Title: Figurative versus operative cues in the acquisition of conservation of number /
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Title: Figurative versus operative cues in the acquisition of conservation of number /
Physical Description: xi, 106 leaves : ill. ; 28 cm.
Language: English
Creator: Clifton, Charles Russell, 1946-
Publication Date: 1976
Copyright Date: 1976
 Subjects
Subject: Number concept   ( lcsh )
Conservation (Psychology)   ( lcsh )
Cognition   ( lcsh )
Psychology thesis Ph. D   ( lcsh )
Dissertations, Academic -- Psychology -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 103-105.
Statement of Responsibility: by Charles Russell Clifton, Jr.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098113
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000178641
oclc - 03122971
notis - AAU5153

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FIGURATIVE VERSUS OPERATIVE CUES IN THE
ACQUISITION OF CONSERVATION OF NUMBER








By

CHA;RLES:RUS~SELL CLIFTON, IR.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHTILOSOPHY

















UNIVERSITY OF FLORIDA




















































































~


To my children, Ashley and Joshua.

May your world facilitate your development through
developmental research such as this.















ACKNOWLEDGEMENTS


Several people have earned my special appreciation for their

support during the preparation and execution of this research.

Dr. Jacquelin Goldman, my chairperson, has provided both warm,

supportive counsel and firm suasion throughout my graduate education

and particularly during the long course of this study. Her sensitively

measured portions of support and discipline have sustained my investi-

gatory spirit. I value her friendship, wisdom, and clinical skills

highly. She is a worthy and beloved mentor.

Dr. Harry Grater has also provided inspiration and support for

me, particularly during my earlier years in graduate school. His

teaching, supervision, and modeling, especially in the practice of

psychotherapy, have shaped my perceptions and skills in important ways.

His cooperation and support despite a very busy schedule are deeply

appreciated.

Another very busy person who nonetheless has been supportive

throughout this process is Dr. Betty Siegel. Although my contact with

her has been somewhat more limited than with other committee members,

her easy, genuine enthusiasm and the strong interest which she has

shown in the optimum development of children have influenced me more

than she knows.

Dr. Franz Epting has been one of the most important sources of

inspiration regarding the potential humanness and the humaneness of


iii










research in my graduate car-eer. He has encouraged my initial interest

in understanding and describing people as they live their lives without

manipulating them extensively and artificially as so much research

seems to do.

Dr. Barry Lester was a godsend after Dr. Madeline Ramey departed

from the University and from my committee. Stepping in before I had

completed my proposal for this research, he offered many valuable

criticisms and suggestions based on his own experience with Piagetian

research.

Dr. Judith Phillis and Dr. Douglas Starr, who supervise and support

my work at the North Central Florida Community Mental Health Center,

have offered continued encouragement and have been very understanding

when my energies for work were sometimes sorely taxed by the demands of

this task.

The patience and love and very hard work of my wife Sandra were

absolutely essential factors in the completion of this research. She

has provided unwaivering support for me in this task in spite of the

demands of her job and her roles as wife and mother.

Finally, in moments of greatest despair, the joyful smile and

trusting love of my daughter Ashley have warmed and energized me and

renewed my sense of the worth of children in human life and its

struggles.

I would be seriously neglectful if I did not express my deep

gratitude to the principals, teachers, children, and parents of

Littlewood Elementary School, Metcalfe Elementary School, P.K. Yonge

Laboratory School, Andromeda School, and Mother Goose Nursery. Without

their participation, the research would not have been possible.

















TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS .. .. .. .. .. .. . .. . iii


LIST OF TABLES .. .. .. .. .. .. .. . .. . vii


ABSTRACT ... .. .. ... .. .. .. .. .. .. .. ix


INTRODUCTION .. .. .. .. .. .. .. . ... .. .. .. 1


Operative Aspects of Knowledge. ... .. ... .. 3

Sensorimotor period. .. .. .. .. .. .. .. .. 3


Preoperational period. ... .. .. .. .. .. 7

Period of concrete operational thinking. . .. ... 12


Figurative Aspects of Knowledge .. .. .. .. . .. 16

The nature of perception ... .. .. ... .. 17


The origin and development of perception .. .. .. 19


The function of perception .. .. .. .. .. .. 23


Comparison of Operative and Figurative Aspects of
Knowledge .. .. .. .. .. .. .. ... . . 24

Conservation of Number. . .. .. .. .. ... .. .. 26


Criteria for judging the presence of conservation. . 28


Criteria for judging the level of initial cognitive
development. . .... ... .. . ... . 30


Components of conservation . .. .. .. .. ... 31


Hypotheses. .. .. .. .. .. ... .. .. . . 38

METHOD ... .. .. .. ... .. .. .. .. .. . 41


Subjects. .. ... .. .. .. .. . . . . 42

Procedures and Materials. .. .. .. .. .. .. . 44


Screening. ... .. .. .. .. . . . 44





TABLE OF CONTENTS
(Continued)


Page

Pretests for conservation. .. ... .. .. .. 48

Criteria for determining the initial level of
cognitive development. .. .. .. ... .. .. 49

Posttesting for conservation ... .. .. .. .. 51

Planned experience in conservation of number .. .. 52

Controls for experimenter bias .. ... .. .. 56

Judgment of the adequacy of responses. .. .. .. 57

RESULTS. .. .. .. .. .. .. .. ... . . .. 5

Interjudge Agreement in Classifying Subjects' Responses . 58

Analysis of Dependent Variables ... .. .. . . 59

Rationale for nonparametric analysis .. .. .. .. 59

Experience group effects .. .. ... . . 63

Effects of initial level of cognitive development. . 79

Summary of hypothesis testing. .. . .. .. .. 79

DISCUSSION .. .. .. .. ... . . . . . . 81

APPENDIX A -- MATERIALS AND PROCEDURES FOR SCREENING TO ASSESS
BASIC CONCEPTS . ... .. .. ... .. .. 90

APPENDIX B -- DISPLAYS FOR RECOGNITION OF NUMBER EQUIVALENCE . 94

APPENDIX C -- DISPLAYS FOR CONSTRUCTION OF NUMBER EQUIVALENCE. 96

APPENDIX D -- MATERIALS AND PROCEDURES FOR PROTESTS OF CON-
SERVATION. ... .. .. .. .. .. .. .. 97

APPENDIX E -- PROCEDURE FOR THE HIGH OPERATIVE CUES EXPERIENCE 100

REFERENCES .. .. .. .. .. .. .. ... . . . . 103

BIOGRAPHICAL SKETCH. .. .. .. ... ... .. .. . 106
















LIST OF TABLES


Table Pg

1 Demographic Characteristics of the Sample by Level of
Cognitive Development and by Planned Experience Group. . 43

2 Items Used in Planned Experience Groups for Conservation
of Number. .. .. . .. . . .. .. .. .53

3 Means (M) and Standard Deviations (SD) of the Number
of Correct Judgment and Explanation Conservation
Responses. . .. .. .. . . . .. . 60

4 Means (M) and Standard Deviations (SD) of the Number of
Correct Judgment Only Conservation Responses .. .. .. 61

5 Means (M) and Standard Deviations (SD) for Number of
Trials to Acquisition of Conservation of Number. .. .. 62

6 Frequency Distributions of Raw Scores for the Conserva-
tion of Number Posttes;ts -- Judgment and Explanation .. 64

7 Comparison z Scores for Experience Groups for Conserva-
tion of Number -- Judgment and Explanation .. .. .. 66

8 Frequency Distributions of Raw Scores for the Conserva-
tion of Number Posttests -- Judgment Only. .. .. .. 68

9 Comparison z Scores for Experience Groups for Conserva-
tion of Number -- Judgment Only. .. .. .. .. .. 69

10 Frequency Distributions of Raw Scores for the Conserva-
tion of Length Posttests -- Judgment and Explanation .. 70

11 Comparison z Scores for Experience Groups for Conserva-
tion of Length -- Judgment and Explanation .. .. .. 71

12 Frequency Distributions of Raw Scores for the Conserva-
tion of Length Posttests -- Judgment Only. .. .. .. 73

13 Comparison z Scores for Experience Groups for Conserva-
tion of Length -- Judgment Only. . .. .. .. .. ... 74

14 Frequency Distributions of Raw Scores of the Conserva-
tion of Liquid Posttests -- Judgment and Explanation .. 75





LIST OF TABLES
(Continued)

Table Pg

15 Frequency Distributions of Raw Scores for the Conserva-
tion of Liquid Posttests -- Judgment Only. .. ... 76

16 Frequency Distributions of Raw Scores for Number of
Trials to Acquisition. .................. 77

17 Comparison z Scores for Experience Groups for Number
of Trials to Acquisition .. .. ... .. .. .. .. 78





Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



FIGURATIVE VERSUS OPERATIVE CUES IN THE
ACQUISITION OF CONSERVATION OF NUMBER

By

Charles Russell Clifton, Jr.

December, 1976

Chairperson: Jacquelin R. Goldman
Major Department: Psychology

Sixty children, ages 45 to 88 months, participated first in a

screening procedure designed to ensure their understanding of basic

relational terms and concepts, determine their ability to conserve,

and classify the children into levels of cognitive development. Then,

nonconservers and intuitive conservers were assigned randomly to one

of three planned experience groups. The High Figurative Cues experience

used minimal cognitive cues and enhanced perceptual cues. The Control

group experienced the same low cognitive cues as the High Figurative

Cues group with minimized perceptual cues as well. The High Operative

Cues experience included minimal perceptual cues with enhanced cognitive

(operative) cues in the form of verbal rule instruction and sensori-

notor rehearsal. After participating in their respective planned

experiences, all children were posttested immediately and after 10 to

14 days for conservation of number, length, and liquid.

Three hypotheses were tested. First, intuitive conservers should

have achieved higher scores than nonconservers on both immediate and





delayed posttests. Second, because highly salient perceptual cues have

been identified as distracting and verbal rule instruction and behav-

ioral rehearsal have been identified as facilitative, the High

Operative Cues group should have achieved the highest scores of the

three experience groups; the Control group would have earned inter-

mediate posttest scores; and the High Figurative Cues group would

demonstrate the lowest posttest scores. Finally, it was predicted

that intuitive conservers would require fewer trials to acquire the

conservation concept than nonconservers and that the High Operative

Cues group would require the fewest trials among the planned experience

groups.

A multivariate extension of the Friedman nonparametric analysis

of variance was used to analyze the data with two different criteria

for defining a conserving response -- judgment with explanation and

judgment only. Hypothesis 1 received no support. Hypothesis 2 was

partially supported. The High Operative Cues group produced a distri-

bution of significantly higher scores using judgments with explanations

and using judgments only for conservation of number on the immediate

posttest than did the High Figurative Cues or Control groups and a

distribution of significantly higher conservation.of number scores on

the delayed posttest than the High Figurative Cues group for judgment-

with-explanation conservation. For conserving responses using judgment

only, scores for the High Operative Cues group were significantly

higher than scores of either the High Figurative Cues group or the

Control group on the delayed posttest. There was no significant

difference between the distributions of the High Figurative Cues group

and the Control group for conservation of number responses using either










criterion. The distributions of conservation of length responses using

both judgments with explanations and judgments only were also higher

for the High Operative Cues group than for the other two experience

groups on the immediate posttest. The delayed posttest for conserva-

tion of length demonstrated higher scores for the High Operative Cues

group than for the Control group for both judgment-with-explanation

and judgment-only responses. Again, there were no significant dif-

ferences between the High Figurative Cues and Control groups. Finally,

the first part of the last hypothesis was not supported, but the

second part was supported.

Results confirmed previously demonstrated effectiveness of

experience with relevant cognitive cues. There was no support for the

expectation that with minimal supportive cognitive cues in the Control

group and the High Figurative Cues group, the enhanced perceptual cues

in the High Figurative Cues experience would interfere with acquisition

of the concept of conservation.















INTRODUCTION


In his efforts to define and explain the ontogeny of individual

knowledge, Jean Piaget has described two interdependent systems through

which people acquire knowledge about their universe -- an operative

system of knowing and a figurative system of knowing. The operative

system consists of early sensorimotor behaviors which are the founda-

tion for rational intelligence and of later representational and

rational intellectual operations. The figurative system of knowing

includes perceptual processes, mental imagery, and memory. Research

usually centers on one system or the other. Yet in theory, Piaget and

his colleagues (Piaget, 1967; Piaget, 1969; Piaget & Inhelder, 1969)

have underscored the interdependence of figurative aspects and operative

aspects of knowledge. Recently, American researchers (Youniss &

Dennison, 1971) have successfully demonstrated such interdependence

of figurative and operative aspects of knowing in a single but precise

study. In an attempt to discover the relative contributions of

figurative structures and operative structures to the development of

rational thinking, the present research has been designed to explore

the contributions of figurative and operative systems of knowledge to

the acquisition of conservation of number, one of the early forms of

operational thinking.

Piaget's most concise statement about the relationship between

figurative and operative aspects of knowledge is found in his concluding










chapter of The Mechanisms of Perception (1969):

One might begin by saying that an operative system of
transformations consists -in reciprocal changes between
states and that these states can only be characterized
as configurations: thus the figurative aspect of
knowledge would indicate the existence of, or represent,
the perceived states of objects or events, while the
operative aspect would have to do with their transforma-
tions. Both aspects would be mutually indispensable
at the level where their unity becomes inevitable, but
on the basis of a subordination of configurations to
transformations: if the figurative aspect were left
to itself and had to rely on its own capacities, it
would give rise to all sorts of systematic illusions,
being centred on the states of objects and lacking the
corrective decentrations of transformations. (p. 357)

The summary emphasizes three significant facets of the relation-

ship between figurative and operative aspects of knowledge. First, the

two systems are distinguished as functionally separate. The figurative

phase of knowing represents immediate states using sensory data

organized by perceptual schemes. Operative knowledge, on the other

hand, represents transformations, actions on the object which modify

the object either externally and physically or internally and deduc-

tively. Second, the two aspects of knowing are "mutually indispensable."

Especially during the development of knowledge from birth through young

adulthood, the ontogeny of knowing is dependent on the interaction of

figurative and operative aspects of knowledge (e.g., Youniss &

Dennison, 1971). Prior to the maturation of formal operational thinking

which permits the individual to manipulate thoughts themselves as a

part of thinking, that is, to think about thinking, the individual can

only deal cognitively with those objects or events which are perceived

immediately or have been perceived as objects or events in reality.

In fact, even beyond the full acquisition of propositional (formal

operational) thinking, an individual must continue to deal with concrete










objects and events throughout his lifetime; consequently, the inter-

action between configurational and operative systems continues to occur.

Finally, Piaget's summary statement points out his belief that the

operative aspects of knowledge are superior to the figurative aspects.

For Piaget the superiority is both developmental and structural.

Sensorimotor intelligence, the first phase of cognitive development,

is a necessary precursor and foundation of organized perception.

Intellectual structures, at first similar in some ways to figurative

structures,soon surpass the latter in complexity and flexibility.

Because of the superiority of operative structures, many of Piaget's

descriptions of figurative structures are worded by contrasting and

comparing the latter with the former. Consequently, the first three

(of four) phases of operative development will be presented now with

the hope of providing a more lucid context within which to understand

figurative development.



Operative Aspects of Knowledge


The operative aspects of knowledge are the sensorimotor activities

of infancy, the preoperational representations of early childhood, and

the later logico-mathematical operations; that is,concrete operations

and formal operations. In order to emphasize the operative nature of

the phases of intellectual development, the following section will

present some of the operative features of the first three periods of

the development of intelligence.

Sensorimator period. The first major period of cognitive develop-

ment, lasting from birth to two years, is known as the sensorimotor

period. Sensorimotor intelligence is a practical, behavioral










intelligence primarily functioning without language or representative

concepts. The sensorimotor behaviors are essentially adaptive and are

aimed at getting results rather than communicating truths (Piaget &

Inhelder, 1969). The development of intelligent behaviors is dependent

on the child's manipulation of objects and events in his environment

as well as on his active structuring of the interaction between himself

and the objects or events which he encounters and then works to recon-

struct. These two essential levels of activity are not distinct for

the young child but are united in a hypothetical cognitive structure

known as sensorimotor schema (Piaget, 1952b) or as an action-scheme

(Piaget & Inhelder, 1969) -- a total act of intelligence. In the

beginning the act is the act is the act, that is, an undifferentiated

entity in which personal consciousness (self) is not distinct from the

external world, perception is not separate from movement, and the

incorporation of new behaviors into the action-schemes (assimilation)

is inseparable from the modification of the schema itself to bring it

in line with data from the real world (accommodation). The child is

completely egocentric, seeing the world from a single point of view,

without knowledge of the existence of alternate perspectives and without

awareness that he is imprisoned by his own singular perspective

(Flavell, 1963). However, by the end of the sensorimotor period,

differentiation is well advanced. The progression of sensorimotor

intelligence leads to the child's awareness of the distinction between

objective reality, where his body is one of many distinct elements,

and his personal internal construction or representation of that

reality. With the achievement of greater objectivity near the end of

this initial period of intellectual development, perception and movement





as well as the processes of assimilation and accommodation gain the

capacity to coordinate with and to complement one another in intelli-

gent action.

The young infant's mental life consists of early sensory and motor

behaviors that respond to basic needs such as nutrition. With the

exercise of these early behaviors, change occurs in the form of

-elabor'ation because the behaviors do not merely repeat themselves

exactly. The repeated actions may include an-additional movement or

may focus on a different object in the child's environment. The addi-

tional movement or the new object are then incorporated into the exist-

ing behavior scheme to create broader organized totalities of action

(Piaget, 1967). With increasing behavioral facility, the infant begins

to notice the results of his own behavior. The random movements often

fortuitously produce a result so interesting that he repeats the move-

ments immediately, varying the actions and noticing the results of the

variation. The immediate repetition of behavior has been labeled a

"circular reaction," and it comprises true exploration, the first

"experiment in order to see" (Piaget, 1967, p. 11). As behaviors are

repeated and expanded through variation, the infant's perceptions and

movements become more differentiated and better organized. The pro-

cesses of assimilation and accommodation begin to complement one another.

Action schemes with similar content or focus begin to combine into more

complex action achemes which coordinate and relate the differentiated

and generalized behaviors of the component schemas. This capacity to

form new, complex, and more comprehensive relationships among behavior

even without language is a crucial operative achievement during the

sensorimotor period.










The coordination of action-schemes and the capacity to generate

relationships is a significant milestone in itself, yet it leads to a

second important achievement by the end of the sensorimotor period of

intellectual development, namely, the differentiation of self from

other entities in reality. Piaget summarizes this achievement in the

first of his Six Psychological Studies (1967). The child's impressions

are not clearly associated with a personal consciousness or with ex-

ternal objects. Impressions "exist in a dissociated block ... On the

same plane, which is neither internal nor external but midway between

these poles" (Piaget, 1967, p. 12). The result of this dissociation

is an unconscious egocentricity with "self" as the necessary referent

in experience. But the development of sensorimotor intelligence allows

the child to eventually structure an objective universe in which the

child is one object among others in reality. And with the developing

sense of self, the differentiation of internal and external states

evolves.

From the child's general progression from egocentrism to objec-

tivity, it is a small step to consideration of a facet of sensorimotor

development which is an important precursor of the rational concept of

conservation. Developing objectivity allows a child to recognize the

invariance of some important dimension of an object in the face of

certain changes in other dimensions. This aspect of sensorimotor

development is known by either of two phrases the object equggy: or

obifSA pelagnence. Piaget lists several characteristics which essen-

tially define a mature object concept. First, the object must be seen

as an entity independent of the self as object and observer. Second,

the object must be construed as independent of the immediate activity,









i.e., observing, manipulating, etc., which the child may apply to it.

Finally, full maturation of the object concept implies recognition of

the self as an object sharing the above properties and always existing

among other objects (Flavell, 1963).

Preoperational period. The hallmark of the second major period

of cognitive development, the preoperational period, is the achievement

of thought itself. During this period, which lasts from the second

year until about the sixth or seventh year, the child advances from

time-bound, overt, sensorimotor acts to the inner, symbolic manipulation

of reality which "allows him both to reconstitute the past (to evoke it

in the absence of the objects which were previously acted upon) and to

anticipate future, not yet executed, actions to the point where some-

times actions are replaced by words and are never actually performed"

(Piaget, 1967, p. 22). In an important behavioral achievement of this

period, representational thought can consider action rather than merely

acting immediately or impulsively. In the delineation of the periods

of operative development, the accomplishments of this period stop short

of logical thought. The preoperational period, then,encompasses the

evolution of intelligence from pragmatic, organized, and usually

purposive actions to prelogical symbolic-rep resentational thinking.

Representations replace overt actions as the basis of adaptive

functioning when the~ child achieves what Piaget has termed the symbolic

function (Piaget, 1951, p. 69). The symbolic function is simply the

ability to differentiate a signifier from an object or event that is

being signified (known as the significate). Achievement of the

symbolic function requires an awareness of the separateness of the

signifier and the thing signified. Describing the most important and









most common signifier, Baldwin (1967) notes,

a word does not function as a sign for the object it
represents unless the person using the word realizes
what the word means and also realizes that the word
is not the same as its meaning. It is this .
criterion that excludes perceptions and sensorimotor
schemas from the class of symbolic schemes. (p. 230)

According to Piaget, the child acquires the symbolic function

through particular developments in assimilation and accommodation.

As accommodation matures, one of its most important developmental

products is imitation, an active modeling or replication of an original

external event. During the sensorimotor period this imitation is simply

an overt act mimicking or reproducing the event. With growth and

refinement, however, the child develops the capacity to imitate an

event internally as well as externally. Shortly thereafter, the child

also becomes capable of imitating reality from memory even though the

event has ceased to exist in perception. Piaget refers to this capac-

ity as "deferred imitation." Such an achievement extends the time

boundaries of the child's intellectual functioning beyond the immediate

present. Piaget surmises that accormmodation-as-imitation thus supplies

the very young child with his first signifiers which are capable of

representing an absent significate for him internally. The signifiers

are mental images which he may now evoke in thought, rather than acting

out in reality, as imitations made in the past. According to Flavell

(1963), the capacity to evoke signifiers allows the child to anticipate

events and thus direct future action. Past images are evoked internally

in the present and serve as anticipativee mediators of actions not yet

performed" (p. 153).

The role of assimilation in the symbolic function is somewhat less

complicated though equally significant. With the elaboration and










distillation of the assimilation function, play emerges as its product

and primary mode of expression. Play, like its parent function,

alters reality to suit the history and the needs of the self. Whenever

a child pretends, there is no urgent need either for the objects used

to resemble the pretended objects or for the pretense of behavior to

be convincing or come to some final conclusion. The child may substi-

tute whatever objects are at hand for the represented original objects,

and he may stop the game whenever he likes. Nonetheless, symbolic

play does have a crucial function in the development of thought.

Piaget (1967) asserts, "Practically every form of psychological activity

is initially enacted in play" (p. 23). Through play, reality is

incorporated into schemas to provide external referents and, thus,

meaning for the signifiers supplied through accommodation. Consequent-

ly, representation is a result of a differentiated, complementary

partnership between accommodation (imitation) and assimilation (play).

Being concerned primarily with the nature of operative aspects of

knowledge, the present discussion of the preoperational period would

be inadequate if it ended with the simple summary of the symbolic

function and representation. A number of features of preoperational

thought should be noted carefully in order to clarify the meaning of

operative functioning. These essential features will be presented as

shortcomings of preoperational thought which mark the cognitive

achievements of this period as incomplete forms of reasoning. First,

the intuitive thinking which is the highest intellectual accomplishment

of this period is still bound by particular past or present configura-

tions. The complexity of the configurations represented by preopera-

tional thought has increased over the simple individual image











represented in the most rudimentary thought at the end of the sensori-

motor period. But reasoning is still not free to proceed without a

specific original past or present image. Preoperational thinking is

often in the form of what Piaget terms a mental experiment, "an

isomorphic, step-by-step mental replica of concrete actions and events"

(Flavell, 1963, p. 158). Second, preoperational thought continues to

demonstrate profound egocentricity, that is, the child is unable to

take the role of another person or to see his own perspective as one

of several possible points of view. A third characteristic of pre-

operational thought is its tendency to center attention on a single,

striking aspect of a configuration to the neglect of other important

aspects of the figure or event during the process of reasoning. As

one might infer, the neglect of important aspects of an event or

configuration often will lead to systematic error or distortion. More

advanced operational thinking is able to decenter, that is, to attend

to multiple facets of an object of thought and to coordinate all

features in order to minimize error or distortion in reasoning. Fourth,

the child considers the successive states or configurations of a dis-

play much like viewing motion picture film one frame at a time rather

than the transformations relating the states to one another as a total

process. The child at this point in development cannot consider the

process linking successive states with continuity, thus preoperational

thought is static, immobile,and impressionistic. Fifth, the precon-

cepts of preoperational cognition demonstrate two peculiar character-

istics. The meaning of these preconcepts fluctuates with a changing

context; they have no intrinsic identity which is stable across

situations. And the preconcepts are linked by a primitive reasoning










which moves from one centered, perceptually-compelling feature to

another. The conclusions are thus often based on salient perceptual

features which occur juxtaposed to other perceptual features and which

end the preconceptual reasoning because they appear last in the

sequence. Reasoning is transductive and moves from particulars to

particulars. It is more often incorrect than correct. Finally, and

most significantly for Piaget, preoperational thought is not reversible.

Reversibility is the ability to proceed through a cognitive sequence in

one direction and then reverse direction in thought to retrace the

steps of the reasoning back to the original premises which are con-

stant and unchanged. By simply capturing concrete mental images of

real events which are usually unidirectional in daily experience,

preoperational thought lacks the capacity and initiative to return

logically to the starting point of the event. The quality of irrever-

sibility also means that children in this phase of cognitive development

cannot conserve a particular property of a stimulus array, such as

number, weight, or volume, as invariant in the face of compensating

dimensional changes (e.g., increase in height with decrease in width

of a container). Preoperational reasoning is confined to concrete

images derived from successive perceptual states. Consequently, the

child whose thinking is characterized by preoperational reasoning is

unable to undo mentally the changed state or dimension by using its

inverse to discover the invariance of the crucial property.

Piaget (1951) summarizes the advancement of representational

preoperational thinking by characterizing representation as extending

the field of adaptive behaviors beyond present time and space. Thus

the child can deal with what lies outside of immediate perception and









action. Representation is the product of the signifier, which is

supplied by internal imitation (accommodation) which permits recall,

and the significate, which is supplied by assimilation and provides

meaning for the signifier. "In this respect, the collective institu-

tion of language is the main factor in both the formation and

socialisation of representations" (Piaget, 1951, p. 273).

Period of concrete operational thinking. The highest cognitive

achievement during middle childhood (roughly 7 to 11 years) is concrete

operational thinking. It is worth noting that this period has been

studied more frequently than any other Piagetian period of intellectual

development. This is due at least partly to the fact that increasing

verbal facility and increasing cognitive flexibility render children

at this level of development more testable than younger children. The

present study thus has joined a large number of investigations into

the nature of concrete operational thought and especially the subset

of research investigating the inducement of concrete operational

thought in the form of conservation.

What is the essence of concrete operational thinking? According

to Flavell (1963),

If we look with the Piagetian eye, we see one
higher-order difference which subsumes all the
particulars; and from this one difference stems
most of what Piaget has to say about the subperiod
of concrete operations: It is simply that the
older child seems to have at his command a coherent
and integrated cognitive system with which he
organizes and manipulates the world around him.
Much more than his young counterpart, he gives the
decided impression of possessing a solid cognitive
bedrock, something flexible and plastic and yet
consistent and enduring, with which he can structure
the present in terms of the past without undue strain
and dislocation, that is, without the ever-present
tendency to tumble into the .perplexity and contra-
diction which mark the preschooler. (p. 165)





One significant implication of this indispensable system is that no

operation exists in isolation; any operation is formed by and derives

from a structured system of operations of the same kind such as class

operations, arithmetic operations, geometric operations, etc. The

system is the sine qua non of cognitive operations, whether formal

propositionall) or concrete.

In keeping with the heritage begun during the sensorimotor period

of intelligence, the content of the cognitive system central to opera-

tional intelligence consists of actions performed by the child.

Discussing the mental development of the child, Piaget (1967) has

stated, "Psychologically, an operation is, above all, some kind of

action (the act of combining individual or numeric units, displacing

them, etc.), whose origin is always perceptual, intuitive (representa-

tional), or motoric. The actions which are the starting point for

operations are thus rooted in the sensorimotor schemata, i.e., in

actual or mental (intuitive) experience" (p. 48). Piaget's use of the

term "action" in his theorizing is, of course, generic. Its meaning is

altered slightly with each successive stage. During the sensorimotor

period, action refers to overt, observable behavior. The preoperational

representations are the initial internalized actions; but they are

sporadic and idiosyncratic, not organized into tight ensembles of

related action, and are still bound to immediate perception. The

actions of operational thought are more abstract and internalized than

the actions of the previous periods, but the crucial fact at this point

is that the actions form "increasingly complex and tightly integrated

systems of actions" (Flavell, 1963, p. 165).

Piaget derived his models for explaining how concrete operations










function from the fields of logic and mathematics. These logico-

mathematical structures were postulated as ideal models of human

reasoning. The logical structures which direct concrete operational

thinking are known as groupings. There are nine groupings covering

the concrete operations for logical classes and for relations. Each

of the groupings has in common five major properties. The forn

(equation) in which each property is expressed varies among the

groupings, but the basic properties are universal. The logico-mathe-

matical nature of concrete operations is apparent in the five basic

properties which are described below:

1. Composition. When any element is combined with another
element by means of a defined operation (addition or
multiplication), the product of that combination is itself
an element in the system of classes or relationships.

2. Associativity. The sum or product of a series of elements
is independent of the way they are grouped in the equation.

3. General identity. The identity element is the one and only
element which, when added to or multiplied with any other
element whatsoever, leaves the other element unchanged.

4. Reversibility. For each and every element there is one and
only one inverse element which yields the identity element
when it is added to or multiplied with that unique element.
Two kinds of reversibility operate in the groupings. In-
version or negation, where +A is reversed by -A, is the
inverse operation for classes. Reciprocity, where A < B is
reciprocated by B > A, is the inverse operation of relations.

5. Special identities. Each of the groupings has special
elements which, in special circumstances, may also function
as identity elements. (Flavell, 1963)

Each of these five properties is necessary for construction of an

operational system of actions.

Nevertheless, in Piagetian writings which have been translated

into English and in American research of Piagetian theory, one of

these properties has received an inordinate amount of attention. That






-15-


property is reversibility. As an essential characteristic of operational

thought and, hence, of operative knowledge, several things are known

about the role of reversibility in logical thinking. First, it is

significant to recognize that reversibility allows the child to extend

his cognitive functioning beyond perceptible states of the environment

to include the transformations which produce and relate successive

states. Reversibility is a necessary condition for the child's com-

prehension and use of transformations inherent-in a logical system of

actions. Further, with the advent of reversible operations, the child

achieves the ability to compensate for changes in his environment in

order to recognize its invariant features. Thus he can observe or

imagine actual changes in the environment and also compensate for

complementary changes by imagining the transformations in an inverse

sense. The reversible transformation thus always leaves some feature

of an operational system unchanged; otherwise, it would be irreversible.

In the conservation experiments, the constant invariant feature is the

conserved quality.

The summaries of the first three periods of operative development

are now complete. Sensory-motor intelligence, preoperational repre-

sentational thinking, and concrete operational thought have been

presented briefly in order to provide an explanation for and descrip-

tion of the cognitive functioning of the children studied in the

present investigation. Concrete operational thinking is the upper

limit of cognitive maturity in the children who are tested and trained

in the conservation of number. However, in order to provide some sense

of closure for the discussion of intellectual development, a quotation

will be borrowed from Flavell to summarize the essential achievement





of the ultimate stage of cognitive functioning -- the period of formal

operational thought. According to Flavell's (1963) description,

The most important general property of formal
operational thought ... concerns the real versus
the possible. Unlike the concrete-operational
child, the adolescent begins his consideration of
the problem at hand by trying to envisage all the
possible relations which could hold true in the data
and then attempts, through a combination of experi-
mentation and logical analysis, to find out which
of these possible relations in fact do hold true.
(p. 204)

Thus formal thought is no longer limited to concrete reality and

becomes hypothetico-deductive (allowing "if ... then" reasoning),

propositional (proposing statements about raw data and manipulating

those statements rather than the data themselves), and capable of

systematically isolating all individual variables and assembling these

variables in different combinations for careful consideration, all

without distorting the totality of the system. In short, instead of

thinking of perceptions or motoric actions, the adolescent can think

about thoughts themselves and about the transformations which relate

these abstract "actions."




Figurative Aspects of Knowledge


Piaget's (1969) statement about operative and figurative aspects

of knowledge appearing early in this introduction contains the closest

approximation to a one-paragraph definition of figurative aspects of

knowledge available in English translations of his work. The reader

may recall that he depicts the figurative aspects of knowledge as

indicating "the perceived states of objects of events" (Piaget, 1969,

p. 357). Therefore, perceptual processes serve as the focus of the









discussion of figurative knowledge. In discussing perception as the

primary example of figurative aspects of knowledge, it is useful to

recognize the distinction made by Flavell (1963) between the Piagetian

theory of perception and Piaget's theory about perception. His theory

of perception consists of a precise, probabilistic model of actual

perceptual apparatus and its functioning. His theory about perception

is a general conception of perception and its development, particularly

in relation to the nature and development of intelligence. Because

the present study explored the impact of .figurative and operative

experiences on the development of operative structures, specifically

conservation of number, the discussion centers on Piaget's theory about

perception and mentions the model of perception only as it contributes

to understanding the relationship between perception and intelligence.

The ensuing discussion presents perception by dividing the construct

somewhat arbitrarily into three aspects: the nature of perception,

the origin and development of perception, and the functions which

perception fulfills.

The nature of perception. At the risk of overstating the obvious,

the consideration of the nature of perception should begin by recog-

nizing that Piaget views perception as an aspect of knowledge which is

separate, although not completely autonomous, from intelligence. This

view is reflected both in the abundance of research published by

Piaget and his associates in the series entitled "Recherches sur le

development des perceptions" in the journal Archives de Psychologie

and in his own explicit statements (e.g., Piaget, 1969, p. 338).

Equally important for an adequate understanding of Piagetian theory

about perception is an awareness of the two kinds of visual perceptual





-18-


phenomena -- primary perception and perceptual activities. Primary

perception refers to centered visual fixations "which do not involve

any eye movements and which exist in a single field of focus" (Piaget &

Inhelder, 1969, p. 35). Developmentally, primary perception remains

relatively constant with age and is studied (and operationally defined)

by means of tachistoscopic presentation with exposures lasting from .05

to .20 seconds. Perceptual activities, which develop in quantity and

quality with age, involve shifting one's fixations in space in visual

exploration or comparing two stimuli which may appear in the same place

at different times. With some awareness of these two perceptual

processes, two other characteristics of perception are perhaps more

easily understood. First, Piaget views perception as an active process

rather than a passive reception of raw sensations. The child determines

what he perceives, albeit often unwittingly, by the spot of the stimulus

chosen for fixation in primary perception or by the extent to which he

shifts fixations and the areas of the stimulus viewed via perceptual

activities. Centration and decentration are two terms used to describe

the actions of perception. Centration refers to fixations on a single

salient perceptual aspect of a stimulus array to the exclusion of other

aspects. Decentration is the descriptive label for shifting fixations

in the form of exploration, transposition, or extension. Perception

thus constitutes real activity for the child. Second, perception is

always probabilistic in nature. Piaget and Inhelder (1969) explain,

In looking at a configuration, even a very simple one,
only a small area can be clearly seen in each single
glance. One does not in fact see everything with the
same precision, and one does not see everything at the
same time. The gaze moves from one point to another,
and the "encounters" between the different parts of
the receptive organs and the different parts of the





-19-


stimulus object are distributed with unequal density
according to the regions of the figure, the regions
of the retina, and whether at a given moment these
regions are centered by the foves ... or remain in
the periphery. (p. 36)

Perception is therefore probabilistic in the sense that all elements of

an object will not be viewed with equal precision, frequency, duration,

or ~intensity; consequently, those elements which capture the child's

attention and are observed with greater precision, frequency, duration,

and intensity will be more likely to determine the nature of percep-

tion's often incomplete, global evaluation. Finally, perception is

impressionistic. "Perception provides only instant impressions cor-

responding to a given view point (that of the subject at a given

moment), whereas the concept presupposes a coordination of all viewing

points and an understanding of the transformations leading from one

viewing point to another" (Piaget & Inhelder, 1969, pp. 45-46).

The origin and development of perception. Another index of the

status of perception within Piagetian theory and especially of the

relationship between perception and conception is discovered in the

discussion of the origin and development of perception. As indicated

earlier, Piaget believes that perception is developmentally inferior

to intelligence. Consequently, his discussion of the origin and

development of perception is couched in terms of the comparison-contrast

of perception and intelligence (e.g., Piaget, 1969; Piaget & Inhelder,

1969). Piaget consistently asserts two strong convictions about

perception. First, perception is distinct as an epistemological

process. And second, the perceptual process represents a particular

aspect of sensorimotor organization which serves to enrich the action-

schemes and is dependent on sensorimotor action-schemes for meaning.





The relationship between perception and intelligence is construed in

this way because perception is conceptually a more limited process in

Piagetian theory than it is for many psychologists. Perception involves

instant, impressionistic judgments or inferences. Perception's

organization of successive isolated configurations into perceptual

schemas occurs with essential assistance from sensorimotor actions

which direct the manipulation of the objects (e.g., drawing, construc-

tional games, etc.) to establish multiple perceptual experiences with

the same object and to reveal the correspondence between elements of

perceived objects. Although Piaget repeatedly emphasizes the individual

identity of perception and intelligence, perceptions are generated,

along with new motor skills, from the varied repetition of the earliest

behaviors. In short, perception emerges from the earliest form of

action; and its development subsequently is guided by acts of intel-

ligence from that moment.

Although the origin of perception happens with the occurrence

of the first actions, the development of perception is precocious;

that is, certain perceptual phenomena precede intellectual operations

and forecast their eventual development. Piaget suggests that the

perceptual (figurative) phenomena prefigure the operational notions.

The operational notion of conservation will serve as an example of

prefiguration. The precursor of operational conservation is the

sensorimotor schema of the permanent object which emerges with the

elaboration of the perceptual constancies -- constancy of size and

constancy of form. These perceptual constancies are based on a crude

system of compensations which are also necessary in a more elaborate

form for conservation.










The constancies prefigure conservation in the sense that both are

based on the same functional process of compensation. Moreover, the

common process produces a common product -- the invariance of a

particular quality despite change in salient aspects of objects or

events. The process consists of three factors: a, b, and c. For

both perceptual constancies and conservation, factor c is the invariant

factor that emerges as the product of compensation between a and b.

However, in keeping with the inferior nature of perception as a basis

of knowledge, there are important differences between the nature of

the component factors, a and b, as they function first in the more

primitive perceptual constancies and later in the operational notion

of conservation. For the constancies, the first factor, a, is a

deforming factor, an objective modification of the system such as real

changes in physical distance between objects in perception of size and

real changes in perspective (location) in the perception of form. The

second factor, b, is a deformed factor, a subjective modification of

the system such as apparent size or shape. Factor c, then, is the

perceived invariance of the object in real size or form despite the

change in real distance or location (perspective). The objective

change is thus compensated by the subjective change through a system

of approximate perceptual readjustments. For conservation, both

factors a and b are objective changes in the system, and the invariant

factor c is the product of the compensation between objective factors

a and b. For example, in the case of conservation of number, factor a

is the length of the row, factor b is the density of the elements, and

c is the number of elements in a collection. Length of a row (a) may

increase, but if density (b) decreases, then the number of






-22-


elements (c) will remain the same.

Piaget explains the precocity of perceptual constancies by

observing that the object itself is never physically changed but gives

rise to a subjective modification because it is moved relative to the

observer. According to Piaget (1969), the apparent change (b) is more

easily compensated by the person's perception of the deforming factor

(a) because the perception is a property of the system relating the

object to the subject. The compensation is accessible to perception

and can occur through successive approximate perceptual changes. The

perceptual compensation is based on the experienced invariance of the

object as the person exercises action-schemes in which the apparent

quality (strictly perceptual) is modified during the perception of real

changes in the relationship between the object and the subject because

of the object's movement. Perceptual constancy thus involves only a

single change in the object itself along with a change in the person's

perception. Conservation, on the other hand, involves modification of

two salient qualities of the object itself, simultaneously. Conse-

quently, recognition of the invariance of the constant factor occurs

only when the child can represent each objective modification indepen-

dently and hence can manipulate the qualities in a reciprocal relation-

ship. Piaget (1969) concludes, "the representational readjustments

give rise to a system of genuine operations whose strict reversibility,

in turn, engenders ... strict invariance characterized by its deductive

necessity and not by probabilistic approximations" (p. 328). In

summary then, after a delayed start, perception develops rapidly,

actually reaching its zenith late in the period of representational

thought. Intelligence, emerging from early behaviors, follows a




























































L


-23-


separate, more deliberate course of development which does not achieve
its culmination until early adolescence but which surpasses perception

in accurately guiding adaptive behavior with the achievement of

operational thought.
The function of perception. Finally and very significantly, per-

ception functions as the second of the two fundamental forms of adaptive
human behavior, the other being intelligence. This form of adaptive

behavior prefigures its superior and forecasts its eventual operational

functioning in two ways. First, as transformations eventually relate

objects and events and their changes through complex operations, per-

ceptions connect the early transformations as states between the
transformations. Piaget (1969) states quite clearly that existing in

the here-and-now, perceptions function as connectors which establish

immediate and constant contacts between the child's actions or opera-

tions and the events or objects. These figurative aspects of knowledge
are organized by the person, through perceptual activities, into

configurations which are compatible with the transformations. The
configurations in turn can be used as links between transformations

within a coherent operational system. "A result is the apparent

prefiguration of operational notions in perception" (Piaget, 1969,

p. 360). The second forecast of operational functioning by perception
involves the gradual increase in the child's ability to decenter and
thus to consider successive centrations or fixations as a way of

correcting the distortions or errors of centered perceptions. Although

perceptual decentration does not develop as completely as decentration
in operational thinking and never achieves the accuracy and objectivity

of operational thought, its growth is a necessary condition for maximum





perceptual adaptation and presages the ability of operational thinking

to move from one perspective to another in considering logical alterna-

tives. Piaget sees decentering perception as a constructive process

of gathering as much information as possible about an object. Thus

perceptual decentration expands the amount of information available

and the child's ability to coordinate the increased information in the

process of construing perceptual schemas. In reflecting on the func-

tional role of perception, the reader should realize that prefiguration

only implies earlier appearance and not causation. The developmental

similarities between precocious perception and operational thought are

collateral but not formative. Beginning with sensorimotor actions,

perceptual activity must be integrated with and directed by actions.

Intellectual actions or operations do not replace perceptions, but

they do program the way perceptual data are collected. With the advent

of- operational intelligence, the child often substitutes a science of

measurement for global perceptual evaluations (Piaget & Inhelder,

1969).



Comparison of Operative and Figurative Aspects of Knowledge


In summarizing the discussions of operative and figurative aspects

of knowledge, several characteristics distinguish the two modes of

acquiring knowledge about the world and about oneself in the world.

First, the actions constituting operative knowledge often produce

or participate in change. The active manipulation of elements in the

environment, either physically or mentally, will modify the elements

themselves and often subsequently undo the original modification. In






-25-


the process the child will discover the key to creating new objects

or events by transforming the immediate environment. The essential

quality is the ability to discover the world through environmental

manipulation. Further, operative knowledge at its peak is not limited

by time and space. Formal operational thought functions with past,

present, and anticipated elements, some of which may not even exist in

the present reality. Likewise, formal operational thinking can move

beyond the limits of the senses, however well--organized, to the limits

of outer and inner space. Thirdly, operative aspects of knowledge

enjoy the flexibility of reversibility, that is, the ability to return

to the origin of a concept instantly despite elaboration of the concept

and know that the original remains unaltered. It is interesting to

note that although formal operational thought is able to function well

without immediate perception and is no longer necessarily dependent on

perception, those who have achieved the ultimate operative cognitive

capacity continue to employ perception as the link between new objects

and events and corresponding actions or operations.

In contrast, we find that figurative aspects of knowledge at the

zenith of their development cannot escape their dependence on the

operative mode of knowing for the focusing of attention on relevant

elements and thus the programming of perceptual organization. Viewed

as an active knowledge, figurative activity is nonetheless restricted

to recording and preserving events rather than producing them. The

active figurative processes consist of data-gathering through the

movement of sensors, not objects in the environment or the environment

itself. In addition, figurative aspects of knowledge suffer spatial

and temporal limits in their functioning. Operating most frequently









in the present within the range of human senses, figurative aspects

of knowledge have a restricted capacity to recall past events and no

ability to foresee or anticipate future events. Finally, configurations

are mostly irreversible because retracing perceptions step-by-step to

the original often produces a perception which has been slightly

reorganized by succeeding perceptions.



Conservation of Number


Within the broader theoretical context outlined above, consider

the focal task of the study -- conservation of number. Although it

has served as an example in the discussion of concrete operational

thought, the concept of conservation needs more precise definition.

Conservation denotes the child's realization of the substantial in-

variance of a particular dimension of a stimulus object or a collection

of objects despite change in other perceptible dimensions. In order

for actual invariance of one characteristic to occur and therefore be

recognized by the child, the changes in other dimensions must compensate

for one another, leaving the primary dimension unchanged. The invari-

ant quality is thus independent of the rearrangement of other dimensions

(Lovell & Ogilive, 1960). In the conservation of number, the number of

objects in the stimulus configuration remains the same regardless of

the spatial arrangement of the objects if no objects are added or

taken away during configural transformations. The achievement of the

process of conservation is considered by Piaget (1952b) to be "a

necessary condition for all rational activity" (p. 3). The present

study used the basic conservation of number task as a behavioral index

of cognitive development. The task begins by showing the child two


-26-





-27-


parallel rows of objects equal in number, length, and density. The

objects are placed in one-to-one correspondence. The child then is

asked if both rows contain the same number of objects. When the child

is satisfied that the rows are numerically equivalent, he or she

watches as the experimenter spatially transforms one of the rows by

moving the objects closer together (shortening the row and increasing

the density) or spreading the objects apart (lengthening the row and

decreasing its density). Regardless of the type of transformation,

contraction or expansion, one of the rows is perceptually longer and

less dense than the other, while the number of objects in each row

remains unchanged. The child is asked to judge whether the rows con-

tain the same number of objects and, if not, which row has more.

Finally, the child is asked to explain what led him or her to make

the initial judgment.

A number of validation studies have confirmed the general pro-

gression of cognitive development reported by Piaget (e.g., Braine,

1959; Lovell, Mitchell, & Everett, 1962; Lovell & Ogilive, 1960; and

Uzgiris, 1964). Other studies have verified Piaget's description of

the developmental progression of the concept of number (Elkind, 1964;

Pufall, Shaw, & Syrdal-Lasky, 1973; Wohlwill, 1960). Piaget describes

three stages in the development of the conservation of number. In the

first stage, known as the stage of nonconservation or global compari-

son, the child bases his judgment of the equality or inequality of the

two rows on only one of the two global perceptual qualities inherent

in the array, length of the row or density of the elements, with no

attempt to coordinate these two dimensions. Further, the child at

this stage makes little or no effort to establish one-to-one





correspondence between the elements of the .rows. In the second stage,

called the transitional or intuitive stage, the child spontaneously

but somewhat diffidently establishes numerical correspondence between

the elements of the two sets in response to questions about their

initial equality. According to Piaget (1952b),

The child certainly considers the relationships of
length and density simultaneously, since he can pro-
duce a copy equal in length and density to the model,
but the coordination does not go beyond the plane of
perception; i.e., as soon as the perceived figure,
which made the correspondence possible, is altered,
not only does the correspondence vanish, but also the
coordination between length and density. (p. 81)

For Piaget, when intuitive preoperational coordination of length and

density yields an accurate judgment of conservation but that irrevers-

ible coordination cannot support the judgment with a logical explana-

tion, the child's response is also classified as intuitive conservation.

Finally, in the stage of operational conservation (simply called

"conservation"), the child bases his or her evaluation of equivalence

of number on the reversible coordination of length and density and the

resulting invariant numerical correspondence between elements; per-

ceptual changes become irrelevant to numerical correspondence. With

the achievement of operational conservation, the child can support his

accurate judgment of invariance with a logical, coordinated explanation.

Criteria for judging the presence of conservation. As the meaning

of conservation as a concept of cognitive development is considered,

it is important to note that investigators have expressed differences

in determining the criteria defining an adequate conserving response.

A number of investigators have defined an adequate conserving response

as requiring both an accurate judgment and an explanation of the con-

serving judgment (e.g., Gelman, 1969; Goodnow, 1973; Hamel & Riksen,









1973; Miller, 1973; Hiller & Heldmeyer, 1975; Smedslund, 1961; Uzgiris,

1964; Wallach, Wall, & Anderson, 1967). Other researchers of conser-

vation (Bruner, 01ver, & Greenfield, 1966; Elkind, 1967) have asserted

the adequacy of the conserving judgments ("same," "more," or "less")

without logical verbal explanations. As an example, Elkind (1967)

stated that what Piaget appears to say is not what he means to say.

Elkind asserted that explanations are not necessarily a criterion for

conservation as believed by many of his North ~American colleagues. He

explained, "The ... types of verbal explanations ... are really post

hoc rationalizations rather than veridical reflections of the processes

leading to conservation" (Elkind, 1967, p. 20). Earlier (Elkind,

1964), he asserted that "Piaget used children's predictions, judgments,

and explanations interchangeably as signs of conservation or non-

conservation" (p. 21). Calhoun (1971) omitted the verbal response

altogether and judged conservation of number by observing which row

of candy objects was eaten by his young subjects because the row was

perceived as having more candy. The controversy in effect focuses on

the operational definition of conservation within individual studies.

And in view of basic differences regarding which cognitive schemes

immediately precede the full development of conservation (e.g., Bruner

et al., 1966 versus Elkind, 1967), a final consensus is not likely

soon. Recognition of the invariance of the essential quality of an

event in spite of other changes, the heart of the conservation concept,

does not seem to be in question. Rather the observable behaviors

which demonstrate the certitude of the child's judgment of invariance

are specified differently by different researchers. While precision

in understnading the progression of children's thinking is important









and work should focus on determining valid and reliable behavioral

predictors of the child's confident understanding of the invariance

judgment that is conservation, the present study did not address that

issue specifically. Based on a belief that the risk of assuming

understanding which it is not mature is greater than the risk of asking

for further demonstration of maturity, the present study used a

moderate definition of conservation by requiring an adequate explana-

tion of the conservation judgment in defining a conserving response.

Criteria for judging the level of initial cognitive development.

Many studies of conservation among younger children select only non-

conservers as candidates for training methods. It is much simpler to

operationally define a nonconserver as one who does not conserve on

a conservation task. However, the present study, hoping to understand

the development of conservation as a process which is more continuous

than dichotomous, attempted to distinguish both nonconservers and

intuitive conservers as described by Piaget (1952a) within the larger

group of children who have not acquired the conservation of number

response. A~s noted in an earlier paragraph, in addition to their

inability to conserve number in the face of salient perceptual changes,

nonconservers have been described by Piaget (1952a) as being unable to

consistently establish one-to-one correspondence between the elements

of two sets of objects. Nonconservers attend primarily to the per-

ceptual configuration rather than number. Even when asked to construct

a configuration containing the same number of objects as are present in

a model configuration, nonconservers reportedly reconstruct the can-

figuration with little regard for number, attending instead to shape,

length, or other perceptual cues. Intuitive conservers, on the other





hand, whose representational thought is a step closer to logical

thinking, have clearly acquired the rudiments of cardinal number and

can establish quickly the one-to-one correspondence between stimulus

groups or arrays. However, when a major perceptual change occurs in

the configuration of one of the two rows without altering the number

of elements, the intuitive conserver cannot recognize the invariance

of number. The perceptual change is too compelling and overwhelms the

tenuous cognitive recognition of number invari-ance. Previous observa-

tion has suggested however, that intuitive conservers do demonstrate

more resistance to irrelevant perceptual change than nonconservers.

Consequently, in the present study, children were classified as non-

conservers or intuitive conservers according to three criteria:

(1) performance on the screening tasks for construction of equivalent

sets; (2) judgments of conservation; and (3) explanations of conser-

vation.

Compnents of conservation. Since research has verified the

developmental progression of the concept of number according to

Piagetian theory, second-order questions concerning the mechanisms of

development have become appropriate foci of study. One such question

concerns the experiences or component tasks from which the operation

of conservation of number is ultimately constructed. One method of

answering that question is longitudinal research which provides or

withholds experiences considered to be crucial for the development of

conservation and evaluates the appearance and adequacy of skills of

conservation. To date few, if any, such long term studies controlling

deprivation of or stimulation through crucial experiences have been

performed. Another, more economical method which has been popular










among researchers assessing the processes and experiences relevant to

cognitive development is specific training in or experience with

suspected component skills. The basic paradigm involves a pretest to

determine the child's current level of functioning, a subsequent train-

ing period or experience intended to facilitate relevant learning, and

finally a posttest to assess the changes, if any, effected by the

special experiences. Brainerd and Allen (1971), reviewing research

on the induction of conservation through specific training, noted that

seven of nine reviewed studies reported success in inducing conservation

of number in children. The seven used a variety of training procedures,

including discrimination of relevant cues (e.g., Gelman, 1969), verbal

rule instruction (e.g., Beilin, 1965), or a mixture of both (e.g.,

Wallach, Wall, & Anderson, 1967). Incidentally, their more recent

summary of the research on induction of conservation contrasted with

Flavell's (1963) review which found little or no support for the

induction of conservation through directed training.

Several recent studies have investigated the influence of per-

ceptual cues on the conservation of number. Results of investigations

by Gelman (1969) and by Wallach, Wall, and Anderson (1967) have

demonstrated the Piagetian nation that perceptual cues, which are

powerful during the sensorimotor and the preoperational periods,

actually interfere with the recognition of conservation or invariance

by distracting attention from relevant to irrelevant dimensions of a

stimulus. Gelman (1969) employed a discrimination learning paradigm

of oddity training in order to teach her child subjects to attend to

a relevant stimulus dimension and to ignore irrelevant perceptual

dimensions. Her results supported two conclusions concerning the










influence of perceptual factors on the acquisition of conservation of

number. First, irrelevant, nonquantitative cues, such as length and

density, are salient for young children; they are therefore more likely

to attend to these perceptual cues than to quantitative dimensions.

Second, a larger proportion of errors on those trials where irrelevant

cues opposed relevant cues reflects significant interference by

irrelevant dimensions in the evaluation of conservation. Wallach,

Wall, and Anderson (1967), employing a reversibility training procedure

among others, concluded that their reversibility training may have been

successful in inducing conservation of number primarily because it

taught the children to stop using misleading perceptual cues. According

to these researchers,

Conservation probably never occurs when a situation
provides what the subject takes to be a clear per-
ceptual cue for nonconservation .... making the Ss
stop using such a cue although it is still present
may have been -- with or without the experience of
reversibility training per se -- a crucial factor in
the success of our doll-reversibility training
procedure. (p. 441)

Research by Peters (1970) and by Whiteman and Peisach (1970) has

provided a somewhat different perspective on the influence of perceptual

cues on conservation without contradicting the conclusions cited above

concerning the disruptive influence of length and density. The

results of these two more recent studies suggested that the salience

of perception may be manipulated to improve judgments of conservation

of number, Whiteman and Peisach (1970), avoiding the often limited

transfer or generalization of training effects, investigated the

results of supporting a' basic conservation of number task with in-

creasing degrees of perceptual and sensorimotor support. The six

conservation of number trials were presented in the following invariant





-34-


sequence providing a gradual increase in the number of supports:

(a) the standard conservation task with no supporting perceptual or

sensorimotor cues; (b) a color cue, consisting of two identical sets

of candies with five colors appearing in each row and with each color

in one row aligned opposite its identical color in the second row;

(c) the color cues as above plus sensorimotor experience using one

hand to move the candies from the elongated position to the original

position and back to the elongated configuration; (d) color cues plus

guide lines connecting the colored candies in the elongated row with

their twins in the unaltered row; (e) color cues, guide lines, and the

one-handed reversibility experience, and (f) color cues, guide lines,

and sensorimotor experience using both hands to move the candies from

the elongated configuration to the original position directly opposite

the unaltered row and back to the elongated position. As predicted,

for conservation of number the accumulation of supporting cues resulted

in gradually increasing proportions of conservation among both 5- and

6-year-old and 8- and 9-year-old subjects. Evidence also supported

the authors' hypothesis that younger children would benefit more from

sensorimotor experience while the older children would benefit from

increasing perceptual support consistent with the postulated schemas

in effect for each age group. Although the evidence deserved to be

taken seriously, the conclusions of the study were marred by at least

two weaknesses. First, the invariant sequence of presentation failed

to control for learning or practice effects. The final. trial, providing

the greatest perceptual and sensorimotor support, had the added

benefit of practice on five previous trials. Second, the older

children, ages 8 and 9, whose performances showed the greatest






-35-


consistency, were beyond the age at which Piagetian theory predicts

conservation will first occur despite distracting irrelevant cues.

Piaget, supported by naturalistic observation as well as controlled

experimentation, reports that reversible operational thought supports

conservation of number regardless of the presence of irrelevant per-

ceptual cues by the age of 7 or 8 years. Consequently, although the

critical variable is performance on the conservation trials rather than

age per se, the age of the older children confounds the interpretation

of their performances. Nevertheless, the results of Whiteman and

Peisach (1970) did suggest that certain perceptual cues may facilitate

rather than interfere with conservation.

Using a training procedure, Peters' (1970) study included a

treatment group which participated in discovery of conservation of

number guided by perceptual cues. He used visual cues to assist the

children in making the conservation judgment. ~The first three train-

ing trials used correspondence-accentuated materials. The materials

consisted of two identical nine block subsets in which each block of

the two sets (one for S and one for E) had a distinctive colar.

Peters expected the color-matching cues to facilitate the establishment

of initial correspondence and to maintain the relationship between

the sets throughout the transformations. The second three training

trials in the perceptual-cue guided discovery used number-accentuated

blocks. The blocks in these trials were white with sequins ordered

one through nine in domino fashion on the upper surface. The dots

were assumed to evoke counting and additionally could establish match-

ing across sets. Results indicated that for all training groups the

immediate posttest means were significantly higher than the control





-36-


group mean. On the immediate posttest, the means for the non-cued

discovery group (NCD) and for the perceptual-cue-guided discovery (CG)

group did not differ significantly, but a verbal didactic instruction

(VDI) procedure produced a mean which was significantly higher than

both of the first two training groups. However, "over the interim from

post (sic) to delay posttest, the performance of perceptual cue dis-

covery group improved, of the NCD group decreased, and of the VDI treat-

ment group remained stable" (Peters, 1970, p. _713).

Both Whiteman and Peisach (1970) and Peters (1970) employed

matched colors as a perceptual cue to induce correspondence between

the two rows of stimuli. The correspondence established by the match-

ing colors was intended to continue throughout transformations.

However, the correspondence of matched colors is not purely a per-

ceptual correspondence. The matched colors may well evoke iteration

or counting of the objects using individual color symbols rather than

arabic or roman numeral symbols. Such iteration still is primarily a

cognitive rather than a perceptual task. Further both studies mixed

perceptual cues, pure or not, with cognitive cues. White and Peisach

(1970) included sensorimotor experience among their facilitating

experiences. Sensorimotor experience is clearly defined by Piaget as

a cognitive experience. Peters (1970) used number-accentuated blocks

with distinctive arrays of sequins which could have evoked counting

behavior quite readily. Thus neither study provided clear evidence

of the support or interference of perceptual cues only.

The design used in the present study included a figurative

(perceptual) cue conservation-like experience which more nearly

approximates a perceptual task refined to reduce cognitive actions





or operations. Thw low operative cues-high figurative cues experience

used an array consisting of all-yellow blocks placed on a dark blue

background for high figure-ground contrast. It is significant to note

that even in early childhood, perceptual tasks, except perhaps in cases

of tachistoscopic presentation, cannot be entirely free of intellectual

influence. Given Piaget's theoretical conviction that all perceptual

behavior has its origin in sensorimotor actions, it is not possible to

eliminate or control for the contribution of earlier visual or tactile-

kinesthetic sensorimotor schemas. At best, one must assume that

random sampling will control for any systematic differences in sensori-

motor-guided perceptual experience.

Two studies have reported significant success in the induction of

conservation of number using specific verbal rule instruction conditions

(Beilin, 1965; Peters, 1970). Both of these investigators have worded

their verbal instructions to provide three rules which explain the

invariance of the number of objects in the face of configural trans-

formation. Peters ~(1970) instructions will illustrate. Immediately

following completion of each transformation-and-questioning sequence

during training trails the following statement was made:

I have only moved the blocks. They are in another
place, but there are. just as many as before [equivalence
rule]. See, I can put the whole bunch back the way they
were [reversibility rule]. There are still the same
number as before [equivalence] because I did not put
in any more blocks, I only moved them [addition-subtrac-
tion rule]. (p. 713)

The only significant difference in the procedures employed by Beilin

(1965) and Peters (1970) concerned the timing and therefore the

frequency of the rules. Beilin explained conservation only when the

child's response during training was incorrect. Peters included the









explanation on every trial regardless of success. Results in both

studies showed that the children receiving the verbal rule instruction

produced significantly more conservation responses than children in

the control group. It is interesting to note that Hamel and Riksen

(1973), using verbal rule instruction to induce conservation of liquid,

segregated the explanations into two sets of rules based on identity

and reversibility rather than combining them in one elaborate rule.

The subjects receiving instruction for each or the rules performed

significantly better than subjects in a control group. There was no

significant difference between the two verbal rule instruction groups.

In the present study, the low figurative cues-high operative cues

experience group participated in a procedure which included both

sensorimotor experience and verbal rule instruction. The verbal rule

instruction consisted of the equivalence rule, the reversibility rule,

and the addition-subtraction rule. In order to provide operative

sensorimotor experience, the child was asked to manipulate the objects

in one of the two rows by establishing initial equivalence, by trans-

forming the variable row, and finally by returning the transformed row

to its original configuration.

Finally, the procedure for the control group experience was

intended to minimize both perceptual (figurative) and cognitive

(operative) cues as much as possible in order to evaluate the effects

of high operative cues procedure and the high figurative cues procedure

more effectively.



Hypotheses


In considering a developmental theory of cognition, one cannot .










ignore the impact of past experience on the achieved level of compre-

hension and on the readiness to benefit from particular training. In

the present study two levels of cognitive development were represented

in the planned experiences -- nonconservers and transitional or

intuitive conservers. Even within these two substages, individual

differences in relevant experiences could not be controlled directly.

Nonconservers and intuitive conservers were assigned randomly to the

three experience groups; and the assumption was made that the events

which contributed to the development of perceptual schemes and the

actions which provided increments for intellectual schemes were not

significantly different for the children within the groups prior to

their planned experience.

The intention of this study was to intensify figurative experience

or operative experience in separate groups and to observe the effects

on the children's performance of the concrete operational task of

conservation. Several hypotheses were tested. First, since the more

mature late representational schemas of the intuitive conservers

should assimilate the training more easily than the earlier representa-

tional schemas of nonconservers, the immediate and delayed posttest

scores of intuitive conservers on the conservation of number task

should be higher than those scores of nonconservers. Scores for con-

servation of length and of liquid should likewise be higher for the

intuitive conservers. Second, because perceptual cues have been viewed

as irrelevant and distracting in conservation tasks, the immediate and

delayed conservation of number scores of children in the control group

should be higher than the scores of children in the high figurative

cues group. In other words, if past empirical and theoretical evidence





-40-


were accurate and the present study was derived accurately, the high

figurative cues group should demonstrate the lowest posttest scores;

the control group higher posttest scores; and the high operative cues

group the highest posttest scores. The intuitive conservers as a

group should require fewer trials to reach criterion for acquisition

of the concept of conservation than nonconservers. Finally, the high

operative cues group should require fewer trials to acquire the concept

of conservation than the control and the high figurative cues groups.





METHOD


The basic paradigm for the study is a recurrent design in research

focused on aspects of Piaget's conservation tasks (Beilin, 1965;

Belman, 1969; Gruen, 1965; Hamel & Riksen, 1973; Smedslund, 1961;

Wallach, Wall & Anderson, 1967). The basic paradigm includes a pretest

of conservation ability, a training procedure designed to facilitate

the learning of an important skill in judging conservation, and a

posttest of conservation judgments. The present study elaborated the

basic paradigm slightly by adding a screening procedure before pre-

testing in ~order to control for recently identified methodological

problems (Beilin, 1965; Griffiths, Shantz & Sigel, 1967; Harasym,

Boersma & Maguire, 1971) and by employing both immediate and delayed

posttests of specific and nonspecific transfer. The steps in the

resulting design occurred in the following order: (1) screening for

basic concepts, (2) protesting for conservation, (3) planned experience,

(4) immediate posttesting, and (5) delayed posttesting (1 to 2 weeks).

In the first session, the child was screened and pretested for con-

servation. In the second session, he or she was involved in a planned

experience and in the immediate posttesting. The delayed posttesting

was administered in the third session. For those children who required

more than 16 experience trials to reach criterion performance in the

planned experience, there was an additional training session of up to

16 trials before the immediate posttest.






-42-


Subjects


Sixty children from three public elementary schools and two

private nursery schools participated in the study. The design called

for a 3 x 2 nonparametric analysis of variance with an equal number of

children in each of the six cells. Three separate planned experience

groups -- Control, High Figurative Cues, and High Operative Cues --

made up one factor, and two levels of cognitive development -- noncon-

server and intuitive conserver -- comprised the second factor. When

the children had been assigned to one of the levels of cognitive

development, based on criteria reported later in this section, they

were randomly assigned to one of the planned experience groups.

Assignment to the planned experience group was accomplished using a

random number table. Although previous studies had demonstrated no

sex differences for conservation judgments including conservation of

number (Braine, 1959; Dodwell, 1962; Goldschmid, 1967; Pratoomraj &

Johnson, 1966; Uzgiris, 1964), approximately equal numbers of males

and females were assigned to each experience group, as seen in Table

1. As a result 27 girls and 33 boys participated in the study.

Further, 39 white children and 21 black children comprised the sample,

a ratio of 1.85 white children to 1 black child. As seen in Table 1,

approximations of this overall ratio were represented in each of the

planned experience groups. The balance was not so consistent between

intuitive conservers and nonconservers with respect to race or sex.

Mean age for the sample was 68.6 months with a range of 45 to 88 months

and a median of 68.5 months. Although a variety of socio-economic

backgrounds was represented from children of university professors and

professionals to children receiving welfare assistance, specific data











Table 1

Demographic Characteristics of the Sample by Level of Cognitive
Development and by Planned Experience Group



Age Range Mean Sex Race
in Months Age M F B

Nonconservers 45-82 61.7 22 8 12 18

Intuitive 46-88 70.6 11 19 9 21
Conservers




Cb4-871.4 11 9 6 14



HFCa 45-82 67.88 11 9 8 12

HOCb 46-80 66.33 11 9 7 13



Totals 45-88 68.6 33 27 21 39


aHFC High Figurative Cues group
b
C Control group

cHOC High Operative Cues group





on socio-economic level were insufficient to provide a specific dis-

tribution. The sample excluded children who were identified by

teachers or by school records as having notable physical, perceptual,

emotional, or intellectual difficulties and who were thus in need of

special academic assistance.



Procedures and Materials


Screening. Prior to the administration of pre-experience con-

servation tests, each child was evaluated in order to assess his or

her understanding of the relational terms "same" and "more" and his

or her understanding of cardinal number. The assessment was intended

to increase the cognitive homogeneity of the sample by setting a mini-

mum standard for the children's understanding of the language which is

critical for the conventional conservation procedure.

The significance of children's understanding of the relational

terms has been investigated in two recent studies (Griffiths, Shantz,

& Sigel, 1967; Harasym, Boersma, & Maguire, 1971). These two investi-

gations reported dissimilar findings but share a common admonition.

The results reported by Griffiths, Shantz, and Sigel (1967) suggested

that 4- and 5-year-old children use the relational terms "same,"

"more," and "less" less accurately when judging conservation of number

than when judging either conservation of weight or conservation of

length. The term "same" which is the critical response for accurate

judgment of conservation was used correctly significantly less often

than were "more" and "less." In contrast to the accuracy of the term

"more" reported by Griffiths, Shantz, and Sigel (1967), Harasym,





Boersma, and Maguire (1971) used the semantic differential to analyze

the relational terms "more" and "less" and discovered that "less" was

more stable in meaning during the development of conservation. Their

results indicated that the meaning of "more" shifted with the develop-

ment of conservation. Harasym, Boersmda, and Maguire (1971) conclude

that for children, the meaning of "more" becomes increasingly dif-

ferentiated from the meaning of "less" with the development of conser-

vation. Despite the conflict in their data, the conclusions of these

two studies imply at least one common admonition: "To minimize

confounding linguistic and conceptual abilities, the ability to judge

similarity and to use the term 'same' correctly should be determined

prior to testing for conservation" (Griffiths, Shantz, & Sigel, 1967,

p. 486).

Because phrasing of the conservation question used in the present

investigation included only the terms "same" and "more," these two

terms were the focus of the first part of the assessment prior to

protesting for conservation. A two-part discrimination scheme was

adapted from Miller (1973) to test the child's understanding of the

terms "same" and "more." According to Harasym, Boersma, and Maguire

(1971), "when the child is asked whether two objects 'are the same,

or does one have more?' the full implication of the question is 'are

they the same or different, and, if they are different, does one have

more?'" (p. 786). Accepting the interpretation of Harasym et al. as

possible, the child's understanding of the relational terms was

assessed at two levels. First, the child's ability to discriminate

the concepts of "same" and "different" was tested; immediately fol-

lowing, his or her understanding of "same" and "more" was assessed.





The procedure for assessing the child's understanding of "same" and

"different" consisted of the presentation of two stimuli simultaneously.

The stimuli were two-dimensional or three-dimensional geometric shapes.

Both stimuli in each pair were alike or different on salient dimen-

sions. Six displays were presented in counterbalanced order along an

axis parallel to the subject. The basic question posed to the child

was, "Are these things the same or are they different?" A detailed

description of the stimulus materials and procedures used in the

screening for understanding of "same" and "different" is presented in

Appendix A. If the child completed the six arrays which constituted

this first phase of the assessment of relational terms with no more than two

errors, then the experimenter presented four additional arrays designed to

assess his or her understanding of "same" and "more." The four arrays

again consisted of familiar geometric shapes. The objects were arrayed

in two groups with one or two objects per group, so that the total

number of objects in any complete display ranged from two to four.

Two arrays had equal numbers in each group; two had unequal numbers

in each group. The groups were arranged on the right and left sides

of the stimulus display board. The basic question for these four

arrays was, "Do both sides have the same number of things or does one

side have more things?" The details regarding the stimulus materials

and the procedure for assessing each child's understanding of "same"

and "more" are described in Appendix A.

In order to evaluate understanding of cardinal number, another

potentially confounding basic concept, each child was asked to perform

Beilin's (1965) number production task. The child was given a group

of ten white discs and told, "Give me three of them, please." For the





second trial, he or she was asked, "Now, give me six of them."

Finally, each child was asked, "Would you give me eight of them,

please?" The discs were returned after each question so that the child

began each trial with ten discs. The procedure was a non-feedback

procedure; the child's responses were scored simply as correct or

incorrect. Each child had to give an accurate response on two of the

three number production trials in order to be selected for further

participation in the study. The number product-ion task was the last

of the three procedures for assessing the presence of basic relational

and number concepts as criteria for continued participation in the

study.

The second phase of the screening procedure was intended to pro-

vide data for determination of the child's level of cognitive function-

ing. This phase consisted of the presentation of two number equivalence

tasks. These tasks were also borrowed from Beilin (1965).

The first number equivalence task required the child to recognize

the equivalence or non-equivalence of two sets of discs. Four trials

were presented in counterbalanced order with two equivalent sets and

two non-equivalent sets. One display of equivalent sets consisted of

two arrays of four discs arranged in an equilateral triangular con-

figuration with the fourth disc centered in one of the three sides.

The location of the fourth disc was counterbalanced. The array nearest

the child was located about six inches from him or her, and the second

array was about two inches beyond the first. The other display of

equivalent sets consisted of two arrays of three discs, also arranged

in equilateral triangular form. These two arrays were located in the

same positions described for the four-object arrays. The two non-










equivalent displays consisted of a three-item triangular display and

a four-item triangular array as described above, counterbalanced for

position nearer or more distant from the child and for the side in

which the fourth disc was located. The four arrays are depicted in

Appendix B. When the experimenter had constructed the array, he or

she asked the child, "Do both groups have the same number of circles

or does one group have more?" The responses were recorded. The child's

responses to these four items were originally-intended to serve as one

criterion for classification of an individual as either a nonconserver

or an intuitive conserver; however, the perfect or near-perfect

performance of most children on this task rendered it rather meaningless

in discriminating the two levels of cognitive development. The task

was retained in spite of its shortcomings in order to ensure that all

children would receive the same screening procedure. The second number

equivalence task required the child to construct a set equivalent in

number to a model constructed by the experimenter. He or she began

these trials by positioning three discs in a 110-degree angle. He or

she then told the child, "Here are some circles. Would you please

build a group of circles so that your group has the same number of

circles as my group?" The task was repeated with 5, 6, 7, and 8

circles in the angle. After the design containing three discs was

built, the presentation of the other groups was randomized with the

direction of the angle counterbalanced. (Appendix C depicts the

arrangement of the five arrays.) Performance on this task was a

primary determinant for classifying the children as nonconservers or

intuitive conservers.

Pretests for conservation. Following the screening evaluation,





all children selected for continued participation in the investigation

were tested for three types of conservation -- conservation of number,

conservation of length, and conservation of liquid. The types of

conservation protests were presented in counterbalanced order. All

items assessing a particular type of conservation were presented suc-

cessively of course. With the child seated directly across the table

from the experimenter, one stimulus array was designated as belonging

to him or her and the other as belonging to the child for ease of

identification. The child's stimulus, of course, was the one closest

to the subject. The basic procedure for all three types of conserva-

tion was as follows: (1) The experimenter constructed the initial

array with two configurations exactly alike. (2) The child was asked

to confirm the equivalence of the initial configurations. (3) The

experimenter then transformed one of the arrays so that the essential

quality -- number, length, or amount of liquid -- was unchanged while

perceptual dimensions varied. (4) The experimenter then posed the

basic conservation question to the child, asking if the essential

quality in both post-transformation configurations was the same or if

the quality was changed so that one configuration had more (number,

length, liquid). (5) Finally the experimenter asked the child to

explain his or her response to the question, that is, to justify the

conservation judgment of "same" or "more" ("longer" in the case of

length). A detailed description of the stimulus materials and the

instructions for each type of conservation pretest is presented in

Appendix D.

Criteria for determining the initial level of cognitive develop-

ment. By combining the child's performance on the construction of





I


numerical equivalence tasks with his or her performance on the con-

servation protests, his or her level of cognitive development was

determined. In the present study, a child was considered to be a

nonconserver if his performance was characterized by the following:

(1) not more than three correct responses on the screening procedure

for construction of equivalent sets, (2) no correct judgments on the

conservation of number protests, and (3) no adequate or ambiguous

explanations for conservation of number judgments. Intuitive conservers

were designated on the basis of a performance that included: (1) at

least four correct responses in the construction of equivalent sets

(a performance clearly above chance), (2) no more than two correct

judgments on .the conservation of number items, and (3) no more than one

adequate or ambiguous explanation for conserving judgments. The dis-

crimination criterion for the construction of equivalence tasks was

based on the following rationale. First, Gelman and Tucker (1975)

demonstrated that even young nonconservers age 3, could accurately

estimate numerosity of groups of 2 to 5 objects without an overt count-

ing behavior. They labeled the child's perceptual recognition of the

number of objects without counting as subitizing (p. 167). Based on

their results, it seemed reasonable to assume that with the benefit of

the subitizing response aided by chance, any child in the sample might

have given three accurate responses without the ability to attend to

number consistently and carefully. It seemed unlikely that the children

would produce four or five correct responses if they did not attend to

number readily, that is, if number was not a salient dimension given

the appropriate instructions.

The quality of the child's conservation explanations was judged as





adequate if the child referred directly to former (i.e., pre-trans-

formation) equality, reversibility of the transformation, compensation,

addition or subtraction, or the irrelevance of the transformation for

the quality of number. Inadequate explanations were magical, provided

no information justifying the response, made reference only to visual

perceptual cues, or contained only part of an adequate explanation.

For example, if instead of saying, "You just moved them (the discs),

and if you moved them back they would still be the same (or have the

same number)," a child simply stated, "You just moved them," giving no

further information, then the explanation was judged as inadequate. As

noted in the description of the protesting procedure, the child's

judgment of invariance for each trial itself was recorded as correct

(1) or incorrect (0). The child's explanation of number invariance

followed the same binary scoring form. In summary, the profile for a

nonconserver would be as follows: score for construction of equivalent

sets -- 0 to 3; conservation of number score -- 0; and 0 adequate

explanations of conservation. The profile for an intuitive conserver

would resemble the following: score for construction of equivalent

sets -- 4 or 5; conservation of number score -- 0 to 2; and no more

than one adequate or ambiguous explanation of conservation.

Posttesting for conservation. The procedure and scoring criteria

for posttests of conservation were exactly the same as for the protests.

Each child was tested twice following the planned experience. The

first posttest was administered immediately after completion of the

planned experience items during the second session. The second posttest

was administered not less than one week or more than two weeks after

the first posttest.





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Planned experience in conservation of number. The primary

apparatus for the planned experience groups included two painted hard-

board rectangles, 56 cm by 46 cm, and twenty-eight 2.5 cm cubes. One

of the boards had a non-glossy dark blue color; the other was painted

a flat gray color. Fourteen of the cubes matched the flat gray color

of the rectangle with which they were used, while the remaining fourteen

cubes were painted with a glossy, bright yellow enamel and used with

the dark blue display board. External reinforcement consisted of

paper stars affixed either to the child's hand or arm or to a strip of

colored paper.

The procedure for the Control Group, with low figurative cues and

low operative cues, centered on the low contrast apparatus, that is,

the gray display board and matching gray cubes. The conventional

conservation task was modified in several ways for this experience.

First, the practice of satisfying or convincing the child of the

initial equivalence of the two rows was omitted. Instead the experi-

menter constructed two rows of cubes approximately three inches apart

and parallel to each other and to the edge of the display board nearer

the subject. After a brief pause, he or she transformed one of the

rows according to the procedures outlined in the following paragraph

and listed in Table 2. As with the protests, the position of the row

to be transformed (the experimenter's or the child's row) and the type

of transformation (expansion or contraction) were counterbalanced as

indicated in Table 2. The experimenter asked the conservation of

number question. If the child's response was correct, the experimenter

reinforced the response by repeating the child's response and pre-

senting a star. If the child's response was incorrect, the experimenter




















Number in Each Row
Item in a Starting Positiona Transformation


1 A = B A elongated

2 A = B B elongated

3 A > B B elongated

4 A < B A elongated

5 A = B A shortened

6 A = B B shortened

7 A > B A shortened

8 A < B B shortened



aEqual rows had seven blocks in each row. Unequal rows had seven
blocks in the greater row and six blocks in the lesser row.


Table 2

Items Used in Planned Experience Groups for
Conservation of Number










said, "That's not quite right; you have not discovered the secret about

what happens to the blocks in our game."

In order to prevent response stereotypy, half of the eight distinct

trials displayed an equal number of cubes in each row and the remaining

four trials used two rows with unequal numbers of blocks (one row had

one more cube than the other row). The trials using unequal rows

required conservation of inequality. For the trials using equal rows,

the initial, pre-transformation length and density of the two rows

was equal, so that the cubes in one row were positioned directly

opposite from the cubes of the second row. For the trials using the

unequal rows, the pre-transformation rows were constructed with unequal

lengths and densities. None of the cubes in the longer, more numerous

row was directly opposite from a cube in the shorter, less numerous

row. The items are presented descriptively in Table 2. These were

not the only transformations possible; but in each transformation,

there was a planned inconsistency between the length of the row and

the number of cubes in the row so that the less numerous row was the

longer row after transformation. Thus, the child had to make a con-

serving response in order to answer correctly (Gruen, 1965).

The procedure for the experience group with low operative cues

and high figurative cues was the same procedure used for the Control

group except for stimulus materials. Instead of dealing with low

contrast matching gray displays, the. High Figurative Cues experience

group used the high contrast display with a dark blue display board

and glossy bright yellow cubes.





rhe planned experience proceeded somewhat differently with low

figurative cues and high operative cues. The figurative cues were

minimized, as they were in the control group, by using the low contrast

gray materials. The operative cues, which were accentuated, consisted

of sensorimotor actions and verbal rule instruction. The planned

experience involved the child's sensorimotor schemes by requiring him

or her to participate in the actual construction and transformation of

the rows. The intellectual operations which Piaget asserts are neces-

sary for true conservation were emphasized in the form of a verbal

rule centering on the invariance of the number of objects despite

deformation of the length and density of a row.

Borrowing a procedure from Wallach, Wall, and Anderson (1967),

the High Operative Cues experience preceded in the following sequence:

(1) Using the scheme described in Table 2, the child built his or her

own row of blocks as the experimenter built his or her own. (2) The

child was asked to confirm the equality or inequality of these two

original rows. (3) Then the child transformed one of the rows by

moving the blocks apart or closer together. (4) Finally each child

was asked to make a conservation judgment and support the judgment with

a reason. If both the conservation judgment and the explanation were

accurate, the experimenter reinforced the child's response by giving

him or her a gummed paper star and by repeating the response. If

either the judgment or the explanation were inaccurate or ambiguous,

then the experimenter's feedback to the child consisted of verbal rule

instruction which included the notions of equivalence, reversibility,

and no addition or subtraction of objects as logical reasons for the










invariance of number. Finally, the child was asked to return the

transformed row to its original configuration. Greater details of the

procedure for the High Operative Cues experience are provided in

Appendix E. The verbal rule instruction provided experience in con-

crete operational reasoning. Having the child shift the cubes back

into the original array provided sensorimotor experience with the

principle of reversibility which Piaget asserts is essential for true

conservation.

For all three planned experiences -- Control', High Figurative Cues,

and High Operative Cues -- the criterion for learning to conserve

number was correct responses on six consecutive trials. The maximum

number of trials was 32. Early pilot experience with the children

demonstrated frequent inability to attend carefully for 32 consecutive

trials. Consequently, for the children in the present study, the

planned experience was divided into two 16-trial sessions if the child

did not reach criterion performance within the first 16 trials.

Controls for experimenter bias. For the first 22 children, three

experimenters, two undergraduate psychology students working for course

credit and the investigator, administered the three phases of the study

-- screening and protesting for conservation, planned experience, and

posttesting. Unfortunately, when clearance was finally obtained to

begin screening for the remaining 38 children, the University quarter

was well under way and student volunteers were scarce. Consequently,

for the final 38 children, one experimenter screened the children and

assigned them randomly to planned experience groups. The second

experimenter conducted the planned experience approximately one week

later. Finally, with no indication of the experience in which a child





had participated, the first experimenter conducted immediate and delayed

posttesting.

Judgment of the adequacy of responses. As noted in the introduc-

tion, the definition of a conserving response in the present study

included an adequate explanation for the child's conserving judgment.

Three judges, including the investigator, classified the children's

explanations of conservation as adequate (1) or inadequate (0). The

two judges, having limited familiarity with conservation tasks and con-

serving judgments and explanations, were trained to recognize acceptable

conserving explanations. Criterion for adequate comprehension of the

distinction between adequate and inadequate responses by the judges

prior to classification of the children's responses was nine correct

classifications out of ten statements which were taken from pilot study

responses.

Of the 1,440 total explanations (60 children x 4 trials x 6 post-

test conservation tasks), 87.5% were repetitious. In fact there were

83 unique explanations for conservation of number (out of 480), 50

unique explanations of conservation of length, and 46 unique explana-

tions of conservation of liquid. Judges were only asked to classify

the 179 unique conservation explanations offered by the 60 subjects.

















RESULTS


Interjudge Agreement in Classifying Subjects' Responase


Interjudge agreement on classification of the subjects' responses

on the posttests was reflected in two ways -- percentage of unanimous

agreement and a Cochran Q test. Judges were asked to classify the

unique responses to the three types of conservation tasks as conserving

or nonconserving (ambiguous responses were considered as nonconserving).

For conservation of number, the major dependent variable, the three

judges achieved 97.5 per cent unanimous agreement in classifying the

children's responses for 83 unique explanations. The judges achieved

95.7 per cent agreement in classifying the responses for conservation

of liquid with 46 unique statements and 100 per cent agreement in

classifying responses for conservation of length for 50 unique explana-

tions.

In order to provide some comparison of the judges'classifications,

the Cochran Q test was used to determine if the frequencies of the

different classifications, conserving or nonconserving, differed among

the three samples. Results of the test which is distributed like the

chi-square statistic were not significant (Q = 1.0005; df = 2). It

should be noted that the Cochran test does not assess correspondence

among judges' classifications for individual statements but only a

global comparison of frequencies of each classification for each

judge.


-58-












































































~


Analysis of Dependent Variables


Rationale for nonparametric analysis. Standard analysis of

variance techniques could not be applied the the data because responses

were not normally distributed; specifically, the ranges for posttest

scores of conservation were too compact. This condition was true for

consideration responses which were defined by both accurate judgments

with accurate explanations and those which were defined by accurate

judgments alone. Table 3 presents the means and standard deviations

for the immediate and delayed posttest conservation scores using judg-

ments with explanations and illustrates the failure of the data to ap-

proximate a normal distribution. Table 4 depicts the means and standard

deviations for the posttest conservation scores using judgments only;

these data also fail to approximate a normal distribution. The distribu-

tion of trials to acquisition of the concept was highly skewed, thereby

eliminating the standard analysis of variance technique. Table 5 presents

the means and standard deviations for the number of trials to acquisition.

Noting that the maximum number of trials was 32, the skewness is ap-

parent. Therefore, a multivariate extension of the Friedman test was

used. The Friedman nonparametric analysis of variance makes no assump-

tions about the frequency distribution of scores. It performs on

analysis of the frequency distributions of the ranks of measurements

rather than the measurements themselves. This test statistic is a chi-

square statistic rather than an F statistic. It permits comparisons

of main effects but no interactions. No interactions were predicted

by the hypothesis. The analysis was conducted in two stages. First

a six-variate analysis using the six posttest scores (immediate and

delayed posttest scores for number, liquid, and length) was performed




















Nonconserver Intuitive Conserver

Immediate Posttest HFCa Cb HOCC HFC C HOC


Number M 0.6 1.6 3.0 12 2.0 3.5
SD 1.28 1.96 1.61 1.83 2.0 1.20


Length M 0.0 0.4 0.8 0.6 0.0 1.9
SD 0.0 1.2 1.6 1.28 0.0 1.81


Liquid M 0.2 0.0 0.4 0.4 0.0 0.8
SD 1.98 0.0 1.2 1.2 0.0 1.6



Delayed Posttest HFC C HOC HFC C HOC

Number M 0.8 1.6 2.4 1.2 2.0 3.6
SD 1.6 1.96 1.96 1.83 2.0 1.2


Length M 0.2 0.4 0.8 0.8 0.0 1.6
SD 0.6 1.2 1.33 1.6 0.0 1.96


Liquid M 0.3 0.4 0.0 0.4 0.0 1.2
SD 0.9 1.2 0.0 1.2 0.0 1.83


Table 3

Means (M) and Standard Deviations (SD) of the Number of
Correct Judgment and Explanation
Conservation Responses


aHFC High Figurative Cues group
b
C Control group

CHOC High Operative Cues group




















Nonconservers

Immediate Posttest HFCa Cb


Number M 0.7 1.6
SD 1.27 1.96



Length M 0.2 0.4
SD 0.6 1.2



Liquid M 0.9 0.3
SD 1.3 0.9




Delayed Posttest HFC C


Number M 0.9 1.6
SD 0.3 1.96



Length M 0.3 0.4
SD 0.64 1.2



Liquid M 0.4 0.6
SD 0.92 1.28



aHFC High Figurative Cues group

C Control group

CHOC High Operative Cues group


HOCc


3.3
1.27


1.0
1.61


0.5
1.2




HOC


2.8
1.83


0.9
1.3


0.1
0.3


~


~


-61-




Table 4

Means (M) and Standard Deviations (SD) of. the Number
of Correct Judgment Only Conservation Responses


e Conservers

C HOC


2.0 3.5
2.0 1.2


0.0 1.9
0.0 1.81


0.2 0.9
0.6 1.58




C HOC


2.0 3.6
2.0 1.2


0.0 1.6
0.0 1.96


0.4 1.2
1.2 1.83


Intuitiv

HFC


1.2
1.83


0.6
1.28


0.5
1.2




HFC


1.2
1.83


0.8
1.6


0.4
1.2




















Number of Trials

Nonconservers M SD


HFCa 29.6 7.2

Cb 27.1 9.39

HOCc 19.4 10.25





Intuitive Conservers M SD


HFC 27.0 8.08

C 24.4 9.87

HOC 14.3 9.30


-62-




Table 5

Means (M) and Standard Deviations (SD) for Number
of Trials to Acquisition of
Conservation of Number


aHFC High Figurative Cues group
b
C Control group

cHOC High Operative Cues group










for both conservation responses including judgment and explanation and

for judgment-only conservation responses to determine the effects of

the factors on the subjects' responses. The factors included the two

levels of initial cognitive development -- nonconserver and intuitive

conserver -- and the three experience groups -- Control, High Figurative

Cues and High Operative Cues. In the second stage, the analysis was

performed using trials to acquisition of the concept as the single

variable.

Experience group effects. An inspection of the distributions of

posttest scores for the experience groups in Tables 6, 8, 10, 12, 14

and 15 reveals the differences in the performances of subjects in each

group. Holding the initial level of cognitive development constant as

a block variable, the results of the overall tests of the experience

group factor were significant for both judgment-with-explanation conser-

vation scores and judgment-only conservation scores (X2 = 27.39;

df = 12; p < .007 and X2 = 32.95; df = 12; p < .001 respectively).

Once the global analysis determined that the overall effects of the

three experience groups were significant in the degree to which they

improved posttest performance, that is, in improving the scores in the

respective distributions, more detailed analysis was reasonable. A

separate global analysis for the effects of planned experiences on the

distributions of posttest scores for each of the three types of con-

servation was performed. If this second analysis revealed significant

effects, then performance of each experience group was compared with

that of the other experience groups to discern the pattern of relation-

ships among the distributions. In order to facilitate comparisons of

the distributions of the posttest performances of the three experience











Table 6

Frequency Distributions of RawScores for the Conservation
of Number Posttests -- Judgment and Explanation



Posttest Scores

0 1 2 3 4


Immediate Posttest-

Level of Cognitive Development

Nonconservers 16 0 2 0 12

Intuitive Conservers 12 0 0 1 16

Planned Experience Groups

Cb 11 0 0 0 9

HFCa 15 0 1 0 4

HOCc 3 0 1 1 15




Delayed Posttest

Level of Cognitive Development

Nonconservers 18 0 0 0 12

Intuitive Conservers 13 0 0 0 17

Planned Experience Groups

C 11 0 0 0 9

HFC 15 0 0 0 5

HOC 5 0 0 15


aHFC High Figurative Cues group

C Control group

cHOC High Operative Cues group





groups, the rank means for each posttest were compared pairwise by

experience group using a z score. The original posttest scores were

transformed into ranks for the nonparametric analysis of variance.

Without further transformation, the rank means reflected the differences

in the distributions, if any, among the different populations repre-

sented by each experience group. The z score provided a standardized

comparison of each pair of experience groups -- Control and High

Operative Cues, High Figurative Cues and High Operative Cues, and

Control and High Figurative Cues. The significance of any differences

between rank means was immediately evident from the cumulative normal

probabilities of z scores.

Analysis of the effects of the experience groups on the posttest

scores for conservation of number, the primary dependent variable,

revealed significant differences in the distributions of the immediate

posttest responses (judgment with explanation -- X2 = 15.37; df = 2;

p < .0005; and judgment only -- X2 = 16.5; df = 2; p < .0003) and of

the delayed posttest responses (judgment with explanation -- X2 =- 10.087;

df = 2; p < .007; and judgment only -- X2 = 6.69; df = 2; p < .004).

Pairwise comparisons of the z scores (see Table 7) provided evidence

that the scores for the High Operative Cues group in the distribution

for judgment-with-explanation responses were significantly higher

(p < .01) than the scores in the judgment-with-explanation distributions

for the High Figurative Cues group or the Control group in the im-

mediate posttest (see Table 6). These same pariwise comparisons for

the distributions of judgment-with-explanation scores also indicated

that the High Operative Cues group tended to score higher (p < .001)

than the High Figurative Cues group on the delayed posttest of




















Immediate Posttest

Cb HFC


HFCa 1.6169

HOCc -2.2839r -3.9008**






Delayed Posttest

C HFC


HFC 1.2620

HOC -1.8930 -3.1550**


Table 7

Comparison z Scores for Experience Groups
for Conservation of Number --
Judgment and Explanation


aHFC High Figurative Cues group

C Control group

CHOC High Operative Cues group

* p < .01

** p < 001





conservation of number. Pairwise comparisons of the z scores for the

judgment-only conservation scores (see Table 9) demonstrated the same

pattern for the distribution of immediate posttest judgment-only scores

for conservation of number (see Table 8) as evidenced by the judgment-

with-explanation scores. For the delayed posttest scores using the

judgment-only criterion, the pairwise comparisons indicated that the

High Operative Cues group tended to score higher (p < .02) than either

the High Figurative Cues or the Control groups for conservation of

number. There was no significant difference between the performances of

the High Figurative Cues group and the Control group an either immediate

or delayed posttests of conservation of number for judgment-with-

explanation or judgment-only responses.

Similarly, analysis of the nonspecific transfer effects of the

number experience groups on posttest scores for conservation of length

demonstrated significant differences among the distributions of the

groups on both the immediate posttests (judgment with explanation --

X2 = 9.682; df = 2; p < .008; and judgment only -- X2 = 10.009;

df = 2; p < .006) and the delayed posttests (judgment with explanation

-- X2 = 5.805; df = 2; p < .05; and judgment only -- X2 =664

df = 2; p < .035). Again pairwise comparisons (see Table 11) provided

data suggesting that the scores in the judgment-with-explanation

distribution for the High Operative Cues group were significantly

higher (p < .003) than those for the High Figurative Cues group and

the Control group on the immediate posttest of length (see Table 10).

On the delayed posttest for conservation of length, scores in the

distribution of judgment-with-explanation responses for the High

Operative Cues group tended to be higher than for the Control group






-68-


Table 8

Frequency Distributions of Raw Scores for the Conservation
of Number Posttests -- Judgment Only



Posttest Scores

0 1 2 3 4


Immediate Posttest

Level of Cognitive Development

Nonconservers 14 1 2 1 12

Intuitive Conservers 13 0 0 1 16

Planned Experience Groups

Cb 11 0 0 0 9

HFCa 14 1 1 0 4

HOCc 2 0 1 2 15




Delayed Posttest

Level of Cognitive Development

Nonconservers 16 1 0 0 13

Intuitive Conservers 13 0 0 0 17

Planned Experience Groups

C 11 0 0 0 9

HFC 14 1 0 0 5

HOC 4 0 0 0 16


HFC High Figurative Cues group

C Control group

cHOC High Operative Cues group




















Immediate Posttest

Cb HFC


HFCa 1.5024

HOCc -2.5172** -4.0195"***






Delayed Posttest

C HFC


HFC 1.0869

HOC -2.2049* -3.2918*n*



aHFC High Figurative Cues group

C Control group

HOC High Operative Cues group

* p <.01

** p <.006

*** p <.0007


Table 9

Comparison z Scores for Experience -Groups
for Conservation of Number --
Judgment Only











Table 10

Frequency Distributions of Raw Scores for the Conservation
of Length Posttests -- Judgment and Explanation



Posttest Scores

0 1 2 3 4


Immediate Posttest

Level of Cognitive Development

Nonconservers 27 0 0 0 3

Intuitive Conservers 22 1 2 0 4

Planned Experience Groups

Cb 19 0 0 0 1

HFCa 18 0 1 0 1

HOCc 12 1 1 0 5




Delayed Posttest

Level of Cognitive Development

Nonconservers 25 1 2 0 2

Intuitive Conservers 24 0 0 0 6

Planned Experience Groups

C 19 0 0 0 1

HFC 17 1 0 0 2

HOC 13 0 2 0 5


aHFC High Figurative Cues group

C Control group

CHOC High Operative Cues group




















Immediate Posttest

Cb HFC


HFCa 0.3945

HOCc -2.8702*** -2.4758**







Delayed Posttest

C HFC


HFC -0.7359

HOC -2.3548* -1.6189


-71-



Table 11

Comparison 2 Scores for Experience Groups
for Conservation of Length --
Judgment. and Explanation


aHFC High Figurative Cues group

C Control group

cHOC High Operative Cues group

* p < .009

** p <.006

***' p < .003






-72-


(p < .009). Pairwise comparisons of z scores for the judgment-only

responses for conservation of length on the immediate posttest also

demonstrated that the scores of the judgment-only distribution for the

High Operative Cues group were significantly higher (p < .01) than for

the High Figurative Cues group and the Control group (see Tables 12

and 13). And for the delayed posttest conservation of number responses

using the judgment-only criterion, scores in the distribution of the

High Operative Cues group tended to be higher-than those of the Control

group (p < .005). There was no significant difference between High

Operative Cues and High Figurative Cues distributions for either

judgment-with-explanation or judgment-only responses.

For conservation of liquid, analysis of the effects of the experi-

ence groups on the posttest performance yielded no significant dif-

ference for the distributions on the immediate posttest (judgment with

explanation -- X2 = 3.006; df = 2; and judgment only -- X2 = 2.885;

df = 2) or the delayed posttest (judgment with explanation -- X2 = 1.052;

df = 2; and judgment only -- X2 = 0.612; df = 2). No further analysis

was appropriate (see Tables 14 and 15).

The second stage of the nonparametric analysis of variance

evaluated the effects of the experience groups on the number of trials

required to acquire the concept of conservation. Global analysis in-

dicated a significant difference among experience groups in the

distributions for number of trials to criterion for acquisition

(X2 = 14.117; df = 2; p < .0009). Further analysis (see Tables 16 and

17) demonstrated that the scores in the distribution for number of

trials for acquisition by children in the High Operative Cues group

was significantly lower than the same distribution for children in the



















Posttest Scores

0 1 2 3 4


aHFC High Figurative Cues group

C Control group

cHOC High Operative Cues group


Table

Frequency Distributions of Raw
of Length Posttests


12

Scores for the Conservation
-- Judgment Only


Immediate Posttest

Level of Cognitive Development

Nonconservers

Intuitive Conservers

Planned Experience Groups



HFCa

HOCc




Delayed Posttest

Level of Cognitive Development

Nonconservers

Intuitive Conservers

Planned Experience Groups



HFC

HOC


0 0

2 0

2 0








3 0

0 0




















Immediate Posttest

Cb HFC


HFCa -0.6784

HOCc -3.0153*** -2.2269*








Delayed Posttest

C HFC


HFC -1.0016

HOC -2.5667** -1.5650



aHFC High Figurative Cues group

C Control group

CHOC High Operative Cues group

* p <.01

** p <.005

*** p <.001


Table 13

Comparison z Scores for Experience Groups
for Conservation of Length --
Judgment Only










Table 14

Frequency Distributions of Raw Scores for the Conservation
of Liquid Posttests -- Judgment and Explanation



Posttest Scores

0 1 2 3 4


Immediate Posttest

Level of Cognitive Development

Nonconservers 28 0 1 0 1

Intuitive Conservers 27 0 0 0 3

Planned Experience Groups

Cb 20 0 0 0 0

HFCa 18 0 1 0 1

HOCc 16 1 0 0 3



Delayed Posttest

Level of Cognitive Development

Nonconservers 28 0 0 1 1

Intuitive Conservers 26 0 0 0 4

Planned Experience Groups

C 19 0 0 0 1

HFC 18 0 0 1 1

HOC 17 0 0 0 3


aHFC High Figurative Cues group

C Control group

cHOC High Operative Cues group





-76-


Table 15

Frequency Distributions of Raw Scores for the Conservation
of Liquid Posttests -- Judgment Only



Posttest Scores

0 1 2 3 4


Immediate Posttest

Level of Cognitive Development

Nonconservers 23 2 2 1 2

Intuitive Conservers 24 2 1 0 3

Planned Experience Groups

Cb 18 1 0 1 0

HFCa 14 2 2 0 2

HOCc 15 2 0 0 3



Delayed Posttest

Level of Cognitive Development

Nonconservers 25 2 1 0 2

Intuitive Conservers 25 0 0 0 5

Planned Experience Groups

C 17 0 1 0 2

HFC 17 1 0 0 2

HOC 16 1 0 0 3


aHFC High Figurative Cues group

C Control group

HOC High Operative Cues group



















Number of Trials

6 8 10 11 12 13 14 17 18 22 30 31 32


Level of
Cognitive
Development

Nonconserver 1 3 1 2 1 1 1 1 19

Intuitive 2 3 3 1 1 1 1 2 1 15
Conserver








Planned
Experience
Groups

Cb 2 1 1 2 1 13

HFCa 1 1 1 1 16

HOCc 1 5 2 3 1 1 1 1 5


Table 16

Frequency~ Distribution of Raw Scores for Number
of Trials to Acquisition


a
HFC High Figurative Cues group
b
C Control group

cHOC High Operative Cues group



















Cb HFC



HFCa -1.0282



HOCc 2.6155* 3.6437**


-78-


Table 17

Comparison z Scores for Experience Groups
for Number of Trials to Acquisition


aHFC High Figurative Cues group

C Control group

HOC High Operative Cues group

* p < .004

** p < .0001





Control group or the High Figurative Cues group (p < .004). Once more,

there was no significant difference between the High Figurative Cues

group and the Control group.

Effects of initial level of cognitive development. Holding the

effects of the three experience groups constant as a block variable,

the overall test of the effect of the initial level of cognitive develop-

ment was not significant (judgment with explanation -- X2 = 4.413;

df = 6; and judgment only -- X2 = 5.835; df = 6). Likewise, comparison

of the number of trials to acquisition revealed no significant dif-

ferences in the distributions of nonconservers and intuitive conservers

(X2 = 2.519; df = 1).

Summary of hypothesis testing. Considering both the more demanding

j udgment-with-explanation criterion as well as the judgment-only

criterion for conserving responses, there was no support for the first

hypothesis which stated that intuitive conservers would score higher

than nonconservers on all posttests. The second hypothesis, which

predicted that children in the High Operative Cues group would score

higher than the children in the Control group, who in turn would score

higher than children in the High Figurative Cues group on the posttests,

was partially supported. The data indicated that the scores in the

distributions of both judgment-with-explanation and judgment-only

posttest scores for the High Operative Cues group exceeded the scores

in the distributions for both the High Figurative Cues group and the

Control group on the immediate posttests for conservation of number.

On the delayed posttest, using the judgment-with-explanation criterion,

the scores in the distribution for the High Operative Cues group only

exceeded the scores in the distribution of the High Figurative Cues





group. Using the judgment-only criterion, the scores in the distribu-

tion for the High Operative Cues group exceeded the scores in the

distribution for both the High Figurative Cues group and the Control

group. Evidence also indicated that for both judgment-with-explanation

and judgment-only criteria situations, the scores in the distribution

for the High Operative Cues group exceeded the scores in the distribu-

tions for both the High Figurative Cues group and the Control group on

the immediate posttest of conservation of length and exceeded the scores

in the distribution for the Control group on the delayed posttest of

conservation of length. There was no evidence of a difference between

the High Figurative Cues group and the Control group on any of the

posttests.

Finally, data indicated that the High Operative Cues group acquired

the concept of conservation of number in significantly fewer trials

than either the High Figurative Cues group or the Control group. How-

ever, since there was no difference in the distributions of the number

of trials to acquisition for nonconservers and intuitive conservers, the

first part of the hypothesis was not supported.
















DISCUSSION


In considering the results of the present study, note that in-

itially, after the protesting for conservation, the investigator

assumed that the error of falsely classifying a response as conserving

when a child had not acquired the concept of operational conservation

was a more serious risk than the error of failing to recognize con-

serving responses which were in fact valid. In other words, to con-

clude falsely that a child had acquired the concept of conservation

and thus ignore his or her need for the relevant experience was more

serious than to mistakenly judge his responses as less mature, that

is, nonconserving, and provide additional experience. This assumption

led to the decision to use a more complete, more complex definition

of a conserving response in order to verify the reasoning process used

to make the conserving judgment. The assumption had the effect of

decreasing the probability of a Type I error, that is, that the results

would suggest the false conclusion that the experience in the various

groups had resulted in significant cognitive progression. Simultaneous-

ly, the probability of a Type II error was increased since a simpler,

less demanding standard of classification might have produced a more

significant difference and the more probable and perhaps erroneous

acceptance of the alternative hypothesis. Later, however, the less

demanding criterion of a correct judgment only was adopted for the

purpose of additional analysis and comparison.






-82-


Comparison of the effects of the three planned experience groups

and of the two initial levels of cognitive development on the posttest

conservation responses comprised of both accurate judgments and

explanations with same effects on the judgment only conservation re-

sponses revealed just one noticeable difference. With the less

stringent criterion, the distribution of the judgment only conservation

responses was significantly higher for the High Operative Cues group

than for both the High Figurative Cues and the Control groups on the

delayed posttest for conservation of number. For the judgment and

explanation conservation criterion, only the High Figurative Cues group

was significantly different (lower) than the High Operative Cues group

on the delayed posttest for conservation of number. These results are

consistent with the argument formulated by Bruner et ~al. (1966) and

Elkind (1967) that there is no essential difference between conservation

judgments with or without adequate explanations. But this argument is

based on the assumption that identity (not reversibility) is the basic

characteristic of conservation. The fact that analyses based on

judgment with explanation as well as judgment only yield compatible

results therefore indirectly supports Piaget's belief that conservation

is characterized not only by the identity operation but also by

reversibility.

Although the mean numbers of conserving responses for the quality

of number across experience groups were in the expected direction for

both judgment-with-explanation and judgment-only criterion, the results

were only partially significant. The more intense cognitive experience

of the High Operative Cues group produced the expected result for

conservation of number and conservation of length on the immediate





-83-


posttest. There~ was a significantly higher distribution of conserving

responses for the High Operative Cues experience children than for the

High Figurative Cues or Control children using both judgment-with-

explanation and judgment-only conservation response criteria on the

immediate posttest. On the delayed posttest, the scores in the dis-

tribution for conserving responses including judgment and explanation

by children in the High Operative Cues group exceeded those scores for

the High Figurative Cues group on conservation of number and those for

the Control group on conservation of length, For the judgment-only

conservation responses on the delayed posttests, the High Operative

Cues group's distribution of scores was significantly higher than both

High Figurative Cues and Control groups for conservation of number.

Further, the results indicated that the children who participated in

High Operative Cues experience achieved acquisition of the conservation

concept significantly quicker than the children participating in the

other two groups individually. Thus, the results were consistent with

Beilin (1965) and Peters (1970) in affirming that verbal rule instruc-

tion as a model for adequate reasoning about conservation of number is

an effective experience. There was limited transfer of learning to

conservation of length and no transfer to conservation of liquid. The

endurance of the conserving response for number over the 10 to 14 day

delay before the second posttest suggested that the acquisition of the

concept was not simply due to specific operant learning.

.As a matter of intention in the present study, the effectiveness

of more intense cognitive experience of the High Operative Cues group

was of secondary .importance. Of greater interest were the effects,

if any, of manipulating certain perceptual cues in the children's





experiences of a conservation-like task. The specific manipulation

involved essentially contrasting the intensity of perceptual cues in

the Control and High Figurative Cues groups. There was no clear

evidence in the results of any significant effects related to the

manipulation of these perceptual cues. Comparison of the posttest

performances of children in the High Figurative Cues group with per-

formances of children in the Control group yielded no significant

differences on any of the three types of conservation. Unfortunately,

the present data provide no clues as to reasons for the absence of

significant differences in the responses of children experiencing the

high contrast materials. Numerous references to size and length among

the children's explanations of their nonconserving responses suggest

that length was a salient cue in the misjudgments in both High

Figurative Cues and Control experiences. The only conclusion which

seems warranted from the present data is that the attempt to diminish

attention to the irrelevant transformation by using a matching

stimulus array in the Control experience was unsuccessful and that

subjects attended to the irrelevant transformations in the High

Figurative Cues and Control experiences with equal intensity. Since

the work of Gelman and Tucker (1975), Peters (1970), and Whiteman and

Peisach (1970) has demonstrated that heterogeneous stimulus objects

support better operative performance, reasoning inversely, it seems

that homogeneous objects should provide less support for operative

functioning. In fact, Gelman and Tucker (1975) have speculated that

the different stimulus objects facilitate subjects' estimations of

number by calling attention to individual objects. However, one bit

of indirect evidence precludes a clear rejection of the notion that






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the manipulation of perceptual cues had no effect. As noted previously,

comparison of the effects of the planned experiences on the acquisition

of conservation of number revealed that during immediate posttesting

using the judgment-with-explanation conservation criterion, the

children in the High Operative Cues group performed significantly

better than the High Figurative Cues group and the Control group.

However, after 10 to 14 days, there was no significant difference

between the performance of children in the High Operative Cues group

and children in the Control group, both of whom had experienced the

low-contrast gray stimulus materials. In contrast, the significant

differences between the distribution of the High Operative Cues scores

for conservation of number and the High Figurative Cues group's dis-

tribution of scores for conservation of number was clearly maintained.

The strength of the question raised by this result was not supported

by the results for judgment-only responses for conservation of number

or the results for judgment-with-explanation responses or judgment-only

responses for conservation of length.

In view of the intention of the present study to contrast figura-

tive (perceptual) cues in two planned experiences and to contrast

basically figurative experiences with intense operative (cognitive)

experiences, conclusions from the study of figurative and operative

aspects of children's logical inference by Youniss and Dennison (1971)

seem appropriate to consider. Based on the results of their study of

transitive relations influenced by figurative cues, operative cues,

or both, they concluded that figurative and operative aspects of

knowing about the world appear to be complementary within developmental

devels and that "the operative aspect, stemming from structural change,









appears to dominate the figurative" (p. 1846). The advantage of the

complementary system as seen by Youniss and Dennison is "that it

takes into account the specificity of the objects' properties as well

as incorporating common properties into a structural system" (p. 1846).

Perhaps the attempts by Whiteman and Peisach (1970) and by Peters

(1970) to study combinations of cognitive and perceptual cues are more

in keeping with actual human ways of knowing.

Finally, another important factor in the present study, the

initial level of cognitive development, produced no significant effects

for conserving responses in the posttesting for the three types of

conservation or for the speed of acquisition of the concept of con-

servation. The results suggest three possible conclusions. First, the

operational definition of the distinction between the two levels of

cognitive development may be inadequate by itself and thus not suf-

ficiently discriminating. Second, the experience groups may have

affected children at each level in the same way. The latter alterna-

tive seems highly unlikely in view of the pattern of significant and

nonsignificant results in the analysis of the effects of the experience

groups. Third, Table 1 clearly indicates a significant imbalance by

sex among children in the two levels of cognitive development.

Although previous research, cited in the Method section, consistently

indicates no sex differences in the ability to conserve, the influence

of the children's gender cannot be ruled out in attempting to account

for the lack of significance between the nonconservers and the intuitive

conservers. Nevertheless, it appears that the present attempt to

classify youngsters as intuitive conservers based primarily on their

ability to establish one-to-one correspondence was insufficient to





define this level of cognitive development.

For those who share the author's concern for assessing and

understanding the processes which children must experience to advance

from global comparisons through a transitional level to operative

(logical) conservation, several options are available. First, one

might decide to discriminate nonconservers from intuitive conservers

by widening the gap in the ability to establish one-to-one corres-

pondence, that is, to define nonconservers as children who are unable

to establish one-to-one correspondence at all and intuitive conservers

as children to establish one-to-one correspondence four or five times

out of five trials. The apparent difficulty with that decision would

be finding children who were absolutely unable to establish one-to-one

correspondence and yet were able to understand the basic concepts and

relational terms. Second, one might follow the lead of Brainerd and

Brainerd (1972) and Miller and Heldmeyer (1975) by defining intuitive

conservation in terms of the ability to predi et conservation prior to

observing the transformation followed by the inability to maintain the

conserving response after seeing the transformation. Third, one might

select intuitive conservers as Harasym, Boersma, and Maguire (1971)

did by classifying children who make conserving judgments but are

unable to explain them adequately as intuitive conservers. The data

from the present study indicate little or no difference between

children who make conserving judgments consistently and those who make

judgments with adequate explanations consistently. Finally, one might

use a combination of the various descriptive behaviors which have been

used singly to define intuitive or transitional conservation. It

remains for some investigator to carefully assess the various definitions









of intuitive conservation to determine if researchers define the

intuitive level too narrowly by using only a single criterion. Al-

though the results of this present study do not directly support the

results of Gelman (1969) and of Wallach, Wall, and Anderson (1967),

they are nonetheless consistent with the conclusion reached in each

of these studies that experience which focuses the child's attention

on relevant stimulus dimensions in making the appropriate conservation

judgment facilitates the acquisition of the concept of conservation.

In summary, the results suggest four major conclusions. First,

consistent with previous findings, the children whose planned experience

included intense cognitive cues, both sensorimotor actions and verbal

rule instruction focusing attention on relevant attributes of the

conservation of number task, demonstrated a significantly higher

distribution of conserving responses on the immediate posttests for

conservation of number and for conservation of length than children

in two other experience groups which minimized cognitive (operative)

cues. The children in the High Operative Cues experience for conserva-

tion of number also displayed higher distributions than children in the

High Figurative Cues experience for delayed posttest scores on con-

servation of number. Likewise, the distribution of the delayed post-

test scores for conservation of length was significantly higher for

High Operative Cues children compared with the same posttest scores

for the youngsters in the High Figurative Cues experience. The sig-

nificant results for the conservation of length posttests imply

generalization of the effects of the cognitive experience with number

to another significant attributes of things in the world. Third,

operational definition of the achievement of intuitive conservation






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of number using the child's ability to construct one-to-one corres-

pondence spontaneously does not appear to be adequate. Finally, there

was no support for the expectation that with minimal cognitive

structural cues in two kinds of brief experience (Control and High

Figurative Cues) the experience with the most pronounced perceptual

cues (High Figurative Cues) would interfere with acquisition of the

concept of conservation to a greater degree.




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