Journal of Educational Psychology
2006, Vol. 98, No. 1, 29 – 43
Copyright 2006 by the American Psychological Association
0022-0663/06/$12.00 DOI: 10.1037/0022-0663.98.1.29
The Cognitive Correlates of Third-Grade Skill in Arithmetic, Algorithmic
Computation, and Arithmetic Word Problems
Lynn S. Fuchs, Douglas Fuchs, Donald L. Compton,
Sarah R. Powell, Pamela M. Seethaler, and
Andrea M. Capizzi
Christopher Schatschneider
Florida State University
Vanderbilt University
Jack M. Fletcher
University 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 on
language, nonverbal problem solving, concept formation, processing speed, long-term memory, working
memory, phonological decoding, and sight word efficiency as well as on arithmetic, algorithmic
computation, 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 word
problems and that inattentive behavior independently predicted all 3 aspects of mathematics performance.
Other independent predictors of arithmetic were phonological decoding and processing speed. Other
independent predictors of arithmetic word problems were nonverbal problem solving, concept formation,
sight word efficiency, and language.
Keywords: mathematics, cognitive correlates, arithmetic, computation, word problems
35 ⫹ 29), and arithmetic word problems (e.g., John had nine
pennies. He spent three pennies at the store. How many pennies
did he have left?). We focused on third-grade performance because
these three aspects of mathematics skill are addressed in first- and
second-grade curricula, creating a range of skill development by
the beginning of third grade. Upon examination of the cognitive
correlates of primary-grade mathematics performance, most prior
work has focused on a limited set of cognitive abilities related to
a single aspect of mathematics skill, rather than studying how these
abilities operate within a multivariate framework. For this reason,
the literature provides the basis for deliberate hypotheses about
which cognitive abilities may mediate a single aspect of thirdgrade mathematics performance. The literature does not, however,
provide the basis for specifying an integrated theory about how
these variables might operate in coordinated fashion to simultaneously explain the three aspects of mathematics skill. Consequently, the uniqueness and importance of the present study were
the effort to rely on a multivariate framework to examine a set of
cognitive abilities for which empirical and theoretical support
exists largely from univariate work. This study is a necessary step
in theory development because it provides information on the
redundancy of constructs and helps integrate results across univariate studies.
At the outset, we caution readers about three limitations to the
present study. First, we emphasize that the concurrent nature of our
data collection precludes conclusions about causation. Second, as
with any study, we operationalized our constructs with specific
measures. Although the measures we chose are widely used, we
remind readers that findings may depend on instrumentation.
Third, the theoretical perspective represented in the present study
Mathematics is a broad domain, addressing the measurement,
properties, and relations of quantities as expressed in numbers or
symbols. As such, mathematics comprises many branches. The
high school curriculum, for example, offers algebra, geometry,
trigonometry, and calculus; even at the elementary grades, mathematics is conceptualized in strands that include (but are not
limited to) concepts, numeration, measurement, arithmetic, algorithmic computation, and problem solving. Nevertheless, relatively
little is known about the relations among the various aspects of
mathematical cognition and whether the cognitive abilities that
mediate different aspects of mathematics performance are shared
or distinct. Such understanding can provide theoretical insight into
the nature of mathematics development and can provide practical
guidance about the identification and treatment of mathematics
difficulties.
The purpose of the present study was to explore the cognitive
correlates of three aspects of third-grade mathematics performance: arithmetic (e.g., 3 ⫹ 2), algorithmic computation (e.g.,
Lynn S. Fuchs, Douglas Fuchs, Donald L. Compton, Sarah R. Powell,
Pamela M. Seethaler, Andrea M. Capizzi, Department of Special Education, Vanderbilt University; Christopher Schatschneider, Department of
Psychology, Florida State University; Jack M. Fletcher, Department of
Psychology, University of Houston.
This research was supported in part by Grant 1 RO1 HD46154-01 and
Core Grant HD15052 from the National Institute of Child Health and
Human Development to Vanderbilt University.
Correspondence concerning this article should be addressed to Lynn S.
Fuchs, Department of Special Education, Vanderbilt University, 328 Peabody, Nashville, TN 37203. E-mail: [email protected]
29
FUCHS ET AL.
30
poses that mathematics difficulties are secondary to basic cognitive abilities that are domain general, such as memory, reasoning,
language, or spatial systems (e.g., Geary, 1993; Siegler, 1988). An
alternative perspective, not addressed in this study, specifies that
mathematics deficits arise when a more specialized capacity for
recognizing and mentally manipulating discrete numerosities fails
to develop normally (e.g., Butterworth, 1999). In the following
section, we summarize prior related work and how it provided the
basis for hypothesizing that the domain-general variables we considered were related to the aspects of mathematics performance we
studied.
Prior Work
Arithmetic
Arithmetic (e.g., 2 ⫹ 3 ⫽ 5) is defined as adding and subtracting
single-digit numbers. In solving these problems, typically developing children gradually develop procedural efficiency in counting: 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); and
eventually, they count from the larger number (i.e., 3, 4, 5).
Finally, as increasingly efficient counting consistently and quickly
pairs a problem with its answer in working memory, the association becomes established in long-term memory, and children abandon 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 of
five child attributes that may mediate arithmetic: working memory,
processing speed, phonological processing, attention, and longterm memory. First, a relatively large body of work implicates
working 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 items
while processing an additional task (Daneman & Carpenter, 1980).
Although the relation between working memory and memorybased retrieval of number combinations has been repeatedly documented, the nature of that relation is unclear. As described by
Geary (1993), working memory, which is likely to be a domaingeneral ability, involves component skills, including, but not limited to, rate of decay (creating difficulties in holding the association between a problem stem and its answer) and attentive
behavior (hence the finding that children with mathematics disabilities monitor problem solving less well than children without
mathematics disabilities [Butterfield & Ferretti, 1987; Geary,
Widaman, Little, & Cormier, 1987]). In addition, memory span
appears 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 simple
cognitive tasks are executed, represents a second promising candidate. Processing speed may dictate how quickly numbers can be
counted. With slower processing, the interval for deriving counted
answers and for pairing a problem stem with its answer in working
memory increases; this increase creates the possibility that decay
sets in before this pairing is effected and thereby precludes the
development of representations in long-term memory. In fact, Bull
and Johnston (1997) found that processing speed was the best
predictor 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 corroborating
data on the importance of processing speed as a correlate of
arithmetic skill while controlling for vocabulary knowledge.
A third empirically derived component is phonological processing. The hypothesis suggests that arithmetic requires encoding and
maintaining accurate phonological representations of terms and
operators in working memory so that representations can be established in long-term memory (e.g., Brainerd, 1983; Logie, Gilhooly, & Wynn, 1994). This is an interesting possibility because
arithmetic deficits often occur in combination with reading difficulty (Geary, 1993), for which phonological deficits are well
established (e.g., Bruck, 1992). Evidence supporting this link is
inconsistent. For example, Fuchs et al. (2005) found supportive
results when predicting the development of arithmetic skill from
fall to spring of first grade—that is, we noted that phonological
processing emerged as the only unique predictor of arithmetic
skill, besides attention, when we controlled for a host of competing
variables, including reading. By contrast, H. L. Swanson and
Beebe-Frankenberger (2004) identified reading, rather than phonological memory, as a correlate of calculation skill among a
sample of first through third graders, but the calculation measure
combined arithmetic and algorithmic computation. Further, in examining development from fourth to fifth grade, Hecht et al.
(2001) did not identify phonological processing as a viable determinant of arithmetic.
The final two possible components are attention and long-term
memory. In previous work with first graders, attentive behavior
emerged as a potentially robust predictor of arithmetic skill (Fuchs
et al., 2005), even when a range of other cognitive characteristics
was controlled. Differences in skill at allocating attention or returning to a task after attention shifts may challenge the development of representations of number combinations in long-term
memory. At the same time, deficits in long-term memory itself
seem like a plausible determinant, given that arithmetic skill transparently depends on automatic retrieval from long-term memory
(Siegler & Shrager, 1984). H. L. Swanson and BeebeFrankenberger (2004), however, failed to substantiate the viability
of deficits in long-term memory; of course, they operationalized
calculation skill by combining arithmetic and algorithmic
computation.
Algorithmic Computation
Algorithmic computation (e.g., 247 ⫹ 196 ⫽ 443) is defined as
adding, subtracting, multiplying, or dividing whole numbers, decimals, or fractions using algorithms and arithmetic. So, difficulty
can arise from faulty procedural knowledge, in which students fail
to master algorithms, or from arithmetic deficits, which result in
errors or create a bottleneck that diverts cognitive resources away
from procedural work. For this reason, arithmetic skill is a possible
determinant of algorithmic computation.
Prior work creates the basis for hypothesizing that four other
child characteristics may help determine algorithmic computation
performance: attention, working memory, phonological processing, and long-term memory. First, because algorithmic computation involves a relatively laborious series of steps, attentive behavior (i.e., low distractibility) may enhance performance. Russell
COGNITIVE CORRELATES OF MATHEMATICS SKILL
and Ginsburg (1984) provided suggestive evidence supporting this
possibility when they compared fourth graders with mathematics
disabilities with fourth graders and third graders without mathematics disabilities. Results indicated that the algorithmic errors of
students with mathematics disabilities were similar to both of the
typically developing groups but that students with mathematics
disabilities more closely resembled their younger typically developing counterparts in the detection of those errors. This result
suggests that inattentive behavior may inhibit algorithmic computation performance, a finding documented by Ackerman and Dykman (1995) for students with reading disabilities who have and do
not have mathematics disabilities (Ackerman & Dykman, 1995)
and by Fuchs et al. (2005) when controlling for a host of competing cognitive predictors.
Second, prior work implicates working memory (e.g., Geary et
al., 1991; Hitch & McAuley, 1991; Siegel & Linder, 1984; Wilson
& Swanson, 2001; Webster, 1979), although studies do not rule out
the possibility that the role of working memory may be a result of
the need for arithmetic in solving procedural computation problems. Third, phonological processing may be required beyond that
which is involved in arithmetic—that is, algorithmic computation
may require individuals to hold phonological representations in
working memory while selecting, implementing, and monitoring
strategies for algorithmic problem solution (e.g., Brainerd, 1983;
Logie et al., 1994). Evidence for this possibility exists. For example, Hecht et al. (2001) provided evidence for the role of phonological processing in the development of general computation skill
while controlling for processing speed, reading, and vocabulary;
however, the general computation measure mixed arithmetic with
algorithmic computation items. By contrast, in controlling for a
host of cognitive variables and exclusively measuring arithmetic,
Fuchs et al. (2005) failed to substantiate such a relation. The final
cognitive candidate is long-term memory, which seems necessary
to consider given that arithmetic is involved in algorithmic computation and that algorithmic rules are stored in long-term
memory.
Arithmetic Word Problems
Arithmetic word problems (e.g., John had nine pennies. He
spent three pennies at the store. How many pennies did he have
left?) are defined as linguistically presented one-step problems
requiring arithmetic solutions. Because arithmetic is transparently
required to find solutions to these problems, it may mediate performance. In addition, because skill in manipulating numbers
procedurally should enhance understanding about numerical relations, algorithmic computation may facilitate skill in the solving of
arithmetic word problems. Moreover, given the involvement of
language as well as the need to construct a problem model before
a solution can occur, a host of possible cognitive characteristics
may 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 arithmetic 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 arithmetic skills manifest working memory deficits. Second, children
with learning disabilities experience concurrent difficulty with
31
working memory (e.g., De Jong, 1998; Siegel & Ryan, 1989;
Swanson, Ashbaker, & Sachse-Lee, 1996) and mathematical problem 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 problems
involve construction of a problem model, which appears to require
working memory capacity. For example, according to Kintsch and
Greeno, in the solution of arithmetic word problems, new sets are
formed online as the story is processed. When a proposition that
triggers a set-building strategy is completed, the appropriate set is
formed and the relevant propositions are assigned places in the
schema. As new sets are formed, previous sets that had been active
in 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, characterized as good or poor problem solvers, differed on working memory
tasks. Other researchers have found corroborating evidence using
similar methods (e.g., LeBlanc & Weber-Russell, 1996; Passolunghi & Siegel, 2004; H. L. Swanson & Sachse-Lee, 2001). At the
same time, other studies have raised questions about the robustness
of the relation. For example, among typically developing third and
fourth graders, H. L. Swanson, Cooney, and Brock (1993) found
only a weak relation between working memory and problemsolution accuracy, and this relation disappeared once reading comprehension was considered. This finding suggests the need for
additional work that considers a larger pool of cognitive processes.
The other leading candidates for cognitive processes that mediate arithmetic word problems are long-term memory, attention,
nonverbal problem solving, language ability, reading skill, and
concept formation. Long-term memory appears to be implicated
given the need to access mathematics facts from long-term memory to solve arithmetic word problems. H. L. Swanson and BeebeFrankenberger (2004) provided provocative data on this possibility, showing that long-term memory explained a statistically
significant 8% of variance in arithmetic word problems. Of course,
long-term memory was indexed as recognition of mathematics
procedures needed for solving word problems, and such a measurement strategy may ensure a strong relation between long-term
memory and arithmetic word problems.
In studies involving attention, most work has focused on the
inhibition of irrelevant stimuli, with mixed results. Passolunghi,
Cornoldi, and De Liberto (1999) ran a series of studies that
suggested the importance of inhibition. For example, comparing
good and poor problem solvers, Passolunghi, Cornoldi, and De
Liberto (1999) found comparable storage capacity but also found
inefficiencies of inhibition (i.e., poor problem solvers remembered
less relevant but more irrelevant information in mathematics problems). Yet H. L. Swanson and Beebe-Frankenberger (2004) found
no evidence that inhibition contributes to arithmetic word problems. Research has, however, rarely studied the role of attention
more broadly. An exception is Fuchs et al. (2005), who found that
a teacher rating scale of attentive behavior predicted the development of first-grade skill with arithmetic word problems. Clearly,
additional work on this possibility is needed.
Nonverbal problem solving, or the ability to complete patterns
presented visually, has been identified as a unique predictor in the
development of arithmetic word problem skill across first grade
(Fuchs et al., 2005), a finding corroborated by Agness and Mc-
32
FUCHS ET AL.
Clone (1987). This finding is not surprising because arithmetic
word problems, in which the problem narrative poses a question
that entails a change, combine, compare, or equalize relationship between two numbers, appear to require conceptual
representations.
Language ability also is important to consider, given the obvious
need to process linguistic information when building a problem
representation of an arithmetic word problem. In fact, Jordan,
Levine, and Huttenlocher (1995) documented the importance of
language ability when they showed that kindergarten and firstgrade children with language impairments (i.e., with receptive
vocabulary and grammatic closure scores below the 30th percentile) performed significantly lower on arithmetic word problems
than peers with no language impairments.
Finally, it is hard to ignore the possibilities that reading skill or
concept formation may underlie skill in arithmetic word problems.
Reading is transparently involved, even when problems are read
aloud to children, because reading skill provides continuing access
to the written problem narrative after the adult reading has been
completed. This transparent involvement potentially reduces the
load 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 basis
of H. L. Swanson and Sachse-Lee (2001), who argued that information activated from long-term memory may mediate the relation
between concept formation (an aspect of executive function) and
solution accuracy for arithmetic word problems.
Considering the Role of Foundational Mathematics Skills
in Higher Order Performance
The literature cited thus far provides the basis for hypothesizing
a set of variables that may mediate each of the three aspects of
mathematics performance explored in this study. At the same time,
a strong assumption in the mathematics literature is that skills
develop hierarchically (cf. Aunola, Leskinen, Lerkkanen, &
Nurmi, 2004). With respect to the skills targeted in the present
study, arithmetic appears foundational to algorithmic computation
and arithmetic word problems. It also seems plausible that skill in
manipulating numbers procedurally, as in algorithmic computation, 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 mathematics
finds support in that some of the same cognitive variables recur in
the literature across these aspects of mathematics skill. Given
assumptions about the hierarchical nature of the mathematics
curriculum, lower order mathematics skills become key targets as
determinants of more complex skills.
Therefore, it is unfortunate that in most studies examining
cognitive determinants, researchers focus on a single aspect of
mathematics performance or explore multiple aspects of mathematics competence without considering how lower order skills
determine subsequent skills and without considering how the inclusion of lower order skills in a model may affect empirical
findings on the role of cognitive correlates in higher order performance. Two notable exceptions are Hecht et al. (2001) and H. L.
Swanson and Beebe-Frankenberger (2004). Working at Grades
2–5, Hecht et al. simultaneously considered the role of arithmetic
in general computation skill. They used structural equation mod-
eling to estimate the relations between phonological processing
and growth in mathematics computation skill, while controlling for
prior mathematics skill, reading, processing speed, and vocabulary. They found that fourth-grade arithmetic performance accounted for a small but significant percentage of variance in
fifth-grade levels of general computational skill. Of course, Hecht
et al. operationalized general computation skill as a combination of
arithmetic and algorithmic computation, making it difficult to
assess the role of arithmetic specifically to algorithmic computation. Moreover, fifth grade may not be the optimal time to assess
the development of arithmetic skill. With younger children in
Grades 1–3, H. L. Swanson and Beebe-Frankenberger explored
calculation as a determinant of arithmetic word problems, but
again, they operationalized calculation by mixing arithmetic items
with algorithmic computation problems. Results demonstrated
how calculation skill helped determine performance on arithmetic
word problems, but the design of the calculation measures precludes conclusions specific to arithmetic or to algorithmic
computation.
Rationale for the Present Study
We sought to extend the current body of literature by incorporating the three mathematics skills simultaneously into path analysis, a form of structural equation modeling that is designed to help
researchers understand how variables interrelate in complex patterns. Such analysis permitted us to estimate the role of foundational mathematics skills in higher order performance while controlling for the role of cognitive abilities and to identify unique
cognitive correlates. We tested the model shown in Figure 1,
which was based on the hypotheses we derived from previous
work (as described in the Prior Work section). Consistent with the
literature we reviewed, the model specified the following: First,
arithmetic would be associated with working memory, processing
speed, phonological decoding (a proxy for phonological processing, given that we targeted third grade), attentive behavior, and
long-term memory. Second, algorithmic computation would be
associated with arithmetic, attentive behavior, phonological decoding, long-term memory, and working memory. Third, arithmetic
word problems would be associated with arithmetic, algorithmic
computation, working memory, inattentive behavior, long-term
memory, nonverbal problem solving, language ability, reading
skill, and concept formation. Explicit in this model is the hierarchy
of mathematics skills, whereby arithmetic skill helps explain competence in algorithmic computation and arithmetic word problems
and whereby algorithmic computation skill mediates arithmetic
word problem performance.
Method
Participants
The data described in this article were collected as part of a prospective
4-year study in which we assessed the effects of mathematics problemsolving instruction and examined the developmental course and cognitive
predictors of mathematics problem solving. The data reported in the
present article were collected with the 1st-year sample at the first assessment wave, when we sampled participants from 30 third-grade classrooms
in six Title 1 schools and one non–Title 1 school (two to six teachers per
school) in a southeastern metropolitan school district.
COGNITIVE CORRELATES OF MATHEMATICS SKILL
33
Figure 1. Hypothesized paths.
Students were identified for participation on the basis of their performance on the Test of Computational Fluency (Fuchs, Fuchs, Hamlett, &
Appleton, 2002), which provides students 3 min to write answers to 25
second-grade addition and subtraction arithmetic and algorithmic computation problems. Of the 494 children from whom we had parental consent
and student assent (from a total of 499 students), we randomly sampled 330
students for individual participation, blocking within three strata (selecting
25% of students with scores one standard deviation below the mean of the
entire distribution, 50% of students with scores within one standard deviation of the mean of the entire distribution, and 25% of students with scores
one standard deviation above the mean of the entire distribution). Of these
330 students, we have complete data on 312 children for the variables
reported here. As measured on the two-subset Wechsler Abbreviated Scale
of 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 school
district, averaged 57.44 (SD ⫽ 18.02) for the reading composite and 59.31
(SD ⫽ 21.47) for the mathematics composite. Their standard scores on the
Woodcock-Johnson Psycho-Educational Battery–Revised (WJ III; Woodcock, McGrew, & Mather, 2001) Applied Problems averaged 100.70
(SD ⫽ 14.88), and their standard scores on the Woodcock Reading Mastery
Test–Revised (WRMT-R; Woodcock, 1998) Word Identification averaged
100.90 (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-identified
disability (i.e., learning disability, speech impairment, language impairment, attention-deficit/hyperactivity disorder, health impairment, or
emotional– 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, three
whole-class assessment sessions occurred, each lasting 30 – 60 min. In
September and October 2003, two 45-min individual testing sessions
occurred. In this report, we describe only the subset of measures on which
we reported data. The whole-class measures were Assessment of Math Fact
Fluency test of the Grade 3 Math Battery (Fuchs, Hamlett, & Powell,
2003), Double-Digit Addition and Subtraction test of the Grade 3 Math
Battery (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 were
Test 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), WoodcockJohnson III Tests of Achievement (WJ III) Concept Formation (Woodcock
et al., 2001), WJ III Visual Matching (Woodcock et al., 2001), WJ III
Retrieval Fluency (Woodcock & Johnson, 1989), Working Memory Test
Battery–Children (WMTB-C) Listening Recall (Pickering & Gathercole,
2001), WJ III Numbers Reversed (Woodcock & Johnson, 1989), WRMT-R
Word 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-
34
FUCHS ET AL.
strated 100% accuracy during mock administrations. All individual sessions were audiotaped, and 17.9% of tapes, distributed equally across
testers, were selected randomly for accuracy checks by an independent
scorer. Agreement was between 98.8% and 99.8%. In October, teachers
completed the SWAN Rating Scale (H. L. Swanson et al., 2004) on each
student.
Measures
Language. We used three tests of language. The first test, TOLD
Grammatic Closure, measures the ability to recognize, understand, and use
English morphological forms. The examiner reads 30 sentences, one at a
time; each sentence has a missing word. Examinees earn 1 point for each
sentence correctly completed. As reported by the test developer, reliability
is .88 for 8-year-olds; the correlation with the Illinois Test of Psycholinguistic Ability Grammatic Closure (Kirk, McCarthy, & Kirk, 1986) is .88
for 8-year-olds.
The second test, WDRB Listening Comprehension (Woodcock, 1997),
measures the ability to understand sentences or passages. The test presents
38 items, and students supply the word missing from the end of each
sentence or passage. WDRB Listening Comprehension begins with simple
verbal analogies and associations and progresses to comprehension involving the ability to discern implications. Testing is discontinued after six
consecutive errors. The score is the number of correct responses. Reliability 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 expressive vocabulary, verbal knowledge, and foundation of information with
42 items. The first 4 items present pictures; the student identifies the object
in the picture. For the remaining items, the tester says a word that the
student defines. Responses are awarded a score of 0, 1, or 2, depending on
quality of response. Testing is discontinued after five consecutive scores of
0. 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 the
Wechsler 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: pattern
completion, classification, analogy, and serial reasoning. Examinees look
at a matrix from which a section is missing and complete the matrix by
saying the number of options or pointing to one of five response options.
Examinees earn points by identifying the correct missing piece of the
matrix. Testing is discontinued after four errors on five consecutive items
or after four consecutive errors. The score is the number of correct
responses. As reported by the test developer, reliability is .94 for 8-yearolds; 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 shown
illustrations of instances and noninstances of the concept. Examinees earn
credit by correctly identifying the rule that governs each concept. Cutoff
points 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 two
identical numbers that appear in a row of six numbers; examinees have 3
min to complete 60 rows and earn credit by correctly circling the matching
numbers in each row. As reported by the test developer, reliability is .91 for
8-year-olds.
Long-term memory. WJ III Retrieval Fluency (Woodcock & Johnson,
1989) measures long-term memory by asking examinees to recall related
items, within categories, for 1 min per category. They earn credit for each
nonduplicated answer. As reported by the test developer, reliability is .78
for 8-year-olds.
Working memory. We used two measures of working memory. With
WMTB-C Listening Recall (Pickering & Gathercole, 2001), the tester says
a series of short sentences, only some of which make sense. The student
indicates whether each sentence is true or false. After hearing all sentences
in a trial (i.e., 1– 6 sentences) and determining them to be true or false, the
student recalls the final word of each sentence, in the order presented. The
student earns 1 point for each sequence of final words recalled correctly in
the right order, and the score is the total of correct sequences. Testing is
discontinued when the student makes three or more errors in any block of
items. As reported by Pickering and Gathercole, test–retest reliability is
.93. With WJ III Numbers Reversed (Woodcock & Johnson, 1989), the
tester says a string of random numbers, and the student says the series
backward. Item difficulty increases as more numbers are added to the
series. Examinees earn credit by repeating the numbers correctly in the
opposite order. As reported by the test developer, reliability is .86 for
8-year-olds.
Attentive behavior. The SWAN (J. Swanson et al., 2004) is an 18-item
teacher rating scale. Items from the Diagnostic and Statistical Manual of
Mental Disorders (4th ed.; American Psychiatric Association, 1994) criteria for attention-deficit/hyperactivity disorder are included for inattention
(largely, distractibility; Items 1–9) and hyperactivity/impulsivity (Items
10 –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 ⫽ far
above). In the present study, we report data for the Inattentive Behavior
subscale as the average rating per item across the nine relevant items. We
selected this subscale to operationalize inattentive behavior, or reduced
ability to maintain focus of attention. The SWAN has been shown to
correlate well with other dimensional assessments of behavior related to
inattention (J. Swanson et al., 2004). Coefficient alpha in the present study
was .97.
Phonological decoding. WRMT-R Word Attack (Woodcock, 1998)
measures phonetic reading ability; it comprises 45 pseudowords (or very
low-frequency words), arranged in order of difficulty. Two practice items
are used to train students. Testing is discontinued after six consecutive
errors. The score is the number of words pronounced correctly. As reported
by Woodcock (1998), split-half reliability is .94.
Reading. In TOWRE Sight Word Efficiency (Torgesen, Wagner, &
Rashotte, 1999), testers assess sight word reading fluency by asking
examinees to read a list of real words in 45 s. Examinees earn 1 point for
each 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 3
Math Battery (Fuchs et al., 2003) incorporates two subtests. The first
subtest, Addition Fact Fluency, comprises 25 addition fact problems with
answers from 0 to 12, presented horizontally on one page. Students have 1
min to write their answers. The second subtest, Subtraction Fact Fluency,
comprises 25 subtraction fact problems with answers from 0 to 12, presented horizontally on one page. Students have 1 min to write their
answers. The score is the number of correct answers across both subtests.
Percentage of agreement, calculated on 20% of protocols by two independent scorers, was 97.9. Coefficient alpha on this sample was .92. Criterion
validity with the previous spring’s TerraNova (CTB/McGraw-Hill, 1997)
Total Math score was .52.
Algorithmic computation. The Double-Digit Addition and Subtraction
test of the Grade 3 Math Battery (Fuchs et al., 2003) comprises two
subtests. The first subtest, Addition, provides students with 5 min to
complete twenty 2-digit by 2-digit addition problems with and without
regrouping. The second subtest, Subtraction, provides students with 5 min
to complete twenty 2-digit by 2-digit subtraction problems with and without regrouping. The score is the number of correct answers across both
subtests. Percentage of agreement, calculated on 20% of protocols by two
independent 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.
COGNITIVE CORRELATES OF MATHEMATICS SKILL
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 problems 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. The
score is the number of correct answers. A second scorer independently
rescored 20% of protocols, with agreement of 99.8%. Coefficient alpha on
this sample was .83. Criterion validity with the previous spring’s TerraNova (CTB/McGraw-Hill, 1997) Total Math score was .66.
Data Analysis and Results
For constructs in which we had more than one measure available, we created weighted composite variables using a principalcomponents factor analysis across the variables in that conceptually related set. This was the case for language (in which we
created a weighted composite score across TOLD Grammatic
Closure, WDRB Listening Comprehension, and WASI Vocabulary) and working memory (in which we created a weighted
composite score across WMTB Listening Recall and WJ III Numbers Reversed; because each principal-components factor analysis
yielded only one factor, no rotation was necessary.) For other
constructs, only one measure was available: attentive behavior
(SWAN), processing speed (WJ III Visual Matching), long-term
memory (WJ III Retrieval Fluency), concept formation (WJ III
Concept Formation), nonverbal problem solving (WASI Matrix
Reasoning), phonological decoding (WRMT-R Word Attack),
sight word recognition skill (TOWRE Sight Word Efficiency), as
well as the three mathematics measures. In Table 1, we show raw
and standard score means and standard deviations along with
correlations.
Because we had only one measure available for all but two
constructs, we could not use latent variable structural equation
modeling and instead used path analysis, another form of structural
equation modeling. To conduct path analysis, we converted variables to z scores. Then, using the statistical software LISREL 8.5
(Jöreskog & Sörbom, 1993), we normalized the data and tested the
model with path analysis. Figure 2 shows the results, with statistically significant paths in bold. Beta and t values are shown along
the arrows. The chi-square was statistically significant, ␹2(11, N ⫽
312) ⫽ 19.97, p ⫽ .046, but the model fit data were supportive of
the hypothetical model shown in Figure 1: root-mean-square residual (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 variance
in arithmetic, algorithmic computation, and arithmetic story problems, respectively. For arithmetic, the significant predictors were
attentive behavior, phonological decoding, and processing speed;
for algorithmic computation, the only significant paths were arithmetic and attentive behavior; and for arithmetic word problems,
the significant predictors were arithmetic, attentive behavior, nonverbal problem solving, concept formation, sight word efficiency,
and language. These results support the hypothesized model because different cognitive skills predict different mathematics competencies, and the mathematics competencies are hierarchically
related.
A surprising finding was that working memory did not emerge
as a significant predictor anywhere in the model. Because previous
35
work suggests that reading or reading-related processes may influence the relations among cognitive abilities and arithmetic
(Fuchs et al., 2005), arithmetic or algorithmic computation (H. L.
Swanson & Beebe-Frankenberger, 2004), and arithmetic word
problems (H. L. Swanson & Beebe-Frankenberger), we ran a
second, nested analysis with the paths for phonological decoding
and sight word efficiency set to zero (see Figure 3). The goal was
to determine whether working memory contributed to arithmetic
skill when these variables were not controlled. With the paths for
phonological 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 model
accounted for 32%, 47%, and 51% of the variance in arithmetic,
algorithmic computation, and arithmetic story problems, respectively. The difference in models, with and without phonological
decoding and sight word efficiency, was significant, ⌬␹2(3, N ⫽
312) ⫽ 12.74, p ⬍ .01, indicating that phonological decoding and
sight word efficiency cannot be removed from the model without
significantly decreasing the overall fit. Moreover, with and without
phonological decoding and sight word efficiency in the model, the
significance and magnitude of the remaining paths were similar
with two exceptions: With the paths for phonological decoding and
sight word efficiency set to zero, working memory emerged as a
significant correlate of arithmetic and of arithmetic word problems
(but not of algorithmic computation). The fact that working memory may not contribute uniquely to mathematics competencies
independent of phonological processing also has been observed in
research on word recognition processes (Shankweiler & Crain,
1986). Of course, it is also possible that working memory is
already captured within some of the cognitive abilities simultaneously entered within the model.
Because phonological decoding and sight word efficiency are
related to oral language (e.g., in the present study, r ⫽ .42 and .47),
we ran a third, nested analysis, this time with phonological decoding and sight word efficiency in the model but with the path for
language 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 arithmetic story problems, respectively. The difference in models, with
and without language, was significant, ⌬␹2(1, N ⫽ 312) ⫽ 10.76,
p ⬍ .01, indicating that language cannot be removed without a
significant decrease in the overall fit. With and without language
in the model, the significance and magnitude of the remaining
paths were similar, and working memory was not a significant
predictor anywhere in the model.
Finally, because the cognitive correlates specified in our hypothesized model were limited to attributes identified as potentially important in prior work, the model did not consider additional processes that seem interesting and viable. With this in
mind, we attempted to extend understanding for algorithmic computation, in which only arithmetic and attentive behavior emerged
as correlates, by assessing an exploratory model that added nonverbal problem solving and concept formation as paths to algorithmic computation. These constructs seem potentially viable
given that conceptual understanding of place value and the base-10
system 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%
Language factor
TOLD Grammatic Closure
WDRB Listening Comprehension
WASI Vocabulary
Concept formation
Nonverbal problem solving
Attention
Processing speed
Long-term memory
Working memory factor
WMTB Listening Recall
WJ III Numbers Reversed
Phonological decoding
Sight word efficiency
Arithmetic
Algorithms
Story problems
0.05
18.94
21.32
28.08
17.00
15.46
36.71
32.14
497.00
0.03
9.76
8.97
23.54
55.11
19.39
24.58
10.03
M
0.99
6.78
3.82
6.49
7.08
6.59
12.66
5.12
3.67
1.00
3.39
2.80
9.46
11.74
8.76
8.70
3.38
SD
91.05
93.54
110.39
103.35
101.04
93.96
86.15
96.89
47.32
94.97
48.24
M
—
—
—
—
—
—
19.79
14.07
61.24
10.84
15.17
14.36
11.83
17.49
10.38
13.77
11.64
SD
Standard scoreb
—
.85
.87
.84
.55
.40
.45
.30
.36
.46
.48
.29
.47
.42
.39
.37
.57
1
—
.62
.55
.44
.34
.36
.22
.24
.43
.46
.28
.45
.34
.28
.28
.47
2
—
.61
.34
.20
.39
.28
.31
.37
.43
.18
.37
.34
.39
.33
.53
3
—
.47
.34
.41
.26
.38
.38
.35
.29
.40
.39
.34
.33
.45
4
—
.41
.43
.30
.22
.43
.45
.28
.37
.34
.38
.42
.52
5
—
.35
.31
.21
.32
.30
.25
.31
.20
.29
.32
.45
6
—
.42
.18
.29
.26
.23
.41
.44
.44
.60
.51
7
—
.35
.19
.19
.15
.16
.35
.48
.39
.28
8
—
.20
.16
.16
.03
.21
.19
.20
.17
9
—
.80
.82
.47
.38
.26
.27
.39
10
—
.35
.39
.35
.26
.28
.39
11
—
.40
.26
.22
.21
.26
12
—
.60
.33
.35
.44
13
—
.38
.41
.38
14
—
.56
.46
15
—
.40
16
—
17
Note. TOLD ⫽ Test of Language Development–Primary; WDRB ⫽ Woodcock Diagnostic Reading Battery; WASI ⫽ Wechsler Abbreviated Scale of Intelligence; WMTB ⫽ Working Memory Test
Battery; WJ III ⫽ Woodcock-Johnson Psycho-Educational Battery.
a
See Measures section for information about how raw scores were calculated. b Standard scores have a mean of 100 and a standard deviation of 15, except for WASI Vocabulary and WASI Matrix
Reasoning (i.e., nonverbal problem solving), where t scores are used with a mean of 100 and a standard deviation of 10.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Variable
Raw scorea
Correlation
Table 1
Means, Standard Deviations, and Correlations Among Cognitive, Reading, and Math Variables (N ⫽ 312)
36
FUCHS ET AL.
COGNITIVE CORRELATES OF MATHEMATICS SKILL
Figure 2.
37
Hypothesized model: Path coefficients (t values), with bold signifying statistically significant paths.
of the variance in arithmetic, algorithmic computation, and arithmetic story problems, respectively (see Figure 5). Because this
model was not nested, we could not compare it with a competing
model. Instead, we included it only for exploratory purposes. We
observed that neither nonverbal problem solving nor concept formation was statistically significant, and with and without these two
additional predictors in the model, the significance and magnitude
of the remaining paths were similar.
In sum, the significant predictors for arithmetic were attentive
behavior, phonological decoding, and processing speed; the significant predictors for algorithmic computation were arithmetic
and attentive behavior; and the significant predictors for arithmetic
word problems were arithmetic, attentive behavior, nonverbal
problem solving, concept formation, sight word efficiency, and
language. Although working memory was not a significant path in
the overall model, it was a significant predictor of arithmetic and
arithmetic word problems when the paths for reading and phonological processing were set to zero, suggesting that reading or
reading-related processes may influence the relations among working memory and at least two aspects of mathematics skill.
Discussion
With respect to the hierarchical nature of mathematics development, we found that the three mathematics skills constituted a
partial hierarchy. That is, arithmetic was a significant path to
algorithmic computation and to arithmetic word problems. At the
same time, although arithmetic skill was independently linked to
algorithmic computation and to arithmetic word problems, algorithmic computation was not a significant predictor of arithmetic
word problem performance. This finding suggests that skill in
manipulating numbers procedurally, as done in algorithmic computation, may not correspond to the capacity to conceptualize
relations among numbers, at least when those relations are conveyed via language, as is the case with arithmetic word problems.
Arithmetic word problems do not require algorithmic computation
for solution, and one might anticipate that with problem solving
that requires algorithmic computation, a hierarchy that includes
algorithmic computation as a predictor of performance may be
tenable. In a similar way, the need for arithmetic skill within
algorithmic computation and arithmetic word problems is transparent. However, empirical demonstration of these paths, even
when controlling for multiple cognitive abilities and reading performance, strengthens the notion that fluency with single-digit
addition and subtraction is foundational for at least some aspects of
subsequent mathematics skill. This finding also suggests the need
for early intervention for promotion of the development of skills
with number combinations.
With respect to the cognitive correlates of arithmetic skill,
results revealed its unique and direct relations with attentive behavior, processing speed, and phonological decoding. In fact,
38
FUCHS ET AL.
Figure 3. Model with phonological decoding and sight word efficiency removed: Path coefficients (t values),
with bold signifying statistically significant paths.
teacher ratings of attentive behavior were the most robust predictor
in this study—as the only variable that independently accounted
for variance in all three aspects of mathematics skill. Moreover, in
the case of algorithmic computation, for which serial execution of
tasks is required, attentive behavior was the only unique correlate
(beyond arithmetic). Although few studies have considered attentive behavior as a correlate of mathematics skill, previous work
does suggest a role. For example, Ackerman and Dykman (1995)
documented differences in attentive behavior among reading disabled students with and without mathematics disability, and Badian (1983) described a boy with an attention deficit who learned
multiplication facts only after receiving pharmacological treatment. Also, Fuchs et al. (2005), using a different scale than the one
used in the present study, showed that teacher ratings of inattentive
behavior in the fall semester uniquely predicted development of a
range of first-grade mathematics skills.
Present findings again suggest the critical role that attentive
behavior may play, and several explanations are possible. Low
distractibility, or the ability to focus attention, may create the
opportunity to persevere with academic tasks, especially those
tasks requiring serial execution, such as some aspects of mathematics (Luria, 1980). Alternatively, instruction may fail to address
the needs of children with poor mathematics potential, which is
determined by other deficits, and this mismatch between needs and
instruction creates inattentive behavior, which teachers observe.
Another possibility is that teacher ratings of attentive behavior are
clouded by students’ academic performance and therefore serve as
a proxy for achievement rather than index attention and/or distractibility. Finally, attentive behavior may in fact represent a
critical cognitive determinant. Our data do not provide the basis for
distinguishing among these explanations but instead for hypothesizing that inattentive behavior may play a critical role and for
exploring the underlying nature of the relation. In any case, findings do suggest that ratings of attentive behavior may serve to
screen children for risk of mathematics difficulty, and future work
should explore this possibility.
Specifically in terms of arithmetic, other unique predictors in
addition to inattentive behavior were processing speed and phonological decoding. For processing speed, our findings corroborate
the work of Bull and Johnston (1997). While controlling for word
reading ability, item identification, and short-term memory, Bull
and Johnston found that processing speed subsumed all of the
variance in 7-year-olds’ arithmetic skill. Processing speed may
facilitate counting speed so that as young children gain speed in
counting sets to figure sums and differences, they successfully pair
problems with their answers in working memory before decay sets
in, 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 to
arithmetic is interesting in that fact-retrieval deficits often occur
concurrently 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
COGNITIVE CORRELATES OF MATHEMATICS SKILL
39
Figure 4. Model with language removed: Path coefficients (t values), with bold signifying statistically
significant paths.
study, given that we targeted third grade, we used phonological
decoding as a proxy for phonological processing. This is a potential limitation of our study, and additional work that includes direct
measures of phonological processing should be pursued. In the
meantime, however, our findings suggest that phonological processing may mediate deficits in word identification and fact retrieval (cf. Geary, 1993). This possibility gains favor given that
phonological systems are engaged when children use phonological
name codes of numbers to count (Logie & Baddeley, 1987) and
that counting skill, in turn, appears critical to the development of
arithmetic 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 of
the development of first-grade arithmetic skill but not of other
aspects of mathematics skill, even when basic reading skill was
controlled. Hecht et al. (2001) demonstrated that phonological
processing almost completely accounted for the association between reading and computational skill in older children. The
present findings show a direct path between phonological decoding and arithmetic among third graders, even when seven other
abilities were controlled, providing additional suggestive evidence
that phonological processing underlies arithmetic skill. In reading,
the transparent connection between phonological processing and
decoding skill provides the basis for instructional design. In arithmetic, 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 the
value of speeded oral practice in counting to derive number fact
answers or to explore whether instruction in word-level skills may
transfer to improved arithmetic competence.
In terms of arithmetic word problems, four measures beyond
arithmetic and in addition to attentive behavior emerged as unique
correlates: nonverbal problem solving, concept formation, sight
word efficiency, and language. With arithmetic word problems,
students work conceptually with numbers: They listen to brief
scenarios while reading along on paper; each story poses a question that entails a change, combine, compare, or equalize relationship between two numbers, involving a sum or minuend of 9 or
less. 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 that
language ability or concept formation should play an important
role, even though relatively few studies have examined these
possibilities (see Jordan et al., 1995, on the relation between
language and arithmetic word problems). At the same time, the
emergence of nonverbal problem solving as a unique correlate of
arithmetic word problems is interesting. This finding corroborates
previous work at first grade (Fuchs et al., 2005) and third grade
(H. L. Swanson & Beebe-Frankenberger, 2004). Clearly, arithmetic word problems, in which the problem narrative poses a
FUCHS ET AL.
40
Figure 5.
Extended model: Path coefficients (t values), with bold signifying statistically significant paths.
question entailing a change, combine, or equalize relation between
two numbers, requires problem solving. In addition, word recognition skill seems transparently involved in arithmetic word problems, even when problems are read aloud to children, because
word recognition skill provides continuing access to the written
problem narrative after the adult reading has been completed.
Therefore, it is not surprising to find that sight word efficiency
mediates competence with arithmetic word problems.
And what about working memory? With all variables in the
model, including the three mathematics skills, phonological decoding, sight word efficiency, and seven cognitive abilities, working memory did not emerge as a significant predictor of any of the
three mathematics skills considered. This finding contradicts previous work showing the importance of working memory to arithmetic 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 examined working memory as it relates to a single mathematics skill,
without simultaneous consideration of other cognitive abilities and
other 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 arithmetic
word problems, including mathematics calculation skill (opera-
tionalized as arithmetic and algorithmic computation), and found
that in both mathematics areas, working memory accounted for
unique variance. Consequently, it is important to note that we
operationalized working memory with a particular set of measures,
assessing memory span for language stimuli as well as for backward digit span. Although these measures are well accepted for the
indexing of working memory, it is possible that different instruments would reveal working memory as a significant predictor.
Moreover, it is interesting to consider that working memory did
emerge as a significant path for arithmetic and for arithmetic word
problems when the paths for phonological decoding and sight
word efficiency were set to zero. This finding corroborates the
hypothesis that reading or reading-related processes may influence
the relations between cognitive abilities and arithmetic (Fuchs et
al., 2005) as well as between cognitive abilities and arithmetic
word problems (H. L. Swanson & Beebe-Frankenberger, 2004).
Phonological decoding or sight word efficiency may serve as a
proxy for phonological processing, and efficient encoding and
maintenance of phonological information in working memory
should enable children to build accurate arithmetic facts in longterm memory (Siegler & Shipley, 1995; Siegler & Shrager, 1984)
and to devote maximum attentional resources to the building of a
problem model for arithmetic word problems (cf. Kintsch &
Greeno, 1985). Swanson and Beebe-Frankenberger (2004) examined the role of phonological memory within the working memory
COGNITIVE CORRELATES OF MATHEMATICS SKILL
system while considering its relation to calculation and arithmetic
word problems and found no significant relation. Therefore, future
work should continue to examine the interplay among phonological processing, phonological decoding, sight word efficiency, and
working memory in determining skill in arithmetic and arithmetic
word problems.
At the same time, even when the paths for phonological decoding and sight word efficiency were set to zero in the model,
working memory was not a unique correlate of algorithmic computation. This finding is at odds with H. L. Swanson and BeebeFrankenberger (2004), even though they did include reading in
their model. It is interesting to consider how three study design
features may help explain these conflicting results. First, whereas
the present study sampled a broad distribution of children on
written arithmetic and algorithmic computation, H. L. Swanson
and Beebe-Frankenberger selected groups that differed on working
memory and orally presented arithmetic facts; this difference may
have increased the salience of working memory to their findings.
Second, in the present study, students had a written copy of the
arithmetic word problems as the examiner read problems aloud
and as they worked (they could also request rereadings). H. L.
Swanson and Beebe-Frankenberger instead read story problems
aloud to participants, who answered without referring back to the
problems. This type of response may require working memory
capacity beyond that which is needed for typical arithmetic word
problem tasks, in and out of school, in which access to the problem
situation remains beyond an initial, oral presentation. Third, the
present study separated arithmetic from algorithmic computation;
by contrast, H. L. Swanson and Beebe-Frankenberger combined
these skills into a single measure, thus creating the possibility that
the results reflected working memory’s salience for arithmetic, not
for algorithmic computation. In any case, additional work is warranted that examines the role of working memory in various
aspects of mathematics performance when multiple cognitive abilities and reading and mathematics skills are simultaneously
considered.
Before closing, we note that the only unique predictors of
algorithmic computation within our hypothesized model were attentive behavior and arithmetic. Therefore, working memory,
long-term memory, and phonological decoding do not appear to
mediate algorithmic computation performance. For this reason, we
considered a competing model that added nonverbal problem
solving and concept formation as possible determinants. These
constructs seem viable given that conceptual understanding of
place value and the base-10 system may enhance performance.
Within this exploratory model, the path for concept formation did
approach statistical significance. Our measure of concept formation asked students to identify the rules for concepts concerning
color, shape, and size when they were shown illustrations of
instances and noninstances of the concept. Future work might
build on present findings by incorporating a related measure that
taps concept formation in a manner better related to place value.
Results of the present study extend knowledge about the relations among three key aspects of third-grade mathematical cognition. Findings suggest a partial hierarchy of skills, with arithmetic
significantly predicting algorithmic computation and arithmetic
word problems and without algorithmic computation accounting
for variance in arithmetic word problems. In addition, our results
highlight the potential importance of attentive behavior across the
41
three aspects of third-grade mathematics competence we studied,
even as the findings reveal critical differences in the cognitive
abilities that mediate these various mathematics skills. In this way,
results suggest that aspects of mathematical cognition may be
distinct. We note that our cognitive and academic correlates left a
fair amount of variance to be explained. Also, we emphasize the
concurrent nature of our data collection, which precludes conclusions about causation, and we remind readers that findings may
depend on instrumentation. Future work might broaden the search
for cognitive determinants and extrastudent variables, even as it
adopts a longitudinal framework and explores additional aspects of
mathematical cognition at varying grade levels. Present findings in
combination with future related work should set the stage for the
development of integrated theory about how cognitive abilities
operate in coordinated fashion to simultaneously explain different
aspects of mathematics skill.
References
Ackerman, P. T., & Dykman, R. A. (1995). Reading-disabled students with
and without comorbid arithmetic disability. Developmental Neuropsychology, 11, 351–371.
Agness, P. J., & McClone, D. G. (1987). Learning disabilities: A specific
look at children with spina bifida. Insights, 8 –9.
American Psychiatric Association. (1994). Diagnostic and statistical manual of mental disorders (4th ed.). Washington, DC: Author.
Aunola, K., Leskinen, E., Lerkkanen, M., & Nurmi, J. (2004). Developmental dynamics of math performance from preschool to grade 2.
Journal of Educational Psychology, 96, 699 –713.
Badian, N. A. (1983). Dyscalculia and nonverbal disorders of learning. In
H. R. Myklebust (Ed.), Progress (pp. 235–264). New York: Grune &
Stratton.
Brady, S. A., & Shankweiler, D. P. (Eds.). (1991). Phonological processes
in literacy. Hillsdale, NJ: Erlbaum.
Brainerd, C. J. (1983). Young children’s mental arithmetic errors: A
working memory analysis. Child Development, 54, 812– 830.
Bruck, M. (1992). Persistence of dyslexics’ phonological awareness deficits. Developmental Psychology, 28, 874 – 886.
Bull, R., & Johnston, R. S. (1997). Children’s arithmetical difficulties:
Contributions from processing speed, item identification, and short-term
memory. Journal of Experimental Child Psychology, 65, 1–24.
Butterfield, E. C., & Ferretti, R. P. (1987). Toward a theoretical integration
of cognitive hypotheses about intellectual differences among children. In
J. G. Borkowski & J. D. Day (Eds.), Cognition in special children:
Comparative approaches to retardation, learning disabilities, and giftedness (pp. 195–233). Norwood, NJ: Ablex Publishing.
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and
subtraction concepts in grades one through three. Journal of Research in
Mathematics Education, 15, 179 –203.
Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Selfregulated strategy development. The Journal of Special Education, 26,
1–19.
Case, R. (1985). Intellectual development: Birth to adulthood. San Diego,
CA: Academic Press.
CTB/McGraw-Hill. (1997). TerraNova technical manual. Monterey, CA:
Author.
Daneman, M., & Carpenter, P. A. (1980). Individual differences in working
memory and reading. Journal of Verbal Learning and Verbal Behavior,
19, 450 – 466.
De Jong, P. F. (1998). Working memory deficits of reading disabled
children. Journal of Experimental Child Psychology, 70, 75–96.
42
FUCHS ET AL.
Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., &
Hamlett, C. L. (2005). The prevention, identification, and cognitive
determinants of math difficulty. Journal of Educational Psychology, 97,
493–513.
Fuchs, L. S., Fuchs, D., Hamlett, C. L., & Appleton, A. C. (2002).
Explicitly teaching for transfer: Effects on the mathematical problem
solving performance of students with disabilities. Learning Disabilities
Research and Practice, 17, 90 –106.
Fuchs, L. S., Hamlett, C. L., & Powell, S. R. (2003). Grade 3 math battery.
[Unpublished paper.] (Available from L. S. Fuchs, Department of Special Education, 328 Peabody, Vanderbilt University, Nashville, TN
37203).
Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.
Geary, D. C., & Brown, S. C. (1991). Cognitive addition: Strategy choice
and speed-of-processing differences in gifted, normal, and mathematically disabled children. Developmental Psychology, 27, 398 – 406.
Geary, D. C., Brown, S. C., & Samaranayake, V. A. (1991). Cognitive
addition: A short longitudinal study of strategy choice and speed-ofprocessing differences in normal and mathematically disabled children.
Developmental Psychology, 27, 787–797.
Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and
arithmetic cognition: A longitudinal study of processes and concept
deficits with children with learning disabilities. Journal of Experimental
Child Psychology, 77, 236 –263.
Geary, D. C., Widaman, K. F., Little, T. D., & Cormier, P. (1987).
Cognitive addition: Comparison of learning disabled and academically
normal elementary school children. Cognitive Development, 2, 249 –269.
Hecht, S. A., Torgesen, J. K., Wagner, R. K., & Rashotte, C. A. (2001).
The relations between phonological processing abilities and emerging
individual differences in mathematical computational skills: A longitudinal study from second to fifth grades. Journal of Experimental Child
Psychology, 79, 192–227.
Hitch, G. J., & McAuley, E. (1991). Working memory in children with
specific arithmetical learning disabilities. British Journal of Psychology,
82, 375–386.
Jordan, N. C., & Hanich, L. (2000). Mathematical thinking in second-grade
children with different forms of LD. Journal of Learning Disabilities,
33, 567–578.
Jordan, N. C., Levine, S. C., & Huttenlocher, J. (1995). Calculation
abilities in young children with different patterns of cognitive functioning. Journal of Learning Disabilities, 28, 53– 64.
Jordan, N. C., & Montani, T. O. (1997). Cognitive arithmetic and problem
solving: A comparison of children with specific and general mathematics difficulties. Journal of Learning Disabilities, 30, 624 – 634.
Jöreskog, K., & Sörbom, D. (1993). LISREL 8.5: Structural equation
modeling with the SIMPLIS command language. Hillsdale, NJ:
Erlbaum.
Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word
arithmetic problems. Psychological Review, 92, 109 –129.
Kirk, S. A., McCarthy, J. J., & Kirk, W. D. (1986). Illinois Test of
Psycholinguistic Abilities–Revised. Champaign: University of Illinois
Press.
LeBlanc, M. D., & Weber-Russell, S. (1996). Text integration and mathematics connections: A computer model of arithmetic work problem
solving. Cognitive Science, 20, 357– 407.
Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change:
Contributions to children’s learning of multiplication. Journal of Experimental Psychology: General, 124, 83–97.
Lewis, C., Hitch, G. J., & Walker, P. (1994). The prevalence of specific
arithmetic difficulties and specific reading difficulties in 9- to 10-yearold boys and girls. Journal of Child Psychology and Psychiatry, 35,
283–292.
Logie, R. H., & Baddeley, A. D. (1987). Cognitive processes in counting.
Journal of Experimental Psychology: Learning, Memory, and Cognition, 13, 310 –326.
Logie, R. H., Gilhooly, K. J., & Wynn, V. (1994). Counting on working
memory in arithmetic problem solving. Memory & Cognition, 22, 395–
410.
Luria, A. R. (1980). Higher cortical functions in man (2nd ed.). New York:
Basic Books.
Mayer, R. E. (1992). Thinking, problem solving, cognition (2nd ed.). New
York: Freeman.
Newcomer, P. L., & Hammill, D. D. (1988). Test of Language Development (Rev. ed.). Austin, TX: Pro-Ed.
Passolunghi, M. C., Cornoldi, C., & De Liberto, S. (1999). Working
memory and intrusions of irrelevant information in a group of specific
poor problem solvers. Memory & Cognition, 27, 779 –790.
Passolunghi, M. C., & Siegel, L. S. (2001). Short-term memory, working
memory, and inhibitory control in children with specific arithmetic
learning disabilities. Journal of Experimental Child Psychology, 80,
44 –57.
Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access
to numerical information in children with disability in mathematics.
Journal of Experimental Child Psychology, 88, 348 –367.
Pickering, S., & Gathercole, S. (2001). Working Memory Test Battery for
Children. London: Psychological Corporation.
Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition
and Instruction, 5, 49 –101.
Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of
children’s problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.),
The development of mathematical thinking (pp. 153–196). New York:
Academic Press.
Russell, R. L., & Ginsburg, H. P. (1984). Cognitive analysis of children’s
mathematical difficulties. Cognition and Instruction, 1, 217–244.
Shankweiler, D. P., & Crain, S. (1986). Language mechanisms and reading
disorder: A modular approach. Cognition, 24, 139 –168.
Siegel, L. S., & Linder, B. (1984). Short-term memory process in children
with reading and arithmetic disabilities. Developmental Psychology, 20,
200 –207.
Siegel, L. S., & Ryan, E. B. (1989). The development of working memory
in normally achieving and subtypes of learning disabled children. Child
Development, 60, 973–980.
Siegler, R. S. (1988). Strategy choice procedures and the development of
multiplication skills. Journal of Experimental Psychology: General,
177, 258 –275.
Siegler, R. S., & Shipley, C. (1995). Variation, selection, and cognitive
change. In G. S. Halford & T. J. Simon (Eds.), Developing cognitive
competence: New approaches to process modeling (pp. 31–76). Hillsdale, NJ: Erlbaum.
Siegler, R. S., & Shrager, J. (1984). Strategy choice in addition and
subtraction: How do children know what to do? In C. Sophian (Ed.),
Origins of cognitive skills (pp. 229 –293). Hillsdale, NJ: Erlbaum.
Swanson, H. L. (1993). An information processing analysis of learning
disabled children’s problem solving. American Educational Research
Journal, 30, 861– 893.
Swanson, H. L., Ashbaker, M. H., & Sachse-Lee, C. (1996). Learning
disabled readers’ working memory as a function of processing demands.
Journal of Experimental Child Psychology, 61, 242–275.
Swanson, H. L., & Beebe-Frankenberger, M. (2004). The relationship
between working memory and mathematical problem solving in children
at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471– 491.
Swanson, H. L., Cooney, J. B., & Brock, S. (1993). The influence of
working memory and classification ability on children’s word problem
solution. Journal of Experimental Child Psychology, 55, 374 –395.
Swanson, H. L., & Sachse-Lee, C. (2001). Mathematical problem solving
COGNITIVE CORRELATES OF MATHEMATICS SKILL
and working memory in children with learning disabilities: Both executive and phonological processes are important. Journal of Experimental
Child Psychology, 79, 294 –321.
Swanson, J., Schuck, S., Mann, M., Carlson, C., Hartman, K., Sergeant, J.,
et al. (2004). Categorical and dimensional definitions and evaluations of
symptoms of ADHD: The SNAP and the SWAN rating scales. Retrieved
December 20, 2004, from http://www.adhd.net
Torgesen, J. K., Wagner, R. K., & Rashotte, C. A. (1999). Comprehensive
Test of Phonological Processing. Austin, TX: Pro-Ed.
Wagner, R. K., Torgesen, J. K., Rashotte, C. A., Hecht, S. A., Barker,
T. A., Burgess, S. R., et al. (1997). Changing relations between phonological processing abilities and word-level reading as children develop
from beginning to skilled readers: A 5-year longitudinal study. Developmental Psychology, 33, 468 – 479.
Webster, R. E. (1979). Visual and aural short-term memory capacity
deficits in mathematics disabled students. Journal of Educational Research, 72, 272–283.
Wechsler, D. (1991). Wechsler Intelligence Scale for Children (3rd ed.)
San Antonio, TX: Harcourt Brace Jovanovich.
43
Wechsler, D. (1999). Wechsler Abbreviated Scale of Intelligence. San
Antonio, TX: Psychological Corporation.
Wilson, K. M., & Swanson, H. L. (2001). Are mathematics disabilities due
to a domain-general or a domain-specific working memory deficit?
Journal of Learning Disabilities, 34, 237–248.
Woodcock, R. W. (1997). Woodcock Diagnostic Reading Battery. Itasca,
IL: Riverside Publishing.
Woodcock, R. W. (1998). Woodcock Reading Mastery Test—Revised.
Circle Pines, MN: American Guidance Service.
Woodcock, R. W., & Johnson, M. B. (1989). Woodcock-Johnson PsychoEducational Battery–Revised. Allen, TX: DLM Teaching Resources.
Woodcock, R. W., McGrew, K. S., & Mather, N. (2001). WoodcockJohnson III Tests of Achievement. Itasca, IL: Riverside Publishing.
Zhu, J. (1999). Wechsler Abbreviated Scale of Intelligence manual. San
Antonio, TX: Author.
Received May 4, 2005
Revision received July 22, 2005
Accepted September 19, 2005 䡲