Estimating the Size of the Gifted/Talented Population … Estimating the Size of the Gifted/Talented...

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Estimating the Size of the Gifted/Talented Population From Multiple

Identification CriteriaJean Bélanger and Françoys Gagné Université du Québec à Montréal

“How many are gifted or talented?” although very important, this question has received little attention from most scholars in gifted education. four major variables are iden-tified that directly impact prevalence estimates: (a) the type of definition endorsed—disjunctive or conjunctive, (b) the number of abilities/characteristics included, (c) the minimum selection threshold chosen, and (d) the correlations observed between the abilities included. a range of values was determined for the last three variables. By combining these values, we produced four tables of prevalence estimates, which reveal how these variables interact and how they affect the “how many” question. the results show that both the type of definition endorsed for the giftedness or talent concepts and the selection ratio have a crucial impact on the prevalence estimates. the authors argue that gifted education will never be recognized as a proper scientific field unless its scholars reach a consensus on both the definition and prevalence issues.

Howmanyindividualscanbelabeledgiftedortalented?Thisques-tion, commonly called the prevalence issue, appears quite simple,deceptivelyso.Aswewillseebelow,somanyconsiderationsneedtobemadethatnosimpleanswercanbegiven.Thepresentarticleaimstoexploresomeofthefactorsthataffecttheanswerandshowhowinteractionsbetweenthemmodifyprevalenceestimates.Commonsense and simple statistical knowledge do reveal basic rules. Forinstance, the higher (more selective) the cutoff value chosen (e.g.,top15%,top5%,ortop1%),thesmallerthenumberofindividu-als labeled gifted and talented. Similarly, if giftedness and talentcantakemanyforms(e.g.,intellectual,social,artistic,athletic),theprevalencewillincreaseasthenumberofdomainsorfieldsincreases.Or, if twocriteriaarehighlycorrelated, thosewhoexcel inoneofJeanBélangerisProfessorintheDepartmentofSpecialEducationandTrainingattheUniversityofQuebecinMontreal(UQAM).ProfessorFrançoysGagnéisaFrenchCanadianfromMontreal,Quebec.Hehasbeenactiveingiftededucationsincetheearly1980sandhaspublishedextensivelyinthatfield.

Journal for the Education of the Gifted.Vol.30,No.2,2006,pp.131–163.Copyright©2006PrufrockPressInc.,http://www.prufrock.com

Journal for the Education of the Gifted132

themwillalsoexcelintheother,thusreducingthetotalnumberofindividuals excelling in either ability, as opposed to uncorrelatedabilities.Beyondthesesimplecommonsenseobservationsliemuchmorecomplexconsiderations.Weintendtoexploresomeofthemandlookathowthemanipulationofrelevantvariables(e.g.,numberof abilities, cutoff value, correlations between abilities) affects theresultingprevalenceofgiftedandtalentedpersons.

Status of the Prevalence Issue

AccordingtoGagné(1993,1998a),theprevalenceissueisextremelyimportant,notonlytheoretically,butalsopolitically,aswellasprac-tically. Theoretically, the concepts of giftedness and talent belongto the class of normative concepts. Such concepts circumscribesubgroups of people who differ from the norm through specificcharacteristics, like income (poverty, affluence), weight (obesity,emaciation), age (adolescents, seniors), and so forth. As originallypointedoutbyFrancisGalton(1892/1962),appropriatedefinitionsof suchconceptsrequireclearoperationalcriteria formembershiporexclusion.Inhisstudyof familyrelationshipsbetweeneminentpeople, he defined eminence as presence among the top 1:4,000Englishmen intermsofcelebrity throughaccomplishments insci-ence,politics,andarts.Politically,the“howmany”questioniscrucialbecauseof its frequency indiscussionswithmediapeopleandthegeneralpublic;manyamongthemwanttoknowifgiftednessandtalentmanifestthemselvesexceptionally(e.g.,geniusesorprodigies)or rather commonly (the “everyone is somehow gifted” assertion).Practically, perceived prevalence directly impacts identificationpoliciesandprocedures,and,consequently,budgetexpendituresforenrichmentprograms.

Inspiteofitscrucialimportance,theprevalenceissuehasreceivedlittle attention in the gifted education literature. Bélanger (1997)notedthattextbookseitherignorethesubjectcompletely(Coleman& Cross, 2001; Gallagher, 1985; Passow, 1979; Piirto, 1994) ormention itonlybriefly (Borland,1989;Clark,1997;Horowitz&O’Brien,1985;Renzulli&Stoddard,1980;Webb,Meckstroth,&

Prevalence of Giftedness and Talent 133

Tolan, 1982). Neither the term prevalence nor any synonyms everappearintheSubjectIndexofanyofthem.Indeed,onlyonescholar,Gagné(1993,1998a)hasdeemedimportantanexaminationofthequestionindepth.Whyhavesofewresearchersandscholarsanalyzedtheprevalenceissue?Webelievethatthecomplexityoftheproblem,aswellasitssophisticatedmathematicalunderpinnings,explaininlargepart thegeneral silenceonthis subject.Moreover,aswewillsee below, every answer requires that its proponent make a some-whatarbitrarychoice,somethingthatveryfewscholarsseemreadytodo.Gagné’s(1993)initialdiscussionoftheprevalenceissuecov-eredverywellthemainhurdlesthatblockaclearandsimpleanswerto the“howmany”question.Thepresent text focuseson the fourmajorparametersthatprofessionalsneedtoconsideriftheywanttoestimatewithsomeprecisiontheprevalenceofgiftedandtalentedindividuals.Itisworthnotingthateachofthesefourparametersisdirectlyinfluencedbythewayscholarsinthefieldofgiftededuca-tiondefinetheconceptsofgiftednessand/ortalent.

Major Parameters of Prevalence Estimation

The four most relevant factors that directly affect prevalence esti-matesare(a)thesizeoftheminimumthreshold,(b)thenumberofdifferentformsofgiftednessandtalentconsideredsimultaneously,(c)theconjunctiveordisjunctivecharacterofthedefinitionadopted,and(d)thecorrelationsbetweenthevariousabilitiesassessed.

The Minimum Threshold

Mostnormativeconceptsareoperationalizedthroughquantitativemeasurementsexpressedoncontinuousscales:IQscores,weightsinpounds, incomeindollars,age inyears,andso forth.Whenfacedwith continuous quantitative measures, professionals must decide,somewhatarbitrarily,where toplace theboundarybetweenmem-bership and nonmembership. For instance, nutritionists decidedsome years ago to adopt the body mass index (BMI) to measurenormalandabnormalweightinpopulations.Onascalethatranges


fromalowofabout16andahighwellabove60,theyagreedtoplacethenormalrangebetweenthevalues20and24.9,usethevalues25to29.9tooperationallydefineoverweight,andtoassociateobesitywithanyvalueequalorsuperiorto30(NationalInstituteofHealth,1998).Thesenutritionistscouldhavechosenothercutoffvalues;theimportantpointisthatagroupofprofessionalsdecidedtoagreeonasetofvaluesandmakethemtheyardstickforalltheiranalyses.TheBMIindexisbynomeansperfect,yetitstillremainsaverypracticalmeasure,allowingallkindsofcomparisons(e.g.,betweenagegroups,genders,nations,andsoforth).Similaragreementshavebeenmadeinotherfields.Thinkofthedecisionbystatisticianstofixat.05theminimum significance level, that of law enforcement agencies tospecify the minimum amount of alcohol in the blood confirmingimpaireddrivingbehavior,or theeffortsofeconomists to identifyminimumlevelsofincomethatmarkthethresholdbetweenpovertyandsustenance.

Unfortunately,thatkindofconsensushasnotyetreachedgiftededucation.Inthecaseofintellectualgiftedness,whichiscommonlymeasured with IQ tests, minimum thresholds proposed over theyears have ranged from lows of approximately 1% (e.g., Terman,1925)tohighsofmorethan20%(e.g.,Renzulli,1986),withmanyvaluesproposedbetweentheseextremes.Sucha largevariability,nolessthan20foldbetweenextremes,seriouslyquestionstheclar-ity of existing definitions of giftedness and talent. According toGagné(1993),twomodaltendenciesappeartostandout:aselec-tiveperspectiveexemplifiedbyvery lowpercentages(below5%),andaliberalapproachwheretheproposedestimateshoveraround10%or15%.Inanefforttorallybothgroupstosomeconsensus,Gagné (1998a) proposed a metric-based system of intensity lev-elswhoseminimumthresholdisfixedat10%andlabeledmildly.Withinthistop10%ofmildlygiftedortalentedpersons,Gagné’sdifferentiatedmodelofgiftednessandtalent(DMGT)recognizesfourprogressivelymoreselectivesubgroups.Theyarelabeledmod-erately(top1%),highly(top1:1,000),exceptionally(top1:10,000),andextremely(top1:100,000).Itistooearlytoseeifthatproposalwillattractalargegroupofadherents.Inthemeantime,wemust


acknowledgeadiversityofviewpointsonthequestionofthemini-mumthreshold.

The Number of Criteria

The complexity of the prevalence question increases significantlywhen definitions of giftedness, and the ensuing identification cri-teria, include multiple abilities or traits. Most published defini-tions identify multiple traits (e.g., Davis & Rimm, 1994; Gagné,1985,2003;Gardner,1983/1994;Marland,1972;Renzulli,1986;Shriner, Ysseldyke, Gorney, & Franklin, 1993; Sternberg, 1986;Tannenbaum,1983).Insomecases,thenumberofcriteriaremainssmall,likeinSternberg’sdistinctionofthreeformsofgiftednessasso-ciated with the three major components of his triarchic theory ofintelligence (metacomponents, performance components, knowl-edge-acquisition components), or Tannenbaum’s four categoriesoftalent(scarcity,surplus,quota,anomalous).Inothercases,theirnumberincreasessignificantly,likeinGardner’s(1983/1994)sevendistinct intelligences, or in Gagné’s (1985, 2003) four domains ofgiftedness(intellectual,creative,socioaffective,sensorimotor),eachofthemsubdividedintomultiplesubdomains.Themorenumeroustheformsofgiftednessortalent,thelargerthepopulationofgiftedand/or talented persons; as mentioned earlier, when more abilitydomains are recognized, it stands to reason that more individualscanexcelinatleastoneofthem.But,asdescribedbelow,thereisonespecialsituationwherethatcommonsenseruledoesnotapply.

The Conjunctive and Disjunctive Perspectives

Multitraitdefinitionsofgiftedness(ortalent)differnotonlywithregard to the specific number of traits covered by each one; theyalsodifferwithregardtothewaythesetraitsaredealtwith.Borland(1989) distinguished two opposite types of multitrait definitions:(a)conjunctivedefinitions,wheregiftednessrequiresthesimultane-ous(AandBand C)possessionofdistinctcharacteristicsorabili-ties, like Renzulli’s (1986) combination of above-average abilities,creativity,andtaskcommitment;and(b)disjunctivedefinitionsof


giftedness,wheregiftednessand/ortalentcantakedifferent forms(AorBorC),likeGardner’smultipleintelligencesorGagné’sfourgiftednessdomains.

Thesetwotypesofdefinitionshaveoppositeimpactsonpreva-lence estimates. Assuming an identical minimum threshold, forinstancethetop10%foreachtraitorcriterion,conjunctivedefini-tionsleadtodecreasingprevalencevaluesasthenumberofcriteriaincreases:mostofthosewhosurvivethefirstcutoffscore(A)willnotsurvivethesecond(B),andmostoftheAandBsurvivorswillfail to exceed the third threshold (C). As the number of criteriaincludedinthedefinitionandidentificationprocessincreases,theresultingnumberofindividualslabeledgiftedortalentedwillsoonbecome uselessly small. In the case of disjunctive definitions, theimpactisreversed;ifgiftednesscanmanifestitselfasAorBorC,thentheresultingpopulationofpersonslabeledgiftedortalentedwillincreasewitheachnewcriterionorcategoryadded.Onebeginswith the top 10% on criterion A, then adds all those not alreadychosenwhoexcelonB,andsoforthwitheachadditionalcriterion.Taylor (1973) applied this reasoning in the context of his theoryof multiple talents; assuming his eight talents to be uncorrelated,he estimated that 99% of the members of a population would beidentifiedasaboveaverage(top50%)onatleastoneoftheseeightabilities.Withamuchstricter10%selectionratio,thecorrespond-ingprevalencewouldstillbe57%.Inotherwords,morethanhalfofthepopulationwouldbeamongthetop10%inatleastoneofhistalents.AsJohnCarroll’s(1993)seminalworkhassoclearlyshown,mostmentalabilitiesarecorrelatedtosomeextent,andthesecor-relationshaveasignificantimpactonprevalenceestimates.Tayloracknowledgedtheimportanceofthatproblem,butdidnotaddressit.

Correlations Between Criteria

Correlations between some abilities can be very low, for instancebetweenphysicalandcognitiveabilities;theyalsocanbequitestrong,especiallywhenwecompareabilitiesbelongingtothesamedomain.Tounderstandtheimpactofcorrelatedcriteriaonprevalencevalues,


letuslookatthemostsimplesituation,thatoftwoabilities.Ifthesetwomeasuresareuncorrelated,thenanidenticaltop10%thresholdappliedtoeachofthemwillleadtoeither1%identifiedasgiftedinboth(conjunctivedefinitions)or19%identified ineitherofthem(disjunctivedefinitions).1If,ontheotherhand,thesetwocriteriaareperfectlycorrelated,thenthesameindividuals—thetop10%—willbeselectedwithcriteriaAand/orB;inthatspecialcase,disjunctiveand conjunctive definitions will lead to the same result. It followsthat,whenabilitiesarepartiallycorrelated,thegapbetweenthetwoprevalenceindices—here18%—willdecreasetowardzeroasthecor-relationincreases.Itiseasytoimaginehowcomplicatedtheestima-tionprocesswillbecomeasthenumberofcriteriaincreases,becauseeachpairofcriteriawithinthegroupwillhaveadifferentcorrelationthantheotherpairs.

Operational Goals

Nowthatthefourmajorparametersforprevalenceestimationhavebeen identified and their individual impact coarsely described, itbecomeseasiertopinpointthepracticalgoalsofthisstudy.First,weaimtoidentifyastatisticalequationthatwillallowthesimultaneousmanipulationofvariousvaluesofthefourparametersandobservehow they affect the theoretical prevalence of gifted and talentedindividuals.Forexample,howdoesaprogressiveincreaseintheaver-agecorrelationsbetweenabilitiesaffecttheprevalencevalueswhenall other parameters are controlled? Or, how does the prevalenceincrease in a disjunctive situation when the number of measuredabilitiesincreases,againcontrollingfortheotherparameters?Thesemanipulationswillleadtofourtablesofstatisticaldata,twoeachfordisjunctiveandconjunctivesituationsrespectively.Weaimtoshow(a)howthesetablescanhelpprofessionalspredicttheimpactofdif-ferentidentificationproceduresontheensuingpercentageofiden-tifiedgifted(ortalented)studentsand(b)whatthesetablesrevealabout thepractical impactonprevalence issuesofdiverseconcep-tionsofgiftednessandtalent.


Method

Wepresentinthissectiontherangeofvalueschosenforeachofthethreecontinuouslyvariableparameters,aswellasthebestavailablestatisticalequationtoeffectthedesiredmanipulations.

Parameter Values

Concerning the minimum cutoff score, the literature review indi-catesarangevaryingbetweenverystrict thresholds(e.g.,Terman’s[1925]IQof135,ortop1%)andverygenerousones(e.g.,Renzulli’s[1986]20%cutoffforhisTalentPoolmembers).Tolimitthesizeandcomplexityofthetables,wekeptonlyfourcutoffpercentagesjudged closer to existing values, namely 2%, 5%, 10%, and 15%.Percentages(orpercentiles)wereadoptedbecausetheyapplytoanytypeofmeasure, includingIQscores(Gagné,1998a).Concerningthe second parameter, the number of abilities, most identificationsystemsdonotexceedfiveorsixcriteria.Wedidfindafewstudiesthatincludedmorethan10variables.Forinstance,Gagné(1998b)used 19 different ability assessments from peer nominations, andFeldmanandBratton(1980)used20abilityandmotivationmea-suresforallstudents.Thesewereresearchsituationsinsteadofrealidentification situations. Consequently, we chose four values (2,3, 5, 10) that cover most practical identification procedures. Thethird parameter, the disjunctive/conjunctive dichotomy, needednoadditionalspecifications.Finally,wechosethevaluesforthelastparameter,namelycorrelationsbetweenpairsofabilitymeasures,bytaking intoaccountCarroll’s(1993)findings.Inhisrefactoringofmorethan500abilitydatabases,hedidnotobservecorrelationval-ueshigherthan.71.Consequently,wechosefourcorrelationlevels,from.1to.7with.2intervals.

Statistical Equation

Beforebeginningoursearchforastatisticalequationcapableofhan-dlingsimultaneouslydifferentvaluesofthefourparameters,wedefinedwhatwouldconstituteamathematicallyadequatesolution.Thisideal


equationshouldhavethefollowingcapabilities:(a)takeintoaccountadifferentcutoffscoreforeachmeasure;(b)handlemultiplecriteria,uptoatleast15;(c)adjustitselftononnormalscoredistributions(e.g.,thenegativelyskewedschoolgradedistributionsorLikert-typescales,thenegativelydeceleratedscalestypicalwithpeer-nominationscores);(d)beapplicabletobothdisjunctiveandconjunctivesituations;and(e)takeintoaccountthediversityofthecorrelationmatricesforallpairsofabilitymeasures.Werapidlydiscoveredthatourlistofrequire-mentsexceededbyfartheflexibilityofanystatisticalequationeverpro-posedtoestimatetheprevalenceofspecialpopulations.Forinstance,Burns(1983)conductedtheonlyknownstudyofprevalencetohaveusedthistypeofmathematicalmodel.However,hecircumscribedhisanalysistothesimplestmultivariatedistribution,thebinormal(twonormallydistributedvariables),including,ofcourse,justonecorrela-tioncoefficient.Wedidfindtheequationofafunctionthatmetmostofourrequirements,exceptnonnormality(item“c”above;Johnson& Kotz, 1972).2 Unfortunately, because some variables were repre-sentedbymatrices(e.g.,cutoffscores,correlationmatrices),therewasnoknownwaytocomputetheintegralofsuchacomplexfunction.And, these integrals were essential to produce prevalence estimates.Consequently,wehadto“cutcorners”andlookforanexistingsolu-tionthatwouldmeetasmanyrequirementsaspossible,hopefullythemostimportantones.

Gupta (1963) offered the only potentially applicable solution.3But,thatequationrequiredustoputasidemostofourinitialrequire-ments.Theequationdidapplytoanynumberofabilitiesandcouldadapt to both conjunctive and disjunctive situations. On the otherhand,itcouldhandleonlysituationswhere(a)thesameselectionratiowasusedforallvariables,(b)thesamecorrelationwasassumedforallpairsofvariables,and,ofcourse,(c)allvariableswerenormallydistrib-uted.Becausenothingelsewasavailable,wedecidedtotrythatequa-tionandcheckitsapplicabilitybycomparingtheestimatesobtainedwithprevalenceindicesgatheredfromarealdatabase.Thedatacamefrompeerandteachernominationsusedtoidentifymorethanadozendifferentabilities(seeGagné,1998b,formethodologicaldetails).Thisempirical validation revealed that the estimated prevalence indicesbasedontheGuptaformulawerealmostperfectlycorrelated(r =.99)


withvaluesobtainedfromthatdatabase.It isworthnotingthatthedata’sstatisticalcharacteristics(e.g.,highpositiveasymmetry,veryhet-erogeneouscorrelations)were far frommeetingtheequation’s strictassumptions. Moreover, that empirical study revealed that a smallbiasappearedonlywhenthenumberofabilitiesexceededsix,arareoccurrence in practical identification situations. Finally, because theslightbiasobservedhaditsorigininnonnormalscoredistributions,itspractical impactwas judgedminimalbecause themostcommonidentificationmeasures(IQtestsandstandardizedachievementtests)aredistributedmoreorlesssymmetrically.

ThatequationbecamethecoreofaBASICprogramdesignedtomanipulatesystematicallyeachofthethreeinterval-scaledparam-eters;itusedSimpson’snumericintegrationmethod(Youse,1978).Theprogramyieldedprevalenceestimatesforconjunctivesituations(A and B and C); we mathematically derived the correspondingprevalencevaluesfordisjunctivesituations.1

Results

The four tables of values described in this section correspond totwopairsofdistinctperspectives.Thefirstpairseparatesdisjunctiveapproachestothedefinitionofgiftednessortalent(Tables1and2)fromconjunctiveones(Tables3and4).Withineachpair,wewilldistinguishsituationsinwhichtheoverallprevalenceisnotprede-termined(Tables1and3)fromsituationswheresomeorganization,usuallythestate,theschoolboard,ortheprogramcoordinator,hasdecidedapriorihowmanystudents“are”giftedand/ortalentedintheschoolsundertheirsupervision(Tables2and4).

Disjunctive Situations

The following two tables of data apply to identification situationsinwhichstudentswillbe labeledgiftedortalentediftheirperfor-manceexceedsthecutoffscoreonanyinstrumentincludedinamul-tipleabilitytestbattery.Forinstance,schooldistrictsthatchoosetoadopteitherGardner’s(1983/1994)multipleintelligencetheoryor


Gagné’s(2003)differentiatedmodelofgiftednessandtalentastheiridentificationframeworkwouldtypicallymaketheirinitialestimateswitheitherofthetwofollowingtables.

table 1: open Prevalence. Table1presentstotalprevalenceestimatesintheformofpercentagesforvariouscombinationsofthethreemajorparameters.Eachofthepercentagesinthattableanswersthefollow-ingquestion:“Howmanyindividuals(students)wouldbeidentifiedasgiftedand/ortalentedinatleastonespecificability(giftortalent)ifweselectedthetopX%oneachofYabilitieswithanaveragecorrela-tionlevelofZ?”AsmentionedintheMethodsection,theselectionratioXvariesbetween2%and15%(4sectionsofthetable),Yvariesbetween2and10abilitymeasures(4columns),andtheaveragecor-relationZvariesbetween.7and.1(4rows).Thus,eachcellinthetablerepresentsadifferentcombinationofthethreeparametersX,Y,andZ,andthetablecoversallpossiblecombinationsofthevalueschosen.

Withinthedisjunctiveperspective,themajorruleinvolvestheimpactofthenumberofassessedabilitiesontheprevalenceofgift-ednessandtalent.Iftherearemorewaysforstudentstobejudgedgiftedortalented(e.g.,intellectually,socially,musically,technically,athletically),thenmoreofthemwilldeserveeitherlabel,assumingthatallotherparametersarekeptconstant.ThatrulemanifestsitselfstronglyineveryrowofTable1.Forinstance,withonlytwoabilitymeasurescorrelatedatthe.5levelandaselectionratioofthetop5%(secondsection), the resultingprevalence is8.8%;addthreemoreabilitymeasureswithapproximately thesamecorrelationsandthepercentagealmostdoublesto16.6%.Doublethenumberofabilitiesassessedfrom5to10andtheprevalenceofidentifiedgiftedand/ortalentedstudentswillreachalmost25%.

Asecondruleinvolvestheimpactoftheselectionratio,whichis thepercentageof those setasideby thecutoffscoreasgiftedortalented.AswemovedownTable1throughthefourcutoffsections,notehowtheprevalenceestimatesassociatedwithanyrow/columnintersectionincrease,anddosorathersteeply.Forinstance,the8.6%estimateforthe.3/5intersectionatthetopbecomes19.3%inthesecond section, then 34.0% and 46.0% for the last two sectionsrespectively.Inotherwords,ifwemaintainconstantat.3theaverage


levelofthecorrelationsbetweenthefivemeasuresused,increasingthecutoffpointfrom2%to15%multipliesbyfivethepercentageofidentifiedstudents.

Table 1 Estimated Prevalence Values (in %) of Gifted/Talented

Individuals Within a Disjunctive Perspective as a Function of Four Selection Ratios (SR), Four Quantities of Ability Measures (N abil.), and Four Average Correlation

Levels (Av. r)

N abil.Av. R 2 3 5 10

SR = 2%.7 3.4 4.4 6.1 8.9.5 3.7 5.1 7.5 12.0.3 3.8 5.5 8.6 14.8.1 3.9 5.8 9.3 17.3

SR = 5%.7 8.0 10.3 13.5 18.5.5 8.8 11.8 16.6 24.7.3 9.3 13.0 19.3 30.7.1 9.6 13.9 21.6 36.9

SR = 10%.7 15.3 19.0 24.0 31.1.5 16.8 21.8 29.2 40.3.3 17.8 24.2 34.0 49.5.1 18.7 26.2 38.7 59.5

SR = 15%.7 22.2 26.9 33.0 41.3.5 24.2 30.8 39.7 52.2.3 25.8 34.1 46.0 62.8.1 27.2 37.2 52.4 74.0


Finally,aswegodownanycolumn,withinanysectionofTable1, we observe that the prevalence estimates increase significantly.It means that, as the correlations between the abilities assesseddecrease,fewerindividualshavehighscoresonmorethanoneabil-ity;consequently,thepercentageof“new”giftedortalentedpersonsincreases.Thateffect,whichhasalimitedimpactwhenonlytwoorthree abilities are assessed, becomes much more significant whenthereisalargenumberofabilitiesascanbeseenbycomparingthetwoleftmostcolumnswiththerightmostone.

Onewaytobecomemore familiarwithTable1 is to simulatea real-life identification situation. Imagine a gifted and talented(G/T) district coordinator who endorses a disjunctive definitionofgiftednessandtalent,likethoseofGardnerorGagné.Sheplanstoobtainthreedifferentabilitymeasuresfromallstudentsingrade3, for instance a group IQ test, a standardized achievement test,andacreativitytest.Oureducatedguesswouldbethatthesethreemeasures would be correlated about .5 on average. Faithful to herdisjunctivebeliefs, theprogramcoordinatoradoptsa5%selectionratioforeachmeasure,thenaddsupthetotal.Howmanychildrenwouldthusbe identified?UsingTable1data, thatscenariowouldgivea totalprevalenceestimateof11.8%(secondsection, secondcolumn,secondrow).Inotherwords,a5%selectionratioappliedtothreemoderatelycorrelatedtestswillgiveadisjunctiveprevalenceestimatemorethantwiceaslarge.

ThedatainTable1canbeappliedtopartsoftests.Forexample,theSurveyBatteryoftheIowaTestsofBasicSkills(ITBS;Hoover,Dunbar, & Frisbie, 2001) is composed of three 30-minute sub-tests: Reading, Language (e.g., spelling, punctuation, usage), andMathematics.Wecanexpectthattheirscoresarecorrelatedclosetothe.7level.Schooldistrictsusuallykeeponlythetotalscoreasagen-eralindexofacademicachievement.But,moststudentsdonotper-formequallywellonallthreesubtests.If,insteadofretainingonlythetop10%ontheglobalscore,wedecidedtoselectallthosewhoachievedamongthetop10%onanyofthethreesubtests,howmanywouldbecome“academicallytalented”accordingtothatdisjunctivecriterion?Theansweris19%(seeTable1),almosttwiceasmany.


table 2: Predetermined Prevalence.Table2typicallyappliestoidenti-ficationproceduresinwhichanadministrativedecisionwasinitiallytakentopredeterminethegiftedandtalentedpopulationasaspe-

Table 2 Estimated Per Ability Selection Ratios (in %) of Gifted/

Talented Individuals Within a Disjunctive Perspective as a Function of Four Target Prevalence Values (TP), Four

Quantities of Ability Measures (N abil.), and Four Average Correlation Levels (Av. r)

N abil.Av. R 2 3 5 10

TP = 2%.7 1.2 0.9 0.6 0.4.5 1.1 0.8 0.5 0.3.3 1.0 0.7 0.4 0.2.1 1.0 0.7 0.4 0.2

TP = 5%.7 3.0 2.3 1.6 1.0.5 2.8 2.0 1.3 0.7.3 2.6 1.8 1.1 0.6.1 2.6 1.7 1.0 0.5

TP = 10%.7 6.3 4.9 3.5 2.3.5 5.7 4.2 2.8 1.6.3 5.4 3.8 2.4 1.3.1 5.2 3.5 2.2 1.1

TP = 15%.7 9.8 7.7 5.7 3.8.5 8.9 6.5 4.4 2.6.3 8.3 5.8 3.7 2.0.1 7.9 5.4 3.3 1.7


cificpercentageofthetotalschoolpopulation.Thereasonsforsucha decision may vary from philosophical, to political, to financial;theimportantconsiderationisthattheprogramcoordinatormuststructureherprocedurearoundthatconstraint.Ifsheendorsesadis-junctiveviewofgiftednessandtalentandwantstoidentifyadiversepopulationintermsofabilities,thenTable2datawillhelpheresti-matethepercentageofstudentstoselectforeachabilityassessedsothat the total will add up to the preestablished target population.ThequestionmostrelevanttoTable2couldthusbestatedas fol-lows: “Considering that local policy requires to identify as giftedand/ortalentedstudentsX%oftheschoolpopulation,ifIwanttoidentify students inYdifferentabilityareas thatarecorrelatedonaverageat theZ level,whatwillbe theapproximatepercentageofstudentstoselectforeachoftheseabilitiessothattheywilladduptoX,mypredeterminedglobalselectionratio?”Offering64differentanswers,eachcellinTable2givestheestimatedcutoffpercentageforeachability,dependingononeoffourassessmentbatterysizes(col-umns),oneoffouraveragecorrelationlevels(rows),andwithregardtooneoffourdistinctprevalencetargets(sections).

BecauseoftheverydifferentperspectiveadoptedinTable2,mostofthecelldatashowoppositetrendsfromthoseinTable1.Forinstance,insteadofincreasing,thevaluesfromlefttorightdecrease.Itstandstoreason: inordernottoexceedthefixedtargetprevalence,themoreabilitieswedecidetoassess,thefewertopscorersweneedfromeachabilityarea.Similarly,thevaluesdecreasedowneachcolumnwithineachofthefoursections.Whenabilitiesarecloselyrelated,oneneedsmoreindividualsperabilitytoattainthetargetbecausemanyofthoseselectedwithAbilityBwillalreadyhavebeenchosenwithAbilityA.Theselectionratiodecreasesasthetestsbecomelesscorrelated.Themostpredictabletrendisthegrowthinpercentagesfromsectiontosection;thelargerthetargetgroup,themoretopscorerswithineachabilitywillcontributetothatoverallgiftedandtalentedpopulation.

Asaconcreteexample, letususeagainthesituationdescribedintheprecedingsection.Ourprogramcoordinator,whoharborsadisjunctiveviewofgiftednessandtalent,wantsagaintoassessthreedistinctabilitieswithteststhatsheknowstobemoderately(r =.5)correlated.Shehasbeengiven—orhasgivenherself—atargetpreva-


lenceof10%.Withthisinformationinhand,shecanestimatefromTable2thatshewilljustreachhertargetbyselectingthetop4.2%scorersoneachabilitymeasure.Thefactthat3x4.2%doesnotaddupto13.6%butto10%followsofcoursefromthefactthat,duetothemoderatecorrelationsbetweenthemeasures,manystudentswillbeselectedonmorethanoneabilitymeasure.

Conjunctive Situations

Thetwotablesofdatapresentedinthissectionincludeessentiallythesameprevalenceestimatesasthoseinthefirsttwotables,withoneexception:Theytargetidentificationsituationsinwhichtheadmin-istrators try to implement a conjunctive definition of giftednessandtalent.ThebestknownofsuchdefinitionsisRenzulli’s(1986)“Three-Ring”definitionofgiftedness, “Giftedbehaviorconsistsofbehaviors that reflect an interaction among three basic clusters ofhuman traits—these clusters being above average general and/orspecificabilities,highlevelsoftaskcommitment,andhighlevelsofcreativity”(p.76).Sternberg(2005)recentlyproposedhiswisdom,intelligence,creativity,synthesized(WICS)model.Hestates

Accordingtothismodel,wisdom,intelligence,andcreativ-ityaresinequanonsforthegiftedleadersofthefuture.Withoutasynthesisoftheseattributes,someonecanbeadecentcontributortosociety,andperhapsevenagoodone,butneveragreatone.(p.327)

AnotherwouldbeGagné’s(2003)intellectuallygifted,academi-cally talented (IGAT) population. Gagné argues that most schooldistricts select such “bright and achieving” students for their giftedprogramsbecausethetwomostcommonidentificationinstrumentsaregroupIQtests(fortheIGpart)andstandardizedachievementtests(fortheATpart).Tables3and4parallelthesamedichotomyasintheprevioussection,namelyopenversuspredeterminedprevalence.

table 3: open Prevalence. The appropriate question for the Table 3dataslightlydiffersfromtheTable1question:“Howmanyindividu-als(students)wouldbeidentifiedasgiftedand/ortalentedin each of


a combined set of abilities ifweselectedthetopX%oneachofYabili-tieswithanaveragecorrelationlevelofZ?”Or,saiddifferently:“IfwemeasureYdifferentabilities(Y=2,3,5)thathaveaZ(.7to.1)averagecorrelationbetweenthem,howmanystudentswillmeetapreselectedcutoffvalueofX(SR=5%,10%,15%)on each of the chosen abilities?”Theoptionsofferedforthethreeparametershavebeenslightlymodi-fied to delete implausible situations. For instance, Parameter X, theselectionratioappliedtoeachabilitymeasure,nowvariesbetween5%and15%(threesections)becauseaselectionratioof2%wouldpro-

Table 3 Estimated Prevalence Values (in %) of Gifted/Talented

Individuals Within a Conjunctive Perspective as a Function of Four Selection Ratios (SR), Three Quantities

of Ability Measures (N abil.), and Four Average Correlation Levels (Av. r)

N abil.Av. R 2 3 5

SR = 5%.7 2.0 1.1 0.6.5 1.2 0.5 0.2.3 0.7 0.2 0.0.1 0.4 0.0 0.0

SR = 10%.7 4.7 3.0 1.7.5 3.2 1.6 0.6.3 2.2 0.7 0.1.1 1.3 0.2 0.0

SR = 15%.7 7.8 5.3 3.3.5 5.8 3.1 1.3.3 4.2 1.6 0.4.1 2.8 0.6 0.0


duceprevalencelevelswellbelow1%,whichwejudgedoverlyselec-tive.Similarly,themaximumnumberofabilitiesmeasured,ParameterY,hasbeenreducedto5(threecolumns)becausenodefinitionexiststhatrequiresconjunctiveexcellenceonmorethanhalfadozendistinctcharacteristicsinordertoreceivethegiftedortalentedlabel.Theaver-agecorrelationZstillvariesbetween.7and.1(fourrows).

Asshouldbeexpected,thecelldatainTable3differmarkedlyfromthoseinTable1bytheirsmallsize.Asarguedintheintroduc-tion,theconjunctiveapproachismuchmoreselectiveinidentifyinggiftedand/ortalentedindividuals;oneneedstoexcelsimultaneouslyinmorethanonearea.Theimpactoftheselectionratioremainsthesame:Thehigheritisperability,themorepeoplewillbefoundwhomeet the other criteria. The growth in values from section to sec-tiondownTable3showsthatphenomenonquiteclearly.Moreover,increasingthenumberofabilitiesmeasuredhastheexpectedeffectofradicallydecreasingtheensuingprevalence.Take,forinstance,apoolofpeoplewhohavebeenidentifiedasthetop15%ononeabil-itymeasure.As shown in thebottompartof the table,only4.2%willremainifwelook,amongthem,forthosewhoalsoexcelatthesame level (15%) on a second, slightly correlated (r = .3) ability.Thatpercentagewilldrop to1.6% ifweadda similarlycorrelatedthirdabilityandtojust0.4%(4:1,000)whenthenumberofabilitiesincreasestofive.Ifweweretostartwithaninitialselectionratioof10%insteadof15%,theresultingpercentagesfor2,3,and5abili-tieswouldbecome2.2%,0.7%,and0.1%respectively.Finally,Table3showsthesignificantimpactoftestcorrelationsontheresultingprevalence.Forinstance,let’strytofindindividualswhoexcel(top15%)simultaneouslyonthreedistinctabilities.Iftheseabilitiesareclosely related (r = .7), the resulting prevalence is 5.3%; in otherwords, a third of the initial group of 15% on any one ability willremainasthosewhoexcelonthesethreecloselyrelatedabilities.But,ifthethreeassessedabilitiesarejustslightlycorrelated(r=.1),thenthe resulting prevalence drops to 0.6%, nine times fewer individu-alsthanwithcloselycorrelatedmeasures.Thepercentagesdropevenmorewhentheinitialselectionratioisreducedto10%or5%;inthislast case, with three slightly correlated abilities, the probability offindingindividualswhoexcel(top5%)simultaneouslyonthreeof


themisabout4:10,000orjustonepersoninaschoolpopulationof2,500students.

Asaconcreteexample,imaginethataschooldistrictcoordinatorforgiftedandtalentedservicesdecidestoapplyrigorouslyRenzulli’s(1986)three-ringdefinitionofgiftedness.Thatpersonwouldneedtomeasureindependentlyeachofthethreecomponentsofhiscon-ception (above-average ability, creativity, task commitment), thenselectthosewhoexceedapredeterminedcutoffpercentageoneachofthesethreecriteria,logicallythesamecutoffforeachmeasure.Wecouldexpectthatmeasuresofthesethreeconceptswouldbemod-estlycorrelated,probablynotmuchmoreonaveragethan.3.Table3showsthatifthatcoordinatorweretouseacutoffof5%foreachability,shewouldendupwithonly0.2%students(1:500)exceedingthecutoffscoresonallthreemeasures.Evenwithamuchmoregen-erous selection ratio of 15%, the resulting prevalence would reachonly 1.6%, about 1:60. Indeed, among the 36 values presented inTable3,onlythreeexceedthetypical5%usedasabenchmarkpreva-lencelevelinamajorityofschooldistricts,andtwoofthemrequirethemeasureofcloselyrelatedabilities(r =.7).

table 4: Predetermined Prevalence.JustlikeinTable2,theidentifica-tionsituationcoveredbyTable4assumesanadministrativedecisiontoselectapredeterminedpercentageofgiftedandtalentedstudents;thetotalprevalenceisnottheoutcomeoftheidentificationprocess,butitsstartingpoint.Thequestionthenbecomes:“Consideringthatlocal policy requires to identify as gifted and/or talented studentsX%oftheschoolpopulation(thetargetprevalence),ifmyconjunc-tivedefinitionimpliesthemeasureofYdifferentabilities(columns)thatarecorrelatedonaverageataZlevel(rows),whatapproximatepercentageofstudentswillweselectforeachoftheseabilities(celldata)sothatallstudentswithinthattargetedpercentageexceedthechosen selection ratio on every Y measure?” Compared with thedatainTable2,wemadeonlyonemodification,namelytheelimina-tionofY=10,againbecauseweknowofnoconjunctivedefinitionrequiringsimultaneousexcellenceinsomanydistinctabilities.

The most striking feature in Table 4 is the large size of mostpercentagesinthecelldata.Ithadtobeexpectedfromtheobserva-


tionsmadeintheanalysisofTable3;whentheabilitiesconsideredincrease(Y)andarenotcloselyrelated(Z),weneedaverylargepoolof participants to pinpoint those few who simultaneously excel in

Table 4 Estimated Per Ability Selection Ratios (in %) of Gifted/

Talented Individuals Within a Conjunctive Perspective as a Function of Four Target Prevalence Values (TP), Three

Quantities of Ability Measures (N abil.), and Four Average Correlation Levels (Av. r)

N abil.Av. R 2 3 5

TP = 2%.7 5.1 7.5 10.9.5 7.1 11.6 18.3.3 9.5 16.8 27.7.1 12.5 23.3 39.2

TP = 5%.7 10.5 14.4 19.5.5 13.5 20.0 28.6.3 16.8 26.2 38.6.1 20.4 33.1 49.3

TP = 10%.7 18.2 22.8 29.8.5 22.0 29.9 39.7.3 25.8 36.4 49.2.1 29.6 43.0 58.5

TP = 15%.7 24.9 30.8 37.9.5 29.1 37.6 47.7.3 33.0 44.0 56.6.1 36.8 50.1 64.6


manyofthem.Multiplegiftednessortalentisnotaverycommonphenomenon.Forinstance,inordertofindinaschoolpopulationa subgroupof10%whosemembersoutperformallother studentsoneachofjustthreemodestly(r =.3)correlatedcriteria,likethoseinRenzulli’s(1986)three-ringdefinition,thecutoffscoreforeachcriterionwouldneedtobeaverygenerous36.4%.Inotherwords,ifweselectthetop36%oneachofthreeabilitymeasures—averygen-erousthresholdbyanydefinition—only10%ofthemwillbelongtothattop36%onallthreemeasures.Evenwithmeasuresthataremorecloselyrelated(r= .5),wewouldstillneedtoselect thetop30%oneachmeasuretobesurethat10%ofthemwillbeamongtheselectedonallthreemeasures.

Discussion

Westatedatthebeginningofthisarticlethattheprevalencequestionlooked deceptively simple but was in fact very complex. Yet, manyscholars regularly give offhand responses to the question, responsessimilartothewell-knownstatementmadebyMarland(1972)morethanthreedecadesago: “Itcanbeassumedthatutilizationof thesecriteria for identification of the gifted and talented will encompassaminimumof3to5percentoftheschoolpopulation”(p.4).Thatstatement left much room for ambiguity, even if we put aside theexpression“aminimumof.”Forinstance,nowhereinthereportwasitclearlyspecifiedifthat“3to5percent”hadtobedividedmoreorlessequallyamongthesixcategoriesofachievementand/orpotentialabil-ityidentified(generalintellectualability,specificacademicaptitude,creativeorproductivethinking,leadershipability,visualandperform-ingarts,andpsychomotorability),givingeachaprevalenceratioofapproximately1%,assumingofcoursesomeoverlapbetweenabilities.Thisisjustoneexampleoftherecurringambiguitiesobservedintheliteraturewhentheprevalencequestionisbroached.

Themessagewehavetriedtoconveyinthistextisnotjustthecom-plexityof theprevalence issuebut,moreusefully,whatmajor issuesneedtobeaddressedinordertogenerateagreementamongamajorityofscholarsandprofessionalsconcerningthisquestion.Theresultsof


ourmathematicalmanipulationsrevealthatthemostimportantissue,intermsofitsimpactonprevalenceestimates,isthedefinitionused.

Impact of Definitions

Two parameters in our study were directly related to definitionalissues:(a)thedisjunctive/conjunctivedichotomy,and(b)thenumberofabilitiescoveredbythedefinition.Thesetwoparameterscannotbedissociatedbecausethefirstonelogicallyimpliesthepresenceofmorethanonecharacteristic.Indeed,whengiftednessisdefinedsimplyashigh-level intelligence, as is the case with many existing definitions(e.g.,Morelock,1996;Piirto,1994;Robinson,2005;Sternberg,1986),thentheprevalenceproblemreducestothechoiceofanappropriateselectionratioorminimumthreshold.Althoughcloselyrelated,thetwodefinition-relatedparameterswillbediscussedhereseparately.

disjunctive/conjunctive Perspectives.Recallthatdisjunctivedefinitions(AorBor C)explicitlyendorsemultipleformsofgiftednessand/ortalent,whereasconjunctiveones(AandBandC)associategiftedness(ortalent)toacombinationofmultiplecharacteristicsmanifestedatahighlevel.Let’slookattheirdivergentimpactataverybasiclevel.Gagné(2003)arguedthatmoststudentsparticipatinginenrichmentactivitieswithintheK–12educationalsystemareidentifiedwithtwomajor instruments: (a) an assessment of their intellectual giftedness(IG),usuallywithagroup-administeredIQtest, and(b)ameasureoftheiracademictalent(AT)usingeitherschoolgradesorstandard-izedachievementtests.Hethuslabeledthe“IGATstudent”thetypi-calparticipantinmostgiftededucationprogramsacrosstheUnitedStatesandCanada.Letusassumethat(a)alargenumberofschooldis-trictsuseonlythesetwoassessmentinstrumentsand(b)thatsomeusethemdisjunctively(highscoresonIGorAT)andothersconjunctively(highscoresonbothIGandAT).Howwouldtheprevalencefiguresdiffer?Researchhasshownthatthesetwomeasurescorrelatemoreorlessatthe .7level( Jensen,1998).Ifeachschooldistrictchosea5%selectionratioforeachtest,theoneadoptingadisjunctivedefinitionwouldendupwithaG/Tpopulationof8%(seeTable1),whereasthe corresponding percentage in the conjunctive district would be


only 2% (see Table 3). In other words, using an identical selectionthreshold,fourtimesasmanystudentswouldbeeitherIGorATthanthosebeingIGAT(IGandAT).Inourview,a4:1ratiorepresentsahugediscrepancyinthesizeofthegiftedorIGATpopulation.Ifothersociallysignificantcharacteristics(e.g.,poverty,obesity,alcoholicallyimpairedbehavior)weremeasuredwithsuchlooseness,therewouldbeageneraloutcryfromthepublicandthemedia.

Onecouldsay:“Whydon’twejustagreeonthesizeofthetargetpopulation,howeveritisassessed?”Inlightofallargumentsintroducedin the present text, this reaction would just skirt the real problems.Forsomeprogresstohappenonthisquestion,webelievethatscholarsandprofessionalsneedtoreachsomeagreement;thequestionis“whatpreciselyshouldtheyagreeon?”Intheaboveexample,anagreementonatargetpopulationof,say5%,wouldonlygivetheappearanceofasolution.Indeed,asshowninTables2and4,theprogramcoordinatordefendingthedisjunctive(IGorAT)definitionwouldreachthat5%targetbyselectingthetop3%oneachtest,whereasthecoordinatorwhobelievesinaconjunctivedefinitionwouldobtainher5%IGATpopulationbyselectingthetop10.5%oneachofthetwotestsandthendiscardingallthosewhodonotbelongtobothselectedgroups.If,forthesakeofcomparison,bothapproacheshadbeenappliedineitherdistrict, would the resulting two groups include the same students?Notatall.Itcanbeestimatedstatistically,aswedidwithaMonteCarlosimulation,thatthetwogroupswouldoverlapbyapproximately60%(3%/5%).Inotherwords,theconjunctiveapproachwouldoverlookabout40%(2%/5%)ofthedisjunctivelyidentifiedgiftedortalentedandsimilarlyfortheconjunctivegroup.Poolingbothgroups,7%ofthetotalpopulationwouldbeidentifiedaseitherIGAT(3%),IGonly(2%),orATonly(2%).Saiddifferently,for60%ofallstudentsidenti-fied(4%/7%),theirselectionwoulddependonthedefinitionused.Thisshowshowimportantthedefinitionisindeterminingwhowillbeservedinanenrichmentprogram.

number of abilities/characteristics.Therearemanymoredisjunctivethanconjunctiveapproachestothedefinitionofgiftednessortalent.Thisiswhythissectionwillfocusontheimpactofthesecondparame-terondisjunctivedefinitions.Theruleisquitesimple:Allotherfactors


keptconstant,asoneaddsmoreabilities,theprevalenceofgiftedortalentedindividualsincreasesrapidly,thusproducingamoredemocra-tizedviewofthephenomenonofgiftednessandtalent.Thisisquiteevi-dentinTables1and2.Scholarswhoproposeddisjunctivedefinitionsshouldhavepointedoutthe logicalconsequenceof thatmultiplica-tionofabilities,namelytherapidgrowthofthegifted/talentedpopu-lation.Alas,asmentionedintheliteraturereview,onlyTaylor(1973)andGagné(1993)didso;andGagnéalonehasspecifiedthathisbasicthresholdofthetop10%shouldbeappliedtoeachgiftednessdomainandeachfieldoftalent.Whenonenolongerdefinesgiftednessverynarrowly,with just intellectualabilities inmind,butadoptsamuchbroaderviewofhumanabilities,itbecomeseasytoreachprevalencelevelsof30%or40%.Forinstance,justusingamoderateselectionratioof10%andlowtomoderatecorrelations(averager =.30)betweenpairsofabilities,theG/Tpopulationwillreachalmost1:4(24%)withjust3abilitiesmeasuredand1:3(34%)with5distinctabilities;with10abilitiesmeasured,almost1:2(49%)ofthegeneralpopulationwilldeservethelabelofgiftedortalentedinatleastoneofthem.Thinkofit.Acoordinatorofagiftedprogramsstartswithonlythetopthreestu-dentsinagroupofthirty(10%),and,byassessingjustadozendifferentabilities,discoversthatatleasthalfoftheschoolpopulationdeservestobelabeledgiftedortalentedin“something”meaningful.Infact,thesepercentagescloselyparallelthoseobtainedinastudyofmultipleabili-tiesthroughpeerandteachernominations(Gagné,1998b).

Does this mean that with enough different abilities we wouldreach or approach 100%, thus confirming a common belief that“everyoneisgifted?”Bynomeans.Thereisalimittothenumberofdistincthumanabilitiesrecognized,unlessoneconsidersmicrodis-tinctionsbetweenthem(e.g.,Carroll,1993).Gardner(1983/1994)identifiedsevenhuman“intelligences,”andGagné(2003)recognizesfourdifferentgiftednessdomains,leavingsomeroomforsubcatego-rieswithineach.Hedoesmentionthatfieldsoftalentareasnumer-ousashumanoccupations,butexcellenceineachofthemismeasuredinrelationto thepopulationof individualsactive ineachfield. Inotherwords,mostindividualsinanyfieldoftalent(e.g.,carpenter,musician, biologist, basketball player, bus/truck driver, machineoperator,journalist,salesperson)wouldbelabeledcompetent,with


justthetop10%amongthembeingrecognizedastalented.Itmustbepointedout thatas thenumberofmeasuredabilities increases,theproportionofthoseexcellingindomainsorfieldsalreadymea-suredalsoincreases;thus,newadditionstothegroupprogressivelydecreasetowardapracticalceiling.ThisceilingeffectmanifestsitselfveryclearlyinTable1whenonelooksatanyrowfromlefttoright.

The Last Two Parameters

Recallthatthelasttwoparametersusedinthismathematicalstudyweretheselectionratioappliedtoindividualmeasuresandtheaver-agecorrelationbetweenallpairsofmeasurescollected.

The Selection ratio. The selection ratio is the only variable in thepresentstudywhoseimpactaffectsbothdisjunctiveandconjunctivedefinitionsidentically:Thestrictertheselectionratio,thesmallertheprevalenceofgifted/talentedindividualswillbe.Ifaprogramcoordi-natoradoptsaselectionratioof10%,shewillgetfivetimesasmanygiftedortalentedindividualsastheotherprogramcoordinatorwhoadoptsaselectionratioof2%.Theimpactoftheselectionratioiscon-sequentlyquitestraightforward:Withallotherfactorskeptconstant,doubletheselectionratioandyoudoublethepopulation.Inreality,thedatainthetablesshowthatthisruleworkslesswellasthenumberofmeasuredabilitiesgrows.Forinstance,comparesectionsSR=5%andSR=10%inTable1;youwillseethattheestimatesalmostdoublewhenonly2abilitiesareconsidered(leftcolumn),butincreasebyonly60%orsowhen10abilitiesaremeasured(rightcolumn).

Thereisnologicalwaytopinpointpreciselythe“right”valuefortheselectionratio.Asmentionedearlier,somelimitswillreadilyleadtogeneralagreementamongscholarsandprofessionals.Forinstance,fewwouldendorsebasicthresholdslargerthan20%orsmallerthan1%.But,aswemovethesetwomarkersclosertoeachother,thedegreeofagreementwillprogressivelycrumbleatbothends.Inotherwords,thereremainsalargegreyzonewherescholarlyviewpointsroamfree.Still,ifgiftededucationistogainthestatusofascientificallybasedfieldofinquiry,webelievethatthechoiceofabasicidentificationthresholdwillbeamongtheimportantquestionstoaddressandanswer.


correlations Between Measures.Thislastparameterinthestudycanbe dealt with rapidly. The methodological limits imposed by oursearchforasuitableequationaffectedthisvariablemuchmorethanthe three others. Recall that the only practical equation found intheliterature(Gupta,1963)requiredthatweassumeequalcorrela-tionsbetweenallpairsofmeasuresintroducedinthemathematicalcombinations;thatcertainlydoesnotfitwellwithrealmeasurementsituations. On the other hand, using the database of a large iden-tification study with peer nominations (Gagné, 1998b), we foundthatthenonrespectofthatassumptiondidnotbiassignificantlythesimulateddata(Bélanger,1997);theirfittorealprevalencemeasurescreatedfromthedatabaseremainedveryclose.

What thisparameter reveals,basically, followscommonsense:Highlycorrelatedmeasureswilltendtocreateahighoverlapbetweenselectedsubgroups,whereasuncorrelatedmeasureswillcreateverydifferentsubgroupsofgifted/talented individuals.Doesthisaffectsignificantlytheprevalenceestimates?Theanswerseemstodependonthenumberofabilitiesassessed:Themorethereare,thestrongertheimpactofthatvariable.Forinstance,withaselectionratioof2%andjusttwoabilities,variationsoftheaveragecorrelation,from.1to .7,will reducetheprevalenceestimates from3.9%to3.4%(seeTable1); ifthenumberofabilitiesincreasesto10,thesamevaria-tionoftheaveragecorrelationwilllowertheprevalenceestimatebyalmost50%,from17.3%to8.9%.Still,practicallyspeaking,unlessprogramcoordinatorschooseverysimilar(ordissimilar)abilitiestomeasureforidentificationpurposes,theaveragecorrelationbetweenpairsshouldtendtostabilizearoundamedianvaluecloseto.4asthenumberofmeasuredabilitiesincreases.

Limits of This Mathematical Study

Twomajormethodologicallimitsmustbepointedoutconcerningthepresentmathematicalstudy.Thefirstoneconcernsthefactthatwehavedealtinthisstudywithverysimpleandstraightforwardiden-tificationscenarios.ThedailylifeofmostG/Tprogramcoordinatorsrarelyfollowstheconstraintswedefinedtoproduceourfourstatisti-caltables.Inmany(most?)cases,theywillhavetoabidebyallkinds


of hardly measurable constraints: pressures from parents, teachers,oradministrators;transfersfromotherschooldistrictswhereotheridentificationprocedureswereused;pressurestousemorepsycho-metrically fragile—but less expensive—measures, like checklistsgiventoteachersandparents;andsoforth.Becausewearenotawareofrecentsurveysshowingthediversityoflocalidentificationprac-tices, we hesitate to discuss at length these special problems. Oneapproachthatseemssomewhatpopular,calledthe“matrix”approach(Borland,1989;Piirto,1994),isindirectlycoveredinthisstudy.Thematrixapproachusesmultipleidentificationsourcesanddeterminesforthedatafromeachsourceaspecific—anddifferent—scoringgrid(e.g.,5pointsforanIQabove135,4pointsifitisbetween131–135,andsoforth).Thesepointsareaddedtocreateaglobalscoreforeachstudent;administratorsorprogramcoordinatorsthenapplyasimpleselectionratiotothisnewintegratedranking.Althoughtheidenti-ficationprocedurebeginswithmultiplemeasures,theendproductbecomesasinglecompositemeasureandrequiresnothingmorethananappropriate—oftenpredetermined—selectionratio.

The second methodological limit results from our assumptionthroughoutthistextthatallabilitymeasuresusedforidentificationpurposes are psychometrically perfect, both in terms of their reli-abilityandvalidity.Unfortunately,nota singlemeasureofhumanabilities comes close to this ideal. Even with the best instrumentsavailable,namelyIQtestsandstandardizedachievementtests,eventheshort-termreproducibilityofscoresfromthesameinstrumentbarely reaches .9.As forvaliditymeasures, theconvergentvaliditycoefficients between similar instruments from different publisherswillrarelyexceed.7.Hereisaninterestingexample.Pearce(1983)comparedthescoresofaselectedgroupof59fifthgradersonthreewell known IQ tests: Wechsler’s Intelligence Scale for ChildrenRevised(WISC-R),Raven’sStandardProgressiveMatrices(SPM),andMeeker’sStructure-of-IntellectScreeningformforGifted(SOI-SFG).Ifweselectthetop17%(10/59)oneachofthethreetests,only two students from her table of individual scores (Table 1 inPearce,1983)“makethecut”onallthreetests,whereasnolessthan22(37%)emergeamongthattop17%onatleastoneofthem.Thissmall-scale example indicates that even our best instruments will


giveonlypartlyreplicableresults.Twopossible—andcomplemen-tary—explanations come to mind for these very divergent results.First,theimperfectreliabilityofeachinstrumentwillautomaticallylower the correlations between the three sets of test scores, thusincreasingthedisjunctiveprevalenceandreducingtheconjunctiveone.Second,thefactthatthethreetestsdonotmeasureexactlythesameconstructinthesamewayreducesthecorrelationsbetweenthetests, thusproducingthesameresultasthefirstexplanation.Bothexplanationsadduptoincreasethegapbetweentheconjunctiveanddisjunctivepercentages.

Conclusion

Thisstudybeganwithaharshjudgmentonthewaytheprevalenceissuehas(not)beendealtwithinthefieldofgiftededucation.Then,wepresentedaseriesofmathematicalcombinations,whichdemon-stratedthatthisquestioncouldbeanalyzedanditsparameterscir-cumscribed.Theresultsofthesemathematicalcombinationsrevealedthattheanswertothe“Howmany?”questionwasfarfromsimpleand relied first and foremost on preliminary decisions concerningthenatureof thegiftednessandtalentconstructs.Thesequestionscouldbesummarizedasfollows:

1. Isgiftedness(ortalent)aunitaryconcept?2. Ifso,whatminimumselectionratiowouldrallyamajority

ofscholarsandprofessionalsinthefield?3. Ifnot(unitary),shouldtheconceptsofgiftednessandtal-

entbedefineddisjunctivelyorconjunctively?4. Ifaconjunctivedefinitionofgiftednessand/ortalentwas

preferredbymost,whichessentialcontributivecompo-nentsshouldbeincluded?

5. Ifadisjunctivedefinitionwaspreferredbymost,whatwouldbethedifferentmanifestationsofthegiftedness/tal-entconcepts?

6. Afteransweringthetwoprecedingquestions,whatmini-mum selection ratio should apply to the components/formsofthetypeofdefinitionchosen?


Becausenoefforthasyetbeenmadetobringscholarsandprofes-sionalstoaddresstheabovequestions,itisdoubtfulthatanswerswillbeproposedinthenearfuture.Still,werestateourconvictionthattheprevalenceissueisattheheartofascientificexaminationofthegiftednessandtalentconceptsandthatthefieldofgiftededucationwillnotriseaboveitspresentfolkloricstatusuntilsuchquestionsareaddressed—andanswered.

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Author Note

ThestudyreportedhereispartofaPh.D.dissertationconductedbythefirstauthor.Correspondenceconcerningthisarticle shouldbesenttoJeanBélanger,DepartmentofSpecialEducationandSpecialTraining, UQAM, P.O. Box 8888, Station Downtown, Montreal,QC,Canada,H3C3P8;E-mail:[email protected].

End Notes

1 Theconjunctiveprevalence(%Cj)ofidentifiedindividualscorrespondstotheproductoftheselectionratios[Cj%=SR*SR*SR...].Tocomputethedisjunctiveprevalence(%Dj),wejustsubtractfromthewholepopulationthosewhoarenotidentifiedinasingleabilitydomain[%Dj=1–((1–SR)*(1–SR)*(1–SR)...)].

2 The general function of the multinormal distribution( Johnson&Kotz,1972)ispresentedinEquation1.

fk X1

Vexp - 1

2X-M V X-M-1( ) =

( )( ) ( )

2 2πk

t (1)

The “k” parameter the number of abilities, the X matrix allows adifferentselectionratioforeachvariable,andtheVand|V|matrixrepresentrespectivelythematrixofcovariancesbetweenallvariables(similartocorrelations)anditsinverse.Theconjunctive/disjunctiveparameter would be taken into account after computing the con-junctiveestimate(seeEndNote1).

3 Hereisasimplificationofthefunctionofthemultinormaldistribution(Gupta,1963).Thelackofmatricesallowsitsintegra-tion,thusthecomputationofprevalenceestimates.

F CP; F a f x dxNANAρ( ) = ( ) ( )

−∞

∞

∫ (2)


whereCP=thecutoffpointselectedinz scoresforthesetofNAdimen-sions;r=thecorrelationbetweentheabilities;itisassumed>0andsimilarforallpairsofabilities;NA=thenumberofabilitiesconsidered.

a x CP

1-

12

=+

( )ρ

ρ1

2

f(x) = the density function of the normal univariate distribution,thatis:

f xx-

( ) =−

1

2

12

2

σ π

µσe

and

F a f x dxNA

-

a NA

( ) = ( )

∞

∫

Estimating the Size of the Gifted/Talented Population … Estimating the Size of the Gifted/Talented...

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