. Chi-square and sign tests yielded the same pattern of results as for strict transformations only. Thus, in subsequent analyses, only transformations scored under the strict criteria were examined. Tusk rjfjcrcts. Table 2 shows the frequencies of first- and fourth-grade children who transformed each of the I5 tasks. The table shows that frequencies varied widely across the I.5 tasks, but were highly correlated for first- and fourth-graders in both conditions, r( 13) = .%, p < .OOl for the one-state condition, and 1.(13) = .65, p < .Ol for the logical twostate condition. In the one-state condition, fourth-graders transformed 14 of the IS tasks more frequently than first-graders; and in the logical two-state condition, fourth-graders transformed 13 of the I5 tasks more frequently than first-graders. Thus, grade differences in propensity to transform were general across tasks.

DEAN.

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.One Firzt grade

T2tS-k Crawling Shape Running Arc Sliding

snail elongntion people

\late Fourth grade

7 h 5 IO

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IO 0

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Rotating

circle

0

I,

Rotating Rotating

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II

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1

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II

cube

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II) s

Paper folding Figure completion Size progression

RopAting

square

IO

I

Mean

The IS tasks in this study we.re not selected with the goal of contrasting transformation types. Nevertheless. some interesting trends can be noled in Table 2. Excluding the crawling snail5 task. the tasks most frequently transformed involved either a shape transformation (shape elongation. arc, paper folding. and figure completion) or a transposition movement (running people, sliding squares). With the exception of the peg board task, the tasks least frequently transformed involved either a rotation movement (rotating circle. triangle. sticks. squares. and cubes) or a logical mathematical transformation (size progression and additive classification). Possible reasons for these differences include the rclativc complexities of rotation versus transposition movements-i.c., tran5positions that involve change with respect to only one axis. comparccl to two axes for rotation movements-and the relative familiarity or rcallife quality of the materials-i.e.. crawling snails. clay. running people. and bent wires (arc task) all involve somewhat familiar ob.iect% OI- event\ which may cue particular transformations. The most important finding for this study, however, is that specific task variables had similar effect\ for children in both grade>. tr&zg f<[#~~(.t.r. Given the cuing effect of logical two-s(;~lc [ashs. one might argue that childrcn‘h transformation\ of one-state t:il;ks M crc ncjl

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REPRESENTATION

175

truly spontaneous, since, for every child. most one-state tasks were preceded in the presentation order by at least one logical two-state task. The relationship between children’s transformations of one-state tasks and presentation order was examined in two ANOVAs. First, a 2(grade) x 3tpresentation order) ANOVA examined the proportions of one-state tasks transformed as a function of whether they were among the first 5 tasks, second 5 tasks, or third 5 tasks presented. Mean proportions of transformed one-state tasks in these three presentation orders were 27, 23, and 21 for first-graders, and 51, 37, and 45 for fourht-graders. The ANOVA yielded a significant grade main effect, E‘( I. 58) = 7.78, p < .Ol, but no significant effects involving presentation order. Second, a 3tgrade) x 3tpresentation order) ANOVA examined the proportions of transformed one-state tasks as a function of the number of logical twostate tasks transformed prior to the presentation of the one-state task (0, I to 2, 3 to 4). Mean proportions of transformed one-state tasks in the three presentation orders were 19, 29. and 30 for first-graders, and 44,45. and 56 for fourth-graders. The ANOVA yielded a significant main effect for grade, F( I, 15) = 4.30, p < .05, but no significant effects involving presentation order. These results suggest. therefore, that onestate task transformations were not attributable to the cue provided by logical two-state tasks, for either grade.

In the first memory phase. first-graders remembered 47%, 47%, and 30% of one-state, logical two-state, and illogical two-state tasks. respectively, compared to 64%. 75%, and 29% for fourth-graders. In the second memory phase. percentages were 47%. 44%. and 18%~for firstgraders. compared to 65%, 74%. and 25% for fourth-graders. A 2fgrdde) x 3tcondition) x Ztmemory phase) ANOVA performed on numbers of remembered tasks yielded a significant grade x condition interaction, Ft?. 116) = 4.21, p < .05. Newman-Keuls tests indicated that fourthgraders remembered more tasks than first-graders in the one-state condition, p < .Ol, and in the logical two-state condition, p < .05. Memory scores for first- and fourth-graders did not differ on illogical tasks. Grade differences in memory, however, were a reflection of the relative numbers of children in the two grades who transformed tasks in the state construction phase. Table 3 shows the proportions of transformed and nontransformed one-state and logical two-state tasks remembered by first- and fourth-graders at the two memory phases. Proportions for both conditions were based on the numbers of children who had memory scores for both transformed and nontransformed tasks. In the one-state condition. 1I first-graders were excluded, 1 who transformed all five tasks and IO who transformed none. Comparable frequencies for fourthgraders were I and 3. In the logical two-state condition, 3 first-graders

276

DEAN.

GROS.

AND

TASLE

PROPORTIONS OF TRANSFORMED

KUNEN

3

AND NONTKANSFOKMED Memory First

TrXl5 One state Grade I (n Grade Logical

T~slts phase

grade Nontran\

RI-MI:MH~RI:I)

Second Trail\

grade Nontrans

=- 19)

4 tn = 26) two state

were excluded, I who transformed all and 2 who transformed no tasks. Comparable frequencies for fourth-graders were 5 and I. A 2(grade) x 2(memory phase) x 3transformation. no transformation) ANOVA performed on proportions of remembered one-state tasks indicated that transformed tasks were remembered significantly more often than nontransformed tasks by children in both grades. F( I, 43) = 226.97. 11 < ,001. A comparable result was obtained for logical two-state tasks. E’( I. 49) = 778.08, p < ,001, A significant memory phase main effect was also obtained for logical two-state tasks. I;( 1 j 4Y) -7 T.73, p : .Oi. with more tasks remembered immediately after the state-construction phase than I week later. This latter result contrasts with the finding 01‘ no memory phase effect when numbers of tasks remembered were analyzed independently of state construction strategies. This discrepancy can bc explained by the elimination in Table 3 of children who either transformed all five tasks. or who transformed no tasks. In effect. this procedure eliminated children who had extreme memory scores ,tt both mcmor! phases. Children whose scores were le\s extrcmc tended to remcmbermore tasks at Phase I than at Phase 2. One question not answered by the previous analyses wa\ whether children’s memory for transformed or nontransformed task> differed in the one-state and logical two-state conditions. We expected that children would remember more one-state than logical two-state taskx. since oni), one state was to be remembered in the one-state condition. compared to two in the logical two-state condition. To address this question. WC” performed a 2(grade) i< 3transformation vs no transformation) ). 7(onc state vs logical two-state condition) ‘* 2(memory phase) ANOVA. using only those children who had memory hcorcs for both transformed and nontransformed tasks in both conditions (II == I6 in both grades). Again. the results indicated that more transformed tasks were remembered tO3Y 1 than nontransformed tasks t27’;0, F‘( I. 30) mr~295./7 1’ 001 ;lnA illa!

MENTAL

MFAN

PERCENTAGES

277

REPRESENTATION

TABLE REMEMBERED

4 AS A FUNCTION

EXPLANATION

CATEGORY

OF TASKS

Not

OT: TRANSFORMATION

AND

transformed

Explained

Not .-

Transformed, explained

explained ~~~

First memory Grade 1 (12 = 24) Grade Second Grade Grade No/<,. columna.

4 (n = 23) memory I (n 4 (n Children

= 24) = 23) included

in

this

table

are

those

who

had

vzores

in

each

of the

three

tasks were remembered more often in the first memory phase (637~) than in the second memory phase (57%). F( 1, 30) = 4.93, p < .05. NO significant grade or condition effects were obtained. These results, therefore. suggest that when children transformed. they could remember two slates as easily as one. When children did not transform. they had as much difficulty remembering one state as two states. The relationship hetn~cen \,erbal c~xplunations crr~I ttlcttlory. Besides making states to “go best” with the experimenter’s, a second form of elaborative activity performed by children in this study was to verbally explain why their own constructed states went best with the experimenter’s To the extent that explaining a task makes the basis for relating states more explicit in the child’s mind. explained tasks might be remembered better than tasks not explained. Alternatively. children who had a better idea of what they were making on the state construction tasks may have been better able to explain and to remember. To examine the relationship between explaining and memory, each state construction was scored as explained or not explained. Explanations were defined as any statement describing how the child’s construction was related to the experimenter’s. Examples from one first-grade child’s protocol are, (paper folding) “It gets smaller and smaller,” (rotating squares) ‘-the blue one turns around,” (cube rotation) “All have orange tops,” (figure completion) “Little L on (Line) 3. big L on (line) 2.” Statements not scored as explanations were don’t knows, “they’re all different,” “I just knew.” “I guessed.” and so on. Three scorers categorized children’s explanations with 100% agreement. First-graders explained an average of 99%’ of their transformed tasks and 67% of their nontransformed tasks. Corresponding percentages for fourth-graders were 98 and 79%. Table 4 shows percentages of tasks remembered as a function of whether they were explained or transformed.

278

DEAN,

GROS.

AND

KUNEN

Since almost all transformed tasks were explained. memory scores for transformed, not explained tasks are not included in the table. For nontransformed tasks, a 2(grade) x ‘(memory phase) x Z(explanation, no explanation) ANOVA yielded only a significant main effect for explanation category, F( I, 45) = 10.91, p < .01. Children in both grades remembered more nontransformed tasks that they explained than tasks they did not explain, at both memory phases. An ANOVA comparing explanation categories for transformed tasks was not possible. However. for both grades, significantly higher proportions of tasks wet-e remembered that were both transformed and explained than tasks that were explained but not transformed, t;(l. 45) = 474.98, 1I < ,001. Fourth-graders remembered more explained tasks than first-graders, regardless of whether or not they were transformed. F( I, 4.5) = 5.17. 1’ J .05. In the present paper, we hypothcsired that fourth-grade children arc more likely than first-graders to spontaneously conceptualize movements and transformations as ordered series of beginning, middle. and end states. This hypothesis was supported by children’s performances on a conceptual preference task, on which they constructed states to go best with experimenter-constructed states. Fourth-graders predominantly interpreted the experimenter’s states as component parts of movements and transformations. First-graders attended to the perceptual feature5 ot the experimenter’s states. Although some tasks were easier to transform than others for both grades, fourth-graders transformed virtually all tasks more often than first-graders. Thus. although task variables do affect performance. they do not account for performance differences between grades. That first-graders attended to perceptual t’eatures of states on the stateconstruction task does not necessarily mean that their representations of transformations are static. The findings of several studies (Dean si Deist, 1980: Marmor. 1975. 1977: Piaget & Inhelder. I9711 have shown that first-graders and even younger children can often gcncrate a correct end state of a transformation or movement. This result M’:I~ replicated in our third control study. Some clues as to the quality of childrcns’ representations that can produce correct end state:, but not ;I logically ordered series of middle and end states. comes from research in two other domains of cognitive development. Nelson ( 1983). for example. proposes that even very young children (e.g.. 2-year-olda) can represent everyday, familiar event sequences or scripts such as taking a bath OI playing ball. These events, in the mind of an adult. consist of parts such as actors, objects. roles. and so on. Nelson argues. however. that for the young child. events are first rcpresentcd as wholes rather than part constructions. Bath always implies going into the bathroom, taking off clothes. and turning on the water. According to Nelson (p. 1361..

MENTAL. Conceptualizing objects. actions. be manipulated) component parts the beginning of

REPRESENTATION

the parts of the script and properties in order requires a further step and the relations that the conceptual system

379

as separate entities (Le.. separating out to treat them as mental objects th:at can of analyzing the original script into its hold between them. This process marks per SC.

Similarly, Fuson, Richards, and Briars (1982) describe development in children’s knowledge of the count-word sequence as a process of differentiating separate words from an undifferentiated string, and then reintegrating them into a logically ordered series. When children are first learning to count, number word sequences are produced as global, unfrom which individual words cannot differentiated wholes or “strings” be extracted. An early indication of children’s differentiation of the count sequence is their ability to generate “Comes After” relations--i.e.. to name a number word that comes after a given number word within their counting range. At first, however. this task must be accomplilshed by beginning at “1” and counting up until the given number word is reached: the sequence cannot be entered at a point other than “I”. The count sequence at this level thus retains the quality of an unbreakable chain, with the exception that stopping points or final words in the s,equence retain a privileged status. Later in development, the chain becomes breakable; children can begin counting at any word in a forward or backward direction, and given any word in the sequence, can name the words that come before and the words that come after. In both Nelson’s and Fuson et al.‘s descriptions, as in the present study, children’s earliest representations are depicted as dynamic, in that a sequence can be produced from start to finish, but holistic. in that component parts are not separable from the sequence itself. In all three descriptions. later development is a process of differentiating and reintegrating the component parts into a structure. With respect to the representation of movements and transformations, Piagetian theory (Piaget et al., 1977: Piaget & Inhelder I971) suggests a mechanism underlying this dcvclopmental process, i.e.. the shift from functional to reversible logic. Functional logic-i.e., knowledge that a particular action products one or more outcomes-may be sufficient for the production of holistic representations of movements or transformations with defined end states. In contrast, concrete operational logic-i.e., knowledge of the reciprocal relations among outcomes-may be necessary for the construction of logically ordered series of middle and end-states. Perhaps a comparable mechanism underlies the development of event and number sequence representations. That children develop more differentiated and integrated representational knowledge helps to explain why preschool and older children often differ in their performances on a wide range of tasks. Many studies (Gelman 1978: Mandler 1983) have shown that. under optimal conditions, pre-

280

DEAN.

GKOS.

AND

KUNEN

schoolers are often able to execute efficient problem-solving strategies. However, left to their own devices, preschoolers are less likely to spontaneously choose efficient strategies than older children. At least part of the reason for this difference. as Kuhn (1984) suggests. could be that spontaneous strategy selection requires knowledge not only of how to execute the strategy. but also knowledge about the strategy itself-i.c.. its component parts and how they arc interrelated. This latter type ot metaknowledge can provide a basis for choosing a strategy over other well-practiced, but less efficient. strategies. In the present study, we argue that first-graders’ failure to spontaneously choose transformational strategies that they were demonstrably capable of executing reflects a lack of differentiated explicit understanding of transformations. Similarly, Michie (1984) demonstrated that development from global to elaborated conceptions of the count-word sequence is accompanied by a shift from reliance on spatial or perceptual cues in making equivalence judgments to the spontaneous use of counting to make such judgments. Further, the well-known hnding that preschool children do not spontaneously catcgorizc objects taxonomically could reflect a developmental level in which object concepts arc still undifferentiated from the event structure in which they arc originally cmbcddcd (Nelson 1983). Each of these examples suggests a relationship between level of differentiation in children’s representations of strategies or processing modes, and the extent to which strategies or processing modes are spontaneously applied to problem-solving tasks. The effectiveness of spontaneously selected transformational encoding strategies for memory has been amply demonstrated in many studies (Brown et al.. 1983; Piaget & Inhclder. 1973). That children in the present study who spontaneously transformed states remembered them significantly more than children who did not. thcreforc. only scrvcs to validate the qualitative difference between constructions scored as transformations and nontransformations, and to confirm the findings of previous research. Of particular interest in the present study. however. was the degree to which transformational strategies facilitated children’s memory for states. Transforming greatly increased children’s abilities to remember over and beyond the facilitating effects of verbally explaining constructions. Almost all children who transformed tasks explained their constructions. thereby making it impossible to assess individually the contributions of transforming and verbal explanations to memory. However. the finding that children explained nearly all of their transformations. but did not explain many of their nontransformations. in itself lends support to the argument that children who spontaneously transformed states had explicit understanding of the transformations and movements they depicted.

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