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The relationship between working memory, IQ, and mathematical skills in children
Tracy Packiam Alloway
⁎, Maria Chiara Passolunghi
University of Stirling, Italy
University of Trieste, Italy
Received 5 October 2009
Received in revised form 21 September 2010
Accepted 26 September 2010
The aim of the present study was to investigate the contribution of working memory and verbal ability
(measured by vocabulary) to mathematical skills in children. A sample of 206 seven- and eight-year-olds was
administered tests of these cognitive skills. A different pattern emerged that was dependent on both the
memory task and the math skill. In the seven-year olds, visuo-spatial and verbal memory uniquely predicted
performance on the math tests; however, in the eight-year olds, only visuo-spatial short-term memory
predicted math scores. Even when differences in vocabulary were statistically accounted, memory skills
uniquely predicted mathematical skills and arithmetical abilities. This pattern of ﬁndings provides a useful
starting point that can add to existing research on the contributions of working memory and vocabulary to
different mathematical skills.
© 2010 Published by Elsevier Inc.
contributions of working memory–the ability to process and
remember information–and verbal ability (vocabulary) to mathemat-
ical skills. Baddeley's working memory model provides a useful
framework for understanding the role of the different memory
components in mathematical skills. The central executive is a
domain-general component responsible for the control of attention
and processing of information from long-term memory (Baddeley,
Emslie, Kolodny, & Duncan, 1998). The temporary storage of
information is mediated by the phonological loop for verbal material
and the visuo-spatial sketchpad for visual and spatial representations
(Baddeley & Logie, 1999). The fourth component, the episodic buffer,
is responsible for binding information into integrated chunks
(Baddeley, 2000). As measurement tasks have yet to be standardized
for children, this component was not considered in the present study
(but see Alloway, Gathercole, Willis, & Adams, 2004; Alloway &
Gathercole, 2005, for the links between the episodic buffer and
learning in children).
Although it is well-established that working memory is closely
linked to mathematical skills, this relationship is mediated by the task
as well as the child's age. Visuo-spatial memory (represented by
the visuo-spatial sketchpad) functions as a mental blackboard to
support number representation, such as place value and alignment in
columns, in counting and arithmetic tasks (D'Amico & Guarnera,
2005; Geary, 1990; McLean & Hitch, 1999). Speciﬁc associations have
also been reported between visuo-spatial memory and encoding in
problems presented visually (Logie, Gilhooly, & Wynn, 1994; Trbovich
& LeFevre, 2003), and in multi-digit operations (Heathcote, 1994).
Visuo-spatial memory skills uniquely predict performance in nonver-
bal problems, such as sums presented with blocks, in pre-school
children (Rasmussen & Bisanz, 2005), as well as problem-solving
(Passolunghi & Mammarella, 2010).
Verbal short-term memory (represented by the phonological
loop) has been linked to solving single-digit addition problems
(Hecht, 2002; Seyler, Kirk, & Ashcraft, 2003) and maintaining operand
and interim results in multi-digit problem (Fürst & Hitch, 2001;
Heathcote, 1994; Noël, Désert, Aubrun, & Seron, 2001; Seitz &
Schumann-Hengsteler, 2000, 2002). It is possible that verbal working
memory (represented by the central executive in Baddeley's model) is
a reliable indicator of mathematical disabilities in the ﬁrst year of
formal schooling (Gersten, Jordan, & Flojo, 2005; also Bull & Scerif,
2001), but not in older children (Reuhkala, 2001), as other factors,
such as number knowledge and strategies, play a greater role
(Thevenot & Oakhill, 2005).
The present study extends previous research by including a range
of working memory measures. On the basis of the differential links
between the memory components and arithmetical abilities, we
included measures of verbal and visuo-spatial short-term memory
and working memory (see Alloway, Gathercole, & Pickering, 2006;
Bayliss, Jarrold, Gunn, & Baddeley, 2003; for support of this theoretical
structure of working memory in development). This allowed us to
systematically investigate the links between the various memory
skills and mathematical skills and arithmetical abilities.
There is evidence that working memory tasks measure something
different from general ability tests, such as IQ and vocabulary (Cain,
Oakhill, & Bryant, 2004; Siegel, 1988). While these tests measure
Learning and Individual Differences 21 (2011) 133–137
⁎Corresponding author. Department of Psychology, University of Stirling, Stirling,
FK9 4LA, UK. Tel.: + 44 (0) 1786 467639.
E-mail address: email@example.com (T.P. Alloway).
1041-6080/$ –see front matter © 2010 Published by Elsevier Inc.
Contents lists available at ScienceDirect
Learning and Individual Differences
journal homepage: www.elsevier.com/locate/lindif
Author's personal copy
knowledge that the child has already learned, working memory tasks
are a pure measure of a child's learning potential (Alloway & Alloway,
2010). Thus, working memory skills are able to predict a child's
performance in learning outcomes, even after their general ability has
been statistically accounted in reading and language skills (Gathercole,
Alloway, Willis, & Adams, 2006; Nation, Adams, Bowyer-Crane, &
Snowling, 1999; Passolunghi, Vercelloni, & Schadee, 2007; Stothard &
Hulme, 1992; for a review see Swanson & Saez, 2003). The present
study explored whether the same pattern of dissociation in the
contributions of verbal ability, measured by vocabulary, and working
memory would also be evident in tests that assessed a range of
mathematical and arithmetic abilities.
Mathematical skills were measured by tasks involving ranking
numbers, translating numbers from one representation to another
(e.g., words to numbers), quantity discrimination, as well as more
complex number skills, such as arithmetic computation (Butterworth,
2005). Also of interest was whether working memory and vocabulary
would be differentially associated with mathematical skills as a
function of age (Holmes & Adams, 2006; Passolunghi et al., 2007). For
example, in the present study, verbal working memory may play a
greater role in supporting arithmetical ability in younger children (7-
year-olds), while visuo-spatial memory may be more closely linked to
such skills in older children (8-year-olds) since some arithmetic tasks
(e.g., more complex additions and subtractions) could require an
elevated demand of visuo-spatial processing. The link between
vocabulary and mathematical skills may be greater in the younger
cohort as the individual is learning new information, rather than in
the older group when gains made are likely the result of practice (see
There were 206 typically developing children (109 boys) recruited
from four mainstream schools located in the north-west of Italy. The
majority of parents came from professional homes that were predom-
inantly middle class but included families from across the social
spectrum. For the statistical analyses, participants were divided
into two age groups: 7-year-olds (n=100; M=88 months,
SD=3.5 months; 50 boys) and 8-year-olds (n=106; M=103 months,
SD=3.6 months; 46 boys). None was receiving special education
services or had documented brain injury, or behavioral problems.
None of the assessed children belonged to families with socio-cultural
1.2.1. Working memory
All 12 tests from the Automated Working Memory Assessment
(AWMA, Alloway, 2007), a computer-based standardized battery that
provides multiple assessments of verbal and visuo-spatial short-term
memory, and of verbal and visuo-spatial working memory. There
were three measures of verbal short-term memory where the child
immediately recalls a sequence of information: digit recall, word
recall, and nonword recall. Test–retest reliability is .89, .88, .69 for
digit recall, word recall, and nonword recall respectively.
There were three verbal working memory measures: listening
recall, backward digit recall, and counting recall. In the listening recall
task, the child veriﬁes a series of sentences by stating ‘true’or ‘false’
and recalls the ﬁnal word for each sentence in sequence. In the
backwards digit recall task, the child recalls a sequence of spoken
digits in the reverse order. In the counting recall task, the child counts
the number of circles in an array and then recalls the tallies of circles.
Test–retest reliability is .88, .84, .86 for listening recall, counting recall,
and backward digit recall respectively.
Three measures of visuo-spatial short-term memory were admin-
istered. In the dot matrix task, the child is shown the position of a red
dot in a series of four by four matrices and has to recall this position by
tapping the squares on the computer screen. In the mazes memory
task, the child is shown a maze with a red path drawn through it for
three seconds. S/he then has to trace in the same path on a blank maze
presented on the computer screen. In the block recall task, the child
views a video of a series of blocks being tapped and reproduces the
sequence in the correct order by tapping on a picture of the blocks.
Test–retest reliability is .85, .86, .90 for dot matrix, mazes memory and
block recall, respectively.
Three measures of visuo-spatial working memory were adminis-
tered. In the odd-one-out task, the child views three shapes, each in a
box presented in a row, and identiﬁes the odd-one-out shape. At the
end of each trial, the child recalls the location of each odd one out
shape, in the correct order, by tapping the correct box on the screen. In
the Mr. X task, the child is presented with a picture of two Mr. X
ﬁgures. The child identiﬁes whether the Mr. X with the blue hat is
holding the ball in the same hand as the Mr. X with the yellow hat. The
Mr. X with the blue hat may also be rotated. At the end of each trial,
the child recalls the location of each ball in the blue Mr. X's hand in
sequence by pointing to a picture with six compass points. In the
spatial recall task, the child views a picture of two arbitrary shapes
where the shape on the right has a red dot on it and identiﬁes whether
the shape on the right is the same or opposite of the shape on the left.
The shape with the red dot may also be rotated. At the end of each
trial, the child recalls the location of each red dot on the shape in
sequence by pointing to a picture with three compass points. Test–
retest reliability is .88, .84, .79 for the odd-one-out, Mr. X, and spatial
recall, respectively. Raw scores for all tests are reported in the present
Test reliability of the English AWMA is reported here and test
validity is reported in Alloway, Gathercole, Kirkwood, and Elliott
(2008). The tests were translated and voice recorded into Italian by
native speakers. As normative data for Italian is currently being
collected, we report the intercorrelations between working memory
composite scores in Table 1 based on the present sample. The
between-construct coefﬁcients were high (rs ranging from .35 to .66),
suggesting good internal validity of the measures purportedly tapping
four subcomponents of working memory. On this basis, the following
analyses were based on the four memory components as reported in
Alloway et al. (2006). The range of scores for all 12 memory tests is
provided in Table 2 and the skewness and kurtosis values represent
normal distributions for both age groups.
1.2.2. Mathematical skills
All students were administered Italian AC-MT test, which consists
of four tasks with different levels of difﬁculty depending on the age of
the student (Cornoldi, Lucangeli, & Bellina, 2002). In the ﬁrst task,
Number Operations, the student solves basic operations such as
single-digit addition (maximum score= 8). The second test measures
Quantity Discrimination, and the student makes number comparison
Correlations between all memory scores; partial correlations (controlling for age in
months) in upper triangle (n=206).
Measures VSTM VWM VS-STM VS-WM Vocabulary
Verbal short-term memory
1 .44 .33 .21 .35
Verbal working memory (VWM) .50 1 .50 .50 .36
.43 .58 1 .59 .30
Visuo-spatial working memory
.32 .57 .68 1 .33
Vocabulary .46 .48 .53 .50 1
Note: All correlations are signiﬁcant at the .005 level.
134 T.P. Alloway, M.C. Passolunghi / Learning and Individual Differences 21 (2011) 133–137
Author's personal copy
and identiﬁes bigger and smaller numbers in a set (maximum
score= 6). In the third test, Number Production, the student
translates numbers from one representation to another (e.g., 4
dozen= 48; maximum score = 6). In the ﬁnal test, Number Ranking,
the student orders number sequences from the smaller to the higher
and vice versa (maximum score = 11). The reliability is: Number
Operations .74 (7 years) and .68 (8 years); Quantity Discrimination
.69 (7 years) and .65 (8 years); number production .75 (7 years) and
.72 (8 years); and .88 (7 & 8 years) for number ranking. The
concurrent validity, reported in the AC-MT manual is also good: .84
(Cornoldi et al., 2002).
Arithmetical abilities were also assessed using the Numerical
Operations subtest from the Wechsler Objective Numerical Dimen-
sions (WOND, Wechsler, 1996). It consists of 10 four-item tests. The
ﬁrst set assesses the ability to write dictated numerals. The
subsequent sets include computational problems addition, subtrac-
tion, multiplication and division. The correlation the Arithmetic
subtest from the WOND and Number Operations test from the AC-
MT is .42 (with age partialed out), which suggests that they assess
different mathematical components. As with most standardized
assessments, there is a discontinue rule, which means that not all
sets were presented to the children in the present study and the
younger children only completed a smaller number of sets compared
to the older children. As there are currently no Italian standard scores
are available for this standardized test, raw scores are reported in the
In order to assess general ability, the children were administered
the PMA (Thurstone & Thurstone, 1968) vocabulary subtest, which
consists of 60 items. In half the items, the child indicates which word
has the same meaning as the target word. In the other half, the child
indicates which picture corresponds to the target word. Raw scores
were converted into standard scores, with a mean of 100 and a
standard deviation of 15 based on an Italian sample. The test-
reliability is .95. The correlations between vocabulary and the
memory measures are provided in Table 1 and the partial correlation
coefﬁecients indicate a moderate association (rs ranging from .30 to
Descriptive statistics for the cognitive measures as a function of
age-group are shown in Table 2. The following patterns emerged: the
7-year-olds scored lower than the 8-year-olds in all measures. In order
to compare the working memory proﬁle between age-groups, a
MANOVCA was performed on the four memory composite scores,
with age (in months) as a co-variate. The overall group term associated
with Hotelling's T-test was not signiﬁcant (FN1, η
that the memory proﬁle did not differ signiﬁcantly between the 7 and
8-year-olds once age was partialed out.
2.1. Working memory, vocabulary, and mathematical skills
In order to investigate which memory component was linked to
mathematical skills in 7 and 8-year olds, a series of stepwise
regression analyses were conducted on the raw scores on the four
sub-tests from the AC-MT and the Arithmetic subtest from the WOND.
The vocabulary raw score and all four memory composite scores were
entered simultaneously with a stepwise function. This approach
allowed us to identify the best predictive variables for various
mathematical skills as a function of age. Model statistics, as well as
standardized beta values and t-statistics, are provided in Table 3.
For the 7-year olds, vocabulary accounted for signiﬁcant propor-
tion of variance (13%) across all mathematical skills: Quantity
Discrimination (10%); Number Ranking (23%); Number Production
(16%); Number Operations (12%); Arithmetic (13%). Of the four
memory measures, visuo-spatial short-term memory accounted for
signiﬁcant additional variance to Quantity Discrimination (14%) and
Number Production (21%). Verbal short-term memory accounted for
signiﬁcant additional variance in the Arithmetic test from the WOND
(17%); and verbal working memory uniquely predicted Number
For the 8-year olds, vocabulary accounted for signiﬁcant propor-
tion of variance (13%) across some mathematical skills: Quantity
Discrimination (9%); Number Ranking (30%); Number Operations
(27%); Arithmetic (26%); but not Number Production. Of the four
memory measures, only visuo-spatial short-term memory accounted
for signiﬁcant additional variance in predicting scores in Number
Descriptive statistics of raw scores for cognitive measures as a function of age group (7 and 8 years).
7 yrs (n=100) 8 yrs (n=154)
Min Max Skewness Kurtosis Mean SD Min Max Skewness Kurtosis Mean SD
Digit recall 16 32 .260 −.380 23.02 3.58 13 35 −.303 .800 25.22 3.55
Word recall 8 24 −.678 −.198 18.91 3.51 12 28 −.759 .423 20.29 3.20
Nonword recall 7 26 −.318 −.314 16.32 4.55 9 31 .302 .786 18.30 3.94
Verbal STM 12 26 .171 −.479 19.42 3.0 12 30 0 .157 21.27 2.74
Listening recall 3 16 .324 −.364 8.57 2.77 5 18 .416 −.058 10.29 2.66
Counting recall 7 24 .011 −.133 15.01 3.72 6 25 .370 −.768 16.45 4.15
Backward digit recall 3 18 .400 −.018 9.69 2.93 6 28 .111 .897 11.31 3.71
Verbal WM 7 17 .211 −.175 11.09 2.24 7 21 .549 .274 12.69 2.65
Dot matrix 10 29 .211 .444 17.95 3.35 12 33 .278 .143 21.73 4.0
Mazes memory 5 26 .206 −.413 15.98 4.70 8 27 −.911 .083 20.14 4.67
Block recall 7 26 −.300 .177 17.47 3.91 11 33 .176 .262 20.50 4.16
Visuo-spatial STM 10 26 .398 .192 17.13 2.93 12 29 −.132 −.059 20.79 3.51
Odd one out 6 23 .119 .016 13.75 3.47 9 26 .358 −.135 15.90 3.53
Mister X 0 17 .428 −.479 7.47 3.72 1 19 .284 −.377 9.65 3.82
Spatial recall 0 23 −.154 −.439 11.18 4.84 1 24 −.365 −.032 14.00 4.28
Visuo-spatial WM 4 20 .148 −.369 10.80 3.19 6 19 −.106 −.642 13.18 3.03
Number skills tests
Number Operations 0 4 2.21 1.15 1 8 5.59 1.64
Quantity discrimination 1 6 5.38 1.22 0 6 5.42 1.16
Number production 0 6 3.60 1.63 0 6 5.58 1.02
Number ranking 0 11 8.36 2.44 1 11 8.78 1.83
Arithmetic (WOND) 1 12 10.18 1.66 12 22 19.10 2.23
Vocabulary 27 57 44.91 6.82 36 59 52.55 5.02
135T.P. Alloway, M.C. Passolunghi / Learning and Individual Differences 21 (2011) 133–137
Author's personal copy
Ranking (23%); Number Production (10%); Number Operations (19%);
and Arithmetic test from the WOND (35%); but not Quantity
Discrimination. Verbal short-term memory and working memory
did not uniquely predict scores in any scores in the mathematical
tests. The ﬁndings suggest an age-related difference in the contribu-
tion of memory to arithmetical abilities.
The aim of the present study was to investigate the contributions
of working memory and vocabulary to mathematical skills in children.
The comparisons of the younger (7 years)and older (8 years) children
showed slightly different patterns in the contribution of memory to
mathematical skills and arithmetical abilities. Verbal memory pre-
dicted Number Ranking and Arithmetic skills in 7-year-olds, while
visuo-spatial short-term memory predicted these same skills in 8-
year olds. The latter also predicted performance in Quantity
Discrimination and Number Production in the younger group.
In line with previous research, verbal short-term memory was an
important predictor of performance in single-digit addition and
subtraction problems (e.g., Hecht, 2002; Seyler et al., 2003). However,
it did not predict scores in Number Operations, which may be due to
the fewer number of items that the 7-year-olds solved (4 items).
These items were also relatively easy and may have involved
information that was automatically activated as a result of frequent
repetition. Verbal working memory was uniquely linked to Number
Ranking, which required the student to order number sequences from
the smaller to the higher and vice versa. This task appeared to tap
executive resources as they had to hold the items in mind while
placing them in the correct numerical order.
The pattern of association between memory scores and mathe-
matical skills was slightly different for the eight-year-olds. In
particular, only visuo-spatial short-term memory accounted for
signiﬁcant additional variance in predicting the mathematical tests
(except for Quantity Discrimination). The importance of visuo-spatial
short-term memory ﬁts well with evidence that it functions as a
mental blackboard to support number representation particularly
when problems are presented visually (Trbovich & LeFevre, 2003).
One issue worth addressing is the unique contribution of visuo-spatial
short-term memory, but not visuo-spatial working memory. One
possibility is this age group (7–8 years) may have drawn more on
executive resources when performing the visuo-spatial short-term
memory tasks (see Alloway et al., 2006; Cowan et al., 2005).
Inspection of the association between these two constructs for the
present sample conﬁrms that they share almost 50% of their variance,
which is larger than any other of the memory constructs. Thus, given
the close relationship between these two constructs, it is possible that
the visuo-spatial short-memory tasks captured any additional
variance of the visuo-spatial working memory ones.
Vocabulary scores uniquely predicted performance on all mathe-
matical tests across the age groups, with the exception of Number
Production in eight-year-olds. There was not an age-difference in the
contribution of vocabulary to mathematical skills, which may be due
to the nature of the math tests used in the present study. As the older
cohort were presented with more items in some of the tests (e.g.,
Number Operations and Arithmetic) compared to the younger group,
they were exposed to new information which likely tapped general
ability. Indeed, the variance that the Vocabulary scores accounted for
in both these math tests was twice as much for the older group. A key
point is that short-term and working memory signiﬁcantly predicted
mathematical skills and arithmetical abilities after the variance
associated with vocabulary was accounted, which indicates that
working memory is not a proxy for intelligence and measures a
dissociable cognitive construct (Gathercole et al., 2006; Passolunghi,
Mammarella, & Altoè, 2008).
In summary, this exploratory study provides a useful starting
point that can add to existing research on the contributions of
working memory and vocabulary to different mathematical skills. A
novel and signiﬁcant ﬁnding was that even when differences in
vocabulary were statistically accounted, memory skills uniquely
predicted mathematical skills and arithmetical abilities. The pattern
of ﬁndings reported in the present study can provide the ﬁrst step of
a series of subsequent investigations on underlying mechanisms
related to mathematical skills, such as phonological processing skills
(Hecht et al., 2001) and strategy use (Geary, Hamson, & Hoard,
Alloway, T. P. (2007). Automated Working Memory Assessment. London: Harcourt
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validity of the Automated Working Memory Assessment. Educational Psychology,7,
Stepwise regression analyses predicting numerical skills as a function of age group.
Dependant variables Age group Independent variables R2 ΔR2 df ΔFΒt
Quantity Discrimination (AC−MT) 7 1 Vocabulary .10 –1, 98 11.43 * .32 3.38*
2 Visuo-spatial STM .14 .04 1, 97 4.84* .21 2.12*
8 1 Vocabulary .09 –1,104 10.44* .30 3.23*
Number Ranking (AC-MT) 7 1 Verbal WM .18 –1, 98 21.53* .42 4.64*
2 Vocabulary .23 .05 1, 97 6.87* .25 2.62*
8 1 Visuo-spatial STM .23 –1,104 31.15* .48 5.58*
2 Vocabulary .30 .07 1,103 10.28* .29 3.21*
Number Production (AC-MT) 7 1 Vocabulary .16 –1, 98 18.07* .40 4.25*
2 Visuo-spatial STM .21 .05 1, 97 6.75* .25 2.60*
8 1 Visuo-spatial STM .10 –1,104 11.74* .32 3.43*
Number Operations (AC-MT) 7 1 Vocabulary .12 –1,98 12.73* .34 3.57*
8 1 Visuo-spatial STM .19 –1,104 25.02* .44 5.00*
2 Vocabulary .27 .08 1,103 10.72* .30 3.27*
3 Verbal STM .30 .03 1,102 4.78* .20 2.19*
Arithmetic (WOND) 7 1 Vocabulary .13 –1, 98 15.07* .37 3.88*
2 Verbal STM .17 .04 1, 97 4.26* .21 2.07*
8 1 Vocabulary .26 –1,104 37.12* .51 6.09*
2 Visuo-spatial STM .35 .09 1,103 14.41* .33 3.80*
Note: STM=short-term memory; WM =working memory; * pb.05; Β=standardized beta values.
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