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The relationship between working memory, IQ, and mathematical skills in children

Tracy Packiam Alloway

a,

⁎, Maria Chiara Passolunghi

b

a

University of Stirling, Italy

b

University of Trieste, Italy

abstractarticle info

Article history:

Received 5 October 2009

Received in revised form 21 September 2010

Accepted 26 September 2010

Keywords:

Working memory

Arithmetical abilities

Vocabulary

Mathematical skills

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.

Themainaimofthepresentstudywastocomparethe

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: t.p.alloway@stir.ac.uk (T.P. Alloway).

1041-6080/$ –see front matter © 2010 Published by Elsevier Inc.

doi:10.1016/j.lindif.2010.09.013

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

Jensen, 1980).

1. Method

1.1. Participants

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

disadvantage.

1.2. Measures

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

study.

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

Table 1

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

(VSTM)

1 .44 .33 .21 .35

Verbal working memory (VWM) .50 1 .50 .50 .36

Visuo-spatial short-term

memory (VS-STM)

.43 .58 1 .59 .30

Visuo-spatial working memory

(VS-WM)

.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

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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

present study.

1.2.3. Vocabulary

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

.36).

2. Results

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

2

p=.02), suggesting

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

Ranking (18%).

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

Table 2

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

Memory tests

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

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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.

3. Discussion

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,

2000).

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Table 3

Stepwise regression analyses predicting numerical skills as a function of age group.

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