ERIC ED529593: Predicting First Graders' Development of Calculation versus Word-Problem Performance: The Role of Dynamic Assessment

2012·0.09 MB·English

by  ERIC

JournalofEducationalPsychology ©2011AmericanPsychologicalAssociation
2012,Vol.104,No.1,224–234 0022-0663/11/$12.00 DOI:10.1037/a0024968
Predicting First Graders’ Development of Calculation Versus
Word-Problem Performance: The Role of Dynamic Assessment
Pamela M. Seethaler, Lynn S. Fuchs, Douglas Fuchs, and Donald L. Compton
VanderbiltUniversity
Thepurposeofthisstudywastoassessthevalueofdynamicassessment(DA;degreeofscaffolding
requiredtolearnunfamiliarmathematicscontent)forpredicting1st-gradecalculations(CAs)andword
problems(WPs)development,whilecontrollingfortheroleoftraditionalassessments.Among1841st
graders,predictors(DA,QuantityDiscrimination,TestofMathematicsAbility,language,andreasoning)
wereassessednearthestartof1stgrade.CAandWPwereassessedneartheendof1stgrade.Planned
regressionandcommonalityanalysesindicatedthatforforecastingCAdevelopment,QuantityDiscrim-
ination, which accounted for 8.84% of explained variance, was the single most powerful predictor,
followedbyTestofMathematicsAbilityandDA;languageandreasoningwerenotuniquelypredictive.
By contrast, for predicting WP development, DA was the single most powerful predictor, which
accountedfor12.01%ofexplainedvariance,withTestofMathematicsAbility,QuantityDiscrimination,
andlanguagealsouniquelypredictive.Resultssuggestthatdifferentconstellationsofcognitiveresources
arerequiredforCAvs.WPdevelopmentandthatDAmaybeusefulinpredicting1st-grademathematics
development,especiallyWP.
Keywords:mathematics,dynamicassessment,kindergarten,calculations,wordproblems
Dynamicassessment(DA)involvesstructuringalearningtask, Campione,1986)ortheamountofscaffoldingrequiredduringthe
providing feedback or instruction to help the examinee learn the assisted learning experience to reach criterion performance (e.g.,
task,andindexingresponsivenesstotheassistedlearningexperi- Murray,Smith,&Murray,2000;Spector,1992).Interactionstyle
enceasameasureoftheexaminee’scapacitytoprofitfromfuture is another dimension along which DAs vary. With standardized
instruction.BeginningwithVygotsky’s(e.g.,1934/1962)proposal DAs (e.g., Ferrara et al., 1986), testers rely on a fixed series of
morethan75yearsago,discussionshavecenteredonwhetherDA prompts;otherDAs(e.g.,Tzuriel&Feuerstein,1992)areindivid-
might serve as an alternative to the conventional assessment par- ualized, with testers addressing the specific obstacles examinees
adigm, in which examinees respond without assistance. The con- reveal.YetanotherdimensionalongwhichDAsdifferisthenature
cernisthatsuchstaticassessmentsrevealonlytwostates,unaided ofthetasksusedfortheassistedlearningexperience,whichmay
success or failure (Sternberg, 1996; Tzuriel & Haywood, 1992), focus on domain-general cognitive abilities (e.g., Budoff, 1967;
which masks distinctions among children who cannot perform a Feuerstein,1979)oroncognitiveabilitiespresumedtounderliethe
task independently but can succeed with varying levels of assis- academicdomaintobepredicted(e.g.,Swanson&Howard,2005)
tance.
or on domain-specific abilities such as reading or mathematics
TheliteratureonDAisdiffuse.Studiesvarywithrespecttohow
tasks(e.g.,Bransford,Delclos,Vye,Burns,&Hasselbring,1987;
DAsarestructured.Intermsofscoring,DAsmayquantifyrespon-
Campione,1989;Campione&Brown,1987;Spector,1992).
sivenesstotheassistedlearningexperienceasimprovementfrom
TheDAliteraturealsovariesintermsofresearchquestionsand
unassisted pretest to unassisted posttest (e.g., Ferrara, Brown, &
methodological features. Researchers who index pretest–posttest
improvement typically investigate whether the DA score distin-
guishes between individuals with and without a preestablished
diagnosisassociatedwithpoorlearning(e.g.,Tzuriel&Feuerstein,
ThisarticlewaspublishedOnlineFirstAugust22,2011.
1992). By contrast, researchers who index degree of scaffolding
Pamela M. Seethaler, Lynn S. Fuchs, Douglas Fuchs, and Donald L.
typically examine the value of that score in predicting a learning
Compton,DepartmentofSpecialEducation,VanderbiltUniversity.
ThisresearchwassupportedbyU.S.DepartmentofEducationAward outcomeexternaltotheDA(e.g.,Spector,1992).Thissecondtype
NumberR324A090039andbyEuniceKennedyShriverNationalInstitute of study can be further categorized in terms of whether static,
of Child Health & Human Development Award Number R01HD059179 competing predictors of outcome are considered and whether the
andCoreGrantNumberHD15052awardedtoVanderbiltUniversity.The externallearningoutcomeisassessedconcurrentlywiththeDAor
contentissolelytheresponsibilityoftheauthorsanddoesnotnecessarily at a future time. Studies that control for competing predictors or
represent the official views of the U.S. Department of Education or the
measure the external learning outcome at a later time impose a
Eunice Kennedy Shriver National Institute of Child Health & Human
morestringenttestofDA’svalue.Forthesereasons,inthepresent
DevelopmentortheNationalInstitutesofHealth.
study, we examined the contribution of DA in forecasting future
CorrespondenceconcerningthisarticleshouldbeaddressedtoPamela
M.Seethaler,228Peabody,VanderbiltUniversity,Nashville,TN37203. externallearningwhileconsideringthecontributionofcompeting
E-mail:[email protected] predictors.WewereinterestedinDA’scontributioninpredicting
224
PREDICTINGFIRSTGRADERS’DEVELOPMENT 225
two transparently different forms of mathematics development: year,studentswereassessedonWPmeasuresproximalanddistal
calculations(CAs)andwordproblems(WPs). to instruction. Structural equation measurement models showed
that DA measured a distinct dimension of pretreatment ability.
Prior DA Studies Predicting Mathematics Learning Structuralequationmodelingshowedthatthenatureofinstruction
External to the DA While Considering Competing (validated vs. conventional) was sufficient to account for WP
development proximal to instruction; yet, language, pretreatment
Predictors
mathperformance,andDAwereuniquelypredictiveinforecasting
Toestablishthecontextandrationaleforthepresentinvestiga- WPdevelopmentmoredistaltoinstruction.
tion,wedescribepriorstudiesthathaveexploredDA’scontribu- Inthepresentstudy,weextendedL.S.Fuchs,Fuchs,Compton
tioninpredictinglearningexternaltotheDAwhilecontrollingfor etal.(2008)inthreeways.First,weexaminedtheroleofDAin
competingpredictors.Weconsideredinvestigationsthatpredicted predicting future mathematics learning using a different domain-
concurrentorfutureoutcomesaswellasDAsofvaryingstructure specific DA: solving four types of nonstandard expressions. We
and design, while limiting our search to studies that focused on selectedthisdomainbecause(a)wecouldassumeitwasunfamil-
mathematics. This netted three relevant investigations. Speece, iar and sufficiently difficult that most first graders would not be
Cooper,andKibler(l990)measuredfirst-gradestudentsonaDA abletosolvenonstandardexpressionswithoutassistance,butcould
task associated with overall cognitive ability: solving matrices. learn the content with varying amounts of support; (b) we could
Usingastandardizedstyleofinteraction,theyindexedthenumber assume that beginning first graders would have the prerequisite
of prompts students required during the assisted learning experi- skillstosupporttheassistedlearningexperience—representations
ence.Thisscoreaccountedforuniquevariance,beyondverbalIQ, of,Arabicnumeralnamesfor,andcountingskillsassociatedwith
pre-DAmatricesperformance,andlanguageability,inexplaining smallquantities(1–10);(c)wecoulddelineatestrategiesforsolv-
individual differences on the Wide-Range Achievement Test- ingthenonstandardexpressions,whichweusedtoconstructclear
Arithmetic subtest (WRAT; Wilkinson, 1993), with 2% of the explanationswithinagraduatedsequenceofprompts;and(d)via
varianceinWRATuniquetotheDA. pilot work, we had established that the DA’s four types of non-
Swanson and Howard (2005) extended Speece et al.’s (1990) standard expressions were increasingly difficult, with later types
studybycenteringtheirDAoncognitiveresourcesmorespecifi- building on earlier types, such that transfer across the four DA
cally presumed to underlie reading or math performance: phono- equationtypesmightfacilitatehigherDAscores.
logicalworkingmemory(i.e.,rhymingtasksthatrequiredrecallof BeyondfocusingonadifferentDA,asecondandmoreimpor-
acoustically similar words) and semantic working memory (i.e., tant extension to L. S. Fuchs, Fuchs, Compton, et al. (2008) was
digit/sentence tasks that required recall of numerical information that,inthepresentstudy,weassessedtheutilityofDAatthestart
embeddedinshortsentences).Theinteractionstylewasindividu- of first grade, when forecasting learning has proved especially
alized,withDAtesterschoosingamongfourstandardizedhintsto challenging (e.g., Compton et al., 2010; Johnson, Jenkins, Pet-
select the least obvious hint that aligned best with the student’s scher, & Catts, 2009). This is due to difficulty in distinguishing
errors.ThreeDAscoresweregenerated:gainscore(highestscore between two types of students who score poorly on static mea-
obtainedwithassistance);maintenancescore(stabilityofthehigh- sures: those with poor learning potential who require special
estlevelobtainedafterassistancewasremoved);andprobescore intervention versus those whose low score is due to limited prior
(numberofhintstoachievehighestlevel).DAscoresforphono- learning experience but who have good potential to learn in re-
logicalworkingmemorywerecombinedintoafactorscore,aswas sponse to generally strong classroom instruction. DA, which in-
doneforDAsemanticworkingmemory;bothwereusedtopredict dexes how much instructional scaffolding is required to produce
concurrent WRAT performance. Among students averaging learning,maybemoreusefulthanstaticmeasuresformakingsuch
10–12 years of age, static measures of verbal IQ and pre-DA distinctions.
phonological working memory as well as the semantic DA score The final and most important extension to L. S. Fuchs, Fuchs,
uniquely accounted for individual differences in WRAT perfor- Compton,etal.(2008)wasthatthepresentstudyfocusedonDA’s
mance; the variance uniquely attributable to the semantic DA value as a predictor of learning as a function of type of mathe-
factorwas25%. matics development: CA versus WP performance. These two
Therefore,SwansonandHoward(2005)foundstrongersupport forms of mathematics development are transparently different.
for concurrent relations with calculations outcomes for a DA Whereas CA problems are set up for solution, WPs require stu-
centered on cognitive abilities presumed to underlie mathematics dents to use linguistic information to construct a problem model:
performancethandidSpeeceetal.(1990),whoseDAaddresseda identifying missing information, constructing a number sentence,
task associated with more general cognitive ability. However, andsettingupaCAproblemforsolution.Beyondthetransparent
neither study assessed mathematics development at a future time differences between CA and WP, prior work suggests that the
or used a DA that involved a domain-specific, mathematics task. cognitive characteristics underlying development in CA versus
Weidentifiedonlyonesuchstudy.L.S.Fuchs,Fuchs,Compton, WP differ (e.g., L. S. Fuchs, Fuchs, Stuebing, et al., 2008; L. S.
etal.(2008)developedadomain-specificDAdesignedtobenovel Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010;
to the third-grade participants by focusing on early algebraic Swanson&Beebe-Frankenberger,2004).Forexample,processing
cognition tasks. In the fall, students were assessed on cognitive speed(L.S.Fuchs,Fuchs,Stuebing,etal.)andworkingmemory
resources associated with word-problem performance, initial CA (Bull & Johnston, 1997) seem to contribute to CA development,
and WP performance, as well as DA. On the basis of random whereas WP skill appears to be uniquely predicted by concept
assignment, students received 16 weeks of validated WP instruc- formation,nonverbalreasoning,sight-wordproficiency,language,
tion or conventional WP instruction. Near the end of the school andreading(L.S.Fuchsetal.,2006;L.S.Fuchs,Geary,Compton,
226 SEETHALER,FUCHS,FUCHS,ANDCOMPTON
Fuchs, Hamlett, Seethaler, et al., 2010; Swanson, 2006). More- Booth, 2004), we distinguished variance associated with knowl-
over, skill with WP is significantly linked to CA skill, as L. S. edgeofsmallquantities,whichistransparentlyprerequisitetoand
Fuchs et al. (2006) showed with path analysis of arithmetic, shouldsupportperformanceontheDA,fromvarianceassociated
arithmeticcomputation,andarithmeticWPperformanceofthird- withwhattheDAwasdesignedtoindex:scaffoldingrequiredto
grade students, making CA skill necessary but not sufficient for learn novel mathematics content. Given that QD predicts CA
solvingWPs. outcomes(Chardetal.,2005;Lembke&Foegen,2005)andthat
Althoughthesetransparentdifferencescreatedifferentdemands CA is required for WPs, we anticipated that QD would explain
on students, we identified no prior studies that examined DA’s varianceinbothtypesofoutcomes.
valueforthese(orother)contrastingsubdomainsofmathematics At the same time, because QD is a speeded assessment that
performance. We hypothesized that DA’s predictive value might focusesonalimitedconceptualizationofearlynumericalcompe-
differonthebasisofthesetransparentdifferencesanddifferences tency, we also included a power test assessing a broader set of
in cognitive correlates. On one hand, the present study’s DA, early mathematical competencies, including informal and formal
which focuses on balancing equations, may reflect conceptual knowledge: the Test of Early Mathematics Ability-3 (TEMA;
understandingofarithmeticortheequalsign,whichmaybemore Ginsberg & Baroody, 2003). In terms of informal knowledge
central for WP than CA, because WP (but not CA) development constructs,TEMAassessesnumbering(e.g.,countingby1,by10s,
seems to be linked with concept formation and reasoning. The or from a number; identifying number before or after), number
present study’s DA incorporates increasingly explicit, conceptu- comparison(e.g.,identifyingsmaller/largerquantitiesfromcollec-
ally based worked examples, which may draw on the same cog- tions of items or from Arabic numerals; selecting the Arabic
nitive resources as does solving WP. On the other hand, the DA numeral closer to a given numeral), CAs (e.g., solving mental,
includes strategies for deriving answers to CA problems, while nonverbal addition problems with sums to 12; demonstrating ad-
avoidinganynarrativeWPcontextandinstruction;inthisway,the dition of one or more objects), and understanding of cardinality
DAmaybetterreflectcapacityforCAthanWPdevelopment. (e.g.,shownacollectionofprintedstars,countingandsayinghow
many)andequalpartitioning(e.g.,givenasetoftokens,showing
howtosharethe“cookies”fairlybetweentwosisters).Thetypes
Competing Predictors
of formal knowledge assessed are numeral literacy (reading/
In considering the value of DA for predicting CA versus WP writing numerals from 1 to 4 digits), number facts (speeded in
development,wewereinterestedincontrollingvarianceassociated answeringadditionormultiplicationfacts),CAs(performingwrit-
with predictors that represent traditional (static) domain-specific tenormentalcalculationsoftwo-digitnumerals),andunderstand-
numericalcompetenciesanddomain-generalcognitiveresources. ingtheadditivecommutativityprinciple(e.g.,9(cid:1)7isthesameas
7(cid:1)9)andbaseten(e.g.,identifyinghowmany$10billsequalone
$100bill).
Domain-Specific Numerical Competencies
In previous predictive validity research with kindergarten or
Okamoto(2000,citedinKalchman,Moss,&Case,2001)iden- first-gradesamples,TEMAhasbeenusedasanoutcomemeasure,
tified the ability to discriminate between quantities as a distinct with coefficients ranging from .33 to .69 (Lembke & Foegen,
dimensionofkindergarteners’mathematicsperformance,requiring 2005;Mazzocco&Thompson,2005;Seethaler&Fuchs,inpress;
children not only to distinguish between numerosities but also to Teisl, Mazzocco, & Myers, 2001). To our knowledge, however,
move across representational systems. Kindergarteners differ in TEMA has not been evaluated as a predictor of outcome. In the
theirabilitytodiscriminatebetweenquantities(i.e.,Whichnumber context of the present study, in which DA is a lengthy, untimed
isbigger,4or6?),evenwhencontrollingforcountingandsimple assessment,wewereinterestedincontrollingforvarianceassoci-
computation (Griffin, Case, & Siegler, 1994). Quantity Discrimi- ated with another lengthy, untimed (but static) assessment of
nation (QD; Chard et al., 2005) is a measure of the speed and numericalcompetencies.Controllinginthiswayfortheroleofa
accuracywithwhichchildrendistinguishbetweenandmapArabic more established, extended, and comprehensive (but static) mea-
numerals onto small numerosities (i.e., students quickly identify sure of early mathematical competencies, which also maps more
thelargerquantityinpairsofArabicnumeralsrangingfrom1to directlythanDAontotheskillsmeasuredinouroutcomes,created
10). astringenttestforthepredictivevalueofDA.BecauseTEMAtaps
Inthepresentstudy,weincludedQDbecauseevidenceindicates multiple forms of mathematical knowledge, we hypothesized it
that, at the beginning of first grade, it is a strong predictor of wouldpredictCAandWPoutcomes.
subsequent mathematics achievement (e.g., Chard et al., 2005;
Clarke & Shinn, 2004; Lembke & Foegen, 2005). We had three
Domain-General Cognitive Resources
additionalreasonsforincludingQD.First,itiscommonlyusedin
schoolsforscreeningriskforpoormathematicsdevelopmentatthe In the present study, we also assessed the contribution of two
start of first grade. Second, in choosing QD, we opted against domain-generalcognitiveabilitiesinpredictingoutcomes.Forthis
measuresthatrequireoperationalmanipulationsofsmallquantities purpose, we included language and reasoning as predictors of
as in the Number Sets Test (Geary, Bailey, & Hoard, 2009) or futuremathematicsability,twoimportantsubtestsofmanytradi-
Curriculum-BasedMeasurement-Computation(L.S.Fuchsetal., tionalIQtests,whichhavealsobeenlinkedtoWPdevelopmentin
2007), because we sought a measure of early numerical compe- firstgraders(e.g.,L.S.Fuchs,Geary,Compton,Fuchs,Hamlett,&
tencythatdidnotrequiretheoperationsrequiredforourCAand Bryant, 2010; L. S. Fuchs, Geary, Compton, Fuchs, Hamlett,
WP outcomes. Finally, by fixing on small quantities (rather than Seethaler,etal.,2010).Languageabilityisimportanttoconsider
larger magnitudes, as in Number Line Estimation; Siegler & given the need to process linguistic information during school
PREDICTINGFIRSTGRADERS’DEVELOPMENT 227
instruction. As Dehaene (1997) suggested, even infants have an guage(.5%)disability;sixstudents(3.3%)wereEnglishlanguage
informalandprimarysenseofnumber,whichmaybeinherent.As learners.
children develop intellectually, however, they rely on symbolic Screeningmeasurestoobtainarepresentativesample. Be-
andverbalcomprehensionofnumbers,necessaryforformalmath- causethetwoscreeningmeasureswereusedtoselectarepresen-
ematical competence (Jordan, Glutting, & Ramineni, 2010). Jor- tativesample,itemsonthescreenersrepresentedarangeofdiffi-
dan,Levine,andHuttenlocher(1995)documentedtheimportance culty.Becausethescreenerswereusedforsampleselections,they
of language ability when kindergarten and first-grade language- werenotusedaspredictorsinthestudy.TheFirst-GradeTestof
impairedchildrenperformedsignificantlylowerthannonimpaired ComputationalFluency(L.S.Fuchs,Hamlett,&Fuchs,1990)is
peers on WPs. In addition to language, reasoning, measured by a single page of computation items representing the first-grade
completing visually presented patterns, has been identified as a curriculum:ninesingle-digitadditionitems,ninesingle-digitsub-
uniquepredictororcognitivecorrelateofvariousaspectsofmath- tractionitems,twodouble-digitadditionitemswithoutregrouping,
ematics development. For example, L. S. Fuchs et al. (2005) three double-digit subtraction items without regrouping, and two
demonstrated the importance of reasoning in WP development single-digitadditionitemswiththreeaddends.Itemsaredisplayed
acrossfirstgrade,afindingcorroboratedbyAgnessandMcLone infiverowsoffiveitems,andstudentshave2mintowriteanswers
(1987). Seethaler and Fuchs (2006) found reasoning to be a sig- nexttoorbeloweachitem.Theyareinstructedtofirsttrytheitems
nificantcorrelateofcomputationalestimationskill,andreasoning that seem easier and then to go back to harder items. The score
againemergedasasignificantpredictoroffifthgraders’develop- is the number of correct digits. Coefficient alpha for this sample
mentofcomputationwithwholeandrationalnumbers(Seethaler, was.87.
Fuchs, Star, & Bryant, 2011). Because the relations between The First-Grade Test of Mathematics Concepts and Applica-
domain-generalabilitiesmaybestrongerforWPthanforCA(e.g., tions (L. S. Fuchs et al., 1990) comprises 25 items, displayed on
L. S. Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; three pages. Items represent the first-grade curriculum, including
numeration, concepts, geometry, measurement, applied computa-
L. S. Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al.,
tion, money, charts, graphs, and word problems. The tester reads
2010), we hypothesized that the domain-general cognitive re-
eachitemoutloud,withoutreadingkeynumbersornumberwords.
sourceswouldcapturemorevarianceforpredictingWPthanCA
Asthetesterreads,studentsfollowalongonapapercopy,while
development.
coveringotheritemsonthepage.Beforemovingtothenextitem,
thetestergivesstudentssufficienttimetorespond(15or20s,as
Method dictated by standard directions, based on field data indicating
adequate response time for almost all first graders). The score is
Participants. Two-hundred two participants (i.e., the maxi-
thenumberofcorrectlyanswereditems.Coefficientalphaforthis
mum number of students our research funds permitted, which
samplewas.85.
corresponded to power analyses indicating adequate sample size)
Predictormeasures. WithQD(Chardetal.,2005;Lembke
were identified. In 61 classrooms in 17 elementary schools (14
& Foegen, 2009; Research Institute on Progress Monitoring,
TitleI;3non-TitleI)inasoutheasternmetropolitanschooldistrict,
2009), students have 1 min to select the larger of two numbers
one-hundred fifteen students were excluded, who, as part of a
(rangingfrom1to10),presentedin56boxesacrosstwopages(28
different study, would be receiving 16 weeks of mathematics
perpage).Itisindividuallyadministered.Test–retestreliabilityis
tutoring. (These students were excluded because tutoring was
.85–.99(Clarke,Baker,Smolkowski,&Chard,2008).
designed to disrupt the predictive value of their initial status on TEMA-3 (Ginsburg & Baroody, 2003) assesses informal and
variablesliketheonesincludedinthepresentstudy.)Theremain- formal mathematics knowledge for children 3 years 0 months
ing 866 students with parental consent were administered two through 8 years 11 months. It comprises 72 items of increasing
screening measures, the First-Grade Test of Computational Flu- difficulty,withmultipletrialsforeachitem.Thetesterscoreseach
ency and the First-Grade Test of Mathematics Concepts and Ap- trial as right or wrong by determining whether the number of
plications(L.S.Fuchs,Hamlett,&Fuchs,1990;seetheMeasures correct trials warrants a point for that item. For example, some
section),intheirgeneraleducationclassroomsbytrainedresearch itemsrequiretwoofthreecorrecttrialstoearnapoint;otheritems
assistants. A latent class approach (combining screening scores requirealltrialsansweredcorrectly.Studentsreachaceilingwhen
into a single latent factor) produced a three-class solution that five consecutive items do not meet criteria to earn a point; the
specifiedhigh,average,andat-riskstrata.Stratifyingbyclassroom testerensuresabasaloffiveconsecutivelycorrectitems.Testing
andstrata,two-hundredtwostudentswererandomlyselected(all takesupto45min.Coefficientalphafor6-year-oldsis.95.
schoolsandclassroomswererepresented).Eighteenstudentswho To assess language, the Wechsler Abbreviated Scale of Intelli-
movedpriortocompletingspringtestingwerecomparabletothe gence(WASI)Vocabularytest(PsychologicalCorporation,1999)
remaining students on all demographics and on all mathematics was used, which measures expressive vocabulary and verbal
performance variables administered in the fall. The present anal- knowledge.Theexaminerpresentswordsforthestudenttodefine
ysesthusincludedthe184studentsforwhomfallandspringdata andimmediatelyscorestheresponseas0,1,or2pointsdepending
havebeencollected.Ofthesestudents,80(43.5%)weremen,and onquality.Forthefirstfouritems,thestudentseesapictureofan
112(60.9%)receivedfreeorreducedlunch.Intermsofethnicity, itemtodefine;fortheremainingitems,theexaminerpresentsthe
74 students (40.2%) were African American, 85 (46.2%) were wordorally.Testingisdiscontinuedafterfiveconsecutivescores
Caucasian,13(7.1%)wereHispanic,eight(4.3%)wereAsian,and of zero. Zhu (1999) reported split-half reliability at .86–.87; the
four (2.2%) were “other.” Nine students (5.5%) received special correlation with the Wechsler Intelligence Scale for Children
education services for a learning (1.1%), speech (3.3%), or lan- (Wechsler,1999)is.72.
228 SEETHALER,FUCHS,FUCHS,ANDCOMPTON
Toassessreasoning,weusedWASIMatrixReasoning(Psycho- thisone?”andpointingtothefirstitem.Ifafter15additionalsthe
logical Corporation, 1999), which measures reasoning skill with studentstillhasnotwrittenanything,thetesterasks,“Areyoustill
pattern completion, classification, analogy, and serial reasoning. working or are you stuck?” If student responds that he or she is
Studentsseeacolorpictureofamatrixwithonepiecemissingand stuck or if 15 additional s elapse with no observable attempt to
select the correct piece to complete the picture from five choices solvetheproblem,thetesterbeginsthenextlevelofinstructional
displayedbelowthepicture.Theexaminerawards1pointforeach scaffolding.
correctanswer;testingisdiscontinuedafterfouroutoffivemissed Eachequationtypeincludesfourorfiveinstructionalscaffold-
items.AsperZhu(1999),reliabilityis.94. ing levels, each of which has two teaching items with which the
Balancing Equations Dynamic Assessment (DA; Seethaler & examiner models and explains a problem-solving strategy. The
Fuchs,2010a)measuresthedegreeofscaffoldingrequiredtolearn scaffolding is scripted to ensure consistency in language and
unfamiliar mathematics content, specifically, solving for missing procedures. Examiners maintain student attention with frequent
variablesinnonstandardadditionandsubtractionexpressions.The questions and participation. The scaffolding levels increase in
DA comprises four types of equations of increasing difficulty. instructionalexplicitness.Withinanequationtype,thefirst(least
Testers present strategies of increasing explicitness to teach stu- explicit) level only defines relevant mathematical terms (e.g.,
dentstobalancebothsidesoftheequation;studentsprogresstothe equalmeansthesameas;aplussignmeanstoaddmore).Withthe
levelofexplicitnesstheyrequiretoachievemasteryofthatequa- second scaffolding level, the examiner uses a balance scale and
tion type, at which time they advance to the next, more difficult 2-in. plastic teddy bear manipulatives to demonstrate balancing
equationtype.Beyondthereasonsalreadydiscussedearlierinthis bothsidesoftheequation.A4-in.(cid:4)2-in.equalsign((cid:2))isprinted
article, balancing equations were chosen as the DA task because onawhitecard,whichisaffixedtothecenterofthescale;student
elementaryschoolstudentsoftenmisinterprettheequalsign((cid:2))as attentionisdrawntonotethesidesoftheequationasparalleltothe
an operational rather than as a relational symbol (McNeil & sides of the scale, with manipulatives used to represent the
Alibali, 2005; Sherman & Bisanz, 2009) and because solving amountsintheequation.Thethirdlevelofscaffoldingprovides
equations with missing numbers is important for higher level instruction in solving the equations while using tally marks
mathematics skills and thus is valuable content for students to drawn on paper to represent the numerical amounts. The next
learn. scaffolding level provides instruction in solving the equations
FourequationtypesconstitutetheDA.EquationTypeArequires in conjunction with an 8-in. number line printed on a piece of
solving for a missing variable in the first or second position in cardstock; students are taught to move their finger to count
equations that use 1 as an addend and for which the sum is not spaces on the number line while solving the equations. This is
greaterthan9(e.g.,__(cid:1)1(cid:2)4or8(cid:1)__(cid:2)9).WithEquation designedtobuildunderstandingoftheinverserelationbetween
TypeB,studentssolveforamissingvariableinthefirstorsecond addition and subtraction (e.g., for ___ (cid:1) 2 (cid:2) 6, students count
positioninequationsthatdonotuse1asanaddendandforwhich on4from2togetto6,revealingthat6–2(cid:2)4).Thefinal,most
thesumisnogreaterthan9(e.g.,__(cid:1)2(cid:2)6or3(cid:1)__(cid:2)5).For explicit scaffolding level increases the support for the student to
EquationTypeC,studentssolveforamissingvariableinthefirst successfully apply the number line strategy for building under-
position in subtraction equations with minuends no greater than standingoftheinverserelationbetweenadditionandsubtraction.
nine(e.g.,__(cid:3)7(cid:2)2).EquationTypeDrequiresstudentstosolve Toward that end, different colored markers on the number line
foramissingvariableinanyoffourpositions,withsumsonboth representdifferentpartsoftheequation.(EquationTypesAandC
sidesoftheequalsign,noneofwhichexceed9(e.g.,__(cid:1)5(cid:2)3(cid:1) have five levels of instruction, whereas Equation Types B and D
4 or 3 (cid:1) 6 (cid:2) 5 (cid:1) __). The equation types are presented so that havefourbecausenewmathematicaltermsarenotintroducedfor
successwithanearlierequationtypeshouldpromoteunderstand- Equation Type B or D.) Worked examples used during instruc-
ingofasubsequentequationtype. tional scaffolding are not displayed during mastery testing; how-
TheadministrationandscoringproceduresfollowL.S.Fuchs, ever, all materials necessary for applying the strategies taught
Fuchs, Compton, et al. (2008). Within each equation type, the duringscaffoldingarealwaysdisplayedonthetestingtable(even
testerbeginsbyassessingmasteryofthatequationtype.Ifmastery duringtheinitialmasterytest).Studentsarenotpenalizedbytheir
isdemonstrated,thestudentadvancestothenextequationtype.If choiceofproblem-solutionstrategies.Anequationtypeisdeemed
not,instructionalscaffoldingbeginswiththeleastexplicitlevelof masteredwhenatleastfiveofthesixitemsareansweredcorrectly,
scaffolding.Masterytestingthenrecurs.Ifmasteryisachieved,the atwhichtimetheexaminerprogressestothenextDAskill.
studentprogressestothenextequationtype.Ifnot,thenextmore DAscoresrangefrom0to22.Zeroindicatesastudentdidnot
explicitlevelofinstructionalscaffoldingispresented,andmastery masteranyequationtype;22reflectsastudentmasteringeachof
testingfollows.Inthisway,amaximumoffourorfive(depending the four equation types on the first administration of the mastery
onequationtype)increasinglyexplicitlevelsofinstructionalscaf- test(i.e.,withoutanyinstructionalscaffolding;EquationTypesA,
folding are used. If the student fails to master an equation type B,C,andDareworth6,5,6,and5points,respectively,because
afterthefinallevelofscaffoldingfortheequationtype,theDAis EquationTypesAandChavefivelevelsofinstructionalscaffold-
terminated. ing, whereas Equation Types B and D have only four levels). A
Eachmasterytestcomprisessixitemsrepresentingthetargeted testersubtracts1pointfromthemaximumof22eachtimealevel
equation type. Items repeat across alternate test forms (used for of instructional scaffolding is required. For example, if a student
successivemasterytestingwithinthatequationtype),butitemsare demonstratesmasteryonthefirstadministrationofthemasterytest
presented in different orders. Mastery test items are not used for for Equation Types A, B, and C (without any instructional scaf-
instructionalscaffolding.Ifastudentwritesnothingonamastery folding), but requires two levels of instructional scaffolding to
testfor5s,thetesterpromptsthestudentbyasking,“Canyoutry masterEquationTypeD,theexaminersubtracts2pointsfromthe
PREDICTINGFIRSTGRADERS’DEVELOPMENT 229
maximumof22,awardingascoreof20.Bycontrast,ifastudent tered simultaneously. In explaining CA development, the combi-
requires three levels of instructional scaffolding to master Equa- nationofQD,TEMA,language,reasoning,andDAaccountedfor
tion Type A, four levels of instructional scaffolding to master 57.7%ofthevariance,F(5,178)(cid:2)48.63,p(cid:5).001.Fourofthe
Equation Type B, and fails to master Equation Type C (thereby fivepredictorsmadeauniquecontributioninexplainingindividual
terminatingtheDAsuchthatEquationTypeDisnotpresented), differencesinCAdevelopment.QDwasthestrongestcontributor,
the student loses 3 points for Equation Type A, 4 points for with TEMA a close second, followed by DA; language and rea-
EquationTypeB,6pointsforEquationTypeC,and5pointsfor soning were not uniquely predictive. By contrast, in explaining
EquationTypeD,forascoreof4.Internalconsistencyreliability WPdevelopment,thecombinationofthefivepredictorsaccounted
wasindexedbycorrelatingthescorefromeachDAequationtype for a larger percentage (71.6%) of variance, F(5, 178) (cid:2) 89.67,
withtheDAtotalscore,usingthesubsetofstudentswhohadnot p (cid:5) .001. The predictors that made a unique contribution in
reachedaceilingonperformancepriortotheadministrationofthat explainingindividualdifferencesinWPdevelopmentalsodiffered
DA equation type. For Equation Type A, r (cid:2) .90; for Equation from those involved in CA. For WP, DA was the strongest pre-
TypeB,r(cid:2).86;forEquationTypeC,r(cid:2).82;forEquationType dictor,followedbylanguage,TEMA,andQD;reasoningwasnot
D,r(cid:2).84.1 uniquelypredictive.
End-of-first-grade mathematics outcome measures. Both To supplement these regressions, we conducted a complete
outcomemeasuresarepowertests,inwhichtimelimitsareample commonality analysis (Beaton, 1973; Capraro & Capraro, 2001;
forallstudentstocompletetheitemstheyarecapableofanswer- Newton&Spurell,1967;Nimon,Lewis,Kane,&Haynes,2008)
ing. To assess CA performance, the Arithmetic subtest of the specifying the unique and shared variance associated with each
WRAT-3 (Wilkinson, 1993) was used, with which students have predictorandeachcombinationofpredictorsforCAdevelopment
10 min to write answers to 40 calculation items of increasing and for WP development. See Table 3, in which we list the
difficulty(kindergartenthroughGrade12).Noneofthestudentsin predictors and all possible combinations thereof in the first col-
the present study used all 10 min. According to the manual, umn. In the second and fourth columns, we show coefficients
reliabilityis.94. expressing the proportion of total variance explained by the pre-
ToassessWPperformance,StoryProblems(Jordan&Hanich, dictor(s).Inthethirdandfifthcolumns,wetranslatedthepropor-
2000) was used, which comprises 14 single-step addition and tions of total variance to percentages of explained variance. To
subtraction WPs of the types most often encountered in the pri- derivepercentagesofexplainedvariance,wetookthecoefficient
marygrades:combine,compare,andchange.Thetesterreadseach expressing the proportion of total variance explained by a given
itemaloudandprovidesoneadditionalreadingifrequestedtodo predictor and divided that coefficient by the total amount of
so. Students have 30 s to answer on their paper copy of the test, explained variance across predictors, then multiplied by 100. For
which they have available throughout testing so they can read example,inexplainingindividualdifferencesinWPdevelopment,
along while the tester reads or refer back to problems as they the coefficient expressing the proportion of total variance ex-
derivesolutions.Then,thetesterreadsthenextproblem.Students plained by DA is .086, and the coefficient denoting the total
completed all work within the 30-s time limit for each problem. proportionofvarianceexplainedbyallthepredictors(individually
Coefficientalphaonthissamplewas.90. andincombination)is.716.Toderivethepercentageofexplained
Procedure. In late September and October, the WASI Vo- variance accounted by DA in WP development, we divided .086
cabulary,WASIMatrixReasoning,QD,andTEMAwereadmin- by.716,whichequals.1201,andthenmultipliedby100:12.01%.
istered. In October and November, the DA was administered. Inthediscussionthatfollows,werelyonpercentageofexplained
CompetingpredictorswereadministeredbeforetheDAsothatthe variancetofacilitatecomparisonsacrossCAandWP.
teachingintheDAwouldnotinfluenceestimatesofperformance
onothermeasures.Thegoalwastocompletethetwosessionsfor
Discussion
eachstudentwithin1schoolweek.Thishappenedforamajority
ofstudents,butsometimestookanadditionalweek(ifthestudent
The purpose of this study was to assess the value of DA in
wasabsentorotherwiseunavailablefortestinginthetargetedtime
predictingtwotransparentlydifferentformsofmathematicsdevel-
frame). In May, the CA and WP tests were administered. Tests
opment:CAandWP.InevaluatingtheroleofDA,wecontrolled
wereadministeredindividuallybygraduatestudentswhodemon-
for traditional assessments, in which examinees respond without
strated100%accuracyduringpracticeadministrationofthemea-
assistance. In this way, we contrasted the assessment of what
sures. All testing sessions were audiotaped, and 16% of the ses-
students already know (static tests) against the assessment of
sions,distributedequallyacrosstesters,wererandomlysampledto
students’capacitytolearn(DA).TocreateastringenttestofDA’s
assess fidelity of administration. Scoring agreement was 99.4%.
value in forecasting development, we included different types of
All data were independently entered into two databases, which
staticassessmentsinourmodel.
werecomparedfordiscrepanciesandresolvedagainsttheoriginal
One of those static measures, QD, assesses the speed and
protocols.
accuracywithwhichchildrendistinguishbetweenandmapArabic
numerals onto small numerosities. Including the QD predictor
Data Analysis and Results permittedustodistinguishvarianceassociatedwithknowledgeof
smallquantities,whichisatransparentprerequisitetoandshould
Table 1 provides means, standard deviations (SDs), and corre- supportDAperformance,fromvarianceassociatedwithwhatthe
lationsamongthepredictorsandoutcomes.Weproviderawscores
as well as standard scores when applicable. In Table 2, we show
results of regression analyses, into which all predictors were en- 1ContactPamelaM.SeethalerformoreinformationontheDA.
230 SEETHALER,FUCHS,FUCHS,ANDCOMPTON
Table1
Means,StandardDeviations,andCorrelationsaAmongPredictorandOutcomeMeasures(n(cid:2)184)
Rawscore Standardscoreb
Measure M (SD) M (SD) L R QD T DA CA WP
Predictors
Language(L) 19.20 (6.51) 43.72 (10.23) —
Reasoning(R) 9.95 (5.78) 50.30 (9.88) .42 —
QuantityDiscrimination(QD) 31.58 (10.13) .39 .40 —
TEMA(T) 39.22 (9.96) 100.21 (14.08) .50 .54 .64 —
DynamicAssessment(DA) 9.20 (8.08) .48 .62 .53 .68 —
Outcomes
Calculations(CA) 18.59 (3.45) 100.70 (15.84) .44 .49 .64 .69 .63 —
WordProblems(WP) 7.01 (4.40) .59 .58 .60 .71 .78 .71 —
Note. LanguageisWechslerAbbreviatedScaleofIntelligence(WASI)Vocabluary;ReasoningisWASIMatrixReasoning.TEMA(cid:2)sTestofEarly
MathematicsAbility,thirdedition.
aAllcorrelationssignificantatp(cid:5).01. bStandardscoresforWASIVocabularyandWASIMatrixReasoningareTscores(M(cid:2)50;SD(cid:2)10);for
Calculations(Wide-RangeAchievementTest-Arithmetic),themeanis100(SD(cid:2)15).
DA was designed to index: ability to learn mathematics. So it is surprising that TEMA also accounted for a sizable percentage of
noteworthy that QD, which takes only 1 min to administer and explainedvarianceinCAdevelopment(6.41%).Evenso,itisalso
involves none of the addition or subtraction demands of the CA impressive that with the two static domain-specific assessments
outcome, was the strongest single predictor of CA development alreadyuniquelyaccountingfor15.25%ofexplainedvariance,DA
across first grade, uniquely accounting for 8.84% of explained made a uniquely significant, albeit smaller, contribution to pre-
variance.ThepowerofQDinforecastingindividualdifferencesin dictingCAoutcome,accountingfor2.77%ofexplainedvariance
CA development underscores children’s appreciation of magni- ((cid:6) (cid:2) .20). At the same time, as revealed in the commonality
tudesasfoundationaltoformalmathematicslearning(e.g.,Berch, analysis,thesethreedomain-specificpredictorsalsosharedasub-
2005;Dehaene,1997;Okamoto&Case,1996),whileillustrating stantial amount of additional variance in predicting CA. So the
theimportanceofprerequisiteknowledgeasaconditionoffuture bulk of explained variance was attributable to domain-specific
learning. predictors,withthedomain-generallanguageandreasoningvari-
QD’s predictive role is especially noteworthy given that we ables, traditionally incorporated in intelligence tests to predict
included in our model another, more comprehensive and lengthy schoollearning,failingtoachievestatisticalsignificance((cid:6)s(cid:2).05
staticindexofincomingmathematicsperformance,TEMA,which and.06).
inpartalsoassessesunderstandingofsmallmagnitudes.Another
Withrespecttothemajorpurposeofthepresentstudy,assessing
component of the TEMA battery is incoming CA skill, which
whetherthecontributionofthesepredictorsdifferedforCAversus
createsbetteralignmentthanQDwiththeCAoutcome.Soitisnot
WP development, findings were interesting. Whereas DA was
overshadowed by QD and TEMA in predicting CA, DA was the
strongestsinglecontributortoWPdevelopment,uniquelyaccount-
Table2
ing for 12.01% of explained variance—nearly one and one-half
RegressionModelsPredictingIndividualDifferencesinFirst-
times as much explained variance as QD accounted for in CA
GradeMathematicsDevelopment
(8.84%). For DA’s prediction of WP development, beta was a
Outcome B SE (cid:6) t(5,178) p sizeble .45. Moreover, although the contributions of QD and
TEMA were significant, betas were substantially smaller (.14 for
Calculations
Constant (cid:3)9.58 0.84 11.47 (cid:5).001 QD;.18forTEMA),aswerethepercentagesofexplainedvariance
Language 0.03 0.03 0.05 0.91 .360 (1.54%forQD;1.96%forTEMA)—despitethatTEMAwasthe
Reasoning 0.04 0.04 0.06 0.98 .329 only predictor to explicitly assess WPs (i.e., nonverbal addition
QD 0.10 0.02 0.30 4.62 (cid:5).001 problemswithsumsto12).
TEMA 0.11 0.03 0.30 3.94 (cid:5).001
ThesecondwayinwhichfindingsdifferedforWPdevelopment
DA 0.08 0.03 0.20 2.61 .010
WordProblems concerns the role of the domain-general predictor variables.
Constant (cid:3)3.43 0.88 (cid:3)3.91 (cid:5).001 Whereastheselanguageandreasoningpredictorsfailedtomakea
Language 0.14 0.03 0.20 4.28 (cid:5).001 significant contribution to CA development, language (but not
Reasoning 0.04 0.04 0.05 1.02 .311
reasoning) was uniquely predictive of WP development, with
QD 0.06 0.23 0.14 2.65 .009
TEMA 0.08 0.28 0.18 2.92 .004 beta equal to .20 (the same as DA in predicting CA develop-
DA 0.24 0.03 0.45 7.35 (cid:5).001 ment) and with 4.05% of the explained variance in WP devel-
opment uniquely attributable to language. Moreover, whereas
Note. LanguageisWechslerAbbreviatedScaleofIntelligence(WASI)
DA shared relatively little variance with language and reason-
Vocabulary;ReasoningisWASIMatrixReasoning.QD(cid:2)QuantityDis-
crimination; TEMA (cid:2) Test of Early Mathematics Ability, third edition; ing in predicting CA (2.27%), the corresponding figure in
DA(cid:2)DynamicAssessment. predicting WP development was more than 3 times larger
PREDICTINGFIRSTGRADERS’DEVELOPMENT 231
Table3
CommonalityAnalysisforPredictingEnd-of-YearMathematicsDevelopment
Calculationsdevelopment Word-problemdevelopment
Coefficient:Proportion Percentageofexplained Coefficient:Proportion Percentageofexplained
Variable ofvarianceexplained variance ofvarianceexplained variance
Uniqueto:
QuantityDiscrimination(QD) .051 8.84 .011 1.54
TestofEarlyMathematicsAbility(TEMA) .037 6.41 .014 1.96
DynamicAssessment(DA) .016 2.77 .086 12.01
Language(L) .002 0.35 .029 4.05
Reasoning(R) .002 0.35 .002 0.28
Commonto:
QD(cid:1)TEMA .062 10.75 .018 2.51
QD(cid:1)DA .010 1.73 .011 1.54
QD(cid:1)L .002 0.35 .003 0.42
QD(cid:1)R .000 (cid:3)0.03 .000 0.00
TEMA(cid:1)DA .027 4.68 .042 5.87
TEMA(cid:1)L .005 0.87 .009 1.26
TEMA(cid:1)R .004 0.69 .002 0.28
DA(cid:1)L .002 0.35 .017 2.37
DA(cid:1)R .008 1.39 .025 3.49
L(cid:1)R .001 0.17 .002 0.32
QD(cid:1)TEMA(cid:1)DA .066 11.44 .057 7.96
QD(cid:1)TEMA(cid:1)L .011 1.91 .011 1.54
QD(cid:1)TEMA(cid:1)R .005 0.87 .002 0.28
QD(cid:1)DA(cid:1)L .002 0.35 .004 0.56
QD(cid:1)DA(cid:1)R .003 0.52 .003 0.42
QD(cid:1)L(cid:1)R .000 0.00 .000 0.00
TEMA(cid:1)DA(cid:1)L .009 1.56 .025 3.49
TEMA(cid:1)DA(cid:1)R .026 4.51 .039 5.45
TEMA(cid:1)L(cid:1)R .002 0.35 .003 0.42
DA(cid:1)L(cid:1)R .003 0.52 .015 2.09
QD(cid:1)TEMA(cid:1)DA(cid:1)L .034 5.89 .048 6.70
QD(cid:1)TEMA(cid:1)DA(cid:1)R .066 11.44 .059 8.24
QD(cid:1)TEMA(cid:1)L(cid:1)R .004 0.69 .003 0.42
QD(cid:1)DA(cid:1)L(cid:1)R .002 0.35 .003 0.42
TEMA(cid:1)DA(cid:1)L(cid:1)R .020 3.47 .047 6.56
QD(cid:1)TEMA(cid:1)DA(cid:1)L(cid:1)R .097 16.81 .127 17.74
Total .577 100.00 .716 100.00
Note. LanguageisWechslerAbbreviatedScaleofIntelligence(WASI)Vocabulary.ReasoningisWASIMatrixReasoning.
(8.02). In these ways, DA appears to invoke the need for the Fuchs,&Barnes,2007;Hodent,Bryant,&Houde,2005),where-
same kinds of language and reasoning skills that help children asSeethaleretal.(2011)showedthatlanguageisauniquepredic-
profit from WP classroom instruction. torofCAwithrationalnumbersamongfifthgraders.Strengthwith
It is therefore interesting to consider that in this study’s DA, oral language may support teachers’ explanations, facilitating in-
studentssolvedformissingnumbersinmathematicalexpressions sightsintonovelconceptsasrequiredintheDA.Italsosuggests
without the need to process text. In this way, the DA is more that the cognitive resources involved in DA may rely on similar
transparently aligned with CA than WP, raising questions about typesofmentalflexibility,manipulationofsymbolicassociations,
why DA was more predictive of WP than CA development. A and maintenance of multiple representations that are reflected in
possibleexplanationisthatDArepresentsameasureofconceptual oral language and reasoning abilities. This may be more true for
understanding of arithmetic or understanding of the equal sign, WPthanforCA,atleastinpart,becauseclassroominstructionis
eitherofwhichmayberequiredmoreforWPthanCA.Giventhat lessexplicitandproceduralforWPthanCA.
DA did not require the processing of text, as in WP, it is also More generally, results suggest that different types of mathe-
curious that DA shared variance with language in predicting WP matics depend on distinct aspects of mathematical cognition, as
development.ApossibleexplanationforthisfindingisthattheDA previouslyshown.Forexample,L.S.Fuchs,Fuchs,Compton,et
nevertheless involves language ability because its instructional al. (2006) documented links among skill with arithmetic, proce-
scaffolding is offered via language as the examiner explains dural calculations, and WPs, even as distinct constellations of
problem-solution concepts and strategies. In fact, much of the predictorsemergedforeachareaofmathematicsperformance.In
schoolcurriculumisdeliveredviaorallanguage.Inmathematics, a related way, Hart, Petrill, and Thompson (2009) found support
evidence suggests that language plays an important role in the for different genetic and environmental influences on students’
acquisitionofearlynumeracyconceptsandskills(Fletcher,Lyon, WP versus CA performance. And, as illustrated in the present
232 SEETHALER,FUCHS,FUCHS,ANDCOMPTON
study,forforecastingmathematicsdevelopment,therelativevalue childrenwithmathematicaldifficulties.JournalofLearningDisabilities,
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