Group Title: measure of meter conservation in music, based on Piaget's theory
Title: A measure of meter conservation in music, based on Piaget's theory
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Title: A measure of meter conservation in music, based on Piaget's theory
Physical Description: viii, 107 leaves, 2 leaves of music : ; 28cm.
Language: English
Creator: Serafine, Mary Louise
Copyright Date: 1975
 Subjects
Subject: Musical meter and rhythm   ( lcsh )
Conservation of substance (Psychology)   ( lcsh )
Child psychology   ( lcsh )
Curriculum and Instruction thesis Ph. D
Dissertations, Academic -- Curriculum and Instruction -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Mary Louise Serafine.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 103-106.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098156
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 - 000162733
oclc - 02717824
notis - AAS9082

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A MEASURE OF METER CONSERVATION IN MUSIC,
BASED ON PIAGET'S THEORY










By
MARY LOUISE SEPJ-INE


A DISSERTATION PRESENTED TO THE GRADnUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREME'-NTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA

1975


__I















ACKNOWLEDGEMENTS


I wish to thank several people who have helped me

throughout the conduct of this study. I am particularly

indebted to my supervisory committee for their continued

encouragement, inspiration, and expertise. I would

especially like to thank Dr. Jacquelin Goldman for

helping me to gain the insights into Piagetian theory

that have served as the basis of this study. I would also

like to thank Dr. Phyllis Dorman, whose help made it

possible for me to clarify the applications to musical

development. Finally, I owe inestimable thanks to Dr.

William Hedges, chairman of the supervisory committee, for

helping me to synthesize many difficult ideas and give them

the necessary focus in this study. His help at every

step in the research is gratefully acknowledged.

In addition, I wish to thank Dr. William Ware, who

offered many valuable suggestions regarding the study,

and Mr. Edward Troupin, who assisted with the preparation

of musical materials.

I also wish to thank P. K. Yonge School, Kirby Smith

School, and Bent Twig Day Care Center for their participation

in this study. The fine cooperation of the faculty and

students of these schools made the research possible. In





iv



addition, I would like to thank Ms. Linda Wagner, my

research assistant, for an uncommon carefulness and attention

to detail in the data collection.

Grateful acknowledgement is given to the Florida

Educational Research and Development Council for the

financial support of this research. The generosity of

the Council's grant is greatly appreciated.

Finally, several friends contributed their support

and encouragement when I most needed it. I would especially

like to thank Madlyn Levine, Kate Gallagher, and Michael

Hanes.
















TABLE OF CONTENTS


ACKNOWLEDGEMENTS................................... ... . 1

LIST OF TABLES......................................... vi

ABSTRACT OF THE STUDY .... ................................. vii

CHAPTER ONE INTRODUCTION.............................. 1

Statement of the Problem............................4
Need for the Study.................................. 5

CHAPTER TWO REVIEW OF RELATED LITERATURE...............7

CHAPTER THREE RESEARCH DESIGN AND METHODOLOGY .........30

Development of the Conservation of Meter Task......30
Validation of the Conservation of Meter Task.......34
Definitions.........................................37
Instrumentation....................................40
Training Procedures. ......................... .... 52
Sampling Procedure and Subject Selection...........55
Design of the Study.................................57
Data Collection.................................... 60

CHAPTER FOUR DATA ANALYSIS AND RESULTS OF THE STUDY...62

Reliability of the Conservation of Meter Task......62
Summary of Descriptive Statistics ..................64
The Questions of the Study........................ 73

CHAPTER FIVE SUMMARY, CONCLUSIONS, AND
RECOMMENDATIONS...........................................94

Summary ............................... ............. 94
Conclusions....... ................................. 95
Recommendations for Further Research ............. 101

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

BIOGRAPHICAL SKETCH..... .............. ................. 107















LIST OF TABLES


Table 1 Composition of Subject Sample...............56

Table 2 Conservation of Meter Task Scores...........66

Table 3 Conservers of Meter (Pretest Scores) .......67

Table 4 Conservation of Meter Task Scores............68

Table 5 Conservation of Meter Posttest Scores.......69

Table 6 Concept Assessment Kit Scores...............70

Table 7 Draw-A-Man Task Standard Scores.............71

Table 8 Correlations Between Measures ...............72

Table 9 Partial Correlation Coefficients ............74

Table 10 Multiple Regression: COM with CAK, DAM,
and Age......................................75

Table 11 Analysis of Variance of COM Scores..........78

Table 12 Analysis of Variance of COM Scores..........79

Table 13 Analysis of Variance of COM Scores..........81

Table 14 Differences Among Mean COM Scores ...........82

Table 15 Analysis of Variance of COM Posttest Scores.84

Table 16 Incorrect Responses to COM Task Items....... 86

Table 17 Duple and Heniola Items on the COM Task.....89

Table 18 Diminution and Augmentation Items on the
COM Task.....................................90

Table 19 Analysis of Variance of COM Posttest Scores.91













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



A MEASURE OF METER CONSERVATION IN MUSIC,
BASED ON PIAGET'S THEORY

By

Mary Louise Serafine

December, 1975


Chairman: William D. Hedges
Major Department: Curriculum and Instruction


The purposes of this study were 1) to develop a

task designed to measure conservation of meter in children

ages four to nine years, and 2) to'test the validity of

the measure in terms of Piaget's principle of conservation.

The main assumption of the study was that the Conservation

of Meter Task could tentatively be considered valid if

the following hypotheses, predicted from Piagetian theory,

were supported: 1) Success on the task is positively

related to age. 2) Success on the task is positively

related to success on conservation tasks involving physical

properties (e.g., space, number, substance, etc.). 3) Improve-

ment in performance on the task is resistant to training

for children at the preoperational level. The procedures

of the study involved the following: Children ages four,






viii


five, seven, and nine years were pretested on the Con-

servation of Meter Task, and on Piagetian tasks of con-

servation of space, number, substance, continuous

quantity, weight, and discontinuous quantity. Subjects

who scored as nonconservers of meter were divided into

two groups--those who received training in conservation

of meter and those who received no training. Following

training, subjects were posttested on the same measures

used in the pretest. The dataware analyzed using multiple

regression analysis and analysis of variance. The

results of the study tentatively supported all three of

the hypotheses, and the Conservation of Meter Task was

considered validated.
















CHAPTER ONE
INTRODUCTION


In recent years, there has been increased interest

in applying research and theory from developmental psycho-

logy to the arts. The graphic arts and music have been

the primary focus of these attempts, and Piaget's theory

of cognitive development, perhaps because it weds human

development and epistemology, has thus far offered the

most fruitful applications. The present study was

concerned with one aspect of the development of musical

thought--how the mind perceives and organizes meter and

rhythm in music. Piaget's theory was chosen as the

framework for investigating this area because one of the

general theses to emerge from his work is that certain

aspects of intelligence are operative irrespective of the

content of thought. Piaget's work has, for example,

uncovered commonalities in child thought on the topics of

space, time, number, language, logic, and other areas.

The assumption on which this study was based is that

those cognitive operations which are common to the physical,

visual, and verbal domains of thought should also be

observable in the musical/aural sphere. This study

attempted to apply one aspect of Piaget's theory--the











principle of conservation--to the perception of rhythm

and meter. The purpose of the study was to develop and

validate a conservation of meter task which represents

a musical analogue of the well known Piagetian tasks in

conservation of substance, number, weight, etc.

Conservation of meter may be described as the aware-

ness of a steady musical pulse concommitant with changes

in rhythmic pattern. Since music is a temporal art, the

relationship between meter (the constant time intervals)

and rhythm (the variable organization of sounds within

those time intervals) is one that seems so necessary and

so obvious that it is largely taken for granted by most

musicians. A rhythmic pattern is, after all, merely a

succession of a certain number of sounds, each of the sounds

having a particular duration. Inasmuch as these rhythms

tend to be grouped by the mind into patterns, and to

take on accents or points of stress, the patterns give

rise to meter--stresses that delineate the equal intervals

of time into which rhythmic patterns tend to fall.

In Western music, certain notational conventions--

the use of barlines, measures, meter signatures, and

metronome markings--attest to the importance of meter as

an invariant temporal organizer for a piece of music.

Other more obvious examples of the effects of music's

inherent meter/rhythm relationship are certain human

physical activities--such as marching and some forms of










dancing--where participants move steadily to a regular

recurring "beat" irrespective of how "fast" or "slow" are

the sounds within the equal time intervals. All musical

activities--listening, performing, or composing--involve

an experience of the invariance of meter superimposed on

variations in rhythm. Composers, for example, may

exploit the invariant meter/variable rhythm relationship

for the purpose of special effects, as when meter remains

the same but a "pseudoritard" is created by increasing

the durations and decreasing the number of notes in a

measure. The following example is a condensation of the

rhythmic changes in 4/4 meter that occur in the last

eight measures of Brahms' Rhapsodie (for Piano), Op. 79

No. 2:


2441)2.7 7 P m .-rm I 2242 iJjjtIj(jj
(C' C) 3 3


^---------^ ^---- ---> t-3 <-?
----------equal time durations---------

The above example is similar to the type of stimulus

that was used in the task administered in this study. It

is an obvious example of the meter/rhythm relationship

exploited for special purposes, although more subtle

technical uses of the relationship are everywhere to be

found in music. Aside from special effects, however, the

meter/rhythm relationship provides in all music the necessary,

on-going temporal organization.










The general issue with which this study was

concerned is whether the awareness of this seemingly

necessary relationship between rhythm and meter is one

that results from learning, or is an inherent quality of

the way the mind conceives sounds through time, or whether

the awareness of this relationship is one that evolves

slowly as the result of cognitive development and exper-

ience. In other words, the concern here was with whether

the construct "conservation of meter" is a valid one. If

it were, then it could be expected that Piaget's principle

of conservation (and his theory of cognitive development)

would be applicable, and the ability to conserve meter

would be shown to be a slowly evolving, developmental

one, rather than an ability that lies in some inherent

quality of the mind, or an ability that is simply the

result of learning.


Statement of the Problem


The purposes of the study, then, were as follows:

1. To develop a task designed to measure conservation

of meter in children ages 4 to 9 years.

2. To test the validity of the measure in terms of Piaget's

principle of conservation.

In accomplishing the purpose of testing the validity

of the measure, the following questions were posed:

1. Is success on the Conservation of Meter Task related

to age?










2. Is success on the Conservation of Meter Task related

to success on tasks of other types of conservation

(e.g., substance, number, weight, etc.)?

3. Does training improve performance on the Conservation

of Meter Task?

Briefly, the procedures involved in carrying out the

above-mentioned purposes were as follows:

1. Children ages 4, 5, 7, and 9 years were pretested on

the Conservation of Meter Task, and on Piagetian tasks

of conservation of space, number, substance, continuous

quantity, weight, and discontinuous quantity.

2. Subjects who scored as "nonconservers" on the Conser-

vation of Meter Task were divided into two groups--

those who received training in conservation of meter

and a control group (no training).

3. Following training, these subjects were posttested

on the same measures used in the pretest (Conservation

of Meter Task and Piagetian conservation tasks).


Need for the Study


While the traditional focus in music education

research has been on the social, affective, perceptual,

and psychomotor responses to music and musical training,

little research until recently has been directed toward

the cognitive aspects of musical experience. There is a

general need to investigate the nature of musical cognition,










and Piaget's theory offers a framework in which to do

this.

A few previous studies have, like the present one,

attempted to develop music listening tasks that purport

to be measures of musical conservation. Primarily these

studies have assumed the existence of a music conservation

schema (e.g., conservation of melody, conservation of

meter, conservation of rhythm, etc.), but have not attempted

to empirically validate the construct. As a result,

musical conservation tasks vary from study to study, with

each new investigator proposing his own without validation

of the measure. The present study attempted to respond to

the need for a validated measure of conservation of meter,

and represents a beginning attempt to develop instru-

ments that will be needed in further research on musical

development.
















CHAPTER TWO
REVIEW OF RELATED LITERATURE


Investigation into the nature of musical thought is

a relatively recent research pursuit. Early attempts in

this area have focused on the study of musical abilities

and the development of standardized tests of musical

aptitude and achievement. What follows is a brief review

of some of the rhythm and meter tasks used in these standard-

ized measures.

Aptitude tests purporting to measure rhythmic abilities

or perception of "time" have used a wide variety of tasks.

The Seashore Measures of Musical Talents (1939) present a

"time" test on which the subject indicates whether the

second of two successive tones is longer or shorter than

the first, and a rhythm test on which the subject indicates

whether the second of two short rhythmic patterns is the

same as or different from the one preceding it. The

Kwalwasser-Dykema Music Tests (1930) utilize very similar

items for time and rhythm discrimination, and the Bentley

Measures of Musical Ability (1966) contain a "rhythmic

memory" test similar to Seashore's rhythm test. The Drake

Musical A-itude Tests (1957) contain items for rhythm

in which the subject hears a metronome, with a voice








simultaneously counting the pulses. When the voice and

metronome cease, the subject is to continue counting at the

same rate, and, when told to stop, to write down the number

of pulses he has counted to that point. The Gordon

Musical Aptitude Profile (1965) contains a test for meter

in which the subject indicates whether the meters of two

short melodies are the "same" or "different."

Unlike aptitude tests, achievement tests purport to

measure musical abilities that are learned rather than

innate, although similar types of items may appear in

both types of tests, and there is much controversy over

whether the abilities measured by such items are the

result of "talent" or achievement. The Colwell Music

Achievement Tests (1969), for example, contain items in

meter discrimination similar to those on the Gordon

Aptitude Profile, except that, given a short melody, the

subject must indicate whether the meter is duple or triple.

In addition to the items involving rhythm and meter on

standardized music tests, other types of individually

administered tasks, as described below, have been designed

to measure children's musical abilities.


Zimmerman's Studies


The first well-known American study which attempted

to apply a Piagetian concept to musical development was

done by Marilyn Pflederer Zimmerman (Pflederer, 1963, 1964,

1966b).









She developed nine tasks of musical conservation and

individually administered six of the tasks in a pilot

study to eight 5-year-olds and eight 8-year-olds. The

tasks required the subjects to conserve one aspect of a

musical stimulus (recognize that it remained the same),

while alterations were made in another aspect (the "foil").

The six tasks were: conservation of meter (with change

of durational values), conservation of rhythm (with change

of tonal pattern), conservation of melody (with change

of durational values), conservation of tonal pattern

(with change of pitch level), conservation of tonal

pattern (with change of rhythm), and conservation of melody

(with change of rhythm and addition of harmonic accom-

paniment). On all tasks, 8-year-olds gave more correct

responses than did 5-year-olds. The results were explained

in terms of preoperational and concrete operational thought,

with preoperational children tending to center their

attention on the perceptual changes, without being able to

conserve the stable property.

Of greatest interest for the present study is Pflederer

Zimmerman's Conservation of Meter task. Here the subject

was presented with a story about "families" of beats in

"twos" or "threes." Examples of each were played on a

drum, and it was explained to the child that meter

remains invariant irrespective of changes in rhythmic

subdivisions. Pairs of rhythms were played to the










child such as duple: J J and J fl and triple:

i J 4 and 4 P 2. and he was told that

both members of each pair were in twos or in threes

"because the same amount of time was taken, but some of

the notes were shorter and so we could use more in order

to take up the time." Following this explanation, the

child was presented with short rhythmic patterns played

on a drum and tape recorded, and was asked to tell whether

these patterns moved in twos or threes. In answering

these items, 29% of the responses of 5-year-olds were

correct, as compared to 54% of those of 8-year-olds. A

second set of items was similar to the first, except

that instead of patterns played on a drum, the items were

short musical phrases played on a piano. Again 5-year-olds

did less well than 8-year-olds, with 44% of responses

correct for the former group, and 75% for the latter.

In this same dissertation Pflederer Zimmerman

described a task of "Conservation of Meter with One to

One Correspondence," but the task was not administered

in the study. One to one correspondence here referred to

the relationship between the rhythm and words that are

chanted representing a "conversation" between two dolls.

In the first part of the task, the subject is told that

all the correct phrases of the dolls will "move in twos"

and that he is to listen to the conversations and identify

which doll makes the mistake of not "taking the same









amount of time." Drum rhythms accompany the conversations,

for example:

"Johnny" "What is it?" "Come and play."



"Not right now."

J J J

The last phrase is wrong because it consumes three beats

instead of two. A second set of examples in this task

was similar to the first set, except that correct phrases

by the dolls must be "in threes" instead of "in twos."

In a later theoretical article, Pflederer (1967)

summarized Piaget's work on conservation and reviewed

various standardized musical aptitude tests. After

discussing what she terms Piaget's "five conservation laws,"

(combinativity, reversibility, associativity, identity,

and tautology and iteration), she proposed her own five

laws of musical conservation: 1) identity: the notion

that repetitions of a theme are identical to the original;

2) metrical groups: an awareness of the temporal aspects

of music, separating meter from rhythm; 3) augmentation

and diminution: an increase or decrease in durational

values; 4) transposition: change of key; 5) inversion:

the substitution of low notes for higher ones and vice

versa.

In a second, extended empirical study, Pflederer

Zimmerman and Sechrest (1968; Pflederer and Sechrest, 1968a,










1968b; Pflederer, 1966a; Zi=merman, 1970) carried out

five experiments involving musical conservation. While

all five experiments are of importance, the concern here

is primarily with the conservation of meter task that was

used in the first experiment.

In Experiment I of this research, tasks similar to

those of the previous 1963 study were used to measure

conservation of duration (with changes in durational values),

of meter (with changes of durational values), of rhythm

(with changes of tonal pattern), of melody (with changes

of tempo), and of tonal pattern (with changes of pitch and

of rhythmic pattern). The six tasks were administered to

80 subjects, 10 boys and 10 girls each at ages 5, 7, 9,

and 13 years. Split-half reliabilities for the shorter

tasks, I, II, and IV, were low, ranging from .02 to .57,

while reliabilities for the longer tasks, III, V, and VI,

ranged from .74 to .81. The results of the study showed

that, with the exception of Task I which was judged

"completely unreliable," the relationship between age

and successful responses to the tasks was positive and

nearly linear. The authors reported that improvements can

be seen at each age level, and that there is thus little

evidence of real stages in the Piagetian sense. The

exceptions were the two tasks on conservation of tonal

pattern, on which there was little improvement after age

nine; however, the authors admitted that this may have been

due to the limited difficulty of these tasks.









Of greatest concern here are Task I and Task II of

this study, Conservation of Duration and Conservation of

Meter. The authors claimed that Task I (Duration) is

analogous to Piaget's "sausage test" where a ball of

plasticene is subdivided into smaller parts. Here two

tones of the same pitch were sounded on a piano ( a and ),

the second tone was subdivided first into two shorter tones

( o and c J ), and then into four shorter tones( o

and J J J J), and the subject was asked whether the

members of each pair took the same amount of time. Following

administration of the task, it was judged to be poorly

constructed and unreliable. No age gradient was produced

by the task, and if anything, the trend was toward an

inverse relationship between age and successful response.

The authors concluded that perhaps this was due to the

fact that a good score could be obtained simply by agreeing

with the examiner, which the younger children would be

most likely to do.

Task II, Conservation of Meter, was the same task

regarding "families" of beats cited in the previous 1963

study where the subject indicates whether a rhythmic

pattern is "in twos" or "in threes." Again scores on the

task (computed by the number of correct responses) increased

with age. It was pointed out, however, that the rate of

improvement on this task was not as great as the rate of

improvement on other tasks, most notably Tasks III, V, and

VI.










In Experiment II of this research 198 subjects,

ages 5, 7, 9, and 13 years, were tested on conservation of

melody under deformation of instrument, mode, tempo, har-

mony, rhythm, contour, and interval. Musical material for

the tasks was taken from Bartok's For Children. Subjects

were asked to determine whether the stimulus and the "foil"

which followed it were the same, different, or both (same

in some ways, different in others), with the last being

the correct answer. It was found that age was a significant

factor in success on the task, and that a short-term

training session in conservation of melody did not

significantly improve scores over thoseof the control

group.

In Experiment III, a Conservation of Melody task was

administered to 141 subjects in grades K, 2, 4, and 8. The

task was similar to the one in the preceding experiment

except that four familiar songs (rather than Bartok) were

used, and were made to undergo changes in instrument,

mode, tempo, harmony, rhythm, contour, and transposition.

Subjects responded via a key press mechanism which was

intended to control for the varying degrees of verbal

ability across this age range. Again, age was a significant

factor in success on the task, but training, this time a

longer-term experience (6 sessions,.20 minutes each) of

classroom instruction,was not.










Experiment IV also involved conservation of melody

(using the same four familiar songs as before) but with dou-

ble rather than single deformations (e.g., changes of

instrument and tempo, of harmony and rhythm, of mode and

contour, etc.). Seventy-one subjects at ages 5, 7, 9, and

13 years responded to the task with the key press mechanism

which was tested under two conditions: Condition I called

for one key representing the subject's awareness that a

variation or deformation was being presented, and another

key for the presentation of entirely different music.

Condition II called for the subject's pressing a key only

when he heard completely different music. On this task,

Condition II resulted in higher scores than did Condition

I, while no significant age effect was found.

Experiment V tested conservation of tonal pattern

(under deformation of rhythm) and conservation of rhythm

(under deformation of tonal pattern) through the use of

a "paper-and-pencil" instrument rather than an individually

administered task. Half of the answer sheets contained

visual aids; the other half did not. One hundred and

sixty subjects in grades 2,3, 4, and 5 were tested. It

was found that each grade significantly improved in

performance over that of the previous grade, with a leveling

off after grade 4. It was found that visual stimuli

aided in performance on the test.










Commentary


The work done by Pflederer Zimmerman, with the help

of Sechrest, has made the contribution of drawing attention

to the need for studies of cognitive musical development,

and to the possibility of applying Piaget's work to

aural cognition. This work has, as well, contributed

to the development of tasks which are more suitable for

young children than the paper-and-pencil methods so often

used in studies of musical ability. There are, however,

problematic areas in the meter and rhythm tasks.

First, the Conservation of Meter task regarding

"families" of beats in twos and threes is more of a discrim-

ination of meter task than one of conservation. The subject's

response is not whether the meter remains the same, or

changes in some way, but whether it is in twos or in threes.

While the manner of presenting the task--an individual

administration in story form--is quite different from

the aptitude and achievement tests discussed earlier, the

actual substance of the task is quite similar to the

familiar paper-and-pencil items requiring the subject to

indicate whether a meter is duple or triple. While it is

true that the items involve changes of rhythm within each

measure (so do the more conventional tests), the subject's

focus is nevertheless centered on identifying and naming

the meter as either duple or triple. In this sense it

is more of a discrimination or identification task than









one of conservation and, in fact, the tern "discrimination"

was used in the author's description of the task. Moreover,

measurement of the subject's awareness of the inverse

relationship between the number and length of notes in a

measure (compensation) is made partially invalid by the

fact that, as part of the explanation of the task, the

experimenter actually tells the subject that "the s me

amount of time was taken, but some of the notes were

shorter and so we could use more in order to take up the

time." This gives the subject, prior to testing, a brief

verbal retraining on the notion of compensation.

The second task described, but not administered to

subjects in Pflederer Zimmerman's 1963 study, offers a

slightly different problem. Here the task on Conservation

of Meter with One to One Correspondence represents a

rather questionable analogy to the Piagetian tasks

involving one to one correspondence of objects such as

eggs and egg cups. On the meter task, the child never

directly compares the words to the drum rhythms, as he

never hears them singly, but always simultaneously, and

the words and drum beats always occur in exactly the same

rhythm, with one never faster, slower, or otherwise

different from the other. There is thus no reason to

believe that a true notion of one to one correspondence

is ever utilized by the child. In fact, it is probable

that, upon hearing them, the words and drum beats become











so intertwined in the mind of the subject that the same

task might just as easily have been administered with the

words alone, and not with thldrum beat.

Finally, an additional weakness affecting both of

these conservation of meter tasks is that correct responses

depend on aural memory as well as discrimination or

"conservation." Particularly in the task of Conservation

of Meter with One to One Correspondence, the subject must

remember the pattern he has just heard in the past, in

order to compare it to the one he is presently hearing.

Especially for young children, the difficulty of relying

on aural memory to make comparisons poses a serious threat

to the validity of a conservation-type task.

The Conservation of Duration task from the 1968

study has already been discussed as poorly constructed and

unreliable, by the authors' admission. An additional weak-

ness of the task (in addition to the problem pointed out

by the authors that a correct response is mere agreement

with the experimenter) is that, like the meter tasks,

accurate aural memory is a prerequisite to correctly

responding to the task. It is the temporal quality of music

that demands auralmemory and which prevents this task from

being, as the authors intend, analogous to what they

referred to as Piaget's "sausage test." That is, the

subject can never directly compare the long single tone

(the standard) and the two or four shorter tones (the









variable stimulus) because they do not exist together,

but only in succession. The comparison of the two durations

must be made in memory.


Jones' Study


A more recent study which attempted to apply

Piagetian principles to the development of the meter

concept has been done by R. L. Jones (1971). Jones

performed a scalogram analysis on a sequence of eleven

musical tasks, beginning with "basic discrimination" of

sounds (such as high/low, fast/slow, and long/short) and

progressing to the actual "meter concept." Again the

meter concept was here defined as the subject's ability

to determine whether short fragments "swing" in groups of

twos or threes. However, unlike Pflederer Zimmerman,

Jones was not concerned with conservation per se, and

did not rely on Piaget's study of conservation of weight,

number, substance, etc., but rather on his study of the

child's conception of time (Piaget, 1969).

For four of his tasks, Jones borrowed from Piaget

a task in which the subject manipulates an apparatus

consisting of two flasks, one directly above the other,

connected by a spigot. At the start of Piaget's experiment,

the top flask is filled with water, and the bottom one is

empty. The subject makes a drawing of this, and then

allows some water to flow from the upper to the lower

flask. Successive drawings are made each time the water








is allowed to flow, so that eventually the subject

has made a series of drawings representing successive

time intervals, during which the top flask is gradually

emptied, and the bottom one is filled with water. The

task continues, with the experimenter questioning the

subject, shuffling the drawings, having the subject

re-seriate them, and so on. In the study by Jones,

four of the eleven scaled musical tasks (Nos. IV, V, VI,

and VII) leading up to the meter concept were directly

derived from this experiment by Piaget. In place of

the subject's drawings of water flasks, however, Jones

presented the subject with a ready-made picture of two

staves, the top one bearing a treble clef and five whole

notes descending stepwise, the bottom staff bearing a

bass clef and five whole notes (directly under the five

treble notes), ascending stepwise. (See example below.)


I1 2 13 4 5


112 113


114 11 5










Thus the following analogy was made to Piaget's

experiment with the water flasks: at interval 1, the top

flask is filled with water, and the bottom one is empty;

likewise, where the musical notation is used, the top note

is at its highest point, the bottom note at its lowest.

At interval 2, the water in the top flask has dropped

slightly, while there is now a small amount of water in the

lower one. Likewise the note on the top staff has moved

down a step, and the bottom note has moved up a step, and

so on for five time intervals. Jones made use of this

picture of musical notation for his Tasks IV, V, VI, and

VII, which, like Piaget's tasks, are intended to measure,

respectively, the child's conception of seriation, double

seriation, simultaneity, and succession.


Commentary


While Jones' task bears some resemblance to Piaget's,

it differs from it by the fact that, in Jones' task,

the child has directly witnessed no actual motion, as he

has in watching the flow of water, and he has not, through

drawing, made his own representation of motion. Jones'

task is, in essence, a non-musical one, because the child

does not hear the pitches as they descend (in the top

staff) or ascend (in the bottom staff). Jones himself

admits that having the subject hear the tones was omitted

from the experiment only "because of an oversight," but










he maintains that "this does not seem to affect the

basic nature of the task" (1971; page 51).

The present experimenter finds it difficult to

justify a task as a musical one when no sound is evident.

The fact that four such non-musical tasks comprise a major

segment of the total scaled sequence (11 tasks) poses a

threat to the face validity of the entire sequence.

Musical notation is only a visual symbol system; it is not

a substitute for music. Jones' subjects were presented

in these four tasks with only static pictures. Having

not heard the sounds which were represented by the notes,

the symbols on paper could not be representative of

sounds previously experienced, as was the case with Piaget's

experiment, in which the subject's own pictures were

symbols of motion previously witnessed. Above all, the

important quality in Piaget's experiment which is lacking

in Jones' is the notion that time is an abstraction which

is always relative to some motion or movement or to a

change in or transferral of energy. Thus all motion or

energy expenditure is temporal--i.e., measures time or

passes through time. In the case of Piaget's experiment,

the flow of water from one flask to another is a "clock"

which measures off units of time, and the pictures

represent this time sequence. In the case of Jones'

experiment, the changes in pitch levels could have

represented different time intervals, but they were not











heard by the subject. Thus no time intervals were exper-

ienced by the subject, and the pictures did not represent

a series of events that were witnessed. To this must be

added the fact that, while the pictures of water flasks

at least look like the actual flasks themselves, symbolic

musical notation does not at all "look like" the sounds

it represents. Musical notation, for a child, is likely

to be relatively unfamiliar and confusing. Even if the

subject had heard the sounds and then viewed the musical

notation, the problem remains that he is required to

engage in the difficult process of translating an aural

experience into visual symbols. To Piaget's favor, on

the other hand, the experience of witnessing the flow of

water in the flasks is a visual/spatial one, and likewise

the pictures (symbols) are visual/spatial.


Other Music Conservation Studies


Because of Pflederer Zimmerman's pioneering work in

the development of music conservation tasks, a few recent

studies have appeared which make use of her tasks as models

for the measurement of musical conservation. Primarily these

studies have not attempted to validate the tasks, but

rather have had as their main purpose the investigation

of relationships between music conservation and other

factors such as musical experience, age, home environment,

etc. Three such studies are briefly described below,










along with a study which assessed the effects of specific

training in conservation of melody.

King (1972) developed a Melodic Pattern Conservation

Test which makes use of non-Western music in an effort

to present a culturally "neutral" stimulus to American

subjects. The subjects were asked to conserve 12 melodic

pitch patterns in the face of deformation of timbre,

pitch, and duration. The study sought relationships between

performances on the music conservation test and the following:

grade level (1, 5, and 9), social class (lower, middle,

upper), social environment (urban, rural), sex, home

music environment (using the Wermuth Questionnaire), and

prior experience with non-Western music. The results of

the study showed that grade level, social class, and

social environment contribute to the ability to conserve

melodic pitch patterns, while home environment and sex

(except for duration items) do not. A plateau in music

conservation was found at the fifth grade level. As

social class and grade level increased, responses became

more consistent and more correct.

Thorn (1973) developed a music conservation task in

which melodic and rhythmic patterns were presented twice:

first in their "original" form and then with some change

in presentation (the "foil"). Thorn tested children

ages 7 to 13 years, and noted correlations between scores

on the music conservation task and age, degree of partici-

pation in music groups, type of private music lessons, and


~










number of years of lessons. She found that, although music

group participation and type of private music lessons

affected the child's music conservation, age had the most

significant effect. The number of years of private

lessons, however, had little effect on the musical conser-

vation scores. She also found (similar to Piaget) three

levels in the development of conservation: nonconservation,

a transitional stage, and conservation. She concluded

that Piaget's theory could be useful in teaching melodic

and rhythmic concepts.

Larsen (1973) developed a four-part task of melodic

variation, which he believed would require formal operational

thought, and administered it to 24 subjects, eight each in

grades 3, 5, and 7. The task required subjects to construct

a melody by ordering a set of bells to fit a model and to

perform variations on the melody. Subjects were also

shown visual diagrams of the variation techniques of inver-

sion, retrograde, and retrograde inversion, and asked how

valid they felt these methods of variation were. Larsen

found it consistent with Piaget's theory that: 1) younger

subjects had a more difficult time than older ones in

simply ordering a set of pitches; 2) as age increases,

so does the awareness of the concept of variation;

3) as age increases, so does the tendency to accept the

modifications as "valid" means of varying the original

pattern; 4) as age increases, so does the ability to

recognize inversion, retrograde, and retrograde inversion

from visual diagrams.










A recent study by Botvin (1974) tested the effective-

ness of two types of training in improving conservation

of melody. "Conservation of melody" in this study was

construed as the subject's ability to recognize that a

simple melody (in this case a familiar song like "Yankee

Doodle" or "Old MacDonald") remains invariant in spite of

changes in tempo (both faster and slower than the original).

First grade children were pretested in Conservation of

Melody, and also in conservation of mass, weight, liquid,

and number. Two types of training in conservation of

melody were then administered to separate groups of

nonconservers:

1. Training that involved behavior-shaping through

successive approximation (SA), and

2. Training that involved the above-mentioned technique,

with the addition of verbal rule instruction (SA-VRI).

Posttests were administered in conservation of melody and

conservation of mass, weight, liquid, and number.

The results of the study indicated that:

1. Both methods of training were equally successful in

improving performance on the conservation of melody

task.

2. Training in conservation of melody also improved

performances on the Piagetian conservation tasks in

mass, weight, and number (but not liquid); unexpectedly,

the SA training was more successful than the SA-VRI










method in effecting this cross-modal transfer (from

the auditory to the visual).

As a result, Botvin made the following conclusions:

1. There isreason to doubt the notion that experimentally-

induced conservation is only a pseudoconcept. Subjects

can apparently not only be successfully trained in

conservation of melody, but there is as well a transfer

effect to other types of conservation.

2. Since both musical and non-musical types of conserva-

tion appear to be mediated by the same cognitive

structures, one may be improved by the other.


Commentary


One difficulty with the conclusion drawn from the

apparent success of the training is that there was no

definite indication of how permanent or stable were the

effects of training in conservation of melody. Depending

on the length of time that elapsed between the subjects'

training and their posttests, retention time remains in

question. Further, the posttest in conservation of melody

followed the same format as the pretest; i.e., no addi-

tional techniques were used on the posttest, such as

probing, attempts at extinction, or countersuggestion by

the experimenter, which might have been used to measure

the depth of the subject's understanding of conservation.

In short, there is no definite proof that true "learning"










of conservation--either of melody or of the physical

properties--resulted from the training in conservation

of melody.

A second difficulty with the Botvin study is that, like

other studies involving musical conservation, there has

been no validation of the task used to measure the construct

in question. In the present case, the conservation of

melody task appears to have face validity, but its validity

as a measure of a "true" Piagetian conservation remains in

question. Moreover, a re-interpretation of the results

of the study could in fact negate the validity of the mus-

ical conservation task. If it is assumed, for example,

that conservation results from developmental rather than

learned abilities, then the fact that successful per-

formance on the conservation of melody task was trainable

could mean that the task itself was not actually measuring

a "conservation" ability at all. In short, "conservation

of melody," as a construct, has not been validated.


Summary of Literature Review


Early attempts to measure the musical abilities of

children have focused on the development of standardized

tests of music aptitude and achievement. These tests

contain a variety of tasks involving meter, most of them

requiring the subject to identify meter groupings (duple,

triple, etc.).









Pioneer work by Marilyn Pflederer Zimmerman resulted

in the application of Piaget's principle of conservation

to the development of children's musical concepts.

Zimmerman's tasks were designed to measure conservation

of meter, as well as conservation of rhythm, melody,

tonal pattern, etc. It was found that successful

performance on these tasks increased with age.

R. L. Jones developed a different type of meter

task, based not on conservation, but rather on Piaget's

study of the child's conception of time.

Various studies continued to investigate children's

performances on music conservation tasks. These studies

corroborated the finding that success on the tasks

improves with age. In addition, Botvin has found that

training can improve performance on a conservation of

melody task and also have a transfer effect on Piagetian

conservation tasks.

After reviewing the tasks that have been developed

to measure conservation of meter (and conservation in

general), it was concluded by the present experimenter

that there is a need for the validation of a measure of

conservation of meter.















CHAPTER THREE
RESEARCH DESIGN AND METHODOLOGY


Development of the Conservation of Meter Task


For this study a Conservation of Meter Task was

developed which presented to young children musical examples

in which meter and rhythm were set in strong relief. In

each musical example, the regular, invariant recurrence

of meter was made obvious by accenting the meter with a

continual aural or visual stimulus (a loud non-pitched

click or a flash of light), while simultaneously the rhythm

underwent obvious changes of either steadily increasing

(to more, shorter notes) or decreasing (to fewer, longer

notes). It should be pointed out that under normal

circumstances, musical activities such as listening or

performing do not involve such an obvious focus on the

invariance of meter and the variability of rhythm. It

is believed by the experimenter, however, that only by

making a strong juxtaposition between rhythm and meter,

by accenting their qualities and setting them in obvious

relief, can the child's grasp of the fundamental nature

of meter and rhythm be assessed.

The task developed for the present study was an

attempt to remedy some of the problems surrounding the










concept of conservation of meter that have been iden-

tified in the review of literature on this topic. The

intent was to devise and test a task of meter conservation

which conforms to the Piagetian notion of conservation,

and yet represents a truly musical and aural phenomenon

rather than either a visual or a psychomotor one. The

task, then purports to be one of true conservation of meter--

not the identification or discrimination of various types

of meters (e.g., duple or triple; the ideas of "swinging"

or "families" of twos or threes), but the awareness that

a meter (of whatever type, whether duple, triple or

otherwise) is conserved, i.e., remains constant. The

essence of the meter concept proposed here, then, is not

the idea of a numerical grouping (e.g., two or three),

but the fact that it remains invariant.

It is believed by the present experimenter that this

latter definition of meter not only represents a truer type

of conservation, but that it also solves one of the diffi-

culties of a meter task which tests for duple vs. triple

discrimination--i.e., the problem of cultural bias. It

is primarily the music of modem Western cultures which

groups meter impulses by twos or threes--a custom not

shared by many other cultures, or indeed even other epochs

in Western history (Sachs, 1953). Other groupings, such

as five or nine for example, are common among so-called

primitive cultures, as well as in some of the very recent

music of the West. It is hoped that the task used in the









present study, by construing meter as a constant impulse

or "beat," however grouped, may have more universal

applications and be less susceptible to cultural bias.

In addition, the task of the present study purports

to be a more accurate analogue to the Piagetian tasks

by embodying the property of compensation, which figures

prominently in all conservation tasks. In the conser-

vation of quantity task, for example, where a liquid is

poured from a tall, narrow container into a low, shallow

one, the decrease in height is compensated for by an

increase in width. Likewise, in the conservation of

number task, where parallel rows of objects are alternately

spread out or condensed, an increase in the length of the

row is compensated for by a decrease in density. Various

studies (Goodnow, 1973; Curcio, Levine, and Robbins, 1972;

Gelman and Weinberg, 1972) have assessed the role of compen-

sation--the awareness of covarying dimensions in a display--

in the development of conservation. The task used in the

present study attempts to embody the property of compen-

sation by the use of rhythms which undergo compensatory

change in a consistent and progressive manner; that is,

in each successive measure, rhythms in diminution appear

to get "faster" by virtue of the fact that the notes

increase in number, but decrease in duration, while rhythms

in augmentation appear "slower," as there are fewer, but

longer notes per measure. Thus a compensatory relationship










exists between the number and durationsof notes comprising

the rhythm.

Further, the task attempts to assess conservation of

meter with a minimum of reliance on aural memory. The

factor of aural memory is one which plagues many musical

studies, since the temporal property of music makes it

impossible to compare two musical stimuli simultaneously,

as can be done with tangible, "before the eye" displays.

The task, then, attempts to diminish the subject's

reliance on aural memory by not requiring a direct "before

and after" comparison of two different stimuli, but

rather a continuous attention to a single stimulus.

Finally, the task developed for the present study

attempts to diminish the effects of the subject's reliance

on a purely aural mode of perception in completing the

task. An alternate form of the task serves as a check

on the possibility that the measurement of conservation

is confounded with aural discrimination. The two forms

of the task are: aural/aural (where steady clicks repre-

senting meter are heard simultaneously with variable

rhythms) and aural/visual (where the same steady meter

clicks and variable rhythms are heard, but with the

addition of a visual stimulus to reinforce the meter in the

form of a light which steadily blinks on and off simultaneously

with the meter clicks). Thus, those subjects who fail

the aural/aural form of the Conservation of Meter Task are










given the aural/visual form to determine whether their

failure resulted from a true lack of conservation (in

which case they would fail both forms of the task) or from

merely an inability to distinguish aurally between the

clicks and the rhythms heard (in which case they should

score as conservers on the aural/visual form).


Validation of the Conservation of Meter Task


The procedures used in this study to test the validity

of the Conservation of Meter Task conform to certain of the

guidelines set forth by the American Psychological Associa-

tion Committee on Psychological Tests in 1952 and 1954

(Cronbach & Meehl, 1955; Helmstadter, 1964). These

procedures involve:

1. Setting forth the hypothesis that the test measures

the construct in question;

2. Making predictions from existing theory regarding

(among other things) group differences, expected

correlations, and changes in performance as a result

of experimental manipulation;

3. Securing data which will indicate rejection or non-

rejection of the hypothesis.

For the purposes of the present study, the hypothesis

was set forth that conservation of meter is a valid construct

derived from Piagetian theory, and is measured by the

Conservation of Meter Task used in this study.











Further, the assumption was made that the task can

tentatively be considered valid if the following hypotheses,

predicted from Piagetian theory, were supported:

1. Success on the task is positively related to age.

2. Success on the task is positively related to success

on conservation tasks involving physical properties

(e.g., space, number, substance, etc.).

3. Improvement in performance on the task is resistant to

training for children at the preoperational level.

In relation to the above hypotheses, data were collected

to answer the following questions:


Major Questions of the Study


1. a) Are scores on the Conservation of Meter Task

related to age?

b) Is there a significant difference in mean scores

on the Conservation of Meter Task among various age

groups (4, 5, 7, and 9 years), or between sexes?

2. Are scores on the Conservation of Meter Task related

to scores on conservation tasks involving physical

properties (e.g., space, number, substance, etc.)?

3. Is improvement in performance on the Conservation of

Meter Task resistant to training for preoperational

subjects? Specifically, is there a significant

difference in mean scores on the Conservation of Meter

posttest between trained and untrained subjects, at











the preoperational (or concrete operational) level?


Secondary Questions


4. If a Conservation of Meter Task item is incorrectly

answered, is the erroneous response to an item involving

diminution more likely to be "faster"? Is the erroneous

response to an item involving augmentation more likely

to be "slower"? Specifically, is there a significant

difference between the number of "faster" and "slower"

responses among the incorrect responses given on each

of the Conservation of Meter Task items?

5. Are Conservation of Meter Task items involving duple

rhythms more difficult than those involving hemiola

rhythms? Specifically, is there a significant difference

between the number of subjects who correctly answer

Conservation of Meter Task items involving duple

rhythms and those involving hemiola rhythms?

6. Are Conservation of Meter Task items involving rhythms

in diminution more difficult than those involving

rhythms in augmentation? Specifically, is there a

significant difference between the number of subjects

who correctly answer Conservation of Meter Task items

involving rhythms in diminution and those involving

rhythms in augmentation?

7. For what age groups) or sex(es), if any, does training

improve performance on the Conservation of Meter Task?










Specifically, is there a significant difference in

the mean scores on the Conservation of Meter posttest

between the trained and untrained subjects, at

various age levels (4, 5, 7, and 9 years) and for each

sex (male, female)?


Definitions


Meter is here defined as the regular recurrence of

accents, at equal time intervals. Rhythmic patterns

occurring within these time intervals may vary with respect

to the number and duration of sounds comprising the pattern.

An inverse relationship exists between the number and

duration of notes in the rhythmic pattern; i.e., within

a set of equal time intervals or measures (meter), an

increase in the number of notes must be compensated for

by a decrease in the duration of all or some of the notes,

and, conversely, an increase in the duration of the notes

must be compensated for by there being fewer of them.

Meter, then, remains constant, while rhythmic patterns

vary, with the patterns embodying Piaget's notion of

compensation (i.e., numbers and durations of sounds vary

in an inverse relationship).

Conservation of meter refers to the understanding

that meter is invariant, irrespective of transformations

in rhythm. In operational terms, conservation of meter

refers to the awareness that the durations of










successive measures of music (the time interval between

meter impulses) are equal, regardless of the distributions

of sounds within the measure. In this context, conservation

of meter has nothing to do with the subject's knowledge of

the number of "beats" in a measure, or whether the meter

is duple or triple.

For the purposes of the present study, a conserver

of meter is a subject who correctly responds to at least

three of the four items on the Conservation of Meter

Task--whether the aural/aural form or the aural/visual

form of the task. Thus a subject who fails the aural/aural

task (with clicks) may still be designated as a conserver

if his performance is satisfactory on the aural/visual

task (clicks simultaneous with blinks of light). A

nonconserver of meter is a subject who correctly responds

to less than three of the four items on the aural/aural

task and less than three on the aural/visual task.

An aural/aural task of conservation of meter, which

was administered to all subjects, here refers to a task

in which sound is the medium for both the variable

rhythmic patterns and the steady impulses representing

meter (e.g., a loud non-pitched click). An aural/visual

task of conservation of meter is one in which the rhythms

are again aurally perceived, but the impulses representing

meter are both aural and visual (i.e., a flash of light

is seen simultaneously with the meter click). The









aural/visual task was administered to those subjects

who did not perform as conservers on the aural/aural

task. The aural/visual task was used to determine

whether failure on the aural/aural task was actually due

to nonconservation (in which case there should be no

significant difference between scores on both tasks) or

due only to the inability to discriminate or distinguish

aurally between the meter clicks and the rhythms (in

which case the subject should perform as a conserver on the

aural/visual task).

Rhythmic diminution here refers to a change in rhythm

such that the sounds are gradually subdivided into smaller

units (e.g., J 1 p etc.), while more tones

are sounded per measure. Rhythmic augmentation is the

opposite--an additive process where the length of tones

increases, and there are fewer tones per measure (e.g.,

P 0 J etc.).
Duple rhythms are those in a proportion of 2:1 or

1:2, as in J = = etc. with 1, 2, 4, 8, 16,

etc. notes per measure. Hemiola rhythms alternate in

proportions of 2:3 or 3:2, as in J J=J j J ==, =
etc. where there may be 2, 3, 4, 6, 8, 9,

etc. notes per measure. The four possible types of

rhythms used in this study are, then: duple in diminution,

duple in augmentation, hemiola in diminution, and hemiola

in augmentation.










In addition to the Conservation of Meter Task,

Piagetian tasks were administered in conservation of space,

number, substance, continuous quantity, weight, and discon-

tinuous quantity. These tasks are scored from 0 to 12

(one point each for judgment and explanation on each of

the six tasks). On the basis of these scores, a preopera-

tional subject (or nonconserver) is defined as any subject

who achieves a score of three or less on these measures.

A conserver or concrete operational subject is one who

achieves a score of six or above on these tasks. Subjects

who score between these points ( a four or five) are

considered to be in a transitional stage.


Instrumentation


Four instruments were administered in this study:

1. The Preliminary Vocabulary Test

2. The Conservation of Meter Task

3. The Concept Assessment Kit--Conservation

4. The Draw-A-Man Task.

The instruments were administered by the experimenter

and three assistants (undergraduate students in Childhood

Education). The order of the tasks varied, with the

exception that the Preliminary Vocabulary Test always

preceded the Conservation of Meter Task. Otherwise, the

order in which the tasks were administered depended on

convenience and the subject's attention span, resistance










to fatigue, and so on. If a child became fatigued by the

aural/aural form of the Conservation of Meter Task, for

example, he was given the Draw-A-Man Task to provide

a break in the activity before returning to the aural/visual

form of the task. All tasks, however, were completed by

a subject in a single session lasting approximately 45

minutes.


Preliminary Vocabulary Test


A Preliminary Vocabulary Test designed by the exper-

imenter was used to screen subjects for knowledge of the

words "faster," "slower," and "stays the same," as applied

to a sequence of sounds, since these terms are used in

the Conservation of Meter Task. In order to control

for possible vocabulary difficulties, only children who

correctly answered all four items on the vocabulary test

were used as subjects for the experiment. All items are

prerecorded on tape, with the sounds (clicks) generated

from an electric metronome with a rheostat dial.

The subject is given the following explanation of the

task:

"You will hear some clicks like these. (Demonstrate
by playing tape of eight steady clicks at tempo of LM = 80.)
I am going to play some more clicks, and you tell me if
the clicks get faster, or slower, or stay the same."

(Play Item 1--Slower. The clicks decrease in tempo
from MM = 120 to 1M = 60 in 15 seconds. If the subject
requested it, the item was repeated, but only once per
item.)









"What about the clicks--did they get faster or slower
or stay the same?" (The subject's response was recorded
on the form. Knowledge of the correctness of the answer
was not given. The experimenter continues by saying:)

"Now listen again and tell me if the clicks get faster
or slower or stay the same." (The same procedure was
followed for the remaining items:)

Item 2: Remains the same. Steady clicks at a tempo
of MM = 80 for 15 seconds.

Item 3: Faster. The clicks increase in tempo from
MM = 72 to MM = 160 in 15 seconds.

Item 4: Slower. The clicks decrease in tempo from
MM = 80 to MM = 40 in 15 seconds.

The order of the experimenter's use of the words "faster,"

"slower," and "stays the same," was varied randomly with

each item.


Conservation of Meter Task (COM)


Aural/Aural Task (COM AA)


The Conservation of Meter Task designed by the

experimenter contains four items. For each item, the

subject hears eight moderately loud non-pitched impulses

Clicks) at a steady tempo (MM = 60) representing meter.

The subject is asked whether the clicks "get faster,

slower, or stay the same." After the subject asserts that

they stay the sane, he is asked to listen to a second

example in which the same steady meter clicks are heard

simultaneously with repeated tones in various rhythms. The

subject is then asked whether the clicks "got faster,









slower, or stayed the same." After his response, he is

asked: "Why? How can you tell?" The subject is not told

to clap his hands, tap feet, etc., but if he does so of

his own accord, these behaviors are noted on the scoring

form, along with any other behaviors or comments. The

subject is not given any knowledge of the rightness or

wrongness of his responses.

All items on the Conservation of Meter Task are

prerecorded using a SONY reel-to-reel stereo tape recorder.

The sounds are generated from a Moog synthesizer, with

nonpitched clicks representing meter (left channel) and

sine wave pitches on a single tone used in the rhythms

(right channel). A different pitch was used in each item

to prevent the subject's saturation with a single tone.

Both clicks and rhythms are heard at the same level of

volume, and a 4/4 meter is used in each item. The

procedures result in each item's having only changes in

rhythm, while pitch, loudness, and timbre remain the same

and are controlled. Thus, the meter/rhythm relationship

is exposed without influence from other sound variables.

The explanation of the task is as follows:

"Now you will hear some clicks again. Tell me if
the clicks get faster or slower or stay the same. (Play
Item 1, part A, 8 clicks alone.) Did the clicks get faster
or slower or stay the same? Now you will hear some clicks
and also some other sounds. Listen very carefully to the
clicks and tell me if they get faster or slower or stay
the same. (Play Item 1, part B, clicks with rhythms.)
What about the clicks--did they get faster or slower or stay
the same? Why? How can you tell? (Record responses on
form.) Okay, now listen again (Item 2). (Same procedure
for Items 2, 3, and 4.)"






44


The following describes the four items: (See

musical examples, pages 45 and 46.)

Item 1: Duple rhythms in diminution: While the meter
clicks continue at a steady tempo, repeated notes (on e)
are heard with one note per click, then two notes per
click, four, eight, etc.

Item 2: Duple rhythms in augmentation: The reverse
of item 1, where the repeated notes (on c) are heard with
sixteen notes per meter click, then eight, four, two, etc.

Item 3: Hemiola rhythms in diminution. Repeated
notes (on d) are heard with the following numbers of notes
per meter click: two, three, four, six, eight, etc.

Item 4: Hemiola rhythms in augmentation: The reverse
of Item 3, with eight, six, four, three, two repeated
notes (on d-flat) per meter click.






45



= clicks




X x )S A n rr



























IK K














A A







46





~s as d2~' br-~ ~
.6 d
















a ~ o ddo qa p do w




















xx
dpd


.. 7-2



so .0 ove~ed a~


&V al










The four items were always administered in the above

order. The order in which the words "faster," "slower,"

and "stays the same" were used was randomly varied with

each item.

The purpose of the four different items is to assess

what difference there is, if any, between perceptions of

meter a) when the rhythms get "faster" (i.e. diminution)

and b) when the rhythms get "slower" (i.e., augmentation),

and what difference there is between perceptions of meter

with c) duple rhythms (where the shift in rhythms is more

pronounced, due to the proportions 1:2:8:16, but where

the "beat" is also more obvious and regular), and d) hemiola

rhythms (where the change is more gradual due to the pro-

portions 2:3:4:6:8, but where the "beat" is less regular

and more obscure). In other words, the intent of the

different items is to assess whether the difficulty

imposed by certain perceptual cues is more influenced by

the gradualness of the changes or the regularity of beats in

the changes.


Aural/Visual Task (COM AV)


For the purposes of this study, a conserver of meter

is defined as a subject who answers correctly at least

three of the four items on the Conservation of Meter

Task. Any subject who is a nonconserver of meter on the aural/

aural task (i.e., incorrectly answers two or more items)

is given the aural/visual form of the task. This









serves as a check, in the event that the subject's failure

on the aural/aural task is due, not to a lack of

conservation, but to the inability to discriminate aurally

between the clicks and the rhythms. The aural/visual form

of the task is the same as the aural/aural form, except that

the clicks are reinforced by the presence of a small blinking

light, with one blink per click. A light mechanism is

connected to the left channel of the tape recorder, and

each click of meter powers a blink of light. Thus the

subject both hears and sees an impulse representing meter.

The procedure for this task is the same as that for the

aural/aural task, except that the subject is asked to

watch the blinks of light. The explanation is as follows:

"Now you will see some blinks from this little light.
(Demonstrate by turning on light.) Tell me if the blinks
get faster or slower or stay the same. (Play Item 1,
part A, blinks and clicks alone.) Did the blinks get faster
or slower or stay the same? Now you will see the blinks
and also hear some other sounds. Watch the blinks very
carefully and tell me if they get faster or slower or
stay the same. (Play Item 1, part B, blinks and clicks
with rhythms.) (The same procedure for Items 2, 3, and 4,
like that in the aural/aural task.)"

The reliability of the Conservation of Meter Task

was established by applying the Kuder-Richardson formula

(Tate, 1955, page 367). The reliability of the aural/aural

form ranged from .77 to .79; that of the aural/visual form

ranged from .58 to .76. (See Chapter Four.)











Conservation of Physical Properties:
The Concept Assessment Kit--Conservation (CAK)


The Concept Assessment Kit--Conservation (Goldschmid

and Bentler 1968) was used to measure conservation of the

following properties: two-dimensional space, number,

substance, continuous quantity, weight, and discontinuous

quantity. Form A was administered as the pretest, while

Form B, a parallel form, was used as the posttest. (Use

of Form C, a test of generalization to conservation of area

and length, was omitted.) The tests of the Concept

Assessment Kit--Conservation were administered according

to the instructions in the manual, a procedure which consumes

approximately 15 minutes. Responses were recorded on the

forms provided, and scored as recommended in the manual

(pages 5-6): correct judgments (behaviors) receive a score

of 1, incorrect judgments, 0; correct explanations (those

involving identity, compensation, and/or reversibility)

receive a score of 1, incorrect explanations (perceptual,

magical, etc.), receive 0. Forms A and B, then, each

yield raw scores of 0 to 12 (one point each for judgment

and explanation on each of six tasks). Percentile

norms were provided for ages four to seven, but these

were not used since the sample of the present study

exceeded that age range. The authors report a reliability

of .94 for Forms A and B (manual, page 11).


1











The Concept Assessment Kit--Conservation is reviewed

in Buros' Seventh Mental Measurements Yearbook (1972) by

J. Douglass Ayers, by Rheta DeVries and Lawrence Kohlberg,

by Vernon C. Hall and Michael Mery, and by Charles D. Smock.

The reviews range from favorable to cautionary, with

controversy over the effects of a well-controlled, stan-

dardized procedure versus Piaget's more informal clinical

method. While the limitations of the Concept Assessment

Kit must be acknowledged, the kit adequately served the

purpose of its role in the conservation of meter study.

Any attempt to standardize Piaget's clinical interview

method must necessarily entail, on the one hand, a compro-

mise of strict Piagetian methods, and on the other, a

compromise of complete standardization. It should be

pointed out that at least one criticism of the kit does

not apply to its use in this study, since the norms

provided by the authors were not used as part of the data.

In short, the Concept Assessment Kit provided for the

present study an easily-administered measure of the major

Piagetian areas of conservation, with appropriate materials

and a standardized interview schedule.


Measure of Intellectual Maturity--The Draw-A-Man Task (DAM)


In order to obtain data on and control for individual

differences in intellectual maturity, the Draw-A-Man Task

(or more correctly, Draw-A-Person Task) was administered.









(Uniform I.Q. scores were not available from school

records.) This task is an adaptation of the Goodenough

Draw-A-Man Test, and its later revision, the Goodenough-

Harris Drawing Test (Harris, 1963). The subject is given

a sheet of plain white paper, 8 1/2 by 11 inches, a No. 2

pencil, and the following instructions (adapted from

Harris, 1963, page 240):

"I am going to ask you to draw a picture for me
today. On this paper I want you to make a picture of a
person. Make the very best picture that you can; take your
time and work very carefully. Be sure to make the whole
person, not just the head and the shoulders."

Harris (1963) cites studies in which the reliability

of the Draw-A-Man technique was .89 by the split-half

method, and ranged from .68 to .91 by test-retest methods.

Evidence for the validity of the Goodenough Draw-A-Man

test is cited by Harris (1963, pages 96-97) in the

form of correlations with other I.Q. measures. Correlations

with the Stanford Binet Test, for example, ranged from

.41 to .65 for the I.Q. score and .26 to .92 for mental

age; correlations with various subtests of the Wechsler

Intelligence Scale for Children ranged from .05 to .77.

The drawings were scored by the experimenter according

to the Point Scale outlined by Harris (1963, pages 248-

292), which uses different scales for male and female

figures in the drawings. The raw scores were converted to

standard scores with a theoretical mean of 100 and a

standard deviation of 15, according to Harris' tables of









norms (1963, pages 294-301). Harris (1963, page 294)

indicates that the norms for four-year-olds were not based

on samples as representative as those for other age groups

and are, therefore, only offered as a "tentative guide"

for use with preschool children. It should also be noted

that these norms contain a bias toward underestimating a

subject's standard score, as indicated by more recently

reported data from the national Health Examination Survey

of 1963-1965 (Harris, Roberts and Finder, 1970), which

also used the Draw-A-Person, rather than Draw-A-Man

technique. The report, based on a collection of drawings

of 7,068 children, states:

Mean scores for children aged 6 to 11 years in the
United States tended to be lower than those from
the Harris norms consistently throughout the age
range on the Man and Woman scales for both boys
and girls. (page 10)

Thus, the Draw-A-Man scores in the present study

likely reflect the bias of the Harris norms.


Training Procedures


All subjects who were nonconservers of meter on the

pretest were randomly assigned to one of two groups:

1) training in conservation of meter, or 2) a control

group (no training whatsoever). Training was administered

by the experimenter to groups of four to five subjects of

similar age (within a two year span), in periods of about

30 minutes.









The training was as follows:

1. Subjects observed an electric metronome set at MM = 80,

which simultaneously emits clicks and blinks from a

small light. They were asked to engage in large-

muscle movements which correspond to the metronome

impulses (moving "to the beat"), first clapping,

then swinging arms, tapping one foot, and finally

marching in a circle. The same was repeated with the

metronome set at MM = 60 and MM = 120.

2. The subjects were then asked to listen to musical

selections and duplicate the experimenter's movements

in "follow the leader" style. When the music began,

the experimenter clapped, tapped one foot, or marched

to the beat of the music, and the subjects followed.

Five musical examples were used, which had been chosen

for their maintenance of a steady meter concurrent with

obvious changes of rhythm. Each of the selections

contained some type of rhythmic change involving either

diminution or augmentation. Under the experimenter's

direction, the subjects attempted to maintain their

steady movements to the meter of the music, in spite of

the rhythmic changes. The selections were prerecorded

on tape and were taken from the record series for

elementary schools Adventures in Music (RCA Records, 1970).

The five selections were:

1. J.S. Bach, Suite No. 2 in B minor: Rondeau










2. Gluck, "Armide" Ballet Suite: Musette

3. Gretry, (Arranged by Mottl), Cephale et Procris:

Tambourin

4. Shostakovich, Ballet Suite No. 1: Petite Ballerina

5. Pierne, "Cydalise" Suite No. 1: Entrance of the

Little Fauns


3. If any of the subjects were unable to maintain steady

movements throughout the piece, the experimenter repeated

the selection, and helped those subjects who were

having difficulty by being in closer contact with them,

saying "watch me" and "step, step, step, step" or "one,

two, three, four," etc. Following each selection the

experimenter posed these questions to the group:

"Did our marching (or our feet or our arms, etc.)
move the same all through the piece? Or did we get
faster or slower? (We stayed the same.) And what
about the music? Did the "beat" of the music change
sometimes or did it stay the same? (The "beat stayed
the same; the rhythms or "other notes" changed, got
faster, slower, etc.)"

As much as was practicable, the experimenter continued

in this fashion with each selection, teaching the

subjects to make steady movements to music with

rhythmic changes and afterwards reflecting on their

movements and the music.

The training was designed to enhance the ability to

conserve meter by having the subjects become aware of their

own steady motor responses (clapping, marching, etc.) in

the face of changes in rhythm. It was intended that this










experience promote cognitive conflict in subjects who

would not assert the invariance of the "beat" (meter) by

pointing out to them the fact that their own movements

"went with the beat" and were remaining the same all through

the piece.


Sampling Procedure and Subject Selection


The population of the present study was children

(ages 4, 5, 7, and 9 years) of one day care center and

two elementary schools (one of which included a preschool

center) in Gainesville, Florida, who met the following

conditions:

1. were of the Caucasian race,

2. had not at any time received private music instruction,

3. had the permission of their parents) or guardian(s)

to participate in the study.

One of the elementary schools is part of a university

laboratory school, which has a large population of children

of university employees. The second elementary school

was chosen to counterbalance this biasing effect, since

the second school is in a different geographic area, and

is believed to serve a different (generally lower)
1
socioeconomic population. The day care center was chosen

on the basis of convenience and willingness to participate



However, no significant differences were found
between the two elementary schools on any of the following
measures: Conservation of :Meter Task (t = .16; df = 83),
Concept Assessment Kit--Conservation (t = .15; df = 83), and
Draw-A-Man Task (t = .22; df = 83).









in the study. This center primarily serves the children

of university students and is church-affiliated.

All of the 4-year-olds at the day care center who

met the conditions outlined above were selected for the

study. Also, all of the 4-, 5-, 7-, and 9-year-olds at

the second school who met the conditions were selected.

(School II contained a preschool center from which two

4-year-olds were drawn.) In the first school, however,

5-, 7-, and 9-year-old subjects who met the conditions

were selected only from those classrooms (two at each

grade level) that were assigned by the assistant director

of the school to participate in the study.

The composition of the sample is presented in Table

1, which shows a total of 103 subjects at ages 4, 5, 7,

and 9 years:


Table 1
Composition of Subject Sample


Total
Age No. Males Females School No. School No.

4 yrs. 18 11 7 Day Care 16 Sch. II-Pre 2

5 yrs. 42 24 18 Sch. I 21 Sch. II 21

7 yrs. 22 12 10 Sch. I 10 Sch. II 12

9 yrs. 21 12 9 Sch. I 10 Sch. II 11



The above table shows the complete sample, after a

few subjects in each age group were lost due to absenteeism,










illness, etc. The age indications refer to one year

ranges; i.e., "age 4" means 4 years, 0 months to 4

years, 11 months. The sample was intentionally selected

to have more subjects at the 5-year-old level, since it

was expected that this age group would provide important

data on the nature of nonconservation and the effects

of training.


Design of the Study


The design of the study is a "Pretest-Posttest Control

Group Design" (Campbell & Stanley, 1963, page 13), which

can be simply represented as:

R 0 X O

R 0 0

where R indicates random assignment of subjects to two

different groups, 0 indicates observations or measurements

made, and X indicates treatment or training.

The steps that were followed in carrying out the

design were:

I. Administration of Preliminary Vocabulary Test

All subjects were given the Preliminary Vocabulary

Test. Any who did not answer all four items correctly

were eliminated from the study and were replaced by

other subjects of the same age and sex drawn from the

pool of subjects available at that school

Subjects who passed the vocabulary test were immediately

given the protests. Subjects who did not pass the vocabulary










test and who were not retained in the study were given

the first few items on each of the protests so that, upon

return to their classrooms, their experience with the ex-

perimenter was not different from that of their peers.

II. Administration of Pretests

All subjects who passed the Preliminary Vocabulary

Tests were given:

a. Conservation of Meter Task--Aural/Aural.

All subjects who did not correctly answer at least three

of the four items were given:

Conservation of Meter Task--Aural/Visual

b. Concept Assessment Kit--Conservation, Form A

c. Draw-A-Man Task.

All subjects who did not correctly answer at least

three of the four items on either the aural/aural or the

aural/visual Conservation of Meter Task were classified

as "nonconservers of meter." There were 45 such subjects.

III. Administration of Training

All "nonconservers of meter" were randomly assigned

to one of two groups:

a. those who received training (experimental group),

b. those who received no training (control group).

IV. Administration of Posttests

Following training (and after an interval of about

one week), all subjects assigned to both the training and

control groups were given:











a. Conservation of Meter Task--Aural/Aural, and, if

necessary, Conservation of Meter Task--Aural/Visual,

b. Concept Assessment Kit--Conservation, Form B.

The posttests were the same as the protests, except

that Form B of the Concept Assessment Kit was used as the

posttest.

Although the training dealt with conservation of meter,

and not with the other Piagetian conservation, the posttests

included a re-administration of the Concept Assessment Kit,

in order to provide control over the effects of history.

In the event that some subjects had achieved conservation

and concrete operational thought in the interim between the

pre and post testing, their scores on the Concept Assess-

ment Kit would reflect this by being significantly higher

on the posttest than they were on the pretest. Changes

in score which indicated an advancement toward conservation

and concrete operational thought on the posttest were

statistically controlled in the analysis of the data. In

addition, cross-modal transfer, had it occurred, would

be evident, e.g., where training in conservation of meter

had the effect of improving scores in conservation of

space, number, or substance, etc..

The following is a list of possible confounding

variables that were controlled by the subject selection

and/or design of the study:

1. race (by subject selection),









2. level of previous music instruction (by subject

selection),

3. level of vocabulary understanding (by administration

of the Preliminary Vocabulary Test),

4. level of intellectual maturity (by administration of

the Draw-A-Man Task),

5. level of ability in aural perception (by administration

of the aural/visual form of the Conservation of Meter

Task),

6. effects of maturation and history (both controlled

by presence of control group; in addition, cognitive

maturation of subjects was controlled by the posttest

administration of the Concept Assessment Kit--

Conservation).


Data Collection


All tests were administered individually (to one

subject at a time) in a quiet room away from the normal

classroom. The experimenter and three trained assis-

tants (undergraduate students in Childhood Education)

administered all tasks, using the same procedures by

referring to carefully-written instruction sheets. Only

two examiners administered tests at any given time;

throughout the conduct of the study, however, the

experimenter was aided by three different assistants.

Training was administered only by the experimenter.






61


Within each school, no changes were made in the room,

physical arrangements, or facilities and equipment used

in testing. Complete data collection consumed approximately

four to six weeks at each of the three sites.
















CHAPTER FOUR
DATA ANALYSIS AND RESULTS OF THE STUDY


Analysis of data was accomplished in part through the

use of subprograms from SPSS, Version 6.00 (Statistical

Package for the Social Sciences, Nie et al., 1975).

This chapter reports the results of the study, and is

divided into three sections: 1) the reliability of the

Conservation of Meter (COM) Task, 2) a summary of descrip-

tive statistics, 3) presentation of data relevant to the

questions of the study.


Reliability of the Conservation of Meter Task


The Kuder-Richardson formula (Tate, 1955, page 367)

was used to provide an estimate of the reliability of the

Conservation of Meter Task based on the number of items,

variance of scores, and sum of item variances, according

to the following formula:


r = (n ) (-4P)


where n = the number of task items, !5 = the variance

of the scores, p = the proportion of subjects who answer

a given item correctly, and q = 1 p.

The estimated reliability of the Conservation of

Meter Task--Aural/Aural (COM AA) ranged from .77 (based










on pretest data) to .79 (based on posttest data). The

estimated reliability of the Conservation of Meter Task--

Aural/Visual (COM AV) ranged from .58 (based on pretest

data) to .76 (based on posttest data).

A possible explanation for the lower reliability of

COM AV (based on pretest data) is that the COM AV task

was only administered to those subjects who were low

scorers on the aural/aural form of the task (COM AA)--i.e.,

those who scored less than three out of a possible total

of four. Of the subjects who took the COM AV task, 71.4%

continued to score low (below three) on the task; thus,

the variance of scores for COM AV (1.727) is lower than

that for COM AA (2.274), while the sum of item variances

remained roughly the same (.973 for COM AV and .965 for

COM AA). In the formula, a lower variance, other things

being equal, results in a lower estimate of reliability.

Given this subject selection factor in the administration

of the COM AV task, and the fact that the task is only
1
four items in length, the obtained estimates showed that

the COM task is sufficiently reliable.



1
Increasing the number of items on the task would
have increased its reliability. Using the Spearman Brown
Prophecy formula (Tate, 1955, page 334), the predicted
reliability of the COM AV task would be increased from .58
to .73 if the number of items on the task were doubled,
and other factors remained the same.











Summary of Descriptive Statistics


The following section presents summary tables of

descriptive statistics on scores obtained on the Conser-

vation of Meter Task (COM), the Concept Assessment Kit--

Conservation (CAK), and the Draw-A-Man Task (DAM), as

well as simple correlations between pairs of measures.


Conservation of Meter Task


As described previously, each subject was given the

COM AA task. Those who achieved a score of three or four

(out of four) were designated conservers of meter. Those

who achieved a score of only zero, one, or two were given

the aural/visual form of the task (CCM AV). Likewise,

those who obtained a score of three or four on the COM AV

task were designated conservers of meter, while those

who scored zero, one, or two were called nonconservers of

meter. For the purposes of the data analysis, an "adjusted"

or final COM score was used for each subject. The

"adjusted" score for a subject who was given only COM AA

(and who thus scored a three or four), was the COM AA

score. The "adjusted" score for a subject who was given

both the COM AA and COM AV (and who thus scored zero, one,

or two on COM AA) was the COM AV score. These COM "adjusted"

scores are, then, not statistically adjusted, but represent

the last form of the task that was given.










To summarize, the following may be noted. The COM

AA pretest was given to 103 subjects, of whom 56 scored

as nonconservers. Of these 56 subjects, 11 scored as

conservers on the aural/visual form of the task (COM AV).

On the posttests, 45 subjects were given COM AA (post),

of whom 38 scored as nonconservers. Of these 38 subjects,

9 scored as conservers when given the aural/visual form

(COM AV). Thus a total of 20 subjects were able to improve

their scores as a result of being given the aural/visual

task. One explanation for this improvement is that the

aural/visual form of the task controls for a possible

weakness in aural discrimination on the part of the subject,

and thus measures a conservation of meter schema that the

subject indeed has, but is masked by difficulties with a

purely aural task. This explanation conforms to the

reasoning behind the use of the visual mode, as well as

the aural mode, in the Conservation of Meter Task. However,

other possible explanations are:

1. that presenting the task twice causes practice effects

or the effects of learning, to improve performance on

the second (COM AV) administration of the task;

2. that reinforcing the meter clicks with a visual

stimulus structures the task in such a way as to make

it more understandable (perhaps more concrete) to the

subject;

3. that improvement on the aural/visual form of the task









is a result of the way in which different areas

of the brain mediate stimuli of different modalities.

The adjusted COM scores, then, reflect the improvement

that some of the subjects made on the second form of the

task. Table 2 shows the means and standard deviations of

adjusted COM scores for each age group. The means of

COM scores increase at each age level. (Subjects at ages

5, 7, and 9 years were drawn from two schools. There

was no significant difference between the total mean COM

scores for School I and School II.) (t = .16; df = 83)


Table 2
Conservation of Meter


Task Scores


Age Group Mean Std. Dev. N

For entire population 2.6990 1.4131 103

4-year-olds--Preschool 1.722 1.406 18

5-year-olds--Kindergtn. 2.619 1.413 42

7-year-olds--Grade 2 2.909 1.411 22

9-year-olds--Grade 4 3.476 0.873 21









Table 3 presents the percentages of conservers of

meter (based on adjusted COM scores) within each age

level. There is an increase at each level, with 33% of

the 4-year-olds and 76% of the 9-year-olds being conservers
1
of meter.


Table 3
Conservers of Meter
(Pretest Scores)


No. of Conservers Percent of Age Group
Age Group of Meter Designated Conservers of Meter

4-year-olds 6 33.3

5-year-olds 27 64.3

7-year-olds 15 68.2

9-year-olds 16 76.2








1
It should be noted that, for the purposes of this
conserver/nonconserver classification, 5 subjects in the
4-year-old group and 1 subject in the 5-year-old group
were classified as conservers of meter with considerable
doubt and reservation. While these children did
achieve criterion scores, they did so only with great
difficulty, needing to have items repeated, taking guesses,
changing their answers several times, and requiring the
examiner to re-question them. Thus the figures above are
exaggerated at the younger age levels; more strict
scoring of the performance of these children would result
in even fewer conservers of meter at the 4-and 5-year-old
level.









Table 4 is a composite table of means and standard

deviations of adjusted COM scores, showing, for each age

group, a breakdown by school and sex. (Differences

between age groups and sexes are discussed later.)


Table 4
Conservation of Meter Task Scores



Age Group Mean Std. Dev. N

For entire population 2.6990 1.4131 103

4-year-olds--Preschool 1.722 1.406 18
Day Care 1.938 1.340 16
Male 1.700 1.418 10
Female 2.333 1.211 6
School II--Preschool 0.0 0.0 2
Male 0.0 0.0 1
Female 0.0 0.0 1

5-year-olds--Kindergtn. 2.619 1.413 42
School I 2.524 1.569 21
Male 2.733 1.438 15
Female 2.000 1.897 6
School II 2.714 1.271 21
Male 3.000 1.000 9
Female 2.500 1.446 12

7-year-olds--Grade 2 2.909 1.411 22
School I 2.400 1.350 10
Male 2.800 1.304 5
Female 2.000 1.414 5
School II 3.333 1.371 12
Male 4.000 0.0 7
Female 2.400 1.817 5

9-year-olds--Grade 4 3.476 0.873 21
School I 3.300 0.949 10
Male 3.600 0.894 5
Female 3.000 1.000 5
School II 3.636 0.809 11
Male 3.429 0.976 7
Female 4.000 0.0 4









Finally, Table 5 presents posttest data, showing the

means and standard deviations of adjusted COM posttest

scores for each age group, with a breakdown by school and

treatment group (experimental or trained group and control

or untrained group). (Differences between these groups

are discussed later.)


Table 5
Conservation of Meter Posttest Scores


Age Group

For entire populations

4-year-olds--Preschool
Day care
Control
Experimental
School II--Preschool
Experimental

5-year-olds--Kindergtn.
School I
Control
Experimental
School II
Control
Experimental

7-year-olds--Grade 2
School I
Control
Experimental
School II
Control
Experimental

9-year-olds--Grade 4
School I
Control
Experimental
School II
Control


Mean

2.0667

1.824
1.800
1.571
2.000
2.000
2.000

2.353
1.889
1.000
2.600
2.875
2.667
3.000

1.143
1.200
2.000
0.667
1.000
2.000
0.0

3.500
3.333
3.500
3.000
4.000
4.000


Std. Dev.

1.5869

1.667
1.740
1.718
1.852
1.414
1.414

1.455
1.269
0.816
1.140
1.553
1.528
1.732

1.574
1.789
2.828
1.155
1.414
0.0
0.0

0.577
0.577
0.707
0.0
0.0
0.0


I----- --------------









Concept Assessment Kit--Conservation


The Concept Assessment Kit--Conservation (CAK) is

scored from 0 to 12, with one point each awarded for

correct judgment and explanation on each of six Piagetian

tasks: conservation of two-dimensional space, number,

substance, continuous quantity, weight, and discontinuous

quantity. Table 6 presents the means and standard

deviations of these scores for each age group. There

was an increase at each age level, as would be expected.

(There was no significant difference between total mean

CAK scores for School I and School II.) (t = .15; df = 83)


Table 6
Concept Assessment Kit Scores



Age Group Mean Std. Dev. N

For entire population 6.2524 4.8481 103

4-year-olds--Preschool 1.556 2.526 18

5-year-olds--Kindergtn. 4.476 4.221 42

7-year-olds--Grade 2 8.364 3.849 22

9-year-olds--Grade 4 11.619 0.865 21



Draw-A-Man Task


Standard scores for the Draw-A-Man Task (DAM), as a

measure of intellectual maturity, are based on norms

reported by Harris (1969) which have a mean of 100 and a

standard deviation of 15. Table 7 shows the means and









standard deviations of the Draw-A-Man scores for each

age group in the present study. (There was no signi-

ficant difference between total mean DAM scores for

School I and School II.) (t = .22; df = 83)


Table 7
Draw-A-Man Task Standard Scores



Age Group Mean Std. Dev. N

For entire population 96.8835 16.0314 103

4-year-olds--Preschool 88.000 13.647 18

5-year-olds--Kindergtn. 96.976 16.107 42

7-year-olds--Grade 2 101.136 15.204 22

9-year-olds--Grade 4 99.857 16.662 21



With the exception of the 4-year-olds, group means ranged

from approximately 97 to 101, with standard deviations of

approximately 16 to 17. This can be taken as an indi-

cation that the sample was representative in terms of

intellectual maturity. The exception may have been the

4-year-old group, whose mean Draw-A-Man score was 88;

however, Harris' tables of norms (1969, pages 294-301)

explicitly state that the norms provided for 3-and 4-year-

olds were not derived from representative samples and are

offered only as a "tentative guide." Further, more recent

research has indicated that Harris' norms for all age

groups are not representative of U.S. children (Harris,

1970), and tend to result in low standard scores.









Correlations Between Measures


Simple correlations between pairs of measures were

computed with the Pearson Product-Moment Correlation

Coefficient. Mathematically this correlation is defined

as (Tate, 1955, page 238) :


r = (X X)(Y Y)

[ ( )2] [ (Y )2]


where X and Y are the two variables in question, X = the

mean of X, and Y = the mean of Y.

Table 8 shows the correlation coefficients between

pairs of variables, the levels of significance (for

one-tailed tests), and the number of subjects on which

the calculations were based.


Table 8
Correlations Between Measures



Pearson
Variables Corr. Coef. Sig. N

COM (adjusted) and CAK .52 .001 103

COM AA and COM AV .31 .009 56

DAM (standard) and CAK .38 .001 103

DAM (standard) and COM (adjusted) .28 .002 103



Of particular importance is the correlation of .52

between the measure of Piagetian conservation tasks (CAK)

and the Conservation of Meter Task. The low but significant

correlation between the aural/aural and aural/visual forms









of the COM task (.31) may be partly explained by the fact

that some of the subjects (perhaps those whose COM AA

score was low due to poor aural discrimination) had an

increase in score on COM AV, presumably because of the

addition of the visual stimulus to the task. A second

factor in this low correlation may be the lower reliability

of COM AV (.58) compared to that of COM AA (.77). Finally,

it should be re-emphasized that, while both forms of the

task purport to measure conservation of meter, they do

so with differences in the sensory modalities usedand

this too may be a factor in the low correlation between

the two forms. Of particular interest also are the low

but significant correlations between the Draw-A-Man

task (standard scores) and both the Piagetian conservation

tasks and the COM task. It is consistent with Piagetian

theory to find no high correlation between conservation

ability and an "intelligence" score.


The Questions of the Study


Below are listed each of the questions of the study,

followed by the results of the data analysis.


Questions la and 2


Are scores on the Conservation of Meter Task related to
age?

Are scores on the Conservation of Meter Task related to
scores on conservation tasks involving physical
properties (e.g., space, number, substance, etc.)?









Questions la and 2 were combined for the purposes of

data analysis. The relationship between COM (adjusted)

scores and the combined effects of age (in months) and

CAK scores were investigated through multiple regression

analysis (SPSS subprogram REGRESSION) with Draw-A-Man

raw scores as an additional variable in the analysis.

Table 9 presents the partial correlation coefficients for

the variables in question:


Table 9
Partial Correlation Coefficients



COM Age CAK DAM (raw)

COM .3766 .5235 .4039

Age ---- .7178 .7726

CAK ---- ---- .6595

DAM (raw)



It is apparent that both CAK and DAM are more highly

correlated with COM than is age.









Table 10 summarizes the results of multiple regression

analysis, using the standard regression strategy in which

variables are entered in the order of correlation, and each

variable is given credit only for its incremental contri-

bution, after the effects of the previous variables have

been assessed.


Table 10
Multiple Regression: COM with CAK, DAM, and Age



Variable Multiple R R Squared RSQ Change B Beta

CAK .52352 .27408 .27408 .14261 .48929

DAM (raw) .52930 .28016 .00608 .02064 .15281

Age .53194 .28296 .00280 -.00614-.09270

(Constant) 1.89105


These results

CAK score, DAM raw

by a multiple R of

level (analysis of

factors yields F =


showed that the combined effects of

score, and age were related to COM

.53 which is significant at the .001

variance of the regression and residual

13.02 with df = 3, 99). Thus, CAK,


DAM, and age together accounted for approximately 28.3%
2 2
of the variance in COM scores (R .53 = .28). By

itself, CAK score accounted for 27.4% of the variance in

COM scores; adding both DAM score and age to this effect

increased the amount of variance accounted for by only .9%,

a negligible amount. This result is explained by the






76

fact that, even though DAM scores and the age variable

had partial correlations with COM of .40 and .38, respectively,

their high correlation with CAK scores (.66 and .72

respectively) allowed them to add only a minute contribu-

tion to the overall correlation. In other words, once

adjustments were made in the DAM and age variables for

the effects of CAK score, little remained that could be

attributed solely to DAM scores or age.

Table 10 also gives regression coefficients for the

prediction equation Y' = BX1+ B2 + B3X3 + A, where

Y' = predicted COM scores

X, = CAK scores

X2 = DAM raw scores

X3 = age in months.

Bl, B2, and B3 are coefficients associated with X1, X2,

and X3 respectively, and A = a constant.

The resulting equation is:

Predicted COM = (.143 X CAK) + (.021 X DAM)

+ (-.006 X Age) + 1.891

(with standard error of estimate = 1.215).

The overall significance of this regression equation was

.001 (analysis of variance of the regression and residual

factors yields F = 13.02 with df = 3, 99). Further tests

for the significance of the individual regression

coefficients were made according to the following formula,

using the standard regression approach (Nie et al., 1975,

page 337):










R2 change/i df

1 R2/(N2 k 1) df


The results showed that the regression coefficient B1

for CAK was significant at the .001 level (F = 36.99

with df = 3, 99), while the regression coefficient B2

for DAM was not significant (F = .83 with df = 3,99),

and the coefficient B3 for age (and its negative sign)

were also not significant (F = .42 with df = 3, 99).

These latter two variables, DAM and age, can thus be

ignored as predictors of COM when and only when CAK is

used as a predictor.

It was concluded that, in answer to Questions la and

2, both CAK and age are positively related to COM. When

the effects of CAK, DAM, and age are combined, they attain

a multiple R of .53 and account for 28% of the variance,

of which such a significant portion is due to CAK that the

contributions of DAM scores and age are superfluous.

These results support the hypotheses of the study that

success on the COM task is related to age and that it

is related to CAK scores.

The results obtained through multiple regression can

be underscored by an additional analysis of the data that

was made using analysis of variance procedures. Table 11

below summarizes the analysis of variance of COM adjusted

scores by age group and CAK scores, using a hierarchal

approach (SPSS subprogram ANOVA; option 10) in which the










sum of squares for the first variable, age group, is

not adjusted for the variable CAK (but the variable

CAK is adjusted for the effects of age group).


Table 11
Analysis of Variance of COM Scores



Source of Sum of Mean Sig.
Variation Squares df Square F of F

Main Effects 73.926 14 5.280 3.582 .001

Age Grp. 31.098 3 10.366 7.031 .001

CAK 42.828 11 3.893 2.641 .006

Residual 129.743 88 1.474

Total 203.669 102 1.997



These results showed that age group was significant at

the .001 level, when the sum of squares was not adjusted

for the effects of CAK, and CAK scores were significant

at the .006 level, even when the sum of squares was

adjusted for the effects of age group. Thus, with the

hierarchal approach, age group was seen as significant.

(This same result was also seen in the analysis of variance

of COM scores by age group and sex for Question lb. For

the results of post hoc tests on the differences among

means of each age group, see that section, page 80.)

However, an alternative analysis of variance procedure

produced different results. Table 12 summarizes the










results of analysis of variance of COM scores by age

group and CAK scores using the classic experimental

approach (SPSS subprogram ANOVA; default option). With

this approach, the sum of squares for each factor is

adjusted for the effects of the other factor.


Table 12
Analysis of Variance of COM Scores



Source of Sum of Mean Sig.
Variation Squares df Sauare F of F

Main Effects 73.926 14 5.280 3.582 .001

Age Grp. 2.964 3 .988 .670 .999

CAK 42.828 11 3.893 2.641 .006

Residual 129.743 88 1.474

Total 203.669 102 1.997



Thus, the results showed that CAK was still significant

at the .006 level, even after adjustments for age were

made, but age group was not significant when its sum of

squares was adjusted for the effects of CAK scores.

To summarize the results of the data analysis on this

question, both age and performance on Piaget's conservation

tasks (as measured by CAK) were significantly related to

performance on the COM task; however, since age and CAK

score were highly correlated, the latter variable was a

stronger predictor, and the former variable was not









significant when and only when adjustments were made for

the effects of CAK. These results were, nonetheless,

taken as tentative support for the hypotheses of the

study that success on the COM task is related to age and

success on Piagetian conservation tasks.

The following section (Question lb) pursues this

issue further and shows that age group was significant

when combined in the analysis with sex. This section

also shows the results of the post hoc comparisons among

the mean COM scores of each age group.


Question lb


Is there a significant difference in mean scores on the
Conservation of Meter Task among various age groups
(4, 5, 7, and 9 years) or between sexes?

Table 13 summarizes the results of analysis of

variance (SPSS subprogram ANOVA; classic experimental

approach) of the adjusted COM scores by age group (4, 5,

7, and 9 years) and sex (male, female).









Table 13
Analysis of Variance of COM Scores



Source of Sum of Mean Sig.
Variation Squares df Square F of F

Main Effects 35.542 4 8.886 5.280 .001

Age Grp. 31.765 3 10.588 6.291 .001

Sex 4.445 1 4.445 2.641 .103

2-way inter-
actions 8.245 3 2.748 1.633 .186

Group Sex 8.245 3 2.748 1.633 .186

Residual 159.882 95 1.683

Total 203.669 102 1.997



Age was significant at the .001 level, while sex and

the sex-by-age group interaction were both nonsignificant.

Differences among the COM task means for each age group

were further tested using Duncan's New Multiple Range Test

(Kirk, 1968, page 93). The formula for this test is:


W = q oc;r, v error
n

where W = the difference between two means that a
r
comparison must exceed in order to be declared significant,

qr = a value obtained from Duncan's table (Kirk, 1968,
page 533), MSerror = mean square error, and n = the number

of subjects in each group. Since there was an unequal

number of subjects in each age group, n in the above was

replaced by weighted average equal to
where k = the number of groups being compared two)
where k = the number of groups being compared (two).









Evidence supporting this procedure is given by Kramer (1956).

Table 14 shows the mean COM score for each age group, and

the differences between each pair of means. The figures

in parentheses give the level at which the differences

are significant (n.s. means not significant).


Table 14
Differences Among Mean COM Scores



Age Group 4 years 5 years 7 years 9 years

1.722 2.619 2.909 3.476

4 years 1.722 .897(.05) 1.187(.01) 1.754(.01)

5 years 2.619 .290(n.s.) .857(.05)

7 years 2.909 .567(n.s.)

9 years 3.476



These results show that the differences in mean

COM scores were all in the expected direction--i.e., in

general older subjects had significantly higher mean

scores than younger ones. The exceptions occurred in

the differences between 5- and 7-year-olds and between

7- and 9-year-olds, which were not significant (although

the difference between 5-and 9-year-olds was significant).


Question 3


Is improvement in performance on the Conservation of Meter
Task resistant to training for preoperational subjects?
Specifically, is there a significant difference in
mean scores on the Conservation of Meter posttest
between trained and untrained subjects, at the
preoperational (or concrete operational) level?






83

Table 15 summarizes the results of analysis of

variance (SPSS subprogram ANOVA; classic experimental

approach and covariates processed prior to factors) of

COM posttest scores by treatment group (trained and

untrained subjects) and level of CAK conservation

score (preoperational subjects with a CAK score of three

or less and concrete operational subjects with a score

of six or higher).- Analysis of covariance was added to

the procedure to provide adjustments for possibly

confounding variables:

1. changes in developmental level during the pre-to-post

test interim (indicated by pre-to-post test changes,

if any, in CAK score),

2. intellectual maturity (Draw-A-Man standard scores),
2
3. pretest scores on the COM AA task.

(Of these covariates, only the COM AA variable achieved

significance.)





1
An initial attempt to partition posttested subjects
into three levels of conservation score--preoperational,
transitional, and concrete operational--failed because
the resulting cell sizes were too small.

2
The aural/aural rather than aural/visual form of
the COM task was used as a covariate because of its
higher estimated reliability.









Table 15
Analysis of Variance of COM Posttest Scores


Source of
Variation

Covariates

Change

DAM Stand.

COM AA Pre.

Main Effects

Level

Treatment

2-way inter-
actions

Level Trtmnt

Residual

Total


Sum of


quares df

15.521 3

.088 1

2.184 1

14.079 1

13.699 2

13.678 1

1.004 1


5.981

5.981

58.773

93.975


The results showed that there was no significant

difference between the nean COM posttest scores of

trained and untrained subjects. There was, however, a

significant difference between the COM posttest scores

of preoperational and concrete operational subjects (as

measured by CAK). Preoperational subjects (CAK = 3 or

less) had a mean COM score of 1.88, while concrete

operational subjects (CAK = 6 or more) had a mean of

2.62, with the difference significant at the .009 level.

(Both of these means were below the level specified for

"conservation of meter," i.e., COM equal to three or four.)


Mean
Square

5.174

.088

2.184

14.079

6.849

13.678

1.004


5.981

5.981

1.781

2.410


F

2.905

.050

1.226

7.905

3.846

7.680

.564


3.358

3.358


Sig.
of F

.048

.999

.276

.008

.031

.009

.999


.073

.073


S






85

The level-by-treatment interaction was not signifi-

cant, except at the .073 level, which almost approaches

significance. Theoretically, this interaction would be

expected to yield significance to the effect that trained,

concrete operational subjects achieve the highest mean

score on the COM posttest. This expectation was not

supported, but should be re-investigated, since a

possible reason for the lack of significance may be the

small cell sizes that result from testing interactions.

Perhaps a larger sample would produce a result different

from this one.

These results support the third hypothesis of the

study. Improvement in performance on the COM task is

apparently resistant to training, for both preoperational

and concrete operational subjects. In addition, the

result that concrete operational subjects achieve higher

COM scores than their preoperational counterparts is

consistent with the hypothesis that conservation of meter

is related to conservation of physical properties (as

measured by CAK),


Questions 4, 5, and 6 involve pretest frequency data

on the items of the Conservation of Meter Task. All three

questions were answered by applying chi squared analysis

to the data. The formula for chi squared is (Tate, 1955,

page 263):






86

X2 = (fo- ft)2




where f = observed frequencies, ft = expected frequencies,

and X2 is referred to a table of the sampling distribution

of this statistic (Tate, 1955, page 561).


Question 4


If a Conservation of Meter Task item is incorrectly answered,
is the erroneous response to an item involving diminution
more likely to be "faster"? Is the erroneous response
to an item involving augmentation more likely to be
"slower"? Specifically, is there a significant
difference between the number of "faster" and "slower"
responses among the incorrect responses given on each
of the Conservation of Meter Task items?

Table 16 presents, for each item on the COM task, a

breakdown of the incorrect responses into "faster" and

"slower" responses, and the level at which differences

are significant.


Table 16
Incorrect Responses to COM Task Items



No. Faster No. Slower
Item Type Rhythm Responses Responses Sig.

Conservation of Meter Task--Aural/Aural

1 diminution 18 18 n.s.
2 augmentation 9 30 .01
3 diminution 22 17 n.s.
4 augmentation 11 23 .05

Conservation of Meter Task--Aural/Visual

1 diminution 20 6 .01
2 augmentation 18 15 n.s.
3 diminution 24 5 .01
4 augmentation 12 19 n.s.









Significant differences, when they occurred, were all

in the expected direction; that is, rhythms in diminution

promoted an incorrect response of "faster" (since the

rhythms are getting "faster"), and rhythms in augmentation

promoted an incorrect response of "slower." While

significant differences are, in fact, in the expected

direction, there is no ready explanation for why, on the

aural/aural form of the task, these expectations were

borne out on the items involving augmentation but not

on those involving diminution, while on the aural/visual

form of the task, the reverse is true. (This result may

be described as an item-by-modality interaction.) Since

each of the diminution items and each of the augmentation

items utilize rhythms in both duple and hemiola propor-

tions, the rhythmic relationships used in the items are

apparently not a factor. It may be concluded that the

results are equivocal, and warrant further investigation

of this question. In the meantime, it is only possible

to speculate on the reasons for these results. One

very tentative hypothesis for this item-by-modality

interaction may be that items involving rhythms in

augmentation are more likely to be erroneously perceived

as "slower" (instead of "staying the same") because the

subject's attention is immediately attracted to the very

fast, dense sounds at the beginning of the item. (It

should be recalled that augmented rhythms gradually









get "slower"after a measure of very fast notes.) In

other words, augmentation items, more so than diminution

items, might be more likely to draw the subject's focus

to the rhythms, and thus cause the subject to give the

response of "slower." This effect, which results from

the initial presentation in augmentation items of a

flurry of fast, dense notes, may operate differently on

the aural/aural and aural/visual forms of the task.

Perhaps the effect has strength only in the completely

aural presentation because in the aural/visual presentation,

the visual stimulus (blinking light) has the higher

priority of the subject's attention.

To summarize, further investigation might be devoted

to how the subject's initial attention is differentially

attracted to rhythms in augmentation and diminution, and

how this effect might be altered when a competing

stimulus Dlinks of light) is presented.


Question 5


Are Conservation of Meter Task items involving duple rhythms
more difficult than those involving hemiola rhythms?
Specifically, is there a significant difference
between the number of subjects who correctly answer
Conservation of Meter Task items involving duple rhythms
and those involving hemiola rhythms?

Table 17 shows the number of subjects who correctly

answered the duple items (Items 1 and 2) and hemiola

items (Items 3 and 4) on the Conservation of Meter Task.








Table 17
Duple and Hemiola Items on the COM Task


Duple Items Hemiola Items
1 2 3 4

COM AA N = 103 62 59 59 64
(total: 121) (total: 123)

COM AV N = 56 28 21 25 23
(total: 49) (total: 48)



There was no significant difference between the

number of subjects who correctly answered the hemiola

items and the number of subjects who correctly answered

the duple items. It may be concluded that both types

of items are of equal difficulty.


Question 6


Are Conservation of Meter Task items involving rhythms
in diminution more difficult than those involving
rhythms in augmentation? Specifically, is there a
significant difference between the number of subjects
who correctly answer Conservation of Meter Task
items involving rhythms in diminution and those
involving rhythms in augmentation?

Table 18 presents the number of subjects who

correctly responded to COM items involving diminution

(Items 1 and 3) and those involving augmentation (Items

2 and 4).








Table 18
Diminution and Augmentation Items on the COM Task



Diminution Items Augmentation Items
1 3 2 4

COM AA N = 103 62 59 59 64
(total: 121) (total: 123)

COM AV N = 56 28 25 21 23
(total: 53) (total: 44)



There was no significant difference between the

number of subjects who correctly answered items involving

rhythmic diminution and those who correctly answered items

involving rhythmic augmentation. It may be concluded

that both types of items are of equal difficulty.


Question 7


For what age groups) or sex(es), if any, does training
improve performance on the Conservation of Meter
Task? Specifically, is there a significant dif-
ference in the mean scores on the Conservation of
Meter posttest between the trained and untrained
subjects, at various age levels (4, 5, 7, and 9
years) and for each sex (male, female)?

Table 19 summarizes the results of analysis of

variance (SPSS subprogram ANOVA; default option with

classic experimental approach and covariates processed

prior to factors) of COM posttest scores by treatment

group (trained and untrained), sex (male, female), and

age group (4, 5, 7, and 9 years). Analysis of covariance

provided adjustment for possibly confounding variables:

1. changes in developmental level (pre-to-post test










changes in CAK score),

2. intellectual maturity (Draw-A-Man standard scores),

3. pretest scores on the COM AA task.

Only one of the covariates, COM AA pretest score,

achieved significance (.03 level).


Table 19


Analysis of Variance



Source of Sum of
Variation Squares


Covariates

Change
DAM Stand.
COM AA Pre.

Main Effects

Group
Sex
Treatment.

2-way inter-
actions

Grp. Sex
Grp. Trtmnt.
Sex Trtmnt.

3-way inter-
actions

Grp Sex Tmt.

Residual


15.172

.103
3.403
12.792

13.600

12.002
.524
.054


11.366

.287
9.298
1.671


2.061

2.061

68.601


of COM Posttest Scores


Mean
df Square

3 5.057

1 .103
1 3.403
1 12.792

5 2.720

3 4.001
1 .524
1 .054


1.624

.096
3.099
1.671


2 1,030

2 1.030

27 2.541


110.800 44 2.518


F

1.990

.040
1.339
5.035

1.071

1.575
.206
.021


.639

.038
1.220
.658


.405

.405


Sig.
of F

.138

.999
.256
.031

.399

.218
.999
.999


.999

.999
.321
.999


.999

.999


Total










The results showed that none of the main effects--

age group, sex, or treatment--were significant. In

addition, none of the interaction effects were significant.

Two of the main effects--sex and treatment--were predictably

nonsignificant; the fact that age group, however, was

also not significant is at first glance a surprising

result, given the fact that level of CAK score has pre-

viously been shown to be a significant main effect, and

CAK and age are highly correlated. A possible explanation

for this result lies in the design of the analysis, which

called for dividing the subjects into their four age

groups. This procedure results in small cell sizes,

especially at the upper age levels where nonconservers

of meter were less numerous, and therefore makes

statistical significance of the age group factor less

likely. It should be recalled that in the analysis for

Question 3, where level of CAK score was one of the main

effects, subjects were divided into only two groups--

preoperational and concrete operational levels. It should

also be recalled that in the analysis for Questions la and

2, CAK score was shown to be a better predictor of COM

score than was age. With these qualifications, then, age

group in this analysis of COM posttest scores was not

significant. With regard to the interactions, theoretical

predictions would call for a significant age-by-treatment

interaction in favor of older, trained subjects. The




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