20896653 Academic vs Creative Ability in Math

Academic Versus Creative Abilities in Mathematics:Two Components of the Same Construct?

Nava L. LivneUniversity of Utah

Roberta M. MilgramTel Aviv University and College of Judea and Samaria

ABSTRACT: Structural equation modeling, hithertoused to examine unidimensional theoretical modelsonly, was used to investigate 2 dimensions, abilitiesand levels, simultaneously. Good evidence for the va-lidity of conceptualizing 2 types of mathematical abil-ity, 1 academic and 1 creative, each at 4 hierarchicallevels, was established in 10th- and 11th-grade stu-dents (N = 1,090). IQ scores, representing general ac-ademic ability, predicted academic, but not creative,ability in mathematics. Creative thinking predictedcreative, but not academic, ability in mathematics.These findings led to an innovative approach to identi-fying mathematical abilities and provided reliable andvalid psychometric tools to make it possible. Based ontwo new instruments, teachers can differentiate curric-ula and individualize instructional strategies to matcheach student’s needs.

A serious gap between the achievement in mathemat-ics of students in the United States and of students inmany other countries has developed in the last decade(Stevenson & Stigler, 1992; Stigler & Hiebert, 1999).According to a recent study, “U.S. eighth-grade stu-dents performed near the international average inmathematics …, and U.S. twelfth-graders scored be-low the international average and among the lowest 38Third International Mathematics and Science Study(TIMMS) nations in either mathematics … or ad-vanced mathematics” (National Center for EducationStatistics, 2004, p. 1). Awareness of this gap is increas-ing in the United States and its leaders are seeking so-lutions to this problem. The No Child Left Behind Act(2001) has called on educators, especially in mathe-

matics, to provide “every child, from every back-ground in any part of America an opportunity to reachhis or her fullest potentials, in mathematics and sci-ence” (Bush, 2001, pp. 1–2).

Numerous authorities in mathematical education,including the National Council of Teachers of Mathe-matics (2004), attributed the mediocre achievement inmathematics that produced the gap cited above to thediscrepancy between the different abilities, needs, andinterests of both mathematically talented and averagestudents and the curriculum that is offered to them.One major reason for this discrepancy is probably thefailure to identify the different types and levels ofmathematical ability of some gifted and talented youngpeople, and consequently, the lack of effort to developand enhance these abilities.

A number of eminent mathematicians (Hadamard,1945; Halmos, 1968; Muir, 1988) noticed that inven-tions and accomplishments in mathematics have re-quired creative talent rather than traditional academicability. They distinguished between two kinds of cog-nitive abilities in mathematics, one academic and onecreative. Academic ability in mathematics is the arith-

Creativity Research Journal2006, Vol. 18, No. 2, 199–212

Copyright © 2006 byLawrence Erlbaum Associates, Inc.

Creativity Research Journal 199

This article was based on the doctoral research of the first authordone at Tel Aviv University under the direction of the second author.This study was supported in part by a grant from the Israel Ministryof Education. We thank Richard Mayer, University of CaliforniaSanta Barbara, Amiram Vinokur, Institute for Social Research, Uni-versity of Michigan, and Oren Livne, Department of Mathematicsand Scientific Computing and Imaging Institute, University of Utah,for their helpful suggestions and comments.

Correspondence and requests for reprints should be sent to NavaL. Livne, College of Education, University of Utah, 227 H Street#106, Salt Lake City, UT 84103. E-mail: [email protected]

metic computational ability that is required to get topschool grades in mathematics. Creative ability in math-ematical thinking is the ability to perceive patterns andrelationships using complex and nonalgorithmic think-ing and being capable of original thinking in mathe-matical symbols that results in more than one solutionstrategy and/or solution (Munro, 2000; Smith & Stein,1998; Stein, Smith, Henningsen, & Silver, 2000).Other researchers understood that, to assess mathemat-ical ability, the conceptual components of academicand creative abilities in mathematics needed to be de-fined operationally (Shavinina & Kholodnaja, 1996;Smith, Kher, & Gifford, 1999; Wagner & Zimmer-mann, 1986). Moreover, in mathematical educationsome researchers were aware of the importance of de-fining different levels of mathematical understandingas well (Hart, 1993; Kinard, 2001; Kinard & Falik,1999; Stein et al., 2000; Van Hiele, 1987; Zorn, 2002).However, attempts to introduce awareness of levels ofmathematical thinking into mathematical educationwere limited to academic abilities.

To the best of our knowledge, neither the differencebetween academic and creative abilities in mathemat-ics nor the postulation of four distinct levels of eachability has been clearly defined theoretically or as-sessed separately. In this study, we developed theoreti-cal and operational definitions of the two ability typesat four levels and investigated the structural aspect ofconstruct validity of those abilities (Messick, 1995, pp.744–746).

The structure of giftedness model (Milgram, 1989,1991) postulates giftedness as the result of the complexinteraction of cognitive, personal–social, andsociocultural, influences. We applied the cognitivecomponent of the model to mathematics. Giftedness inmathematics was defined as a bidimensional construct.The first dimension is type of ability, two academic andtwo creative. The second dimension is level of ability.Each of the four ability types of giftedness is postu-lated to occur at four distinct levels of ability: one levelof nongifted ability, and three levels of gifted abilities(mild, moderate, and profound). These distinct abilitylevels are depicted as hierarchically ordered by theirdegree of difficulty, highlighting the fact that thehigher the level, the fewer the people in society whoachieve it. The four ability types, each at four orderedlevels, are schematically illustrated by a bidimensionaltriangle as presented in Figure 1.

Of the two general types of ability postulated by themodel, general intellectual ability refers to the abilityto think abstractly and to solve problems logically andsystematically. It is generally measured by standard-ized individual and/or group IQ tests. General originalor creative thinking refers to the ability to generate alarge number of ideas or solutions in the problem-solv-ing process that results in a few creative solutions ofhigh quality. Operationally, this ability is defined interms of ideational fluency. This definition of generalcreative thinking was suggested by Guilford (1950,1956), Mednick (1962), Torrance (1962), and Wallachand Kogan (1965) and is frequently used in research oncreative thinking.

The model also postulates two types of domain-spe-cific ability in mathematics. Domain-specific academicability in mathematics refers to general intelligence ap-plied to mathematics. It reflects standard–logical think-ing in mathematics, and is demonstrated by computa-tional ability, knowledge of mathematical concepts,principles, and reasoning. We measured this ability bothby school grade point average and a new academic in-dex, consisting of academic mathematical problemsthat will be described in the Methods section.

Domain-specific creative ability in mathematics isgeneral, nonstandard creative thinking ability appliedto mathematics. In this research, we developed twooriginal indexes to assess domain-specific creativeability in mathematics. One consisted of mathematicalproblems that required creative thinking for solution,and the second of challenging out-of-school activitiesrelated to mathematics. Creative mathematical abilitymay be evident long before adulthood. One way toidentify this ability in children is by examining theirleisure time, intrinsically motivated, challengingout-of-school activities in mathematics. The activitiesare often highly intellectual in nature and are done tosatisfy their own curiosity and interests, rather than toachieve high grades or to please their teachers and par-ents. The two new indexes that measured creative abil-ity in mathematics will be described in detail in theMethods section.

In the 4 × 4 model, each type of mathematical abil-ity is postulated as occurring at four ability levels thatdiffer in degree of difficulty and represent distinct cog-nitive processes. Haberlandt (1997) and Siegler (1991)supported this formulation and suggested that groupsof similar mathematical strategies that differ by their

200 Creativity Research Journal

N. L. Livne and R. M. Milgram

degree of task difficulty represented different cognitivelevels for mathematical problem solving. Opera-tionally, the degree of difficulty of an ability level thatcorresponds to a particular class of cognitive interrela-tionships is defined as a group of mathematical strate-gies (Beevers & Paterson, 2002; National Council ofTeachers of Mathematics, 2004). Based on these defi-nitions we examined the distinction between the levels

of both the academic and creative abilities. General in-telligence and general creative thinking were used onlyto examine their influence on the two domain-specificabilities in mathematics.

Hong and Milgram (1996) used confirmatory factoranalysis techniques (Joreskog & Sorbom, 1988) tocompare six alternative theoretical models of abilitytypes in the domain of literature. They found that the


Creative Mathematical Giftedness

Figure 1. Milgram 4 × 4 Structure of Giftedness Model as applied to mathematics.

best fit was a four-factor model, supporting the fourability types postulated by Milgram (1989, 1991).Based on their findings in literature, it was reasonableto investigate the efficacy of the model in supportingdomain-specific ability types in mathematics.

This study differs from that of Hong and Milgram(1996) in tworespects.First, the4×4modelwasappliedto the domain of mathematics for the first time. Second,the construct validity of the concept of levels of abilitiesin mathematics was also investigated for the first time.Based on the literature presented earlier, the goal of thisstudy was to examine the research question of whetherthe 4 by 4 model is the best model to support the postula-tion of general and domain-specific abilities in mathe-matics, each at four levels in mathematics.

Method

Participants

A sample of 1,090 students (565 males and 525 fe-males), 10th- and 11th-grade students (Mean age =16.50, SD = .59) representing a wide range of intellec-tual abilities, was drawn from 22 public schools in ur-ban and rural areas. The schools were selected from alist of 571 schools provided by the Israel Ministry ofEducation and constituted a nationally stratified andrepresentative sample of students in urban and ruralschools. The sampling process is described in detailelsewhere (Livne, 2002).

Materials

Six measures were administered to each researchparticipant. The measures divided by types of ability,instruments, their indexes, types of measure, and scalerange are presented in Table 1.

School grade in mathematics. The final gradein mathematics recorded by the school at the end of thefirst semester of the academic year in which the datafor this study were collected, served as the index of do-main-specific academic ability in mathematics. Thescores could range from zero to 100.

Multiscale Academic and Creative Abilities inMathematics—MACAM (Livne & Livne, 1999a).This instrument measured domain-specific academicand creative abilities in mathematics at four levels, and

consisted of 16 mathematical items, 8 academic and 8creative. Each academic item required standard-logicalthinking and was operationally defined as having onesolution path to reach one correct answer. For example,“How many distinct three-digit numbers can one buildout of three arbitrary digits (from 1 to 9), e.g., 2, 5 and9? A digit may repeat itself inside the number.” By con-trast, each creative item required nonstandard creativethinking and was operationally defined as having morethan one solution path and/or correct answer. For in-stance, “Try to arrive at the number 4, using preciselyfour times (not two times) the digit 4, which is an inte-ger multiplication of the digit 2. Try to make the largestpossible number of solutions that overall include all ofthe following arithmetic operations: addition, subtrac-tion, multiplication, division, square root, factorial,and so on. In every solution separately, one need notuse all the operations.” Each of the 16 items was in theform of an open-ended question, and was scored on acontinuous 8-point scale (0–7), according to the degreeof completeness of its solution. Sixteen scores werecomputed for each research participant. The range ofscores for each of the eight-item measures of academicand the eight-item measure of creative abilities inmathematics was 0–64 for both. The internal consis-tencies of the eight-item measures of academic andcreative abilities in mathematics were .67 and .68, re-spectively.

Following the example of Guttman (Guttman, 1968,1971, 1991), the test items were developed accordingto a mapping sentence developed especially for thisstudy (Livne, 2002; Livne, Livne, & Milgram, 1999).Detailed criteria for scoring each of the 16 items wereprovided in the Scoring Guide for the Multiscale Aca-demic and Creative Abilities in Mathematics devel-oped by the authors of the instrument (Livne & Livne,1999b). Livne (2002) found a high degree of agree-ment among 12 judges in ranking the eight degrees ofcompleteness of the solutions (0–7) to each of the 16items. The overall Kendall coefficient of concordance(1948) for the judges (N = 12) ranged from .72 to .94, p< .0001.

Tel Aviv Activities and Accomplishments Inven-tory: Mathematic—TAAI: M (Livne & Milgram,1999). Building on the work of Hocevar (1980),Holland (1961), Runco (1986, 1987), Wallach andWing (1969), and Milgram (1973, 1983, 1987, 1990)the Tel-Aviv Activities and Accomplishment Inven-



tory: Mathematics was developed. This instrument wasused to measure domain-specific creative ability inmathematics. It is a self-report biographical question-naire of nonacademic talented activities and accom-plishments in mathematics, and was designed to pro-vide a meaningful external criterion of creative abilityin mathematics (Milgram & Hong, 1994; Milgram,Hong, Shavit, & Peled, 1997). The measure consistedof 36 items, presented in random order. Of the 36 items6, 5, 11, and 14 items represented the nongifted, mild,moderate, and profound levels, respectively. For exam-ple, “Do you compete with your friends in solving sim-ple computerized mathematical problems?” or “Doyou explain physical phenomena with an originalmathematical model?” represented the nongifted andprofound creative levels in mathematics, respectively.Each item required the research participant to indicateby answering yes or no whether he or she had or hadnot participated in the particular extracurricular, chal-lenging activity or attained the accomplishment inmathematics. One point was given for each yes answer.

Four scores indicating four levels of domain-spe-cific creative ability in mathematics were computed foreach research participant. The range of scores for the

ordinary, mild, moderate, and profound levels of do-main-specific creative ability was 0–6, 0–5, 0–11, and0–14, respectively. The Kuder-Richardson internalconsistency of the 36-item measure of domain-specificcreative ability in mathematics was .95 (Kuder & Rich-ardson, 1937). Livne and Milgram (2000) found a highdegree of agreement among 11 academic expert judgesthat indicated good evidence of the content validity forthe instrument. The Kendall coefficient of concor-dance for the judges was .63, p < .0001 (Livne, 2002).The Tel Aviv Activities and Accomplishments Inven-tory: Mathematic was also developed on the basis of amapping sentence that was created specifically for thatpurpose. Details of this process are presented else-where (Livne & Milgram, 2000).

Tel Aviv Creativity Test (Milgram & Milgram,1976a). Two verbal items selected from the “Uses”subtest of the Tel-Aviv Creativity Test (Milgram &Milgram, 1976a) were used as indicators of generaloriginal/creative thinking. The Tel-Aviv CreativityTest is an abbreviated and revised form of the Wallachand Kogan (1965) instrument designed to assessideational fluency. It yields scores that are highly reli-



Table 1. Measures of Mathematical Giftedness Divided by Types of Ability, Instruments, Their Indexes, Types of Measure, andScale Range

Ability Type Instruments IndexType of

MeasureScaleRange

Domain-specific AcademicAbility in Mathematics

School Grade in Mathematics Final Grade in Mathematics One FinalGrade

0–100

Domain-specific AcademicAbility in Mathematics

Multiscale Academic andCreative Abilities inMathematics—MACAM(Livne & Livne, 1999a).

Open-ended AcademicMathematical Problems

Eight0–7 items

0–64

Domain-specific CreativeAbility in Mathematics

Multiscale Academic andCreative Abilities inMathematics—MACAM(Livne & Livne, 1999a)

Open-ended CreativeMathematical Problems

Eight0–7 items

0–64

Domain-specific CreativeAbility in Mathematics

Tel Aviv Activities andAccomplishments Inventory:Mathematic—TAAI: M(Livne & Milgram, 1999)

Creative Out-of-schoolActivities in Mathematics

360–1 items

0–36

General Creative Thinking Tel Aviv Creativity Test—TACT(Milgram & Milgram, 1976a)

Responses on Tasks ofIdeational Fluency

Two0–no limit

0–no limit

General Intelligence (Verbal) Abstract Verbal Thinking Test(Glantz, 1996)

General IntellectualProcesses

1800–1 items

0–180

General Intelligence(Nonverbal)

Advanced Progressive Matrices(Raven, 1962)

Nonverbal Tasks 360–1 items

0–36

Note. N = 917.

able and distinct from intelligence (Milgram, 1983;Milgram & Arad, 1981; Milgram & Milgram, 1976b).The quantity of ideational fluency was considered avalid index of general original thinking, based on thefindings of Milgram, Milgram, Rosenbloom, andRabkin (1978) and Milgram and Rabkin (1980) that in-dicated a strong correlation between the quantity andquality of distinct ideas generated on tasks ofideational fluency. The research participants wereasked to write down as many different uses as theycould for each of two common objects, a newspaperand a shoe. Two scores for creative thinking, one foreach object, with a possible range of from zero to nolimit were computed for each research participant;each consisted of the number of distinct ideas gener-ated in response to the item presented. The internalconsistency of the two-item measure of general cre-ative thinking was .77.

Abstract Verbal Thinking Test (Glanz, 1996).This instrument, a verbal group intelligence test widelyused in Israel, was the measure of general verbal intel-ligence. It consists of 180 items divided into ninesubtests representing nine different cognitive pro-cesses (opposites, synonyms, essential characteristics,identification of oddities, groups, classification, defini-tions, proverbs, analogies, and syllogisms) with 20multiple choice items in each. One point was given foreach correct answer. One total score for verbal intellec-tual ability was computed for each participant. It con-sisted of the sum of correct answers for each of the 180items and ranged from zero to 180. The Kuder-Rich-ardson internal consistency of the 180-item measure ofgeneral intelligence was .98.

Advanced Progressive Matrices (Raven, 1962).This instrument was used as a measure of nonverbalgeneral intelligence. It consisted of 36 advanced andprogressively difficult nonverbal matrix tasks, eachpresented with a section missing and followed by eightalternative replacement sections. Research participantswere asked to select one from among the eight alterna-tives as the best replacement section that would serveto complete the matrix. Each correct response receiveda score of 1. One total score for nonverbal general in-telligence was computed for each participant. It con-sisted of the sum of correct answers and ranged from 0to 36.

The instrument was translated from English intoHebrew using a double-blind translation procedure de-veloped by Elder (1973). The Kuder-Richardson inter-nal consistency of the nonverbal 36-item measure ofgeneral intelligence was .90.

Procedure

The instruments were group-administered in three2-hr sessions, with 1 week between each session, to thestudents in their classrooms according to the standardinstructions provided by the authors of each instru-ment. The student participants were divided into fourequal groups, and the instruments were administered inthe first two sessions to each group in one of four coun-ter-balanced orders. In the third session, the verbal andnonverbal measures of intelligence were administeredto the students in a counter-balanced order. The Ad-vanced Progressive Matrices had a time limit and allthe other instruments were administered with no timelimits.

Results

Structural equation modeling (SEM) analysis(Byrne, 1994; Pearl, 2000) was used to test the interre-lationships postulated by the 4 × 4 Model (Milgram,1989, 1991) among the 16 (4 × 4) conceptual compo-nents of general and domain-specific academic andcreative abilities in mathematics at four levels. Thefirst analysis focused on the discriminant validity ofgeneral and domain-specific ability types, and, there-fore, the data were investigated across the four abilitylevels. The findings of the SEM analysis are presentedin detail in Figure 2 in the Appendix. They providedgood evidence of the structural aspect of construct va-lidity for the ability dimension of the 4 × 4 model(Messick, 1995).

To determine whether the four ability levels(nongifted, mild, moderate, and, profound) representedhierarchical levels that could be empirically supported,additional SEM analyses were conducted in which theproposed 4 × 4 was compared with seven alternativemodels. Because this analysis focused on thediscriminant validity of levels, the data were investi-gated across ability types. The findings of this SEManalysis are also presented in detail in Table 3 in the Ap-pendix. They provided good evidence of the structural



aspect of construct validity (Messick, 1995) of four dis-tinct and hierarchical ability levels.

The two sets of SEM findings presented indicatedbetter overall fit of the 4 × 4 model compared with al-ternative models for the four types and four levels ofabilities. Based on these overall SEM findings, we nextexamined the validation of the postulation that of thefour types of abilities in mathematics; two are generaland two domain specific.

First, we examined the causal relationships betweengeneral creative thinking and each of the two indexesof domain-specific abilities in mathematics, one aca-demic and one creative. These detailed SEM results arealso presented in Figure 2 in the Appendix. Generalcreative thinking predicted domain-specific creativeability in mathematics as measured by a score thatcombined both creative mathematical problems andout-of-school activities in mathematics. The relation-ship between them was β = .22, p < .001. However,when domain-specific creative ability in mathematicswas measured by each of the measures separately, thecorresponding relationships were different from thosepresented in Figure 2.

The relationships between general creative thinkingand the creative mathematical problems and theout-of-school activities in mathematics were β = .57, p< .0001 and β = .22, p < .0001, respectively. The rela-tionship between the two measures of domain-specificcreative ability in mathematics, that is, creative mathe-matical problems and out-of-school activities in math-ematics, was also highly significant β = .25, p < .0001.Taken together, these results provided good evidencefor the convergent validity of the general and the do-main-specific creative abilities in mathematics.

For general intelligence—domain-specific aca-demic ability in mathematics, the relationship was β =.38, p < .001. By contrast, no significant relationshipwas found between general intelligence and either ofthe two measures of domain-specific creative ability inmathematics. General creative thinking was not foundto be related to domain-specific academic ability inmathematics. These data provided good evidence ofdiscriminant validity for the two postulated abilitytypes, that is, domain-specific academic and creativeabilities in mathematics.

Based on the good evidence of four distinct andhierarchical ability levels of academic and creativeabilities when analyzed simultaneously as presentedin Figure 2 in the Appendix, we examined the degree

of distinctiveness between the four levels for each ofthe academic and the creative abilities in mathematicsseparately. We compared the percentage of studentswho demonstrated each ability level. Table 2 summa-rizes the percentage of students divided by do-main-specific abilities in mathematics at four abilitylevels, reflecting the ability type and level combina-tions for those abilities.

As predicted, for each of the two domain-specificabilities in mathematics, the higher the ability level, themore infrequently it occurred. To ascertain that thesefrequencies indeed reflected hierarchical levels, theywere further examined by means of theJonckheere-Terpstra test, a technique designed specifi-cally for a priori descending ordering (SPSS, 2004). Forthe academic ability as measured by the academic math-ematical problems, and the creative ability, as measuredby both creative mathematical problems andout-of-school activities in mathematics, the resultingJonckheere-Terpstra statistic was = 32.09, 19.55, and29.09, ps <.0001, respectively, indicating that the differ-ences among the frequencies of students demonstratingtheiracademicandcreativeabilities inmathematics rep-resented hierarchical levels. These findings constitutedstrong evidence of discriminant validity for the four hi-erarchical levels in both domain-specific academic andcreative abilities in mathematics.

Discussion

The most important finding of this study was theempirical support of the two-ability type and four-levelformulation of giftedness in mathematics, as postu-lated by the 4 × 4 structure of giftedness model



Table 2. Percentage of Students Divided by TwoDomain-Specific Academic and Creative Abilities inMathematics by Ability Levels

Domain-Specific CreativeAbility in Mathematics

MathematicalProblems

Out-of-SchoolActivities

Profound 2 1 1Moderate 9 5 6Mild 31 18 15Nongifted 57 76 78

(Milgram, 1989, 1991). The conceptualization ofgiftedness in mathematics as a bidimensional con-struct, consisting of domain-specific academic abilityand domain-specific creative ability in mathematics,each at four hierarchical levels, constitutes a new per-spective of mathematical ability. These findings lead toan innovative approach to identifying mathematicalabilities and provide reliable and valid psychometrictools to make it possible.

One serious problem frequently cited in evaluatingtheories of creativity is the lack of serious empirical ev-idence demonstrating the construct validity of pro-posed theories. For example, Sternberg (1991) criti-cized Gardner’s highly regarded theory of multipleintelligences (MI; Gardner, 1983) as lacking empiricalevidence showing that the cognitive processes underly-ing the domains postulated in MI theory are indeed un-related. More recent research on MI theory-based as-sessments provided evidence in support of Sternberg’sconcern about the psychometric quality of Gardner’sapproach (e.g., Plucker, 2003; Plucker, Callahan, &Tomchin, 1996). In this study, highly sophisticated,modern research methods were used to demonstratethe construct validity of the theory of creativity inmathematics on which the study was based.

The two new instruments developed for use in thisstudy are characterized by both a high level of reliabil-ity and construct validity. The Multiscale Academicand Creative Abilities in Mathematics (MACAM) andthe Tel Aviv Activities and Accomplishments Inven-tory: Mathematics (TAAI: M) provide usefulpsychometric instruments for identifying differenttypes and levels of mathematical abilities. Screeningfor academic and creative abilities in mathematics us-ing the new instruments enables teachers to differenti-ate curricula and to individualize instructional strate-gies to match each student’s needs. In a recent pilotstudy (Milgram, Davidovitz, Livne, Livne, &Lieberman, 2004), the definitions of each of the fourlevels of academic and creative concepts in mathemat-ics were matched to curriculum units used in highschools, and differentiated computerized units for indi-vidual instruction were developed as matched to typeand level of ability. Milgram and her associates foundthat 87% of the students were able to attain a high levelof mastery of the academic concepts by means of com-puterized units, reaching correct solutions of the prob-lems presented to them in the units. The use of individ-ualized computerized units may well contribute to

teaching large classes of students with different typesand levels of ability in mathematics.

Creative thinking predicted creative, but not aca-demic, ability in mathematics. By contrast, IQ pre-dicted academic, but not creative, ability in mathemat-ics. In fact, the relationship of creative thinking tocreative ability in mathematics was significantly stron-ger than the corresponding relationship of IQ to aca-demic ability in mathematics. These findings indicatethat, by using only measures of intelligence and ofschool grades in mathematics in the process of identi-fying giftedness in mathematics, we are losing impor-tant information. Adding measures that can identifygeneral creative thinking and domain-specific creativeability in mathematics at four levels could lead to theidentification of the mathematical abilities of a muchwider range of students both gifted and nongifted. Aprocess of identification that includes measures of fourabilities rather than just IQ and school grades will pro-vide diverse students with more equal opportunity torealize their special abilities in mathematics to the full-est extent.

Creative thinking was found to be more strongly re-lated to domain-specific creative ability in mathemat-ics, when measured by the ability to produce creativesolutions to mathematical problems than when mea-sured by creative out-of-school activities in mathemat-ics. The difference in these relationships could be ex-plained in terms of the degree to which each indexrepresented real-world behavior in mathematics. Themore the measure of domain-specific creative ability inmathematics resembled real-world problem-solvingbehavior in mathematics, the better it was predicted bygeneral creative thinking. However, because the rela-tionship between the two measures of domain-specificcreative ability in mathematics was highly significant,the activities index might be particularly useful in iden-tifying learners with mathematical ability not yet real-ized in solving problems in mathematics,

Two other applications emerged from this study, onepsychometric and one methodological. We developed atwo-stage technique for developing the twopsychometric instruments used in this research (Livne,Livne, & Milgram, 1999; Livne & Milgram, 2000). Inthe first stage a mapping sentence in which the concep-tual components to be measured were clearly definedand delineated was developed. In the second stage, onthe basis of the mapping sentence, the actual items foreach test were created and theoretical and operational



scoring guidelines were developed. Our findings dem-onstrated that both creative and academic abilities inmathematics could be identified separately by the newinstruments. Such a two-stage technique could consti-tute a prototype for developing valid measures of othercognitive processes in mathematics, and/or for develop-ing other instruments in educational and/or psychologi-cal research. However, whether specific abilities wouldbe equally distinct in other domains, such as science,technology, music, drama, dance, art, or even socialleaderships, remainsaquestionworthyof investigation.

There is also a methodological application possiblefrom the results of this study. As a comprehensiveone-shot solver in Brandt’s (1999) terms, the SEManalysis, as used in this research, provided a way to es-timate two distinct dimensions of a theoretical model,ability types and levels, simultaneously. Traditionalconfirmatory analysis has been used forunidimensional models only (Hair, Anderson, Tathan,& Black, 1995). SEM analysis has been recommendedas the best advanced technique for investigating con-struct validity, because it takes into account the valid-ity, reliabilities, and error measurement of each of thescales (Byrne, 1994; Hair et al., 1995). By contrast, er-ror measurement is not controlled when the traditionalmultitrait-multimethod technique is used to investigateconvergent and discriminant validity and, therefore, isnot often used today (Campbell & Fiske, 1959;Coenders & Saris, 2000). The SEM technique can beapplied to investigate the construct validity of othermultidimensional theoretical models. The techniquecan also be applied to a wide variety of educational do-mains in addition to mathematics, such as science, lan-guages, literacy, arts, and others. However, whetherspecific abilities would be equally distinct in other spe-cific domains remains to be investigated.

Unrecognized talent in mathematics is the failureof an adult to realize the potential mathematical abili-ties that he or she demonstrated in youth. With therapid development of the mathematical and scientificchallenges of the 21st century, the problem of unrec-ognized talent in mathematics has become intolerableboth for the individual and the society (Milgram,1993; Rice, 1999). According to Mercer’s strategicretention analysis (2004), this problem is so seriousthat it puts all industry in the United States at risk.The National Council of Teachers of Mathematics(2004) defined this problem as a challenge andformed a new vision of preparing students who are

capable of drawing on knowledge from a wide varietyof mathematical topics and approaching the sameproblem from different mathematical perspectives tofind methods for its solution. The Council called onthe educational system to enable each individual “tocompute fluently and to solve problems creativelyand resourcefully” (National Council of Teachers ofMathematics, 2004, p. 1). The goal of this study wasto contribute to meeting this challenge. We suggestthat improvement in the theoretical formulation andmeasurement of different kinds of mathematical abil-ity at different levels might contribute to the enhance-ment of achievement in mathematics. It might helpeach individual child realize his or her potential morefully and thus provide society with greater utilizationof the abilities of its citizens.

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AppendixStructural Equation Analysis

Structural equation modeling analysis (Byrne,1994; Pearl, 2000) was used to test the interrelation-ships postulated by the 4 × 4 Model (Milgram, 1989,1991) among the 16 (4 × 4) conceptual components ofgeneral and domain-specific academic and creativeabilities in mathematics at four levels. The EQS 5.7program (Bentler, 1997; EQS, 1997) was used to con-duct a unified analysis. The results in the terminologyof SEM are presented in Figure 2.

Four factors (F1 to F4) represent the four abilitytypes. The single-headed arrows leading from the gen-eral intelligence and the general original creative



thinking to domain-specific academic and do-main-specific creative ability in mathematics factors(β13, β24, respectively) represent the hypothesizedcausal relationships between the factors. Four factors(F5 to F8) represent the four hierarchical ability levelsin mathematics. The hierarchy is expressed by threesingle-headed arrows, denoted β56, β67, and β78 thatrepresent causal relationships between pairs of adja-cent ability levels. All eight latent factors (F1 to F8) arelinked to 17 observed variables or indicators repre-sented in Figure 2 by 17 rectangles labeled V1 to V17.The single-headed arrows leading from each factor tothe rectangles V1 to V17 (denoted λ1 to λ29) representregression paths linking each of the eight factors to itscorresponding set of observed indicator scores, that is,how well the specified indicators load onto the appro-priate factors.

To examine the feasibility of four abilities in mathe-matics, two general and two domain-specific, the pro-posed 4 × 4 model, hereafter denoted “Model P” wascompared with four alternative models (Browne &Cudeck, 1993; Hair et al., 1995; Hoyle, 1995). Themodels suggested theoretical combinations thatmerged general and/or domain-specific ability types,each representing a smaller number than the four pos-tulated by the 4 × 4 Model. A one-factor model (ModelA) combined all four ability types into a single abilitytype (F1, F2, F3, F4). There were two types of two-fac-tor models. Model B combined general intelligenceand domain-specific academic ability in mathematics,and general creative thinking and domain-specific cre-ative ability in mathematics into two ability types(F1+F3, F2+F4). Model C combined general intelli-gence and creative abilities into one factor, and do-main-specific abilities into a second factor (F1+F2,F3+F4). There was a three-factor model (Model D)that represented general intelligence, general creativethinking separately, and combined domain-specific ac-ademic and creative abilities into a single ability type(F1, F2, F3+F4). Other combinations that merged abil-ity types were theoretically unacceptable. These analy-ses focused on the discriminant validity of general anddomain-specific ability types, and, therefore, the datawere investigated across the four ability levels.

Of the four alternative models (A to D), three mod-els (A, B, and D) were statistically unacceptable, be-cause they were based on results that represented lin-early dependent parameters that could not generatereliable and valid results (Bentler, 1997; Byrne, 1994).

Only Model C, provided evidence of valid fit indexesthat were compared with those of the 4 × 4 model, thatis, Model P.

The findings indicated that Model P, representingthe four ability types, provided evidence of a better fitto the data when compared with the alternative ModelC that combined general intelligence and domain-spe-cific academic ability in mathematics, and general cre-ative thinking and domain-specific creative ability inmathematics. Model P, representing the four distinctability types, provided good fit indexes, which weresignificantly higher compared to those of Model C.The overall estimation of the 4 × 4 Model P resulted ina Satorra-Bentler scaled χ2 (100, N = 917) = 390.73, p< .001, and an unadjusted χ2 (100, N = 917) = 411.38, p< .001, compared with a Satorra-Bentler scaled χ2

(100, N = 917) =1210.73, p < .001, and an unadjustedχ2 (100, N = 917) = 4678.09, p < .001 for Model C.

However, because these models were nonnestedmodels, that is, they differed in terms of the numberof ability type constructs, the comparison betweenthem relied only on the goodness-of-fit indexes andnot on the χ2 difference test (Hair et al., 1995; Hoyle,1995). The fit indexes of Model P were Normed FitIndex (NFI) = .91; Nonnormed Fit Index (NNFI) =.91; Comparative Fit Index (CFI) = .93; IncrementalFit Index (IFI) = .93; LISREL Goodness-of-Fit Index(GFI) = .95; Adjusted Goodness-of-Fit Index (AGFI)= .93; and root mean squared error of approximation(RMSEA) = .058, compared to the corresponding in-dices of the alternative model NFI = .73; NNFI = .66;CFI = .74; IFI = .74; LISREL GFI = .85; AGFI = .79;and RMSEA = .111. Because the fit indexes of ModelC were far lower than the minimal acceptable fit in-dex value of .90 and minimal range of .05–.07 ofRMSEA, it was rejected. These findings providedgood evidence of the structural aspect of constructvalidity (Messick, 1995) of four general and do-main-specific academic and creative abilities in math-ematics that were distinct.

To determine whether the four ability levels(nongifted, mild, moderate, and, profound) repre-sented hierarchical levels that could be empiricallysupported, the proposed 4 × 4 model, denoted Model P,was compared with seven alternative models. Becausethis analysis focused on the discriminant validity of thefour ability levels the data were investigated acrossability types. The seven alternative models representedall the theoretically acceptable combinations of merg-



211

Figure 2. Structural validation of the 4 × 4 factor structure of giftedness model as applied to mathematics. χ2(100, N = 917) 390.73; Normed FitIndex = .91; Nonnormed Fit Index = .91; Comparative Fit Index = .93; Incremental Fit Index = .93; Goodness-of-Fit Index = .95; Adjusted Good-ness-of-Fit Index = .93; root mean squared error of approximation = .058.

ing adjacent ability levels, each representing a smallernumber than the four postulated by the 4 × 4 model.Other models that included more than four levels werenot tested due to a lack of a theoretical basis.

Evidence of valid fit indexes was found for three ofthe seven alternative models: a one-factor model(Model E) that combined all the four ability levels (F5,F6, F7, F8), a two-factor model (Model F) that repre-sented the nongifted level, and the mild, moderate, andprofound levels combined (F5, F6+F7+ F8), and athree-factor model (Model G) that represented thenongifted and the mild levels combined, the moderateand the profound levels (F5+F6, F7, F8). These modelswere compared with Model P. Because these modelswere nonnested models, the comparison relied only onthe goodness-of-fit indexes (Hair et al., 1995; Hoyle,

1995). The fit indexes of Model P and the three alterna-tive models are presented in Table 3.

Model P, representing the four distinct ability levels,provided good fit indexes, denoted in bold, which werehigher than those of Model E to Model G. Because thefit indexes of the three alternative models were lowerthan those of Model P and did not satisfy the minimalacceptable value of .90, and the RMSEA’s acceptablerange of .05–.07, they were rejected. These findingsprovided good evidence of the structural aspect of con-struct validity (Messick, 1995) of the four distinct andhierarchical ability levels. Based on the better overallfit of the 4 × 4 model compared with alternative mod-els, the validation of four general and domain-specificacademic and creative abilities in mathematics andfour ability levels was further investigated.



Table 3. Goodness-of-Fit Indexes for Four Nonnested Models of Ability Levels of the Structure of Giftedness in Mathematics

Model NFI NNFI CFI IFI GFI AGFI RMSEA

A. One-factor model:(F5, F6, F7, F8) .87 .86 .88 .83 .90 .88 .071

B. Two-factor model:(F5, F6+F7+F8) .85 .83 .87 .87 .90 .87 .078

C. Three-factor model:(F5+F6, F7, F8) .86 .85 .87 .88 .91 .87 .074

P. Four-factor model:the proposed modelF5, F6, F7, F8 .91 .91 .93 .93 .95 .93 .058

Note. NFI = Normed Fit Index; NNFI = Nonnormed Fit Index; CFI = Comparative Fit Index; IFI = Incremental Fit Index; GFI = Goodness of FitIndex; AGFI = Adjusted Goodness of Fit Index; RMSEA = Root Mean Square Error of Approximation. F5 = nongifted level; F6 = mild level; F7= moderate level; F8 = profound level.

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