The Cognitive Correlates of Third-Grade Skill in Arithmetic, AlgorithmicComputation, and Arithmetic Word Problems

Lynn S. Fuchs, Douglas Fuchs, Donald L. Compton,Sarah R. Powell, Pamela M. Seethaler, and

Andrea M. CapizziVanderbilt University

Christopher SchatschneiderFlorida State University

Jack M. FletcherUniversity of Houston

The purpose of this study was to examine the cognitive correlates of 3rd-grade skill in arithmetic,algorithmic computation, and arithmetic word problems. Third graders (N � 312) were measured onlanguage, nonverbal problem solving, concept formation, processing speed, long-term memory, workingmemory, phonological decoding, and sight word efficiency as well as on arithmetic, algorithmiccomputation, and arithmetic word problems. Teacher ratings of inattentive behavior also were collected.Path analysis indicated that arithmetic was linked to algorithmic computation and to arithmetic wordproblems and that inattentive behavior independently predicted all 3 aspects of mathematics performance.Other independent predictors of arithmetic were phonological decoding and processing speed. Otherindependent predictors of arithmetic word problems were nonverbal problem solving, concept formation,sight word efficiency, and language.

Keywords: mathematics, cognitive correlates, arithmetic, computation, word problems

Mathematics is a broad domain, addressing the measurement,properties, and relations of quantities as expressed in numbers orsymbols. As such, mathematics comprises many branches. Thehigh school curriculum, for example, offers algebra, geometry,trigonometry, and calculus; even at the elementary grades, math-ematics is conceptualized in strands that include (but are notlimited to) concepts, numeration, measurement, arithmetic, algo-rithmic computation, and problem solving. Nevertheless, relativelylittle is known about the relations among the various aspects ofmathematical cognition and whether the cognitive abilities thatmediate different aspects of mathematics performance are sharedor distinct. Such understanding can provide theoretical insight intothe nature of mathematics development and can provide practicalguidance about the identification and treatment of mathematicsdifficulties.

The purpose of the present study was to explore the cognitivecorrelates of three aspects of third-grade mathematics perfor-mance: arithmetic (e.g., 3 � 2), algorithmic computation (e.g.,

35 � 29), and arithmetic word problems (e.g., John had ninepennies. He spent three pennies at the store. How many penniesdid he have left?). We focused on third-grade performance becausethese three aspects of mathematics skill are addressed in first- andsecond-grade curricula, creating a range of skill development bythe beginning of third grade. Upon examination of the cognitivecorrelates of primary-grade mathematics performance, most priorwork has focused on a limited set of cognitive abilities related toa single aspect of mathematics skill, rather than studying how theseabilities operate within a multivariate framework. For this reason,the literature provides the basis for deliberate hypotheses aboutwhich cognitive abilities may mediate a single aspect of third-grade mathematics performance. The literature does not, however,provide the basis for specifying an integrated theory about howthese variables might operate in coordinated fashion to simulta-neously explain the three aspects of mathematics skill. Conse-quently, the uniqueness and importance of the present study werethe effort to rely on a multivariate framework to examine a set ofcognitive abilities for which empirical and theoretical supportexists largely from univariate work. This study is a necessary stepin theory development because it provides information on theredundancy of constructs and helps integrate results across univar-iate studies.

At the outset, we caution readers about three limitations to thepresent study. First, we emphasize that the concurrent nature of ourdata collection precludes conclusions about causation. Second, aswith any study, we operationalized our constructs with specificmeasures. Although the measures we chose are widely used, weremind readers that findings may depend on instrumentation.Third, the theoretical perspective represented in the present study

Lynn S. Fuchs, Douglas Fuchs, Donald L. Compton, Sarah R. Powell,Pamela M. Seethaler, Andrea M. Capizzi, Department of Special Educa-tion, Vanderbilt University; Christopher Schatschneider, Department ofPsychology, Florida State University; Jack M. Fletcher, Department ofPsychology, University of Houston.

This research was supported in part by Grant 1 RO1 HD46154-01 andCore Grant HD15052 from the National Institute of Child Health andHuman Development to Vanderbilt University.

Correspondence concerning this article should be addressed to Lynn S.Fuchs, Department of Special Education, Vanderbilt University, 328 Pea-body, Nashville, TN 37203. E-mail: lynn.fuchs@vanderbilt.edu

Journal of Educational Psychology Copyright 2006 by the American Psychological Association2006, Vol. 98, No. 1, 29–43 0022-0663/06/$12.00 DOI: 10.1037/0022-0663.98.1.29

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poses that mathematics difficulties are secondary to basic cogni-tive abilities that are domain general, such as memory, reasoning,language, or spatial systems (e.g., Geary, 1993; Siegler, 1988). Analternative perspective, not addressed in this study, specifies thatmathematics deficits arise when a more specialized capacity forrecognizing and mentally manipulating discrete numerosities failsto develop normally (e.g., Butterworth, 1999). In the followingsection, we summarize prior related work and how it provided thebasis for hypothesizing that the domain-general variables we con-sidered were related to the aspects of mathematics performance westudied.

Prior Work

Arithmetic

Arithmetic (e.g., 2 � 3 � 5) is defined as adding and subtractingsingle-digit numbers. In solving these problems, typically devel-oping children gradually develop procedural efficiency in count-ing: First, they count the two sets in their entirety (i.e., 1, 2, 3, 4,5); then they count from the first number (i.e., 2, 3, 4, 5); andeventually, they count from the larger number (i.e., 3, 4, 5).Finally, as increasingly efficient counting consistently and quicklypairs a problem with its answer in working memory, the associa-tion becomes established in long-term memory, and children aban-don counting in favor of memory-based retrieval of answers(Geary, Brown, & Samaranayake, 1991; Lemaire & Siegler, 1995).

Previous research provides the basis for hypothesizing a set offive child attributes that may mediate arithmetic: working memory,processing speed, phonological processing, attention, and long-term memory. First, a relatively large body of work implicatesworking memory (e.g., Geary et al., 1991; Hitch & McAuley,1991; Siegel & Linder, 1984; Webster, 1979; Wilson & Swanson,2001), which is the capacity to maintain target memory itemswhile processing an additional task (Daneman & Carpenter, 1980).Although the relation between working memory and memory-based retrieval of number combinations has been repeatedly doc-umented, the nature of that relation is unclear. As described byGeary (1993), working memory, which is likely to be a domain-general ability, involves component skills, including, but not lim-ited to, rate of decay (creating difficulties in holding the associa-tion between a problem stem and its answer) and attentivebehavior (hence the finding that children with mathematics dis-abilities monitor problem solving less well than children withoutmathematics disabilities [Butterfield & Ferretti, 1987; Geary,Widaman, Little, & Cormier, 1987]). In addition, memory spanappears to be related to how quickly numbers can be counted(Geary, 1993).

It is not surprising, therefore, that processing speed, which,according to R. Case (1985), is the efficiency with which simplecognitive tasks are executed, represents a second promising can-didate. Processing speed may dictate how quickly numbers can becounted. With slower processing, the interval for deriving countedanswers and for pairing a problem stem with its answer in workingmemory increases; this increase creates the possibility that decaysets in before this pairing is effected and thereby precludes thedevelopment of representations in long-term memory. In fact, Bulland Johnston (1997) found that processing speed was the bestpredictor of arithmetic competence among 7-year-olds, subsuming

all of the variance accounted for by long- and short-term memory,even with reading performance controlled. More recently, Hecht,Torgesen, Wagner, and Rashotte (2001) provided corroboratingdata on the importance of processing speed as a correlate ofarithmetic skill while controlling for vocabulary knowledge.

A third empirically derived component is phonological process-ing. The hypothesis suggests that arithmetic requires encoding andmaintaining accurate phonological representations of terms andoperators in working memory so that representations can be es-tablished in long-term memory (e.g., Brainerd, 1983; Logie, Gil-hooly, & Wynn, 1994). This is an interesting possibility becausearithmetic deficits often occur in combination with reading diffi-culty (Geary, 1993), for which phonological deficits are wellestablished (e.g., Bruck, 1992). Evidence supporting this link isinconsistent. For example, Fuchs et al. (2005) found supportiveresults when predicting the development of arithmetic skill fromfall to spring of first grade—that is, we noted that phonologicalprocessing emerged as the only unique predictor of arithmeticskill, besides attention, when we controlled for a host of competingvariables, including reading. By contrast, H. L. Swanson andBeebe-Frankenberger (2004) identified reading, rather than pho-nological memory, as a correlate of calculation skill among asample of first through third graders, but the calculation measurecombined arithmetic and algorithmic computation. Further, in ex-amining development from fourth to fifth grade, Hecht et al.(2001) did not identify phonological processing as a viable deter-minant of arithmetic.

The final two possible components are attention and long-termmemory. In previous work with first graders, attentive behavioremerged as a potentially robust predictor of arithmetic skill (Fuchset al., 2005), even when a range of other cognitive characteristicswas controlled. Differences in skill at allocating attention or re-turning to a task after attention shifts may challenge the develop-ment of representations of number combinations in long-termmemory. At the same time, deficits in long-term memory itselfseem like a plausible determinant, given that arithmetic skill trans-parently depends on automatic retrieval from long-term memory(Siegler & Shrager, 1984). H. L. Swanson and Beebe-Frankenberger (2004), however, failed to substantiate the viabilityof deficits in long-term memory; of course, they operationalizedcalculation skill by combining arithmetic and algorithmiccomputation.

Algorithmic Computation

Algorithmic computation (e.g., 247 � 196 � 443) is defined asadding, subtracting, multiplying, or dividing whole numbers, dec-imals, or fractions using algorithms and arithmetic. So, difficultycan arise from faulty procedural knowledge, in which students failto master algorithms, or from arithmetic deficits, which result inerrors or create a bottleneck that diverts cognitive resources awayfrom procedural work. For this reason, arithmetic skill is a possibledeterminant of algorithmic computation.

Prior work creates the basis for hypothesizing that four otherchild characteristics may help determine algorithmic computationperformance: attention, working memory, phonological process-ing, and long-term memory. First, because algorithmic computa-tion involves a relatively laborious series of steps, attentive be-havior (i.e., low distractibility) may enhance performance. Russell

30 FUCHS ET AL.

and Ginsburg (1984) provided suggestive evidence supporting thispossibility when they compared fourth graders with mathematicsdisabilities with fourth graders and third graders without mathe-matics disabilities. Results indicated that the algorithmic errors ofstudents with mathematics disabilities were similar to both of thetypically developing groups but that students with mathematicsdisabilities more closely resembled their younger typically devel-oping counterparts in the detection of those errors. This resultsuggests that inattentive behavior may inhibit algorithmic compu-tation performance, a finding documented by Ackerman and Dyk-man (1995) for students with reading disabilities who have and donot have mathematics disabilities (Ackerman & Dykman, 1995)and by Fuchs et al. (2005) when controlling for a host of compet-ing cognitive predictors.

Second, prior work implicates working memory (e.g., Geary etal., 1991; Hitch & McAuley, 1991; Siegel & Linder, 1984; Wilson& Swanson, 2001; Webster, 1979), although studies do not rule outthe possibility that the role of working memory may be a result ofthe need for arithmetic in solving procedural computation prob-lems. Third, phonological processing may be required beyond thatwhich is involved in arithmetic—that is, algorithmic computationmay require individuals to hold phonological representations inworking memory while selecting, implementing, and monitoringstrategies for algorithmic problem solution (e.g., Brainerd, 1983;Logie et al., 1994). Evidence for this possibility exists. For exam-ple, Hecht et al. (2001) provided evidence for the role of phono-logical processing in the development of general computation skillwhile controlling for processing speed, reading, and vocabulary;however, the general computation measure mixed arithmetic withalgorithmic computation items. By contrast, in controlling for ahost of cognitive variables and exclusively measuring arithmetic,Fuchs et al. (2005) failed to substantiate such a relation. The finalcognitive candidate is long-term memory, which seems necessaryto consider given that arithmetic is involved in algorithmic com-putation and that algorithmic rules are stored in long-termmemory.

Arithmetic Word Problems

Arithmetic word problems (e.g., John had nine pennies. Hespent three pennies at the store. How many pennies did he haveleft?) are defined as linguistically presented one-step problemsrequiring arithmetic solutions. Because arithmetic is transparentlyrequired to find solutions to these problems, it may mediate per-formance. In addition, because skill in manipulating numbersprocedurally should enhance understanding about numerical rela-tions, algorithmic computation may facilitate skill in the solving ofarithmetic word problems. Moreover, given the involvement oflanguage as well as the need to construct a problem model beforea solution can occur, a host of possible cognitive characteristicsmay be implicated. These characteristics include working memory,long-term memory, attentive behavior, nonverbal problem solving,language ability, reading skill, and concept formation.

Prior work examining which cognitive processes mediate arith-metic word problems has focused heavily on working memory,probably for three reasons. First, research (e.g., Hitch & McAuley,1991; Siegel & Ryan, 1989) shows that children with poor arith-metic skills manifest working memory deficits. Second, childrenwith learning disabilities experience concurrent difficulty with

working memory (e.g., De Jong, 1998; Siegel & Ryan, 1989;Swanson, Ashbaker, & Sachse-Lee, 1996) and mathematical prob-lem solving (e.g., L. P. Case, Harris, & Graham, 1992; H. L.Swanson, 1993). Third, theoretical frameworks (e.g., Kintsch &Greeno, 1985; Mayer, 1992) posit that arithmetic word problemsinvolve construction of a problem model, which appears to requireworking memory capacity. For example, according to Kintsch andGreeno, in the solution of arithmetic word problems, new sets areformed online as the story is processed. When a proposition thattriggers a set-building strategy is completed, the appropriate set isformed and the relevant propositions are assigned places in theschema. As new sets are formed, previous sets that had been activein the memory buffer are displaced.

In line with theoretical models that implicate working memory,the literature provides support for its importance. For example,Passolunghi and Siegel (2001) found that 9-year-olds, character-ized as good or poor problem solvers, differed on working memorytasks. Other researchers have found corroborating evidence usingsimilar methods (e.g., LeBlanc & Weber-Russell, 1996; Passol-unghi & Siegel, 2004; H. L. Swanson & Sachse-Lee, 2001). At thesame time, other studies have raised questions about the robustnessof the relation. For example, among typically developing third andfourth graders, H. L. Swanson, Cooney, and Brock (1993) foundonly a weak relation between working memory and problem-solution accuracy, and this relation disappeared once reading com-prehension was considered. This finding suggests the need foradditional work that considers a larger pool of cognitive processes.

The other leading candidates for cognitive processes that medi-ate arithmetic word problems are long-term memory, attention,nonverbal problem solving, language ability, reading skill, andconcept formation. Long-term memory appears to be implicatedgiven the need to access mathematics facts from long-term mem-ory to solve arithmetic word problems. H. L. Swanson and Beebe-Frankenberger (2004) provided provocative data on this possibil-ity, showing that long-term memory explained a statisticallysignificant 8% of variance in arithmetic word problems. Of course,long-term memory was indexed as recognition of mathematicsprocedures needed for solving word problems, and such a mea-surement strategy may ensure a strong relation between long-termmemory and arithmetic word problems.

In studies involving attention, most work has focused on theinhibition of irrelevant stimuli, with mixed results. Passolunghi,Cornoldi, and De Liberto (1999) ran a series of studies thatsuggested the importance of inhibition. For example, comparinggood and poor problem solvers, Passolunghi, Cornoldi, and DeLiberto (1999) found comparable storage capacity but also foundinefficiencies of inhibition (i.e., poor problem solvers rememberedless relevant but more irrelevant information in mathematics prob-lems). Yet H. L. Swanson and Beebe-Frankenberger (2004) foundno evidence that inhibition contributes to arithmetic word prob-lems. Research has, however, rarely studied the role of attentionmore broadly. An exception is Fuchs et al. (2005), who found thata teacher rating scale of attentive behavior predicted the develop-ment of first-grade skill with arithmetic word problems. Clearly,additional work on this possibility is needed.

Nonverbal problem solving, or the ability to complete patternspresented visually, has been identified as a unique predictor in thedevelopment of arithmetic word problem skill across first grade(Fuchs et al., 2005), a finding corroborated by Agness and Mc-

31COGNITIVE CORRELATES OF MATHEMATICS SKILL

Clone (1987). This finding is not surprising because arithmeticword problems, in which the problem narrative poses a questionthat entails a change, combine, compare, or equalize relation-ship between two numbers, appear to require conceptualrepresentations.

Language ability also is important to consider, given the obviousneed to process linguistic information when building a problemrepresentation of an arithmetic word problem. In fact, Jordan,Levine, and Huttenlocher (1995) documented the importance oflanguage ability when they showed that kindergarten and first-grade children with language impairments (i.e., with receptivevocabulary and grammatic closure scores below the 30th percen-tile) performed significantly lower on arithmetic word problemsthan peers with no language impairments.

Finally, it is hard to ignore the possibilities that reading skill orconcept formation may underlie skill in arithmetic word problems.Reading is transparently involved, even when problems are readaloud to children, because reading skill provides continuing accessto the written problem narrative after the adult reading has beencompleted. This transparent involvement potentially reduces theload on working memory and thereby facilitates solution accuracy.At the same time, concept formation, which involves identifying,categorizing, and determining rules, seems plausible on the basisof H. L. Swanson and Sachse-Lee (2001), who argued that infor-mation activated from long-term memory may mediate the relationbetween concept formation (an aspect of executive function) andsolution accuracy for arithmetic word problems.

Considering the Role of Foundational Mathematics Skillsin Higher Order Performance

The literature cited thus far provides the basis for hypothesizinga set of variables that may mediate each of the three aspects ofmathematics performance explored in this study. At the same time,a strong assumption in the mathematics literature is that skillsdevelop hierarchically (cf. Aunola, Leskinen, Lerkkanen, &Nurmi, 2004). With respect to the skills targeted in the presentstudy, arithmetic appears foundational to algorithmic computationand arithmetic word problems. It also seems plausible that skill inmanipulating numbers procedurally, as in algorithmic computa-tion, should enhance understanding about numerical relationships,and this, in turn, may facilitate skill in arithmetic word problems.Moreover, the relation among these three aspects of mathematicsfinds support in that some of the same cognitive variables recur inthe literature across these aspects of mathematics skill. Givenassumptions about the hierarchical nature of the mathematicscurriculum, lower order mathematics skills become key targets asdeterminants of more complex skills.

Therefore, it is unfortunate that in most studies examiningcognitive determinants, researchers focus on a single aspect ofmathematics performance or explore multiple aspects of mathe-matics competence without considering how lower order skillsdetermine subsequent skills and without considering how the in-clusion of lower order skills in a model may affect empiricalfindings on the role of cognitive correlates in higher order perfor-mance. Two notable exceptions are Hecht et al. (2001) and H. L.Swanson and Beebe-Frankenberger (2004). Working at Grades2–5, Hecht et al. simultaneously considered the role of arithmeticin general computation skill. They used structural equation mod-

eling to estimate the relations between phonological processingand growth in mathematics computation skill, while controlling forprior mathematics skill, reading, processing speed, and vocabu-lary. They found that fourth-grade arithmetic performance ac-counted for a small but significant percentage of variance infifth-grade levels of general computational skill. Of course, Hechtet al. operationalized general computation skill as a combination ofarithmetic and algorithmic computation, making it difficult toassess the role of arithmetic specifically to algorithmic computa-tion. Moreover, fifth grade may not be the optimal time to assessthe development of arithmetic skill. With younger children inGrades 1–3, H. L. Swanson and Beebe-Frankenberger exploredcalculation as a determinant of arithmetic word problems, butagain, they operationalized calculation by mixing arithmetic itemswith algorithmic computation problems. Results demonstratedhow calculation skill helped determine performance on arithmeticword problems, but the design of the calculation measures pre-cludes conclusions specific to arithmetic or to algorithmiccomputation.

Rationale for the Present Study

We sought to extend the current body of literature by incorpo-rating the three mathematics skills simultaneously into path anal-ysis, a form of structural equation modeling that is designed to helpresearchers understand how variables interrelate in complex pat-terns. Such analysis permitted us to estimate the role of founda-tional mathematics skills in higher order performance while con-trolling for the role of cognitive abilities and to identify uniquecognitive correlates. We tested the model shown in Figure 1,which was based on the hypotheses we derived from previouswork (as described in the Prior Work section). Consistent with theliterature we reviewed, the model specified the following: First,arithmetic would be associated with working memory, processingspeed, phonological decoding (a proxy for phonological process-ing, given that we targeted third grade), attentive behavior, andlong-term memory. Second, algorithmic computation would beassociated with arithmetic, attentive behavior, phonological decod-ing, long-term memory, and working memory. Third, arithmeticword problems would be associated with arithmetic, algorithmiccomputation, working memory, inattentive behavior, long-termmemory, nonverbal problem solving, language ability, readingskill, and concept formation. Explicit in this model is the hierarchyof mathematics skills, whereby arithmetic skill helps explain com-petence in algorithmic computation and arithmetic word problemsand whereby algorithmic computation skill mediates arithmeticword problem performance.

Method

Participants

The data described in this article were collected as part of a prospective4-year study in which we assessed the effects of mathematics problem-solving instruction and examined the developmental course and cognitivepredictors of mathematics problem solving. The data reported in thepresent article were collected with the 1st-year sample at the first assess-ment wave, when we sampled participants from 30 third-grade classroomsin six Title 1 schools and one non–Title 1 school (two to six teachers perschool) in a southeastern metropolitan school district.

32 FUCHS ET AL.

Students were identified for participation on the basis of their perfor-mance on the Test of Computational Fluency (Fuchs, Fuchs, Hamlett, &Appleton, 2002), which provides students 3 min to write answers to 25second-grade addition and subtraction arithmetic and algorithmic compu-tation problems. Of the 494 children from whom we had parental consentand student assent (from a total of 499 students), we randomly sampled 330students for individual participation, blocking within three strata (selecting25% of students with scores one standard deviation below the mean of theentire distribution, 50% of students with scores within one standard devi-ation of the mean of the entire distribution, and 25% of students with scoresone standard deviation above the mean of the entire distribution). Of these330 students, we have complete data on 312 children for the variablesreported here. As measured on the two-subset Wechsler Abbreviated Scaleof Intelligence (WASI; Wechsler, 1999), these students’ IQ averaged 97.04(SD � 15.13). Their normal-curve equivalent scores on the TerraNova(CTB/McGraw-Hill, 1997), administered the previous spring by the schooldistrict, averaged 57.44 (SD � 18.02) for the reading composite and 59.31(SD � 21.47) for the mathematics composite. Their standard scores on theWoodcock-Johnson Psycho-Educational Battery–Revised (WJ III; Wood-cock, McGrew, & Mather, 2001) Applied Problems averaged 100.70(SD � 14.88), and their standard scores on the Woodcock Reading MasteryTest–Revised (WRMT-R; Woodcock, 1998) Word Identification averaged100.90 (SD � 10.61). Of these 312 students, 149 (47.8%) were male, 189(60.6%) received a subsidized lunch, and 24 (7.6%) had a school-identifieddisability (i.e., learning disability, speech impairment, language impair-ment, attention-deficit/hyperactivity disorder, health impairment, oremotional–behavioral disorder). Race was distributed as 131 (42.0%)

African American, 138 (44.2%) White, 18 (5.8%) Hispanic, 9 (2.9%)Kurdish, and 16 (5.1%) “other”.

Procedure

Students were assessed in the fall of third grade. In September, threewhole-class assessment sessions occurred, each lasting 30–60 min. InSeptember and October 2003, two 45-min individual testing sessionsoccurred. In this report, we describe only the subset of measures on whichwe reported data. The whole-class measures were Assessment of Math FactFluency test of the Grade 3 Math Battery (Fuchs, Hamlett, & Powell,2003), Double-Digit Addition and Subtraction test of the Grade 3 MathBattery (Fuchs et al., 2003), and Story Problems (Jordan & Hanich, 2000,adapted from Carpenter & Moser, 1984; Riley & Greeno, 1988; Riley,Greeno, & Heller, 1983). The individually administered measures wereTest of Language Development–Primary (TOLD) Grammatic Closure(Newcomer & Hammill, 1988), Woodcock Diagnostic Reading Battery(WDRB) Listening Comprehension (Woodcock, 1997), WASI Vocabulary(Wechsler, 1999), WASI Matrix Reasoning (Wechsler, 1999), Woodcock-Johnson III Tests of Achievement (WJ III) Concept Formation (Woodcocket al., 2001), WJ III Visual Matching (Woodcock et al., 2001), WJ IIIRetrieval Fluency (Woodcock & Johnson, 1989), Working Memory TestBattery–Children (WMTB-C) Listening Recall (Pickering & Gathercole,2001), WJ III Numbers Reversed (Woodcock & Johnson, 1989), WRMT-RWord Attack (Woodcock, 1998), and Test of Word Reading Efficiency(TOWRE) Sight Word Efficiency (Torgesen, Wagner, & Rashotte, 1999).Tests were administered by trained examiners, each of whom had demon-

Figure 1. Hypothesized paths.

33COGNITIVE CORRELATES OF MATHEMATICS SKILL

strated 100% accuracy during mock administrations. All individual ses-sions were audiotaped, and 17.9% of tapes, distributed equally acrosstesters, were selected randomly for accuracy checks by an independentscorer. Agreement was between 98.8% and 99.8%. In October, teacherscompleted the SWAN Rating Scale (H. L. Swanson et al., 2004) on eachstudent.

Measures

Language. We used three tests of language. The first test, TOLDGrammatic Closure, measures the ability to recognize, understand, and useEnglish morphological forms. The examiner reads 30 sentences, one at atime; each sentence has a missing word. Examinees earn 1 point for eachsentence correctly completed. As reported by the test developer, reliabilityis .88 for 8-year-olds; the correlation with the Illinois Test of Psycholin-guistic Ability Grammatic Closure (Kirk, McCarthy, & Kirk, 1986) is .88for 8-year-olds.

The second test, WDRB Listening Comprehension (Woodcock, 1997),measures the ability to understand sentences or passages. The test presents38 items, and students supply the word missing from the end of eachsentence or passage. WDRB Listening Comprehension begins with simpleverbal analogies and associations and progresses to comprehension involv-ing the ability to discern implications. Testing is discontinued after sixconsecutive errors. The score is the number of correct responses. Reliabil-ity is .80 at ages 5–18 years; the correlation with the WJ III (Woodcock &Johnson, 1989) is .73.

The third test, the WASI Vocabulary (Wechsler, 1999), measures ex-pressive vocabulary, verbal knowledge, and foundation of information with42 items. The first 4 items present pictures; the student identifies the objectin the picture. For the remaining items, the tester says a word that thestudent defines. Responses are awarded a score of 0, 1, or 2, depending onquality of response. Testing is discontinued after five consecutive scores of0. The score is the total number of points. As reported by Zhu (1999),split-half reliability is .86–.87 at ages 6–7 years; the correlation with theWechsler Intelligence Scale for Children (3rd ed.; WISC–III) Full Scale IQ(Wechsler, 1991) is .72.

Nonverbal problem solving. WASI Matrix Reasoning (Wechsler,1999) measures nonverbal reasoning with four types of tasks: patterncompletion, classification, analogy, and serial reasoning. Examinees lookat a matrix from which a section is missing and complete the matrix bysaying the number of options or pointing to one of five response options.Examinees earn points by identifying the correct missing piece of thematrix. Testing is discontinued after four errors on five consecutive itemsor after four consecutive errors. The score is the number of correctresponses. As reported by the test developer, reliability is .94 for 8-year-olds; the correlation with the WISC–III Full Scale IQ is .66.

Concept formation. WJ III Concept Formation (Woodcock et al.,2001) asks examinees to identify the rules for concepts when shownillustrations of instances and noninstances of the concept. Examinees earncredit by correctly identifying the rule that governs each concept. Cutoffpoints determine the ceiling. The score is the number of correct responses.As reported by the test developer, reliability is .93 for 8-year-olds.

Processing speed. WJ III Visual Matching (Woodcock et al., 2001)measures processing speed by asking examinees to locate and circle twoidentical numbers that appear in a row of six numbers; examinees have 3min to complete 60 rows and earn credit by correctly circling the matchingnumbers in each row. As reported by the test developer, reliability is .91 for8-year-olds.

Long-term memory. WJ III Retrieval Fluency (Woodcock & Johnson,1989) measures long-term memory by asking examinees to recall relateditems, within categories, for 1 min per category. They earn credit for eachnonduplicated answer. As reported by the test developer, reliability is .78for 8-year-olds.

Working memory. We used two measures of working memory. WithWMTB-C Listening Recall (Pickering & Gathercole, 2001), the tester says

a series of short sentences, only some of which make sense. The studentindicates whether each sentence is true or false. After hearing all sentencesin a trial (i.e., 1–6 sentences) and determining them to be true or false, thestudent recalls the final word of each sentence, in the order presented. Thestudent earns 1 point for each sequence of final words recalled correctly inthe right order, and the score is the total of correct sequences. Testing isdiscontinued when the student makes three or more errors in any block ofitems. As reported by Pickering and Gathercole, test–retest reliability is.93. With WJ III Numbers Reversed (Woodcock & Johnson, 1989), thetester says a string of random numbers, and the student says the seriesbackward. Item difficulty increases as more numbers are added to theseries. Examinees earn credit by repeating the numbers correctly in theopposite order. As reported by the test developer, reliability is .86 for8-year-olds.

Attentive behavior. The SWAN (J. Swanson et al., 2004) is an 18-itemteacher rating scale. Items from the Diagnostic and Statistical Manual ofMental Disorders (4th ed.; American Psychiatric Association, 1994) cri-teria for attention-deficit/hyperactivity disorder are included for inattention(largely, distractibility; Items 1–9) and hyperactivity/impulsivity (Items10–18). Items are rated on a scale of 1 to 7 (1 � far below, 2 � below, 3 �slightly below, 4 � average, 5 � slightly above, 6 � above, 7 � farabove). In the present study, we report data for the Inattentive Behaviorsubscale as the average rating per item across the nine relevant items. Weselected this subscale to operationalize inattentive behavior, or reducedability to maintain focus of attention. The SWAN has been shown tocorrelate well with other dimensional assessments of behavior related toinattention (J. Swanson et al., 2004). Coefficient alpha in the present studywas .97.

Phonological decoding. WRMT-R Word Attack (Woodcock, 1998)measures phonetic reading ability; it comprises 45 pseudowords (or verylow-frequency words), arranged in order of difficulty. Two practice itemsare used to train students. Testing is discontinued after six consecutiveerrors. The score is the number of words pronounced correctly. As reportedby Woodcock (1998), split-half reliability is .94.

Reading. In TOWRE Sight Word Efficiency (Torgesen, Wagner, &Rashotte, 1999), testers assess sight word reading fluency by askingexaminees to read a list of real words in 45 s. Examinees earn 1 point foreach correctly read word. As reported by the test developer, reliability is.95 for 8-year-olds; the correlation with WRMT Word Identification(Woodcock, 1998) at third grade is .92.

Arithmetic. The Assessment of Math Fact Fluency test of the Grade 3Math Battery (Fuchs et al., 2003) incorporates two subtests. The firstsubtest, Addition Fact Fluency, comprises 25 addition fact problems withanswers from 0 to 12, presented horizontally on one page. Students have 1min to write their answers. The second subtest, Subtraction Fact Fluency,comprises 25 subtraction fact problems with answers from 0 to 12, pre-sented horizontally on one page. Students have 1 min to write theiranswers. The score is the number of correct answers across both subtests.Percentage of agreement, calculated on 20% of protocols by two indepen-dent scorers, was 97.9. Coefficient alpha on this sample was .92. Criterionvalidity with the previous spring’s TerraNova (CTB/McGraw-Hill, 1997)Total Math score was .52.

Algorithmic computation. The Double-Digit Addition and Subtractiontest of the Grade 3 Math Battery (Fuchs et al., 2003) comprises twosubtests. The first subtest, Addition, provides students with 5 min tocomplete twenty 2-digit by 2-digit addition problems with and withoutregrouping. The second subtest, Subtraction, provides students with 5 minto complete twenty 2-digit by 2-digit subtraction problems with and with-out regrouping. The score is the number of correct answers across bothsubtests. Percentage of agreement, calculated on 20% of protocols by twoindependent scorers, was 99.7. In the present study, coefficient alpha was.93. Criterion validity with the previous spring’s TerraNova (CTB/McGraw-Hill, 1997) Total Math score was .48.

34 FUCHS ET AL.

Arithmetic word problems. Following Jordan and Hanich (2000,adapted from Carpenter & Moser, 1984; Riley & Greeno, 1988; Riley,Greeno, & Heller, 1983), Story Problems comprises 14 brief story prob-lems involving sums or minuends of 9 or less, with change, combine,compare, and equalize relationships. The tester reads each item aloud;students have 30 s to respond and can ask for rereading(s) as needed. Thescore is the number of correct answers. A second scorer independentlyrescored 20% of protocols, with agreement of 99.8%. Coefficient alpha onthis sample was .83. Criterion validity with the previous spring’s Terra-Nova (CTB/McGraw-Hill, 1997) Total Math score was .66.

Data Analysis and Results

For constructs in which we had more than one measure avail-able, we created weighted composite variables using a principal-components factor analysis across the variables in that conceptu-ally related set. This was the case for language (in which wecreated a weighted composite score across TOLD GrammaticClosure, WDRB Listening Comprehension, and WASI Vocabu-lary) and working memory (in which we created a weightedcomposite score across WMTB Listening Recall and WJ III Num-bers Reversed; because each principal-components factor analysisyielded only one factor, no rotation was necessary.) For otherconstructs, only one measure was available: attentive behavior(SWAN), processing speed (WJ III Visual Matching), long-termmemory (WJ III Retrieval Fluency), concept formation (WJ IIIConcept Formation), nonverbal problem solving (WASI MatrixReasoning), phonological decoding (WRMT-R Word Attack),sight word recognition skill (TOWRE Sight Word Efficiency), aswell as the three mathematics measures. In Table 1, we show rawand standard score means and standard deviations along withcorrelations.

Because we had only one measure available for all but twoconstructs, we could not use latent variable structural equationmodeling and instead used path analysis, another form of structuralequation modeling. To conduct path analysis, we converted vari-ables to z scores. Then, using the statistical software LISREL 8.5(Joreskog & Sorbom, 1993), we normalized the data and tested themodel with path analysis. Figure 2 shows the results, with statis-tically significant paths in bold. Beta and t values are shown alongthe arrows. The chi-square was statistically significant, �2(11, N �312) � 19.97, p � .046, but the model fit data were supportive ofthe hypothetical model shown in Figure 1: root-mean-square re-sidual (RMSR) � .024, comparative fit index (CFI) � .99,goodness-of-fit index (GFI) � .99, adjusted goodness-of-fit index(AGFI) � .92, normed fit index (NFI) � .99, nonnormed fit index(NNFI) � .96, accounting for 33%, 47%, and 52% of the variancein arithmetic, algorithmic computation, and arithmetic story prob-lems, respectively. For arithmetic, the significant predictors wereattentive behavior, phonological decoding, and processing speed;for algorithmic computation, the only significant paths were arith-metic and attentive behavior; and for arithmetic word problems,the significant predictors were arithmetic, attentive behavior, non-verbal problem solving, concept formation, sight word efficiency,and language. These results support the hypothesized model be-cause different cognitive skills predict different mathematics com-petencies, and the mathematics competencies are hierarchicallyrelated.

A surprising finding was that working memory did not emergeas a significant predictor anywhere in the model. Because previous

work suggests that reading or reading-related processes may in-fluence the relations among cognitive abilities and arithmetic(Fuchs et al., 2005), arithmetic or algorithmic computation (H. L.Swanson & Beebe-Frankenberger, 2004), and arithmetic wordproblems (H. L. Swanson & Beebe-Frankenberger), we ran asecond, nested analysis with the paths for phonological decodingand sight word efficiency set to zero (see Figure 3). The goal wasto determine whether working memory contributed to arithmeticskill when these variables were not controlled. With the paths forphonological decoding and sight word efficiency set to zero,�2(14, N � 312) � 32.71, p � .003, RMSR � .036, CFI � .99,GFI � .98, AGFI � .90, NFI � .98, NNFI � .94, the modelaccounted for 32%, 47%, and 51% of the variance in arithmetic,algorithmic computation, and arithmetic story problems, respec-tively. The difference in models, with and without phonologicaldecoding and sight word efficiency, was significant, ��2(3, N �312) � 12.74, p � .01, indicating that phonological decoding andsight word efficiency cannot be removed from the model withoutsignificantly decreasing the overall fit. Moreover, with and withoutphonological decoding and sight word efficiency in the model, thesignificance and magnitude of the remaining paths were similarwith two exceptions: With the paths for phonological decoding andsight word efficiency set to zero, working memory emerged as asignificant correlate of arithmetic and of arithmetic word problems(but not of algorithmic computation). The fact that working mem-ory may not contribute uniquely to mathematics competenciesindependent of phonological processing also has been observed inresearch on word recognition processes (Shankweiler & Crain,1986). Of course, it is also possible that working memory isalready captured within some of the cognitive abilities simulta-neously entered within the model.

Because phonological decoding and sight word efficiency arerelated to oral language (e.g., in the present study, r � .42 and .47),we ran a third, nested analysis, this time with phonological decod-ing and sight word efficiency in the model but with the path forlanguage set to zero (see Figure 4). The model, �2(12, N � 312) �29.34, p � .004, RMSR � .026, CFI � .99, GFI � .98, AGFI �.90, NFI � .99, NNFI � .94, accounted for 33%, 47%, and 50%of the variance in arithmetic, algorithmic computation, and arith-metic story problems, respectively. The difference in models, withand without language, was significant, ��2(1, N � 312) � 10.76,p � .01, indicating that language cannot be removed without asignificant decrease in the overall fit. With and without languagein the model, the significance and magnitude of the remainingpaths were similar, and working memory was not a significantpredictor anywhere in the model.

Finally, because the cognitive correlates specified in our hy-pothesized model were limited to attributes identified as poten-tially important in prior work, the model did not consider addi-tional processes that seem interesting and viable. With this inmind, we attempted to extend understanding for algorithmic com-putation, in which only arithmetic and attentive behavior emergedas correlates, by assessing an exploratory model that added non-verbal problem solving and concept formation as paths to algo-rithmic computation. These constructs seem potentially viablegiven that conceptual understanding of place value and the base-10system might enhance performance. The model, �2(9, N � 312) �14.34, p � .111, RMSR � .019, CFI � 1.00, GFI � .99, AGFI �.93, NFI � .98, NNFI � .97, accounted for 33%, 48%, and 51%

35COGNITIVE CORRELATES OF MATHEMATICS SKILL

Tab

le1

Mea

ns,

Stan

dard

Dev

iati

ons,

and

Cor

rela

tion

sA

mon

gC

ogni

tive

,R

eadi

ng,

and

Mat

hV

aria

bles

(N�

312)

Var

iabl

e

Cor

rela

tion

Raw

scor

eaSt

anda

rdsc

oreb

12

34

56

78

910

1112

1314

1516

17M

SDM

SD

1.L

angu

age

fact

or0.

050.

99—

—2.

TO

LD

Gra

mm

atic

Clo

sure

18.9

46.

7886

.15

11.8

3.8

5—

3.W

DR

BL

iste

ning

Com

preh

ensi

on21

.32

3.82

96.8

917

.49

.87

.62

—4.

WA

SIV

ocab

ular

y28

.08

6.49

47.3

210

.38

.84

.55

.61

—5.

Con

cept

form

atio

n17

.00

7.08

94.9

713

.77

.55

.44

.34

.47

—6.

Non

verb

alpr

oble

mso

lvin

g15

.46

6.59

48.2

411

.64

.40

.34

.20

.34

.41

—7.

Atte

ntio

n36

.71

12.6

6—

.45

.36

.39

.41

.43

.35

—8.

Proc

essi

ngsp

eed

32.1

45.

1210

1.04

15.1

7.3

0.2

2.2

8.2

6.3

0.3

1.4

2—

9.L

ong-

term

mem

ory

497.

003.

6793

.96

14.3

6.3

6.2

4.3

1.3

8.2

2.2

1.1

8.3

5—

10.

Wor

king

mem

ory

fact

or0.

031.

00—

.46

.43

.37

.38

.43

.32

.29

.19

.20

—11

.W

MT

BL

iste

ning

Rec

all

9.76

3.39

91.0

519

.79

.48

.46

.43

.35

.45

.30

.26

.19

.16

.80

—12

.W

JII

IN

umbe

rsR

ever

sed

8.97

2.80

93.5

414

.07

.29

.28

.18

.29

.28

.25

.23

.15

.16

.82

.35

—13

.Ph

onol

ogic

alde

codi

ng23

.54

9.46

110.

3961

.24

.47

.45

.37

.40

.37

.31

.41

.16

.03

.47

.39

.40

—14

.Si

ght

wor

def

fici

ency

55.1

111

.74

103.

3510

.84

.42

.34

.34

.39

.34

.20

.44

.35

.21

.38

.35

.26

.60

—15

.A

rith

met

ic19

.39

8.76

—.3

9.2

8.3

9.3

4.3

8.2

9.4

4.4

8.1

9.2

6.2

6.2

2.3

3.3

8—

16.

Alg

orith

ms

24.5

88.

70—

.37

.28

.33

.33

.42

.32

.60

.39

.20

.27

.28

.21

.35

.41

.56

—17

.St

ory

prob

lem

s10

.03

3.38

—.5

7.4

7.5

3.4

5.5

2.4

5.5

1.2

8.1

7.3

9.3

9.2

6.4

4.3

8.4

6.4

0—

Not

e.T

OL

D�

Tes

tof

Lan

guag

eD

evel

opm

ent–

Prim

ary;

WD

RB

�W

oodc

ock

Dia

gnos

ticR

eadi

ngB

atte

ry;W

ASI

�W

echs

ler

Abb

revi

ated

Scal

eof

Inte

llige

nce;

WM

TB

�W

orki

ngM

emor

yT

est

Bat

tery

;W

JII

I�

Woo

dcoc

k-Jo

hnso

nPs

ycho

-Edu

catio

nal

Bat

tery

.a

See

Mea

sure

sse

ctio

nfo

rin

form

atio

nab

out

how

raw

scor

esw

ere

calc

ulat

ed.

bSt

anda

rdsc

ores

have

am

ean

of10

0an

da

stan

dard

devi

atio

nof

15,e

xcep

tfo

rW

ASI

Voc

abul

ary

and

WA

SIM

atri

xR

easo

ning

(i.e

.,no

nver

bal

prob

lem

solv

ing)

,w

here

tsc

ores

are

used

with

am

ean

of10

0an

da

stan

dard

devi

atio

nof

10.

36 FUCHS ET AL.

of the variance in arithmetic, algorithmic computation, and arith-metic story problems, respectively (see Figure 5). Because thismodel was not nested, we could not compare it with a competingmodel. Instead, we included it only for exploratory purposes. Weobserved that neither nonverbal problem solving nor concept for-mation was statistically significant, and with and without these twoadditional predictors in the model, the significance and magnitudeof the remaining paths were similar.

In sum, the significant predictors for arithmetic were attentivebehavior, phonological decoding, and processing speed; the sig-nificant predictors for algorithmic computation were arithmeticand attentive behavior; and the significant predictors for arithmeticword problems were arithmetic, attentive behavior, nonverbalproblem solving, concept formation, sight word efficiency, andlanguage. Although working memory was not a significant path inthe overall model, it was a significant predictor of arithmetic andarithmetic word problems when the paths for reading and phono-logical processing were set to zero, suggesting that reading orreading-related processes may influence the relations among work-ing memory and at least two aspects of mathematics skill.

Discussion

With respect to the hierarchical nature of mathematics develop-ment, we found that the three mathematics skills constituted apartial hierarchy. That is, arithmetic was a significant path to

algorithmic computation and to arithmetic word problems. At thesame time, although arithmetic skill was independently linked toalgorithmic computation and to arithmetic word problems, algo-rithmic computation was not a significant predictor of arithmeticword problem performance. This finding suggests that skill inmanipulating numbers procedurally, as done in algorithmic com-putation, may not correspond to the capacity to conceptualizerelations among numbers, at least when those relations are con-veyed via language, as is the case with arithmetic word problems.Arithmetic word problems do not require algorithmic computationfor solution, and one might anticipate that with problem solvingthat requires algorithmic computation, a hierarchy that includesalgorithmic computation as a predictor of performance may betenable. In a similar way, the need for arithmetic skill withinalgorithmic computation and arithmetic word problems is trans-parent. However, empirical demonstration of these paths, evenwhen controlling for multiple cognitive abilities and reading per-formance, strengthens the notion that fluency with single-digitaddition and subtraction is foundational for at least some aspects ofsubsequent mathematics skill. This finding also suggests the needfor early intervention for promotion of the development of skillswith number combinations.

With respect to the cognitive correlates of arithmetic skill,results revealed its unique and direct relations with attentive be-havior, processing speed, and phonological decoding. In fact,

Figure 2. Hypothesized model: Path coefficients (t values), with bold signifying statistically significant paths.

37COGNITIVE CORRELATES OF MATHEMATICS SKILL

teacher ratings of attentive behavior were the most robust predictorin this study—as the only variable that independently accountedfor variance in all three aspects of mathematics skill. Moreover, inthe case of algorithmic computation, for which serial execution oftasks is required, attentive behavior was the only unique correlate(beyond arithmetic). Although few studies have considered atten-tive behavior as a correlate of mathematics skill, previous workdoes suggest a role. For example, Ackerman and Dykman (1995)documented differences in attentive behavior among reading dis-abled students with and without mathematics disability, and Ba-dian (1983) described a boy with an attention deficit who learnedmultiplication facts only after receiving pharmacological treat-ment. Also, Fuchs et al. (2005), using a different scale than the oneused in the present study, showed that teacher ratings of inattentivebehavior in the fall semester uniquely predicted development of arange of first-grade mathematics skills.

Present findings again suggest the critical role that attentivebehavior may play, and several explanations are possible. Lowdistractibility, or the ability to focus attention, may create theopportunity to persevere with academic tasks, especially thosetasks requiring serial execution, such as some aspects of mathe-matics (Luria, 1980). Alternatively, instruction may fail to addressthe needs of children with poor mathematics potential, which isdetermined by other deficits, and this mismatch between needs andinstruction creates inattentive behavior, which teachers observe.Another possibility is that teacher ratings of attentive behavior areclouded by students’ academic performance and therefore serve as

a proxy for achievement rather than index attention and/or dis-tractibility. Finally, attentive behavior may in fact represent acritical cognitive determinant. Our data do not provide the basis fordistinguishing among these explanations but instead for hypothe-sizing that inattentive behavior may play a critical role and forexploring the underlying nature of the relation. In any case, find-ings do suggest that ratings of attentive behavior may serve toscreen children for risk of mathematics difficulty, and future workshould explore this possibility.

Specifically in terms of arithmetic, other unique predictors inaddition to inattentive behavior were processing speed and pho-nological decoding. For processing speed, our findings corroboratethe work of Bull and Johnston (1997). While controlling for wordreading ability, item identification, and short-term memory, Bulland Johnston found that processing speed subsumed all of thevariance in 7-year-olds’ arithmetic skill. Processing speed mayfacilitate counting speed so that as young children gain speed incounting sets to figure sums and differences, they successfully pairproblems with their answers in working memory before decay setsin, thus establishing associations in long-term memory (e.g.,Geary, Brown, & Samaranayake, 1991; Lemaire & Siegler, 1995).

At the same time, phonological decoding’s unique relation toarithmetic is interesting in that fact-retrieval deficits often occurconcurrently with reading difficulty (e.g., Geary, Hamson, &Hoard, 2000; Jordan & Montani, 1997; Lewis, Hitch, & Walker,1994), for which phonological deficits are well established (e.g.,Brady & Shankweiler, 1991; Wagner et al., 1997). In the present

Figure 3. Model with phonological decoding and sight word efficiency removed: Path coefficients (t values),with bold signifying statistically significant paths.

38 FUCHS ET AL.

study, given that we targeted third grade, we used phonologicaldecoding as a proxy for phonological processing. This is a poten-tial limitation of our study, and additional work that includes directmeasures of phonological processing should be pursued. In themeantime, however, our findings suggest that phonological pro-cessing may mediate deficits in word identification and fact re-trieval (cf. Geary, 1993). This possibility gains favor given thatphonological systems are engaged when children use phonologicalname codes of numbers to count (Logie & Baddeley, 1987) andthat counting skill, in turn, appears critical to the development ofarithmetic skill (Aunola et al., 2004; Geary & Brown, 1991;Lemaire & Siegler, 1995). Also, prior work (Fuchs et al., 2005)showed that phonological processing was a unique determinant ofthe development of first-grade arithmetic skill but not of otheraspects of mathematics skill, even when basic reading skill wascontrolled. Hecht et al. (2001) demonstrated that phonologicalprocessing almost completely accounted for the association be-tween reading and computational skill in older children. Thepresent findings show a direct path between phonological decod-ing and arithmetic among third graders, even when seven otherabilities were controlled, providing additional suggestive evidencethat phonological processing underlies arithmetic skill. In reading,the transparent connection between phonological processing anddecoding skill provides the basis for instructional design. In arith-metic, deriving the instructional implications of a possible link

with phonological processing may be more challenging. It does,however, warrant attention, with research conducted to assess thevalue of speeded oral practice in counting to derive number factanswers or to explore whether instruction in word-level skills maytransfer to improved arithmetic competence.

In terms of arithmetic word problems, four measures beyondarithmetic and in addition to attentive behavior emerged as uniquecorrelates: nonverbal problem solving, concept formation, sightword efficiency, and language. With arithmetic word problems,students work conceptually with numbers: They listen to briefscenarios while reading along on paper; each story poses a ques-tion that entails a change, combine, compare, or equalize relation-ship between two numbers, involving a sum or minuend of 9 orless. The language within the story determines the relationships,which must be deciphered to build a problem model (cf. Kintsch &Greeno, 1985). Given these demands, it is not surprising thatlanguage ability or concept formation should play an importantrole, even though relatively few studies have examined thesepossibilities (see Jordan et al., 1995, on the relation betweenlanguage and arithmetic word problems). At the same time, theemergence of nonverbal problem solving as a unique correlate ofarithmetic word problems is interesting. This finding corroboratesprevious work at first grade (Fuchs et al., 2005) and third grade(H. L. Swanson & Beebe-Frankenberger, 2004). Clearly, arith-metic word problems, in which the problem narrative poses a

Figure 4. Model with language removed: Path coefficients (t values), with bold signifying statisticallysignificant paths.

39COGNITIVE CORRELATES OF MATHEMATICS SKILL

question entailing a change, combine, or equalize relation betweentwo numbers, requires problem solving. In addition, word recog-nition skill seems transparently involved in arithmetic word prob-lems, even when problems are read aloud to children, becauseword recognition skill provides continuing access to the writtenproblem narrative after the adult reading has been completed.Therefore, it is not surprising to find that sight word efficiencymediates competence with arithmetic word problems.

And what about working memory? With all variables in themodel, including the three mathematics skills, phonological de-coding, sight word efficiency, and seven cognitive abilities, work-ing memory did not emerge as a significant predictor of any of thethree mathematics skills considered. This finding contradicts pre-vious work showing the importance of working memory to arith-metic and algorithmic computation (Geary et al., 1991; Hitch &McAuley, 1991; Siegel & Linder, 1984; Webster, 1979; Wilson &Swanson, 2001) as well as to arithmetic word problems (e.g.,LeBlanc & Weber-Russell, 1996; Passolunghi & Siegel, 2004;H. L. Swanson & Sachse-Lee, 2001). Most prior work has exam-ined working memory as it relates to a single mathematics skill,without simultaneous consideration of other cognitive abilities andother aspects of mathematics performance. The work of H. L.Swanson and Beebe-Frankenberger (2004) is a notable exception,because they examined the role of multiple abilities for arithmeticword problems, including mathematics calculation skill (opera-

tionalized as arithmetic and algorithmic computation), and foundthat in both mathematics areas, working memory accounted forunique variance. Consequently, it is important to note that weoperationalized working memory with a particular set of measures,assessing memory span for language stimuli as well as for back-ward digit span. Although these measures are well accepted for theindexing of working memory, it is possible that different instru-ments would reveal working memory as a significant predictor.

Moreover, it is interesting to consider that working memory didemerge as a significant path for arithmetic and for arithmetic wordproblems when the paths for phonological decoding and sightword efficiency were set to zero. This finding corroborates thehypothesis that reading or reading-related processes may influencethe relations between cognitive abilities and arithmetic (Fuchs etal., 2005) as well as between cognitive abilities and arithmeticword problems (H. L. Swanson & Beebe-Frankenberger, 2004).Phonological decoding or sight word efficiency may serve as aproxy for phonological processing, and efficient encoding andmaintenance of phonological information in working memoryshould enable children to build accurate arithmetic facts in long-term memory (Siegler & Shipley, 1995; Siegler & Shrager, 1984)and to devote maximum attentional resources to the building of aproblem model for arithmetic word problems (cf. Kintsch &Greeno, 1985). Swanson and Beebe-Frankenberger (2004) exam-ined the role of phonological memory within the working memory

Figure 5. Extended model: Path coefficients (t values), with bold signifying statistically significant paths.

40 FUCHS ET AL.

system while considering its relation to calculation and arithmeticword problems and found no significant relation. Therefore, futurework should continue to examine the interplay among phonolog-ical processing, phonological decoding, sight word efficiency, andworking memory in determining skill in arithmetic and arithmeticword problems.

At the same time, even when the paths for phonological decod-ing and sight word efficiency were set to zero in the model,working memory was not a unique correlate of algorithmic com-putation. This finding is at odds with H. L. Swanson and Beebe-Frankenberger (2004), even though they did include reading intheir model. It is interesting to consider how three study designfeatures may help explain these conflicting results. First, whereasthe present study sampled a broad distribution of children onwritten arithmetic and algorithmic computation, H. L. Swansonand Beebe-Frankenberger selected groups that differed on workingmemory and orally presented arithmetic facts; this difference mayhave increased the salience of working memory to their findings.Second, in the present study, students had a written copy of thearithmetic word problems as the examiner read problems aloudand as they worked (they could also request rereadings). H. L.Swanson and Beebe-Frankenberger instead read story problemsaloud to participants, who answered without referring back to theproblems. This type of response may require working memorycapacity beyond that which is needed for typical arithmetic wordproblem tasks, in and out of school, in which access to the problemsituation remains beyond an initial, oral presentation. Third, thepresent study separated arithmetic from algorithmic computation;by contrast, H. L. Swanson and Beebe-Frankenberger combinedthese skills into a single measure, thus creating the possibility thatthe results reflected working memory’s salience for arithmetic, notfor algorithmic computation. In any case, additional work is war-ranted that examines the role of working memory in variousaspects of mathematics performance when multiple cognitive abil-ities and reading and mathematics skills are simultaneouslyconsidered.

Before closing, we note that the only unique predictors ofalgorithmic computation within our hypothesized model were at-tentive behavior and arithmetic. Therefore, working memory,long-term memory, and phonological decoding do not appear tomediate algorithmic computation performance. For this reason, weconsidered a competing model that added nonverbal problemsolving and concept formation as possible determinants. Theseconstructs seem viable given that conceptual understanding ofplace value and the base-10 system may enhance performance.Within this exploratory model, the path for concept formation didapproach statistical significance. Our measure of concept forma-tion asked students to identify the rules for concepts concerningcolor, shape, and size when they were shown illustrations ofinstances and noninstances of the concept. Future work mightbuild on present findings by incorporating a related measure thattaps concept formation in a manner better related to place value.

Results of the present study extend knowledge about the rela-tions among three key aspects of third-grade mathematical cogni-tion. Findings suggest a partial hierarchy of skills, with arithmeticsignificantly predicting algorithmic computation and arithmeticword problems and without algorithmic computation accountingfor variance in arithmetic word problems. In addition, our resultshighlight the potential importance of attentive behavior across the

three aspects of third-grade mathematics competence we studied,even as the findings reveal critical differences in the cognitiveabilities that mediate these various mathematics skills. In this way,results suggest that aspects of mathematical cognition may bedistinct. We note that our cognitive and academic correlates left afair amount of variance to be explained. Also, we emphasize theconcurrent nature of our data collection, which precludes conclu-sions about causation, and we remind readers that findings maydepend on instrumentation. Future work might broaden the searchfor cognitive determinants and extrastudent variables, even as itadopts a longitudinal framework and explores additional aspects ofmathematical cognition at varying grade levels. Present findings incombination with future related work should set the stage for thedevelopment of integrated theory about how cognitive abilitiesoperate in coordinated fashion to simultaneously explain differentaspects of mathematics skill.

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Received May 4, 2005Revision received July 22, 2005

Accepted September 19, 2005 �

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