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MIQBEGLKN STATE UNIVERSETY
Charles E. Brainerd
1970

 

J

' LIBRARY ‘9

THESIS

Michigan State
University

     

This is to certify that the

thesis entitled

THE C(NSTRUCTION OF THE FOIMAL OPERATIONS 01“ IMPLICATION-
REASONING AND PROPORTIONALITY IN CHJIDREN AND ADOLESCENTS

presented by

Charles J. Brainerd

has been accepted towards fulfillment
of the requirements for

Pho Do degree in PsyCh'Dlggy

,2 :7
ac 2L— 4 I%O{Ml~z/l‘_

Major professor

Date 6/10/70

0-169

mm
THE CUSTRUCTIU 01" m POM OPBMTIWS OF mama-mans
AND PROPORTIGALITI Ill cmm AND Edi-$-13

By
Charles J. Brainerd

Tito experiments are reported that are concerned with Piaget‘s theory
of the developnent of formal-ope rational intelligence . For the most part .
the experiments were designed as tests of Specific predictions derived
fran Piaget' s assertions concerning his two categories of formal Operations
(“propositional operations“ and "formal-ope rational schemata” ) . The first
experiment was focused on one of Piaget' s ”propositional operations“ (impli-
cation), while the second experiment was focused on one of Piaget's Pfomal-
Operational schemata" (proportionality). Subjects from three age-groups
were employed in both experiments: eight-nine year-olds; 11-12 year-olds;
and 14-15 year-olds.

In experiment I, the ”propositional ope ration” of implication was
studied via its transitivity property (i.e. , if A implies B and B implies
C, then A implies C). A 2 x 3 x 2 design was employed with the factors
being propositional syntax, age of subject, and propositional semantics.
Two dependent measures were considered: correctness-incorrectness of
implication-reasoning answers and the latencies of these same answers.
The results indicated that} subjects‘ implication-reasoning abilities
gene rally improved with age; the improvements with age in implication-
reasoning tend to be linear only; propositional semantics strongly affected
the correctness-incorrectness of implication-reasoning answers; proposi-

tional syntax affected answer latencies; and syntax affected younger

Charles J. Brainerd

younger subjects more than older subjects.

In experiment II, three indices of Piaget' s ”proportionality schema"
were investigated: density conservation; solid volume conservation; and
liquid volmne conservation. Both conservation answers and explanations
were analyzed as dependent variables . The results of eaqaeriment II indi-
cated that: subjectsI abilities to conserve the three indices improved
with age; the age increases in conservation ability were almost entirely
linear; the conservations of solid and liquid volume tend..to precede the
conservation of density invariably; the three indices intercorrelate sig-
nificantly; and older subjects learned density conservation more readily
than younger subjects.

Since the same subjects were employed in both experiments, a predic-
tion derived from Piagetian theory to the effect that the dependent vari-
ables of the two experiments should be related within subjects was tested
but not confirmed. The results of the experiments were discussed in terms
of general implications for the Genevan theory of cognitive development
and Kagan' s reflectivity-impuls ivity dimension of cognitive style .

1;”de

Date ’7’ May /7’)0

 

 

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THE CONSTRUCTION OF THE FORMED OEERATIONS OF IIFLICAIION-REASONING
AND PROPOREIONILITY'IN CEILDBEN AND'ADOEBSIENTS

By

Charles J? Brainerd

A THESIS

:hflnflifled to
Michigan State university
in.partial fulfillment of the requirements

for the degree of

DOCTOR.OF‘PEILOSQEHI

Department of Psychology
1970

Ge (Awe

I .,- a7... ‘7!

ACKNWTS

Preparation of the present thesis and the research reported herein
were supported by Public Health .Grant M32736-Ol to the author from the
National Institutes of Mental Health. The author also wishes to acknow-
ledge the many helpful comments and criticisms of the thesis committee:
Ellen A. Strommen, Chairman; Hiram E. Fitzgerald; Donald M. Johnson; and
Charles P. Hanley. Special thanks also are due the faculty, staff, and
students of the Holt Public Schools .

ii

TAEEE OF CONTENTS

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Method...........................................................20
Results..........................................................28
Discussion.......................................................hl
ZEIEERIHENT II: THE PROPORTIONALITI’SCHEHA............................#fl
Hbthod...........................................................h5
Results..........................................................5O

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APPENDIX II: ASSESSMENTVQUESTIONS FOR THE NEGATIVE SEMINTICS CONDI-.

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LIST OF TAEEES

Tahle 1: Truth Conditions for the Propositional ”Operation" of
Implication.......................................................8
Tahle 2: The Transitivity of Truth-Functional Implication............ll
Tahle 3: Implication-Reasoning Propositions..........................23
Table a. Summary of Analysis of variance for Implication Answers.....30
Table 5: Summary of Analysis of variance for Implication Latencies...3l
Tahle 6: Orthogonal Polynomials Analysis for Implication Answers.....33
Tahle 7: Orthogonal Polynomials Analysis for Implication.Latencies...3h
Tahle 8: Reliabilities of Implication-Reasoning Dependent variables..35
Tahle 9: ‘t Tests for all Possible Points in Figure l.................38
Table 10: _t_ Tests for all Possible Points in Figure 2................l+0
Table 11: Age_Trend;§ Ratios for Proportionality One‘way Analyses of
variance.........................................................51
Table 12: Orthogonal Polynomials Analyses for Density Conservation...53
Table 13:: Orthogonal Polynomials Analyses for Solid Volume Conserva-
tion.............................................................5h
Table 1“: Orthogonal.Polynomials Analyses for} Liquid volume Conser-
vation...........................................................55
Table 15: Percent of Explanations Falling in each Category.........-.56
Table 16: Simple and Multiple Correlations Between the Three Depenv
dent variables...................................................58
Table 17: Relationship Between the Conservations of Density and

iv

Solid Volume......................................................59
Table 18: Relationship Between the Conservations of Density and

Liquid Volmne.....................................................6O
Table 19: Age Differences in the Learning of Density Conservation.....63
Table 20: Correlations of Implication Answers with Proportionality

Variables.........................................................69

 

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INTRCDUCTIGU

Piaget has proposed a theory of the ontogeny' of adult intelligence
(e.g., Piaget, l9llr9, 1953: Inhelder 8: Piaget, 1958) which describes the
11-12 year-old child' s acquisition of two hypothesized features of mature
thought. The first of these (both ontogenetically and in terms of importance)
is the acquisition of propositional logic as a dominant mode of thought.
with the advent of these "propositional operations" , Piaget holds that
the emphasis of thought shifts from the Lee; to the possible and that
there is a consequent increase in the ability to reason by hypothesis.
The second of the proposed features of mature thought involves the acqui-
sition of several "formal-operational schemata” (Piaget, 19149) which owe
their origins to the aforementioned coordination of propositi onal logic
in the young adolescent. These operational schemata are said to corres-
pond roughly to general equilibrium laws that characterize physical sys-
tems (e.g. , mechanical equilibrium, 'action-reaction') and to be manifest
in the child's increased ability to solve problems which involve such
relations (Inhelder, 1953: Inhelder 8: Piaget, 1958).

The research to be reported herein was concerned with these two gen-
eral features of the deve10pment of adult thought and their relation to
each. other. Twn experiments were conducted to investigate the develop-
ment of one of Piaget's "propositional operations“ (implication) and one
of his ”formal-ope rational schemata" (proportionality). The first experi-
ment was designed to examine the effects of age of subject, semantic con-
tent of prepositions , and syntactic order of propositions on the proposi-
tional operation of implication. The second eiqae riment concerned

1

develOpmental trends in three proposed (Inhelder & Piaget, 1958) indices
of Piaget' s "preportionality schema" (density conservation, solid volume
conservation, liquid volume conservation) and the possible existence of
an invariant acquisition sequence among the reSpective indices of the
”preportionality schema." A final major objective of the present research
was to determine the magnitude of the relation (11' any) between the propo-
sitional Operation and Operational schema of interest. Consonant with
this latter aim, the same children and adolescents served as subjects

in both experiments.

Theoretical Background

TO continue with Piaget' s substantive notions , his theory of intel-
lectual development is well known as a stage theory and it is difficult
to justify predictions pertaining to the final stage of cognitive deveIOp-
ment (formal operations) without some consideration of the preceding stage
(concrete operations). During the concrete-Operational period, the child
presumably acquires the ability to classify objects simultaneously accord-
ing to one or more criteria and to comprehend simple relations among ob-
jects and events. (Hence, Piaget refers to the indigenous logic of this
stage as the "logic of classes and relations”--Inhelder 8: Piaget, 196‘!)
Another distinctive feature of this concrete-Ops rational stage is that
the child's thought operations are now held to be reversible (i.e., there
exists the permanent possibility of any thought returning to its point
of departure). As the present writer previously has noted (Brainerd,
1970: Brainerd 8: Allen, 1970) , Piaget advocates two distinct forms of
Operational reversibility, viz . inve rs ion-ne gation (a singing Ope ra-
tion analogous to negating a single affirmation) which applies exclusively

to the concrete logic of classes and reciprocity (a P231921 Operation analo-
gous to canpensating changes in one affirmation with equal and Opposite
changes in a related affirmation) which applies exclusively to the con-
crete logic of relations .

Piaget argues (1949, 1953) that these classificatory, relational,
and reversibility features of concrete-operational thought allow for the
coordination of eight "elementary groupements” (four for classes and four
for relations). Although these "elementary groupements” are limited in "
scOpe , they are said to facilitate some forms of intelligent behavior
(e.g. , seriation of asymmetrical transitive relations, conservation of
simple quantitative invariants) and they may be described as. follows:

1. Each "elementary groupement" is either a §_l_a_s_§_ or a relation.

2. Each "elementary groupement" is either smtrical or asmtrical.
3. Each "elementary groupement" is either multiplicative or additive .
Since each “elementary groupement" is classified in terms of three fac-
tors with two levels each, there are exactly eight possible "elementary
groupements" (2 x 2 x 2 = 8).

It is out of these eight "elementary groupements" that the preposi-f
tional Operations ("the sixteen binary Operations") , referred to earlier
as characteristics of formal intelligence , are thought to be coordinated
(Piaget, 19142, 1949, 1953). This coordination presumably produces the
lattice structure of formal thought--the ultimate equilibrium. The prin-
ciple reason why this coordination does not come about sooner in life
is also the reason given for the limited generality of the eight ”elemen-
tary groupements," viz. the two reversibilities of classes (inversion-
negation) and relations (reciprocity) are themselves not coordinated and
they only can be applied successively, not simultaneously.

The crucial feature of the consolidation of formal-Operational in-
telligence, then, is said to be the coordination of the two forms of re-
versibility into a single schema--the INRC group--capable of simultane-
ous application to prepositions . The two general advances of formal Opera-
tions mentioned previously (prepositional logic and dependent Operational
schemata) can be thought of as 'bonuses' entailed by the effect that this
coordination has on the structures of concrete thought (the "elementary
mutants”) -

The '3' and 'B' of 'INRC' denote the two forms of reversibility
(inversion-negation and reciprocity) which the group unites , while '1'
denotes antidentity element and ' C ' denotes a correlative element (the
inverse of the reciprocal). As Parsons (1960) notes, Piaget's INRC
group is isomorphic with the well-known four-group of mathematics . In
the case of group DIRC, the defining equalities are: OR = N: RR = C;

NC = R; and I = NRC. Although a precise explication of the mathematics

of the four-group is beyond the scOpe of the present report, it is important
to consider some of the qualitative characteristics of Piaget's applica-
tion of the four-group to the structure of adult cognition. Indeed, Pia-
get's general statements about the group INRC constituted the sole basis

for a. major hypothesis of the present research to the effect that the
ontogenies of a particular prOpos itional Operation (implication) and a
particular Operational schema (prOportionality) would be intertwined.

Pm the perspective of the present research, the most important
characteristic of the group INRC is that it supposedly comes in two varie-
ties: a 'logical' form and a 'physical' form. The logical form of the
group INRC refers to the fact that the series of prOpositional Operations
which are acquired aprOpos formal thought ("the sixteen binary operations ")

are said to be structured according to the rules of the four-group (compo-
sition, associativity, identity, inversion). Similarly, the 'phys icsl '
four of the group INFO refers to the fact that Special systems of physi-
cal transformations (such as those gathered under the rubric "Operation-
al schemata”) also are said to be governed by the rules of the four-group.
Hence, the group INRC is held to structure both the 'logical' Operations
of abstract thought and the interpOlative cognitions about immediate phy-
sical experience. 1
Although these Speculations about the nature and functions of the
four-group may seem somewhat academic, the implications of these assertions-
in terms of experimental predictions-~are exceedingly concrete. In so
far as the group INRC structures the transformations of both the intern-
alized Operations of propositional lOgic and our cognitions about systems
of physical transformations, there comes to mind an ObVious hypothesis
to the effect that if behavioral manifestations of one Of these two OOg-
nitive 'systems' are present in a given subject, then behavioral manifestar
tions of the other also should be present. Hence, in addition to the pre-
dictions of Specific developmental changes in the child's abilities to
reason via.prOpositional operations and to apply the dependent operation-
al schemata, Piagetian theory also seems to authorize the conclusion that
these two groups of skills should be related linearly within subjects.
ngpggitional Operations
Of the several principles of Piaget‘s theory of the development of
formal intelligence, the most basic is the child's supposed acquisition
of the system.of "propositional Operations" (Piaget, l9fi9, 1953. Piaget &
Inhelder, 1969; Inhelder & Piaget, 1958). The coordination of these propo-

sitional Operations is thought to precipitate the realization of a

generalized lattice-structure . The propositional operations which consti-
tute the elements of this lattice are isomorphic to the 16 binary rela-
tions of formal pox-positional logic. However, Piaget substitutes "opera-
tions“ for ' relations ' .

The difference between Piagetian "operations" and fomal 'relations'
is, in part, the difference between psychological activity and passin-
ty. The metamathematical notion of ' relation’ is completely abstract
and may be defined only by resorting to certain 'formal characteristics'
(symmetry, reflexivity, correlativity, transivity, etc. ). Hence , by de-
finition, the notion of relation is nonactive and only refers to the pos-
sible permutations and combinations observed when conjoining numbers or
prOpositions . Conversely, Piaget' s concept of "operations" derives from
action--as does everything else in his system. These operations amount to
internalized, reversible actions which have been organized according to
principles that insure the equilibrium of the cognitive system. Although
there apparently are no immediate behavioral consequences of this distinc-
tion-~empirically the results of the "Operation" of disjunction (p v q)
appear to be the same as the results of the 'relation' of disjunction
(also p v q)--Piaget probably is justified in drawing the distinction
since it is difficult to conceive of any psychological isomorph of the
metamathematical concept of ' relation' .

To return to the main thread of Piaget's argument, the formal-
Operational child is said to acquire the ability to think hypothetically
and propositionally according to the rules of that species of generalized
mathematical IOgic known as propositional logic. (It is important to
note that there are infinitely many other formalized mathematical languages-

sane of which, as Parsons, 1960, has suggested, are more powerful than

 

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the one employed by Piaget.) Formal discussions of propositional logic
include the fact that if one considers only propositions EEEEEHEQEEE.2£
2'3 mtg glues, then there are precisely two singgarlz relations among
such.propositions (negation and affirmatiod and.16 bi§a§z_relations among
such.propositions. This latter fact serves as the source of the 16 ele—
ments of Piaget's lattice model of formal thought. The full compliment
of’l6 binary operations is tabled on p. 103410“ of Inhalder and Piaget
(1958)-

Experiment I is concerned with Piaget's logical operation of impli-
cation. An implication consists of an initial proposition (called an
antecedent) and a second preposition (called a consequent). Depending
on which of the two base propositions one wishes to consider as implying
the other, the truth conditions for this operation.may be found in either
column three or column four of Table l. The affirmation that this
antecedent-consequent relation (implication) Obtains between any two ar-
bitrary propositions is equivalent to asserting ‘if the antecedent is
true, then the consequent.must be true}. The truth-falsity of the con-
sequent when the antecedent is false may vary.

The propositional operation of implication has been offered as the
logical counterpart of what we commonly refer to as the cause-effect re-
lation (Inhelder & Piaget, 1958)--i.e., the set of all possible cause-
effect relations is a.proper subset of all possible implications. There-
fore, cause-effect is a sufficient but not necessagy condition for imp
plication. Alternatively, there is no conceivable cause-effect relation
whose truth conditions are not Specified adequately by the formal-
implication.model. Hence, in the initial experiment reported here, rea-

soning tasks were employed whose elements were propositions placed in

 

Table 1

Truth Conditions for the PrOpositional "Operation" of Implication

 

 

P Q P30 0)?
True True True True
True False Fblse True
False True True False
False False True True

 

a cause-effect ralation to one another and whose solutions required that
subdects of three chronological ages be able to reason on the basis of such
relations. The effects of certain.propositionrspecific treatments (syn-
tactic order of propositions and semantic content of propositions) on the
same subjects' abilities to reach valid conclusions and their response
latencies also were considered.

Piaget points out that there are cognitive skills present at the con-
crete level which are deceptively similar to fommal-operational implicae
tion. These concrete skills constitute inherent sources of error in any
investigation Of the ontogeny of reasoning via implication. ‘Without con-
sidering his assertions in depth, it suffices to say Piaget's premise is
that the concrete-operational child can come close to solving tygrelement
implications (p:)q or q:)r) by employing simple concrete correspondences.
Rather than study such two—element implications with their built-in error
factor, it seemed.more apprOpriate to study‘a.pggpg£§z,of the implication
relation which can be arrived at only if the ability to handle the impli-
cation relation is well developed. Like most other asymmetric relations,
the implication relation is transitive and it is precisely this transi-
tivity property that was employed in experiment I. This approach to the
study of implication via its transitivity property is roughly analogous
to previous studies of the concrete concepts of '1ess than' and 'more
than' via their transitivity prOperty (e.g. , Piaget, Inhelder, 8: Szemin-
ska, 1960; Inhelder & Piaget, 1964; Smedslund, 1963a; Murray 8: Youniss,
1968).

To say that a binary relation R is transitive on or among a set of
elements (for’present purposes, the set of elements is restricted to a

series of verbal propositions conjoined by cause-effect relations) is

10

to assert that for any arbitrary elements 1:, y, and z, B. relates x to
z whenever R relates x to y and y to 2. In the case of a simple relation
such as 'less than' , the transitive property is intuitively as well as
logically evident (if A>B and B) C, then trivially A) C). Although the
transitivity property of implication is perhaps not so intuitively apparent ,
it is nonetheless logically evident and is specified clearly by Table 2.
The fact that implication is a transitive relation suggested that
one might devise three-element implications similar to the three-element '
problems employed in the previously mentioned studies of concrete transi-
vity. Thus if one has three base propositions (in lexicographical order:

p, q, and r) and if one links the first with the second and the second

 

with the third by the operation offimplication, then it is valid to con-
clude (by Table 2) that the first and third also are linked by the same
operation. Therefore, if it is the case that 'p implies q and q implies
r', then it also must be the case that 'p implies r'--i.e., (qu) ‘ (q)
r))(p) r) is valid. On this point Piaget's theory of formal thought
reduces to the hypothesis that if the formal-Operationfl. child is pre-
sented with the first two implications, he will infer that the third
implication also holds. The general procedure for assessing children's
abilities to reason in terms of implication thus becomes somewhat clear-
er, viz. to present subjects with the first two implications and to de-
termine the relative presence of the remaining implication through appro-
priate interogation. Reasonably enough, the "appropriate interogation"
correSponds to the fundamental truth conditions for implication set down
in Table 1.

As previously mentioned, the truth conditions of the cause-effe ct

relation are representable in terms of the implication relation. Therefore ,

 

OH

Table 2

The Trans itivity of Truth-mmctional Implication

 

 

p q r (13);) - (quDfiDr)
True True True True
True True False True
True False True True
False True True True
False False True True
False True False True
True False False True
False False False True

 

12

to study the cause-effect relation is to study a 'pure' example of an en-
vironmental isomorph of the logical operation of implication (Inhelder a
Piaget, 1958). In conformity with the above model, these considerations
suggested a series of problems that presents subjects with three base prOpo-
sitions of which the first and second, and the second and third are con-
joined in terms of cause and effect. The problems were designed to dis-
cern the extent to which subjects inferred that the cause-effect relation
also obtained between the first and third propositions . Thus the proce-
dure for investigating reasoning by implication consisted of a series of
eight problems for which the following example served as a prototype:
1. Witions: p = Jack washes the family car; q = Jack's father is
very pleased; r = Jack receives 50¢.

a. (qu): If Jack washes the family car, then Jack's father will

be very pleased.

b. (qu): Whenever Jack's father is very pleased, Jack receives
50¢-

 

2. Evaluation: Subjects' inferences that the third implication (if Jack
washes the car, then he will receive 50¢) held were assessed by four ques—
tions of the following form:
a. (p . r): If Jack washes the family car, then what else will hap-
pen?
b. (is . r): If Jack doesn't wash the family car, is it possible that
he still might receive 50¢?
c. (P . i"): If Jack washes the family car, is it possible that he
won't receive 50¢?

d. (5 . r): If Jack doesn't wash the family car, then what else will

 

13

happen?

Each of the preceding questions corresponds to one of the four cells
of the fundamental truth table for implication (Table l) . It should be
noted that this method did not yield an ' all-or-none' judgment vis>a-vis
implication; rather, incremental evidence was adduced about the m
to which the implication relation is graSped in each problem. Obviously,
this feature of experiment I facilitated parametric analyses of the data.

The first line of evidence deriving from erqaeriment I concerns the
ontogeny of inplication-reasoning. Data indicating an increasing abili-
ty to reach these conclusions which are authorized by the premisses of
the proposed problems would be supportive of Piaget' s hypotheses about
the development of propositional operations . The ontogenetic question
aside, experiment I also was concerned with the effects of two further
variables on subjects' abilities to reason in terms of implication and
the rapidity of their implication responses , viz . the semantic content
of the base propositions and the presentation order of the two initial
implications-41030 . (qu) vs. (CDr) - (19(1).

The work of De Vries (1969) suggests that the semantic content of
simple Piagetian reasoning tasks has a pervasive effect on the conclusions
of three to six year-olds. In erqaeriment I an attempt was made to discover
whether or not this effect extends to more complex reasoning tasks such
as implication. The semantic manipulation was quite simple: the result-
ant effect (r) of each three-element implication problem was pleasant for
(randomly) one-half the subjects and aversive for the remaining half of
the subjects. The second influence to be investigated was the order in
which the two initial implications were presented v tea subjects. logically,

the presentation order has absolutely no effect on the ultimate validity

 

I'D

ll!-

of the third implication (thanks to the associativity property of the logic
of propositions). Empirically, however, the order in which facts or state-
ments are proposed has been shown to affect subjects' judgments (Inhelder a
Piaget, 1958, 1961+). The order manipulation consisted of reversing the
order of the two initial implications for (randomly) one-half of the sub-
jects. Hence, one-half of the subjects considered the formula (qu) .
(qu) and the other half considered the formula (qu) . (qu). These
two formuli are logically equivalent--the question remains whether or not
they are psychologically equivalent.

If a syntactic (order) treatment were effective, one might expect
a decrease in the influence of order reversal in favor of older subjects.
This prediction derives fran Piaget's numerous assertions that the concrete-
Operational child is influenced strongly by time-space correspondence phe-
nomena such as serial-orde r. Conversely, the abstract operations of formal
thought, with their relative independence from concrete phenomena, are
said to approximate more closely the rules of 'pure ' mathematical relations
(such as, in this case, associativity).
Formal-Operational Schemata

Piaget argues (19149, 1953: Piaget 8: Inhelder, 1969; Inhelder 8: Pia-
get , 1958) that the coordination of the previously mentioned proposition-
al operations produces an n-by-n combinatorial system (structure d' ensem-
ble) whose structure is that of the lattice and whose laws of composition
are those of the mathematical group. This formal structuring of thought
in turn precipitates the acquisition of several “conceptual instrumentali-
ties" (Flavell, 1963) which Piaget has called (e.g., 19149, 1953) “formal-
operational schemata. " If the propositional operations and their inher-

ent structure are the most general features of formal thought and if imediate

15

sense impressions constitute the least general feature of formal thought,
then these formal-ape rational schemata are at some intermediate level of
generality. To continue the analogy, the formal-ope rational schemata are
supposed to reconcile the most general features of formal intelligence with
the least general features thereof.

Several formal-operational schemata have been referred to by Piaget
and among these are: proportionality; mechanical equilibrium: ”all-other-
things-being—equal"; 'action-reaction': and others. The develorment of
these operational schemata is said to be necessitated by the occurence
and reoccurence of certain general 'foms' or types of problems in every-
day life .

(If. the several operational schemata, the schema of pmortionalitz
probably has received the most extensive consideration in Piaget's theo-
retical expositions. The proportionality schema is of particular interest
when considered in relation to experiment I, because Piaget offers the sche-
ma as a physical realization of the group lIRC--the same group that is
held to be influential in the coordination of propositional operations
such as implication.

As Piaget defines the notion, the schema of proportionality refers
to a qualitative structure which facilitates the understanding of complex
physical systems that contain many factors or forces which compensate each
other. The intended range of application of the proportionality schema
may be illustrated for the case of the conservation of volume. As a spa-
tial concept, volume conservation requires that subjects be able to place
the three relevant spatial dimensions (length, width, depth) of two dif-
ferent containers (or objects) into a compensating proportionate relation

to one another (i.e., given 112 = X'Y'Z', subjects must be capable of

 

16

inferring n/x'r' = Z'/Z)if they are to conserve volume. In addition,
Inhelder and Piaget (1958) assert that it follows that the concept of d_en-
_s_i_.tz also requires application of a proportionality schema, since density
is a second-order concept based on a weight per unit y_ol_u_n£ relation.

In view of their presumed representativeness as applications of the
proportionality schema to the real world, the concepts of volume and den-
sity were examined in the second experiment reported here. As was the
case for implication-reasoning, the age changes in children's understand-
ing of the concepts of volume and density were assessed. Previous research
(ELkind, 1961) tentatively suggests that the median age for acquisition
of volume conservation is greater than 11 years. There is a complete ab-
sence of objective data relative to the ontogeny of density conservation.
Further, the linearity of the acquisition functions also was of interest.
Since three equally spaced age-groups were studied (third, sixth, and ninth
graders), the possibilities of positively or negatively accelerated acqui-
sition functions existed.

An additional aim of experiment H was to investigate a hypothesized
. (Inhelder & Piaget, 1958) invariant sequence in the acquisition of volume
and density concepts. In so far as density is a concept that is dependent
upon the notion of volume, Inhelder and Piaget assert that an adequate
conception of volume must antedate an adequate conception of density.

To briefly characterize the relevant procedures of experiment II,
the concept of volume was assessed by a typical conservation: method which
takes account of Piaget‘s (1968) warnings about ”psuedo-conservation" and
Rothenberg's (1969; Rethenberg & Courtney, 1969) warnings about "False
positives." The general. method for evaluating conservation of volume has

been described correctly by E‘lkind (1961) and incorrectly described by

l7

Trabasso (1968). The reality of whether or not one actually is investi-

gating M turns on whether or not one asks a spatial question as op-

posed to a quantity question (Trabasso's error) of the subjects. Obvious-

ly, the spatial question (e.g., "Which one takes up more space or room”)

is the appropriate choice. Conservation of density was assessed via a

new technique generated from the results of a broader investigation of

density concepts conducted by Inhelder and Piaget (1958). The density '
technique employed in experiment II makes use of the fact that concrete ' ‘
nonconservers of density think of the concept in terms of absolute weight

rather than weight-pe r-unit-volume .

Smary

Broadly speaking then, the aim of the two experiments reported here
was to investigate some of the hypotheses offered by Piaget as part of his
theory of the ontogeny of formal thought. Specifically, experiment I was
concerned with the ability of children and adolescents to reason in terms
of formal-logical implication. kperinent II was concerned with the articu-
lation of the concepts of volume and density, in the same children and
adolescents , as a means of adducing data pertinent to Piaget' s proportion-
ality schema.

It also was of interest to consider just what relation obtained be-
tween subjects ' abilities to reason via implication and their abilities
to apply the proportionality schema to concrete situations (i.e. , to con-
serve density and volume). As previously mentioned, the prediction that
these two cognitive skills should be related within subjects follows di-
rectly from Piaget's explication of the four-group INFO as a structuring

agent of formal thought. In sum, if the rules and properties of the group

18

INRC organize the cognitive structuring of both the more general proposi-
tional operations (of which implication is one) and the less general opera-
tional. schemata (of which proportionality is one), then the ontogenies

of implication-reasoning and the proportionality schema should be similar
within subjects .

EXPERIMENT I: IMPLICATIm-EASMING

The initial experiment was focused on subjects' (_S_s') implication-
reasoning abilities. A 2 x 3 x 2 factorial design was employed with the
factors being semantic content of component propositions , age of g (third
grade, sixth grade , ninth grade), and syntactic order of component propo- '
sitions. TflD dependent variables were measured, viz. the answers to the V ‘
implication-reasoning problems and the latencies of these same implication-

reasoning answers. Five major questions were of interest in experiment

 

I.

1. Does the implication-reasoning facility of S_s increase linearly
from, say, age eight to age 15 as Piagetian theory predicts (Piaget, 1949,
1953: Inhelder & Piaget, 1958)?

2. Does the implication-reasoning facility of _S_s vary with the seman-
tic content of component propositions ?

3. Does the implication-reasoning facility of _S_s vary with the syn-
tactic order of the antecedent propositions?

1+. If semantic content and/ or syntactic order affect implication-
reasoning facility, does the effect or effects vary in intensity depend-
ing on the age of _S_‘I

5. If implication-reasoning facility does increase from age eight
to age 15, is this increase entirely linear or are there some nonlinear
features about it?

In the preceding five questions, the term 'facility' denotes both
correctness-of-answers and response latencies. In addition to these major

questions , some related though miner questions , such as the reliabilities
19

20
of the dependent variables also were of interest.
Method

Sub sets

The experimental. _S_s were drawn from three age-levels c eight-nine
year-olds (third graders); 11-12 year-olds (sixth graders); and lit-15 year-
olds (ninth graders). These age-ranges were chosen because they corres-
pond to Piagetian levels of cognitive development that were particularly
relevant to the present research. Piaget has reported (e.g., Piaget a.
Inhelder, , 1969) that the initial group of _Ss (eight-nine year-olds) is
well into the period of concrete operations, the second group of gs (ll-
12 year-olds) is in a time of transition from concrete to formal thought,
and the final group of Se (111-15 year-olds) is well into the period of
formal thought. In so far as one of the objectives of erqreriment I was
to study the transition from concrete to formal thought via implication-
reasoning (cf. questions 1, 1+, and 5), these three age-levels were logi-
cal choices.

Twelve boys and 12 girls were selected from each level for a total
of 72 §_s. All §_s were pupils of the Holt Public Schools, a moderately-
sized semirural school system. For purposes of generality, it seemed more
appropriate to study ‘ average ' children than either 'bright' or 'dqu'
children. Therefore , only _S_s within the 90-110 IQ range and with an aca-
demic grade-point in the 2.0-2. 5 range (on a four-point scale) were in-
cluded in the present experiment.

All §_s had to meet three other criteria to be included in experiment
I: conservation of number (Piaget, 1952); conservation of length (Piaget,

Inhelder, 8: Szeminska, 1960); and minimal reading ability. The classic

 

21

tests of number and (length conservations that are so ubiquitous in the
developmental literature were used as the first two criteria. To meet
the third criterion, it was necessary for S to read six sentences that
were similar in difficulty to those he would encounter in the form of
implication-reasoning prepositions. All _S_s met the nutrber criterion, only
one §_ (a third grader) failed length conservation and only one §_ (another
third grader) failed to display minimal reading ability. These two _S_s
were replaced by two other third graders who met all criteria. In theory,
the pretests of number and length conservation served as an assurance that
all §s--by Piagetian criteria--had attained at least the period of concrete
operations.

(hie-half of the 21+ §_s in each age-group participated in erqaeriment
I first. The remaining half participated in experiment II first.
m aratus and Hate rials

A Sony 230 tape recorder was used throughout the experiment. Upon
entering the experim9ntaliroom, each _S__ was fitted with Sony DR-6A headphones
which were not removed until §_ left the room. Instructions were p_r_e__r_e;-
995139 on the left channel track. _S_s' reSponses were recorded on the right
. channel track. Thus, all gs heard E's instructions and comments through
the left headphone, while their own responses simultaneously were being
recorded on the right channel track. This particular record-playback-
record technique is referred to by the audio industry as “sound-with-sound. "
Sony PR-l50 recording tape was used exclusively and the recording Speed
was a constant 7% i.p.s.

The other important materials employed in the first experiment were
white stimulus cards by which the implication-re asoning problems were pre-

sented to the So. All cards were three inches by five inches and the

22

canponent propositions of an implication-reasoning problem were typed in
large red letters on each card.
Treatment Conditions

Age; As noted above, three age-levels were studied in the present
erqaeriment (eight-nine year-olds, ll-lZ year-olds, lib-15 year-olds). Also
as noted, these three age-levels presumably correspond to Piaget's periods
of concrete operations , concrete-formal transition, and formal operations.

M. The 48 propositions employed in the present experiment are
enumerated in Table 3. The assessment procedure focused on the extent
to which gs concluded that the cause-effect relation held between the propo-
sitionsof columns A and C of Table 3. Obviously, there are two possible
orders in which the propositions of columns A, B, and C may be conjoined
to necessitate the transitive inference 'A causes C', viz. 'A causes B
and B causes 0' or 'Bcauses, Czsand licenses B'. For convenience, the form-
er ordering is referred to as the forward syntactic treatment (FS) and
the latter ordering is referred to as the reverse syntactic treatment (BS).
One-half of the SS received as implication problems and the other half
received RS implication problems.

Semantics . Examination of the propositions of Table 3 will reveal that
each implication-reasoning problem centered on a single central character
(either Jack or Jill depending on the sex of §_). The semantic manipula-
tion consisted of varying the reinforcement consequences of the implica-
tion problems for this central character. are-half of the §_s were given
implication-reasoning problems in which the transitive inference (A causes
C) involved some pleasant consequence for this central character. The
other half of the _S_s were given implication-reasoning problems in which

the transitive inference involved some unpleasant consequence for this

 

23

Table 3
Implication-Reasoning Propositions

 

 

Jack (Jill) sweeps Mother is very pleased Jack (Jill) gets all the
the kitchen floor dessert he (she) wants

Jack (Jill) washes Father is very pleased Jack (Jill) gets 50¢
the dishes

Jack (Jill) gets all Both of Jack's parents Jack (Jill) gets to stay

'A's' on his (her) are very pleased up later as a reward
report card

Jack (Jill) does well Teacher is happy Jack (Jill) gets less
on an English test schoolwork to do

Jack (Jill) news the Jack (Jill) works hard Jack's (Jill's) father
lawn at something treats him (her) to an
ice cream cone

 

Jack (Jill) has a All of Jack's (Jill's) Jack (Jill) gets a lot
birthday relatives come to see of money

him (her)
Jack (Jill) helps Mother does not have Mother fixes Jack's
mother with the much work to do (J 111' 5) favorite food
shopping for dinner
Jack (Jill) plays a Jack (Jill) does a Jack (Jill) gets an
part in a school very good job award from the school
play
Jack (Jill) throws a A window breaks Mother sends Jack (Jill)
rock at a window to bed without supper
Jack (Jill) complains Jack's (Jill's) Jack's (Jill's) friends
about how bad things friends get mad won't talk to him (her)
are
Jack (Jfll) plays Jack (Jill) starts a Jack's (Jill's) father
with matches fire in his (her) takes away .his (her)

House allowance

24

Table 3 (cont'd. )

 

 

Jack (Jill) leaves
his (her) bicycle in
the driveway

Jack (Jill) forgets
something he (she) is
supposed to do

Jack (Jill) wiSpers
in class

Jack (Jill) breaks
one of mother's
favorite dishes

Jack (Jill) doesn't
cane home right after
school

Father hits Jack's
(Jill's) bike with
his car

Father tells Jack
(Jill) what to do

Teacher gets mad

Mother is unhappy

Jack's (Jill's)
parents worry about
him (her)

Father takes away Jack's
(Jill's) bike

Jack (Jill) feels
ashamed

Jack (Jill) has to stay
after school

Jack (Jill) cannot go
outside and see his
(her) friends

Jack (Jill) cannot watch
television

 

25

central character. The pleasant outcome problems are referred to as the
positive semantic treatment (+8) and the unpleasant outcome problems are
referred to as the negative semantic treatment (-S). The eight +8 prob-
lems appear in the top half of Table 3, while the eight -s problems appear
in the bottom half of the same table.

Wations. Equal numbers of _S_s from the three age-groups were
assigned randomly to the two syntactic conditions and to the two semantic
conditions with the single provision that the treatment levels be divided
equally with respect to sex. From the 8! ways in which the eight implica-
tion problems might have been ordered, 24 problem orders were selected
at random and randomly assigned to _S_s within each age-group. Finally,
the order in which the four assessment questions were asked was varied
randomly for each randomly ordered problem.

Procedure

Each g was presented with either the eight implication-mas oning prob-
lems appearing in the top half of Table 3 or the eight implication-reasoning
problems appearing in the bottom half of Table 3. The elements of these
problems appearing in columns A. B, and C of Table 3 were three base assertions.
As a means of controlling for possible age changes in short term memory,
the problems were presented one-at-a-time on 3 x 5 cards and E read each
one aloud. _S_s were allowed to retain and reread each card during the inter-
im during which E asked them questions concerning it.

In reading the problem 1; connected the column A proposition with the
column B proposition and the column B proposition with the column 0 propo-
sition by means of a cause-effect relation (i.e.. 'A causes B and B causes
0'). Following the reading of each 3 x 5 card, 3 assessed the extent to

which §_s inferred that a cause-effect relation also obtained between the

26

column A and C propositions (i.e., 'A causes C') via four questions of the
following general form:

a. g: "If A occurs, then what else will occur?m (correct answers = C or
both B and C)

b. E: “If A occurs, then is it possible that c 329:; occur?" (correct
answer = no)

c. E: "If C occurs sometime, is it possible that i didn't occur?" (cor-
rect answer = yes) i

d. _B_: "If A doesn't occur, then what else could occur?" (correct answer =
Cmayormaynot occur or Bandeay ormaynot occur)

The first and last questions obviously required greater extemporiza-
tion on the part of §_ than did questions b and c. whenever such a situa-
tion obtains , there is always the possibility that individual differences
in motivation or anxiety may increase error variance. Hence, it seemed
advisable to institute some precautionary measures to minimize the chances
that failure to answer questions a and d might have been due to reticence
on the part of s_. These precautionary measures were a maximum of two prompt-
ings that were provided whenever § failed to answer (i.e., gave an, I'I
don't know") either the first or last question.

Prior to seeing the initial problem, §_s heard the following instruc-
tions: ”I am going to show you some white cards one-at-a-time. Cm each
card there is a short little story about a boy named Jack (girl named Jill).
I shall read each story to you as you look at the card. I shall then ask
you some questions about the story on the card. When you have answered
thequestions, wewillgo ontoanewcardanddothe same thingagain
until we are finished."

To sunmarize the procedural details of eaqaeriment I, gs from each

2?

age-group were assigned randomly to either condition FS or condition BS
and to either condition +5 or condition -S. Next, they were read the a-
bove instructions. Finally, the eight cards with the implication-reasoning
problems were presented and read aloud one-at-a-time; gs were asked four
questions pertaining to each card.
Mndent Variables

The dependent variables of interest were the content of §_s' answers .
to E's implication-reasoning questions (i.e., the correctness-incorrectness
of §s' responses) and the latencies of _S_s' answers (i.e., the length of
the intervals between E's questions and _S_s' resPonses). It should be noted
that Piagetian theory focuses exclusively on the former of these two de-
pendent variables and does not include any consideration of response la-
tencies as a reasoning parameter.

Mication-Reasoning Answers. The complete set of 32 implication
reasoning questions that were posed to the 36 §s in condition +S appears
in Appendix I, and the alternative set of 32 questions posed to the 36
§_s in condition -S appears in Appendix II. The four assessment questions
for each of the problems were scored in the following manner:

1. Question a ("If A occurs, then what else will occur‘t"): _S_s received
a score of '3' if they gave a correct answer on the first try; .23 received
a‘ score of '2' for a correct answer following a single prompting; gs re-
ceived a score of '1' for a correct answer following a. second prompting.

2. Question b ("If A occurs, then is it possible that C m occur7”):
§s received a score of '2' for a correct 'no' answer.

3. Question c ("If C occurs sometime, is it possible that A didn't
occur-1’"): §s received a score of '2' for a correct 'yes' answer.

’4. Question d ("If A doesn't occur, then what else could occur?"):

 

28

ga received a score of '3' if they gave a correct answer on the first try;
88 received a score of '2' for a correct answer following a single prompt-

ing; §_s received a. score of 'l' for a correct answer following a second
prwpting-

The four scores within individual problems were sumed for each _s_
and these totals were in turn sumed across the eight problems to yield
a single ~ 'implication—reasoning' score. This single score represented
each §'s status on the present dependent variable and it was used in all
analyses of the variable save reliability estimates .

m Latencies . The second dependent variable was the elapsed
time between the last words of _E_'s questions and the first words of the
_S_s' reaponses. All estimates of the second dependent variable were made
' from the tape recordings after _S_s had participated in both experiment I
and experiment II.

The latency of each of the _S_s' 32 neponses was estimated three times
to the nearest 1/10 of a second. The average of the three estimates then
was taken as the latency of a particular reaponse. As was the case for
the first dependent variable , latencies for individual §_s were sumed with-
in problems and then across problems to yield a single reaponse latency
value. This value represented a §_'s status on the second dependent vari-
able and was used in all analyses of that variable save reliability esti-

mates .
Beeults

Separate 2 x 3 x 2 analyses of variance for fined-effects were per
formed on the data pertaining to the two dependent variables , the factors

being syntax (A), age (B), and semantics (C). A summary of the analysis

29

of variance for the first dependent variable (correctness-incorrectness

of implication-reasoning answers) appears in Table 4 and a summary of the
analysis of variance for the second dependent variable (response latencies)
appears in Table 5.

It is apparent from Table ’4 that the adequacy of fis' answers was in-
fluenced strongly by both their age and the semantic content of the prob-
lens to which they were exposed. Alternatively, the order in which _S_s
received the initial propositions ('A causes B and B catses C' vs. 'B
causes C and A causes 3') did not affect the adequacy of gs' answers.
Finally, the effects of the age and semantic factors were simple and ad-
ditive with no interactive tendencies being noted.

(I: the other hand, it is apparent from Table 5 that the latency of
_S_s' answers was influenced by their age and the syntactic order of the
component propositions of the implication-reasoning problems. The seman-
tic content of the problems did not appear to affect the rapidity with
which §_s answered the various questions . In addition, there was a Sigri—
ficant tendency for the factors of age and syntax to interact. Post hoc
analysis revealed that this was attributible to the fact that the syntac-
tic manipulation was more effective with the two younger groups than with
the older groUp. It is also apparent that the three significant 2. ratios
of Table 5 account for a much smaller portion of the total variance than
do the two significant _F._‘_ ratios of Table 1+.

Since three age-groups were studied, the possibility existed that
the age increases in the correctness and rapidity of implication answers
were characterized by certain nonlinear features. For example, the dif-
ference between the third and sixth graders might have been proportionate-

ly larger than the difference between the sixth and ninth graders (or vice

Summary of Analysis of Viriance for'Iiplication Answers

30

Table #

 

 

 

39m 38 9.1.. HS .13. a

Syntax (1) 66.12 1 66.12 (1 J,
Ago (B) 32%.33 2 1617.17 19.98 (.0005

Semantics (c) 3029.01 1 3029.01 144.67 (.0005

A x B 316.3% 2 158.17 1.95

A x c 5.02 1 5.02 (1

B x c 1:44.11 2 222.06 2.71» <.10

A x a x c n10.11 2 205.06 2.53 (.10

Error ”857.83 60 80.96

Total 12362.87 71

 

Table 5

Summary of Analysis of Variance forkplication Latencies

 

 

W 35 9!. MS I 2
Syntax (A) 7.93 1 7.93 8.35 .01
Age (B) mm 2 7.01 7.38 (.005
Semantics (c) 1A8 1 1.48 1.56

A x B 7A5 2 3.73 3.93 = .025
A x C 1.94 1 1.914 2.01;

B x c 5.32 2 2.66 2.80 (.10
A x a x c 2.75 2 1.38 1A5

Error 56.81% 60 .95

Total 97.73 71

 

 

32

versa). This nonlinear possibility was explored via an orthogonal poly-
nmials analysis of both dependent variables . Summaries of these two analy-
ses appear in Tables 6 and 7. Neither analysis provided any evidence of
nonline arity. While the linear component is large and highly significant
in the case of both answers and latencies, the quadratic ca‘nponent is less
than one for both dependent variables .

In so far as the analyses of variance demonstrated clear and consis-
tent effects of the various factors , it is likely that the two dependent
measures are characterized by adequate reliabilities. In addition, Cron-
bach's coefficient alpha was used to calculate exact reliability estimates
for each dependent variable . These total reliability estimates , as well
as the separate components contributed by each age-group, appear in Table
8. It can be seen that the rather short (eight problem) evaluation of
implication-reasoning facility was quite reliable and that the reliabili-
ty estimates do not vary notably with age of _S_. Hence even though eight-
nine year—olds tend to answer implication-reasoning questions incorrect-
ly, they nonetheless answer these same questions reliably. Since total
reliability estimates for the two dependent variables are identical, the
large difference in portion of variance accounted for in the separate analy-
ses of variance (Tables 1+ and 5) of the two variables cannot be attributed
to differences in reliability.

The average number of points that gs received for each problem are
plotted against problem order for each of the age-groups in Figure 1.

It can be seen that there was some improvement in the quality of the an-

swers as §_s proceeded through the eight problems. As is typical of 'wam
up' effects, the improvement was more pronounced for tle first through the
fourth problems than for the fifth through the eighth problems. The three

Orthogonal Polynomials Analysis for Implication Answers

33

Table 6

 

 

 

mores SS 9; MS g p
theen 3234- 33 2
Linear 3117.09 1 3117.09 22.81 (.0005
Quadratic 117.24 1 117.2» <1
Error 9128. 54 69 132. 30
Total 12362.87 71

 

Orthogonal Polynomials Analysis for Implication Latencies

Table 7

 

 

 

Source 38 1; HS _1: p
Emu 11". 03 2
Quadratic .03 1 .103 <1
Error 83.70 69 1.21
Total 97.73 71

 

 

35

Table 8
Reliabilities of Implication-Reasoning Dependent Variables

 

 

 

 

 

Ase-Group
Variable
8-9 yearsa 11-12 yearsa lib-15 years“ all _S_sb .1
Answers .89* .88“ .85“ .9o"'
“fancies o 88* o 85* e 9"* e m*

 

aN= ther cell
bfl= 72 per cell
'p<.0005

'7

 

POINTS

ANSWER

CORRECT

36

 

 

Figure l
8
Vi 4\
9TH GRADE
7
/6m GRADE
6
3RD GRADE
5
4

 

 

l 2 3 4 5 6 7 8
PROBLEM IN ORDER OF APPEARANCE

37

possible paired answer comparisons for each of the eight problems plotted
in Figure 3 are given in Table 9. The generally large and highly signi-
ficant 3 ratios of Table 9 authorize the conclusion that each of the three
inswerebyiproblem.curves of Figure 3 tends to be significantly different
frun the other two curves. In line with the significant linear component
reported in Table 6, paired inepections of the three columns of Table 9
indicate that the differences between corresponding points on the ninth
and third grade curves tend to be larger and more highly significant than
the differences between corresponding points on either the third and sixth
grade curves or the sixth and ninth grade curves.

The average total reaponse latencies are plotted against problem or-
der for each of the age-groups in Figure 4. As was the case for problem
answers, there is a general improvement in the latencies across problems
with the dechement being most pronounced for the first few problems. The
three possible paired reSponse comparisons for each of the eight problems
plotted in Figure 1+ are given in Table 10. Careful inspection of Figure
h and Table 10 reveals the reason why the age main effect of Table 5 is
so much smaller than the age main effect of Table 4. It is apparent that
only the latter halves of the third and sixth grade reaponse latency curves
differ significantly from each other. It also is apparent that the sixth
and ninth grade curves differ significantly only with reapect to the first
problem. In short, sixth grade latencies started out at a level not signi-
ficantly different from the third grade latencies and 'warmed up' to a
level not significantly different from ninth grade latencies.

Thus, the smaller age main effect evidenced in Table 5 as opposed
to Table 4 is almost exclusively the result of the large and highly sig-

nificant differences between the correSponding points on the third and

33 Tests for all Possible Pairs of Points in Figure l

38

Table 9

 

 

 

 

 

Age Pairings
Problem
Number Third Grade vs. Sixth Grade vs. Third Grade vs.
Sixth Grade Ninth Grade Ninth Grade
First .87 3. 6 2“" 5.29"”
Second 1+.36**** 2.73*** 7.99****
Third it. 96"" 3. 00*" 8. Mun
M 3.57“" 2.08“ 5. Beat-tn
Fifth 2. 33" 1+. 81"“ 6. 4 2M"
Sixth 1.00 4.69““ 5.66"“
Seventh .30 3.71“" ##6##
Eighth 1.90”“ 2.82"”: u.u9**"
‘2 (~05
“2 <025
at“! 4505

«at? <, 0005

 

 

RESPONSE LATENCIES IN SECONDS

39

Figtuez

3RD GRADE

6TH 6 RA DE

9TH GRADE

 

 

l 2 3 4 5 6 7 8
PROBLEM IN ORDER OF APPEARANCE

40

Table 10
33 Tests for all Possible Pairs of Points in Figure 2

 

 

 

 

58° P91191183
Problem
lumber Third Grade vs. Sixth Grade vs. Third Grade vs.
Sixth Grade Ninth Grade Ninth Grade
First .114 2.16" 3.69""
Second 1.69”: .81 236*“
Third .80 1.21 2.99""
Fourth 3.26“" .46 3.86””
Fifth 1.72" .93 238*"
Sixth 2.9V” .92 3.62"”
Seventh 3.18’" .78 n.19""
Eighth 2.96"" .70 3.87""
*2 <05
“‘2 <.025
“*2 <,oo5

area-v2 <.ooo 5

 

 

 

 

1+1

ninth grade curves. These data are also consistent with the significant
linear component reported in Table 7. The fact that the linear term of
Table 7 is roughly one-half the size of the linear term of Table 6 also

is accounted for by the preceding considerations.
Discussion

To return to the first of the five questions posed earlier in rela-

. tion to experiment I, the data clearly support Piaget's contention that ‘I
the implication-reasoning facility of children and adolescents increases

linearly between the ages of eight to 15. This contention was supported

by the age main effect of the analysis of variance for implication answers

and by the age main effect of the analysis of variance for implication

response latencies. I

Concerning the second question, the very large E ratio of Table 4
corresponding to'.the semantic main effect aggests that §s' abilities to
reach appropriate transitive conclusions based on the implication relation
is affected strongly by this variable. In the present experiment, those
§s receiving the +3 treatment produced reliably better answers than those
_S_s receiving the -S treatment. Alternatively, the insignificant _1: ratio
of Table 5 corresponding to the semantic main effect suggests that although
semantic content clearly affects correctness-incorrectness of answers,
this manipulation has a negligible impact on the rapidity of answer pro-
duction.

The conclusions regarding the third question are essentially the in-
verse of those for the second question. The syntax main effects of Tables
1+ and 5 suggest that the order in which implication-reasoning propositions
are presented influences the rapidity with which implication answers are

produced, but such order does 13% affect the correctness of implication—

1+2

reasoning productions. Those gs receiving the F8 treatment responded in
reliably shorter periods of time than those §_s receiving the RS treatment.
The absence of a reliable syntax by semantics interaction may imply that
propositional sequence affects the production Speed of incorrect and cor-
rect answers equally.

There was not a tendency for the semantic effect reported in Table
4 to be more effective for one age-group than for the others. Thus, the
semantic content of the propositions did not affect the correctness of the
implication-reasoning productions of the three age-groups differentially.
Alternatively, the syntax by age term of Table 5 demonstrates that the
order manipulation affected the latencies of the three age-groups differ-
entially. Post hoc examination revealed that, asswectefl. propositional
order produced the least latency variance in the oldest gs (lit-15 year-
olds). Unexpectedly, however, propositional order produced the most la-
tency variance in sixth grade gs. In so far as Piaget argues that the
Ill-12 year-old child is in a period of transition from concrete to form-
al operations (specifically, substage IIIA of Inhelder 8: Piaget, 1958),
this significant syntax by age interaction is certainly open to cognitive
dissonance interpretations (e.g., Fastinger, 1957; Smedslund, 1961b).

Regarding the fifth question, no quadratic (i.e . , positively or
negatively accelerated) tendencies were noted for either the age;~trends:*~in
correctness of implication answers or the age trend in the rapidity of
answer production. Instead, the quadratic component was less than one
for both dependent variables . This fact coupled with the high reliabili-
ties of the dependent variables might be construed as a dim forcast indeed
for those who would opt for quadratic trends within the specified age-

ranges on such parameters.

 

43

Given the identical reliabilities of the two dependent variables ,

the discrepancies between the results of the analysis of variance for inp-

lication answers and the results'of the analysis of variance for inplica-

tion latencies imply something about the interchange ability of these two

measures. Experiment I identified at least three differences between these

two variables vis>a-vis implication-reasoning: semantic content of propo-

sitions affects correctness of answers but not answer latency; proposition- I
al order affects answer latency but not the correctness of answers; propo-
sitional order affects the three age-groups differentially. It is import-
ant to note these empirical discrepancies, because they indicate that the

two dependent variables are in no wise equivalent measures of implication-

 

reasoning in particular and perhaps of deductive reasoning in general.
Since it typically is not a simple matter to record data pertaining to
both or ' these dependent variables in a given experiment, experimenters
frequently choose to study one at the expense of the other. The implica-
tion of the noted discrepancies is that any choice between response accu-
racy and reSponse latency must result from carefully reasoned preference
rather than simple expediency or caprice .

Finally, the fact that the main effect 2: ratios noted in relation
to implication-reasoning answers were four to five times larger than those
noted in relation to the latencies of these same answers suggests that
g the latter of the two variables probably is fixed earlier in life and re-
mains relatively more resistant to manipulations such as those examined

in the present experiment.

EXPERIMENT II: THE PRCPORTIGALITI SCHEMA

The second experiment was designed to investigate ontogenetic changes
in We cogritive skills that Inhelder and Piaget (1958) claim are indices
of the ability to handle qualitative proportions , viz. volume conservation
and density conservation. The Genevan school holds that children' s abili- '
ty to reason in terms of a "proportionality schema” increases during the "
ages studied in the present experiment. Experiment II was conducted in
the hope that five major questions pertaining to Piaget's specualtions
about the ”proportionality schema” would be answered.

 

1. Does the frequency of the conse rvations of volume and density in-
crease fran, say, age eight to age 15 as Piagetian theory predicts?

2. To what extent is it true--as Piaget claims—that the indices of
volume conservation (both solid and liquid) and the index of density con-
servation employed in the present experiment tap the gem; cognitive skill
(the ”proportionality schema” ) 7

3. Is the Genevan school correct in its assumption that an adequate
conception of volume must invariably precede an adequate conception of
density?

1+. Are there any nonlinear features about the age trends in answers -
and/ or explanations given by §_s in relation to volume conservation and
density conservation?

5. Do older §s tend to 'learn' density conservation more readily than
younger gs?

Minor questions concerning the. relation between conservation answers
vs . conservation eaquanations and possible differences in age trends for

141+

1+5
these two estimates of conservation also were of interest.

Method

Subjects
The same 72 _S_s employed in experiment I also participated in experi-

ment II. (he-half of the §s (randomly) participated in experiment I first

and then in experiment II. The remaining §s did the reverse.

Apparatus "
The same recorder, headphones, and microphones employed in experiment

I were retained for experiment II. A large (5000 m1) water-filled beaker

and three 50g: balls of- rubber-base clay (blue, red, yellow) were used

 

to assess density conservation. Each of the clay balls was more dense
than water. The pieces of clay, a medium-sized rubber band, and an amber-
colored glass 11+.5cm high and 6.0cm in diameter were used to assess solid
volume conservation. Two uncolored glasses 14.5cm high and 6.0cm in diame-
ter, one uncolored glass 11+.5cm high and 3.0cm in diameter, and one un-
colored glass 11+.5cm high and 9.0cm in diameter were used to assess liquid
volume conservation.
Procedure

Three basic assessments were executed in experiment II: density con-
servation; solid volume conservation; and liquid volume conservation. Since
Inhelder and Piaget (1958) do not separate liquid and solid volume conser-
vations when making predictions , it was deemed advisable to measure both
aSpects of volume conservation.

Each of the six possible orders of presentation was assigned random-
ly to four _S_s from each of the three age-groups. Because the assessment

technique varied somewhat depending on the conservation concept, the

46

details of the three assessment procedures are given separately.
Donsitz. § first was shown the 500ml water-filled beaker and one
of the 50m clay balls was placed in _S_..‘s hand. S was asked to predict
whether or not the ball would fall to the bottom of the beaker if it was
placed in the water and also to justify the prediction. The ball then was
placed in the water and §_ observed that it sank to the bottm. Follow-
ing this initial demonstration, the four steps of the assessment were exe-
cuted: ' ‘.
Step 1: The ball was flattened into a "pancake“ and _S_ was asked whether
or not the clay would float now that its shape had been altered. _E_ re-

quired that S explain the basis for his prediction. The ”pancake" then

 

was placed in the water and _S_ observed that it sank to the bottom.

Step 2: The "pancake” was removed from the water and dried. Q then cut
off approodmately two-thirds of the clay and asked _S_ if the remaining piece
would float. Again, E required that _S_ explain the basis for his predic-
tion. The smaller piece of clay was placed in the water and _S_ observed
that it sank to the bottom.

Step 3: The reduced piece of clay was removed from the water and dried.
Fm this _E; cut an even smaller piece that was approximately the size of

a dime and _S_ was asked whether or not this very small piece would float.
(hoe again, §_ had to justify his prediction. The very small piece of clay
then was placed in the water and §_ observed that it sank.

Step 4: After _S_ had observed that the clay sank when it was in the four
different sizes and shapes, guasked _S_, ”Do you think we could ever get a
piece of this clay small enough so that it would float?” _S_ was not required
to justify this last reSponse.

Solid Volume. _S_ first was shown an amber-colored glass which was

a?

approximately Wo-thirds full of water and had a rubber band around it.
Q then was sham one (randomly selected) of the three balls of clay. g
placed the ball in the glass and asked _S_ to mark the new higher water level
with the rubber band. After _S_ had marked the new water level, _S_ removed
the ball, dried it, and rolled it into a "sausage.” 5 then asked g the
following (randomly ordered) three questions: I

1. E: ”Iprlacethesausageintheglass,willthe watergo_a_b_9_v_e '
the rubber band now?" (_S agreed or disagreed)

2. _E_: ”If I place the sausage in the glass, will the water go M
the rubber band now?” (a agreed or disagreed)

3. g: "If I place the sausage in the glass, will the water go right

 

back to the rubber band?” (_S_ agreed or disagreed)

Obviously, the correct sequence of answers involved disagreeing with
the first two assertions and agreeing with the final assertion. S was
given each question segrately and subsequent questions were not posed
until the previous question had been answered. Following the three ques-
tions, _S_ was asked to explain the basis for his responses.

The method of requiring that _S_s both ggr_e_e_ and disagree with E's as-
sertions during conservation assessment orginally was developed by Rothen-
berg (1969: Rothenberg 8: Courtney, 1969). Briefly, Rothenberg argues that
the data of studies such as those of number conservation in very young
children (Mahler 8e Bever, 1967) are confounded by the fact that _S_s tend
to agree with an 13 more frequently than they disagree. Since this state
of affairs ushers in the possibility of ”false positives ," Rothenberg sug-
gests that the ideal sequence of conservatism assessment questions is
structured so that _S_ must both agree and disagree with _E_ to be judged a

conserver. The separate treatment of each of the three questions

 

1:8

enumerated above was precipitated by Rothenberg's contentions.

M m. S_s first were shown two uncolored glasses 14.5cm high
and 6.0cm in diameter. Q then put one liter of water in each glass and
g asked _S_ if the water in the first glass took up the same amount of space
or row as the water in the second glass and vice versa. If _S agreed
(and all _Ss did), the water in one of the two identical glasses was poured
into an uncolored glass 14.5cm high and 3.0cm in diameter. _S_ then was
asked two questions .

l. Q: "Does the water in these two glasses take up the sore amount
of space or roan?" (_S_ agreed or disagreed)

2. Q: ”Does the water in one of the glasses take up more space or
roan than the water in the other glass?" (§_ agreed or disagreed)

_S_ also was required to eiquain the rationale for his reaponses. As
was indicated for solid volume , the procedure of using questions that in-
volved both agreeing and disagreeingwith E was modeled after Rothenberg's
technique (Rothenberg, 1969; Rothenberg 5. Courtney, 1969).

The water in the 11+.5cm x 3.0cm glass then was returned to the origin-
al 1&5 x 6.0 glass. _S_ was asked whether or not the water in the two
original glasses still. took up equal amounts of space. If _S_ agreed (and
. all _S_s did), the water from one of the glasses was poured into a glass
lb.5cm x 9.0cm and _S_ was asked the same two questions as before. §_ again
was required to justify his answers.

Dependent Variables
It has been noted elsewhere (Brainerd & Allen, 1970) that there ex-

ists some controversy concerning the appropriate dependent variables to

measure as part of conservation assessment. Gmen (1966) , Griffiths, Shanta, ,-

and Sigel (196?), and Rothenberg (1969; Rothenberg 8: Courtney, 1969) have

 

 

49

contributed papers to this controversy. The most important psychometric
feature of this debate concerns; whether or not '_S_'s explanations of his
agreements-disagreements with E's assertions should be necessfl to the
'conserver-nonconserver' judgment. Some investigators (e.g., Bruner, 196%
have made the conserver-nonconserver judpent only on the basis of gs'
agreements-disagreements. Others (Snedslund, 1961a) also have required
that So adequately explain their agreements—disagreements to be judged
conservers. "
Regardless of which of the preceding criteria is chosen, one is left
with an ' all-or-none' model of conservation. Rather than choose either
alternative, the data of both answers and explanations were combined and

 

analyled in the present experiment. Most generalizations are based on
g ratios deriving from the combined answer and explanation data: however,
the answer and explanation data also were analyzed separately to provide
psychometric justification for employing a 'combined' criterion. The
scoring procedure for both answers and explanations is given below.

Answers. In all three conservation assessments, appropriate agree-
ments and disagreements were assigned a 'l' , while inappropriate agree-
ments and disagreements were assigned a '0'. This method yielded a five-
point range (04+) for density and liquid volume and a four-point range
. (0-3) for solid volume.

autism. The rationales given by the So were sorted into five

categories which are summarized below. Explanations were sorted by two
judges and an overall interjudge agreement of .93 was noted. The first
of the five categories below (inversion reversibility) was not deemed rele-
vant to density conservation and was used only in relation to volume con-
servation explanations .

50

l. Inversion reversibility: § made reference to the fact that percep-
tual deformations performed by E can always be reversed. (e.g. . ”You could
always pour it back. ")

2. Reciprocity reversibility: _S_ made reference to the fact that changes
in certain dimensions of an object are compensated by changes in related
dimensions (e.g., "This one is now taller but it is also skinnier"). I-
dentity explanations (e.g. , "It's the same") also fall in this category.

3. Conceptual irrelevant: These are erquanations that, like categor- "
ies l and 2, are not based on simple perceptual features. Nonetheless,
they are irrelevant to why conservation actually obtains . For example ,
to say, ”It's the weight that makes it so,” is not relevant to either vol—

 

ume or density.

4. Perceptual Irrelevant: These are the explanations that typically
follow incorrect conservation answers. Reference is made to deceptive
perceptual features of the stimuli (e.g., "It's skinnier so it must take
up more Space ").

5. Don' t know: This category is self-explanatory.

Explanations falling in the first and second categories were assigned
a '1' and explanations falling in the remaining three categories were as-
signed a '0'. _S_s' total scores for explanations were added to their to-

tal answer scores for purposes of the preliminary analyses.
Results

General Ag: Trends
One way analyses of variance were performed on the dependent variable
data. The 3 ratios for each of the three proportionality schema estimates

appear in Table ll. Separate E ratios were calculated for each of the

 

Table Ill
Age Trend _1: Ratios for Proportionality (he Way Analyses of Variance

 

 

 

Criterion
Concept
Combined Ansuers Explanations
Density
“ma 19.90" 10.71* 16.31"
Consem
Solid Volume
*
cons u ‘1 16.99 * 15.52“ 10.50"
erva on
Liquid Volume * *
tron‘ 8.89 11.56 2A0
Conserve

 

 

‘9; = 1/69 for all cells

‘2 4%
"2 <o°°°5

 

52

three possible 'conservation criteria' (combined, answers only, explana-

tions only). In general and by all criteria, the age differences in den-

sity and volume conservations were large and highly significant. For both

density and solid volume, the _1: ratio was larger for the combined criteri-

on than were the separate _F_'s for answers and explanations: this suggests

superior reliability of the combined criterion. This generalization does

not hold for liquid volume , because explanations failed to discriminate

the age-groups. '
The age trends for density and volume conservations reported in Ta-

ble 11 also were subjected to orthogonal polynomials analysis to determine

whether or not there were any nonlinear features about” these trends. The

 

results of these analyses appear in Tables 12, 13, and 14 for density,
solid volume, and liquid volume reapectively. As was the case in experi-
ment I, it was the linear terms of the orthogonal polynomials analyses
that were the most highly significant and accounted for most of the vari-
ance. In addition, small but significant quadratic trends were noted for
both density and solid volume conservations. In both instances, however,
the quadratic trend was limited to eflanations and failed to show up for
answers. For liquid volume conservation, complete linearity was the rule
with all quadratic components being smaller than one .

The proportion of conservation explanations falling in each of the
previously mentioned categories is reported by age-group and by conserva-
tion in Table 15. In line with the significant _F; ratios of Table ll, it
is apparent that §_s' conceptual justifications for density and solid vol-
ume conservations improved steadily with age. Alternatively, it also is
apparent that the quality of §_s ' rationales for liquid volume conserva-
tion do not vary appreciably between age-groups studied in the present
experiment.

Orthogonal Polynulials Analyses for Density Conservatim

53

Table 12

 

 

 

 

 

Source 33 g; as _1: 2
Combined Criterion
theen 133.86 2
Linear 117.18 1 117.18 37.81» (.0005 ‘I
Quadratic 16.68 1 16.68 4.96 (.05
Error 232.09 69 3.36
Total. 365.95 71
Answers
Between 38.68 2
Linear 35.01 1 35.00 19.38 (.0005
Quadratic 3.67 1 3.67 2.03
Error 124. 64 69 1 . 81
Total 163.32 71
Explanations
Btween 28. 78 2
Linear 28.10 1 24.10 27.31» (.0005
Quadratic 4.68 1 8.68 5.19 (.05
Error 60. 88 69 . 88
Total 89.66 71

 

5“

Table 13
Orthogonal Polymials malyees for Solid Volume Conservation

 

 

 

 

Source 33 gr; us 1; 2

Cabined Criterion

Btween 35.58 2
Linear 33- 35 1 33-35 32-92 (.0005
Quadratic 2.23 1 2.23 2.21

Error 69. 92 69 1. 01

Total 105. 50 71

Answers

Between 20.86 2
Linear 20.00 1 20.00 29.77 (. 0005
Quadratic .86 1 . 86 1.27

Error 46,42 69 . 67

Total 67. 28 71

Explanations

Between 3. 36 2
Linear 2.72 1 2.72 17.00 (.0005
Quadratic .6h 1 .68 8.01 (.05

Error 11.08 69 .16

Total 18.114 71

 

 

55

Table 1“

Orthogonal Polynmials Analyses for Liquid Volume Conservation

 

 

 

 

Source ss 9; us 1: p_

Cabined Criterion

Between 88.11 2
Linear 85.33 1 85.33 17. 95 (.0005
Quadratic 2.78 1 2. 78 < 1

Error 341.84 69 4. 95

Total 429.95 71

Answers

Between 52. 78 2
Linear 52.12 1 52.12 22.83 (. 0005
Quadratic .66 l .66 < 1

Error 157. 54 69 2.28

Total. 210. 32 71

Explanations

Etween 1+.08 2
Linear 3.53 1 3.53 4.14 <-05
Quadratic .55 1 . 55 < 1

Error 58. 79 69 . 85

Total 62.87 71

 

 

56

Table 15

Percent of Explanations Falling in each Category

 

Explanation Cate gory

 

 

 

 

Grade and
Concept 1 2 3 1+ 5
mnSity e a 0 [+606 5701'" 12.0 26.0
Third
Solid 0.0 13.8 51.7 20.7 13.8
Graders
Liquid 2!+.l 35.6 0.4 35.9 0.0
mnSj-ty e e a 7.8 "401'" 29.6 18.2
Sixth
Solid 10.0 13.3 146. 7 16.7 13.3
Graders
Liquid 23.1 42.3 9.6 25.0 0.0
Density 0 a a 3307 5508 608 "'02
Ninth
Solid 28.0 28.0 . 8. .
Graders 35 7 3 o 0
Liquid 16.7 50.3 22.9 2.1 0.0

 

 

57

The simple and multiple coefficients of correlation between the three
dependent variables of the present experiment are given in Table 16. A11
multiple coefficients are significant and moderate to moderately high por-
tions of the variance are accounted for by these values. Table 16 suggests
that it is true-mas Inhelder and Piaget (1958) contend--that the conserva-
tions of density and volume tap the same or related cognitive skills .
Invariant Sequences

For purposes of evaluating Inhelder and Piaget's (1958) contention
that volume conservation invariably precedes density conservatim, each
of the 72 _S_s was classified as either a 'conserver' or a 'nonconserver'
of density, solid volume, and liquid volume. This judgnent was based en-
tirely on _S_s ' answers to conservation questions . The criterion for:‘=.inclu-
sion in each 'conserver' group was a perfect sequence of answers (agreements-
disagreements) made in relation to the particular conservation under consi-
deration. Thus , liquid volume conservers had given five consequtive correct
agreements-disagreements , density conservers had given four consequtive
correct agreements-disagreements, and solid volume conservers had given
three consequtive correct agreements-disagreements.

These data were placed in four-fold contingency tables for Sigrifi-
cance testing (Tables 17 and 18). As McMenis (1969) has pointed out, the
upper right-hand and lower left-hand cells of such four-fold tables are
the crucial cells pertaining to the question of invariant sequence. For
the assumption of invariant sequence to be supported, the number of _S_s
appearing in the lower left-hand cell should be significantly larger than
the number of _S_s appearing in the upper right-hand cell. Theoretically,
£12 _S_s should appear in the upper right-hand cell if the invariant sequence
assumption is veridical. However, such a result would be expected only

 

58

Table 16

Simple and Multiple Correlations Between the Three Dependent Variables

 

Conservation Concept

 

 

 

 

 

Grade and
Concept Density Solid Liquid Multiple
DOnS‘lty e e e .1102‘ 01-18 OMB.
Solid .402" . . . .810” 512”"
Graders
Liquid .118 .410” .. . .412"
Density . . . .481""‘ . 353* . 596"“
Solid .1481’" 001» 513*”
Graders . ,t O . *
Liqu-id e 353 cow 0 e e Ono]-
Density . . . . 352‘ . 530“" . 596""
Ninth 352.: 375* 513*,"
Graders «an a . . use
Liam . 53“ . 375 .579
*‘p (.05
"p <. 025
an! <,01

at“? <, 005

59

Table 17
Relationship Between the Conse rvations of Density and Solid Volume

 

 

 

 

Solid Volume
Density 3*
Present Absent
*
Present 25 2
Absent 16* 29

 

*p = b.87 x 10'“

60

Table 18
mlationship Between the Conservations of Density and Liquid Volume

 

 

 

 

 

Liquid Volume
Density
Present Absent
a
Present 26 2
Absent 26* 18

 

*2 = Lid 1: 10"6

61

in the event that both measures were perfectly or near-perfectly reliable.
Given the typically ave rage to moderate reliabilities of conservation prob-
lems (Almy, Chittenden, 8. Miller, 1966), such a 'textbook' result was not
anticipated.
Siegel's (1956) binomial test was used to test the sigrificance of
the differences between the upper right-hand and lower left-hand cells
of Tables 17 and 18. The significance level was quite high for both den-
sity vs. solid volume and density vs. liquid volume. In light of the pre- '
viously mentioned fact that the reliabilities of conservation problems
frequently are marginal, Tables 17 and 18 would seem to provide strong

support indeed for a volume-density invariant sequence .

 

Ag: M in Conservation Learning

Here we shall consider only the density conservation data of the third and
sixth grade _S_s. The question that is of interest is whether or not there
was a difference in the extent to which S5 of different ages 'learned'
the density conservation concept in the course of the assessment procedure.
A ceiling effect for ninth grade _S_s (i.e., almost perfect agreement-
disagreement performance) restricted the investigation of this question
to third and sixth grade _S_s.

It will be remembered that in the final step of the density conser-
vation assessment 5 posed the following question to _S_, "Do you think we
could ever get a piece of this clay small enough so that it would float?"
Since E had shown §_ that the clay did not float in the preceding three
steps of the assessment, it was thought that the extent to which §_s an-
swered 'no' to this final question represented some estimate of how much
they had been influenced by the preceding steps. Hence, third and sixth

grade §_s were classified according to two criteria. First, they were

62

divided into those gs who made at least one error during the first three
steps and those who had performed perfectly during the first three steps.
Second, the former or critical group was divided into those §_s who had
answered 'no' to the final question and those who had answered 'yes'.

These data appear in Table 19. It is apparent from the significant
chi-square value that sixth graders who made one or more errors during
the first three steps were more likely to '1earn' by their experiences
than the third graders who also made one or more errors during the first
three steps.

Discussion

Returning to the first of the five questions posed as a prelude to
experiment II, the data indicate that _S_s' performances in relation to the
three indices of the proportionality schema do improve linearly between
the ages of eight to 15. This prediction of Piagetian theory is supported
by the large and highly sigrificant _1: ratios of Table 11 for both conser-
vation answers and explanations .

Second, the simple and multiple correlation coefficients obtaining
between density, solid volume , and liquid volume generally were signifi-
cant and, therefore, suggest that these variables are, to some extent,
all estimates of the same cognitive skill. This seems to be a sigrifi-
cant result in that it represents the first clear empirical support for
Inhelder and Piaget's (1958) assertion that density and volume both are
conserved as a result of a single cognitive developnent, viz. the child's
acquisition of the formal-ope rational "proportionality schema. " Although
Inhelder and Piaget present data from different groups of _S_s which suggest

that the ontogenies of density and volume concepts are roughly parallel ,

 

 

63

Table 19
Age Differences in the learning of Density Conservation

 

_S_s flaking some Density Conservation Errors

 

 

 

 

 

_S_s Answering 'yes' §_s Answering 'no'
Age-Group to to

the Final Question the Final Question
Third Graders‘ 9 10
Sixth Graders“ 15 0

 

hf :- 8.79: 2 (.005

64

such evidence is hardly an adequate substitute for the intercorrelation-
a1 analyses that are necessary to establish that these measures are as-
sociated reliably w S_s. Thus, to a significant degree, the three
dependent variables seem to measure the same underlying ability; whether
or not this 'ability' is an incipient "proportionality schema" is, of
course, another matter.

The argument that an adequate concept of volume invariably precedes
an adquate concept of density also receives strong support from the data
of the present experiment. This result is important from the general per-
spective of Piagetian theory in that it represents a point in favor of

'logicsl' methods of predicting concept acquisition. Prima facie, volume

 

seems to be a more complex concept than density. While gs £2113 conserve
density merely by thinking of it as a simple invariant characteristic of
each substance, volume remains a property that is not observed or measured
easily. The logic of physics, on the other hand. dictates that density
is a more complex concept than volume, since density is defined as weight-
per—unit-volume. In so far as the Genevan school typically employs such
‘1ogical' approaches, the data of the present experiment might well be
construed as support for a general method of making predictions as well

as for a specific prediction.

There were some nonlinear features about the age trends for density
and solid volume comservations. In general, however, the linear components
were overwhelmingly larger than the quadratic components. In the two in-
stances where quadratic components were found to be significant, the sig-
nificance level was not great (p (.05) and the corresponding linear com-
ponents were four to five times as large as the quadratic terms. In ad-

dition, the fact that significant quadratic terms were noted only in

65

relation to conservation eglanations but not conservation answers suggests
that these nonlinear results do not merit more general consideration.

The significance of the fact that older _S_s tended to 'learn' density
conservation more readily than did younger _S_s also should be considered
briefly. As previously noted, Piagetian theory predicts and developent-
a1 psychologists gene rally have believed that the extent to which conser-
vation concepts are trainable is partly a function of the age of §_. Given
a particular conservation concept and a particular training technique ,
then, the prediction is that older _S_s should be afle’cted more by their ex-
periences than younger §_s. Pinard and Larendeau (1969) argue that‘ this
prediction follows directly from the fact that among nonconservers of a
particular concept, older _S_s are more likely to possess already m‘opera-
tional basis for the concept and, therefore , are nearer to acquiring the
conservation ’naturallly' than are younger §_s (i.e., the closer §_ is to
'natural' acquisition, the more he or she presumably profits from the learn-
ing experience).

As Brainerd and Allen (1970) have pointed out, however, this predic-
tion has not been supported as yet by studies of concrete conservation
training. Some of the least successful conservation training experiments
(e.g., Wohlwill 8: Lowe, 1962: Smedslund, 1963b) involved _S_s that were old-
er than the §s used in some of the most successful of these experiments
(e.g., Gelman, 1969: Wallach 8: Sprott, 1961+). This result is not deemed
to be particularly damaging to the preceding hypothesis by the present
author, because of the relatively narrow age ranges used in previous con-
se rvation training experiments . Since preschoolers typically are more
difficult to obtain than elementary schoolers and since a large number

of second graders have acquired one or more of the concrete conse rvations ,

 

 

66

kindergarten and first grade §_s have been employed almost exclusively.
Given this constricted age-range (one year), perhaps it is not surprising
that age has not been shown to be associated with amount of conservation
learning.
Alta rnatively, the data of the present experiment clearly support
the presumed association between age and conservation learning. This re-
sult is not seen as descrepant with the previously mentioned lack of sup- '
port from earlier conservation training experiments. Rather, it is like- .
1y that this result can be attributed to the fact that the age difference
between the groups of Table 19 is three times greater than has been the
rule-for similar emeriments.

 

Finally, when taken together the data of Table 11, 12, 13, and 14
suggest that the 'controversy' concerning whether or not conservation
explanations should be included as necessary criteria to be considered
when judging gs to be conservers or nonconservers may well be a pseudocon-
troversy. In so far as conservation answers and explanations manifest
similar age trends, there seems to be no good reason for not combining
those two criteria to yield a single criterion. This combined criterion
is characterized by both higher reliability and susceptibility to para-
metric analyses. Such a combined criterion eliminates all-or-none conser-

vation judgments with their large built-in random error.

 

~ .
,

I‘

r r

,
.
r
r

c
a

 

GENERAL DISCUSSICN

In this section it remains to consider the within _S_s relations between
the results of the two experiments. As indicated earlier, the same §_s
were employed in both experiments so that implication-reasoning and the
three indices of the "proportionality schema" could be compared within
_S_s. It will be remembered that on the basis of Piagetian theory one would
predict that implication-reasoning ability should be associated positive-
ly with each of the three indices of the "proportionality schema." To
reiterate, Piaget argues (e.g., 191+9, 1953: Inhalder & Piaget, 1958) that
the "sixteen propositional operations" (of which implication is one) and
the "formal-operational schemata" (of which proportionality is one) are
alternative manifestations of the same underlying cognitive structure,
tie group INRC. If two measurable cognitive skills are manifestations of
the same incipient cognitive structure, it certainly seems legitimate to
expect that estimates of two such cognitive skills should be related sig-
nificantly related within _S_s.

To ascertain the strength of the relation between implication—
reasoning ability and the proportionality schema indices, the scores of
individual §s on these variables were correlated within each age-group.
Because the semantic treatment of experiment I affected implication-
reasoning ability significantly, the total scores of all the §s in the
-8 condition were corrected so that their scores did not differ signifi-
cantly from the other 12§s in their particular age-group. To accomplish
this, each -S score was transformed linearly by adding to it the differ-
ence between the means of conditions +8 and -S for the particular age-group

6?

68

that it was from. These corrected implication-reasoning values were cor-
related separately with density, solid volume , and liquid volume scores

and also with a summed proportionality schema estimate based on gs' total
scores on all three variables of experiment II. The 12 coefficients appear
in Table 20.

It is obvious from Table 20 that the predicted intrasubject associa-
tion between the dependent variables of experiments I and II did not ma-
terialize. While the predicted correlations would be positive and signi-
ficant, the obtained values range from negative and insignificant through
zero to positive and insignificant. Clearly, if Piaget's assertions are
correct, we must wait for future date to bear them out.

Of course, the inferences that can be made based on Table 20 are
limited. ASide from the obvious problem of arguing for the null hypothe-
sis , there is always the limitation of reliability to consider. As Alni
et al. (1966) have pointed out, the respective reliabilities of Piaget's
conservation problems are not overpowering. If the 'true' relation be-
tween implication-reasoning and the proportionality schema estimates is
a small to mode rate one , the reliability factor may be reSpons ible for
the absence of positive and significant validity correlations . However,
the relatively large _F_' ratios obtained in relation to the age trends for
density and volume conse rvations , as well as the gene rally significant
correlations of these variables with each other, lead one to suspect that
the reliabilities of the proportimality schema estimates are quite ade-
quate . Since the implication-reasoning reliability estimates are high
(Table 8), reliability should be a prohibitive factor only in the event
that the relation between implication-reasoning and the proportionality
schema variables is indeed a precise one.

 

 

 

‘69

Table 20

Correlations of Implication Answers with Proportionality Variables

 

Proportionality Schema Variables

 

 

 

 

A89-
Greup‘ Density Solid Liquid Conbined
Third

”0230 .12“ -0102 -.1’+9
Grade rs
Sixth

“OW -I037 "0115 “.124
Graders
Ninth

-0166 ”.302 '0035 -0139
Grade rs

 

8N = 2.4 for all cells

70

Even if it is true--as the data suggest--that there is no significant
relation between the dependent variables of experiments I and II, this fact
does only limited damage to the Piagetian assumption that the "proposition-
al operations" and the "formal-operational schemata" are manifestations
of the same structural reality, the group INRC. The present experiments
were focused on only one propositional operation and one formal-Operational
schema. Fifteen propositional operations and an indeterminant number of
formal-operational schemata remain to be studied and interrelated. Unless
similar data are forthcoming from studies of these other variables, the
importance of the present data vis>a-vis the assumption of a single under-
lying determinant of the two categories of formalhoperations will remain

limited.
Conclusions and Implications

In general, it can be said that the predictions pertaining to each
experiment were wellsupported. In experiment I, age of §, semantic con-
tent of propositions, and syntactic order of propositions each was shown
to have some influence on the dependent variables. In addition, a first
order interaction between age and syntax suggested that--as Piagetian
theory predicts--an effective order manipulation may have a greater influ-
ence on some age-groups than on others. The data of experiment II con-
firmed the major age trend and invariant sequence predictions offered by
the Genevans in relation to the proportionality schema variables.

0n the other hand, the Mimeriment prediction concerned with the
presumed relation between the dependent variables of the two experiments
clearly was not supported. This result highlights one of the dangers of

the clinical method employed by Piaget and his collaborators. Conclusions

71

based on probabilistic inference frequently are called for when such a
method is employed. For example, members of the Genevan school typical-
ly consider empirical evidence indicating that two or more aspects of cog-
nitive development are characterized by similar age trends to be a suffi-
cient basis for inferring that the cognitive variables are manifestations
of the same underlying structure. This technique has been used by Inhelder
and Piaget (1958) to bridge the gap between the parallel developnental
trends they identified for the ”propositional operations" and the "formal-
operational sonnets.” However, the data of Table 20 do not support their
inference.

This conflict is particularly unfortunate because it probably is un-
necessary. When one wishes to establish an unequivocal relation between
variables displaying similar developmental trends, there is but a single
additional step involved in appropriately quantifying the variables for
correlation. If Inhelder and Piaget had “plated the analysis of their
data in this matter, the status of the results in Table 20 would be clear-
er. The implications of these considerations, of course, are that probab-
ilistic inference and statistical probability are two different (though
related) things, and the latter is always preferrable to the former where
choice is possible .

Some final mention should be made of the significance of the age trends
in implication-reasoning reSponse latencies noted in erqaeriment I. It
will be remembered that the analysis of variance revealed that response
latencies decreased linearly between the ages of eight to 15. The surpris-
ing feature of this result is that it is clearly at odds with the devel-
omental trends identified by Kagan and his associates (e.g., Kagan, Moss,
3. Sigel, 1963) in relation to their reflectivity-impulsivity dimension

 

 

72

of cognitive style . Kagan' s reflectivity-impulsivity results , like those
appearing in Tables 5‘ and 7 of experiment I, are based on response laten-
cy measures. Contrary to the results of experiment I, however, Kagan finds
that reaponse latencies increase as a function of age. This trend is thought
to be a function of /a generalized tendency to ”reflect" on all matters as
one ages (a little generation gap here). Fortunately, there are at least
two differences between Kagan's studies and experiment I of the present
report that may be reSponsible for these inverse age trends.

First and perhaps least important, Kagan and his associates typical-
ly have studied _S_s that were much younger than those employed in the pre-
sent research. While _S_s from up to and including the first year of high

 

school were investigated in experiment I, the reflectivity-impulsivity -
age trends tend to be based on early elementary school _S_s . Although much
older _S_s were used to study implication-reasoning, the present author finds
it difficult to believe that a single cognitive variable characterized by
a clear and consistent trend within one age-range would be characterized
by the inverse trend within a slightly older age-range.

Second and seemingly more important, there are marked differences
between the cognitive requirements of the implication-reasoning problems
and the cognitive requirements of the problems employed in studies of
reflectivitybimpulsivity. The emphasis of the implication-reasoning prob-
lems clearly was on deductive Species of reasoning. Alternatively, both
Siegelman (1969) and Ward (1968) suggest that the emphasis of reflectivity-
impulsivity studies ison perceptual—inductive reasoning. Reflectivity-
impuls ivity studies usually involve some variety of the matching-to-s ample
technique (e.g., the Matching Familiar Figures Test).

It does not seem unreasonable to conclude that latency intervals

73

characteristic of matching-to-s ample tasks are largely ' search times' --
i.e. , intervals during which §_ overtly scans a perceptually present array.
01 the other hand, no additional external information is required as part
of the implication-reasoning problems and the latencies characteristic
of such problems should be largely 'think times'-i.e., intervals during
which _S_ covertly combines and recombines given propositions. In so far
as these requirements involve different or even reciprocal cognitive pro-
cesses , it perhaps is not farfetched to expect nonequivalent developnent-
al trends. At the very least, the latency data of experiment I suggest
that the reflectivity-imprfls ivity dimension probably is less general and
more task-dependent than previously has been acknowledged. (e.g. , Kagan.
Rosman, Day. Albert, 8. Phillips, 196+).

 

 

 

 

 

 

 

LIST OF REFERENCES

 

LIST OF murmurs

Almy. H;-.?-Chittenden, E., 8: Miller, P. _I_o_y_ng' children's M. New York:
Columbia University Teacher's College Press, 1966.

Brainerd, C. J. Continuity and discontinuity hypotheses in studies of con-
servation. Develgwntal Psycholog, 1970, _2_, in press.

Brainerd, C. J ., 8: Allen, T. U. Experimental inductions of the conserva-
tion of "firs t-order" quantitative invariants . Pszghological____ Bulle-
t__in, 1970,.715 in press.

Bruner, J. The course of cognitive growth. American Psychologist, 1964,
18.1.15.

De Vries, R. Constancy of generic identity in the years three to six.
‘ Monographs__ of the Societ f2}; Research in Child pavelopment, 1969,
3,“!1919 NOT-127 e

Elkind, D. Children's discovery of the conservation of mass, weight, and
volume: Piaget replication study II. J ournal 93 Genetic PsEholog,
1961, 28, 219-227.

Festinger, L. g theogz 21; cognitive dissonance. New York: Harper 8: Row,
1957..

Flavell, J. H. The developmental 5 hole of Je__a_n_ Piat.g_e_ .Princeton,
New Jerseyxi V. Nostrand, 1963.

Griffiths, J. A., Shantz, C. A., 8: Sigel, I. E. A methodological problem
in conservation studies: the use of relational terms . Child Develop-
ment, 1967, 2_8_, 1229-12146.

Gruen, G. Note on conservation: methodological and definitional considera-
tions . Child pevelopment, 1966, 22, 977-983.

Inhelder, B. Le raisonnement experimental de 1' adolescent. Proceedings
93 _t_h_e_ 12th International Congress of; Psmhology.

Inhelder, B., 8: Piaget, J. _T_h_e_ growth__ of logical thinking___ from childhood
to adolescence. New York: Basic Books; 1958.

Inhelder, B., 8: Piaget, J. _Thg early growth 2.1:. logic in the child. London:
Boutiedge 8: Kegan Paul , 1961}.

74

 

75

Kagan, J .. Moss, E. 1., 8: Sigel I. E. Psychological significance of styles
of conceptualization. In J. C. Wright 8.- J. Kagan (Eds. ) , BaBic cog-
nitive processes in children. Mono r hs p_f_ the Societ for Ibsearch
3; Child Development, 1963, _gg—(E'Efil’, e Nan-8'6), pp. 3:12..

Kagan, Jo. m, Be Le, Day, De. Albert. Jo. & P111111”, We Momtion
processing in the child: significance of analytic and reflective
attitudes . Psychological Monographs , 196+, 28 (1, whole No. 578).

McHanis , D. L. Conservation and transitivity of might and length by norm-
als and retardates. Develgmental Psygholog, 1969, 1, 373-382.

Meheler, J ., 8: Dover, T. G. Cognitive capacity in very young children. }
Science , 1967, 1.2, 1141-142.

Murray, J. P. , & Youniss , J. Achievement of inferential transitivity and
itsésrelation to serial. order. Child Develmnt, 1968, 22, 1259-
12 .

Parsons, C. Inhelder and Piaget's & growth 93 logical thinkin : II.
a logiciarb viewpoint. British Journal 9; Pszgholog, l9 0, 2);, 75-

 

Piaget, J. Classes, relations, 93 nunbres: essai §_u_r; .133 ”groments"
iq .1_a- logistgue _e_t_ _la reversibilite 9 la; pgnsee. Paris: Vrin, 191+2.

Piaget, J. Traits/93 logigue. Paris: Colin, 1949.

Piaget. J. Logic and psycholog. Manchester: University of Manchester
Press, 19530

Piaget, J. 9; the; develgment pf memogz g identit . Barre, Mass: Clark
Univeristy Press with Barre Publishers, 9 .

Piaget, J ., 8e Inhelder, B. _T_h_e_ psychologz 21; _th_g child. New York: Basic
Books, 1969.

Piaget, J ., Inhelder, B., 8. Sminska, A. 33.; child's conception 22 gems-
Me New YDrkI 38510 39135, 19600

Pinard, A. , 8: Laurendeau, M. "Stage" in Piaget's cognitive-developmental
theory: exegesis of a concept. In D. Elkind a J. H. Flavell (Bds.),
Studies _in M development. New York: Ozdord University Press,
1%9.

Rothenberg. B. B. Conservation of number among four and five year-old child-

ren: some methodological considerations . Child Develgmnt , 1969 ,
22,. 383-406. ""

Bothenberg, B. B., 8: Courtney, R. G. Conservation of number in very young
children. Developmental Psycholog, 1969, 1, 493-502.

Siegel , S. _N_o_nparametric statistics £31; 3.3 behavioral sciences . New York:
McGraw-Hill, 1933.

 

76

Siegelman, E. Reflective and impulsive observing behavior. Child Devel-
M, 1969, 49, 1213-1222. ‘

Smedslund , J . The acquisition of conservation of substance and weight in
children. II. External reinforcement of conservation of weight and
of the operations of addition and subtraction. Scandanavian Journal

of Psycholog, 1961, 2, 71-84. (a)

Smedslund, J. The acquisition of conservation of substance and weight in
children.V. Practice in conflict situations without reinforcement.
Scandanavian Journal of PsychologI 1961, _2_, 156-160. (b)

Smedslund, J. Development of concrete transitivity of length in children. '
Child Development, 1963, 24, 389-405. (a) .

medslund , J . Patterns of experience and the acquisition of conservation
of length. Scandanavian Journal of Psygholog, 1963, 4, 257-264. (b)

Trabasso, T. Pay attention. PsEholog Today, 1968, _2 (5), 30-36.

Wallach, L. , a Sprott, R. L. Inducing number conservation in children.
Child Development, 1964, 35, 1057-1071.

 

Ward, W. C. Reflection-impulsivity in kindergarten children. Child Davel-
mt,1968,29, 867-874.

Wohlwill, J. F., 8: Lowe, R. C. An emperimental analysis of the development
of the conservation of number. Child Develomgnt, 1962 , 33, 153-167.

APPENDICES

 

APPENDIX I
ASSESSMENT QUESTIONS FOR THE POSITIVE SEHANTICS CONDITION

1. If Jack (Jill) sweeps the kitchen floor, then Jack's (Jill's) mo-
ther will be very pleased. Whenever Jack's (Jill's) mother is very pleased,
he (she) gets all the dessert he (she) wants.

a. If Jack (Jill) sweeps the kitchen floor, then what else will occur?
Anything else? Could it be that Jack (Jill) won't get all the dessert

that he (she) wants?

 

b. If Jack (Jill) sweeps the kitchen floor, then could it be that Jack
(Jill) 122...}. get all the dessert that he (she) wants?

c. If Jack (Jill) gets all the dessert he (she) wants sometime, could it
be that Jack (Jill) didn't sweep the kitchen floor?

d. If Jack (Jill) doesn't sweep the kitchen floor, then what else will
occur? Anything else? Could it be that Jack (Jill) could still get all
the dessert that he (she) wants?

2. If Jack (Jill) washes the dishes, then Jack's (Jill's) father will
be very pleased. Whenever Jack's (Jill's) father is very pleased, Jack
(Jill) gets 50¢.

a. If Jack (Jill) washes the dishes, then what else will occur? Anything

else? Could it be that Jack (Jill) won't get 50¢.

b. If Jack (Jill) washes the dishes, then could it be that Jack (Jill)

3.99.1.2 set 50¢-

c. If Jack (Jill) gets 50¢ sometime, could it be that Jack (Jill) M
77

78

Mb the dishes?
d. If Jack (Jill) doesn't wash the dishes, then what else will occur?
Anything else? Could it be that Jack (Jill) could still get 50¢.

3. If Jack (Jill) gets all 'A's' on his (her) report card, then both
of Jack's (Jill's) parents will be very pleased. Whenever both of Jack's
(Jill's) parents are very pleased, Jack (Jill) gets to stay up later as
a reward.

a. If Jack (Jill) gets all 'A's' on his (her) report card, then what else
will occur? Anything else? Could it be that Jack (Jill) won't get to
stay up later?

b. If Jack (Jill) gets all 'A's' on his (her) report card, then could it
be that Jack (Jill) 3.9.9.3.? get to stay up later as a reward?

c. If Jack (Jill) gets to stay up later as a reward sometime, could it

be that Jack (Jill) ;__didn_'_t_ get all 'A's' on his (her) report card?

d. If Jack (Jill) doesn't get all 'A's' on his (her) report card, then
what else will occur? Anything else? Could it be that Jack (Jill) could
still stay up later?

4. If Jack (Jill) does well on an English test, then Jack's (Jill's)
teacherwill be very pleased. Whenever Jack's (Jill's) teacher is very
pleased, Jack (Jill) gets less schoolwork to do.

a. If Jack (Jill) does well on an English test, then what else will occur?
Anything else? Could it be that Jack (Jill) won't get less schoolwork?

b. If Jack (Jill) does well on an English test, then could it be that Jack
(Jill) w get less schoolwork to do?

c. If Jack (Jill) gets less schoolwork to do sometime, could it be that
Jack (Jill) £13113 do well on an English test?

d. If Jack (Jill) doesn't do well on an English test, then what else will

 

 

79

occur? Anything else? Could it be that Jack (Jill) could still get less
schoolwork to do?

5. If Jack (Jill) news the lawn, then Jack (Jill) works very hard
at it. Whenever Jack (Jill) works very hard at something, Jack's (Jill's)
father treats him (her) to an icecream cone.
a. If Jack (Jill) news the lawn, then what else will occur? Anything else?
Could it be that Jack (Jill) won't get an ice cream cone? ‘ 1
b. If Jack (Jill) mews the lawn, then could it be that Jack's (Jill's)
father wo__n__'t treat him (her) to an icecream cone?
c. If Jack's (Jill's) father treats him (her) to an icecream cone sometime,
could it be that Jack (Jill) M mow the lawn?

 

d. If Jack (Jill) doesn't mow the lawn, then what else will occur? Anything
else? Could it be that Jack (Jill) will still get an icecream cone?

6. If Jack (Jill) has a birthday, then all of Jack's (Jill's) rela-
tives come to see him (her). Whenever all of Jack's (Jill's) relatives
come to see him (her), Jack (Jill) gets a lot of money.
a. If Jack (Jill) has a birthday, then what else will occur? Anything
else? Could it be that Jack (Jill) won't get a lot of money?
b. If Jack (Jill) has a birthday, then could it be that Jack (Jill) 3.9.9.13
get a lot of money?
c. If Jack (Jill) gets a lot of money sometime, could it be that Jack (Jill)
didn't have a birthday?
(1. If Jack (Jill) doesn't have a birthday, then what else will occur?
Anything else? Could it be that he (she) could still get a lot of money?

7. If Jack (Jill) helps his (her) mother with the shopping, then Jack' 5
(Jill's) mother does not have much work to do. Whenever Jack's (Jill's)

mother does not have much work to do, she fixes Jack's (Jill's) favorite

80

food for dinner.

a. If Jack (Jill) helps his (her) mother with the shopping, then what else
will occur? Anything else? Could it be that Jack (Jill) won't get his
(her) favorite food?

b. If Jack (Jill) helps his (her) mother with the shopping, then could it
be that Jack's (Jill's) mother 222:3 fix Jack's (Jill's) favorite food?

c. If Jack's (Jill's) mother fixes his (her) favorite food for dinner some-
time, could it be that Jack (Jill) 93.13;; help mother with the shopping?
d. If Jack (Jill).doesn't help mother with the shopping, then what else
will occur? Anything else? Could it be that Jack (Jill) could still get
his (her) favorite food for dinner?

 

8. If Jack (Jill) plays a part in a school play, then he (she) will
do a very good job. Whenever Jack (Jill) does a very good job at some-
thing, Jack (Jill) gets an award from the school.

a. If Jack (Jill) plays a part in a school play, then what else will occur?
Anything else? Could it be that Jack (Jill) won't get an award?

b. If Jack (Jill) plays a part in a school play, then could it be that
Jack (Jill) 323113 get an award from the school?

c. If Jack (Jill) gets an award from the school sometime, could it be that
Jack (Jill) 51319;; play a part in a school play?

d. If Jack (Jill) doesn't play a part in a school play, then what else

will occur? Anything else? Could it be that Jack (Jill) could still get

an award from the school?

APPENDIX 11
ASSESSENT QUESTICWS FOR THE NEGATIVE SEHANTICS CGDITICN

1. If Jack (Jill) throws a rock at a window, then the window will.
break. Whenever Jack (Jill) breaks a window, Jack's (Jill's) mother sends
him (her) to bed without supper.

a. If Jack (Jill) throws a rock at a window, then what else will occur?
Anything else? Could it be that Jack (Jill) won't be sent to bed without

supper?
b. If Jack (Jill) throws a rock at a window, then could it be that Jack's

 

(Jill's) mother 3333 send him (her) to bed without supper?

c. If Jack's (Jill's) mother sends him (her) to bed without supper some-
time, could it be that Jack (am) 9399;; throw a rock at a window?

d. If Jack (Jill) doesn't throw a rock at a window, then what else will

occur? Anything else? Could it be that Jack (Jill) could still be sent
to bed without supper?

2. If Jack (Jill) complains about how bad things are, then Jack's
(Jill's) friends get mad. Whenever Jack's (Jill's) friends get mad, they
won't talk to Jack (Jill) anymore.

a. If Jack (Jill) complains about how bad things are, then what else will
occur? Anything else? Could it be that Jack (Jill) will still be spoken
to by his (her) friends?
b. If Jack (Jill) complains aboutlhow bad things are, then could it be
that Jack's (Jill's) friends will still talk to him (her)?

81

 

82

c. If Jack's (Jill's) friends won't talk to him (her) sometime, could it
be that Jack (Jill) _d_di.d__n_'_t complain about how bad things are?

d. If Jack (Jill) doesn't complain about how bad things are, then what
else will occur? Anything else? Could it be that Jack's (Jill's) friends
could still not talk to him (her)?

3. If Jack (Jill) plays with matches, then he (she) will start a fire
in his (her) house. Whenever Jack (Jill) starts a fire in his (her) house,
Jack's (Jill's) father takes away his (her) allowance.

a. If Jack (Jill) plays with matches:;, then what else will occur? Anything
else? Could it be that Jack (Jill) could still have his (her) allowance?
b. If Jack (Jill) plays with matches, then could it be that Jack's (Jill's)

 

father m take away Jack's (Jill's) allowance?

c. If Jack's (Jill's) father takes away his (her) allowance sanetime, could
it be that Jack (Jill) M play with matches:

d. If Jack (Jill) doesn't play with matches, then what else will occur?
Anything else? Could it be that Jack's (Jill's) father could take away
Jack's (Jill's) allowance anyway?

’4. If Jack (Jill) leaves his (her) bicycle in the driveway, then Jack' 3
(Jill's) father will hit the bike with his car. Whenever Jack's (Jill's)
father hits Jack's (Jill's) bike with his car, he takes the bike away fran
Jack (Jill).

a. If Jack (Jill) leaves his (her) bicycle in the driveway, then what else
will occur? Anything else? Could it be that Jack (Jill) could still have
his (her) bike?

b. If Jack (Jill) leaves his (her) bike in the driveway, then could it be
that Jack's (Jill's) father 3211:}; take the bike away from Jack (Jill)?

c. If Jack's (Jill's) father takes away Jack's (Jill's) bike sometime,

83

could it be that Jack (Jill) M leaVe the bike in the driveway?

d. If Jack-.(Jill.) doesn't leave- his (her) bicycle in the driveway, then
what else will occur? Anything else? Could it be that Jack's (Jill's)
father could still take the bike away?

5. If Jack (Jill) forgets something he (she) is supposed to do, then
Jack's (Jill's) father tells him (her) what to do. Whenever Jack's (Jill's)
father tells him (her) what to do, Jack (Jill) feels ashamed.

a. If Jack (Jill) forgets something he (she) is supposed to do, then what
else will occur? Anything else? Could it be that Jack (Jill) won't feel
ashamed?

b. If Jack (Jill) forgets something something he (she) is supposed to do.
then could it be the Jack (Jill) _w3_n_'_t feel ashamed?

c. If Jack (Jill) feels ashamed sometime, could it be that Jack (Jill)
9.39.9.2 forgetsmnething he (she) was supposed to do?

d. If Jack (Jill) doesn't -: forget something he (she) is supposed to do,
then what else will occur? Anything else? Could it be that ‘Jack (Jill)
could feel ashamed anyway?

6. If Jack (Jill) wi5pers in class, then Jack's (Jill's) teacher will
get": mad. Whenever Jack's (Jill's) teacher gets mad, Jack (Jill) has to
stay after school.

a. If Jack (Jill) wiSpers in class, then what else will occur? Anything
else? Could it be that Jack (Jill) won't have to stay after school?

b. If Jack (Jill) wispers in class, then could it be that Jack (Jill) Mt.
have to stay after school?

c. If Jack (Jill) has to stay after school sometime, could it be that Jack
(Jill) _c_l_i_c_l_n__'__t wisper in class?

d. If Jack (Jill) doesn't wisPer in class, then what else will occur?

84

Anything else? Could it be that Jack (Jill) could still stay after school?

7. If Jack (Jill) breaks one of his (her) mother's favorite dishes, then
Jack's (Jill's) mother will be unhappy. Whenever Jack's (Jill's) mother is
unhappy, Jack (Jill) cannot go outside and see his (her) friends.

a. If Jack (Jill) breaks one of his (her) mother's favorite dishes, then

what else will occur? Anything else? Could it be that Jack (Jill) could
still go outside and see his Gler) friends?

b. If Jack (Jill) breaks one of his (her) mother's favorite dishes, then
could it be that Jack (Jill) could still go outside and see his (her) friends?
c. If Jack (Jill) cannot go outside and see his (her) friends sometime,

coxfld it be that Jack (Jill) didn't break one of mother's favorite dishes?

 

d. If Jack (Jill) doesn't break one of mother's favorite dishes, then what
else will occur? Anything else? Could it be that Jack (Jill) still could
not go outside and see his (her) friends?

8. If Jack (Jill) doesn't come home right after school, then Jack's
(Jill's) parents worry about him (her). Whenever Jack's (Jill's) parents
worry about him (her), Jack (Jill) cannot watch television.

a. If Jack (Jill) doesn't come home right after school, then what else
vfill occur? Anything else? Could it be that Jack (Jill) could still
watch television?

b. If Jack (Jill) doesn't come home right after school, then could it be
that Jack (Jill) could still watch television?

c. If Jack (Jill) cannot watch television sometime, could it be that Jack
(Jill) did come home right after school?

d. If Jack (Jill) do_es come home right after school, then what else will
occur? Anything else? Could it be that Jack (Jill) could not watch tele-
vision still?

 

 

 

    

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