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http://dx.doi.org.ezproxy.utm.my/10.1016/j.jecp.2004.07.001
Journal of Experimental Child Psychology
Volume 89, Issue 2, October 2004, Pages 140–158
120043||
Working memory and literacy as predictors of performance on
algebraic word problems
Kerry Leea, ,
, SweeFong Ngb
, EeLynn Nga
, ZeeYing Lima
Show more
Abstract
Previous studies on individual differences in mathematical abilities have shown that
working memory contributes to early arithmetic performance. In this study, we extended
the investigation to algebraic word problem solving. A total of 151 10yearolds were
administered algebraic word problems and measures of working memory, intelligence
quotient (IQ), and reading ability. Regression results were consistent with findings from
the arithmetic literature showing that a literacy composite measure provided greater
contribution than did executive function capacity. However, a series of path analyses
showed that the overall contribution of executive function was comparable to that of
literacy; the effect of executive function was mediated by that of literacy. Both the
phonological loop and the visual spatial sketchpad failed to contribute directly; they
contributed only indirectly by way of literacy and performance IQ, respectively.
Keywords
Shortterm memory; Reading comprehension; Executive functions; Cognitive
processes; Mathematical ability; Problem solving
Introduction
Previous studies have shown that there are significant age and individual differences in
mathematical abilities. In an early study, Cockcroft (1982) reported sameage differences
that varied by the equivalence of a 7year achievement range. A more recent study
showed that the magnitude of individual differences varied across countries, with
variation in Singapore being smaller than that in most other countries (Singapore Ministry
of Education & Research & Testing Division, 2000).
Investigations into the causes of individual differences have considered a wide variety of
contributory factors such as biological (for a review, see Geary, 1993) and motivational
(e.g., Ashcraft, Kirk, & Hopko, 1998). Recently, an area of active research has focused on
the role of working memory. Working memory is involved in shortterm memory storage,
reasoning, problem solving, and other higher cognitive tasks that require simultaneous
representation and manipulation of information. In this study, we examined the relation
among working memory, reading abilities, intelligence, and children’s abilities to solve
algebraic word problems.

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Working memory
Depending on one’s theoretical perspective, the structures and processes that make up
working memory differ (for a review, see Miyake & Shah, 1999). In this study, we adopted
the model and terminology first used by Baddeley and Hitch (1974). The latest version of
Baddeley’s working memory model consists of four components: central executive,
phonological loop, visual spatial sketchpad, and an episodic buffer (Baddeley, 2000).
Both the phonological loop and the visual spatial sketchpad are shortterm storage
systems. The former is responsible for storing and rehearsing auditory information,
whereas the latter maintains visual spatial information. Episodic buffer is the newest
addition to the model. It is postulated as a structure that facilitates exchange of
information between the central executive and longterm memory. Although the functions
of the central executive are contentious (for a review, see Baddeley, 1996), it is often
conceptualized as a manager of attentional resources. It is also understood to be
involved in the execution of other higher cognitive tasks such as inhibition of
inappropriate response and planning.
Phonological storage and arithmetic performance
Some studies have suggested that phonological storage plays an important role in
arithmetic processing. For example, crosscultural studies suggest that Chinese
speakers’ superior arithmetic performance could be partially attributed to shorter sound
duration and, thus, greater memory spans for numbers (Geary, BowThomas, Liu, &
Siegler, 1996; Hoosain & Salili, 1987; Lau & Hoosain, 1999). Recent findings on the role
of phonological storage in mental calculation are consistent with these findings. Fürst
and Hitch (2000), for example, found mental addition to be disrupted by phonological
suppression (see also Adams & Hitch, 1997; Logie, Gilhooly, & Wynn, 1994).
In contrast, studies using academic or standardized measures of arithmetic find less
evidence of direct phonological contribution. Bull and Johnston (1997), for example,
administered both word and digit span tasks to 7yearolds with high versus low
arithmetic abilities. Although the two groups differed on both memory measures, these
differences were not reliable when data were adjusted for differences in reading abilities.
Gathercole and Pickering (2000a) examined the contribution of all three components of
working memory to arithmetic performance among 7yearolds. The phonological loop
measure correlated reliably with arithmetic performance. However, when variance
attributable to age and executive function were controlled, contribution from the
phonological measure was rendered unreliable. A similar pattern was found in Lehto
(1995); digit span correlated with arithmetic performance but was rendered unreliable
when variance attributable to executive function measures was controlled.
In summary, these findings suggest that the relation between phonological storage and
arithmetic performance is weaker when participants are tested using mental calculation
than when they are tested using standard mathematics achievement tests. One
explanation for this difference is that in mental calculation participants have a larger
memory load. They must remember the problem, the interim solutions, and any relevant
strategies or previous knowledge. In contrast, in standard achievement tests,
participants are generally allowed to use paper and pencil, which act as external memory
aids.
Executive function and arithmetic performance
Recently, several studies have focused on the contribution of executive function to
arithmetic performance. Bull and Scerif (2001) used a battery of executive function
measures: perseveration, inhibition efficiency, working memory span, and dual task
performance. All measures were highly correlated with performance on the Group
Mathematics Test. Working memory span provided information independent of that
provided by reading and intelligence quotient (IQ) measures. Sikora, Haley, Edwards,
and Butler (2002) examined performances on the Tower of London Test among children
and adolescents with or without arithmetic or reading difficulties. Participants with
arithmetic difficulties had poorer test scores than did those with either reading difficulties

or no difficulties. Bull, Johnston, and Roy (1999) examined the inhibition aspect of
executive function. Primary school children with poorer arithmetic performance had
higher perseverative scores on the Wisconsin Card Sorting Test than did their peers with
higher arithmetic scores. Findings from these studies suggest that executive function
plays a contributory role in arithmetic performance.
Visual spatial memory and arithmetic performance
Investigations that have examined the role of visual spatial shortterm memory have
yielded less consistent findings. Using the Corsi blocks, Bull et al. (1999) failed to find
differences among children with good versus poor arithmetic abilities. In other studies,
differences were found, but only across specific contrasts. Using a standardized working
memory battery, Gathercole and Pickering (2000b) found that children who were poor in
both English and mathematics had lower visual spatial memory spans than did their
peers from a normal achievement group. These differences extended over a number of
visual spatial tasks. However, when children who were poor only in mathematics were
compared with children in a normal achievement group, differences were found in only
one of four visual spatial tasks. McLean and Hitch (1999) found that children with poorer
arithmetic abilities had lower spatial memory span when compared with their age
matched peers but not when compared with their abilitymatched peers.
One reason for the inconsistency in findings is that there may be multiple causes for
arithmetic difficulties. In the clinical literature, it has been shown that visual spatial deficits
are implicated in only some instances of arithmetic disabilities. Rourke (1993), for
example, reviewed several of his earlier studies that contrasted children affected by
arithmetic disabilities alone with those affected by both arithmetic and reading
disabilities. Children affected by arithmetic disabilities alone exhibited visual spatial
deficits (cf. Shafrir & Siegel, 1994).
The current study
The current study examined the relation among working memory, reading abilities, and
mathematical performance. Previous studies have concentrated on numeracy skills such
as counting, number knowledge, and basic arithmetic (e.g., addition, subtraction). Less is
known about the contributions of working memory and reading abilities to the
performance of more complex mathematical problems. We extended the investigation to
algebraic word problems.
How do algebraic and arithmetic word problems differ? The most salient difference is the
complexity of relations specified in them. Take, for example, Questions 1 and 4 in the
Appendix. For the arithmetic problem (Question 1), the number of pupils in Dunearn
Primary School is a known state (Bednarz & Janvier, 1996). Enrollments at Sunshine
Primary School and Excellent Primary School are unknown. For this type of problem, the
known state allows easy entry into the problem because the unknown can be easily
found by using the stated relations to link the unknowns with the known.
In algebraic problems, the known state is a combination of the unknowns. In Question 4,
the total weight of the three animals is the known state. The individual weights of the
three animals, related to each other by comparative relations, are the unknowns. In this
problem, two types of relations must be processed simultaneously: addition and
subtraction. A fundamental difference with arithmetic problems is that entry into the
problem is by way of an unknowns. Although any of the unknown can be chosen as the
entry point into such problems, there is usually one, identified as the “generator”
(Bednarz & Janvier, 1996), that enables easier entry into the problem. An important task
faced by problem solvers is to determine the most appropriate generator. This is not a
task encountered in arithmetic word problems. Given the added complexity of algebraic
word problems, they should impose more cognitive demands than do arithmetic word
problems.
A number of studies in the psychological and educational literatures have examined
issues associated with learning and solving word problems (for reviews, see Kintsch &
Greeno, 1985; Mayer & Hegarty, 1996). However, the number of studies examining the

role of working memory is small, with most studies concentrating on arithmetic word
problems. In an earlier study, Johnson (1984) showed spatial ability, as indexed by a
mental rotation task, to be positively related to arithmetic word problemsolving ability. In
a more recent study, Swanson, Cooney, and Brock (1993) administered a number of
domain knowledge (problem classification and knowledge of operations), cognitive
(executive function and sentence recall), and reading comprehension tasks to 9year
olds. Reading comprehension and knowledge of operations provided the best prediction
of problemsolving accuracy. Executive function capacity was correlated with problem
solving accuracy but failed to provide unique contributions when other variables were
taken into account. Kail and Hall (1999) replicated these findings. They found both
phonological and executive function measures to be correlated with number of errors on
a word problem task. Again, these measures failed to provide unique contributions when
other variables—processing speed and reading skill—were taken into account. In
contrast, Passolunghi and colleagues (Passolunghi, Cornoldi, & De Liberto, 1999;
Passolunghi & Siegel, 2001) showed that even after vocabulary was controlled, poor
problem solvers exhibited lower executive function scores, as indexed by poorer
inhibitory abilities, than did good problem solvers.
Given the dearth of information directly relevant to algebraic word problems, we used a
standardized measure, the Working Memory Test Battery for Children (WMTBC)
(Pickering & Gathercole, 2001), to examine the role of working memory span. The
contribution of reading ability was measured using a comprehensive test that has been
modified and normed for the local population. Similar to previous studies, we controlled
for differences in general intellectual functions by using an abbreviated intelligence
measure.
Working memory, reading ability, and mathematical performance
A second aim of this study was to examine the interrelations among working memory
capacities, reading ability, and mathematical performance. As discussed above, several
studies have found working memory measures to be strongly correlated with
mathematical performance. However, they explained only a small amount of
performance variation when language ability or IQ was taken into account. In Bull and
Scerif (2001), for example, executive function measures correlated with mathematical
performance at the 20% level. When variances attributable to reading ability and IQ were
taken into account, executive function explained only 3% of variation in mathematical
performance. Because working memory is assumed to play a central role in information
processing, it is surprising that it had such minimal contribution to mathematical
performance.
Some recent findings suggest that the contributions of working memory may have been
underestimated. In particular, the contributions of working memory to mathematical
performance may be both direct and mediated by language abilities. The possibility of a
mediated effect was prompted by a recent review in which Baddeley, Gathercole, and
Papagno (1998) argued that the phonological loop played a critical role in language
acquisition. They found moderate to strong correlations between phonological memory
and vocabulary across early to middle childhood (Baddeley et al., 1998). Further
evidence came from crosslagged correlational data showing that preschool vocabulary
development was dependent on phonological memory (Gathercole, Willis, Emslie, &
Baddeley, 1992). The central executive is also likely to be involved in aspects of
language processing. In a review, Daneman and Merikle (1996) showed that complex
verbal working memory span tasks provided better prediction to reading than did simple
phonological span tasks.
A question of interest in the current study was whether the contributions of working
memory to mathematical performance were direct, indirect and mediated by language, or
both. Because regression analyses model only direct effects, we examined these
relations using path analyses. In addition to its theoretical interests, this question has
applied implications. Findings on the relative importance of direct and indirect
contributions can offer guidance on where best to focus educational effort. If the findings

show that language ability alone contributes directly to mathematical performance,
instructions should place more emphasis on how mathematically related terms are
mapped onto mathematical operations (e.g., “more than” does not necessarily translate
into the use of additive or multiplicative functions).
Method
Participants
A total of 151 Primary 5 (i.e., Grade 5) children (77 boys and 74 girls, average age = 10.7
years, SD = 0.65) were recruited from public schools located in the western zone of
Singapore and participated with parental consent. An a priori power analysis showed that
this sample size yields more than 95% power (statistical parameters: α = .05, medium
effect size f2
= .15, 5 predictors). Most children in these schools were drawn from middle
to lower middleclass areas.
In Singapore, Primary 5 classes were streamed according to pupils’ performances in the
languages and mathematics. The sample was roughly representative of the national
distribution. Approximately one fifth (21%) of the children were from the top stream, and
the remainder were from the middle stream. The lowest stream was not recruited
because of low enrollment and because it followed a different mathematics syllabus.
Apart from language classes, all other lessons in government schools were conducted in
English. All pupils were functionally bilingual. Apart from English, the second language
was usually Mandarin Chinese, Malay, or Tamil; for approximately half of all pupils, the
second language was actually their native language. The majority of children in this study
would have had at least 7 years of English instruction.
Materials and procedure
Children were administered a mathematical test, the WMTBC, an abbreviated IQ test,
and a reading ability test. With the exception of the mathematical test, the other three
measures were administered on a oneonone basis. Because of time constraints
imposed by the schools, we administered the working memory, reading ability, and IQ
measures according to one of two orders: (a) the reading ability and IQ measures on the
same day, with the WMTBC administered on a separate day; or (b) the WMTBC
administered first, followed by a 5min break, the reading ability test, and the IQ
measures. Approximately half of the children were administered the tests using the first
order; for the remainder, the second order was used.
Mathematical performance
The mathematical test contained 10 word problems. Question selection was guided by
the school curriculum. Because testing was conducted during the middle of the academic
year, we selected 2 questions from the Primary 4 curriculum, 6 questions from the
Primary 5 curriculum, and 2 questions from the early Primary 6 curriculum. The
Singapore mathematical curriculum can be characterized as having a spiral nature, with
material covered in this study being taught and retaught at various points of the senior
primary school years. Selecting material from Primary 4 to 6 allowed us to fully capture
individual differences in attainment. To give children an easier entry point into the test, the
first question was arithmetic in nature. The remaining questions were algebraic in nature
and tested children’s understanding of relational concepts: specifically, more than, less
than, as many as, older than, and concepts involving proportional reasoning (see the
Appendix for one full set of questions). All questions contained three objects and were of
a form identical to the questions described in the introduction.
Each child completed three parallel versions of the test. In each version, children were
asked to solve the questions in one of three ways: (a) using a pictorial heuristic called the
model method, (b) using any method but the model method, or (c) using any method.
Children were given 1 h to answer each set of 10 questions. One version was
administered each week over 3 consecutive weeks. The sequence in which the three
versions were administered was counterbalanced across schools. In this study, we
focused on averaged accuracy across the three tests. Test scores ranged from 0 to 10.

Working memory capacity
To assess children’s working memory capacity, the WMTBC (Pickering & Gathercole,
2001) was used. This battery contained nine separate measures. Three domains of
working memory were assessed: central executive, phonological loop, and visual spatial.
Each domain was measured using tasks that employed a span approach. For example,
in the Counting Recall task, which assessed central executive capacity, children were
asked to count arrays of dots presented on a series of cards. Once the dots were
counted, children were asked to recall the total number of dots on each card in the order
in which they were presented. The Digit Recall task was designed to assess the capacity
of the phonological loop. Children were asked to recall sequences of digits spoken by the
examiner in the same order as they were heard. The Mazes Memory task was designed
to assess the capacity of visual spatial storage. Children were asked to remember a route
traced by the examiner through a maze and to reproduce an identical route by drawing it
in a response booklet.
Pickering and Gathercole (2001) reported test–retest reliabilities for each subtest. Based
on a sample of 10 to 11yearolds tested on two occasions separated by an interval of 2
weeks, the coefficients for each subtest ranged from a high of .82 (Digit Recall) to a low of
.38 (Listening Recall). In this study, we reported component scores for the central
executive, phonological loop, and visual spatial domains, each derived from total scores
on the nine separate measures.
Reading ability
The Wechsler Objective Reading and Language Dimensions (Rust, 2000) was a locally
normed instrument designed to measure reading and listening skills. Because we were
primarily concerned with children’s ability to decode written information, only the reading
component was used. This component had three subtests: reading, spelling, and reading
comprehension. The reading subtest consisted of a series of printed words, and children
were required to read each word aloud. In the spelling subtest, a series of words were
dictated, and children were told to spell out each word by writing their responses on
paper. The reading comprehension subtest consisted of a series of short printed
passages and orally presented questions designed to tap skills such as recognizing
detail and making inferences. Children were asked to respond orally to each question.
Verbal and performance IQ
To control for individual differences in intelligence, we used two subtests from the
Wechsler Intelligence Scale for Children—third edition (WISCIII) (Wechsler, 1991):
vocabulary and block design. Previous work showed that among all subtests, these two
have the highest correlations with the verbal and performance IQ subscales, respectively
(Sattler, 2001).
Results
Preliminary analyses
Accuracy scores from the three versions of the mathematics test showed high
intercorrelations, rs > .82. The Kuder–Richardson20 coefficient (KR20) for each version
was more than .80, showing acceptable levels of internal reliability. Children’s
performances were less accurate when they were asked to use the model method
(M = 2.50, SD = 2.38) than when they were asked to use any method (M = 2.97,
SD = 2.63) or any method but the model method (M = 2.96, SD = 2.60), F(2, 300) = 10.24,
p < .01, η2
= .06. Details regarding differences in error patterns across the three versions
are beyond the scope of this article. Here, we reported the averaged accuracy data.
Data screening showed that the averaged accuracy score was positively skewed. This
was corrected using a logarithmic transformation. One case with an extremely low score
on the performance IQ measure was found to be a univariate outlier. This outlier was set
to a value 3 standard deviations below the mean. This allowed us to retain the extreme

a
b
*
**
observation without it having an adverse effect on either the distribution or the covariance
estimation.
Further screening showed that verbal IQ was highly correlated with the reading measure.
Tests for multicollinearity showed that their inclusion in a regression analysis would have
resulted in unreliable solutions. To attenuate this problem, a principal components
analysis was conducted with verbal IQ and the various reading subtests. A literacy factor
(eigenvalue 2.96) was derived, and factor scores were computed. This factor captured
74% of the total variance. Variable loadings for vocabulary, reading, spelling, and reading
comprehension were .87, .87, .87, and .81, respectively.
Table 1 presents the means, standard deviations, and intercorrelations among the
variables. The central executive component of the WMTBC, performance IQ, and
literacy all were positively correlated with mathematical performance. The highest
correlation was observed between the literacy and mathematical performance
measures.
Table 1.
Means, standard deviations, and intercorrelations among the study variables
Variable M SD 1 2 3 4 5
1. Mathematics abilitya
0.49 0.26
2. Phonological loop component (maximum
145)
109.26 17.84 .370**
3. Visual spatial component (maximum
145)
98.85 15.27 .395**
.206*
4. Central executive component (maximum
145)
107.66 18.84 .517**
.511**
.371**
5. Performance intelligence (maximum 69) 44.45 10.03 .534**
.293**
.397**
.349**
6. Literacy factorb
0.00 1.00 .592**
.574**
.398**
.561**
.390**
6 6a 6b 6c 6d
a. Reading (maximum 55) 40.27 5.50 .866**
b. Spelling (maximum 50) 35.28 5.07 .887**
.782**
c. Reading comprehension (maximum 38) 25.52 4.49 .809**
.550**
.571**
d. Vocabulary (maximum 60) 24.48 8.40 .872**
.630**
.680**
.685**
Note. Correlation coefficients for the last four variables were not reported fully because they were not
individually included in the regression or path analyses. Their means, standard deviations, and
intercorrelations were included here to give some indications of the children’s literacy levels. The working
memory scores are derived from standardized scores from the WMTBC norms.
The mathematics score was averaged across three tests and transformed using a base 10 logarithmic
function. The raw mean and standard deviation for the averaged score were 2.81 and 2.39,
respectively.
The literacy factor score was derived from the reading, spelling, reading comprehension, and
vocabulary tests using principal components analysis. Statistical parameters of factor scores are
typically constrained to M = 0 and SD = 1.
p < .05.
p < .01.
Regression analysis
To examine the unique contributions of each predictor variable to mathematical
performance, a standard multiple regression analysis was performed. The predictors
were the three component measures from the WMTBC, performance IQ, and the
literacy scores. Table 2 presents the results from the analysis. The predictors accounted
for a reliable amount of variation in mathematical performance (R2 = .49, adjusted
R2
= .48), F(5, 145) = 28.09, p < .01. Inspection of the regression coefficients showed that
only the central executive, performance IQ, and literacy measures predicted reliably to
mathematical performance. The semipartial correlations showed that these predictors
uniquely accounted for 2.6, 7.3, and 6.8% of variation in mathematical performance,
Table options

**
respectively. To examine whether the findings would replicate using only data from the
algebraic questions, we reran the analysis using accuracy data from the nine algebraic
questions. The overall pattern was the same. The only exception was that the unique
contribution of executive function was reduced.
Table 2.
Summary of regression analysis for variables predicting mathematical word problemsolving ability
Variable β sr
Phonological loop component −0.04 −0.03
Visual spatial component 0.06 0.05
Central executive component 0.21**
0.16
Performance intelligence 0.31**
0.27
Literacy factor 0.35**
0.26
Note. sr, semipartial correlation.
p < .01.
Path analyses
Results from the standard regression analysis showed that the direct contribution of
working memory to mathematical performance was modest but reliable. This pattern of
results is similar to that found in Bull and Scerif (2001), where executive function
accounted for 3% of variation in a numeracy and arithmetic test when reading ability and
IQ were controlled. Because it has been postulated that language development and
comprehension are dependent on working memory processes, a series of path models
were tested to examine the interrelation among working memory, literacy, and
mathematical performance. The data were analyzed using maximum likelihood
estimation by way of LISREL 8.30 (Jöreskog & Sörböm, 2000). First, we tested whether
working memory contributed directly to mathematical performance.
Model 1
For this model, it was postulated that working memory had both direct and indirect effects
on mathematical performance (Fig. 1). Each of the three working memory components
was modeled with direct pathways to mathematical performance. In addition, as
suggested by Baddeley et al. (1998) and Daneman and Merikle (1996), both the central
executive and the phonological loop were expected to contribute indirectly to
mathematical performance by way of literacy.
Fig. 1.
The path diagram for Models 1, 2, and 3, showing standardized path coefficients and correlation ratios. In
Model 1, all paths were included. In Model 2, all direct paths from working memory to mathematical
performance (dashed and dotted lines) were constrained to zero. In Model 3, direct paths from the
phonological loop and the visual spatial sketchpad to mathematical performance (dashed lines) were
constrained to zero.
Table options
Figure options

Regarding the role of performance IQ and its relation to visual spatial storage, less
guidance is available. In the verbal domain, the assignment of a pathway from the
phonological loop measure to literacy was guided by Baddeley et al. (1998). One
possibility is that a similar domainspecific relation exists between visual spatial storage
and performance IQ. We tested this proposition by including pathways that flowed from
the visual spatial sketchpad to performance IQ.
According to Baddeley’s (2000) model, the phonological and visual spatial storage
systems are independent. Nevertheless, we specified a correlational pathway between
them. This was guided by recent work arguing that even with the performance of
relatively pure visual spatial tasks, such as the Corsi blocks, participants were likely to
use some of the same mechanisms that contributed to the performance of phonological
tasks (Pickering, 2001).
Model 1 accounted for 47% of total variance in mathematics performance. Because a
number of test statistics were available and there was no general agreement on which
statistics should be reported, we reported those recommended by Hu and Bentler (1999).
With the exception of the chisquare ratio (χ2
/df), both the comparative fit index (CFI) and
the standardized root mean square residual (SRMR) indicated a satisfactory fit ( Table 3).
χ2/df was higher than the criterion level of 2. All modeled paths were significant with the
exception of direct paths from the phonological loop and visual spatial components to
mathematical performance.
Table 3.
Fit indexes for all models
Model χ2
/df CFI SRMR
1. Direct and indirect paths 5.37 0.95 0.07
2. Indirect paths only 4.26 0.93 0.08
3. Indirect paths and direct central executive to mathematics 3.51 0.95 0.07
Note. CFI, comparative fit index; SRMR, standardized root mean square residual. For χ2
/df, values
smaller than 2 indicate good fit. For CFI and SRMR, corresponding values are ⩾.195 and ⩽.08,
respectively.
Model 2
In this model, we tested the hypothesis that working memory has no direct effect on
mathematical performance. This model is similar to Model 1 with the exception that
coefficients for direct pathways from each of the three working memory components to
mathematical performance were constrained to zero. Model 2 accounted for 42% of total
variance in mathematical performance. Fit indexes indicated a moderate fit. Although
differences were small, the χ2
/df ratio indicated a better fit than did Model 1, but the other
indexes indicated a poorer fit. A chisquare test for differences between the two models
showed that Model 1 provided a better fit than did Model 2, χ2(3) = 9.45, p < .05.
Model 3
Model 1 showed that executive function contributed both directly and indirectly to
mathematical performance. In contrast, both storage measures contributed only
indirectly to mathematical performance. Model 3 was derived from these findings. It
differed from Model 1 in that path coefficients from the phonological loop and visual
spatial components to mathematical performance were constrained to zero. This
resulted in a slight improvement in the χ2
/df ratio. As shown in Table 4, all modeled paths
were significant. A chisquare test revealed no differences between Models 1 and 3,
χ2(2) = 1.42, p > .05.
Table 4.
Standardized parameter estimates for Models 1, 2, and 3
Path Model 1 Model 2 Model 3
PL → Literacy .39**
.39**
.39**
Table options

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**
Path Model 1 Model 2 Model 3
CE → Literacy .36**
.36**
.36**
CE → Performance IQ .23**
.23**
.23**
VS → Performance IQ .31**
.31**
.31**
PL → MP −.04 — —
CE → MP .21**
— .21**
VS → MP .06 — —
Literacy → MP .36**
.47**
.36**
Performance IQ → MP .32**
.37**
.33**
Note. PL, phonological loop component; CE, central executive component; VS, visual spatial component;
MP, mathematical performance.
p < .01.
In summary, Model 3 provided the most parsimonious fit for the data. Again, we reran this
model using only data from the nine algebraic questions. The overall pattern was the
same. Taken together, these analyses showed that the only working memory component
that contributed directly to mathematical performance was the central executive
(standardized direct effect = .21). In addition, it contributed to mathematical performance
indirectly by way of literacy and performance IQ. Apart from the central executive, both
literacy and performance IQ contributed to mathematical performance. The standardized
total effect (i.e., the direct effect combined with the two indirect effects) of the central
executive (.42) was larger than that of both literacy (total effect = .36) and performance IQ
(total effect = .33).
Discussion
The findings showed reliable and moderate to strong correlations between all predictors
and mathematical performance. Children who had greater storage and greater executive
function capacities were better able to perform the mathematical problems. Similarly,
children with higher performance IQ and better reading and vocabulary abilities
performed better on the mathematical problems. As expected, many of the predictor
variables were intercorrelated. Results from the standard regression showed that after
such intercorrelations were controlled, the central executive, performance IQ, and
literacy measures provided unique contributions to the prediction of mathematical
performance. When all predictor variables were used, they accounted for 49% of
variation in mathematical performance.
The contribution of executive function
Findings on the contribution of executive function resemble those reported in earlier
studies. Compared with Bull and Scerif (2001), for example, both studies found that
executive function measures exhibited strong bivariate correlation with mathematical
performance. When literacy and IQ were taken into account, the contribution of executive
function was attenuated but remained reliable. In both studies, 2 to 3% of total variation in
mathematical performance could be uniquely attributed to central executive capacity.
Although reliable, direct executive function contribution was modest and smaller than
that of literacy. Given the complexity of algebraic word problems, a greater reliance on
executive function was expected. This is especially the case when most children found
the mathematical test challenging; the mean performance score was quite low.
There are several explanations for the contribution of working memory being lower than
expected. First, the WMTBC employed a span approach to measure capacity across the
three components. For executive function, three tasks were used: listening span,
counting span, and backward digit span. All three measures indexed capacity to store
and process simultaneously. In contrast, Bull and Scerif (2001) used measures that
targeted four aspects of executive function: perseveration, inhibition efficiency, capacity,
and coordination. Although the various measures were intercorrelated, the first three
contributed uniquely to mathematical performance. One possible implication of their
Table options

findings is that, in addition to functional capacity, algebraic problem solving might also
rely on other aspects of executive function not captured in this study. A preliminary error
analysis suggested that some children failed to pay due consideration to the relations
specified in the word problems and applied correct procedures to the wrong operands.
The propensity to focus on salient but inappropriate aspects of a problem is perhaps one
aspect of perseveration and inhibition. We are currently looking at the relation between
perseveration and the transition from arithmetic to algebraic problem solving.
An alternative explanation relates to the age of participants. In this study, we sampled
children who were several years older than those sampled in Bull and Scerif (2001).
Concomitant developmental growth in various aspects of working memory, knowledge
base, and processing strategies (for a review, see Cowan, 1997) may have compensated
for the increase in mathematical task complexity. Swanson et al. (1993), for example,
showed that knowledge of mathematical processing operations was a reliable predictor
of mathematical performance. Although speculative, one explanation of the similarity
between our data and Bull and Scerif’s data is that much of the difference between
simple arithmetic and algebraic word problems can be attributed to knowledge
requirement. In other words, although algebraic word problems are more complex, there
is little additional working memory involvement because solving such problems is largely
supported by increases in content and procedural knowledge. Further studies are
needed to ascertain the developmental relation among mathematical performance,
knowledge, working memory, and other general processing abilities.
A third explanation is a variant of the knowledge base explanation. It is possible that in
the context of word problems, literacy serves as an obstacle to further processing. All
problems used in this study contained relational information. The algebraic question cited
in the introduction, for example, specifies both quantitative (150 or 130 kg) and
qualitative (more or less) relationships. Before children can compute a solution, they
must understand and assign the numerical relations appropriately. If they are unable to
understand and decode the questions, any further processing is unlikely to result in a
correct solution. Thus, rather than mathematically specific knowledge, it may be reading
abilities that account for similarities between the two studies. What is unclear from the
regression analysis are the extent to which working memory contributes to this initial
decoding process and the extent to which it contributes to subsequent processing. In the
next subsection, we discuss findings from our path models that examined the relation
between literacy and working memory.
The role of working memory in literacy, mathematics, and performance IQ
Working memory, particularly the central executive, has been found to contribute to
language comprehension and development (Baddeley et al., 1998; Daneman & Merikle,
1996). Furthermore, both working memory and language have been found to predict
mathematical performance. These findings can be explained using either Baddeley’s
theory or a more functional view of working memory such as Engle’s theory of executive
controlled attention (Engle, 2002; Engle, Conway, Tuholski, & Shisler, 1995). However, a
typical finding is that the contribution of working memory is reduced when the effect of
language is partialed out. One way to account for these findings is to assume that the
contribution of working memory is mediated by literacy. Results from the path analyses
are consistent with this assumption. The central executive measures predicted literacy,
which in turn predicted mathematical performance. We tested an alternative model in
which executive function had a direct effect on literacy but not on mathematical
performance. Fit indexes showed that this alternative model was less consistent with the
data.
The path analyses also showed that although the direct contribution of central executive
to mathematical performance was smaller than the contribution of literacy, its total
contribution—consisting of both direct and indirect effects as mediated by literacy and
performance IQ—was substantial. A comparison of standardized total effects showed
that the contribution from the central executive was no less than that from either literacy
or performance IQ. This pattern suggests that children with higher central executive

capacity are likely to possess better literacy skills. Although having better literacy skills
will itself result in better mathematical performance, children’s superior central executive
capacity also contributes to better mathematical performance. In contrast to previous
findings, the current findings suggest a larger and more fundamental role for the central
executive.
Other aspects of note include the contributions of the phonological loop and the visual
spatial sketchpad. These two components contributed directly to literacy and
performance IQ, respectively, but failed to contribute directly to mathematical
performance. These findings are consistent with previous findings showing that pure
storage measures contribute little to general mathematical performance. This is in
contrast to findings from the mental calculation literature that typically show a unique
contribution from the phonological loop. An explanation for this difference is that in
studies where children are allowed to show their work, they have less to remember; thus,
there is less demand on the phonological loop. Adams and Hitch (1997) tested a variation
of this explanation. In their study, children’s mental addition spans were measured when
numbers to be added were made either visible or hidden. Consistent with a storage
constraint hypothesis, addition spans were higher when the numbers were visible.
Visual spatial storage and performance IQ
In the path models, a similar structure as that imposed on phonological memory and
literacy was imposed on spatial memory and performance IQ. The assumption was that
just as phonological loop was a general storage processor that contributed to literacy, the
visual spatial sketchpad contributed to performance intelligence. We also tested an
alternative model in which visual spatial memory and performance intelligence were
assumed to be correlated, but no dependency was assumed. This alternative model fit
the data poorly. (Details of this analysis are available from the authors.)
Although the precise nature of the relation between visual spatial storage and
performance IQ is unclear, previous studies have shown strong correlations between
similar tasks. In Shah and Miyake (1996), for example, a spatial working memory span
task correlated highly with a number of spatial visualization tasks. They argued that
visualization tests often require participants to transform stimuli while maintaining and
making judgments on intermediate products of their transformation. As such, these tests
impose both storage and processing demands; just as executive function span tasks do.
In our study, similar to visualization tasks used in Shah and Miyake’s study, participants
performing the block design task were required to compare visual stimuli. The difference
was that in block design, participants had to construct the stimuli from the blocks
provided to them. Although the target design was constantly visible as participants were
considering which blocks to use, it was likely that they used both a mental representation
of the target design and the actual design to guide selection. Many children who had
difficulties with this task seemed to have difficulties in processing the target design into
constituent parts that correspond to the blocks. If this is correct, children with larger visual
spatial span would be likely to outperform because larger spans would aid in both the
remembrance and processing of patterns that are more complex.
Conclusions
From a theoretical perspective, the key finding from this study is that central executive
span plays a more prominent role than was thought previously. When both direct and
indirect effects were taken into account, central executive span was found to have as
much influence on mathematical performance as did literacy. Consistent with previous
findings, the storage components of working memory—phonological loop and visual
spatial sketchpad—did not contribute directly to mathematical performance.
From an applied perspective, if one were to administer just one test to predict
mathematical performance, the literacy tests have a slight advantage over other tests
used in this study. However, the data showed that even when the effects of literacy and
performance IQ were controlled, executive function still provided modest improvement in
the prediction of mathematical performance. These findings suggest that the use of

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Adams and Hitch, 1997
Ashcraft et al., 1998
executive function tests may be a useful adjunct in highstakes streaming examinations.
From a pedagogical perspective, the current findings highlight the importance of literacy.
Although the amount of variation that can be attributed uniquely to literacy was still
relatively small, it was the best predictor. Furthermore, in the context of this study, it is the
ability that is perhaps most amenable to educational intervention. This is not to say that
knowledge specific to mathematics is unimportant. Mathematical knowledge was not
measured in this study. In other studies, it has been shown to play a role equal to, if not
larger than, that of linguistic ability (Sakamoto, 1998; Swanson et al., 1993).
Acknowledgments
This study was supported by a grant from the Education Research Fund (EP 2/02 KL).
We thank Sean Kang and MeiYin Wong for assistance in data collection. We also thank
all of the children who participated in this study and the schools for giving us access.
Some of these data were reported at the International Neuropsychological Conference,
Hong Kong, December 2003.
Appendix.
Dunearn Primary School has 280 pupils. Sunshine Primary School has 89 pupils
more than Dunearn Primary. Excellent Primary School has 62 pupils more than
Dunearn Primary. How many pupils are there altogether?
A school bought some mathematics books and four times as many science books.
The cost of a mathematics book was $12, while a science book cost $8. Altogether,
the school spent $528. How many science books did the school buy?
A tank of water with 171 liters of water is divided into three containers: A, B, and C.
Container B has three times as much water as Container A. Container C has one
fourth as much water as Container B. How much water is there in Container B?
A cow weighs 150 kg more than a dog. A goat weighs 130 kg less than the cow.
Altogether, the three animals weigh 410 kg. What is the mass of the cow?
At a sale, Mrs. Tan spent $530 on a table, a chair, and an iron. The chair cost $60
more than the iron. The table cost $80 more than the chair. How much did the chair
cost?
Mae Ling bought a new box of cat biscuits. In the first week, she gave the cat half of
the biscuits and 3 more. In the second week, she gave the cat half of the remaining
biscuits and 3 more. In the third week, she gave the cat half of the remaining biscuits
and 3 more. There was only 1 biscuit left. How many biscuits were there in the new
box?
Mr. Raman is 45 years older than his son, Muthu. In 6 years’ time, Muthu will be one
fourth his father’s age. How old is Muthu now?
Vincent bought a total of 62 $3 and $5 stamps. Altogether, Vincent spent $254. How
many $3 stamps were there?
Each week, Yah Hui gets $6 more pocket money than Philip. Each week, Yah Hui
and Philip each spend $19 on books. After some weeks, Yah Hui saved $98, while
Philip saved $56. How much pocket money does Philip get each week?
During a class game, Peter threw a ball to four times as many boys as girls. Mei Lin
threw the ball to five times as many boys as girls. If Peter and Mei Lin threw the ball to
every pupil in the class, how many boys were there in the class?
References
J.W. Adams, G.J. Hitch
Working memory and children’s mental addition
Journal of Experimental Child Psychology, 67 (1997), pp. 21–38
[SD008]
M.H. Ashcraft, E.P. Kirk, D. Hopko

Baddeley, 1996
Baddeley, 2000
Baddeley et al., 1998
Baddeley and Hitch, 1974
Bednarz and Janvier, 1996
Bull and Johnston, 1997
Bull et al., 1999
Bull and Scerif, 2001
Cockcroft, 1982
Cowan, 1997
Daneman and Merikle, 1996
On the cognitive consequences of mathematics anxiety
C. Donlan (Ed.), The development of mathematical skills, Psychology Press, Hove, UK (1998), pp.
175–196
[SD008]
A. Baddeley
Exploring the central executive
Quarterly Journal of Experimental Psychology A, 49 (1996), pp. 5–28
[SD008]
A. Baddeley
The episodic buffer: A new component of working memory?
Trends in Cognitive Sciences, 4 (2000), pp. 417–423
[SD008]
A. Baddeley, S.E. Gathercole, C. Papagno
The phonological loop as a language learning device
Psychological Review, 105 (1998), pp. 158–173
[SD008]
A. Baddeley, G.J. Hitch
Working memory
G.A. Bower (Ed.), The psychology of learning and motivation, Academic Press, New York (1974), pp.
47–90
[SD008]
N. Bednarz, B. Janvier
Emergence and development of algebra as a problemsolving tool: Continuities and
discontinuities with arithmetic
N. Bednarz, C. Kieran, L. Lee (Eds.), Approaches to algebra, Kluwer, Dordrecht, Netherlands (1996),
pp. 115–136
[SD008]
R. Bull, R.S. Johnston
Children’s arithmetical difficulties: Contributions from processing speed, item identification,
and shortterm memory
[SD008]
R. Bull, R.S. Johnston, J.A. Roy
Exploring the roles of the visualspatial sketch pad and central executive in children’s
arithmetical skills: Views from cognition and developmental neuropsychology
Developmental Neuropsychology, 15 (1999), pp. 421–442
[SD008]
R. Bull, G. Scerif
Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching,
and working memory
[SD008]
W.H. Cockcroft
Mathematics counts: Report of the Committee of Inquiry Into the Teaching of Mathematics in
Schools under the chairmanship of W.H. Cockcroft
Her Majesty’s Stationery Office, London (1982)
[SD008]
N. Cowan
The development of working memory
N. Cowan (Ed.), The development of memory in childhood, Psychology Press, Hove, UK (1997), pp.
163–199
[SD008]
M. Daneman, P.M. Merikle
Working memory and language comprehension: A metaanalysis
Psychonomic Bulletin and Review, 3 (1996), pp. 422–433
[SD008]

Engle, 2002
Engle et al., 1995
Fürst and Hitch, 2000
Gathercole and Pickering, 2000a
Gathercole and Pickering, 2000b
Gathercole et al., 1992
Geary, 1993
Geary et al., 1996
Hoosain and Salili, 1987
Hu and Bentler, 1999
Johnson, 1984
Jöreskog and Sörböm, 2000
Kail and Hall, 1999
R.W. Engle
Working memory capacity as executive attention
Current Directions in Psychological Science, 11 (2002), pp. 19–23
[SD008]
R.W. Engle, A.R.A. Conway, S.W. Tuholski, R.J. Shisler
A resource account of inhibition
Psychological Science, 6 (1995), pp. 122–125
[SD008]
A.J. Fürst, G.J. Hitch
Separate roles for executive and phonological components of working memory in mental
arithmetic
Memory & Cognition, 28 (2000), pp. 774–782
[SD008]
S.E. Gathercole, S.J. Pickering
Assessment of working memory in six and sevenyearold children
Journal of Educational Psychology, 92 (2000), pp. 377–390
[SD008]
S.E. Gathercole, S.J. Pickering
Working memory deficits in children with low achievements in the national curriculum at 7 years
of age
British Journal of Educational Psychology, 70 (2000), pp. 177–194
[SD008]
S.E. Gathercole, C.S. Willis, H. Emslie, A.D. Baddeley
Phonological memory and vocabulary development during the early school years: A
longitudinal study
Developmental Psychology, 28 (1992), pp. 887–898
[SD008]
D.C. Geary
Mathematical disabilities: Cognitive, neuropsychological, and genetic components
Psychological Bulletin, 114 (1993), pp. 345–362
[SD008]
D.C. Geary, C.C. BowThomas, F. Liu, R.S. Siegler
Development of arithmetic competencies in Chinese and American children: Influence of age,
language, and schooling
Child Development, 67 (1996), pp. 2022–2044
[SD008]
R. Hoosain, F. Salili
Language difference in pronunciation speed for numbers, digit span, and mathematical ability
Psychologia, 30 (1987), pp. 34–38
[SD008]
L.T. Hu, P.M. Bentler
Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new
alternatives
Structural Equation Modeling, 6 (1999), pp. 1–55
[SD008]
E.S. Johnson
Sex differences in problem solving
[SD008]
K. Jöreskog, D. Sörböm
LISREL (Version 8.30) [computer software]
Scientific Software International, Chicago (2000)
[SD008]
R. Kail, L.K. Hall
Sources of developmental change in children’s wordproblem performance

Kintsch and Greeno, 1985
Lau and Hoosain, 1999
Lehto, 1995
Logie et al., 1994
Mayer and Hegarty, 1996
McLean and Hitch, 1999
Miyake and Shah, 1999
Passolunghi et al., 1999
Passolunghi and Siegel, 2001
Pickering, 2001
Pickering and Gathercole, 2001
Rourke, 1993
Rust, 2000
[SD008]
W. Kintsch, J.G. Greeno
Understanding and solving word arithmetic problems
Psychological Review, 92 (1985), pp. 109–129
[SD008]
C.W. Lau, R. Hoosain
Working memory and language differences in sound duration: A comparison of mental
arithmetic in Chinese, Japanese, and English
Psychologia, 42 (1999), pp. 139–144
[SD008]
J. Lehto
Working memory and school achievement in the ninth form
Educational Psychology, 15 (1995), pp. 271–281
[SD008]
R.H. Logie, K.J. Gilhooly, V. Wynn
Counting on working memory in arithmetic problem solving
[SD008]
R.E. Mayer, M. Hegarty
The process of understanding mathematical problems
R.J. Sternberg, T. BenZeev (Eds.), The nature of mathematical thinking, Lawrence Erlbaum, Mahwah,
NJ (1996), pp. 29–53
[SD008]
J.F. McLean, G.J. Hitch
Working memory impairments in children with specific arithmetic learning difficulties
[SD008]
A. Miyake, P. Shah
Models of working memory: Mechanisms of active maintenance and executive control
Cambridge University Press, Cambridge, UK (1999)
[SD008]
M.C. Passolunghi, C. Cornoldi, S. De Liberto
Working memory and intrusions of irrelevant information in a group of specific poor problem
solvers
[SD008]
M.C. Passolunghi, L.S. Siegel
Shortterm memory, working memory, and inhibitory control in children with difficulties in
arithmetic problem solving
[SD008]
S.J. Pickering
The development of visuo–spatial working memory
Memory, 9 (2001), pp. 423–432
[SD008]
S.J. Pickering, S.E. Gathercole
Working memory test battery for children
Psychological Corporation, Kent, UK (2001)
[SD008]
B.P. Rourke
Arithmetic disabilities, specific and otherwise: A neuropsychological perspective
Journal of Learning Disabilities, 26 (1993), pp. 214–226
[SD008]
J. Rust
Wechsler objective reading and language dimensions (Singapore)
Psychological Corporation, London (2000)

Sakamoto, 1998
Sattler, 2001
Shafrir and Siegel, 1994
Shah and Miyake, 1996
Sikora et al., 2002
Singapore Ministry of Education and Research and Testing Division, 2000
Swanson et al., 1993
Wechsler, 1991
[SD008]
M. Sakamoto
Existing knowledge which determines performance in arithmetic word problems
Psychologia, 41 (1998), pp. 102–113
[SD008]
Sattler, J. M. (2001). Assessment of children: Cognitive applications (4th ed.). San Diego:
Author
[SD008]
U. Shafrir, L.S. Siegel
Subtypes of learning disabilities in adolescents and adults
Journal of Learning Disabilities, 27 (1994), pp. 123–134
[SD008]
P. Shah, A. Miyake
The separability of working memory resources for spatial thinking and language processing: An
individual differences approach
Journal of Experimental Psychology: General, 125 (1996), pp. 4–27
[SD008]
D.M. Sikora, P. Haley, J. Edwards, R.W. Butler
Tower of London test performance in children with poor arithmetic skills
[SD008]
Singapore Ministry of
Education and Research and Testing Division. (2000). Third International Mathematics and Science
Study 1999 (TIMSS 1999) national report for Singapore. Singapore: Ministry of Education, Research
and Testing Division
[SD008]
H.L. Swanson, J.B. Cooney, S. Brock
The influence of working memory and classification ability on children’s word problem solution
[SD008]
D. Wechsler
Wechsler Intelligence Scale for Children—third edition
Psychological Corporation, San Antonio, TX (1991)
[SD008]
Corresponding author. Fax: +6568969410
Copyright © 2004 Elsevier Inc. All rights reserved.
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