ED 110 464

DOCUMENT RESUME

93 TM 004 724

AUTHOR Klausmeier, Herbert J.; And OthersTITLE First Cross-Sectional Study of Attainment of the

Concepts "Equilateral Triangle", "Cutting Tool" and"Noun" by Children 5 to 16 of City A. Report from theProgram on Children's Learning and Development.Technical Report No. 287.

INSTITUTION Wisconsin Univ., Madison. Research and DevelopmentCenter for Cognitive Learning.

SPONS AGENCY National Inst. of Education (DHEW), Washington,D.C.

REPORT NO WRDCCL-TR-287PUB DATE Feb 74CONTRACT NE-C-00-3-0065NOTE 70p.; For rebated documents, see ED 103 481-485

EDRS PRICE MF-$0.76 HC -$3.3 PLUS POSTAGEDESCRIPTORS Abstraction Levels; hild Development; *Cognitive

Development; *Cognitive Measurement; *ConceptFormation; Conceptual Schemes; *Cross SectionalStudies; Elementary Secondary Education; *FundamentalConcepts; Hypothesis Testing; Knowledge Level;Language Learning Levels; Models; Prediction;Tests

IDENTIFIERS Cutting Tool; Equilateral Triangle; *Model ofConceptual Learning and Development; Noun

ABSTRACTTheory and research regarding four levels of concept

attainment and three uses of concepts as specified by the conceptuallearning and delitlopment (CLD) model are described. Assessmtnt,batteries were developed to assess each child's level of conceptattainment and alSo the related use of the concepts equilateraltriangle, cutting tool, and noun. Each of the three batteries wasdesigned as a paper-and-pencil task and administered to from 60 to100 children enrolled in each of four grades: kindergarten, third,sixth, and ninth. The three populations receiving the three batterieswere not independent; more than 80 percent of the children receivedall three batteries. Predictions based on the model about children'sconceptual development were strongly supported across all threeconcepts: (1) The three concepts were attained in an invariantsequence according to four successive levels: concrete, identity,classificatory, and formal. (2) As the concepts were attained tohigher levels they were used increasingly (a) ih cognizingsupraordinate-Subordinate relationships in a hierarchy where theattained concept was an element of the hierarchy; (b) inunderstanding principles that stated a relationship between theattained concept and one or more other concepts; and (c) in solvingproblems that required the use of the particular concept. (3) Havingthe labels of the concept and of its defining attributes facilitatedattainment of the concept, and mastery of the three uses of,theconcept. (Author)

Or4

Technical Report No. 2_87C=)

LL1 FIRST CROSS-SECTIONAL STUDY OF ATTAINMENT OFTHE CONCEPTS EQUILATERAL_ TRIANGLE , CUTTING TOOL AND NOUN

BY CHILDREN AGE 5 TO 1.6 OF CITY A

0

by

Herbert J. Klausmeier, Tl mas S. Sipple,and Patricia 5. /Allen

Report from the Program onChildren's Learning and Development

Herbert J. KlausmeierPrincipal Investigator

U S DEPARTMENT OF HEALTH,EOUCATION I WELFARENATIONAL INSTITUTE OF

EDUCATION

THIS DOCUMENT HAS BEEN REPRODuCE0 EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIGINATING IT POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRE,SENT OFFICIAL NATIONAL INSTITUTE OFEDUCATION POSITION OR POLICY

Wisconsin Research and DevelopmentCenter for Cognitive LearningThe University of Wisconsin

Madison, Wisconsin

February 1971

2/3

it

r:,

Published by the Wisconsin Research and Development Center for Cognitive Learning,

supported in part as a research and development center by funds from the National

Institute of Education, Department of Health, Education, and Welfare. The opinions

expressed herein do not necessarily reflect the position or policy of the National

Institute of Education and no official endorsement by that azency should be inferred.

Center Contract No. NE-C-00-3-0065

4

Statement of Focus

Individually Guided Education (IGE) is a new comprehensive system ofelementary education. The following components of the IGE system are invarying stages of development and implementation: a new organization forinstruction and related administrative arrangements; a model of instructionalprograming for the individual student; and curriculum components in prereading,reading, mathematics, motivation, and environmental education. The-develop-ment of other curriculum components, of a systm for managing instruction bycomputer, and of instructional strategies is needed to complete the system,Continuing programmatic research is required to provide a sound knowledgebase for the components under development and for improved second generationcomponents. Finally, systematic implementation is essential so that the prod-ucts will function properly in the IGE schools..

The Center plans and carries out the research, development, and imple-mentation Components of its IGE program in this sequence: (1) identify theneeds and delimit the component problem area; (2) assess the possible con-straintsfinancial resources and availability of staff; (3) formulate generalplans and specific procedures for solving the problems; (4) secure and allo-cate human and material resources to carry out the plans; (5) provide foreffective communication among personnel and efficient management of activi-ties and resources; and (6) evaluate the effectiveness of each activity andits contribution to the total program and correct any difficulties through feed-back mechanisms and appropriate management techniques.

A self-renewing system of elementary education is projected in eachparticipating elementary school, i.e., one which is less dependent on externalsources for direction and is more responsive to the needs of the children attend-ing each particular school. In the IGE schools, Center-developed and othercurriculum products compatible with the Center's instructional programing modelwill lead to higher student achievement and self-direction in learning and inconduct and also to higher morale and job satisfaction among educational per-sonnel, Each developmental product makes its unique contribution to ICE asit is implemented in the schools : The various research components add to theknowledge of Center Practitioners,: developers, and theorists.

ii:5

II

Acknowledgments

The authors wish to acknowledge the cooperation and assistanceprovided by Donovan L. Richards, Assistant Superintendent for Instruc-tional Services; Laurence Baker, junior high school principal; CharlesBruce, Joseph Checota, Martin Kerwin, and Wayne J. Simon, elemeh-t ry school principals;: and the staff and students of Riverside Junior

igh School, Douglas, Lit ..;o1n, Webster, and Schurz elementary schoolsof the Watertown., Wisconsin Public School System.

We thank Mary Montgomery,. Assistant Scientist at the WisconsinResearch and Development Center for Cognitive Learning; ProfessorRoy A. Lindberg, Department of Mechanical Engineering, Universityof Wisconsin; and Sister 'Suzanne Lorentz,, Logewciod High School,for their consultation with us regaruing the content of the assessmentoatteries d1ing with equilateral triangle, cutting tool, and noun,respecly. Assistance regarding statistics and measurement wasprovided by Professors Laurence Hubert, Joel Levin, and iviicnaelSubkoviak, Department of Educational Psychology, University ofWisconsin;: computer programming services by Geneva Diluzio andEd Haertel, Wisconsin Research and Development Center; and art anddesign for tne assessment batteries by Karen McLellan and Vera Meyer,Wisconsin Researcn and Development Center.

Table of Contents4

Acknowledgments iv

List of Tables and Figures vii

Abstract xi

I. Introduction:' Concepts and Concept Learning 1

The Nature of Concepts 1

An Overview of the Conceptual Learning and Development Model 2Operations Related to Levels of Concept Attainment 4Acquiring Appropriate Labels 8Concept Extension and Utilization 8Additional Features of the CLD Model 10

II. Purposes and Methods of the Study 11

Overview of the Longitudinal Study of Children's Conceptual Development . . . . 11Design of the Present First-Year Cross-Sectional Study 12

III. Results of CLD Assessment Series If Equilateral Triangle 17

Overview 1 7Child Population

1 17Criteria for Full Attainment 18Proportion of Each Grade Group Conforming to the Predicted

Invariant Sequence 18Predicted Sequence of Concept Attainment and Difficulty of the Levels 19Proportion of Grade Groups Attaining the Four Levels 2 1Relationship Between Full Attainment of Various Levels and Full

Attainment of the Various Uses 23Difficulty of the Three Uses 23Relationship Between Vocabulary Development and Attainment of

Concept Levels and Uses 25

IV. Results of CLD Assessment Series II: Cutting Tool 29

Overview 29Child Population 29Criteria for Full Attainment 29Proportion of Each Grade Group Conforming to the Predicted

Invariant Sequence 29Predicted Sequence of Concept Attainment and Difficulty of the Levels 32Proportion of Grade Groups Attaining the Four Levels 33Relationship Between Full Attainment of Various Levels and Full

Attainment of the Various Uses 33Difficulty of the Three Uses 36Relationship Between Vocabulary Development and Attainment of

Concept Levels and Uses 35

I

7 v

V. Results of CLD Assessment Series III:' Noun 39

Overview 39

Child Population 39Criteria for Full Attainment 39

Proportion of Each Grade Group' Conforming to thePredicted Invariant Sequence 39

Predicted Sequence of Concept Attainment and Difficulty of the Levels. 40

Proportion of Grade Groups Attaining the Four Levels 42

Relationship Between Full Attainment of Vag lous Levels andFull Ptainment of the Various Uses 43

Difficulty of the Three Uses 44

Relation Between Vocabulary Development and Attainmentof Concept Levels and Uses 44

VI. Summary and Conclusions 49

Summary 49

Conclusions 50

References 57

vi

a

List of Tables and Figures

,Table Page

1 Sampling Design for the CLD Tests 13

2 Number of Males and Females, Mean Age, and Age Range at EachGrade Group 1 7

3 Number of Items and Criteria Defining Full Attainment for eachConcept Level and Use . 18

4 Number and Proportion of Four Grade Groups Conforming and Not ,Conforming to Predicted Sequence of Attainment 20

5 Frequencies of Subjects According to Pattern of Exception and ItemsCorrect at Each Concept Level Not Attained 2 1

6 Number and Proportion of Eacn Grade Group That Fully MasteredEach Level of Attainment 22

7 Relationship Between Full Mastery of Various Levels and FullMastery of Uses

8 Number and Proportion of Each Grace Group That Fully MasteredEach of tne Three Concept Uses 25

9 Means and Standard Deviations for Combined Concept Levels, ConceptUses, Combined Levels and Uses, and Vocabulary at Each Grade Group . . . 26

10 Pearson Product-Moment Correlations Bets4een Mean Vocabulary Scores andMean Scores on Concept Levels, Concept Uses, and CombinedLevels and Uses 27

11 Number of Males and Females, Mean Age, and Age Range at EachGrade Group 30

12 Number of Items and Criteria Defining Full Attainment for Each ConceptLevel and Use 30

13 Number and Proportion of Four Grade Groups Conforming and NotConforming to Predicted Sequence of Attainment 31

14 Frequencies of Subjects According to Pattern of Exception and ItemsCorrect at Each Concept Level Not Attained 32

15 Number and Proportion of Each Grade Group That Fully MasteredEach Level of Attainment 34

16 Relationship Between Full Mastery of Various Levels and Full Masteryof Uses 35

17 Number and Proportion of Each Grade Group That Fully MasteredEach of the Three Concept Uses

9

37

vii

viii

List of Tables and Figures (con't.)

18 Means and Standard Deviations for Combined Concept Levels,Concept Uses, Levels and Uses, and Vocabulary at EachGrade Group 37

19 Pearson Product-Moment Correlations Between Mean VocabularyScores and Mean Scores On Concept Levels, Concept Uses, andCombined Levels and Uses 38

20 Number of Males and Females, Mean Age, and Age Range atEach Grade Group 40

21 Number of Items and Criteria Defining Full Attainment for EachConcept Level and Use 40

22 Number and Proportion of Four Grade Groups Conforming and NotConforming to Predicted Sequence of Attainment 41

23 Frequencies of Subjects According to Pattern of Exception and ItemsCorrect at Each Concept Level Not Attained 42

24 Number and Proportion of Each Grade Group that Fully Mastered EachLevel of Attainment 43

25 Relationship Between Full Mastery of Various Levels and FullMastery of Uses 45

26 = Number and Proportion of Each Grade Group that Fully MasteredEach of the Three Concept Uses 46

27 Means and Standard Deviations for Combined Concept Levels, CpnceptUses, Combined Levels and Uses, and Vocabulary at Each Gr'rde Group 47

28 Pearson Product-Moment Correlations Between Mean Vocabulary Scoresand Mean Scores on Concept Levels, Concept Uses, and CombinedLevels and Uses 47

29 Proportion of Total Subject Population Conforming to Predicted Pass-FailPatterns of Attainment: Comparing Equilateral Triangle, Cutting Tool,and Noun 5 1

30 Frequencies of Total Subject Population Showing Pass-Fail Patterns ofException:, Comparing Equilateral Triangle, Cutting Tool, and Noun . 5 1

31 Proportion of Each Grade Group that Fully Mastered Each Level ofAttainment: Comparing Equilateral Triangle, Cutting Tool, and Noun 53

32 Relationship Between Full Mastery of Levels and Full Mastery of Uses:Comparing Proportions of Total Subject Population Passing Uses forEquilateral Triangle, Cutting Tool and Noun . 54

33 Proportion of Each Grade Group that Fully Mastered Each of the ThreeConcept Uses: Comparing Equilateral Triangle, Cutting Tool, and Noun. . 55

34 Pearson Product-Moment Correlations at Each Grade Group Between MeanVocabulary Scores and Mean Scores on Concept Levels, Concept uses,and Combined Levels and Uses:, Comparing Equilateral Triangle,Cutting Tool, and Noun 55

10

List of Tables and Figures (con't.)

35 Correlations for Total Subject Population Between Mean VocabularyScores and Mean Scores on Levels, Uses, and Combined Levels

' and Uses: Comparing Equilateral Triangle, Cutting Tool, and Noun . . 56

Figure Page

1 Cognitive operations in concept learning 3

;

11 IX

Abstract

Theory and research regarding_four levels of concept attainmentand three uses of concepts as specified by the conceptual learningand development (CLD) model are described. The strategy and ob-jectives of a longitudinal assessment of Children's conceptual learningand development are presented. Perspective is provided regarding therole of the pre-Sent cross-sectional study in the longitudinal assessment.

For tne present study, assessment batteries were developed toassess each child's level of concept attainment and also the relatedwe of the concepts equilateral triangle, cutting tool, and noun. Eachof the three batteries was designed as a paper-and-pencil task andadministered to from 60 to 100 children enrolled in each of four grades:,kindergarten, third, sixth, and ninth. The three populations receivingthe three batteries were not independent;, more than 80r/q of the childrenreceived all tnree batteries.,

Predictions based on th4 model about children's conceptual develop-ment were strongly supported across all three concepts:,

1. The three concepts were attained In an invariant sequence ac-cording to four successive levels: concrete, identity, classifi-catory, and formal.

2. As the concepts were attained to highe levels they were usedincreasingly (a) in cognizing supraordinate-suborainate relation-ships in a hierarchy where the attained concept was an elementof the hierarchy;: (5) in understanding principles that stated arelationship between the attained concept and one or more otherconcepts; and (c) in solving problems that required the use ofthe particular concept.

3. Having the labels of the concept and of its defining attributesfacilitated (a) attainment of tne concept,and (b) mastery of thethree uses ,)f the concept.

12

IIntroduction: Concepts and Concept 1L,aming

Individuals at all levels of human devel-opment are constantly learning new conceptsand extending and using old concepts in newsituations. It is apparent, however, thatan individual's level of mastery of a partic-ular concept will differ depending uponhis experiences with concept instances andhis ability to perform the cognitive operatiOns.For example, a four- year -old child and abiotogist may both have a concept of tree:although both ,may perform equally well whenasked to identify a few obvious examples andnonexampies of tree, their concepts differmarkedly. Despite the large difference inlevelxof understanding, concepts are thefundamental agents of thought for humanbeings from early childhood through adult-hood.

A substantial amount of research onconcept learning has been completed duringthe past two decades. Two types of researchhave been conducted, one dealing with theinternal and external conditions of conceptlearning and another type involving the be-havioral analysis of learning concepts re-lated to various subject- matter fields. Suf-ficient knowledge has accrued so thatKlausmeier, bhatala, and Frayer (1974)were able to formulate a model that speci-fies and describes the cognitive operationsinvolved in tne attainment of concepts atspecifiable levels of mastery by individualswhose abilities change in predictable wayswith age. (We use "age" as a shorthandterm to indicate the product of learningand maturation; age, per se, is not consid-ered a determining factor of how well in-dividuals can perform.)

An analytic*clescriptive model of con-ceptual learning and Jevelopment (CLI)model) was initially formulated by Klaus-meter (1971) and Jescribed more fully byKlausmeier Ghatala and Prayer (1974).,

The model defines four levels of conceptattainment and the possible uses and exten-sions of attained concepts, specifies thecognitive operations involved in learning con-cepts at each of the four levels, and postulatesinternal and external conditions of learningrelated to the specified levels. The levelsof concept mastery, the operations, and theconditions of learning have been identifiedthrough behavioral analyses of conceptlearning tasks and through empirical researchin laboratory and sChool settings carriedout at the Wisconsin Research an Develop-ment Center for Cognitive Learning and ',-other research laboratories.

The Nature of Concepts

The word concept is used by Klausmeier,Ghatala and Frayer (1974) to designate men-tal constructs of individuals and also identi-fiable public entities that comprise part ofthe substance of the various disciplines. Thus,concept is used appropriately in two differ-ent contexts just as many other English wordsare. A concept is defined as ordered infor-mation about the properties of one or morethings -- objects, events, or processes- -thatenables any particular thing or class of thingsto be differentiated horn, and also relatedto, other things or classes of things.

In connection with concepts as mentalconstructs it is noted that eac,: maturing in-dividual attains concepts according to hisunique learning experiences and maturationalpattern. In turn, the concepts ne attains areused in his thinking about the pnysical andsocial world.

Concepts as public entities are definedas organized information corresponding tothe meaning of woris. Carroll (1964) relate,:concepts,, words, and word meanings in the

13

following way.; Words in a language can bethought of as a series of spoken or writtenentities. There are meanings for words thatcan be thought of as a standard of communica-tive behavior that is shared by those who speaka language. Finally, there are concepts- -that is, tne classes of experiences forme inindiviuuals either independently of languag,processes or in close dependence on languageprocesses. Putting the three together,Carroll stated: "A 'meaning' of a word is,therefore, a societally standardized concept,and when we say that a word stands for ornames a concept it is understood that we arespeaking of concepts that are shared amongmembers of a speech community [1964, p. 1871."

At the inception of a large programmaticresearch.effortdealing with concept learningand instruction, Klausmeier, Davis, Ramsay,Fredrick, and Davies (1965) formulated a con-ception of concept in terms of defining attri-butes common to many concepts from variousdisciplines. Klausmeier, Ghatala, and Frayer(1974) further refined the definition by speci-fying eight attributes of concepts:: learnabil-ity, usability, validity, generality, power,structure, instance numerousness, and in-stance perceptibility. Other researchers andsubject-matter specialists are also treatingconcepts in terms of defining attributes.For example, ,l'lavell (1970) indicated that aformal definition of concept in terms of itsdefining attributes is useful in specifying whatconcepts are and -ire not and also in identi-tying the great variability among concepts.Markle and Tiemann (1969) and Tennyson andBoutwell (1971) have shown that the externalconditions of concept learning can be deline-ated throu,'gli research that starts with a sys-tematic analys.s of the attributes of the par-ticular concepts used in the research., Schol-ars at the Wisconsin R & D Center demon-strated that analysis of concepts in terms oftheir relevant and irrelevant attrioutes isuseful in clarifying the meanings of the con-cepts drawn from four disciplines: languagearts -Golub,, Fredrick, Nelson, and Frayer(1971):: mathematicsRomberg, Steitz, andFrayer (1971);, scienceVoelker, Sorenson,and Frayer (1971);, and social studies--Tabacn-nick, Weible, and Frayer (1970).

The CLD model deals primarily withconcepts represented by words that can beleaned m terms of dttributes , although someconcepts -are tefined on otner basf s, includ-ing through the use of synonyms and antonyms.Further, not all words potentially sofinablein terms of attributes are So delined even inunabridgeu ,Therefore, tne re-searcher and also tne developer of curriculummaterials must ascertain the defining attributes

2

independently or cooperatively with schdlarsfrom the various desciplines.

An Overview of the Conceptual Learningand Development Model

Figure 1 shows the structure of the model.Four levels in the attainment of the same con-cept at successively higher levels are outlined.The four successive levels are concrete, iden-tity, classificatory, and formal., As a con-cept is attained by an individual at the suc-cessive levels it becomes increasingly usableand valid, as defined earlier.

A sec- "igure 1 shows the waysin whict of be extended and usedConcepts ..ired at only the concrete andidentity levels can be used to solve simpleproblems that require only the relating ofobvious sensory perceptions. For example,to save time, or for some other reason, achild may walk diagonally across a rectangu-lar block rather than remaining on the side-walk and walking around a corner of the block.He need not have attained the concepts of dis-tance, angle, diagonal,, or straight line atthe classificatory level.

Concepts acquired at the classificatoryand formal levels may be generalized to newlyencountered instances, related to other con-cepts, and used in problem-sol ing situations.Here we are concerned both with transfer oflearning and the use of concepts in thinking.

Figure 1 also indicates thi operationsinvolved in attaining a concept at each level.Attending to and discriminating objects andthen remembering what was discriminatedare involved in attaining a concept at the con-crete level. The same operations are alsoinvolved at each subsequent le,el -and aresupplemented with the higher-level opera-tions of generalizing, hypothesizing, andevaluating.

Although some of the same operationsare postulated to occur at various levels,what is operated on and remembered changeswith the attainment of the successively higherlevels. That is, the operations are carriedout on more sharply differentiated and ab-stracted stimulus properties at the four suc-cessive levels.

By focusing, on the attainment of succes-sively higher levels cf the same concept,, weare able to clarify the snort-term learningconditions at each level and to describe con-ceptual devi-lopment over long time inter-vals. Thus, the model provides\ a basis fororganizin.1 knowledge and carrying out research 4,

related to both the external and internal con-ditions of learning at each of the four levels.

14

Acquiring

and

remembering

the names

of the

concept

and its

attributes

LEVELS OF CONCEPT ATTAINMENT

CONCRETE LEVEL

Attending to things

Discriminating one thing from other

things

Remembering the discriminated thing

IDENTITY LEVEL

(Three prior operations and)

Generalizing that two or more forms

of the same thing are equivalent

CLASSIFICATORY LEVEL

(Four prior operations and)

Generalizing that two or more in-

stances are equivalent in some way

CON="1"XTENSION

SSE

I Using the concept in solving

1

1simple problems that can

be solved on the basis of

1

1perceptible elements of

the situation

FORMAL LEVEL

(Five prior operations and)

Discriminating the defining

attributes of the concept

Hypothesizing the rele-

vant attributes and/

or rules

Remembering hypotheses

Evaluating hypotheses

using positive and

negative instances

Cognizing the common

attributes and/or'

rules from

positive instances

Inferring the concept

Figure 1,

Cognitive operations in concept learning.

Generalizing to positive

instances of the concept

and discriminating

noninstances

Cognizing supraordinate,

coordinate, and sub-

ordinate relationships

involving the concept

and other concepts

Cognizing cause-and-effect,

correlational, prob-

ability, and other

relationships of the

.attained concept with

other concepts

Using the concept in

solving problems

The fourth part of the model snows thatacquiring and remembering the name of theconcept may come at any of the fur levels.The solid line indicates that being able toname the concept and its relevant attributesis essential to attaining concepts at the formallevel. The broken lines indicate that anindividual may acquire the name at ai,out thesame time he first attains the concept atlower levels but that this is not requisite.For example , a yOung child might attain aconcept at all three lower levels but not theconcept name. The younger the child is uponattaining the concept, the less likely he is tohave the name for it.

At this time, we shall delimit the sub-stantive domain that we are treating. Themodel in its totality describes the four le',elsof concept attainment and uses of the sameconcept rather than each of four kinds of con-cepts. The four levels apply to the manyconcepts that are or can be defined in termsof attributes and which have actual percepti-ble instances or readily constructed repre-sentations of instances. We have already citeda few examples of this kind, including all theconcepts comprising the plant kingdom andthe animal kingdom. However, the operationsat each level are intended to be applicable alsoto different kinds of concepts, some of wi,ich,because of tneir nature, are not attainable atall four levels. We can specify these kindsof concepts and the levels at which they canbe attained.

There are some concepts for which thereis only one instance such as the earth's moonand Abraham Lincoln and some that have manyidentical instances, for example, inch andpound. Related to Figure 1, such single-in-stance or identical-instance concepts which,have defining attributes can be attained atthe concrete, identity, and formal levels, butnot at the classificatory level. By our def-inition of classificatory level, there must beat least two nonidentical instances that canbe placed in the same class. Therefore, someconcepts cannot be attained at the classi-ficatory level.

Other concepts are of such low validitythat there may not be agreement as to the de-fining attributes, for example, beauty andmorality. Concepts such as these might belearned at the three lower levels but not atthe formal level.

Finally, there are concepts with no per-ceptible instances, suer: as infinity and atomThese cannot be learned at the three loWerlevels but mignt be learned at the formallevel.

Returning to the four levels given inFigure 1' we postulate that attaining a concept

4

at the four successively higher levels isthe normative pattern for large numbers ofindividuals under two conditions. First, theconcept is, of the kind for which there areactual perceptible instances or readily con-structed representations; and second, theindividual has experiences with the instancesor representations starting in early child-hood. Furthermore, in order to proceed tothe formal level, individuals muss acquirelabels for the concept and for its attributes.For example, the individual will have succes-sively attained the concrete, identity, andclassificatory levels of the concept plant be-fore he describes and treats plant formallyin terms of its defining attributes.

Children have direct experiences duringpreschool years with many things and attainconcepts of these things at the first twolevels. They also attain many concepts atthe classificatory level and learn the soci-etally accepted names for the concepts andtheir attributes through formal and informalinstruction.

Earlier we indicated that some individu-als, because of environmental conditions,may not encounter actual instances of aconcept;' rather, they experience instancesonly in verbal form. Thus, these individualsmay attain a concept at either the classifi-catory or the" formal level at the outset.It is also noted that the mature person, al-though capable of attaining a concept at theformal level, may stop at a lower level ofattainment because of the way in which theperceptible instances are encountered orother conditions' of learning.

Operations Related to Levelsof Concept Attainment

Having considered the overall featuresof the model, we may take up the operationsIn more detail, starting with those pertain-ing to the concrete level.

Concrete Level

Attainment of a concept at the concretelevel is inferred when the individual cognizesan object that he has encountered on a prioroccasion. We use the term "operations" cisGuilford (1967) does. Guilford has defined(he operations of cognition, memory, pro-ductive thinking, and evaluation in terms oftest performances. lie stated that cogni-tion must be related to the products cognizedand he formally defined cognition as follows:,

Cognition is awareness, immediate

16

discovery or rediscovery, or recog-nition of information in variousforms; comprehension or understand-ing . . . The most general 'term,awareness, emphasizes having activeinformation at the moment or in thepresent . . . the term, recognition,is applied to knowing the sameparticular on a second encounter. . . if cognition is practicallyinstantaneous, call it recognition;if it comes with a slight delay, callit "immediate discovery;" [pp. 203-2041."

The first step in attaining this level is attend-ing to an object and representing it internally.Woodruff (1961) pointed out that:

All learning begins with some.- form of personal contact with actual

objects, events, or circumstances.. . The individual gives attention

4o some object. . . . Through a lightwave, or a sound wave, or, someform of direct contact with a sen-sory organ in the body, an impresssion is picxed up and lodged in themind [p. 66J.

Gagrg (1970) indicated that as tne individ-ual attends to an object, he discriminates itfrom other objects. Woodruff (1961) called theoutcome of these attending and discriminatingoperations a concrete concept, a mental imageof some real object experienced directly by thesense organs. The infant, for example, attendsa large red ball and a white plastic mottle,discriminates each one, maintains a mentalimage of each, and cognizes each of theobjects when experienced later

The discrimination of objects involves at-tending to distinctive features that serve todistinguish the objects from one another.Thus, very early the child learns to respondto gross differences in such features of objectsas size, shape, color, and texture. As thechild matures, he becomes capable of makingfiner discriminations involving these andother features.

The attainment of a concept at the con-crete level'', thus requires attending to the dis-tinctive features of an object and forming amemory image which represents the abjectas a unique bundle of features. The conceptat this level may or may not be associatedwith the concept label, depending on whetherthe label has been learned and remembered,and whether it has been associated ith theconcept.

The prece ling analysis of the op-tenonsis attaining concepts at the concrete level issufficiently comprehensive to include motoricexperiencing of objects. That is, a,, object

17

may be manipulated physically and repreientedenactively, as well as explored visually andrepresented iconically, to use Bruner's (1964)terminology. The model postulates that attend-ing, discriminating, and remembering areinvolved in sgnsorimotof -experiencing (to usethe terms of Piaget, 1970) as well as in thevisual perception of objects.

Identity Level

Attainment of a concept at the identity '

level is inferred when the individual cognizesan object as the same one previously encoun-tered when observed from a different perspec-tive or sensed in a different modality. Forexample, the child's making the same responseto the family poodle when seen from straightahead, from the side, and from various anglesis evidence of his having attained the conceptof poodle at the identity level. Whereas con-cept attainment at the concrete level involvesonly the discrimination of an object fromother objects, attainment at the identy levelinvolves both discriminatirg various formsof the same object from other objects andalso generalizing the forms as equivalent.Generalizing is the new operation postulatedto emerge as a result of learning and matura-tion that makes attainment at the-identityle.uel possible. -

As noted earlier, there are some validand powerful concepts, such as the Englishalphabet, for which there is only 'one instancebut which can be represented in differentwayk, e.g., aurally and in printed form.These concepts are typically learned at theconcrete and identity levels but not at theclassificatory level. Therefore, individualsproCeed directly from the identity to theformal level with this kind of concept.

Bruner, Goocinow, and Austin (1p56) havepointed out that identity responsesloccur veryearly in life and that the capability to recog-nize identity may be innate and merely ex-tended to new events through learning. Vernon(1970) indicates that infants have to learnby experience that objects and el/kits in theenvironment are permanent even *ugh theymay change their appearance from! time totime as their distance and orientation changes.Clearly,, the capacity to recognieldentity,indeed the expectation of the continuity ofobjects and events in the environment, iswell developed in the perception of adults.

Recognition of object identity is centralto Piaget's formulations. According to El-kind (1969), Piaget's conception of conceptemphasizes the variability that occurs withinthingschanges in state, form,, and appearance

5

which can occur to any entity.Elkind pointed out further that American

psychologists have tended to ignore thiswithin-instance variability of concepts andhave emphasized the discriminative responseaspect of concept attainment by which posi-tive instances are cognized and discriminatedfrom noninstances. Elkind summarized thetwo points of view thus:

From the discriminative responsepoint of view, the major functionof the concept is the recognitionor classification of examples.The Piagetian conception, how-ever, assumes that a major func-tion of the concept is the discrim-ination between the apparent andthe real. This discrimination,in tarn, can be reduced to thedifferentiation of between- andwithin-things types of variabil-ity. Here again, a comprehensiveconception of a concept must in-clude both functions because, infact, every concept does serveboth purpose-s [1969, p. 1871.'

The present model proposes that a con-cept is attained at the identity level tempor-ally before it is attained at the classifactorylevel. Stated differently, the individual mustbe able to cognize various forms of the sameobjects as equivalent before ne is able to gen-eralize that two or more different objectsbelong to the same class.

Classificatory Level

The lowest level of mastery at the classi-ficatory level is inferred when the individualresponds to at least two different instances ofthe same class as equivalent, even though hemay not be able to describe the basis for hisresponse. For example, when the child treatsthe family's toy poodle and the neighbor's min-iature poodle as poodles, although he may notname the attributes of poodles, he has attaineda concept at the classificatory level.

While jeneralizing that at least two dif-ferent instances are equivalent in some way isthe. lower limit of this level of concept learning,,the is still at the classificatory lev-el of concept learning when he can correctlyclassify a larger number of instances as exam-ples and nonexamples, but cannot accuratelydescribe the basis for his grouping in terms ofthe defining attributes. Henley (cited in Deese,1967), like many other reseel-chers, has observedthis phenomenon. Many of ne,r subjects wereable to sort cards correctly Ito eyamples andnonexamples of the concepts being learned, yetgave totally erroneous definitions of the concepts.

6

Formal Level

A concept at the formal level is inferredwhen the individual can give the name of theconcept, can discriminate and name its in-trinsic or societally accepted defining at-tributes, can accurately designate instancesas belonging or not belonging to the set,and can state the basis for their inclusionor exclusion in terms of the defining attri-butes. For example, the maturing childdemonstrates a concept of dog at the formallevel if, when shown dogs, foxes, andwolves,of various sizes and colors, he pro-perly designates the dogs as such, callsthem "dogs," and names the attributes thatdifferentiate the dogs from the foxes andwolves. The distinctive aspect of this levelof concept mastery is the learner's abilityto specify and name the defining attributesand to differentiate among newly encountered ,;

instances and noninstances on the basis ofthe presence or absence of the defining at-tribukes.

As noted in Figure 1, the labels for theconcept and the defining attributes may belearned at any of the three lower levels,but are not essential at those levels. Sim-ilarly, the discrimination of the definingattributes may occur prior to the formallevel, but this is not essential. Thus, dis-crimination of things on their global anddiffuse stimulus properties which is essen-tial at the concrete level changes to dis-crimination of more specific and abstractproperties at the identity and classificatorylevels. However, at the formal level theindividual must be able to discriminate andlabel all the defining attributes of the con-cept.

The operations involved in the learningof concepts at the formal level are alsoshown in Figure 1. The first,operationgiven atrthe formal level is that of dis-criminating the attributes.. As already noted,for some concepts with obvious attributessuch as color and form, the discriminationsmay have occurred at earlier levels. However,making the discriminations and having thclabels for the attributes are both essentialat the formal level. This is true whetherthe individual infers the concept 1-e hypothe-sizing and evaluating relevant attributesor cognizing the attributes common to posi-tive instances, as shown in Figure 1.

Individuals differ in their ability to ana-lyze stimulus configurations into abstradimensions or attributes. There is eviden e(Gibson, 1969) that this ability develops wiage. Retarded children may have difficultywith simple concept learning tasks because

isN

of the difficulty in learning to select out andattend to specific dimensions (Zeeman &House, 1963). Even among children of ade-quate intelligence, there are those whocharacteristically analyze the stimulus fieldand apply labels to attributes while otherstend to categorize on the basis of a relativelyundifferentiated stimulus (Kagan, Moss &Sigel, 1963). .-

Orienting instructions may be given tomake explicit the attributes of the stimuli(Klausmeler & Meinke, 1968). These instruc-tions facilitate the learning of concepts at theformal level by assuring that the learnerknows all of the attributes thatmay be relevantto the concept.

Having discriminated and named the attri-butes, an individual may infer the formal levelof a concept in either of the two ways shownin Figure 1. One way involves formulatingand evaluating hypotheses and the other in-volves cognizing the common attributes inpositive instances. Which strategy a/learneruses,depands on the instructions he has beengiven, his age, and the kind of concept in-stances he experiences.

Levine (1963) defined a hypothesis as thesubject's prediction of the correct basis forresponding. In the hypothesis-testing ap-proach, the learner guesses a possible de-fining attribute or combination of attributes.He then compares this guess with verifiedexamples and nonexamples of the concept tosee whether it is compatible with them. Ifthey are not compatible, he makes anotherguess and evaluates it against further exam-ples and nonexamples. Eventually, ne com-bines the information he has obtained fromtesting his hypotheses so as to infer all thedefining attributes and thereby the concept. ,

Essential-to the hypothesis-testing ap-proach are the operations of rememberingand evaluating hypotheses. There is support(Levine, 1963; Williams, 1971) for the ideathat the subject formulates and remembers apopulation of aypotheses, remembers thehypotheses that were rejected, and also re-members the last one accepted as correct.In connection with evaluating hypotheses,Bruner, Goodnow and Austin (1956) indicatedtact an individual determines whether or nothis hypothesize,' concept is valid by recourseto an ultiMate criterion, test by consistency,,test by consensus, or test by affective con-gruence. Inherent in all four procedures isestablishing a criterion for Judging the cor-rectness of a hypothesis, In the presentmodel, the validity of an individual's conceptmay be assessed in terms of how nearly itcorresponds to expert agreement concerningthe concept. Our experiments have shownthat instructions to subjects whicn include a

decision rule for evaluating hypotheses facil-itate concept attainment.

The operations involved in the hypothesis-testing approach to inferring concepts appearto characterize indivuddals who cognize theinformation available to them in laboratoryand classroom settings from both positive andnegative instances. These individuals appar-ently reason like thisf Instance 1 had landsurrounded by water. It is a member of theclass. Instance 2 has land but is not sur-rounded by water. It is not a member of theclass. Therefore, lands surrounded by waterbelong to the class and lands not surroundedby water do not. Surroundea by water is adefining attribute of the concept. This indi-vidual has .attained a partial and possiblycomplete definition of the concept based onexperiences with only one positive and onenegative instance.

A second way of inferring the concept isby noting the commonalities in examples ofthe concept. The commonality approabh isused more often than the hypothesizing ap-proach by children, either because tney areincapable of carrying out the hypothesizingand evaluating operations or because theyfor other reasons pursue the commonalitystrategy (Tagatz, 1967). In this connectionthe commonality approach 1s entirely ap-propriate for use when only positive instancesof the concept are available. Thus, it isprobably employed in situations where the

It individual is given only positive instances orverbal descriptions of positive instances.

Our moc.;:l is considered appropriate forlearning concepts at the formal level by a didac-tic method of information presentation as well asan inductive one. We agree with Ausubel (1966)that many concepts are attained at the classifi-catory and formal levels by upper elementary,high school, and college students through beinggiven the names of concepts, verbal definitions,and verbal examples, but no actual instances ofthe concepts. Ausubel has designated this kindof learning "concept assimilation," an exampleof meaningful reception learning, to contrast itwith "concept formulation," an example ofmeaningful discovery learning.

We should consider briefly what takesplace when the learner is given the conceptname, its defining attributes and a verbaldescription of an instance or two, asquently done in classroom settings. Theindividual may attain a concept at a low levelof mastery through this brief instructionalsequence. However, nis`main task there-after is to generalize properly to newly en-counterea positive instances and to discrim-inate noninstances. The basic operationsentailed in this identification of newly

19 7

encountered instances are hypothesizing wheth-er the instance does or does not belong tothe concept and evaluating the hypothesis interms of the defining attributes given in thedefinition. Prerequisite to these two opera-tions are discriminating the attributes of theconcept and knowing their labels.

Acquiring Appropriate Labels

The importance of language in conceptlearning is widely acknowledged by American(Bruner, 1964) and Russian (Vygotsky, 1962)psychologists. Having the labels of conceptsenables the individual to think in symbolsrather than in images and to attain other con-cepts through language experiences in the ab-sence of perceptible instances. Carroll (1964)as noted earlier, has outlined the close rela-tionships among concepts, meanings, andwords. However, the purpose here is not todeal with the relationships between languageand concept learning, which will be dealtwith in a later chapter, but to show at whatpoint: labels may be learned and associatr-Uwith the various levels of concepts.'

Figure 1 indicates that a concept label maybe associated with an instance of the concept atany of the four levels--concrete, identity, clas-sificatory, or formal. For example, a childmight manifest a sequence like this:: The youngchild first encounters a dog. The child's motherpoints to the Iog and says "dog." The childthen says "dog," and associates the name withhis concrete concept of the dog. Next, thechild develops the concept of the same dog atthe identity level through experiencing it indiffrent locations and situations. His motherrepeats the name at various times in the pres-ence of the dog; the child says the word re-peatedly. The word "dog" now comes to repre-sent the child's concept of the dog at the iden-tity level. Subsequently, the child encountersother dogs and observes that they, too, arecalled "dogs." He generalizes the different dogs,as equivalent in son* way and associates thename "dog" with whatever similarities he hasnoted. The -word thus 'comes to represent hisclass of things called 'dogs." At the formallevel, the more mature hild discriminates andlearns the societally ac epted attributes of theclass of things called "dogs" and also learnsthe names of the attributes. Now the child'sconcept of dog approaches or becomes identicalto the societally accepted definition of the word"dog." As Carroll (1964) pointed out, the con- t

cepts held by individuals and the meanings ofthe words representing the concept'sare thesame for mature individuals who share similarcultural experiences and the same language.

In connection with language and concept

8

attainment, we recognize that deaf individualsand others who lack normal speech develop-ment may attain concepts at the formal level.By our definition, the individual must knowthe defining attributes of the concept and must\be able to communicate this knowledge. Ver-balizing is normally used in this kind of com-munication. Other types of symbolic com-munication, for example, sign language, mayalso be employed. Speech,per se, is notnecessary for the attainment of concepts, butsome means for symbolizing and communicatingthe concept in the absence of examples isnecessary at the formal level.

Conlept Extension and Utilization

The individual who has formed a conceptmay extend and use it as shown in Figure 1.As noted earlier, a concept attained only to

',the concrete or identity level may be usedin solving simple perceptually based prob-.le s Concepts-learned at the classificatoryan formal levels can betted in generalizingto zl instances-, cognizing supraordinate-subor inate relations, cognizing cause-and-effect and other relations among concepts,and in solving problems.

Ausubei (1963) and Gagne (1970) havetheorized concerning the use and extensionof attained concepts; however, very littleempirical research has been done., In thisregard Ausubel (1963} formulated the con-structs of cognitive structure, advance or-ganizer, correlative subsumption, and deri-vative subsumption to show how previouslyattained and newly encountered concepts arerelated, while Gagne has indicated that at-tained concepts are prerequisite to the learn-ing of rules.

Generalizing to New Instances andDiscriminating Noninstances

The attainment of concepts at the classi-ficatory and formal levels reduces the needfor additional learning and relearning, pri-marily because the individual is able togeneralize to new instances of a concept andto discriminate noninstances. Having a con-cept also provides the individual with ex-pectations which nelp him deal effectivelywith new instances of it. Once he identifiesa plant as poison ivy, he may treat it gin-gerly. One test of concept attainment in ourexperiments is the individual's ability toproperly categorize instances not previouslyencountered as instances or noninstancesof the particular concept. We find that both

20

school children and college-age studentsgeneralize to new instances readily. Further-more, the use of instances and noninstancesin instructional materials to teach conceptscan be manipulated so that errors of over-generalization and undergeneralization canbe reduced (Feldman, 1972; Swanson, 1972).

Not only does having a concept enable thelearner to identify new instances and actappropriately toward them, but direct andverbal experiences with the new instancespossibly increase the validity and power ofthe concept for the individual. For example,the Canadian visiting Kenya during January,when it is summer there, may attain morevalid and powerful concepts of flower andplant. Similarly, by being told that a whaleis a mammal, an individual comes to realizethat mammals can live in the water as wellas on land. Hence, his concept of mammalhas greatei validity.

Cognizing Supraordinate-SubordinateRelationships

Besides generalizing to new instances,individuals can also use their concepts at-tained at the formal level, and possibly atthe classificatory level, in cognizing co-ordinate, supraordinate, and subordinaterelationshios among classes of things. Thelowest level of cognizing these relationshipsis inferred when the individual, accordingto verbal instructions, puts instances ofhierarchically arranged concepts in theirproper groups. For example, an individualupon request puts all instances of red andblue eqUilateral triangles and of right tri-angles in a giouping of triangles', and all in-stances of triangles and of rectangles in agrouping of polygons. Furthermore, he justi-fies each group formed on the basis of the de-fining attributes of the group. For example,he states that equilateral triangles includeall the triangles \that have three equal sides,triangles include all the polygons that havethree sides, and polygons include all theclosed, planar figures that have three ormore sides. More precise terminology mightbe required such as "an equilateral triangleis a simple, plane, closed figure with threesides of equal length."

Possible higher levels of attaining thesupraordinate-coordinate-subordinate relation-'ships include what Kofsky (1966) aesignatedas the "whole is the sum of the parts" and

some but not all. Agatha, merely being ableto group a few instances properly accordingto verbal instructions is not a sufficient testof cognizing the sets of relationships; an ade-

quate justification for the actions is required.According to Kofsky (1966), knowledge con-cerning supraordinate-subordinate relation-ships increases with age.

The understanding of supraordinate-subordinate relationships increases the valid-ity and usability of the individual's concepts.For example, knowing the attributes of acidand also that vinegar is an acid leads to theinference that vinegar has the attributes ofall acids, as well as the attributes peculiarto vinegar. Thus, all of the things knownabout acids--for example, how they reactwith bases--are true for vinegar also. In thisway, learning that acid is a concept supra-ordinate to vinegar increases the validity andusability of the concept of vinegar for the in-

..dividual.

Cognizing Other Relationships

In the model, statements of relations be-tween or among concepts involving cause andeffect, correlation, probability, and otherlawful relations such as contained in axioms

rare treated as different from relations amongsupraordinate and subordinate concepts. Thesefirst three kinds of relationships are referredto by Marx (1970) as laws and by Gagne (1966,1970) as principles or rules. Mathematiciansparticularly specify established lawful rela-tions of "givens" in axiomatic statements.

Bruner, Goodnow and Austin (1956) havepointed out that understanding lawful relation-ships between or among concepts permits therelating of classes of things instead of individ-ual things . in this connection, Gagne (1970) hascited the example of the rule "round thingsroll," and indicated that this rule enables theindividual to predict what will happen to allround things under certain circumstances. Or,consider the more complex relationship: "Whentwo substances at different temperatures comeinto contact, the temperatures of the substan-ces tend to equalize." This relationship per-mits us to infer what will happen in such di-verse situations as putting ice cubes in warmsoda pop or being lost in a snowstorm.

In all cases, being able to understand anduse a lawful relationship depends on knowingthe concepts that are related. Only thencan the principle or axiom be understoodand possibly applied to the appropriate phe-nomena.

Using Concepts in Problem-SolvingSituations

Woodruff (1967) discussed the role of con-

219

cepts in higher-level mental activities, in-cluding problem-solving. Also, Gagne (1970)indicated that one way in which conceptsare called into play in solving problems isby the application of principles to the prob-lem-solving situations. For example,principles underlying the concepts of pres-sure,- volume, gravity, and distance can beutilized to determine the height of a mountainusing a barometer.

Additional Features of the CLD Model

The CLD model is more heavily orientedtoward learning than toward development inthat it implies that all the concepts held byany individual are learned; they do notemerge simply with maturation. In this con-text it is similar to four theories of conceptlearning generated by American experimen-tal piychologists and reviewed by Bourne,Ekstrand, and Dominowski (1971): theoryof associations (Bourne & Restle, 1959),theory of 'hypotheses (Levine, 1966; Tra-bas.,o & Bower, 1968), theory of mediation(Osgood, 1953), and theory of informationprocessing (Hunt, 1962). Also, in agree-ment with these theories, the model speci-fies that the attainmeriL of concepts ispotentially explainable in terms of principlesof learning. Despite some differences interminology, the CLD model, like Hunt's,represents an information-processing ap-proach to learning. The CLD model differsfrom the four theories )ust mentioned inthat it describes different levels in the at-tainment of the same concept and specifiesthe operations essential to attaining con-cepts at the successively 4igher levels.While some of the operations are postulatedto be common to more thanone level, theseoperations at the successively,tiigner levelsare carried out on more highly differentiatedand abstract properties of actual conceptinstances or on verbal descriptipns of in-stances and attributes.

The CLD model is similar to Gagne's(1970) cumulative learning model in thatboth provide a framework for studying theinternal and external conditions of learning.It also differs in two regards. WhereasGagne describes seven forms of learning,ranging from the simplest learning throughrule learning and problem solving, in theCLD model only one form of learning, concept

10

learning, is analyzed according to its sev-eral constituent cognitive behaviors ateach of four levels. Gagne also postulated alinear vertical learning hierarchy extendingfrom signal learning through problem solving.The CLD.model, as shown in Figure 1, indi-cates that a concept when learnedtat the classi-

'ficatory or the formal level may be used incognizing supraordinate-subordinate relationsamong the concept and other attained concepts,in understanding relations among concepts suchas incorporated in principles and laws, andin problem solving. Thus, the CLD modeldeparts from the straight linear learning hier-archy postulated by Gagne.

Possibly different from the precedinglearning theories and more in agreement withPiaget (1970), the CLD model presumes thatthe new operations at each successive levelinvolve qualitative changes in operating oninstances and attributes of concepts, notmerely additions to or modification of prioroperations. Further, the operations thatcontinue from one level to the next are car-ried out on more highly differentiated and abTstracted concept attributes. While the modeldoes not postulate a stage concept associatedwith age levels as does Piaget, qualitativedifferences in thinking of the' kinds pointed toby Kagan (1966) and Bruner, Olver, Green-field, et al. (1966) are recognized. Also,Bruner's (1964) conceptualization of enactive,iconic, and symbolic representation is acceptedas a satisfactory global explanation of how ex-periences are represented arid stored.

The roles of language and directed learn-ing experiences are recognized as of centralimportance in attaining concepts at the classi-ficatory and formal levels. The cross-culturalstudies of Bruner, Olver, Greenfield, et al.(1966) support the directed-experiences pointof view (cf. Goodnow, 1969). Also, Bruner's(1964) intermediate position that specifies howlanguage facilitates thinking, rather than be-ing essential to thinking (Luria, 1961) orbeingdependent on thought (Inhelder & Piaget,1964), appears valid for the present model.Accepting directed experience as critical inconcept attainment deempha sizes a maturation-al'readiness viewpoint, such as that expressedby Gesell (1928, 1945). While it is acceptedthat certain cognitive operations emerge witheducational experience, this conception doesnot espouse a behaviorist-environmentalistpoint of view regarding learning to the extentthat either Gagne (1970) or Staats (1971) does.

22

IIPurposes and Methods of the Study

This chapter is intended to provide someperspective regarding the role of the presentstudy in a longitudinal assessment program andto delimit the specific Purposes of the pres-ent study. The plan of the longitudinal re-search is briefly outlined. Then,the purposesof the present study, the specific predictionsthat are evaluated, and the research designare described in detail.

Overview of the Longitudinal Study ofChildren's Conceptual Development

The primary objective of the longitudinalresearch is to chart thiccnceptual develop-ment of children from about age 4 to 18.This goal will be accomplished primarily fromanalyses of longitudinal data that are collectedonce each year over a period of several years.The rationale and strategy for this program-matic research on children's conceptuallearning and development from preschool tothe high school years have been outlined inan earlier paper by Hooper and Klausmeier(1973). The theoretical framework for thestudy reported in this paper is the CLD modelwhich has been described in Chapter I.

The data collected annually as part ofthe longitudinal study may be treated as across-sectional assessment of children'sconceptual learning, The present report,based on the first year assessment of thelongitudinal study, is the first such reportingof cross-sectional findings from the largerprogram. Performances of children on threeCLD assessment batteries are compared overfour age groups in order to obtain informationabout the course of children's conceptualdevelopment. This information is evaluatedin terms of various predictions that are de-rived from the CLD model.

The Strategy for the LongitudinalAssessment

The plan of the longitudinal investigationis to study a sample of children from flatage groups at four consecutive times wringsligntly more than three calendar years.The grade groups at the Wile of first-yearassessment in 1973 were kindergarten, third,sixth, and ninth. Each group will be testedin the'spring of 1974; 1975, and 1976. Thus,over three calendar years data will be gath-ered that include the entire age range of 5to 18 years with 100% overlap of the firstand final assessments for the four agegroups.

Essential control groups are incorporatedin the longitudinal design to permit an evalu-ation of possible confounding effects corn-monly associated with long-term repeated-measurement designs. Among these methodo-logical concerns is the possible role of re-,peated testing effects. In the present in-stance a variation of the Campbell andStanley (1963) posttest-only control groupdesign will be employed to evaluate the roleof repeated test administrations. Since thedesign Ito be used does riot provide for thedisentanglement of the effects of repeatedtesting and selective drop-out, specialattention will be directed toward the possiblechanging characteristics of the survivingcore longitudinal samples.

Sampling Design of theLongitudinal Study

Selective sampling problems and theassociated constraints upon external validityare difficult to avoid in any-investigation oftic present type. -While generalization of

2311

the resultant developmental norms will ob-viously be confined to similar age-gradelevels and demographic classifications,attempts will be made to ensure representa-tive sampling among classes within two dif-ferent school populations. Cohort biasesare not expected to be a major concern but

' will be controlled for in one school district.The target field locations for the longi-

tudinal study are Watertown, Wisconsin, andBeloit, Wisconsin. The public schools ofWatertown provide the locale for the initialtryout and validation of the CLD model assess-ment batteries related to the concepts' ofequilateral triangle, cutting tool, noun, andtree. At the same time, the four age groupsthat participate in these studies will alsobq followed for each of the successive years.

The Watertown and Beloit-stildiez com- ,

prise a simultaneous replication of the longi-rtudinal study. Beloit has been designatedas the major source of longitudinal datahowever, because its population better re-,fleets the distribution of socioeconomiclevels in the U.S. The overall sampling de-sign for the research in Beloit is shown inTable 1. The Watertown sample includes .a

asmaller number of children at each grade andno control groups.

Objectives of the Longitudinal Study

The major data analyses will utilize thelongitudinal and cross sectional assessmentsunder the following general guidelines. Thecross-sectional data collections will permitthe specification of relative talk difficulties(e.g., group means comparison, intercorrela-tions, and pass/fail contihgency,analyses)and the suggestion of the probable order ofacquisition of these concept domains. Thelongitudinal data collections will enable usto (1) specify the order of attainment of thevarious levels and uses of concepts by chil-dren in the various grade groups of the twoschool districts; (2) describe the form of thedevelopmental curve fox- each level of attain-ment, !concept use, and vocabulary acqui-sition 1r from first partial attainment throughfinal full mastery; and (3) relate the mastery

,'of each level to the mastery of each use, andto vocabulary development.

Thus, the primary objectives of the longi-tudinal study are (a) to chart the course ofchildren's attainment of selected conceptsin various subject fields during their schoolyears; (b) to chart the course of children'suses of the same concepts during theirschool years; (c) to chart the course of chil-drekils development of crucial terminology

12

related to the selected concepts; and (d) torelate the three preceding areas of develop-ment. The general propositions and morespecific predictions pertainingto these mat-ters that can be tested in a cross-sectionalstudy are treated later in this chapter. ,

Other goals of the longitudinal study,cross-sectional studies, and various controlledexperiments are as follows: (1) to determinemore explicitly the internal conditions of (.

learning associated with children's mastery ofthe various levels of concept attainment andtheir uses; (2) to determine more explicitlythe external conditions of learning that facili-tate children's attainment and use of conceptsin school settings; (3) to relate children'sperformances on the four CLD batteriesdealing with equilateral triangle, cutting tool,noun, and tree; (4) to relate children's levelsof conceptual development as assessed byOLD model batteries to their school achievementin various subject matters; and (5) to validatethe CLD model in terms of its robustness inproviding a framework for research in conceptlearning, concept development, and relatedinstruction.

Design of the Present First-YearCross-Sectiodal Study

(The data collected Bach year as part ofthetiongitudinal study may provide usefulinformation about patterns of conceptual learn-ing and development. Therefore the presentstudy serves both as the initial data collectionin the longitudinal program and also as across-sectional study of conceptual develop-ment.

Purposes of the Study

The CLD model embodies three majorpropositions. Related to each propositionthere are several specific predictions. Thesemajor propositions and the related predictionsconcern hypothesized patterns of children'sconceptual learning and development. Thepurpose of the present study is to test these'predictions, thereby clarifying presumedsequencing in conceptual development.'

A. Many concepts are attained in aninvariant sequence according to four succes-sive levels: concrete, identity, classificatory,and formal. Each level is presumed to be in-creasingly difficult to attain because of thenew operations which are essential to attainingthe particular level. It is further presumedthat to attain a concept at any particular levelan individual must be capable of all of the

TABLE 1

SAMPLING DESIGN FOR THE CLD TESTS

Cohort 197619 73

Time of Mearurement

19 74 19 75

19671968

6* (N=100) 78

9[Kindergarten] 6 4 0

7** 8** (Cohort effect group)1967 719 69 6** 7**19 6 7' 8**19 70 6**1967 9 (Test effect group)

19 64 9 10 11 12(N =100)[Third Grade] 9 u**19 65 10*

19 64 101966 9** - 10**1964 11 **19 67 9**19 64 12

1961 12 (N=10 13 140)[Sixth Grade] 12 13**

---14**19 62

1961 1319 63 12** 13**1961 14**1964 12**1961 15

195 8 15 (N =100) 16 17 18[Ninth Grade] 15 16**1959 17**

1958 161960 15** 16*-*195 8 17**1961 15**1958 18**

* Table entries are approximate mean ages

**These groups will not be continued if cohort and practice effects are not foundafter the first y ar. If effects are found, decisions about continuing will bemade after data are analyzed.

25 13

operations at that level and at the prior leveland must also have attained the particularconcept at the preceding level. These, then,are the prerequisite internal conditions forattaining each consecutive level.

The major proposition also indicatesthat many, but not necessarily all, conceptsare attained in an invariant sequence. Threeconditions are essential for concepts to beattained according to the invariant sequence.First, many actual instances or readily con-structed instances are present in the immediateenvironment-Chat children experience. Second,the child must have experiences with theactual instances or the representations thereofstarting early in childhood. Finally, thechild-must be developing normally, free ofsevere handicaps of speech,'language devel-opment, brain injury, etc.

The.preceding proposition concerning theinvariant sequence can be evaluated defin-itively only through longitudinal study.However, there are a number of predictionswhich follow from the proposition that canbe tested in a cross-sectional study in whichchildren of various age levels or grade groupsparticipate. The three specific predictionstested in the present cross-sectional studyare as follows:

1. All children of all grade groups willconform to five acceptable patternsof mastery of the four concept levels.',These acceptable patterns are to(a) fail all four levels (FFFF), (b) passthe concrete and fail the next threelevels (PFFF), (c) pass the concreteand identity lev:As but fail the nexttwo levels (PPFF), (d) pass the firstthree levels but fail the formal level(PPPF), and finally, (e) pars all fourlevels (PM).

2. The number and proportion of childrenwithin a single grade group who passeach successive level of conceptattainment will decrease, For example,fewer th.rd-grade children will passthe classificatory level than pass theidentity level.

3. The number _and proportion of childrenof successively nigner grade groupsmastering each concept level willincrease. For example, more sixth-grade children than third-grade chil-dren will pass each of the four levels.

B. Concepts attained to various levels:nay be used in (I) cognizing supraordinate-su`pordinate relationships in a 'nierarchy wherethe attained concept is an clement of thehierarchy, (2) in understanding principlesthat state a relationsnip between_thp attainedconcept and one or more other concepts, and

14

(3) in solving problems that require use of theparticular concept.

The specific predictions which followfrom the preceding proposition and which weretested in the present study are as follows:

4. A higher proportion of children whoattain a concept at'the formal level, incomparison with those who attain -itthe classificatory level, will alsomaster each of the three concept uses.

5, Children who attain a concept to only'the concrete andjor identity level willbe able to use that concept only in ---understanding simple perceptual relatetionships with other object coriceprs andin solving simple perceptuerrp-roblems.

6. The number and propoftion of childrenof successivelY-higher grade groupswho_ma-ster each concept use will in-

-crease.C. Having the labels of the concept and

its attributes (1) facilitates attainment of theconcept at the classificatory level and possiblythe'other levels, (2) is_reguis_ite for attainingthe concept at the formal level, and (3) facili-tates mastery of the three uses of theconcept.This proposition emphasizes the importance oflanguage in attaining concepts at the classifi-catory and the formal levels and also in beingable to use ,the concept in various ways.

The two specific predictions related tothis proposition which were tested in the pres-ent study may be stated as follows:

7. Vocabulary scores and scores based onattainment of the four levels and thethree uses will correlate positivelywithin grade groups. The correlationsmust be positive to support the predic-tion; however, within some gradegroups they may be low due to verylittle variability in either mastery ofthe vocabulary or in attainment of thevarious levels and uses.

8. Vocabulary scores and scores based onthe levels and uses will correlatepositively for the combined grade groupscorrelations should be higher thanthose obtained within grade groups.These correlations should be of greatermagnitude since large variations amongthe children both in vocabulary attain-ment and in attainment of the levelsand uses in expected when all thechildren of tne combined grade groupsare included.

Assessment Batteries Used

In addition to the usual criteria of re:ia-batty, , objectivity, and usability, several

26

other criteria guided development of the bat-teries. First, the mate:iais and instructionshad to permit assessment Of subjects of pre-ithool through high-school age. It was pre-sumed that not all subjects of preschool agewould attain a given concept at the concretelevel and that not all high-school subjectswould attain it at the formal level. Second,to test for attainment at the concrete, iden-tity, and classificatory levels the particularconcept selected had to have perceptibleinstances or representations thereof. Third,the concept had to be definable by publiclyaccepted attributes in order to test attainmentat the formal level., (It should be noted thatmany concepts are definable in terms of at-tributes even though this method of definitionis often not used, even in unabridged diction-aries.) Fourth, the concept selected for abattery should be relatable to the subjectmatter which pupils encounter in school.This is in keeping with the supposition thatdirected experience, incloding instruction in/school, is a powerful determinant of the par-"ticular concepts attained by individuals andalso of their level of attainment and use.Further, since much instruction in schooldeals with concepts, the CLD model shouldbe applicable to the design of instruction,and the subtests, when fully validated," shouldbe usable in assessing the level of conceptualdevelopment in school-age children. Fifth,the particular concept had to be part of ataxonomy in order to test its use in cognizingsupraordinate-subordinate relationships.Finally, the concept had to be usable in cog-

. nizing principles and in problem solving.(A concept may be usable in solving simpleproblems that can be solved on a perceptiblebasis without being used first in understandinga principle, or it may be used first in under-standing a principle and then in solving morecomplex problems.)

In the present study, assessment bat-teries were developed for each of the followingthree concepts: eqUilateral triangle (Klaus-meier, Ingison, Sipple, and Katzenmeyer,1973a) from the field of mathematics; cuttingtool , (Klausmeier, Bernard, Katzenmeyer,and Sipple, 1973) which is a general conceptprobably related more to science than to othercurriculum areas; and noun (Klausmeier,Ingison, Sipple , and Katzenmeyer, 1973b)from the field of English language arts.

To develop the tests of concept attain-ment and utilization, the behaviors involvedin attaining the concept were analyzed andthen test items and administrative proceduresto assess the behaviors were developed. Foreach of the three batteries, a subtest wasdeveloped to assess each of the four levels

of concept attainment. Each subtest wasconstructed specifically to assess the par-ticular operations involved in attaining a con-cept at each of the four levels. A subtest wasalso developed to ascertain the extent towhich a child could apply the concept in eachof the three uses that have been described.Thus, a total of 21 subtests was developed,seven for each of the three concepts. Allsubtests in each of.the three batteries weredesigned as paperand-pencil tasks thatcould be group administered.

For each of the batteries, item withinthe concrete, identity, and classif eaterylevels were constructed to be mor aifficultas nonexamples (1) increased in umbdr and(2) shared more relevant andjor rrelevantattributes with the target examples. Specificinformation regarding the exact number of itemsused in each subtest of a battery is availablein Tables 3, 12, and 21 which are includedin the three chapters dealing with results ofthe assessment of equilateral trianglk, cuttingtool, and noun. All test items went throughexpert review and empirical validation whileunder development.

Participating Children

Over 300 sonool children from Watertown,Wisconsin participated in eacn of the threeassessment series. The total number of children-tested in the three assessments varied slightly:324 children took part in the eguila_teral trianglestudy, 363 to the cutting tool study, and 32; inthe noun, study. The three populations were notindependent; that-is, 289 children received allthree batteries. Cnildrenat four grade levels- -kindergarten,, third, "sixth, and ninth grade--tookpart in each of the assessment series. Childrenin the three lower grades were enrolled in fourdifferent elementary schools. The ninth-gradestudents were enrolled in a high school.These schools and classrooms were judged tobe typical of the particular school system andalso of a large number of classrooms in smalltowns in Wisconsin. Each of the three subjectpopulations is described in greater detail in thechapters presenting results.,

Data Collection

The appropriate subtests of the batterywere administered to children in intact class-rooms, with the exception of kindergartenchildren. These youngest children were testedin small groups of about five to eight in orderto reduce distractibility and, in general, toenable the test administrators to supervise the

2715

test-taking situation more closely. Each ofthe three batteries was administered at a dif-ferent time within a six-month period. Twotest administrators, one male and one female,were responsible for giving all batteries inthe three assessment studies. The classroomteacher usually remained in the classroomduring the test administration to assist inmonitoring the test situation. The same ad-ministrators were responsibl- for scoring thetests and for coding the tes JAL, -nation forsubsequent data analyses.

Treatment of the Data

On each subtest of an assessment bat-tery a subject was scored as having eitherpassed or failed; that is, results of eachassessment battery were treated as dichoto-mous data. Criteria set for each subtest

16

determined passing, or attainment of a sub-test. Specific 'criteria for attainment of eachsubtest in each of the assessment batterieswill be explained in Chapters III, IV, and V,which present, respectively, results of theassessment of equilateral triangle, cuttingtool, and noun.

In general, data were quantified by com-puting frequencies ind proportions of subjectsat each grade group who attained each conceptlevel and each uses subtext. In addition,certain post tog statistical tests were usedwhere appropriate to obtain more specificinformation about differences in frequenciesand proportions. The predicted relationshipsbetween vocabulary and performance on thesubtests dealing with the concept levels anduses were evaluated by computing correlationcoefficients. Each of the specific predictionsoutlined in this chapter is evaluated in termsof these descriptive analyses.

28

1

IIIResults of CLD Assessment Series I:

Equilateral Triangle

Ovarviaw

In this chapter the child population isbriefly described. Specific criteria employedin the study for full attainment of each sub-test of the battery are also reported. Theresults, as they pertain to various predictionsbased on the CLD model, comprise the majorportion of the chapter,

Child Population

Table 2 provides a summary description,according to age and sex, of the 324 children

1

TABLE 2

who participated in the first assessmentseries, equilateral triangle.' Inspection ofthe table shows that 62 children from kinder-garten, 86 from third grade, 92 from sixthgrade, and 84 from the ninth grade took partin this study. Since intact classrooms wereused, the number of boys and girls withineach grade is unequal, although the totalnumber of boys and total number of girls, 161and 163 respectively, is very close. The agerange for each grade varied from 13 to 16months. The smallest range, 13 months, wasin the kindergarten group and the largest, 16months, occurred among the ninth graders.

NUMBER OF MALES AND FEMALES, MEAN AGE, AND AGE RANGE AT EACH GRADE GROUP

Grade Males FemalesMean Age

(if years and months)Age Range

(in years and months), _

K 32 30 5-10 5-4 to 6-53rd 48 38 8-10 8-3 to 9-56th 40 52 11-10 11-3 to 12-6 i

9th 41 43 14- 7 14-2 to 15-6

2917

Criteria for Full Attaintment

It was reported earlier that one test wasused for each level of concept attainment andone test for each concept use. For each ofthe levels and uses subtests specific criteriadetermined full attainment. In general, thecriterion established for each subtest was thatall items, e.:cept one, had to be passed.Permitting one error was presumed co providea reasonable degree of flexibility for possibleerror in measurement. The criteria establishedfor full attainment are especially importantinasmuch as passing the four levels consecu-tively is considered critical for describing thecourse of children's conceptual developmentas presumed by the CLD model.

Table 3 summarizes information describingthe seven subtests, including the total numberof items in each subtest and the criterion forfull attainment of each particular level anduse. There are two exceptions to the criteriaconvention; the classificatory subtest consistedof three items, all of which were required forattainment, and the labels test consisted ofseven items, five of which were required forattainment. The criteria for the other levels ofattainment and the three uses can be reviewedin Table 3. (Note that the model requires twoother kinds of items in the assessment of theformal level:, however, at the time of this ad-ministration these items were not yet developed.)

Proportion of Each Grad* GroupConforming to the Predicted

-Invariant SeqUence

Each pucce level of concept attain-ment is postulated by t D model to re-quire the use of one or mere neW cognitiveoperations. It is hypothesized that eachsubtest will be more difficult than the previoussubtest because it requires the use of anadditional cognitive operation. Thus, theCLD model itself constrains the number ofacceptable patterns of success and failure forthe four concept levels. Five patterns ofpassing and failing the four successive levelsof attainment are consistent with the model.These are: to fail all four levels (FFFF), topass the concrete level and fail the otherthree (PFFF), to pass both the concrete andidentity and to fail the last two (PPFF), topass the first three levels and fail formal(PPPF), and finally to pass all four levels(PPM.

The first five rows of Table 4 show thenumber and proportion of each grade groupthat attained the successive levels accordingto each of the five patterns predicted by themodel. Two hundred ninety-seven of the324 subjects, or approximately 92%, per-formed as predicted. More specifically,the following numbers conform to each of thefive predicted patterns: 9 FFFF, 5 PFFF,

TABLE 3

NUMBER OF ITEMS AND CRITERIA DEFINING FULL ATTAINMENT

FOR EACH CONCEPT LEVEL AND USE

Subte st Number of Items Criteria Foi Full Attainment

1. Concrete 8 7 correct

2. Identity 8 7 correct

3. Classificatory 3 3 correct

4. Formala. Discriminating Attributes:,b. Labels

3

7

2 correct5 correct

5. Principle 5 pairs 4 correct

6. Problem Solvincr 5 4 correct

7. Supraorclinate-Suboidinate 4 pairs 3 correct

18 30

37 PPFF, 120 PPPF, and 126 PPPP. The rankorder of grade groups conforming to predictedpatterns are ninth, third, sixth, and kinder-garten.

There are 11 patterns of performance thatare inconsistent with the model. The numberand proportion of children who displayed non-conforming patterns appear at the bottom ofTable 4. The 27 children (8%) who did notadhere to the predicted patterns were tiis-tributed over nine of the eleven deviatingpatterns. However, 17 of the exceptions areconcentrated in three patterns. Six childrenPassed the identity level who did not pass theconcrete level and five individuals passedboth the identity and classificatory level whodid not pass the concrete level. Finally,six individuals passed the concrete, failedthe identity, and then passed the classifica-tory level.

Inspection of Table 4 also reveals con-tributions of the four grade groups to non-conforming patterns. A total of 10 kindergartensubjects, six third graders, nine sixth gradersand two ninth graders deviated from predictedsequences. Nearly half of the 17 cases thatcluster in three patterns of exception, asnoted earlier, originate in the kindergartengroup.'

The large percentage of children whoconformed to the predicted patterns is highlysupportive of the major proposition that in-dividuals attain concepts in a highly predict-able sequence. Although the percentage,of,children whose performance was consistentwith the predicted sequence was very high, ,7

it may be informative to examine the proto-cols of children who showed non-conformingpatterns to ascertain how near each child cameto passing.at the criterion specified forconcept attainment.

Table 5 presents frequencies of subjectsaccording to each pattern of exception andnumber of items correct at each conceptlevel for which attainment criterion was notmet. Since non-conforming patterns are thosein which a subject fails a level and thengoes on to attain one or more higher levels,performances which barely miss the criterion

'''for attainment are the most easily explained.For example, Table 5 shows that about 50%of the subjects who failed the criterionfor attainment at the concrete level, but whodid attain at a more difficult level, fell shortof the criterion (seven items correct) by onlyone item. Similarly, half the subjects whofailed the identity level missed criterion forattainment by only one item, and all subjectswho failed the classificatory subtest, butwho went on to attain the formal level, fellshort by only one item. These "bare miss"

deviations from the sequence' of predictedattainment are probably most simply interpretedas errors of measurement inevitably associatedwith each subtest, or as consequences of thestringent criteria established for full attain-ment. Explanations for protocols that indicatea subject attained a more difficult conceptlevel after falling far short of criterion at alower level are more problematic. However,measurement error may be again invoked toexplain these cases, in addition to suchspecific situational contingencies as inatten-tiveness during test-taking and misunder-standing of directions.

Predicted Sequence ofConcept Attainment andDifficulty of the Levels

The CID hypothesis is that the sequenceof attainment is invariant because each suc-cessively higher concept level requires theuse of one or more increasingly complex cog-nitive operations. As a consequence the itemsand the total subtest at each successive levelare more difficult. It might be argued that 2the invariant sequence, of attainment is not afunction of difficulty determined by increasinglycomplex cognitive operations at the successiveconcept levels, concrete through formal, butthat it is 'simply a function of increasing testitem difficulty unrelated to the operations,.In order to ensure that the number of subjectsconforming and not conforming to the predictedsequence was not merely due to increasingdifficulty of the successive subtests unrelatedto the more complex /operations, a statisticalprocedure accounting for independent difficultylevel was applied to data from the presentassessment.

Computations were performed using theoverall grade-group proportions passing andfailing each of the four subtests so that awide range of conceptual attainment would beobtained. These computations yielded expectednumbers of subjects following each of the 16 ,possible patterns Of attainment (five acceptable,11 unacceptable to the model). It was notedthat fewer than five subjects were expectedto follow each of ,eight of the 16 patterns. Tomeet the requirements of the Chi-square test,patterns were combined so that the minimumexpected number of subjects in each cellwould approximate five. Ten patterns andcombinations of patterns resulted, and wereused for the test. A Chi-square goceiness-of-fit test was used to determine whether theobtained number of subjects whei followed thesepatterns differed significantly from the slumberof subjects expected to follow these patterns.

31 19

TABLE 4

NUMBER AND PROPORTION OF FOUR GRADE GROUPS CONFORMING ANDNOT CONFORMING TO PREDICTED SEQUENCE OF ATTAINMENT

Pass-Fail SequenceK

(n=62)3rd

(n =86)6th

(n=92)9th

(n=84)All Grades

(n=324)

FFFF 9 0 0 0 9

.15 .00 .00 .00 .03

PFFF 3 2 0 0 5

.05 .02 .00 .00 l .02

PPFF 20 13 3 1 37.32 .16 .03 .01 .11

PPPF 20 52 36 12 120.32 .61 1. .39 .14 .37

PPPP 0 13 44 69 126.00 .15 .48 .82 .39

Subtotal Conforming 52 80 83 82 297.84 .93 .90 .98 .92

FFFP 0 0 0 0 0

.00 .00 .00 .00 -.00

FFPF 2 0 0 0 2

.03 .00 .00 .00 .01

FFPP 0 1 0 0 1

.00 .01 .00 .00 .00

FPFF 4 0 2 0 6.07 .00 .02 .00 .02

FPFP 0 0 Of: 0 0

.00 .00 .00 .00 .00

FPPF , 3 1 0 1 5

.05 .01 .00 .01 .02

FPPP 0 0 1 0 1

.00 .00 .01 .00 .00

PFFP 0 0 0 t 1 ' 1

.00 .00 .00 .01 .00

PFPF 1 3 2 0 6

.02 .04 .02 .00 .02

PFPP 0 0 3 0 3

.00 .00 .03 .00 .01

PPFP 0 1 1 0 2

.00 .01 .01 .00 .01

Subtotal Not Conforming 10 6 9 2 27.16 .07 .10 .02 .08

20 32

TABLE 5

FREQUENCIES OF SUBJECTS ACCORDING TO PATTERN OF EXCEPTION ANDITEMS CORRECT AT EACH CONCEPT LEVEL NOT ATTAINED

N Pattern of Exception Number of Items Correct on Concrete Subtest (7 required)

2 3 4 5 6

6 FPFF 2 1 32 FFPF 1

1 FFPP 1

S FPPF 1 2 21 FPPP 1

Numb( r of Items Correct on Identity Subtest (7 required)

0 1 2 3 4 5 6

2 FFPF 1

I FFPP 1

1 PFFP1,

3 PFPP6 PFPF 5

Number of Items Correct on Classificatory (3 required)0 1 2

2 PPFP 21 PFFP 1

The resulting Chi-square provided convincingevidence that the number of subjects followingand not following acceptable patterns wasnot a function of increasing difficulty oftest items unrelated to the operations (X2 =70.64, d.f. = 5, p< .001).

Proportion of Grade GroupsAttaining the fOur Levels

Two related and important predictionsregarding sequence of attainment of conceptlevels can be tested in a cross-sectic.naistudy. The first prediction holds chat withina grade group the percentage of childrenpassing each successive level should de-crease. At the same time, the percentage ofchildren passing a given level should increaseas a function of increasing grade group.

Table 6 presents the number and propor-tion' of each grade group that fully attainedeach concept level and thus provides infor-mation relevant to both predictions. Reflect-ing the accuracy of the first prediction,which asserts that the percentage of hildrenwithin a single grade passing each successivelevel will decrease, Table 6 reveals thatthere is indeed a gradual decrement in tne

percentage of children at each grade groupwho pass the four successive concept levels.(One exception to the predicted directionoccurred in the kindergarten group where 71%mastered the concrete level and 76%, theidentity level. Hereafter, only reversals topredicted directions that are 5% or greaterwill be specifically noted.) The patternwithin each grade group is also observablein the overall percentages for all grades. Asshown in the bottom row of Table 6, 93% of thechildren passed the concrete level; 92%, theidentity level; 81%, the classificatory level;and 41%, the formal level.

Cochran Q tests were used to find out ifthe proportions of subjects fully attaining thefour concept levels differed significantlywithin each of the four grade groups. Signifi-cance of the differences among the proportionsfor each of the four grade groups was beyondthe .001 level (Q = 91.72, kindergarten; 155.99,third grade;, 89.13, sixth grade;, 30.25, ninthgrade ki.f. = 3J). McNemar tests were run atthe .05 level of significance to determine wherespecific differences in attainment among thefour levels occurred within each of the fourgrade groups. Six comparisons were possible:concrete with (1) identity, (2) classificatoryand (3) formal; identity with (4) classificatory

33 ----' -21

--H

TABLE 6

NUMBER AND PROPORTION OF EACH GRADE GROUP THATFULLY MASTERED EACH LEVEL OF ATTAINMENT

Grade Concrete Identity Classificatory Formal

K(n=62)Number 44 47 26 0

Proportion .71 .76 .42 .60

3rd (n=86)Number 84 80 70Proportion .98 .93 . 8 1 ----;

6th (n=92)Number 89 87 86 49Proportion .97 .95 .93 .53

9th (n=84)Number 83 83 82 70Proportion .99 .99 .98 .83

All Grades (N=324)Number , 300 297 264 134Proportion .93 .92 .81 .41

and (5) formal f finally, classificatory with(6) formal. Five of the six comparisons dif-fered significantly for the kindergarten gradegroup, the exception being concrete withidentity. In each set of comparisons that diddiffer significantly, fewer kindergarten chil-dren passed the higher concept level. Forthe third-grade group, the comparison betweenidentity and classificatory differed signifi-cantly. For the third-, sixth-and ninth-gradegroups, differences between concrete andformal, identity and formal, and classificatoryand formal were significant. In each compari-son, fewer children attained the higher con-cept level.

Information in the columns of Table 6 isdirectly relevant to the second prediction - -thepercentage of children passing any particularlevel of concept attainment should increaseas a functi in of increasing grade group. Forexample, 42% of kindergarten children,81% of the third graders, 93% of the sixthgraders, and 98% of the ninth graders fullyattained the concept at the classificatorylevel.

Chi-square tests were used to ascertainwhether the proportions of individual gradegroups passinj each of the four levels differej

22

significantly from the proportions of the com-bined grade groups passing each of the fourlevels of concept attainment. The differencein proportions attaining each of the levelswas significant beyond the .001 level (Chi-square = 52.55, concrete; 27.25, identity;87.53, classificatory; 130.41, formal [d.f. =3]). A Chi-square analog to Scheffe's theoremwas performed to determine where differencesbetween grade groups in attainment of eachof the four levels were significant at the .05level. Significant results were as follows:at the concrete, identity, and classificatorylevels, the third-, sixth; and ninth-gradegroups each surpassed attainment of the kin-dergarten group. At the classificatory level,performances of sixth and ninth graders werealso superior when each was compared withperformance of third graders. All pair-wisecomparisons of grade groups were signifi-cantly different at the formal level; that is,the higher the grade group, the greater theproportion of children attaining the formallevel.

Quite clearly, both of the predictionsdealing with difficulty of the levels receivestrong support from the data assessing fullattainment of the concept, equilateral triangle.

34

Relationship Between Full Attainmentof Various Levels and Full Attainmentof the Various WV

The CLD model specifies that individualswho have attained a concept only to the con-crete or to the identity level may be able touse that concept in cognizing simple percep-tual retions among concepts and in solvingsimple pr blems of a perceptualthey will not be el.:_le_to-tise-t` concept inus: Lerstanding-iiipraordinate-subordinate re-

------Tations, understanding more complex prin-ciples,' or in solving more complex problems.It also,implies that individuals who attainthe formal level will perform the uses subtestsmore effectively than those whose conceptualattainment is only at the classificatory level.

Inspection of Table 7 discloses informa-tion bearing on the first prediction. Forthose five individuals whose highest levelof concept attainment was concrete, nonepassed any concept use subtest. Forty-threechildren from kindergarten through grade ninepassed the identity level as their highestlevel of attainment. Of these, only threethird graders were able to pass a uses sub-test (supraordinate-subordinate). That is, illthe present research, only 7% of the 43 sub-jects achieving the identity level successfullymastered any of the uses of the conceptequilateral triangle.

Table 7 presents information of specialint9re-sf for comparing the performances ofindividuals attaining the classificatory leveland those attaining the formal level. Accord-ing to the CLD model, each of the conceptuses should be mastered by e higher percen-tage of children who perform at the formallevel than by children who attain only theclassificatory level.

A total of 133 subjects mastered theclassificatory level as their highest leveland another 134 subjects mastered the formallevel, as can be seen in the bottom rows ofTable 7. Comparisons between childrenperforming at these two levels in terms oftheir performance on the three uses subtestsare of special interest for both learningtheorists and educators.

The supraordinate-subordinate subtestwas passed by 8% of those who passedthe classificatory level and by 34% of thosewho passed the formal level. The p:inciplessubtest was mastered by 5% of thos whopassed the classificatory level comaaredto 43% of those who passed the formal level.Problem solving was passed by only 3%of the subjects whose highest attainmentwas the classificatory level, whereas34% of the subjects who passed the formal

level were also able to solve problems suc-cessfully.

Statistical tests also revealed the over-all superiority on uses subtests for childrenoperating at the formal level of conceptattainment. For each of the three cons,ot.--uses, Chi-square testsoveralladyantage-WYOnd the .001 level) for

performing at the formal level whencompared to those performing at the classi-ficatory level (Chi-square = 27.72, supra-ordinate- subordinate; 55.54, principle;41.6e, problem-solving [dd. =

In summary, the data obtained fromassessment of equilateral triangle stronglysupport the prediction that attainment of aconcept at the formal level, in comparisonwith attainment only at the classificatorylevel, facilitates uses of the concept.

Difficulty of the Three Uses

Table 8 presents the number and propor-tion of subjects who fully_ attained each ofthe three concept uses: supraordinate-sub-ordinate, principle, and problem solving.The prediction was that performance would im-prove as a function of increasing grade group.Of special interest is the marked improvementthat occurred in mastery of the three conceptuses between sixth and ninth grade. Thelarge increase in numbers of children whosuccessfully mastered the uses subtests isattributed to some combination of instructionregarding equilateral triangle, as well as tothe emergence of cognitive operations speci-fied by the model.

Chi-square tests were run to ascertainthe significance of the difference betweenthe proportions of individual grade groupspassing each of the three uses and theproportion of the combined grade groupspassing each of the three concept uses. Thedifference in the proportion of subjects attain-ing each of the uses was significant beyondthe .001 level (Chi-square = 48.92, supra-ordinate-subordinate; 117.88, principle;81.43, problem-solving (d.f. = 3)). A Chi-square analog to Scheffe's theorem was per-formed between all pairs of grade groups todetermine where the differences in the useswere significant at the .05 level. Statisti-cally significant results were as follows:the ninth grade's mastery of the three conceptuses was superior to that of each of the othergrade groups. In addition, performance ofsixth graders surpassed that of third gradersand kindergarteners on the subtest involvingprinciples. Sixth graders were superior tothird graders on the problem-solving subtest.

3523

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TABLE 8

NUMBER AND PROPORTION OF EACH GRADE GROUP THAT FULLYMASTERED EACH OF THE THREE CONCEPT USES

Grade Supraordinate- Subordinate Principle Problem Solving

K (n=62)Number 0 0 0

Proportion .00 .00 .00

3rd (n=86)Number 12 0 0

Proportion .14 .00 .00

6th (n=92)Number 11 14 12Proportion .12 .1k* .13

9th (n=84)Number 35 49 37Proportion .42 .58 .44

All Grades (N=324)Number 58 63 49Proportion .18 .19 .15

Examination of Table 8 reveals anotherpoint of interest. Eighteen percent of all sub-jects passed the supraordinate-subordinatesubtest, 19% the principle subtest, and 15%the problem-solving subtest. These rathermodest attainments must/be interpreted interms of the items used in the subtests, thecriteria established for passing the subtests,and the nature or the particular concept,equilateral triangle. Obviously, it should notbe assumed that these same results for fullattainment would obtain for other concepts.

The prediction that the higher grade groups,compared to the lower, would demonstrategreater mastery of concept uses has receivedstrong support from these data.

Relationship Behveen VocabularyDevelopment and Attainment ofConcept Levels and Uses

As described earlier, language has beenaccorded a central role in the learning ofconcepts by the CLD model as well as severalother theories. Figure 1 makes explicit thatverbal labels or otner symbols may be learnedat any of the first three levels and are essen-tial at the formal level of concept attainment.It is presumed also that the acquisition of

labels facilitates using the concept in solvingproblems, understanding principles, and cog-nizing supraordinate-subordinate relations.This section is specj,fically addressed tothe prediction that having the verbal labelsfor the concept of equilateral triangle and itsattributes will be positively correlated withattainment of the levels and performance onuses subtests.

In order to compute correlation coefficients,,a special scaling system was used. Foreach subject, a point score of 1 was assignedto full attainment of each concept level andeach use, and a score of 0 to each whenmastery was not attained. The second vari-able for all computations was the mean per-formance on the seven-item vocabulary test inwhich a score of 1 was again assigned toeach correctly answered item. Therefore,for each individual, scores on the four conceptlevels could vary from 0-44, scores on thethree concept uses could vary from 0-3; andscores combining levels and uses could varyfrom 0-7. (Combining levels and uses sub-tests provided a measure of overall task per-formance.) .Similarly, the scores on the vo-cabulary test varied from 0, for no labelscorrect, to a perfect score of 7. For eachsubject, then, overall performances on con-cept level subtests, uses subtests, combined

3725

levels and uses subtests, and vocabularywere calculated. (Oilier means of scoringthe levels and uses are under empirical study.)

This scoring systdm generated Table 9which presents means 4nd standard deviationsfor levels, uses, levelt and uses, and vocab-ulary scores at each grade group. It can beobberved that performance on the vocabularyteimproved considerably with increasing

d group. The predicted, improvement inconcept attainment with increasing gradegroup is again demonstrated.`,, In addition,older children, compared to iounger, obtainedhigher mean scores on uses subtests, as wellas on a combination of levels arid uses subtests.

Pearson product-moment correlations (r)were then calculated in order to discover therelationship between vocabulary comprehensionand task performance. Table 10 presents,for each grade group, the correlations be-tween scores on the vocabulary test andscores on (1) concept level, (2) concept uses,and (3) combined levels and uses.

Inspecting this table,shows that thecorrelations within each grade group arefairly modest, especially among the youngestsubjects; the low correlations reflect, nodoubt, a limited range of performance amongthe younger children. The vocabulary scoresof the kindergarten group, for example, showed

little comprehension of verbal labels; therange of attainment on concept levels wassmall; and no uses subtests were passed.For this grade group, correlations were, eitherzero or of a very low order. Third, sixth,and ninth graders, by contrast, demonstratedan increasing competence with verbal labels,and a wider range of performance on conceptattainment and concept uses. Thus, the ,

correlations for these grade groups, althoUghnot extremely high, do indicate a positive krelationship between test performance andvocabulary scores. In fact, seven of the ninecorrelations obtained for third, sixth, andninth grade groups were statistically signifi-cant from zero at or beyond the .05 level.

For the total subject population, correla-`tions were considerably higher. Betweenoverall performance on concept level subtestsand overall performance on the vocabularysubtest, the correlation was .57. The corre-lation between overall performance on threeconcept uses and vocabulary scores was .56and between ovell performance on combinedlevels and uses and vocabulary scores, .70.These three correlations were statisticallysignificant from 0 at or beyond the .01level. The predicted relation between vocab-,ulary proficiency and concept attainment anduse is clearly supported by the obtained cor-relations.

TABLE 9

MEANS AND STANDARD DEVIATIONS FOR COMBINED CONCEPT LEVELS,CONCEPT USES, COMBINED LEVELS AND USES,

AND VOCABULARY AT EACH GRADE GROUP

Grade ConceptLevels:(Maximumscore, 4)

ConceptUses:(Maximumscore, 3)

Levelsand Uses:(Maximumscore, 7)

Vocabulary:,(Maximumscore, 7)

M S.D. M S.D. M S.D. M S.D.62 2.27 1.19 .00 .00 2.27 1.19 .44 .80

3rd 86 3.50 .72 .141 .35 3.64 .82 2.83 1.65

6th 92 3.80 .50 .40 .74 4.21 .96 4.40 1.74

9th 84 3.90 .40 1.44 1.09 5.35 1.28 5.92 1.36

All Grades 324 3.46 .93 .52 .90 3.98 1.49 3.61 2.41

26 38

TABLE 10

PEARSON PRODUCT-MOMENT CORRELATIONSBETWEEN MEAN VOCABULARY SCORES AND MEAN SCORES ON

CONCEPT LEVELS, CONCEPT USES, AND COMBINED LEVELS AND USES.

Grade Group N Four ConceptLevels

Three ConceptUses

Combined Levelsand Uses

K 62 .14 .00 .15

3rd 86 , 43** .06 .40**

6th 92 .18 .384* .39** 0

9th 84 .25* .42** .44**

All Grades 324 .57** .56** .70**

* p < .05** p < .01

39

,

i

27

fi

IvResults of CLD Assessment Series H:

Cutting Tool

Overview

This chapter begins with a description ofthe child population and the criteria fOr fullattainment of each of the seven subtests usedin assessing cutting tool. Results of theassessment, as they bear on specific predic-tions of the CLD model, are recoiled in themajor part of this chapter.

Child Population

Three hundred and sixty-three children tr

participated in assessment of the, concept,cutting tool. Table 11 shows the compositionof the total number of subjects according toage and sex. Seventy-six children from kin-dergarten, 93 from third grade, 101 from sixthgrade, and 93 from ninth grade took part in

the study. The total number of boys andgirls is very close, 181 and 182 respectively;within each grade group boys and girls are notequally represented because intact classroomswere used.. Age ranges varied from 13 months,

Ifor the third graders, to 16 months, for bothsixth and ninth graders:

Criteria for Full Attainment

Criteria were again determined to definefull attainment of each of the four conceptlevels and three uses, In general, attainmenton a subtest was permitted only when thesubject missed no more than one item, asshown in Table 12. In order to provide someflexibility for possible error of measurement,,one' error was permissible. For several ofthe subtests, however, the criterion waslowered somewhat when the number of itemson a subtest was fairly large. For example,assessment of the concrete concept levelincluded 12 items, 10 of which were requiredfor full attainment. It should be noted that

40

in addition to tests for discriminating attri-butes and for knowledge of labels, a defi-nition (required for attainment) was includedat the formal level. Items comprising eachof the remaining subtests and specificcriteria for attainment may be found in Table 12.

ProportioA of Each Grade GroupConforming to the PredictedInvariant Sequence

The descriptive data included in thissection are addressed to the prediction thatthe sequence of attainment of the four conceptlevels is invariant. Only five patterns ofpassing and failing the four successivelevels are compatible with the CLD model(FFFF, PFFF, PPF', PPPF, and PPPP). Logicgenerates an additional eleven pass-failpatterns that are deviant, but must be in-cluded for an evaluation of the model's power.

The first five,rows ;, of Table 13 presentthe permissible pass-f it patterns and thenumber and proportion of each grade groupfalling within each pattern. From the totalnumber of 363 subjects, 340 or about 94%demonstrated attainment in a sequence con-sistent with the model. Nearly 90% of the340 subjects who did conform to acceptablepatterns can be observed in either the ?PPPor the PPPF sequence. The rank order ofgrade groups conforming to predicted patternsare third, ninth, kindergarten, and sixth.

The number and proportion of childrenwho displayed non-conforming patterns appearin the remainder of Table 13. About 6% (23subjects) of the total population deviated fromthe acceptable sequence; all cases fall intoone of three deviating patterns, but most (18out of 23) displayed the PPFP pattern. Thatis, 78% of the subjects who showed non-conforming patterns passe the concrete and

29'

TABLE 11

NUMBER OF MALES AND FEMALES, MEAN AGE, AND AGE RANGE AT EACH GRADE GROUP

Grade Males Females Mean Age Age Range(in years and months) (in years and months)

K 38 38 6-0 5-6 to 6-9

3rd 53 40 9-2 8-6 to 9-7

6th 44 , 57 12-2 1174 to 12-8

9th 46 47 15-1 ^14-4 to15-8

NUMBER OF ITEMS AND CRITERIA DEFININp FULL ATTAINMENTFOR EACH CONCEPT LEVEL AND USE

1

SubtestNumber

ofItems

driteria For Full Attainment

I

1.; Concrete 12 10 correct

2, Identity 12 10 correct

3. Classific'atory 6 5 correct

4. Formal

a. Discriminating Attributes '5 4 correctb. Labels 6 5 correctc. Definition 1, 1 correct

5. Principle 5 pairs 4 correct

6. Problem Solving 5 4 correct

7. Supraordinate-Subordinate 4 pairs 3 correct

identity levels, failed classificatory, andwent on to attain the formal level.

Sixth graders contributed to'eleven ofthe 23 deviating cases and all of these weresubjects who attained the concrete and iden-tity levels, but who passed the formal levelafter failing the classificatory. S.x non-con-forming cases showed up among ninthgraders, also in the PPFP pattern of exception.Five subjects in the kindergarten group didnot conform; only one case of deviation wasdisplayed by the third-grade group.

Although the percentage of children whose

30

performance was consistent with predictedsequence of attainment was very large,examination of the protocols of children whoseperformance violated the prediction may beinstructive; explanations for these anomalousrases may lie in a more thorough scrutiny oftheir test performance.

Table 14 shows frequehcies of subjectsaccording to each pattern of exception andnumber of items correct at each concept levelfor which attainment criterion was not met.Three subjects passed the-itity level afterfalling far short of criterion for the concrete

41

TABLE 13

NUMBER AND PROPORTION OF FOUR GRADE GROUPS CONFORMING ANDNOT CONFORMING TO PREDICTED SEQUENCE OF ATTAINMENT

Pass-Fail SequenceK

(n=76)3rd

(n=93) _6th

(n=101)9th

(n=93)AD Grades

(n=363)

FFFF 1 0 0 0 1

.01 .00 .0i. .00 .00FsFF 0 0 0 0 0

.00 .00 .00 .00 .00PPFF 15 15 4 0 34

.20 .16 .04 , 00 .09PPPF 54 42 26 8 130

.71 .45 .26 .09 .36PPPP 1 35 60 79 175

.01 .38 .59 .85 .48Subtotal Conforining 71 92 90 87 340

.93 .99 .89 .94 .94FFFP 0 0 0 0 0

.00 .00 .00 .00 .00FFPF 0 0 0 0 0

.00 .00 .00 .00 .00FFPP 0 0 0 0 0

.00 .00 .00 .00 .00FPFF 3 0 0 , 0 3

.04 .00 .00 .00 .01FPFP 0 0 0 0 0

.00 .,00 .00 .00 .00FPPF 0 0 0 0 0

.00 .00 .00 .00 .00FPPP 0 0 0 0 0

.00 .00 .00 .00 .00_P-F-PF--- 0 0 0 0 0

.00 .00 .00 .00 .00 _PFPF 2 0 0 0 2

.03 .00 .00 .00 .01PFPP 0 0 0 0 , 0

.00 .00 .00 .00 .00PPFP 0 1 11 6 18

.00 .01 .11 .07 .05

Subtotal Not Conforming 5 1 11 6 23.07 .01 .11 .07 .06

42 31

TABLE 14

FREQUENCIES OF SUBJECTS ACCORDING TO PATTERN OF EXCEPTION ANDITEMS CORRECT AT EACH CONCEPT LEVEL NOT ATTAINED

Pattern of Exception

3 FPFF

2 PFPF

18 PPFP

Number of Items Correcton Concrete Subtest (10 required)

0 1 2 3 4 5 6 7 8 9

Number of Items Correcton Identity Subtest (10 required).

2 3 4 5 6 7 8! 9

Number of Items Correcton Classificatory Subtest (5, required)

0 I 2 3 4

18

2

subtest (10 of 12 items were required for at-tainment) . These three subjects were all inthe kindergarten group. Explanation for suchdeviating cases probably, must rest, at thispoint, in presuming inattention to directionsor misunderstanding test instructions. Twoindividuals showed the PFPF pattern of excep-tion and 18 the PPFP pattern. All of thesecases failed a level that should have beenattained, according to the CLD model, byfalling just one item short of criterion,, ascan be obser"ed in Table 14. Thus, whenprotocols are examined, it is clear that apredicted pattern was just barely missed.Errors of measurement and stringency of cri-teria 'are probably both responsible for sudhperformances.

Predicted Sequence ofConcept Attainment andDifficulty of the Levels

The CLD hypothesis is that the sequenceof attainment is invariant because each suc-cessively higher concept level requires theuse of one or more increasingly complexcognitive operations. As a consequence theitems and the total subtest at each successivelevel are more difficult. It might be arguedthat the invariant sequence of attainment isnot a function of lifficulty Jetermine,i byincreasingly complex coinitive operationsat the successive concept levels, concrete

32

through formal, but that it is simply a functionof increasing test item difficulty unrelatedto the operations. In order to ensure that thenumber of subjects conforming and not con-forming to the predicted sequence was notmerely due to increasing difficulty of thesuccessive subtests unrelated to the morecomplex operations, a statistica' procedureaccounting for independent difficulty levelwas applied to data from the present assess-ment.

Computations were performed using theoverall grade-group proportions passing andfailing each of the four subtests so that awide range of conceptual attainment would beobtained. These computations yielded ex-pected numbers of subjects following eachof the 16 possible patterns of attainment(five acceptable, 11 unacceptable to themodel). It was noted that fewer than fivesubjects were expected to follow each of12 of the 16 patterns. To meet the requirementsof the Chi-square test, patterns were combinedso that the minimum expected number of sub-jects in each cell would approximate five.Six patterns and combinations of patterns re-sulted and were used for the test. A Chi-square goodness-of-fit test was used to de-termine whether the obtained number of sub-jects who followed these patterns differedsignificantly from the number of subjectsexpected to follow these patterns. The re-sulting Chi-square provided convincing evi-dence that the number of subjects following

43

GrO e0,-/rO-0

and not following acceptable patterns wasnot a function of increasing difficulty oftest items unrelated to the operations (X2 =9.53, d.f. =1, p < .01).

Proportion of Grade GroupsAttaining the Four Levels

The interrelated predictions that will beevaluated in this section state that withina given grade group the percentage of chil-dren passing each successive level shoulddecrease and, at the same time, the percent-age of children passing a given level shouldincrease as a function of increasing gradegroup.

The number and proportion of each gradegroup that fully attained each concept levelare presented in Table 15. Inspection ofthese data shows that within each gradegroup, as predicted, fewer children attainconcept levels as the levels become moredifficult. This trend is especially markedwhen proceeding from identity to classifica-tory and from classificatory to formal. Forexample, about 95% of the kindergarten sub-jects attained both the concrete level andthe identity level; but the percentage attainingthe classificatory level declined to 75% andonly 1% attained the formal level. Similarpatterns occur with each of the other gradegroups, as well as in the proportions re-sulting from combining all grade groups.

Cochran Q tests were used to find outif the proportions of subjects fully attainingthe four concept levels differed significantlywithin each of the four grade groups. Signifi-cance of the differences among the propor-tions for each of the four grade groups wasbeyond the .01 level (Q = 173.76, kinder-garten; 137.95, third grade; 58.46, sixthgrade; 14.57, ninth grade [def., = 3]).McNemactests were run at the .05 level ofsignificance to discover where specificdifferences in attainment among the fourlevels occurred within each of the four gradegroups. Six comparisons were possible:concrete with (1) identity, (2) classificatoryand (3) formal; -identity with (4)-classificatoryand (5) formal; finalliclassificatory with(6) formal. In the kindergarten-, third; andsixth-grade groups , every comparison showeda significant difference except that betweenconcrete and identity levels, That is, everycomparison within each of these grade groupsindicated fewer children attained the higherconcept level. In the ninth-grade group,four comparisons were significantly differentin attainment of levels: concrete and classi-ficatory; concrete and formal; identity and

classificatory; and identity and formal. Again,,the higher the concept level, the fewer thechildren who attained it.

According to the CLD model, the percent-age of children passing any particular levelof concept attainment should increase as afunction of increasing grade group. Thecolumns of Table 15 reveal a consistent trend,at each concept level, for the percentagesattaining a level to increase as grade groupincreases. This increment is especiallypronounced at the formal level in-.which at-tainment was demonstrated by 1% of thekindergarten grade group, 39% of the thirdgraders, 70% of the sixth graders and 91%of the ninth graders.

Chi-square tests were used to determinewhether the proportions of individual gradegroups passing each of the four levels ofconcept attainment differed significantly fromthe proportions of the combined grade groupspassing each of the four levels. The differencein proportions attaining each of the levelswas significant beyond the .05 level (Chi-square = 15.27, concrete;. 11.42, identity;11.33, classificatory; 156.36, formal [d.f. = 3]).In order to discover where differences amonggrade groups in attainment of each of thefour levels were significant at the .05 level,a Chi-square analog to Scheffe's tneorem wasused. Statistically significant results wereas follows: at the concrete and identitylevels, third-, sixth; and ninth-grade groupseach surpassed attainment of kindergartners.At the classificatory level, the ninth-gradegroup was superior to the kindergarten group.At the formal level, all pair-wise comparisonsof grade groups were significantly different;that is, the higher the grade group, the greaterthe proportion of children attaining the formallevel.

The predictions that deal with increasingdifficulty of the levels received considerablesupport from the data assessing full attain-ment of the concept, cutting tool.

Relationship Between Full Attainmentof Various Levels and Full Attainmentof-the-Various. -User

This section is concerned with the twopredictions that deal with the relation betweenlevel of concept attainment and use of theconcept in cognizing supraordinate-subordinaterelations, understanding principles, and insolving problems, First, the CLD modelpredicts that individuals who are able to attaina concept only at the concrete or identitylevel are restricted in their effective use ofthe concept; simple problenis of a perceptual

4433

TABLE 15

NUMBER AND PROPORTION OF EACH GRADE GROUP THAT FULLYMASTERED EACH /LEVEL OF ATTAINMENT

Grade Concrete Identity Classificatory Formal

K (n=76)Number 72

// 73 57Proportion .95 .96 .75 .01

3rd (h=93)Number 93 93 77 36Propottion 1.00 1.00 .83 .39

6th/(n=101)Number 1 0 1 1 0 1 86 71'Proportion 1.00 1.00 .85 .70

9th (n=93)Number 93 93 87 85Proportion 1.00 1.00 .94 .91

All Grades (n=363)Number 359 360 307 193Proportion .99 .99 .85 .53

kind may be solved and similar principles maybe understood, but more complex problemsolving and understanding principles is notpredicted.

Consider Table 16. No children attainedthe concrete level as their highest level.Among those 37 children whose highest levelof attainment was identity, there were 16instances of attainment on uses (nine, supra-ordinate-subordinate; six, problem-solving;one, principles). Such a large number is, ofcourse, discrepant with the prediction andrequires some further attention.

Twelve subjects were responsible for these16 instances of mastery of uses by identitylevel attainers. That is, in several casesthe same subject attained criterion on morethan one uses subtest. A more detailed ex-amination of the protocols of these 12 childrenmay provide some explanation for their perfor-mance. Ten Jf the 12 subjects barely missedcriterion for attainment at the classificatorylevel. Five out of six ttems were required formastery; these children passed four out of six.Eleven of these subjects missed the same twospecific items of the six included at the class-ificatory level. These two items, in particu-lar, were difficult because item instructionscould have been legitimately interpreted toinclude responses that were, in fact, 'scoredas incorrect. These ambiguous classificatoryitems have been revised or discarded for future

34

use in the cutting log battery. Proficiencyon the uses subtests by identity-level children,then, is attributed to their barely missing ,

attainment at the classificatory level, whichin turn was due to several ambiguous items.

Comparing performances of individualsattaining the classificatory level with thoseattaining the formal level permits an evalu-ation of the second prediction to be consid-ered in this section: concept uses shouldbe mastered by a higher percentage of chil-dren attaining the formal level compared tothose attaining at the classificatory level.Table 16 shows that the supraoedinate-subcir-dinate subtest was passed by 27% of thosewho attained at the classificatory level and75% of those at the formal level. The prin-ciple subtest was passed,by 7% of classi-ficatory attainers compared to 48% of thosewho attained at the-formal level. Finally,20% of those individuals who attained atthe classificatory level, compared to 73%of those attaining at the formai level, wereable to solve problems successfully.

For each of the three concept uses,Chl-square tests also showed a significantadvantage (beyond the .001 level) for chil-dren performing at the formal level whencompared to those performing at the classi-ficatory level (Chi- square = 71.09, supra-ordinate-subordinate 61. 07 , principle; 87.59,

45

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problem solving [d.f. = 11).Data obtained from assessment of cutting

tool_ provide strong support for the predictionthat, in comparison with attainment at theclassificatory level, attainment of a conceptat the formal level has a facilitative effecton ability to use the concept.

Difficulty of the Three Uses

In order to evaluate the prediction thatperformance on uses subtests would improveas a function of increasing grade group,Table 17 presents the number and proportionof subjects at each grade who fully attainedeach of the three concept uses: supraordinate-subordinate, principle, and problem solving.Inspection shows that with each successivelyhigher grade group there was a marked in-crement in percentage of subjects meetingcriteria for attainment. For example,. 8% ofthe kindergarten group, 43% of the thirdgraders, 66% of the sixth graders, and 82%of the ninth graders fully attained the supra-ordinate-subordinate subtest. The samepattern of progression occurred for the othertwo uses subtests.

Examination of Table 17 also shows that52% of all subjects passed the supraordinate-subordinate subtest, 28% , principle, and47%, problem solving. These overall attain-ments on uses are fairly impressive, butmust be considered as specific to the par-ticular concept, subtest items, and criteriafor mastery.

Chi-square tests were used to find thesignificance of the difference between theproportions of individual grade groups passingeach of the three uses and the proportion ofthe combined grade groups passing each ofthe three concept uses. The difference in theproportion of subjects attaining each of the useswas significant beyond the .001 level (Chi-square = 103.48, supraordinate-subordinate;112.15, principle; 122.32, problem solving [d.f.= 3J). A Chi-square analog to Scheffe's theoremwas performed between all pairs of grade groupsto discover where the differences in performanceon the uses subtests were significant at the.05 level. Statistically significant resultswere as follows:, for the supraordinate-subor-dinate subtest, the performances of third-,sixth-, and ninth-grade groups were each supe-rior to that of kindergarteners; both sixth- andninth-grade groups surpassed the performanceof third graders. For the principle subtest, theperformance of ninth graders was superior tothat of each of the other grade groups; attain-ment of sixth graders also surpassed thirdgraders and kindergarteners. For the problem-solving subtest, performance of the ninth-

36

grade group was superior to that of each ofthe other grade groups; the third-and sixth-grade groups also each surpassed the per-formance of the kindergarten group.

Data obtained from performance on usessubtests of the concept, cutting tool, stronglysupport the prediction of an increased abilityto use the concept with increasing gradegroup.

Relationship Between VocabularyDevelopment and Attainment ofConcept Levels and Uses

This section of the present report isconcerned with the predicted relation betweenvocabulary proficiency and attainment of thelevels and performance of the uses. Inorder to compute correlation coefficients,the same scaling system was employed.forcutting tool as'that described in Chapter IIIfor enullaieral triangle. The reader is re-ferred to the preceding chapter for a completedescription of the scaling system.

Table 18 presents means and standarddeviations, based on the scaled scoringsystem, for levels, uses, combined levelsand uses, and vocabulary scores at eachgrade group. Performance on the six-itemvocabulary test of the cutting tool assessmentbattery was quite proficient at each of thefour grade groups, although Table 18 doesshow some gradual improvement over grades.The predicted improvement in concept attain-ment with increasing grade group is, ofcourse, apparent in these data based onmean scaled scores as it was in the databased on proportions. Table 18 also showsthat, compared to younger children, olderchildren obtained higher mean scores on theuses subtests and on overall test performance(i.e., combining levels and uses).

Pearson product-moment correlationswere calculated between scores on the vo-cabulary test and scores on (1) concept level,(2) concept uses, and (3) combined levelsand uses. These correlations for each gradegroup and over all grade groups are presentedin Table 19. Although fairly modest, eightof the correlations within grade groups weresufficiently high to show statistical signi-ficance from zero at or beyond the .01 level.Interestingly, the correlations at the ninth-grade group are very low (and include onenegative correlation). Explanation for theselow correlations may be found by examinationof Table 18 which indicates little variabilityin obtained mean scores for the ninth graders;variability on the vocabulary test was ex-tremely small. The low correlations obtained

47

for the ninth-grade group reflect this markedlack of variability.

Compared to within-grade group correla-tions, those obtained for the total subjectpopulation were of a greater magnitude, aspredicted, and all were statistically signifi-cant from zero at or beyond the .01 level.The correlation between vocabulary and over-

TABLE 17

all performance on concept level subtestswas .51. Vocabulary correlated .43 withoverall performance on concept uses,.and.52 with overall performance ,on levels anduses. The predicted relationship betweenvocabulary proficiency and concept attain-ment and use receives strong support fromthese data.

NUMBER AND PROPORTION OF EACH GRADE GROUP THAT FULLYMASTERED EACH OF THE THREE CONCEPT USES

Grade Supraordinate-Subordinate Principle Problem Solving

K (n=76)Number 6 0 0Proportion .08 .00 .00

3rd (n=93)Number 40 10 38Proportion .43 .11 .41

6th (n=101)Number 67 30 56Proportion .66 .30 .55

9th (n=93)Number 76 62 78Proportion .82 .67 .84

All Grades (n=363)Number 199 102 172Proportion .52 .28 .47

TABLE 18

_ MEANS AND STANDARD DEVIATIONS FOR COMBINED CONCEPT LEVELS,CONCEPT USES, LEVELS AND USES, AND VOCABULARY AT EACH GRADE GROUP

ConceptLevels:

(MaximumGrade Group N Score. 4)

-ConceptUses:

(MaximumScore, 3)

Levelsand Uses:, Vocabulary:

(Maximum (MaximumScore, 7) Score, 6)

M S.D. M S.D. M S.D. M S.D.K 76 2,67 .67 .08 ,27 2.75 .73 4.72 1.39

3rd 93 3.22 .70 95 .86 4,16 1.23 5.22 1.22

6th 101 3.55 .57 1.5i 1.00 5.07 1.31 5.65 .939th 93 3.85 .36 2.32 .81 6.17 1.01 5.96 .20

All Grades 363 3.36 .72 1.28 1.13 4.63 1.64 5.42 1.11

4837

TABLE 19

PEARSON PRODUCT-MOMENT CORRELATIONS BETWEEN MEAN VOCABULARY SCORESAND MEAN SCORES IN CONCEPT LEVELS, CONCEPT USES, AND COMBINED LEVELS AND USES

Grade Groiip NFour

Concept Levels''Three

Concept Uses,

K 76 .41 ** .20

3rd 93 .35** .30**

6th 101 .44** .3 1**

9th 93 -.09 .15

All Oracles 363 .51 ** .43**

CombinedLevels and Uses

38

** < .01

49

VResults of CLD Assessment Series III: Noun

Overview

The present chapter firtt describes thechild population, and then presents the spe-cific criteria determining full attainment ofeach of the seven subtests in the battery.Results of the assessment of noun as theybear on specific predictions of the CLD model,comprise the major portion of. this chapter.

Child Population

Table 20 shows that 325 children par-ticipated in this assessment: 59 kindergar-teners, 88 third graders, 91 sixth graders,and 87 ninth graders. The total number ofboys and girls, 163 and 162 respectively, isvery close although within grade groups thenumber of boys and girls is not equal becauseintact classrooms were used. Age rangevaried from 14 months for the two youngestgrade groups to 17 months for the nintn-grade.group.

Criteria for Full Anointment

The number of items included yiffachsubtest and the criterion detelinining fullattainment on each subtest are presented inTable 21. In general, attainment of a subtestwas permitted only when the subject missedno more than one item; permitting one errormade some allowance for possible error ofmeasurement. The subtest at the formallevel also included a definition which wasrequired for attainment of that level. Compo-sition of the remaining subtests, along withcriteria for passing, can be found in Table 21.

Proportion of Each Grade GroupConforming to the PredictedInvariant Sequence

In this section, descriptive data are con-

sidered in order to evaluate the predictionthat the sequence of attainment of the fourconcept levels is invariant. The number andproportion of subjects at each grade groupwho attained the successive levels accordingto the five patterns that are consistent withthe model are presented in the first five rowsof Table 22. Three hundred and nineteensubjects, or about 98%, conformed to themodel. More specifically, the five predictedpatterns included the following numbers42 FFFF, 16 par, 95 PPFF, 141 PPPF, and25 PPPP. Raak order of grade groups conform-ing to predicted patterns are sixth, third,ninth, and kindergarten.

The number and proportion of subjectswho performed according to the 11 patternsof performance that are inconsistent withthe model are presented in the remainder ofTable 22. Of the six children (2%) whodeviated, three showed the FPFF and threethe PPFP patterns of exception. Two kinder-garten children, one third grader, one sixthgrader, and two ninth graders failed to con-form to predicted patterns.

Table 23 presents frequencies of subjectsfor each pattern of exception observed in thepresent assessment. Examination of thesefive protocols may be helpful in determiningwhy performance did not conform to the pre-dicted sequence. Three subjects (all fromthe two youngest grade groups) passed theidentity level after failing the concrete level.Inspection of Table 23 shows that these threechildren just missed criterion for attainmenton the concrete subtest. One sixth graderand two ninth graders failed the classificatorylevel, but went on to pass formal. Exami-nation of Table 23 indicates that two of theseindividuals barely missed the criterion forattainment on the classificatory subtest, andone individual fell further short of criterion:It seems most reasonable to diagnose allsix cases .of non-conformity to the model asdue to either measurement error, stringencyof criteria, or both.

5039

TABLE 20

NUMBER OF ALES AND FEMALES, MEAN AGE, AND AGE RANGE AT EACH GRADE GROUP

Grade Males

K 31

3rd 49

6th 39

9th 44

Females Mean Age Age Range(in years and months) (in years and months)

28 6-3 5-8 to 6-10

39 9-4 8-7 to 9- 952 12-4 11-6 to 12-10

43 15-2 14-5 to 15-10

TABLE 21

NUMBER OF ITEMS AND RITERIA DEFINING FULL ATTAINMENTFOR EACH ONCEFIT LEVEL AND USE

SubtestNumber

of1 Items

Criteria for Full Attainment

1. Concrete 8 7 correct

2. Identity 8 7 correct

3. Classificatory 8 7 correct

4. Formala . Discriminating attributes 4 3 correctb. Labels 9 7 correctc. Concept definition 1 1 correct

5. Principle 4 pairs 3 correct

6. Problem Solving 4 3 correct

7. Supraordinate-Subordinate 4 pairs 3 correct

Predicted Sequence ofConcept' Attainment andDifficulty of the Levels

The CLD hypothesis is that the sequenceof attainment is invariant because each suc-cessively higher concept level requires theuse of one or more increasingly complex cog-nitive operations._ As a consequence theitems and the total subtest at each successivelevel are more difficui:. It might be arguedthat the invariant sequence of attainment isnot a function of difficulty determined by

40

increasingly complex cognitive operationsat the successive concept levels,, concretethrough formal, but that it is simply a functionof increasing test item difficulty unrelatedto the operations. In order to ensure thatthe number of subjects conforming and notconforming to the predicted sequence wasnot merely due to increasing difficulty of thesuccessive subtests unrelated to the r.orecomplex operations, a statistical procedureaccounting for independent difficulty levelwas applied to data from the present assess-ment:

PO tO7 -744-5

TABLE 22

NUMBER AND PROPORTION OF FOUR GRADE GROUPS CONFORMING ANDNOT CONFORMING TO PREDICTED SEQUENCE OF ATTAINMENT

Pass-Fail SequenceK

(n=59)3rd

(n=48)6th

(n=91)9th

(n=87)All Grades

(n=325)

FFFF 40 1 1 0 42.68 .'01 .01 .00 .13

PFFF 16 0 0 0 16.27 .00 .00 .00 .05

PPFF 1 57 24 9513

.02 .65 .26 .15 L .29

PPPF 0 29 64 48 141.00 .33 .70 .55 .43

PPPP 0 0 1 24 25.00 .00 .01 '.28 .08

Subtotal Conforming 57 87 90 85 319.97 .99 .99 .98 .98

FFFP 0 0 0 0 0.00 .00 .00 .00 .00

FFPF 0 0 0 0 0.00 .00 .00 .00 .00

FFPP 0 0 0 0 0

.od .00 .00 .00 .00

FPFF 2 1 0 0 3.03 .01 .00 .00 .01

FPFP' 0 0 0 0 0.00 .00 .00 .00 .00

FPPF 0 0 0 0 0.00 .00 .00 .00 .00

FPPP 0 0 0 0 0

.00 .00 .00 .00 .00

PFFP 0 0 , 0 0 0.00 .00 .00 .00 .00

PFPF 0 0 0 0 0.00 .00 .00 .00 .00

PFP 0 0 0 0 '0.00 .00 .00 .00 .00

PPFP 0 0 1 2 3

.00 .00 .01 .02 .01

Subtotal Not Conforming 2 1 1 2 6.03 .01 .01 .02 .02

52 41

TABLE 23

FREQUENCIES OF SUBJECTS ACCORDING TO PATTERN OF EXCEPTION ANDITEMS CORRECT AT EACH CONCEPT LEVEL NOT ATTAINED

N Pattern of Exception

3 FPFF

3 PPFP

Number of Items Correcton Concrete Subtest (7 required)

0 1 2 3 4 5 6

1 2

Number of Items Correcton Classificatory Subtests (7 required)

0 1 2 3 4 5 6

1 2

Computations were performed using theoverall grade-group proportions passing andfailing each of the four subtests so that awide range of conceptual attainment would beobtained: These computations yeilded ex-pected numbers of subjects following eachof the 16 possible patterns of attainment(five acceptable , 11 unacceptable to the model).It was noted that fewer than five subjectswere expected to follow each of the 16 pat-terns. To meet the requirements of the Chi-square test, patterns were combined so thatthe minimum expected number of subjects ineach cell would approximate five. Ten pat-terns and combinations of patterns resultedand were used for the test. A Chi-squaregoodness-of-fit test was used to determinewhether the obtained number of subjects whofollowed these .patterns differed significantlyfrom the number of subjects expected tofollow these patterns. The resulting Chi-square provided convincing evidence thatthe number of subjects following and notfollowing acceptable patterns was not a func-tion of increasing difficulty of test itemsunrelated to the operations (X2 = 350.94,d.f. p < .01).

Proportion of Grade GroupsAttaining the Four Levels

Tne number and proportion of each gradegroup that fully attained each concept levelare presented in Table 24. These data willbe employed to evaluate two interrelatedpredictions: within a given grade group thepercentage of children passing each succes-

42

siVe level should decrease and, at the sametime, the percentage of children passing agiven level should increase as a function ofincreasing grade group.

Inspection of Table 24 shows that withineach grade group, as predicted, fewer chil-dren attain concept levels as the levelsbecome more difficult. The pattern obtainedwithin each of the four grade groups is re-flected also in the data for combined grades;86% of the subjects attained the concretelevel; 82%, identity; 51%, classificatory; and9% , formal.

Cochran Q tests were used to find outif the proportiqps of subjects fully attainingthe four concept levels differed significantlywithin each of the four grade groups. Sig-nificance of the differences among the propor-tions for each of the grade groups was beyondthe .01 level (Q = 40.97, kindergarten;211.51, third grade; 213.47, sixth grade;14863, ninth grade [d.f. = 3]). McNemartests were run at the .05 level of significanceto discover where specific differences inattainment among the four levels occurredwithin each of the four grade groups. Sixcomparisons were possible: concrete with(1) identity, (2) classificatory and (3) formal;(4) identity with classificatory and"(5) formal;finally, (6) classificatory with formal. Forthe kindergarten group, three comparisonsshowed a significant difference: fewerkindergarten children attained the identity,classificatory,, and formal levels when eachwas compared with the concrete level.. Inthe third-, sixth-, and ninth-grade groups,every comparison showed a significant dif-ference except that between concrete and

S3

TABLE 24

NUMBERAND PROPORTION OF EACH GRADE GROUP THAT FULL!MASTERED EACH LEVEL OF ATTAINMENT

Grade Concrete Identity Classificatory Formal

K (n=59)Number 17 3 0 0Proportion .29 ,05 .00 - .00

3rd (n=88)Number 86 87 29 0Proportion .98 .99 .33 \ .00

6th (n=91)Number 90 90 65 2.

Proportion .99 .99 .71 .02

9th (n=87)Number 87 87 72 26Proportion 1.00 1.00 .83 .30

All Grades (n=325)Number 280 267 166 28Proportion .86 .82 .51 .09

identity levels. That is, every comparisonwithin each of these three grade groups in-dicated significantly fewer children attainedthe higher concept level.

According to prediction, the percentageof children passing any given level of con-cept attainment should increase as a function6f increasing grade group. information in thecolumns of Table 24 is relevant to this pre-diction. For example, 0% of the kindergartengroup, 33% of the third grade, 71% of the sixthgrade, and 83% of the ninth graders attainedthe concept at the classificatory level. Asimilar pattern of progression exists also forthe formal level, although at the concreteand identity levels any pattern is obscuredby the uniformly high level of attainment byall grade groups except kindergarten.

Chi-square tests were used to determinewhether the proportions of individual gradegroups passing each of the four levels differedsignifica,:tly from the proportions of the com-bined grade groups passing each of the fourlevels of concept attainment. Tne differencein proportions attaining each of the levelswas significant beyond the .001 level (CM-square = 198.88,, concrete; 292.09, identity;,1')3.19, classificatory; 68.61,, formal = 3D.,1- -rier to discover where differences betweengrace groups in attainment of eacn of the four

levels were significant at the .05 level, aChi-square analog to Scheffe's theorem wasused. Statistically significant results wereas follows: at both the concrete and identitylevels, third-, sixth; and ninth-grade groupseach surpassed the attainment of kinderear-,tellers. Third-, sixth; and ninth-grade groupseach surpassed the attainment of kindergar-teners at the classificatory level as well; inaddition, classificatory attainments of sixthand ninth graders were each superior to thatof third graders. At the formal level, theninth-grade group surpassed attainments ofeach of the three lower-grade groups.

The predictions that deal with increasingdifficulty of the levels of concept attainmentreceived strong support from data assessingmastery of the concept, noun.

Relationship Between Full Attainmentof Various Levels and Full Attainmentof the Various Uses

This section will deal with two specificpredictions that were stated more fully inChapter II: first, a higher proportion of chil-dren who attain a concept at the formal level,in comparison with those who attain at theclassificatory level, will also master each

5443

of the three concept uses. Second, childrenwho attain a concept to only the concrete and/or identity levels will be able to use that con-cept to solve only simple problems of a per-ceptual type and to understand similar prin-ciples.

1

(

Data in Table 25 indicate that 16 chil-dren passed the concrete level as their highestlevel of attainment; no uses subtests werepassed by these subjects. Ninety-eight chil-dren attained the identity level as theirhighest attainment level; 24 of these childrenwere able to pass the problem-lolving subtest,one the principle subtest, and none the supra-ordinate-subordinate subtest. These dataattest to the limited use of a concept when a .

concept is attained only at the lower levels.Comparing performances on uses subtests

of subjects attaining at the classificatorylevel and subjects attaining at the formallevel addresses directly the presumed efficacyin using a given concept that accompaniesattainment at the formal level. A total of141 subjects attained the classificatory levelas their highest level and 28 subjects attainedthe formal level, as can be observed at thebottom of Table 25. The supraordinate-sub-ordinate subtest was passed by 2% of those atthe classificatory level, compared to 11% ofthose at the formal level. The principle sub-test was passed by 18% and 86%, respectively,of classificatory and formal attainers. Problemsolving was passed by 55% of those whosehighest level was classificatory, whereas93% of the subjects who passed the formallevel were also to solve problems success-fully.

For each of the three concept uses, Chi-square tests also revealed a significantsuperiority (beyond the .05 level) in thosechildren performing at the formal level com-pared to those performing at the classificatorylevel (Chi-square = 5.03, supraordinate-subordinate; 50.75, principle; 13.91, problemsolving [d.f. = 11).

In summary, the data obtained from assess-ment of noun provide strong support for theprediction that attainment of a concept at theformal level, compared with attainment at theclassificatory level, greatly facilitates useof the concept.

Difficulty of the Three Uses

It has been predicted that the numberand proportion of children who master eachconcept use will increase as a function ofincreasing grade group. Table 26 shows thatperformance on uses 6ubtests did improve

'44

with increasing grade group. Although theprogression is weak for the supraordinate-subordinate subtest, substantial increasesare apparent in the proportion of childrenwho pass both the principle and problem-solving subtests as grade group increases.Between sixth and ninth grade, considerableimprovement occurred in mastery of thesetwo concept uses.

Examination of Table 26 also revealsthat 2% of all subjects passed the supra-ordinate-suboruinate subtest; 16%, principle;and 39%, problem solving. These overallattainments are fairly modest and must beevaluated within the context of the particularconcept being assessed and battery used.

CM-square tests were run to ascertain thesignificance of the difference between the pro-portions of individual grade groups passingeach of the three uses and the proportion of thecombined grade grotips passing each of the threeconcept uses. The difference in the proportionof subjects attaining each of the used was sig-nificant beyond the .05 level (Chi-square =10.35, supraordinate-subordinate; 86.91, prin-ciple; 129.88, problem solving [d.f. = 3]). AChi-square analog to Scheffe's theorem was per-formed between all. pairs of grade groups to dis-cover where the differences in performance onthe uses subtests were significant at the .05level. Statistically significant results were asfollows: for the supraordinate- subordinate sub-test, the ninth-grade's mastery was superior tothat of the third-grade group. For the principlesubtest, performance of the ninth-grade groupsurpassed that, of each of the three lower gradegroups. Performance of the ninth-grade groupwas also superior to that of each of the threelower-grade groups on the problem-solvingsubtest. In addition, the sixth-grade groupshowed a significant superiority when com-pared to both third grade and kindergartengroups on the problem-solving subtest.

Data obtained from performance onuses subtests of the concept, noun, provideconsiderable support for the prediction of anincreasea ability, to use the concept withincreasing grade group.

Relation Between VocabularyDevelopment and Attainmentof Concept Levels and Uses

Vocabulary scores have been predictedto correlate positively with attainment on thefour concept levels and three u for bothindividual grade groups and for combinedgrade groups.

In order to address this prediction, a

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TABLE 26

NUMBER AND PROPORTION OF EACH GRADE GROUP THAT FULLYMASTERED EACH OF THREE CONCEPT USES

Grade Supraordinate-Subordinate Principle Problem Solving

K (n=59)Number 0 0 0

Proportion .00 .00 .00

3rd (n=88)Number 0 1 11

Proportion .00 .01 .13

6th (n=91)Number 1 10 48Proportion .01 .11 .53

9th (n=87)Number 5 40 69Proportion .06 .46 .79

All Grades (n=325)Number , 6 51 128Proportion .02 ,16 ,39

special scaling system was devised to obtaina score for each-subject on each of fourmeasures:: a nine-item vocabulary test,concept levels, uses, and combined levelsand uses. A detailed account of the scoringsystem is provided in Chapter III which de-scribes this method as it was applied todata from the equilateral triangle concept.The same procedure was used for the presentconcept, noun.

Based on this scoring system, meansand standard deviations for the levels, uses,levels and uses, and vocabulary scores ateach grade group are shown in Table 27.Inspection reveals that vocabulary becamemuch more proficient with increasing gradegroup. The same strong trend, snowing in-creasingly high scores with increasing gradegroup can be observed for concept attain-ment, concept uses, and overall task perform-ance scores (i.e., levels and uses combined).

Pec, son product-moment correlations (r)were calculated to determine the relationshipbetween vocabulary proficiency aria taskperformance. For each grade group,, the cor-relations obtained between scores on thevocabulary test and scores on (1) conceptlevel. (2) concept uses, and (3) combinedlevels and uses are presentee in Table 28,

46

It can be observed that correlationswithin grade groups are moderately low orzero for the youngest children. Absence ofa statistical relationship between vocabularyand task performance for the kindergartengrade group,reflects a very limited range ofperformance among these youngest subjects.Table 27 shows that kindergarten childrenpassed no uses subtests and no vocabularyitems;; the range of attainment on conceptlevels was very small. The correlations,in general, gain in magnitude with increasinggrade group for third-, sixth-, and ninth-gradegroups. The older children not only demon-strated an increasing competence with vocab-ulary, but also exhibited a wider range ofperformance on concept attainment and con-cept uses. All correlations obtained forthese grade groups were sufficiently highto reach statistical significance from zeroat or beyond the .05 level.

As predicted the correlations obtainedfor the total subject population were higherthan those obtained within grade groups andal! three were statistically significant fromzero at or beyond the .01 level. Correlationbetween overall concept attainment and vo-cabulary was .67,, between overall performance'on uses and vocabulary .75,, and between

57

combined levels and uses and vocabulary. hension of labels and concept attainment.79.

. and use is supported by the correlatiohalThe predicted relation between compre- data obtained for the concept, noun,

TABLE 27

MEANS AND STANDARD DEVIATIONS FOR COMBINED CONCEPT LEVELS,CONCEPT USES, COM3INED LEVELS AND USES, AND VOCABULARY AT EACH GRADE GROUP

Grade Group N

ConceptLevels:

(MaximumScore, 4)

ConceptUsesf

(MaximumScore,3)

Levelsand Uses:(MaximumScore, 7)

Vocabulary:,(MaximumScore,9)

M S.D. M S.D. M S.D. M S.D.K 59 .34 .51 .00 .00 .34 .51 .00 .00

3rd 88 2.30 .55 .19 .35 2.43 .71 2.08 1.52

6th 91 2.71 .54 .65 .66 3.36 .96 2.68 2.06

9th 87 3.12 .64 1.31 .81 4.49 1.22 5.52 2.62

All Grades 325 2.28 1.12 .57 .77 2.85 1.67 2.79 2.68

TABLE 28

PEARSON PRODUCT-MOMENT CORRELATIONS BETWEEN MEAN VOCABULARY SCORESAND MEAN SCORES ON CONCEPT LEVELS, CONCEPT USES, AND COMBINED LEVELS AND USES

Grade Group-N.Four

Concept LevelsThree

Concept UsesCombined

Levels and Uses

K 59 .00 .00 .00

3ru 88 ,34** .26* .40**

6th 91 .29** .55** .54**

9th 87 .55** .67** .72**

All Graies375 .67** .75** .79**

* p< .05p< .01

r erD » R6--0

58 47

VISummary and Conclusions

In this chapter a brief summary of thestudy precedes a discussion of results of allthree assessment batteries as they bear onour conclusions concerning each of the eightCLD predictions.

Summary

The CLD model encompasses three majorpropositions dealing with patterns of children'sconceptual learning and development. Re-lated to each proposition there are severalspecific predictions. Inasmuch as the majorpropositions and predictions are discussedin some detail in prior chapters, the predic-tions are stated in summary fashion as follows::

1. All children of all grade groups willconform to five acceptable patterns ofmastery of four concept levels.

2. The number and proportion of childrenwithin a grade group who pass eachsuccessive level of concept attain-ment will decrease.

3. The number and proportion of childrenof successively higher grade groupsmastering each concept level willincrease.

4. A higher proportion of children whoattain a concept at the formal level incomparison with those who attain atthe classificatory level will alsomaster each of the three concept uses

"5. Children who attain a concept toonly the concrete and/or identitylevel will be able to use that conceptonly in understanding simple percep-tual relationsnips with other objectsand in solving simple perceptualproblems.

6. The number and proportion of childrenof successively higher grade groupswho master each concept use willincrea se .

7. Vocabulary scores and scores based

on attainment of the four levelsand the three uses will correlatepositively within grade groups.

8. Vocabulary scores and scores basedon the levels and uses will correlatepositively for the combined gradegroups.

The preceding predictions were-tested,using three specially constructed assessmentbatteries. One battery was used for eachconcept: equilateral triangle, cutting tool,and noun. Each battery also had one subtestfor each of the four levels of concept attain-ment and for each of three uses of an attainedconcept. Thus, a total of 21 tests was de-veloped, seven for each of the three concepts.

Children at four grade levels--kinder-garten, third, sixth, and ninth--participatedin the study. Children in the three lowergrades were enrolled in four different elemen-tary schools. The ninth-grade students wereenrolled in a junior high school. The schoolsand classrooms in which the children wereenrolled were judged to be typical of theparticular school system and also of a largenumber of classrooms in small towns ofWisconsin and other states.

The tests of the various batteries wereadministered to children in intact classroomgroups;- except that, kindergarten childrenreceived the tests in small groups of aboutfive to eight. On each subtest a child'sresponses were scored as passing or failing.Criteria were set to determine passing andfailing as has been reported in chapters III,IV, and V. The data were quantified by com-puting frequencies and proportions of subjectswithin each grade group who attained eachconcept level and each use. Post hoc statis-ticaLteSts were used where appropriate toobtain more specific information about dif-ferences in frequencies and proportions.The predicted relationship between vocab-ulary and performance was evaluated bycorrelation coefficients.

5949

Conclusions

Results of the three assessments havebeen presented and discussed separately inthe preceding chapters. Now we will con-sider the concepts simultaneously both inorder that our conclusions regarding the eightpredictions can be stated with a greater degreeof generality and confidence and so thatrelevant cross-concept comparisons can bemade. Each prediction is stated, and theevidence based on all three concepts is sum-marized and discussed.

1. All children of all grade groups willconform to five acceptable patternsof mastery of the four concept levels.This prediction was supported byresults obtained for the individual

. concepts assessed in the presentstudy. Table 29 summarizes therelevant information for all three con-cepts. Inspection of this table showsthat 92% of all the children conformedto the predicted pass/fail patternsfor equilateral triangle, 94% for cuttingtool, and 98% for noun. Presentedtogether, these large percentagesprovide strong support for the pre-diction of invariant sequence of attain-ment.

Table 30 presents the frequenciesof the total subject population who dianot conform to the predicted patterns.Twenty-seven children (8%) deviatedfrom the predicted sequence in theassessment of equilateral triangle.Similarly, 6% did not conform oncutting tool, and 2% did not conformon noun. Those subjects not con-forming on the equilateral triangletask were distributed over m5st ofthe 11 pass/fail patterns. Those notconforming on cutting tool are con-centrated in three patterns and theexceptions on noun in two patterns.Further examination of protocols fromchildren who did not conform to thefive predicted patterns revealed nooverlap among the three concepts.That is, all 27 children who wereexceptions on the equilateral triangletask conformed to acceptable patternson the other two tasks:: the 23 indi-viduals who were the exceptions oncutting tool were conformers on theother two tasks. This independencesuggests that the exceptions to thepredicted patterns are probably theresult of errors of measurement orproblems associated with the criteria

50

established for concept level attain-ment rather than true exceptions interms of sequential development. Hadthe same children deviated on allthree concepts we would, of course,conclude that these children werenot in fact conforming to the hypothe-sized invariant sequence of conceptualdevelopment.

2. The number and proportion of childrenwithin a single grade group who passedeach successive level of conceptattainment will decrease. We haveseen that this prediction was uphelduniformly for the four grade groupswhen each of the three concepts wasexamined individually. Table 31enables us to make cross-conceptcomparisons for the proportion of eachgrade group that fully mastered eachlevel of attainment. It may be seenthat there could have been 48 excep-tions to this prediction. In factsonly three minor reversals exist. Inthe kindergarten group 71% passed theconcrete level of equilateral triangleand 76% passed the identity level.In the kindergarten group also 95%passed the concrete level of cuttingtool and 96% passed the identity.The other exception to the predicteddirection involves noun; 98% of thethird graders passed the concretelevel and 99% passed the identitylevel. That these few minor excep-tions occurred at all is most probablydue to error of measurement and prob-lems associated with the criteriaestablished for passing the variouslevels.

3. The number and proportion of succes-sively higher grade groups masteringeach concept level will increase.This prediction was also upheld foreach of the three concepts. Thecolumns of Table 31 summarize therelevant data. Again, 48 exceptionsmight have occurred. In fact, therewas only one minor reversal to thepredicted direction. Ninety-eightpercent of the third grade children,compared to 97% of the sixth grade,passed the concrete level of equi-lateral triangle.

4. A higher_ roportion of children whoattain a concept at the formal levelin comparison with those who attainit at the classificatory level willalso master each of the three uses.This prediction was upheld.

60

TABLE 29

PROPORTION OF TOTAL SUBJECT POPULATION CONFORMING TO PREDICTEDPASS-FAIL PATTERNS OF ATTAINMENT:

COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

Pass-Fail Sequence Equilateral Triangle(N=324)

Concept

Cutting Tool(N=363)

Noun(N=325)

FFFF .03 .00 .13

PFFF .02' .00 .05

PPFF .11 .09 .29

PPPF .37 .36 .43

PPPP .39 .48 .08

Total .92 94 .98

TABLE 30

FREQUENCIES OF TOTAL SUBJECT POPULATION SHOWING PASS-FAIL PATTERNS OFEXCEPTION: COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

Pass-Fail Patternof Exception

Equilateral Triangle(N=324)

Cutting Tool(N=363)

Noun(N=362)

Total

FFFP 0

FFPF 2 2

FFPP 1 1

12FPFF

FPFP------FPPF

6

5

3 3

0

5

1FPPP

PFFP--1

1 1

PFPF 6 2 8

PFPP

PPFP

Total

Percx,nto.-Te ofTotal Population

3 3

18 232

27

.08

23

.06

3

6

.02

61 51

Summarizing data for, all three con-cepts, Table 32 shows the proportionof children who passed the four levelsand who also passed various uses.There are nine possible exceptionsto the prediction that children whoattain a concept at the formal level,compared to those attaining at theclassificatory level, demonstratesuperior performance on the conceptuses. - Inspection of the bottom halfof Table 32 shows there were no ex-ceptions to this prediction. Moreover,the differences in the actual size ofthe percentages are consistentlylarge. For example, data for equi-lateral triangle show that a markedadvantage in mastery of uses occurredfor individuals attaining at the formallevel: 34% passed the supraordinate-subordinate subtest, compared to 8%of classificatory attainers; 43%, passedthe principles subtest, compared to5% of classificatory attainers; and34% passed the problem solving sub-test, compared to 3% cf classificatoryattainers.

5. Children who attain a concept to onlythe concrete and/or the.identity levelwill be able use that concept onlyin understan.ii.ig simple perceptualrelationships with other oblect con-cepts and in solving simple perceptualproblems. This prediction was sup-ported in general, although some ex-ceptions occurred for two of the con-cepts. Table 32 shows that the ex-ceptions were as follows: 24% (9 outof 37) of the children who passed theidentity level of cutting tool and 7%(3 out of 43) who passed the identitylevel of equilateral triangle alsopassed the supraordinate-subordinatetest.

6 The. number and proportion of childrenof successively higher grade groupswho master each concept use willincreasee. This prediction was upheld.Table 33 shows that 36 exceptionsmight have occurred. Inspection showsthere was only one minor reversal.Fourteen percent of the third gradersmastered the supraordinate-subordinatetest for equilateral triangle, whereasonly 12% of the sixth graders passed.In addition tos'the fact that only oneminor exception occurred, it is notedthat improvement in mastery of thevarious uses markedly increased wan

52

increasing grade group. For example,for the supraordinate-subordinatesubtest of cutting tool the progressionwas as follows: kindergarten .08,third grade .43, sixth .66, and ninthgrade .82.

7. Vocabulary scores and scores basedon the attainment of the four levelsand the three uses will correlate posi-tively within grade group. This pre-diction was supported by data for eachconcept assessed in the present study.Table 34 summarizes, for all threeconcepts, the 36 correlations obtainedbetween vocabulary scores and scoreson levels, uses, and combined levelsand uses at each grade group. Table 34shows one exception to the prediction:the correlation between vocabularyand attainment of concept levels was-.09 for the ninth graders who performedon the cutting tool assessment battery.In general, however; the size of thecorrelations within the various gradegroups was considerably higher thananticipated. Twenty-four of the total36 correlations were statisticallysignificant from zero at or beyond the.05 level; actual values ranged from.26 to .72. For both equilateral triangleand noun concepts, relatively low orzero order correlations occurred inthe kindergarten group; ranW of per-formance on vocabulary scores and onthe levels and uses subtests was verysmall for this grade group. For cuttingtool, another set of low correlationswas found among the ninth graders.Here also, as explained in chapter IV,lack of variability was responsible;the ninth graders demonstrated uniformlyhigh scores on the vocabulary testfor cutting tool and also on the levelsand uses subtests.

8. Vocabulary-scores and scores basedon the attainment of the four levels andthe three uses will be positively cor-related and higher across combinedgrade 'groups. This prediction receivedstrong suppory from all three sets ofdata, as-summarized in Table 35. Thehighest set of correlations for combinedgrade groups was obtained for noun andthe lowest for cutting tool. In general,the nine correlations entered in thistable are notably larger in magnitude thanthose obtained within grade groups andare statistically significant from zero ator beyond the .01 level.

62

TABLE 31

PROPORTION OF EACH GRADE GROUP THAT FULLY MASTERED EACH LEVEL OF ATTAINMENT:COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

Grade Group Concrete Identity Classificatory Formal

K

N = 62 Equilateral Triangle .71 .76 .42 .0076 Cutting Tool .95 .96 .75 .0159 Noun .29 .05 .00 .00

3rd

N = 86 Equilateral Triangle .98 .93 .a1 .1793 Cutting Tool 1.00 1.00 .83 .3988 Noun .98 .99 .33 .00

6th

N = 92 Equilateral Triangle .97 .95 .93 .531 0 1 Cutting Tool 1.00 1.00 .85 .7091 Noun .99 .99 .71 .02

9th

N = 84 Equilateral Triangle93 Cutting Tool

.991.00

.991.00

.98.94

.83,

.9187 Noun 1.00 1.00 .83 .30

63 53

cn 14.4

a

TA

BL

E 3

2

RE

LA

TIO

NSH

IP B

ET

WE

EN

FU

LL

MA

STE

RY

OF

LE

VE

LS

AN

D F

UL

L M

AST

ER

YO

F U

SES:

CO

MPA

RIN

G P

RO

POR

TIO

NS

OF

TO

TA

L S

UB

JEC

T P

OPU

LA

TIO

N P

ASS

ING

USE

SFO

R E

QU

ILA

TE

RA

L T

RIA

NG

LE

, CU

TT

ING

TO

OL

, AN

D N

OU

N

Gra

deC

oncr

ete

as H

ighe

stId

entit

y as

Hig

hest

Cla

ssif

icat

ory

as H

ighe

st.

Form

al a

s H

ighe

st

Supr

a-Su

bPr

in.

Prob

.So

lv.

Supr

a-Su

bPr

in.

Prob

.So

Iv.

Supr

a-Su

bPr

in.

Prob

.So

lv.

Supr

a-Su

bPr

in.

Prob

.So

Iv.

N P

assi

ng L

evel

543

133

134

N P

assi

ng U

se0

00

30

010

64

4557

45E

quila

tera

l Tri

angl

e.0

0.0

0.0

0.0

7.0

0.0

0.0

8.0

5.0

3.3

4.4

3.3

4

N P

assi

ng L

evel

037

132

193

N P

as s

ing

Use

00

09

16

369

2614

492

140

Cut

ting

Too

l.0

0.0

0.0

0.2

4.0

3,.1

6.2

7.0

7.2

0.7

5.4

8.7

3

N P

assi

ng L

evel

1698

141

28N

Pas

sing

Use

00

00

124

326

783

2426

Nou

n.0

0.0

0.0

0.0

0.0

1.2

4-.0

2.1

8.5

5.1

1.8

6.9

3

1

TABLE 33

PROPORTION OF EACH GRADE GROUPTHAT FULLY MASTERED EACH OF THE THREE CONCEPT USES:

COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

/

Grade Group Supraordinate-Subordinate Principle Problem Solving

K

N = 62 Equilateral Triangle .00 .00'' .0076 Cutting Tool .08 .00 .0059 Noun .00 .00 .00

3rdN =-- 86.Equilateral Triangle .14 .00 .00

93 Cutting Tool .43 . i 1 .4188 Noun .00 .0,1 .13

6thN = 92 Equilateral Triangle .12 .15 .13

1 0 1 Cutting Tool .66 .30 .5591 Noun .01 .11 .53

9thN = 84 Equilateral Triangle .42 .58 .44

93 Cutting Tool .82 . .67 .8487 Noun 06 .46 .79

TABLE 34

PEARSON PRODUCT-MOMENT CORRELATIONS AT EACH GRADE GROUP BETWEEN MEANVOCABULARY SCORES AND MEAN SCORES ON CONCEPT LEVELS, CONCEPT USES, AND

COMBINED LEVELS AND USES: COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

Grade Group FourConcept Levels

ThreeConcept Uses

CombinedLevels and Uses

K

N = 62 Equilateral Triangle .14 .00 .1576 Cutting Tool .41 ** .2.0 .43**59 Noun .00 .00 .00

3rdN = 86 Equilateral Triangle .43** .06 .40**

93 Cutting Tool .35** .30,-* .41 **88 Noun .34** .26* .40**

6thN = 92 Equilateral Triangle .18 .38** .39**

101 Cutting Tool .44** .31 ** .43**91 Noun .29** .55** .54**

9thN = 84 Equilateral Triangle .25* .42** .44**

93 Cutting Tool -.09 .15 .0987 Noun .55** .67** .72**

*p < .05**p.< .01

65 55

TABLE 35

CORRELATIONS FOR TOTAL SUBJECT POPULATION BETWEEN MEAN VOCABULARY SCORESAND MEAN SCORES ON LEVELS, USES, AND COMBINED LEVELS AND USES::

COMPARING EQUILATERAL TRIANGLE, CUTTING TOOL, AND NOUN

Concept FourConcept Levels

ThreeConcept Uses

CombinedLevels and Uses

N = 324 Equilateral Triangle

N = 363 Cutting Tool

N = 325 Noun

57*

, .5 I*

.67*

.56*

.43*

75*

.70*

.52*

.79*

*p < 01

56 66r,P0 n0' -'88 -3

References

Ausubel, D. P. The psychology of meaning-ful verbal learning., New York:. Grune& &ration, 1963,

Ausubel,, D. P. Meaningful reception learn-ing and the acquisition of concepts. InH. J. Klausmeier & C. W. Harris (Eds.),Analyses of concept learning., New York:,Academic Press, 1966., Pp. 157-175.

Bourne, L. E.., Jr., Ekstrand B. R., &Dominowski R. L, Ihepgyqhfthinking. Englewood Cliffs , N. J.:,Prentice-Hall, 1971.

Bourne, L. E, Jr., & Rest le F. Mathe-matical theory of concept identification.Psychological Review, 1959,, 16(5)278-296.

Bruner, J. S.; The course of cognitive growth.American Psychologist, i964,, 19,, 1-15:

Bruner, J. S. Goodnow J. J. & Austin, G. A.A sway of thinking. New York:, Wiley,,1956.

Bruner, J. S. Giver,, R. R.,,, Greenfield,P, M., et al. ,Studies in cognitivegrowth. New York:: Wiley, 1966.

Campbell, D., & Stanley, J. Experimentaland quasi-experimental designs forresearch on teacning. In N. Gage(Ed.), Handbook of research on teaching.Chicago:, Rand McNally, 1963.; Pp. 171-246,

-Carroll, J. B. Words, meanings, and concepts.Harvard Educational Review, 1964, 34,178-202.;

Deese, J. Meaning and change of meaning.American Psychologist, 1967, 22, 641-651.,

Elkind, D. Conservation and concept for-mation. In D. Elkind & J.: H., Flavell(Eds.), Studies in cognitive development:Essays in honor of Jean Piage't. Londonand New York: Oxford University Press,1969, Pp. 171-189.

Felornan, K., V: Tne effects of number ofpositive and negative instances, conceptdefinition, and emphasis of relevantattributes on tile attainment of-mattie-

mAtical concepts . Technical ReportNo. 243, Madison:, Wisconsin Researchand Development Center for CognitiveLearning, 1972.

Flavell, J. H. Concept development. InP. H. Mussen (Ed.), Cannichael'smanual of child psychology. Vol. 1.New York: Wiley. 1970, Pp. 983-1059.

ne , R. M. The learning of principles.In H. J. Klausmeier & C. W. Harris(Eds.), Analyses of concept learning.New York: Academic Press, 1966,Pp. 81-95.

Gagne,, R. M. The conditions of learning.(2nd ed.) New York: Holt, Rinehartand Winston, 1970.

Gesell, A. Infancy and human growth.New York:, Macmillan, 1928.

Gesell, A. The embryology of behavior:The beginnings of the human Mind.New York: Harper, 1945.

Gibson, E. T. Principles of perceptuallearning and development. New York:,Appleton, 1969.

Golub L. S., Fredrick, W., Nelson, N.:,& Prayer, D. A. Selection andarolumof language arts concepts for inclusionin tests of concept attainment. WorkingPaper No. 59. Madison: WisconsinResearch and Development Center forCognitive Learning, 1971.

Goodnow, J. J. Problems in research onculture and thought. In D. Elkind &J. H. Flavell (Eds.), Studies in cog-nitivt. New York: OxfordUniversity Press, 1969,, Pp. 4,39-462.

Guilford, J. P. The nature of human intel-ligence. New York1967.

Hooper, F. H, & Klausmeier,, H., J.Description and rationale for a longi-tudinal assessment of children's cog-nitive development and concept learningWorking Paper No 113. Madison:.Wisconsin Research and DevelopmentCenter for Cognitive Learning, 1973.

6757

Hunt, E. B, informa-tion- processing problem. New York:Wiley, 1962.

Inhelder, B., & Piaget, J. The early growthof logic in the child: Classification andseriation. New York: Harper & Row,1964,

Kagan, J. A developmental approach to con-ceptual growth. In H. J. Klausmeier &C. W. Harris (Eds.), Analyses of conceptlearning. New York: Academic Press,1966, Pp. 97-116.

Kagan, J., Moss, H, A., & Sigel , I. E.Psychological significance of styles ofconceptualization. Monograph of theSociety for Research in Child Develop-ment, 1963, 2.1 (Whole No. 86) , Pp. 73-112.

Klausmeier, H., J. Cognitive operations inconcept learning. Educational Psycholo-gist, November 1971,,,Pp. 1 and 3-8.

Klausmeier, H. J., Bernard, M. E., Katzen-meyer, C. G., & Sipple, T. S.Development of Conceptual Learning andDevelopment Assessment Series II:Cutting Tool. Working Paper No. 120.Madison: Wisconsin Research andDevelopment Center for Cognitive Learn-ing, 1973.

Klausmeier, H. J., Davis, J. K., Ramsay,J. G., Fredrick, W. C., & Davies, M.H.Concept 'taming aria Problem solving:A bibliography. 19501964. WisconsinResearch and Development Center forCognitive Learning, Technical ReportNo. 1, 1965.

clidausmeier, H. J., Ghatala, E. S., & Frayer,D. A. Concentual_iearning_and devel-opment: A cognitive view. New York:Academic Press, 1974.

Klausmeier, H. J., Ingison, L. J., Sipple,T. S., & Katzenmeyer, C. G.Development of Conceptual Learningand Development Assessment Series I:Equilateral 'Triangle. Working PaperNo. 119. Madison: Wisconsin Researchand Development Center for CognitiveLearning, 1973. (a)

Klausmeier, H. J., Ingison, L. J., Sipple,T. S., & Katzenmeyer, C. G.Development of Conceptual Learning andDevelopment Asseament Series III: Noun.Working Paper No. 121. Madison:Wisconsin Research and DevelopmentCenter for Cognitive Learning, 1973. (b)

Klausmeier, H, J., & !Minket D. L. Con-cept attainment as a function of instruc-tions concerning the stimulus material,,a strategy, and a principle for securinginformation. Journal of EducationalPsychology,, 1968,, 59, 215-222,

58

Kofsky, E. A scalogram study of classi-ficatoFy development. Child Develop-ment, 1966, .3,/, 191-204.

Levine, M. Mediating processes in humansat the outset of discrimination learning.Psychological Review, 1963, 70, 254-276.

Levine, M. Hypothesis behavior by humansduring discrimination learning. ,Journalof Experimental Psychology, 1966, IL331-336.

Luria,, A. R. The role of speech in the regu-lation of normal and abnormal behavior.New York: Liveright, 1961.

Markle, S. M., & Tiemann, P. W. Reallyunderstanding concepts: Or in frumiousplirsuit of the jabberwock: Champaign,

Stipes, 1969.Marx, M. H. (Ed.) Learning: Theories.

New York: Macmillan, 1970.Osgood,, C. E. Method and theory in

experimental psychology. New York:Oxford University Press, 1953.

Piaget, J. Piaget's theory. In P. H Mussen(Ea.) , Carmichael's manual of childpsychology. New York:: Wiley, 1970,Pp. 703-732.

Romberg, T. A., Steitz, J., & Frayer,D. A. Selection and analysis ofmathematics concepts for inclusion intests of concept attainment. WorkingPaper No. 55. Madison: WisconsinResearbh and Development Center forCognitive Learning, 1971.

Staats, A., Child learning, intelligence andpersonality. New York: Harper & Row,1971.

Swanson,, J. E. The effects of number ofPositive and negative instances,concept definition, and emphasis ofrelevant attributes on the attainmentof three environmental concepts bysixth-grade children. Technical ReportNo. 244. Madison:: Wisconsin Researchand Development Center for CognitiveLearning, 1972.

Tabachnick,, B. R., Weible, E. , & Frayer,13 A. Selection and analysis of socialstudies concepts for inclusion in testsof concept attainment. Working PaperNo. 53., Madison: Wisconsin Researchand Development Center for CognitiveLearning, 1970,

Tagatz, G. E. Effects of strategy, sex, andage c,n conceptual behavior of elementaryschool children. journal of EducationalPsychology, 1967, 58, 103-109.

Tennyson, R. D. & Boutwell, R. C.- A qual-ity control design and evaluation modelfor hierarchical sequencing of programmedinstruction. National Society_forPro-

68

arammed Instruction Togrnal, 1971, 10,5-10.

Trabasso, T. , & Bower, G. Attention inLearning. New York: Wiley, 1968.

Vernon, M. D. Perception through experi-ence. New York: Barnes & Noble, 1970.

Voelker, A. M. , Sorenson, J. S., & Frayer,D. A. Selection and anglvsis of clas-sificatory science coaceRts by inter-medikte-arade children. WorkingPaper No. 57. Madison: WisconsinResearch and Development Center forCognitive Learning, 1971.

Vygotsky, L. S:- (translated by E. Hanfmann& G. Vakar) Thought and language.Cambridge, Mass.: M.I.T. Press, 1962.

(

GOO .07-7,*-2.

Williams, C. F. A model of memory in con-cept learning. Cognitive Psychology,1971, 2, 158-184.

Woodruff, A. D. Basic concepts of teaching.(Concise ed.) San Francisco: ChandlerPublishing, 1961.

Woodruff, A. D. Cognitive models of learn-ing and instruction. In L. Siegel (Ed.),Instruction: Some contemporary vim=Points, San Francisco: Chandler Pub-lishing, 1967, Pp. 55-98.

Zeaman, D. , & House, B. J. The role ofattention in retardate discriminationlearning. In N. R. Ellis (Ed.), Handbookof mental deficiency. New York:McGraw-Hill, 1963, Pp. 159-223.

69 59

National Evaluation Committee

Helen BainPast PresidentNational Education Association

Lyle E. Bourne, Jr.Institute for the Study of Intellectual BehaviorUniversity of Colorado

Sue BuelDissemination and Installation ServicesNorthwest Regional Educational Laboratory

Francis S. ChaseProfessor EmeritusUniversity of Chicago

George E. DicksonCollege of EducationUniversity of Toledo

Chester W. HarrisGraduate School of EducationUniversity of California

Hugh J. ScottConsultantNational Evaluation Committee

H., Craig SipeDepartment of InstructionState University of New York

G Wesley SowardsDean of EducationFlorida International University

Joanna WilliamsPr'pfessor of Psychology azirEducationColumbia University

Executive Committee

William R. BushDirector, Program Planning and ManagementDeputy Director, R & D Center

M. Vere DeVault.ProfessorSchool of Education

Herbert J. KlausmeierPrincipal InvestogatorR & D Center

Joel R. LevinPrincipal InvestigatorR & D Center

Donald N. MclsaacAssociate Dean, School of EducationUniversity of Wisconsin

Richard A. :tossmiller, Committee ChairmanDirectorR & D Center

Len VanEssAssociate Vice ChancellorUniversity of WisconsinMadison,

Dan WoolpertDirector, Management Systems

& D Center

Faculty of Principal Investigators

Vernon L. AllenProfessorPsychology

B. Dean BowlesAssociate ProfessorEducational Administration

Frank H. FarleyAssociate ProfessorEducational Psychology

Marvin J. FruthAssociate ProfessorEducational Administration-

John G HarveyAssociate ProfessorMathematics

Frank II.. HooperAssociate ProfessorChild Development

Herbert J. KlausmeierV. A. C. Hermon ProfessorEducational P,ycnology

Gisela Laboi,vieAssistant ProfessorEducational Psychology

Joel R. LevinAssociate ProfessorEducational Psychology 70

L. Joseph LinsProfessorInstitutional Studies

James LiphamProfessorEducational Administration

Wayne OttoProfessorCurriculum

Robert PetzoldProfessorCurriculum

and Instruction

and InstructionThomas A. Romberg

Associate ProfessorCumculum and Instruction,

Dennis W. SpuckAssistant ProfessorEducat ional Administration

Richard L. VenezkyAssociate ProfessorComputer Science

Larry M. WilderAssistant ProfessorCommunication Arts