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This is to certify that the
thesis entitled
THE ANALYSIS OF THE OPERATIONS UNDERLYING
CONSERVATION OF NUMBER AND THEIR
ORDER OF ACQUISITION

presented by

Paulette Margaret Valliere
has been accepted towards fulfillment

of the requirements for

_M.._A.____degree in PSYCh0109Y

rig/@VMI

Major professor

Date June 9, 1978

0.7639

 

LIBRARY

hfx‘. "33F. P753

Lt: LlVCt‘Sl Cy

 

THE ANALYSIS OF THE OPERATIONS UNDERLYING
CONSERVATION OF NUMBER AND THEIR

ORDER OF ACQUISITION

BY

Paulette Margaret Valliere

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Psychology

1978

 

concep
serve

induce

 

fl \
\ 03“)

ABSTRACT

THE ANALYSIS OF THE OPERATIONS UNDERLYING

CONSERVATION OF NUMBER AND THEIR
ORDER OF ACQUISITION

BY

Paulette Margaret Valliere

Researchers have assumed that certain number
concepts are necessary before a child is able to con-
serve number and that training in these concepts can
induce number conservation.

The present study investigated eight number
concepts assumed by previous researchers to underlie
number conservation, testing for their presence in
conserving and non-conserving children. They were given
to 75 children along with the standard number conserva-
tion task. The data were analyzed by Guttman scalogram
and linear multiple regression analyses to provide
information on the developmental ordering of the tasks
and their predictability of number conservation.

Results showed that the obtained sequence signi-
ficantly correlated with the predicted outcome. The
reliability of the number conservation task itself was

found to be high. However, the eight tasks were poor

 

 

predic
regres

factor

 

 

Paulette Margaret Valliere

predictors of number conservation when added to the
regression equation, indicating the presence of other

factors not tested in the present study.

DEDICATION

To my parents, Ray and F10. Without
your faith and love, I could never
have accomplished so much. Keep
telling me that I can do it,

because you know I can.

ii

 

chair“
guidan
few ye

in the

 

skepti
Withou!
thesis
in dev
awaken
and Li

The li

aPPlau

ACKNOWLEDGMENTS

My thanks go to Dr. Ralph Levine, my thesis
chairperson. His comments, criticisms, and helpful
guidance were much appreciated during these past
few years. Dr. Terrence Allen was also very helpful
in the preparation of this thesis with his comments and
skepticism. Special thanks are due to Dr. Ellen Strommen.
Without her tremendous help and encouragement, this
thesis might never have been completed and my interests
in developmental psychology might never have been
awakened. Thanks also to my good friends Patti R. Hawn
and Liz Gordon, who helped me get through this hassle.
The little kids in my study also deserve a round of

applause for being so cooperative and normal.

iii

LIST OF TABLES . .

INTRODUCTION . . .

Piaget' s Theory

Number Development Studies

Training Studies
Hypotheses . .

METHOD . . . . .
Subjects . . .
Apparatus . -
Procedure . .

RESULTS . . . .

Analyses . . .
Sample Analyses

TABLE OF CONTENTS

Correlational Analysis
Developmental Sequence

Predictability of Tasks

DISCUSSION . . .

APPENDIX A: STIMULUS CARDS

APPENDIX B: DATA SHEET

REFERENCE NOTES . .

LIST OF REFERENCES .

FOR TASKS 4, 5, AND

iv

14
36

43
43
43
44
50
50
50
52
55
58
61
69
71
74

76

 

Table

 

l. T

LIST OF TABLES

Table Page

1. Total Scores, Means, and Variances of Age
Groups . . . . . . . . . . . . . . 51

2. Intercorrelations Between Independent,
Dependent, and Age Variables . . . . . . . 53

3. Percentage and Number of Subjects Passing
Each Task and Number of Errors Made on
Each Task . . . . . . . . . . . . . 57

4. Results of Multiple Regression Analysis
Predicing Number Conservation from Eight
variables 0 O O O O O O O O I O O O 59

 

{I

 

develc

child'

Much d

 

 

Cognitl

 

and re

operatj
childre
COnserv
Spite c
PerfOrm

which p

INTRODUCTION

Initially, investigators in the field of number
development in children were interested mainly in the
child's mastery of number concepts and their development.
Much of the early work looked at how children count,
perceive, and discriminate number. They also assessed
children's knowledge of number combinations which
provided them with normative charts showing which of
these number facts children knew. These earlier studies
(e.g., Brownwell, 1941) simply catalogued number facts
which children of various ages knew, rather than
question what children understood about addition and
subtraction. They were not designed to determine the
Operations which children believe to change number.

Piaget, in the formulation of his theory of
cognitive development, began the shift in methodology
and reasoning to the investigation of these underlying
operations in looking at number deve10pment in young
children. Piaget (1952, 1953, 1968) defined the term
conservation as the invariance of a characteristic in
spite of the fact that transformations have been
performed on the objects (or a collection of the objects)
which possess this characteristic. More specifically,

I

 

conse
of ob
distr
tion

of thi
shift
were

where

of the

origin
Ordina
researc

number

 

Oflmr]
behaVic
have at
the COr

“Umber

abilits
0f Seve
congel.V
traini:
COHCEPt

conservation of number is the concept where a collection
of objects "conserves" its number when the shape or
distribution of a collection is changed or the collec-
tion is separated into smaller subsets. With the onset
of this newer approach, the framework of the research
shifted from one where a child's responses to number
were thought to simply indicate number mastery to one
where responses were believed to reveal the functioning
of the child's logical, or cognitive, operators.

Another theoretical viewpoint considers the
origins of number concepts and their relation to the
ordinal, cardinal, and natural theories of number. This
research bases itself on the mathematical theories of
number concepts put forth by Peano, Russell, Frege, and
other logicians and attempts to link these to the
behavioral foundations of these concepts. Researchers
have attempted to determine the order of emergence of
the concepts of cardination, ordination, and natural
number in the young child.

The literature review suggests that children's
ability to conserve number is preceded by the attainment
of several concepts generally thought to underlie number
conservation. Researchers have variously found, through
training studies, that sometimes only one of these
concepts or, at other times, several of them, is

necessary for the child to become a conserver. The

 

preser

should

 

 

order‘

text On
preoper
2 to .7

present study examined number conservation and the
number concepts that have been assumed to be necessary

for its attainment. The purpose of the present study was

 

to determine whether or not certain mathematical concepts
are present in young children before or after conserva-

tion of number has been attained. If these concepts are

 

present before number conservation is achieved, it
should be possible to place them in a develOpmental
ordering in hOpes of establishing a sequence of logical
prerequisites to number conservation. If these concepts
can be placed in developmental sequence, it then might
be possible to analyze children's correct or incorrect
responses to the number conservation task in terms of
prerequisite concepts that have or have not been
obtained. These concepts might also be analyzed in
terms of their predictive power. That is, if a child
has mastered a certain number concept, is it possible
to predict his conservation ability from the knowledge

that we have about his numerical abilities?

Piaget's Theory

 

Ginsburg and Opper (1969), in their explanatory
text on Piaget's theory, indicate that the child in the
preoperational period, that is between approximately ages

2 to 7, supposedly is not able to conserve number until

towards the end of the stage when the child is about to

 

 

 

becom«
a chi.
lengtl
the rc
to use
rows a
the re
child
and de

constr

which
Conserx
nOncons

unfaili

 

Sistent
exact,
Ceding
nor do

matiOn .

equival
set equ
COrreSPI
conSErv‘

maintail

become concrete operational. This is to say that when

a child is presented two rows of equal number but unequal
length and spacing, the child under 7 will most often say
the rows are unequal in number. The child may be able

to use the concept of one-one correspondence when the
rows are lined up exactly but tends to concentrate on

the row length when they are uneven. In doing so, the
child fails to coordinate the two dimensions of length
and density simultaneously and therefore is unable to
construct sets equivalent in number.

Piaget described three distinct stages through
which children pass on their way to attaining total
conservation ability. Children who are consistent
nonconservers are labeled as being in Stage I. They
unfailingly make incorrect judgments and give incon-
sistent explanations for their answers. To be more
exact, the children neither predict conservation pre-
ceding the transformation of one of the stimulus arrays
nor do they maintain conservation after the transfor—
mation.

Children in Stage II can easily notice the
equivalence of two sets. They are able to construct a
set equal in number to another or can show one-one
correspondence between two sets. They also can predict
conservation prior to deformation of a set but fail to

maintain conservation after the deformation has taken

 

 

place
of th
or in'
child
times
focus'

referr

true c
conser
afterwI
With t}
rely or

Element

 

SOphis

n0nCOns
°nali
know th
unchang
fEel th

place. This indicates that the child is somewhat unsure
of the reasoning or criteria used to establish equality,
or inequality, of number. At times, as in Stage I, the
child centers on the length of the set, and, at other
times, centers on the density of the sets. This
focusing on only one of the relevant dimensions is

referred to as the principle of centration.

 

The final stage, Stage III, is that during which
true conservation emerges. Children are able to predict
conservation prior to deformation and maintain it
afterwards. They now have become concrete operational.
With the emergence of this stage, the children need not
rely on the perception of spatial proximity between the
elements of each set. They either count them or use a
saphisticated method of one-one correspondence. In the
nonconserving of number, younger children center only
on a limited amount of information available. They
know that the compressed row will be equal to the
unchanged row if spread out but perceptually, they
feel that the number of a set changes when its appear-
ance is altered.

Concrete operational children decenter their
attention and begin to attend to the relevant dimensions
of the stimuli. They recognize length and density as
two separate dimensions of the stimulus array and

associate the two. They then begin to understand the

 

fact
dens'
the y
rever
densi

equal

princ;
the rc
the rd
Opposel
empiri

act of

    
    

transf

 

identh
Since 1

addEd (

which 1
Child.E
begun l
estimat
Cognitj

C
onseqt

fact that as length of one line increases, the perceived
density of the other line will also increase. This is

the principle of reciprocity, which is a form of

 

reversibility. The increase in length counteracts the
density increase which results in a reversal and an
equal number.

Conserving children are also capable of using the
principle of negation. They realize that the changes to
the rows can be annulled or negated. By spreading out
the row again, the contraction is negated. This is
opposed to Stage II children who are capable of
empirical reversibility but can not focus on the reverse
act of rearrangement. They attend to states, not
transformations. They sometimes use an argument of
identity, reasoning that the numbers must be the same
since the same objects are involved; nothing has been

added or taken away.

Number Development Studies
There have been attempts to provide a means by
which the early development of number concepts and the
child's conception of number can be studied. Some have
begun by defining the basic concepts of operators and
estimators. Gelman (1972b) defines operators as the
cognitive processes by which children determine the

consequences of transforming a quantity in various ways.

 

Estim
by th

numer

opera
perce
invol'
the a]
amounl
the a:
consic

integ:

  
   
  

They a
number
Childr

they h

 

counti
Child.
to con
when e
hYPoth

a mult

Estimators are then defined as the cognitive processes
by which children determine some quantity, such as the
numerosity Of an array.

It is important tO differentiate between
operators and estimators. Estimators are linked to
perceptual input, whereas Operators are not, the former
involving a "lower" level Of processing. An outcome Of
the application Of a particular operation on a given
amount can be specified by an operator whether or not
the amount is actually present. Operators are also
considered "more cognitive" than estimators, providing
integrative connections between successive estimates.
They are also more central to mature conceptions Of
number. Conservation is considered an Operator task.
Children are said not to have a concept Of number until
they have attained conservation. In previous scalogram
analyses of number concepts, Operational tasks were
assigned a higher developmental ranking than estimation
tasks (Siegel, 1971b; WOhlwill, 1960).

In summarizing the literature Of estimators and
counting in young children, Gelman (1972b) felt that a
child's ability tO abstract number appears to be related
to counting abilities - a young child will count even
when estimating small numbers. In her own research, she
hypothesized that young children think about number in

a multidimensional way where there exists a hierarchy Of

 

 

cues
contr«
Exper
of an
lengtl
abstra
as the

itself

thesis
they a
basis
likely
not es
SUggeS

based .

—i—
_

0f the

later,

effeCt

 

integra
preSEnt
lationS
(1972b)

I'EIatic

 

cues such as length, density, and numerosity that
controls their attention to an array's numerosity.
Experiences that set children to attend to the number

Of an array and not tO the other features, such as
length and density, increase the expectation Of number
abstraction. Children will develOp better number skills
as the use Of number estimators decreases and number
itself becomes the dominant factor.

In an experiment designed to test this hypo-
thesis, Gelman (1972b) found that as children get Older,
they are more likely to estimate linear arrays on the
basis Of number. Younger children (4 to 7) are more
likely to estimate on the basis Of length, but they dO
not estimate on the basis Of density. This finding
suggests that the young child's concept Of number is
based on length and number first with the understanding
Of the relevance Of density to numerosity developing
later.

Operators mediate the ability to classify the
relevancy Of a set's transformation, anticipate the
effect Of a manipulation of numerical arrays, and
integrate perceptions from two or more successive
presentations Of a set through inferences about manipu-
lations that change or do not change number. Gelman
(1972b) considers Piaget's conservation task and its

relation to Operators. If a correct answer is given

 

(clas
under
the e
trans
dent

use 0.

alwayl

 

 

 

(classifed as a conserver), the child could be said to
understand Operators. However, a child who responds to
the equivalent number array (time 1) and then to the
transformed array (time 2) might be making two indepen-
dent responses which could therefore also indicate the
use Of estimators. The presence Of Operators cannot
always be inferred in such cases.

Gelman (1972b) devised an experiment using young
children that would test for the presence Of Operators
by requiring them to keep track Of quantitative rela-
tionships. This was done by the use Of the surprise
reactions as an index Of cognitive capacity, in which
the surprise reaction occurred when the subject's
perception failed to conform with his expectation. By
using a magic show, Gelman changed the number Of mice
in an array previously seen by the subject and desig-
nated as the winner (the larger Of two arrays). The
subject was instructed to point out the winner after
the two non-transformed arrays had been shown and then
hidden. The experimenter transformed the larger row
covertly and the reaction Of the subject to the
transformed "winner" row was noted when it was revealed.

The results Of the study provided evidence that
children as young as 3 years Of age have number invar-
iance Operators, and can use the Operators on small

numbers (1, 2, 3) which they can estimate. NO evidence

 

was 1
years
that
numbe
was 4
who t
spout
that

has I

which
& Tucl
the g]
in ove
Count:

Older

 

10

was found Of these Operators in younger children (2 1/2
years). With the use Of larger arrays, it was found
that the hypothesis that whether or not children use
number estimation predicts the use of number Operators
was supported. There was a tendency for those children
who treated number as invariant to count out loud
spontaneously. This was attributed to the possibility
that children will count an array when a transformation
has rendered the array discrepant to their expectation.

Similar findings showed that there are conditions
which enhance young children's tendency tO count (Gelman
& Tucker, 1975). They found that the younger the child,
the greater the tendency tO count overtly.~ The decrease
in overt counting with age may be attributed tO covert
counting being more prevalent in Older children or the
Older children may rely more on estimation.

Using the magic show method again, Gelman (1972a)
looked at the possibility Of young children (mean age,
3-6) revealing a capacity to correctly classify the
Operations that are irrelevant to number. This magic
paradigm showed the young children treating number as
invariant under displacement and correctly explaining
decreases or increases in number as the effects Of
subtraction or addition. This is in direct contradiction
Of the conclusions derived by the conservation paradigm.

It is assumed that the conservation task tests only a

 

 

 

chilc
The 1
evalL
trol,
vatic
of co

are C

of nu;
numbeJ
that c
numbe
addit'
Piage
neces
EXPer'
Seque

Cardin

criter
mental
Criter
mEntal
tasks,
is net

°°mpet

 

11

child's logical capacity to deal with number invariance.
The results Of this study, however, indicate that it
evaluates other skills as well, such as attention con-
trol, correct semantics, and estimation skills. Conser-
vation ability is therefore a more SOphisticated level

Of cognitive development in which many separate abilities
are coordinated.

Another viewpoint concerning the child's concept
Of number is that Of cardinal, ordinal, and natural
numbers and their developmental sequence. Piaget holds
that conservation Of number is the criterion Of natural
number competence, whereas mathematicians claim that
addition and subtraction Of integers is the criterion.
Piaget also claims that ordination and cardination are
necessary precursors to natural number competence.
Experimenters have therefore attempted to establish a
sequence among the emergences Of natural, ordinal, and
cardinal numbers.

Brainerd (1973b) has found indications that (l)
criterial performance on ordination tasks is the develOp-
mental antecedent Of natural number competence, (2)
criterial performance on ordination tasks is the develOp-
mental antecedent Of criterial performance on cardination
tasks, and (3) criterial performance on cardination tasks
is not the developmental antecedent Of natural number

competence. This therefore gives us the sequence

 

 

 

ordi
SUPP
conc
unde.
as tl
any 1
1975)
from
cardi
prove:

ordine

report
Fraser
concep'
0rdina1
result
in ordi
aVerage

also fO

 

suPGIiO
that th
Cardina
gr°uP's

tion th

 

12

ordination + natural number + cardination. This is
supportive Of the ordinal theory which states that the
concept Of natural number originates primarily in a prior
understanding Of ordination, where ordination is defined
as the quantification of series Of three terms united by
any transitive-asymmetric relation (Brainerd & Fraser,
1975). Piagetian predictions do not receive any support
from this vieWpOint. Piaget's (1952, 1953) claim Of
cardination preceding natural number competence was
proven to be unsupported by the results. Only in the
ordination + natural number was it confirmed.

The finding Of this specific sequence has been
reported by Brainerd (1973c, Note 1) and Brainerd and
Fraser (1975). It has also been reported that these
concepts can be trained in children (Brainerd, 1973c).
Ordination and cardination were shown to improve as a
result Of feedback training, but the average improvement
in ordination performance was much greater than the
average improvement Of cardination principles. It was
also found that children trained in ordination were
superior to those not trained in number performance, but
that there was no significant difference between the
cardination-trained group and the untrained control
group's number performance. This confirms the sugges-

tion that natural number competence is preceded by a

13

prior understanding Of ordination but not by an under-
standing Of cardination.

Brainerd and Fraser (1975), in comparing Piaget's
number conservation as natural number criterion versus
the mathematicians' addition and subtraction Of integers
criterion, showed number conservation tO be an adequate
estimate Of natural number competence. This substitution
for integer manipulation makes no significant difference.
Brainerd and Fraser did find that children can acquire
facility with the integers without prior knowledge of
number conservation which is consistent with Piaget.

Schaeffer, Eggleston, and Scott (1974) consid-
ered number development in terms Of skill acquisition.
They postulated a model Of number develOpment consisting
Of six skills: (1) acquisition Of more x's (where x is
a number Of Objects), (2) judgments Of relative
numerosity, (3) pattern recognition Of small numbers,

(4) the counting procedure, (5) the cardinality rule,
and, (6) one-one correspondences. These are listed in
the order in which they were found to emerge. An
emphasis was placed on counting skills. They hypothe—
sized that, with the learning Of the last two skills,
the cardinality rule and one-one correspondences, which
occur simultaneously, children then learn number conser-
vation. This model assumes that number development is

determined more by the application Of number skills to

l4

Object arrays than by thoughts about numbers in the
absence Of arrays or by "spontaneous cognitive reorgan-
izations, i.e., equilibration." It is interesting to
note that Schaeffer et a1. (1974) used counting as an
indication Of the presence Of the cardinality rule, which
states that the last number named during counting denotes
the number Of Objects in the array. Dodwell (1960) found
that even though children counted two arrays out loud
(equal number, unequal length), they then judged that
arrays to be of differing number. He concluded that
counting is not a guarantee of knowledge about

cardinality.

Training Studies

 

In view Of Piaget's developmental stage hypo-
thesis which states that children should progress to
the next stage or level only after they have acquired
all concepts necessary and in the prescribed order,
experimenters went about trying to disprove Piaget.
They began by theorizing that these concepts could
indeed be induced in children who did not possess them
by intensive training in the cunncept or concepts deemed
necessary to induce number conservation in those children
who were still at the preoperational stage Of cognitive

development by training in the number conservation task

15

itself, or by training Of concepts thought by the
experimenter to be necessary for conservation attainment.

As Brainerd (1978) states: "A subject can be
taught a concept belonging tO some given stage only if
he is already at that stage or in a transition phase
between that stage and the preceding stage. This is
because learning a Piagetian concept is supposed to
consist not Of learning the concept per se but learning
to generalize the mental structures appropriate to a
given stage." He states that, according to Piaget,
learning occurs only when two conditions are met. First,
the training treatment must incorporate laws Of spon-
taneous develOpment. Second, the to-be-trained subjects
must already possess the to-be-learned concept to some
measurable extent. Brainerd feels that, at most,
concept learning experiments consist Of teaching children
to generalize spontaneously acquired mental structures
rather than to acquire concepts per se.

It also follows that, given an effective train-
ing procedure, Stage II children should learn more than
Stage I children (Brainerd, 1977a). Concept learning
is supposed to be a process where existing structures
are generalized to new content. Stage I children are
not capable Of learning because they lack the grouping
structures necessary for conservation. Stage II children

possess them so therefore can be trained. Stage II is

16

an unstable state during which learning experiences are
maximally effective. The children are in a state called
"cognitive disequilibrium." This prOposal was found to
be consistent with the findings Of Brainerd (1972). He
discovered that not only was trainability dependent
on the present stage Of the child but that it was
independent Of the subjects' age.

It is interesting to note that Mehler and Bever
(1967) found children as young as 2-4 showed a type of
number conservation. When showed two unequal rows Of
M&M's, 81% Of the children under 2-8 correctly chose the
bigger row. The appearance Of conservation was found
to be less frequent in the Older children with a minimum
in the group between 3-8 and 3-11. Conservation began
tO appear again at 4-6.

According to Mehler and Bever, the fact that
very young children showed this conserving ability is
an indication that they do have capacities which depend
on the logical structure Of the cognitive Operations.
Eventually they then develop an explicit understanding
Of these Operations. They theorized that this inability
to conserve is due to an overdependence on perceptual
strategies. These perceptual strategies can be overcome
given sufficient motivation. Eventually they develop

more sophisticated integration of logical Operations Of

17

their perceptual strategies, allowing them to count
members Of an array.

Piaget (1968) believes that the data Of Mehler
and Bever (1967) were inadequately analyzed with regards
tO the child's perception Of length as a sufficient
suppressor Of a correct notion Of number. When children
evaluate number with regards to length, this process is
actually based on the concept Of ordinal quantification
which is more complex than is shown in a number experi-
ment. Piaget feels that Mehler and Bever's results
actually show a simpler more primitive type Of ordinal
evaluation based on tOpOlogical structures (proximity,
separation, etc.) that the young child possesses before
considering length. This topological evaluation is one
based on the notions Of "crowding" or "heaping." The
child gets the impression that the short row is more
crowded or bunched and therefore contains more than the
longer one. In this way, children as young as 2 1/2
years gave correct answers almost 100% Of the time in
Mehler and Bever's (1967) study without really under-
standing the numerical relations with which they dealt.

Mehler and Bever (1968) redid their original
study, this time using equal rows and found that there
was no tendency for children at any age to choose the
denser row and again owed this result tO correct quanti-

tative decisions on the part Of the children without

18

referring tO relative densities Of the rows. They felt
that 2-year-Olds have certain cognitive capacities that
are limited, especially by memory and attention, and in
overcoming these limitations, the children form intuitive,
perceptual generalizations that extend their capacities
beyond behavioral limits. When these generalizations
fail, they are integrated into a system, formed by
children through experience, that includes both the
perceptual generalizations and their basic logical
capacities. In this way, the children are able to deal
with external reality, i.e., the array Of counters in
this case.

There have been several attempts at replicating
Mehler and Bever (1967) with differing results.
Rothenberg and Courtney (1968), in repeating Mehler and
Bever's technique, found no indication Of the U-shaped
trend they reported but did find the high percentages
Of conserving responses among 2 1/2 to 4 1/2 year Olds
consistently. When Rothenberg and Courtney compared
these results with those found in the same subjects by
another more stringent method (Rothenberg, 1969), it
was found that this method classified fewer conservers
and a more consistent age increase was found. Rothenberg
and Courtney attributed these differences to methodolo-
gical flaws, especially language difficulties, in the

Mehler and Bever study. Higgins-Trenk and LOOft (1971)

19

found virtually the same results as Rothenberg and
Courtney (1968) in their replications and similarly
concluded that the problems were due tO limited language
ability.

Willoughby and Trachy (1971) found similar
results. They concluded that their failure to replicate
was due to only a partial understanding Of the term
"more" in their subjects. They also felt that the study
was not actually measuring number conservation but some
partial concept Of the stimuli being used. It was shown
that the type Of stimuli used, either M&M's or clay
pellets, affected the subjects' responses.

Calhoun (1971), on the other hand, basically
replicated Mehler and Bever (1967) successfully.
However, positive results could not be Obtained with
the youngest group (2-4 tO 2-7) due to problems in
their understanding Of the procedural instructions.

They were found tO conserve however when the method
employed did not require verbal encoding by the youngest
subjects. Calhoun did find a U-shaped age trend similar
tO the one found by Mehler and Bever and generally con-
cluded that younger children could indeed conserve
number. -

Pufall and Shaw (1972) compared the positions Of
Mehler and Bever (1967) and Piaget (1952) on the struc-

tures Of children's thought and their perception Of

20

number. Children (ages 3 to 6) were presented with six
number problems in which length and density varied. In
analyzing the performance Of the children, it was found
that only 4% made correct judgments on all problems. NO
child judged all problems with a relative density hypo-
thesis, that is, judging that the rows are numerically
equal because densities are equal or that the more dense
row is numerically larger than the less dense row. A
curvilinear relationship between number judgment and

age on number equivalence tasks was found. This

reasult questions Mehler and Bever's conclusion that
young children's performances reflect cognitive intui-
tions analogous to Operational structures in Older
children.

Early studies in the area Of concept training
were thought to be sufficient given what was known about
the trainability Of number conservation at that time
(Flavell, 1963). In a later review Of the existing
training studies, Brainerd and Allen (1971) noticed
that in all the successful studies, there was a common
feature. All exposed the subjects to situations which
specified the Object bound form Of Operational reversi-
bility, namely reversibility by inversion-negation.

This fact is noteworthy in that Piaget attributes the

onset Of first-order thought and conservations to the

21

acquisition Of inversion-negation and reciprocity
which are the two modes Of cognitive reversibility.
Wohlwill (1959) was one Of the first researchers
who attempted to induce number conservation in noncon-
servers. Using subjects with a mean age Of 5-8, he
trained them in addition and subtraction facts with only
three brief training sessions. Wohlwill theorized that
the comprehension Of addition and subtraction in the
young child might be sufficient tO induce conservation.
However, the training proved to be unsuccessful and its
lack Of success was attributed to the brevity of the
training sessions. It was concluded that even though
addition and subtraction comprehension was not sufficient
for number conservation, it did appear to be necessary.
Wohlwill and Lowe (1962) considered reinforce-
ment, differentiation, and inference as possible alterna-
tive interpretations Of the acquisition Of number
conservation. Three hypotheses were formulated and
independent groups were trained in accordance with each
hypothesis (mean age, 5-10) along with a control group.

The reinforcement hypothesis proposed that, as children

 

learn to correctly count and have experience with
different types Of arrays, they gradually learn that
perceptual transformations Of the arrays do not change

the number Of items in the array. SO, systematic

22

reinforced practice in counting rows Of Objects prior to
and after transformation should induce conservation.

The second differentiation hypothesis interprets
inability to conserve in young children as a response tO
an irrelevant but highly visible cue, length, which
tends to correlate with number. That is, when a row is
longer, it usually indicates an increase Of number.
Repeated experience designed to neutralize the cue Of
length and weaken any association between length and
number should therefore be expected to facilitate conser-
vation acquisition.

The inference hypothesis is based partly on

 

Piaget's own analysis on the role Of learning in the

develOpment Of logical operations. He maintained that

experience can be effective only in that it builds upon
previously developed structure Of thought. In the case Of
conservation, children might be led tO infer it as a result
Of a change involving neither addition nor subtraction

per se but rather the cumulative effects Of watching the
adding or subtracting Of an element to a collection.

The results revealed that the reinforced practice
trials, which were expected to produce the best results,
in fact showed poor results. This showed the ineffec-
tiveness Of continued reinforced practice in eliciting
conservation responses. The greatest amount Of improve-

ment tOOk place in the Addition and Subtraction group,

23

This result is consistent with the possible role Of a
process Of inference, that is, conservation as the
end-product Of changes involving neither addition nor
subtraction. NO learning was found in the dissociation
group suggesting that the act of counting interfered
with the children's attention to the cue Of length.

The fact that the Addition and Subtraction group yielded
the most improvement in conservation appears to suggest
that a process Of inference may be Operative in the
development Of number conservation.

Reversibility was recognized to be a factor in
conservation attainment. Piaget (1952) stressed the
necessity Of reversibility on number conservation
acquisition. With this in mind, Wallach and Sprott
(1964) attempted tO induce conservation Of number by
showing children (mean age, 6-11) the reversibility Of
rearrangements, which they regarded as implying changes
in number prior tO conservation.

Dolls and beds were used with nonconserving
subjects (classified by a pretest) in situations where
a row Of dolls and a row Of beds were shown to each
child. The eight situations showed equal numbers Of
beds and dolls with the rows unequal or unequal numbers
of dolls and beds with equal rows. The child was asked
tO predict if there were enough beds for the dolls.

This procedure was repeated until the child correctly

 

 

- .._ _ I...“- W ..._V..~._. .. _ ,

 

24

responded to each Of the eight situations. The child
was then given two posttests, one immediately after
training and the other 14 to 23 days later. The results
showed the training procedure to be effective in
inducing number conservation in nonconservers. It was
suggested that the training in reversibility contri-
buted tO conservation by providing subjects with the
information that rearrangements are reversible, and
by inducing subjects tO think Of reversal's possibility.
Cognitive conflict was also thought to be an
essential condition for the development of conservation
in children. It supposedly induces a reorganization Of
children's intellectual actions, which proceeds along
the lines postulated by Piaget's equilibration theory.
This reorganization then leads to the conservation
strategy. Gruen (1965) compared the effectiveness Of
two types Of training procedures: cognitive conflict
and learning through reinforced practice. The former
induces internal cognitive conflict which brings about
number conservation. The latter provides external
feedback through counting with the reinforcement being
a "knowledge Of results." The intention Of this study
was to isolate some Of the variables that play a role
in the acquisition of conservation. However, the
results showed that neither training methods was

successful in inducing number conservation in over half

25

Of the subjects. There were a number Of subjects who
did acquire conservation and when compared tO some Of
the earlier studies, this was impressive. There was a
better performance by the cognitive conflict group when
compared to the reinforcement group, but this difference
did not reach statistical significance.

A similar study was done by Winer (1968) where
he induced addition and subtraction sets and investigated
whether these sets could determine responses to conflict
trials and to tests Of conservation. It was hypothe-
sized that practice in responding to either additions
and subtractions or to changes in length would induce
in subjects a set tO resPond to either Of these
manipulations. This set would be brought forward on
conflict trials where changes in length Opposed additions
and subtractions. Those children (mean age, 5-11) who
received addition-subtraction training would continue
tO respond to them in conflict trials while children who
received practice in responding to length would continue
to respond to changes in length. Winer also predicted
conservation inducement through addition-subtraction
practice.

The results showed that continued response to
simple additions and subtractions can induce subjects

to respond to these manipulations when they are made tO

26

contrast with changes in the length of a line and subse-
quently to acquire conservation. Although conflict
trials can probably contribute somewhat to the acquisi-
tion Of conservation, the effect is minimal and accom-
plished without the presence Of overt conflict.
Mermelstein and Meyer (1969) compared four types
Of training conditions: cognitive conflict, multiple
classification, verbal rule instruction, and language
activation. In the first condition, subjects were
shown arrays in one-one correspondence several times for
5 sessions. In the final three additional sessions,
one-one correspondence was established, and then, one
row was collapsed into a pile. The subjects were then
questioned about the numerical equality Of the two rows.
Multiple classification training established the
possibilities Of the multiple labeling, classification,
and relations Of the poker chips used. Each child was
then trained in reversibility. They were shown two
rows Of chips in one-one correspondence and questioned
about the row equality. One row was then collapsed
into a pile and the subjects were again asked if there
was the same number in each row. The verbal rule
instruction group received the same reversibility train-
ing as this but one row was immediatley collapsed and
the subject was questioned as to the equality Of the

two rows. The difference between the two methods

27

stemmed from the fact that the experimenter said he was
moving the row as he did so and commented that the
number was not being changed, only its appearance. The
language activation training was a technique based on
the standard Piagetian conservation Of discontinuous
quantity task. Two equal boxes were filled with equal
amounts Of poker chips in a one-one correspondence
technique. One box's contents were poured into a third
taller box. The subject was asked to verify their num-
ber equality.

Results Of the posttesting with various conser-
vation tasks showed that none Of the four training
conditions were successful. All had been previously
found to have been successful in number conservation
training. It was felt that a point should be made that
specific training differs from cumulative life exper-
iences which Piaget suggested are partly the determining
factors for acquiring number conservation.

Cognitive dissonance has been thought to be
quite similar to Piaget's idea Of equilibration and has
been used to induce conservation in transitional con-
servers and nonconservers (Murray, Ames, & Botvin, 1977).
It was felt that the need for establishing cognitive
consistency was similar to the process that Piaget feels
children undergo in cognitive development. Murray et a1.

(1977) found that conservation could in fact be induced

28

permanently in nonconservers and transitional conservers
by the use Of a technique based on cognitive dissonance
theory. They also found that conservers were not at

all affected by the experimental treatment received.
This study is quite important due to the fact that it
ignored the training Of concepts thought tO be necessary
for conservation and concentrated on the conflict that
is thought to gO on during cognitive development,

namely disequilibrium.

Reversibility was again used as a training
procedure. Wallach, Wall, and Anderson (1967) hypothe-
sized that number conservation could be brought about by
reversibility training, basically a replication Of an
earlier study (Wallach & Sprott, 1964) and also be
induced by training in addition and subtraction, some-
thing that.WOhlwill (1959) had suggested. It was thought
that number conservation could be brought about by
reversibility training without addition-subtraction
experience. This was found to be true which suggests
that number conservation is not affected by training
in addition and subtraction. Training in addition and
subtraction without reversibility experience was not
found to be successful in inducing number conservation.
The main effect of the reversibility training procedure
may have been to lead the subjects (mean age, 6-11) to

stOp using a misleading perceptual cue. They felt that

29

with this interpretation, reversibility training success
cannot be regarded as providing evidence for the role Of
the recognition of reversibility in conservation. This
procedure, while appearing to induce conservation,
probably did so by leading the children to stOp using
misleading cues. However, reversibility was still
looked upon as necessary for conservation.

Gelman (1969) hypothesized that young children
fail to conserve because Of inattention to relative
quantitative relationships and attention to irrelevant
features in classical conservation tests. Children
(mean age, 5-4) were assigned tO learning set training
groups, with or without feedback. It was shown that,
given appropriate training, one can elicit conservation
in children who initially fail to conserve. Appropriate
training seemed to involve twO factors: (1) interaction
with many different instances Of quantitative equalities
and differences, and (2) feedback, telling the subject
what is and what is not relevant to the definition Of
quantity.

Whereas most Of the previous studies focused
on the training of one or two concepts tO induce number
conservation, Rothenberg and Orost (1969) trained their
subjects in a sequence Of concepts. They presented
their subjects (mean age, 5-5) with a sequence Of

concepts derived from analysis Of the components

 

 

 

30

assumed tO underlie number conservation attainment.

It was not stated how these concepts were chosen or
analyzed. They were: (1) rote counting, (2) counting
attached to Objects, (3) "same" number, (4) the "same"
versus "more" distinction in terms Of number, (5) addi-
tion and subtraction representing a change in number,
(6) one-one correspondence, (7) reversibility, and

(8) the distinction Of "more" referring to the actual
number Of Objects versus "longer" referring tO their
arrangement iJi space. Each child was given three
training sessions in all concepts. The above sequence
Of concepts was used, implying a rough estimation Of
their develOpmental occurence. .

Results Of this study implied that number
conservation could be induced (or taught) to noncon-
serving children. The sequence Of concepts provided a
reasonable working effort for teaching number conser-
vation. It was not proved that this sequence was the
only sequence acceptable to the hypothesis. There was
no way Of noting if one or several Of the concepts was
excessive. Perhaps a combination Of only a few would
have been as successful as the use Of all eight.

Murray (1972) theorized the number conservation
could be taught to nonconservers through social inter-
action. Basing his hypothesis on Piaget's theory that

the occurence Of repeated communication conflicts

31

between children was a necessary condition for the
movement from preoperational thinking to concrete
Operations, Murray placed a nonconserver along with

two conservers (mean age, 6-7). Each group had tO come
tO an agreement on answers to various conservation tasks.
Afterwards, each nonconserver was posttested and a
significant number were found to have attained number
conservation. It was most successful with Stage II
children. The results indicate that social conflict or
interaction is an important mediator Of cognitive
growth.

These findings were replicated by Botvin and
Murray (1975) who also found that modeling was also
effective in eliciting conservation responses. Those
nonconservers who watched the social interaction groups
give their correct conserving responses were found to
conserve in a posttest and were able to justify their
answers with reasons other than those given by the
group they Observed. This indicates that modeling is
as effective as social conflict in eliciting conserva-
tion responses in nonconservers.

The effects Of training children (ages 6 to 8)
in the conservation task itself were examined by
Figurelli and Keller (1972). Social class was also
investigated as a variable. It was found that this type

Of training (repeated demonstrations with feedback)

32

was successful, but differences across socioeconomic
class were not significant.

Mpiangu and Gentile (1975) trained nonconservers,
4- to 6-years-Old in rote and rationale counting (number
recitation), number recognition (finding a missing
number in a given sequence), relations (before, just
before, after, just after, between), and number synthe-
sis and analysis (giving correct answers and justifica-
tions to simple mathematics problems). These concepts
include some Of those singled out by Rothenberg and
Orost (1969), but also include different mathematical
concepts. The results showed dramatic increases on
arithmetic performance and a significant effect on
number conservation attainment. This result, while
showing that training did facilitate the acquisition Of
number conservation, was interpreted by the authors to
mean that conservation was not a crucial concept for
mathematical understanding. Number conservation and
mathematics were considered as concepts that develop
simultaneously. ‘

Emphasis was placed on discrimination by Halford
and Fullerton (1970). TO conserve number in the classi-
cal type Of task, children must discover that number is
determined by correspondence, potential or actual, to
the standard array and not by length or spacing, unless

these two compensate each other. They must discover

33

that if number is constant, an increase or decrease in
length is compensated by the Opposite change in spacing,
so length and spacing cannot be ignored as cues.

With this in mind, Halford and Fullerton trained
children (mean age, 6-3) in a discrimination task. A
row Of beds was shown along with several differently
arranged rows Of dolls. The child was asked which row
Of dolls would fit the beds. Five individual training
sessions were held. The principle Of one-one corres-
pondence was also shown as each child was shown the row
Of beds with a doll placed in each one. Results showed
that two-thirds Of the children had acquired a stable
type Of number conservation through this manner of
training. 4

A similar line Of research in the area Of
perceptual cues Of number and length in number judgments
has also been investigated by several researchers
(LaPOinte & O'Donnell, 1974; Lawson, Baron, & Siegel,
1974; Smither, Smiley, & Rees, 1974). Children 2- to
S-years-Old were trained in the concept Of a "bunch" by
LaPOinte and O'Donnell (1974). They were shown a small
group Of houses and told this was a bunch and then were
asked to identify similar "bunches" Of houses, referring
to the concepts Of "more" and "same.” Each subject was

then tested on the standard number conservation task and

34

on a variety Of conservation-like tasks designed tO show
changes in density, length, and number.

Results showed that children between the ages
Of 2 and 5 do not employ a consistent hypothesis in
judging number. It appears that they attend to
differences between the rows and give a response based
on the most salient cue. The child understands number
in terms Of perceived correspondence and differences in
arrays. The length cue appears to be the most salient.

Counting also was found to be related to con-
serving number. Four-year-Old children who counted and
conserved counted very carefully before and after
transformation. It was suggested that this helps
children to direct their attention to numerical corres-
pondence between the two rows. In this way, a conserving
response would be facilitated. It was also prOposed
that counting can increase the number Of conservation
responses without implying the lasting equivalence
required by Piaget, indicating Operational thought.

Lawson et al. (1974) on the other hand, found
that number, given small arrays, was the salient cue
rather than length as LaPOinte and O'Donnell (1974) and
Pufall and Shaw (1972) had found. Pufall and Shaw used
larger arrays that perhaps might have been beyond the
children's estimating abilities, making number less

salient. Thus the rule might be: if the number arrays

35

are beyond estimating, length is the cue and when the
array is within estimating range, number is the salient
cue.

Similar results were Obtained by Smither et a1.
(1974). Children, ages 3, 4, and 6, were tested on a
variety Of number conservation-like tasks on which
number, length, and density were varied. Results
showed no support for three stages Of development of
number judgments. It was suggested that accurate number
judgments develop in a continuous fashion and that the
use Of length and density cues depends on age, salience
Of the dimensions and also the magnitude Of number and
number differences. When the length differences and
number were small, and the density and number difference
large, the children tended to make accurate number
judgments. However, when the numbers are large and the
number difference small, the judgments were based on
length. This seems to agree with the hypothesis and
rule put forth by Lawson et al. (1974).

There have been several attempts at trying to
establish the developmental sequence Of number concepts.
Wohlwill (1960) had one of the first studies attempting
to establish a sequential pattern of these concepts
with the use Of scalogram analysis. After training
children to match numbers to stimulus arrays Of dots,

they were tested on 7 tasks designed to assess certain

36

number concepts. These concepts were found to occur
in this order: (1) the ability to match two arrays on
the basis Of numerosity, (2) the ability tO abstract
number from irrelevant cues, (3) memory for numbers,
(4) understanding addition and subtraction changes,
(5) matching larger arrays Of Objects by number, (6)
number conservation attainment, and, (7) ordinal
correspondence.

Siegel (1971a, 1971b) administered 7 tasks tO
3- tO 5-year-Old children. They were tested on
continuous and discrete magnitude discrimination, equiva-
lence, conservation, ordination, seriation, and
addition. She found that equivalence and magnitude
discriminations can both be understood as concepts at
approximately the same time. Number conservation
develops at a slightly later time with ordination,
seriation, and addition developing later. Addition
concepts came quite a bit later in Siegel's study than
in Wohlwill's (1960) but the difference appears to be

in the relative difficulties Of the task type each used.

Hypotheses
From the review Of the literature, eight concepts.
or tasks, have been chosen as best representing the
viewpoints Of the current research as to what is

necessary to induce number conservation in nonconserving

37

children. All eight concepts were found either to be
present in conservers or were able tO induce conservation
through training in them. However, not all were examined
at the same time in any given subject in any Of the
reviewed studies. The present study differed from the
others in that it combined those concepts which were
individually thought tO be necessary precursors tO

number conservation, and tested for their presence in
both conserving and nonconserving children.

The questions raised in the present study were
whether these concepts are all present in number con-
servers and whether or not all, or only several, are
also present in nonconservers. The present study also
examined the possibility Of establishing the presence
Of a developmental sequence Of these concepts. In this
way, it was possible to judge whether the concept was a
logical prerequisite for number conservation as thought
by previous researchers.

With the selection of these eight concepts, it
was then possible to make predictions as to the possible
developmental sequence Of emergence prior to number
conservation. They are listed in order Of predicted
outcome, from earliest to the last emerging before
number conservation.

1. Rote counting: Rote counting has been found by

many to be present in conservers and in nonconservers

38

alike (Piaget, 1952; Pufall, Shaw, & Syrdal-Lasky, 1973;
Rothenberg & Orost, 1969; Schaeffer et al., 1974;
Wohlwill & Lowe, 1962). Rote counting, or reciting the
number names, is in nO way an indicator Of true number
understanding. It simply means that children are able
to recite the number names in their proper order. Even
so, counting has been hypothesized to be necessary for
number conservation (Halford & Fullerton, 1970;

LaPOinte & O'Donnell, 1974; Mpiangu & Gentile, 1975).

2. Object counting: The next stage should be the
ability Of children to attach these memorized number
names tO Objects and count an array (Rothenberg & Orost:
1969; Schaeffer et al., 1974). This ability has been
found experimentally to come after rote counting
(Pufall et al., 1973) and be necessary for number
conservation (Halford & Fullerton, 1970). It was also
found to precede one-one correspondence (Pufall et al.,
1973).

3. Numberjudgments Of two equal groups: The

 

ability Of children to count groups and make judgments
on their equality was hypothesized tO occur next. After
the children are able to count and connect the counting
with Object arrays, they should next be able to make
numerosity judgments. Schaeffer et a1. (1974) theorized
that in number skill development, equality judgments

will emerge before the concept Of inequality. This also

39

has been thought tO occur in other training studies
(LaPOinte & O'Donnell, 1974; Rothenberg & Orost, 1969).

4. Number judgments Of two unequal groups:
Rothenberg and Orost (1969) thought that the training in
this type Of number concept would help to bring out
number conservation in young children. Once children
can count arrays and make judgments on the numerical
equality, they should be able tO make judgments on
inequality. It was theorized that the knowledge that
x + l is greater than x is developmentally later than
equality recognition but occurring before one-one
correspondence and number conservation (Schaeffer et al.,
1974).

5. Addition/subtraction changes in a row: Many
researchers have hypothesized as tO the necessity Of
addition and subtraction skills in the acquisition Of
number conservation. It has been found to be necessary
but not sufficient for conservation (WOhlwill, 1959),
unsuccessful in bringing it about through training
(Wallach et a1, 1967), but more Often its training has
been successful in inducing number conservation (Gruen,
1965; Mpiangu & Gentile, 1975; Rothenberg & Orost, 1969;
Winer, 1965; Wohlwill 8 Lowe, 1962). Gruen (1965) also
found his experimental design showed addition and

subtraction concepts to logically and developmentally

40

precede number conservation as also found by Rothenberg
and Orost (1969).

6. One-one correspondence: The concept Of one-one
correspondence is a crucial one in children's under-
standing Of number concepts. The understanding that if
the density and length Of two rows are both equal, then
the number Of the two rows is equal is a necessary
concept in number conservation (Inhelder et a1, 1974;
Lawson et al., 1974). It has been found to be one Of
the later concepts to develop in children (Wohlwill,
1960) and theorized tO be developmentally occurring just
before conservation (Schaeffer et al., 1974). Training
in one-one correspondence has both been found to be
successful (Halford & Fullerton, 1970; LaPOinte &
O'Donnell, 1974; Rothenberg & Orost, 1969) and unsuccess-
ful (Mermelstein & Meyer, 1969).

7. Reversibility: Reversibility is a concept that
Piaget (1952) feels is a necessary part in the logical
Operations necessary to become concrete Operational and
a number conserver. Children must be able to understand
that an Operation performed on a numerical array can be
reversed or undone to conserve number. Training in
reversibility problems to induce number conservation has
been attempted by Mermelstein and Meyer (1969), and
Wallach et a1. (1967) unsuccessfully, and by Wallach

and Sprott (1964) and Rothenberg and Orost (1969)

41

successfully. Rothenberg and Orost (1969) placed
reversibility developmentally and logically after
one-one correspondence as Piaget (1952) and Inhelder et
a1. (1974) do.

8. Number versus length Of arrays (Spatial arrange-
mgggl: Overcoming perceptual cues to focus on the
numerosity Of an array is one Of the last and possibly
one Of the more important concepts that children must
attain. Children must learn to respond according tO
numerosity rather than by length or density. Pufall
et al. (1973) showed that judging number using relative
length increased with age. Wallach et a1. (1967)
identified reversibility training as the factor which
permdtted their subjects to overcome misleading cues,
recognizing reversibility as well as not using misleading
perceptual cues as necessary for conservation. These
misleading perceptual cues (length, density) have been
found to interfere with correct number judgments at all
ages and also have been shown to be developmentally
occurring very late, before the acquisition Of number
conservation (LaPOinte & O'Donnell, 1974; Lawson et al.,
1974; Rothenberg & Orost, 1969; Smither et al., 1974;
Wohlwill & Lowe, 1962).

TO summarize, the predicted developmental

ordering of the eight tasks and number conservation is:

42

Rote counting

Object counting

Number judgments Of two equal groups
Number judgments Of two unequal groups
Addition/subtraction changes in a row
One-one correspondence

Reversibility

Number versus length Of arrays (spatial arrange-
ment

Number conservation

METHOD

Subjects

 

The subjects were 75 kindergarten students who
were enrolled at the Williamston Memorial School in
Williamston, Michigan, a predominantly lower-middle-
class area. There were 42 males and 33 females, ranging
in ages from 4-11 tO 6-4, with a mean age of 5-6.

The testing was conducted very early in the
school year, approximately four weeks after it began.
Very little instruction in number concepts had been
given to the children by this time, mainly counting
numbers had been taught in each class. Each Of the
eight number concept tasks was administered to every
child as was the standard number conservation assessment

task.

Apparatus
Red and black wooden checkers and 13 white index
cards with blue dots were used as stimuli. Each checker
measured approximately 3.2 cm. x 0.8 cm. The index
cards (12.8 cm. x 20.4 cm.) each had a varying number Of
dots (1.5 cm. in diameter) drawn on them (see Appendix

A).

43

44

Procedure

 

Subjects were met individually at their class-
room and escorted tO the testing area by the experimenter
to be individually tested. During this time, an effort
was made to establish rapport to make the child feel
more at ease with the experimenter. The experimenter
also answered any questions the children had about what
they were expected to do during the experiment. The
child was then seated directly Opposite the experimenter
at a small table in a study area Of the school.

Each child was first given the number conserva-
tion assessment task followed by rote counting. The
following 7 tasks were given in a random order tO each
child to control for learning. Randomization was
achieved through the use Of a calculator with a random
number generator. This series Of tasks were then
followed by a readministration of the number conserva-
tion assessment task.

Criterion for a subject to pass a given task
was two correct responses in three trials, with a score
Of 1 for a pass, and 0 for a fail. This was the scoring
procedure for all tasks except for Task 7, where 3 Of
4 correct responses served as criterion. NO feedback as
to correct or incorrect responses was given to subjects

and all responses given were recorded (see Appendix B).

45

Task 1: Number conservation assessment

The subject was shown two equal rows Of 5
checkers each in one-one correspondence. The eXperi-
menter then asked, "Does your (my) row have the same
number Of checkers as my (your) row?" After the subject
responded, the experimenter then transformed one Of the
rows, either by changing the density or length and
repeated the question previously asked. This task was
given a maximum Of three times with the row tO be pointed
out first randomly changed each time. The row on which
the transformation was to occur and the type Of trans-
formation tO be performed were also randomly assigned.

This procedure is similar tO that which Piaget
(1952) designed. Two main differences did exist,
however. Piaget required both correct judgment and
explanation as criteria for passing the number conser-
vation task. The present study required a judgment-only
criterion in assessing number conservation. Brainerd
(1973a, 1977b) stated that, in the current literature,
the chances Of an erroriJiusing a judgment-only
criterion were fewer than if using a judgment plus
explanation criterion. Errors can be made in two ways:
(1) classifying a child as conserver when in fact she
is not; and, (2) classifying a child as a nonconserver
when in fact she is. Therefore, it was decided to use

a judgment-only criterion based on these findings. Also,

46

Piaget labeled his subjects as Stage I, II, or III
conservers. Since we are using judments-only, it is
not possible to classify subjects in this way, according
tO their responses, and will not be done in the present
study.
Task 2: Rote counting

TO ascertain whether the subject was able to
recite the natural or counting numbers, the experimenter
said, "I would like you to count from one to ten for

me. The subjects were given additional trials if they
stumbled over certain numbers or omitted any Of them;
otherwise they were scored 1 after one correct trial.
Rothenberg and Orost (1969), Wohlwill and Lowe (1962),
and Schaeffer et al. (1974) all used a task similar to
this one in previous studies.
Task 3: Object counting

Schaeffer et a1. (1974) and Rothenberg and Orost
(1969) both used similar tasks to train or assess
Object counting abilities. Their procedures were used
to devise the present task. The subjects were shown
a group Of checkers and were asked, "How many checkers
are there in this group?" The task was repeated, each

time with a randomly assigned number Of checkers, from

4 to 7.

47

Task 4: Number judgments Of two equal groups

The subjects were shown an index card with two
rows Of dots with equal length and number on it. They
were asked, "Does each row have the same number Of
dots?" With each trial, the number Of dots on the card
that was shown was randomly assigned with either 4, 5,
or 6 dots per row. A similar task has been used in
previous studies (LaPOinte & O'Donnell, 1974;
Rothenberg & Orost, 1969).
Task 5: Number judgments Of two unequal groups

Schaeffer et al. (1974) and Rothenberg and Orost
(1969) used similar procedures to those described here
to train or test their subjects on their ability tO
perceive number inequality. The subjects were shown
two index cards, each with a row Of dots, the twO being
unequal in number. They were then asked, "Does each
row have the same number Of dots in it?" The two cards
used for each trial were randomly chosen, with the
numbers varying from 4 tO 7 dots.
Task 6: One-one correspondence

The present study used a one-one correspondence
task quite similar to those in the studies by Halford
and Fullerton (1970), Inhelder et a1. (1974), LaPOinte
and O'Donnell (1974), and Rothenberg and Orost (1969).
A row Of checkers (5, 6, or 7 randomly ordered) was

constructed before the subjects. They were then given

48

a pile Of chekcers and were told, "I would like you tO
make a row just like mine with the same number of
checkers." Each trial was repeated with a different
number Of checkers.

Task 7: Addition/subtraction changes in a row

Two rows Of 5 checkers were constructed in
one-one correspondence in front Of the subjects. The
experimenter ascertained that the subjects understood
that the two rows were numerically equal by asking, "DO
these two rows have the same number Of checkers in
each?" If the child said yes, the experimenter then
began the task. If no, the experimenter then counted
each row out loud to show their equality.

This task differed from others in that there
were 4 trials rather than 3. The experimenter then
manipulated either her row or the subject's, twice on
each in a random order. A checker was then added or
subtracted (twice each in a random order) from or tO
one of the rows while the subject watched, with the
manipulated row's length adjusted to match that of the
non-manipulated row. The subject was then asked, "Who
has more checkers, you or I, or do we have the same
number?"

Pufall et a1. (1973), Wallach et a1. (1967),
Rothenberg and Orost (1969) and others used a technique

similar to the one developed for use in the present study.

49

Task 8: Number versus length Of arrays (spatial arrange-
ment)

Pufall et a1. (1973) used several tasks to test
judgments Of numerosity in spite of misleading cues.
These tasks were incorporated into one for the present
study. The subjects were shown a card with two rows Of
dots on it. The rows were either numerically equal but
unequal in length or equal in length but numerically
unequal. The order Of presentation was randomly
assigned with nO card presented twice to a subject. The
experimenter then asked, "DO these twO rows have the
same number Of dots in them?"

Task 9: Reversibility

The experimenter constructed two equal rows Of
checkers (6, 7, or 8). She then said to the subject,
"These two rows Of checkers have the same number in
them. Now watch me." The experimenter then proceded
to collapse, or "bunch up" one Of the rows while the
subject watched. The subject was then asked, "Does this
row have the same number Of checkers as the other row?"
The number Of checkers in the two constructed rows and
the row tO be manipulated were randomly chosen for each
trial. This technique was similar to that used by
Inhelder et a1. (1974), Wallach et a1. (1967),
Rothenberg and Orost (1969), and Wallach and Sprott

(1964).

RESULTS

Analyses

 

The data were analyzed in several different ways.
The major analyses performed were a Guttman scalogram
analysis and a linear multiple regression analysis. The
Guttman scalogram was performed to give a developmental
ordering Of the eight number concept tasks and also Of
the number conservation task. This analysis also gives
the number Of errors committed by the subjects and the
reproducibility coefficient Of the scale. Multiple
regression analysis was used to establish the ability Of
each number concept task to predict number conservation.
This method also gives the predictability when these
variables are combined in a forward stepwise fashion. In
this manner, we are able to determine the percentage Of
variance accounted for by each variable. The .05 level of
statistical significance was selected for all statistical

analyses in the present study.

Sample Analyses

 

Table 1 lists the distribution Of total scores
(total passed) Of the 75 subjects. The sample was
divided into three age groups: (1) 4-11 to 5-4,
(2) 5-5 to 5-9, and (3) 5-10 to 6-4. The mean total

score and variances for each group are also listed.
50

51

Table 1

Total Scores, Means, and Variances
Of Age Groups

 

 

 

Total Score Age Grogp
(Passed) 4-11 to 5-4 5-5 to 5-9 5-10 to 6-4

2 1 0 0
3 0 2 0
4 3 5 1
5 8 3 2
6 6 5 3
7 6 10 5
8 2 3 6
9 l 2 1

Mean Total Score 5.8 6.1 6.9

G2
2.15 2.69 1.65

 

52

A chi-square analysis was computed to assess any
differences in scores by age group. The analysis showed
that there were no differences in the way the total
scores were distributed across the three age groups
(212 (14) = 16.49, B< .28). It was not expected that the
distributions would vary owing tO the small age range Of
the sample.

A Levene test for homogeneity of variance
(Keppel, 1973, p. 81) was performed on the data to
assess sample homogeneity with regards to within-group
variance. It was found that the sample variance was
indeed homogeneous (3(2, 72) = .105, n.s.).

The data was not analyzed to check for sex
differences. In general, it has been found that sex
differences do not occur in studies Of number conserva-

tion in children (Maccoby & Jacklin, 1975).

Correlational Analysis

The phi correlation coefficients between the
8 number concept tasks, number conservation and age are
listed in Table 2, along with the phi max values. As
can be seen, there was very little relationship between
the variables themselves or between number conservation
or age. The correlation coefficients were quite small,
most very close to zero. However, a few were signifi-

cantly correlated. Reversibility was found to be

53

 

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umm» Hmowumflumum “when map How mo. v we

 

 

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ea. 4mm. so. me. we. se.- ~o.u «mN. we. aueaenemum>mm .m
mucmfimmcmuhe
me. ea. me. me. me. ea. «mm. mm. Heeumem .m
eoeuoeuunsm
saw. me. me.uee~m. «mm. me. om. \eoeeeeee .e
mocmocommouuou
se.- oe.u Hm. se.- oa.u em. meoumao .e
mucmfimosh
ea. «emu. ea. NH. mm. amass: Heaven: .m
mucosmosb
mo. mo.- ea. He. amass: Hesse .e
seem. ma. oo.e meanesoo nomflno .m
mo. mm. meanesoo muom .m
cowum>ummcoo
. . eonssz .H
m m e e m e m N H xmz flee ewes

 

mmHQMHHo> mom one .ucoocmmmo

.pcoocomoocH coozumm mcofluoamuuoououcH

N wanna

54

related significantly with age (r = .27, E.< .05),

number conservation (r = .23, E < .05), and addition/
subtraction in groups (r = .25, p,< .05). Spatial
arrangement was correlated with number conservation

(r = .22, 2_< .05). These number concept tasks are

those which are usually thought to occur later than

other number concepts. Therefore, they appear relatively
close to the appearance Of number conservation with the
probability Of their appearance increasing with age.

The correlation between rote counting and Object
counting was highly significant (r = .32, p_< .01). This
is understandable owing that rote counting ability is a
skill a child needs to count an array Of Objects effec-
tively, and the teaching Of this skill Often involves
arrays Of Objects. Object counting was also correlated
with the counting Of unequal groups (r = .29, p < .01),
and the addition/subtraction changes in arrays (r = .32,
p,< .01). Both are tasks which are also dependent on
counting ability.

The maximum phi was calculated for each correla-
tion between criterion, number conservation, and each Of
the eight number task. Phi max indicates the maximum
level each phi coefficient can attain given the marginal
distributions (Guilford, 1954). Only when the two
variables being correlated are Of equal difficulty can

phi coefficients be at their positive maximum. As can

 

55

be seen on Table 2, most Of the phi coefficient were
less than 1.0, indicating that the difficulty levels
between number conservation and the eight number tasks
varied.

The number conservation task was given at the
beginning and at the end Of the session to each child tO
check for the test-retest reliability Of the conserva-
tion task. Of the 65 children who were initially
classified as nonconservers, only six conserved on the
second trial and Of the ten children who initially
conserved, none failed the second time. The phi
coefficient between conservation scores before and
after the testing was .753 ($_max = .753), showing that

the number conservation task is reliable.

Developmental Sequence

TO determine whether there was a certain sequence
in which the subjects responded to the tasks, the
Guttman scalogram analysis technique (Green, 1954, 1956;
Torgerson, 1958) was employed. This technique is one
commonly used in studies looking at developmental
sequences (KOfsky, 1966; Siegel, 1971b; WOhlwill, 1960).
The tasks were ranked in order Of difficulty on the
basis Of the number Of subjects who passed each task.
This will give us an ordering Of the logical prerequi-

sites Of number conservation and determine whether a

56

subject who has passed a given task has also passed the
logical prerequisites tO that task.

Scalogram analysis assumes that, to achieve a
perfect scale, subjects who pass a given task mp§3_pass
all the logical prerequisites and fail all those tasks
above the subjects' maximum level Of achievement (Green,
1954). An error was made if a subject passed a task
but failed a logical prerequisite tO the task. A
coefficient Of reproducibility, Rep, can be calculated
to determine the proportion Of responses which fit this
"perfect" sequence by each subject. The formula for
Rep is:

E
Rep-1'1?

where N is the number Of subjects, K, the number Of tasks,
and E, the number Of errors made. A total Of 88 errors
were made in 675 responses by the subjects. The value

Of the coefficient for the present study was .87 which

is high and indicates that the scale was fairly gOOd

and could be reproduced.

Table 3 shows the percentage Of those subjects
who passed each task. The order Of difficulty Of the
tasks as given by the scalogram analysis from easiest
to most difficult was: (1) rote counting, (2) number
judgments Of equal groups, (3) one-one correspondence,
(4) Object counting, (5) addition/subtraction changes in

a row, (6) number judgments Of unequal groups, (7)

57

Table 3

Percentage and Number Of Subjects
Passing Each Task and Number
Of Errors Made on Each Task

 

 

Pre- Per- Guttman
dicted Task Passed cent errors
Order
1. Rote counting 72 96.0 3
3. Equal number judgments 70 93.3 5
6. One-one correspondence 66 88.0 8
2. Object counting 65 86.7 7
5. Addition/subtraction 61 81.3 8
changes
4. Unequal number judgments 49 65.3 17
7. Reversibility 46 I 61.3 20
8. Spatial arrangement 25 33.0 14
9. Number conservation 10 13.3 6

 

Note: N = 75

Total errors = 88

58

reversibility, (8) spatial arrangements, and (9) number
conservation. The Obtained results were somewhat
different from the predicted order but were highly
correlated with it. The Spearman rank-order correlation
(Siegel, 1956) Of the predicted logical sequence with

the Obtained sequence Of tasks was .85 (E < .01).

Predictability Of Tasks

Predictability Of number conservation by the
eight number concept tasks was analyzed by the use Of
linear multiple regression. In this way, we were able
tO see to what extent each variable predicted number
conservation both alone and in combination with other
variables. The eight variables were entered into a
forward stepwise multiple regression procedure with
number conservationas the predicted variable. A summary
Of the results Of the analysis is presented in Table 4.

As can be seen, only the first 5 variables, or
steps, added significantly to the regression equation
(F(5,69) = 2.41, p|< .05). Reversibility, spatial
arrangement, one-one correspondence, Object counting,
and addition/subtraction changes were found to be
significant predictors Of number conservation when
added in a stepwise procedure to the multiple regression
equation. One-one correspondence received a negative

beta-weight which could indicate that it is acting as

59

 

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med. NH. mam.n oo. mm. meanesoo ouom m
was. me. «am. co. mm. assessesn amass: Heaven: e
meo. NH. mom. oo. mm. assessese sense: Hesse e
mmmcmau
emeo. me. ems. so. am. eoeuomuunsm\eoeueee« m
ammo. me. and. mo. mm. meeueaoo nomflno e
«wmo. NH. NmH.I no. em. mocoocommmuuoo occloco m
same. we. mes. so. Hm. uemsmmemuum anaemem m
«see. me. see. me. mm. mueaenemum>mm H
e no memm
cocoowm Hound mumm omcmso m m omumuco manmwum> moum

neemem cumcceum m

 

mmHQMHHm> “swam
EOHM cowum>ummcou Honsdz mcquHomHm
mflmwamcd OOflmmmumom mamfluaoz mo muasmmm

e manna

60

a suppressor variable. However, it is relatively small
(-.l92) and therefore most likely is not contributing

as a suppressor (Darlington, 1968). The three remaining
variables did not improve predictive ability a signifi-
cant amount. However, the 5 prediction variables only
accounted for 15% Of the variance with the remaining

3 variables contributing no variance. It appears that
the tasks themselves are not especially gOOd predictors
Of number conservation in spite Of their significance.
This fact will be discussed further in the following

section.

DISCUSSION

The present study's major purpose was the
determination of a develOpmental ordering Of certain
number concepts thought to be logical prerequisites Of
number conservation. These number concepts were among
those prOposed by numerous researchers as necessary for
children's attainment Of number conservation. In
general, an ordering Of these eight concepts was esta-
blished, but evidence for their necessity was not found.

Rote counting was found tO be the easiest, or
first, concept that children acquired before number
conservation. This finding upheld Prediction 1 and
concurred with similar findings (Rothenberg & Orost,
1969; Schaeffer et al., 1974; WOhlwill & Lowe, 1962).

The next concept predicted to develop in
children was the ability to count Objects in an array.
It was thought that once children possessed rote
counting skills, the next skill to develop would be the
application Of the rote counting in determining number
in arrays. This, however, was not found tO be the
case. Number judgments of two equal groups was shown
to be the second concept to develop. Basically,

counting skills were probably necessary to respond tO

61

62

this task, so it might have indirectly assessed this
ability besides the recognition Of equality. It is
also possible that this task was easier than thought by
other researchers (LaPOinte & O'Donnell, 1974;
Rothenberg & Orost, 1969).

One-one correspondence emerged as the third
number concept acquired by the children. This concept
had been predicted tO emerge as the sixth concept, which
would have made it a much more difficult one than has
been shown here. With the prior emergence of correct
number judgments Of equal groups as a concept, it might
be that the next logical concept to develop would be
the ability to construct two such equal groups. This
concept seemed to develop without the understanding Of
inequality as was thought necessary for one-one corres-
pondence by Inhelder et a1. (19%”, Wohlwill (1960), and
Schaeffer et a1. (1974).

The fourth concept, Object counting, had been
predicted tO develop in second position, immediately
after rote counting. This indication Of increased
difficulty than predicted could be attributed tO the
manner in which the Objects were displayed. The checkers
were not presented to each subject in a straight line,
but usually in a bunched group. The children might have
had difficulty remembering if they had already counted

a checker. Several subjects did show signs Of confusion

 

 

 

63

when counting the group Of checkers. Perhaps, if the
checkers had been arranged in a straight line, a higher
percentage Of subjects would have passed this task,
placing it closer to its predicted position. This is
also an indication Of how perceptual factors and memory
play an important part Of children's number judgments.

Addition and subtraction changes in a row was
shown tO emerge as the fifth number concept to develop
before number conservation. This confirmed the predicted
order Of its emergence.

Number judgments Of unequal groups appeared in
most children as the sixth concept to develop. This
concept had been predicted tO appear earlier (fourth)
than the results showed. It is possible that children
must understand the effects Of addition and subtraction
changes in equal rows before they are able to recognize
that two groups Of Objects are unequal in number.

The last three concepts, reversibility, spatial
arrangements, and, finally, number conservation, were
found to appear in this order. Their order Of emergence
was in agreement with the predicted order where the first
two were assumed tO be logical prerequisites Of number
conservation; the Observed order also agreed with the
findings Of other researchers (Inhelder et al., 1974;

Pufall et al., 1973; Wallach et al., 1967).

64

It is interesting tO note that the order Of the
eight concepts could have been changed very easily if a
small number Of subjects had responded differently. In
looking at Table 3, it can be seen that there is only a
spread Of 11 correct responses among the first five tasks.
They could very easily have come out in a different
ordering, but not necessarily in the predicted order.

Apparently, one must be careful when interpreting
the results Of a Guttman scalogram analysis. This type
Of analysis is relatively sensitive to the feasibility
Of the chosen tasks as a scale in themselves. However,
our data for the present study indicate that this
analysis is insensitive to individual differences and
leaves these differences unaccounted for. ”Use Of the
Guttman scale technique has Often served to reinforce
findings Of literature reviews. That is, the researcher
predicts a sequence of tasks and uses the scalogram
analysis merely tO confirm or reject the hypothesized
ordering. NO information is available to the researcher
other than the "goodness," or unidimensionality, Of the
tasks as a scale. It is therefore usually necessary to
use follow-up analyses in hOpes Of gaining more infor-
mation about the tasks.

Each concept's ability to predict number conser-
vation was also analyzed by a stepwise linear multiple

regression procedure. The results of the analysis

 

65

showed that reversibility, spatial arrangements, one-one
correspondence, Object counting, and addition/subtraction
changes in a row were significant predictors of number
conservation and accounted for 15 percent Of the variance
with the remaining variables contributing none Of the
variance.

How might the 85 percent of the variance missing
be accounted for? It appears that there are other
factors behind number conservation acquisition which
are not taken into account when speaking Of logical
prerequisites. Factors such as memory, attention span,
linguistic abilities, and also general reasoning or
problem-solving skills might play an important part in
the acquisition Of number conservation. Results Of
the Object counting task in the present study point to
the part memory might play in conservation acquisition.
NO study has ever trained subjects in these skills
together with number skills to determine their impact
on conservation acquisition.

The number conservation task itself might also
be responsible for the variance differences. It is
assumed that the task tests for children's logical
capacities in dealing with number invariance. It is
possible that this task, while indeed testing number
judgments, also taps other skills as well. In fact,
Gelman (1972a) showed that this task possibly evaluated

 

66

other skills such as attention control, correct seman-
tics, and estimation skills. These results point tO
number conservation as a more SOphisticated type Of
cognitive functioning where numerous skills are inte-
grated into a holistic cognitive system. As far as the
test-retest reliability Of the task, the phi coefficient
was quite high (£.= .753). Therefore, unreliability Of
the task can be ruled out as a factor.

Another question which should be considered is
Of each number concept's necessity for conservation
acquisition. The previous research naturally assumed
that if training in a number concept induced number
conservation, the concept was a necessary one for its
attainment. There are problems in this reasoning. It
is not at all possible to determine what skills each
individual subject possessed before this training and
how these skills affected the training and conservation
acquisition. Are we to assume that the concept training
itself was responsible for the acquisition Of number
conservation or did this training just add to the skill
already possessed by the child creating adequate cogni-
tive structures for conservation? It has been suggested
that most concept learning experiments merely instruct
subjects in the generalization Of mental structures which
are spontaneously acquired instead Of acquiring the

concepts themselves (Brainerd, 1978). The children learn

67

to use spontaneously the trained concepts without true
understanding Of them as is assumed by the researchers
when conservation is induced.

It cannot be assumed that the eight number con-
cepts assessed in the present study are either necessary
2; sufficient for number conservation. It is Obvious
that several children conserved number without passing
all eight tasks and that others who passed all eight
tasks did not conserve. It is probable that those
children who conserved but failed several logical
prerequisites possessed other cognitive skills which
were adequate for number conservation but were not
assessed. The same reasoning may be applied to those
who passed all tasks but who failed the conservation
task. They perhaps lacked other necessary skills not
tested by the present study.

In the present study, age was not a significant
factor in the response patterns Of the subjects. The
age range used was not wide enough to show different
patterns Of responses or in the total correct scores
in the different age groups. It could be expected that
there would be a great deal more variability between age
groups had the age range been extended in both direc-
tions (Older and younger).

It was perhaps a failure on the part Of the

present study in relying on past findings as to the

68

logical prerequisites Of number conservation. Other
aspects should have been investigated and included in
the series Of tasks used. Future research should
investigate this area more fully. There are undoubtably
other skills necessary for number conservation. In
fact, by looking solely at number concepts, researchers
have been ignoring vital information. Mpiangu and
Gentile (1975) feel that number conservation and
mathematical concepts develop simultaneously but inde-
pendently of one another. This fact lends to the hypo-
thesis that other factors play an important role in
number conservation acquisition.

Some attention should be paid to memory and
attentional factors and the part they playin cognitive
skill acquisition. Researchers should investigate this
area more closely, focusing less on the individual
number concepts as necessary prerequisites. It is
Obvious that we are dealing with a concept which is
more complex than that Of a series Of number concepts to

be acquired before number conservation can be attained.

 

APPENDIX A

STIMULUS CARDS FOR TASKS 4, 5, AND 8

69

 

APPENDIX A

Stimulus cards for Task 4.

 

 

 

 

 

 

 

 

Stimulus cards for Task 5.

 

 

 

 

 

STIMULUS CARDS FOR TASKS FOR 4,5, AND 8

 

 

 

 

 

 

 

 

 

 

 

 

Stimulus cards for Task 8.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70

 

 

 

 

 

 

 

 

 

 

 

 

APPENDIX B

DATA SHEET

71

Score 1

Task 1

(1)

Task 2

Task 3

Task 4

APPENDIX B

DATA SHEET

Name Of child

Date Of birth

for pass, 0 for fail.

Conservation Of number assessment.

Trial 1
Trial 2

Trial 3

(2) Trial 1
Trial 2

Trial 3

Rote counting.

Trial 1
Trial 2

Trial 3

Object counting.

Trial 1
Trial 2

Trial 3

Number judgments Of two equal

groups.

Trial 1

Trial 2

Trial 3

NO. used

NO. used

NO. used

4 5 6
4 5 6
4 5 6

72

 

 

(1)
(2)

Score

Score

Score

Score

Score

Task 5

Task 6

Task 7

Task 8

Task 9

73

Number judgments of two unequal
groups.

Trial 1 4 5 6 7
Trial 2 4 5 6 7
Trial 3 4 5 6 7

One-one correspondence.

Trial 1 5 6 7
Trial 2 5 6 7
Trial 3 5 6 7

Addition/subtraction changes in
a row.

Trial l____ A 8
Trial 2__ A S
Trial 3____ A 3
Trial 4_____ A S

Number vs. length Of rows (spatial
arrangement).

Trial l____ 4 5 6 = 7
Trial 2____ 4 5 6 =
Trial 3__ 4 5 6 =
Reversibility.

Trial l;____ 6 7 8
Trial 2____ 6 7 8

Trial 3 6 7 8

Score

Score

Score

Score

Score

REFERENCE NOTES

74

 

 

REFERENCE NOTES

Brainerd, C. J. Analysis and synthesis Of research
on children's ordinal and cardinal number concepts.
TResearch Bulletin #394). London, Canada: Univer-
sity Of Western Ontario, Department Of Psychology,
November 1976.

75

LI ST OF REFERENCES

76

I?

LIST OF REFERENCES

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