working memory as a predictor of written arithmetical skills

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Working memory as a predictor of writtenarithmetical skills in children: The importanceof central executive functions

Ulf Andersson*Department of Behavioural Sciences, Linkoping University, Sweden

Background. The study was conducted in an attempt to further our understandingof how working memory contributes to written arithmetical skills in children.

Aim. The aim was to pinpoint the contribution of different central executivefunctions and to examine the contribution of the two subcomponents of children’swritten arithmetical skills.

Sample and method. A total of 141 third- and fourth-graders were administeredarithmetical tasks and measures of working memory, fluid IQ and reading. Regressionanalysis was used to examine the relationship between working memory and writtenarithmetical skills.

Results. Three central executive measures (counting span, trail making and verbalfluency) and one phonological loop measure (Digit Span) were significant andpredictors of arithmetical performance when the influence of reading, age and IQ wascontrolled for in the analysis.

Conclusions. The present findings demonstrate that working memory, in general,and the central executive, in particular, contribute to children’s arithmetical skills. It washypothesized that monitoring and coordinating multiple processes, and accessingarithmetical knowledge from long-term memory, are important central executivefunctions during arithmetical performance. The contribution of the phonological loopand the central executive (concurrent processing and storage of numerical information)indicates that children aged 9–10 years primarily utilize verbal coding strategies duringwritten arithmetical performance.

Empirical studies show that working memory is an important factor in children’smathematical abilities (Adam & Hitch, 1997, 1998; Gathercole, Pickering, Knight, &

Stegmann, 2004; Kaye, DeWinstanley, Chen, & Bonnefil, 1989). Working memory

deficits have also been implicated as an underlying factor to mathematical difficulties in

children (e.g. Hitch & McAuley, 1991; McLean & Hitch, 1999; Passolunghi & Siegel,

2001; Siegel & Ryan, 1989).

* Correspondence should be addressed to Dr UIf Andersson, Associate Professor, Department of Behavioural Sciences,Linkoping University, SE-581 83 Linkoping, Sweden (e-mail: [email protected]).

TheBritishPsychologicalSociety

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British Journal of Educational Psychology (2008), 78, 181–203

q 2008 The British Psychological Society

www.bpsjournals.co.uk

DOI:10.1348/000709907X209854


One model of working memory that has frequently been used to examine the

connection with mathematical ability is Baddeley’s multicomponent model (Baddeley &

Hitch, 1974; see Baddeley, 1986, 1990, 2000 for revisions). The model consists of three

components, a central executive and two slave components: the phonological loop and

the visuospatial sketchpad. The central executive is the main component assumed to be

an attentional-controlling system that coordinates the activities in the working memorysystem. Baddeley (1996) proposed four other functions of the central executive:

(1) coordinating performance on two separate tasks or operations (e.g. simultaneous

storage and processing of information); (2) switching between tasks, retrieval strategies

or operations (i.e. sequencing); (3) attending selectively to specific information and

inhibiting irrelevant information; and (4) activating and retrieving information from

long-term memory (see also Baddeley & Logie, 1999). However, the concept of the

central executive is not uncontroversial. For example, Parkin (1998) is critical to the

construct as such, due to the lack of good empirical support for this construct and thatthe construct has the form of a homunculus, impossible to falsify (Parkin, 1998).

Nonetheless, a recent developmental study by Zoelch, Seitz, and Schumann-Hengsteler

(2005) provides empirical support for the central executive and its division into the four

separate but interrelated functions proposed by Baddeley (1996; see also Engle,

Tuholski, Laughlin, & Conway, 1999; Lehto, 1996; Miyake et al., 2000). In the model, the

central executive is supported by the phonological loop and the visuospatial sketchpad

which are specialized in storing and processing verbal information and visuospatial

information, respectively. In 2000, Alan Baddeley (2000) proposed a revised version ofthe original three-component model in which he added a fourth episodic buffer

component to the model. This component comprises a system that can integrate

information from the other two slave components and long-term memory, and can

temporarily store this information in the form of an episodic representation. Due to

limited research related to the episodic buffer, the present study employed the three-

component model. In addition, Gathercole, Pickering, Ambridge, and Wearing (2004)

have demonstrated that the three-component structure of working memory is present

from as early as 6 years of age, suggesting that the structure is well established in the 9-to 11-year-old children who participated in the present study.

Although Baddeley’s model is the most influential account of working memory

to date, several other models of working memory exist (e.g. Daneman & Carpenter,

1980; Engle, Cantor, & Carullo, 1992). One interesting alternative account, from a

developmental point of view, is proposed by Pascual-Leone (2000). This model contains

information-bearing schemes and content-free processing resources called hardware

M-operators. Working memory capacity is determined by the capacity of the M-

operators, which is the maximum number of schemes that can be concurrentlyactivated within a single operator. The capacity of the operators increases with age, as a

consequence of biological maturation.

The present study was conducted in an attempt to further our understanding of how

working memory contributes to written arithmetical ability in children, by examining

the contributions of the three components of written arithmetical skill, something

which was not done in many of the previous studies. The study will also pinpoint further

the different contributions of the central executive to written arithmetical ability in

children, which is not well understood yet.A review of the research literature in the context of Baddeley’s multicomponent

model demonstrates that measures of the central executive are particularly strong

predictors of children’s mathematical ability (Fuchs et al., 2005; Gathercole & Pickering,

182 Ulf Andersson


2000; Gathercole et al., 2004; Henry & MacLean, 2003; Holmes & Adams, 2006; Keeler

& Swanson, 2001; Lee, Ng, Ng, & Lim, 2004; Lehto, 1995; Noel, Seron, & Trovarelli,

2004; Swanson, 1994; Swanson & Beebe-Frankenberger, 2004; Wilson & Swanson,

2001). The majority of these studies have assessed the central executive by traditional

memory span tasks that require concurrent processing and storage of information, thus

suggesting that the specific central executive function of coordinating and monitoringsimultaneous processing and storage of information is important during performance of

arithmetical and mathematical tasks. However, a few recent studies have attempted to

pinpoint further the contribution of different central executive functions to children’s

mathematical ability (Bull, Johnston, & Roy, 1999; Bull & Scerif, 2001; McLean & Hitch,

1999; Rasmussen & Bisanz, 2005). McLean and Hitch found significant correlations

between written computation and two central executive functions, shifting (trail-

making task), and the ability to hold and manipulate information accessed from long-

term memory, in a sample of 33 third- and fourth-graders. One-third of the children hadspecific arithmetic difficulties. Performance on the written computation task was also

correlated with measures tapping the phonological loop (Digit Span) and the

visuospatial sketchpad (Corsi-block span), but the correlations were stronger with the

central executive tasks. In a study with a larger sample, Bull and Scerif used the Stroop

task, counting-span task and Wisconsin card sorting test to examine the contribution of

a number of different central executive functions to written mathematical performance

in children. After controlling for IQ and reading, they found that the ability to process

and store (numeric) information concurrently, inhibition control and switching arecentral executive functions which contributed variance to the prediction of children’s

mathematical performance. The importance of shifting in mathematics was also

demonstrated by Bull et al. (1999). Rasmussen and Bisanz found, similar to previous

studies, that tasks tapping simultaneous processing and storage of numeric information

(counting span, backward Digit Span) were significant predictors of mental arithmetic

in preschool children and grade 1 children, but the measure of inhibition control (the

sun–moon Stroop task) was, in contrast to Bull and Scerif (2001; see also Houde, 2000),

not a significant predictor.In sum, available evidence concerning the central executive suggests that

mathematical performance does not only require the capacity to process and store

information simultaneously. The ability to inhibit task-irrelevant information from

gaining access to working memory and the ability to shift from one strategy or operation

to another are also critical central executive functions during mathematical and

arithmetical performance (Bull et al., 1999; Bull & Scerif, 2001; McLean & Hitch, 1999).

A number of studies have reported correlations between measures of the

phonological loop and the visuospatial sketchpad and mental arithmetic in children(Adams & Hitch, 1998; Geary, Brown, & Samaranayake, 1991; McKenzie, Bull, & Gray,

2003; Noel et al., 2004; Rasmussen & Bisanz, 2005). In addition, Rasmussen and Bisanz

found, by means of multiple regression analysis, that these two components accounted

for variance in children’s mental addition. Studies have also found correlations between

the phonological loop and the visuospatial sketchpad and written arithmetical

calculation and school marks in mathematics (Gathercole et al., 2004; Holmes & Adams,

2006; Lehto, 1995; Maybery & Do, 2003; McLean & Hitch, 1999; Swanson, 1994;

Swansson & Beebe-Frankenberger, 2004). However, few of the studies have examinedand found independent contribution from the phonological loop and the visuospatial

sketchpad. That is, the correlations have usually been eliminated after controlling for the

contribution from reading, IQ or the central executive (Bull et al., 1999; Lehto, 1995;

Working memory and arithmetic 183


Swansson & Beebe-Frankenberger, 2004), suggesting that the contribution of the two

slave systems to mathematical performance is mainly indirect by way of reading and

IQ (Lee et al., 2004). Maybery and Do found that a letter span task and a

computerized Corsi-block span task accounted for variance in a curriculum-based

arithmetic test in 9- and 10-year-old children independent of skill in single word reading

and central executive processing (see also Kyttala, Aunio, Lehto, Van Luit, & Hautamaki,2003; Henry & MacLean, 2003). The stronger relationships found between the two slave

components and mental arithmetic compared with written arithmetical calculation and

school marks in mathematics are most probably due to the fact that mental arithmetic

tasks impose a larger memory load than general mathematical tests that are carried out

as paper-and-pencil tests (Heatcote, 1994; Lee et al., 2004; Logie, Gilhooly, & Wynn,

1994; Noel, Desert, Aubrun, & Seron, 2001).

Although quite a number of studies have demonstrated relationships between the

components of working memory and arithmetical skills in children, most of thesestudies have examined mental arithmetic (e.g. Adams & Hitch, 1997; Geary et al.,

1991; McKenzie et al., 2003; Rasmussen & Bisanz, 2005). Moreover, the majority of

researchers focusing on written arithmetical skills have assessed a number of skills

(arithmetic word problem solving, single and multi-digit arithmetic computation and

algebra problems) and combined them into a general measure of written arithmetical

skill (Bull et al., 1999; Bull & Scerif, 2001; Holmes & Adams, 2006; Gathercole,

Alloway, Willis, & Adams, 2006). Thus, only a few studies have used relatively ‘pure’

measures of written arithmetical calculation when studying the relationship withworking memory (e.g. Mayberry & Do, 2003; McLean & Hitch, 1999; Swanson &

Beebe-Frankenberger, 2004; Wilson & Swanson, 2001). Unfortunately, these studies

have not combined the use of large samples, tasks tapping the different functions of

the central executive, and tasks tapping reading skill and IQ. This study sought out to

address these limitations by using a large sample of children, measures of IQ, reading,

fact retrieval, relatively ‘pure’ measures of written arithmetical calculation and more

extensive working memory tasks than previous research. As such, the present study

has the potential to contribute to the research literature not only by coordinatingsimultaneous demands of storage and processing but also by demonstrating that the

central executive is important for children’s written arithmetic skills in a number of

ways. Thus, the aim of the study was to pinpoint the contribution of different central

executive functions and to examine the contribution of the two subcomponents of

children’s written arithmetical skills.

Taking into account the findings from previous studies, the following predictions

were stated:

(1) It was predicted that all four central executive functions should contribute to

written arithmetical calculation, independent of the contribution of the two slave

systems, IQ, reading and age (cf. Swanson & Beebe-Frankenberger, 2004). That is,

the association between each specific central executive function should remain

significant even when measures related to IQ, reading, age, the phonological loop,

the visuospatial sketchpad, and the other three central executive functions are

included in the analysis.

(2) It was also predicted that the phonological loop and the visuospatial sketchpadshould contribute to written arithmetical calculation, independent of the

contribution of the central executive, IQ, reading and age (cf. Henry & MacLean,

2003; Kyttala et al., 2003; Maybery & Do, 2003).

184 Ulf Andersson


To examine these two hypotheses, tasks tapping the phonological loop, the visuospatial

sketchpad and the different central executive functions, proposed by Baddeley (1996),

were employed. The selection of the tasks was guided by previous studies providing a

theoretically motivated battery of relatively simple tasks that have been commonly used

to assess different working memory functions in children, and at the same time being

easy to administer to children (cf. Bull et al., 1999; Bull & Scerif; 2001; Gathercole et al.,2004; McLean & Hitch, 1999; Rasmussen & Bisanz, 2005; Zoelch et al., 2005). The trail-

making task was used to assess the ability to switch between operations, or retrieval

strategies (Baddeley, 1996; Lehto, Juujarvi, Kooistra, & Pulkkinen, 2003; McLean &

Hitch, 1999; Miyake et al., 2000). Semantic verbal fluency was included to tap

controlled retrieval of information from long-term memory (Baddeley, 1996; Ratcliff

et al., 1998; Riva, Nichelli, & Devoti, 2000). Focused attention and inhibition control

were assessed by the colour Stroop task (Stroop, 1935; Bull & Scerif, 2001; Rasmussen &

Bisanz, 2005). The ability to coordinate performance of two separate operations (e.g.concurrent storage and processing of information) was tapped by the counting-span and

visual-matrix span tasks. These complex dual tasks capture processes that tax both the

central executive and the two slave components (Daneman & Carpenter, 1980;

Gathercole et al., 2004; Swanson, 1992). These two tasks impose high demands on the

central executive because they require a shift in attention between the storage and

processing aspects of the tasks (Baddeley, 1996; Engle et al., 1999; Gathercole et al.,

2004; Towse & Hitch, 1995). The Digit Span and the Corsi-block span tasks were used to

tap the capacity of the phonological loop (Baddeley, Thomson, & Buchanan, 1975) andthe visuospatial sketchpad (Logie, 1995), respectively. As almost all working memory

tasks share variance with IQ and reading, and the aim was to examine the independent

contribution of working memory to mathematical performance, measures of IQ and

reading were included in the study (Engle et al., 1999; Swanson & Beebe-Frankenberger,

2004). An arithmetic fact-retrieval task was also used to assess the ability to solve simple

addition problems (e.g. 7 þ 6) by direct retrieval from long-term memory (Russell &

Ginsburg, 1984). This task was included in order to control for this specific process

component of arithmetical skill (see Dowker, 2005).

Method

ParticipantsA total of 141 children in grades 3 and 4 attending 21 public schools in the south-east

part of Sweden participated in this study. In total, 73 children (29 boys) were third-

graders, and 68 children (29 boys) were fourth-graders. The total sample had a mean age

of 124 months (SD ¼ 6:98 months). All children were fluent speakers of Swedish, had

normal or corrected-to-normal visual acuity, and no hearing loss. The 21 schools were

primarily located in middle-class areas. Thus, the sample is fairly homogeneous in

relation to socio-economic status and, since the children were drawn from many

schools, possible school effects should be reduced to a minimum.

General procedureThe tests were administered in two separate sessions, a group test session and an

individual working memory test session. The arithmetical test, reading test and

Raven’s progressive matrices test (Raven, 1976; sets B, C and D) were administered in

groups of four or five children. The group test session started with the arithmetical



test, followed by the reading test, and finished with the Raven’s progressive matrices

test. All group tests were preceded by at least two practice trials before the actual

testing started, to ensure that the children understood the task. Approximately 1–4

weeks after the group test, the working memory tests were administered. All working

memory tasks and the task of arithmetic fact retrieval were performed individually

and the test order was the same for all children. The tasks were conducted in thisorder: Digit Span, verbal fluency, visual-matrix span, arithmetic fact retrieval, Stroop

task, trail-making task, Corsi-block span and counting span. In all the tasks, there was

at least one practice trial before the testing phase to ensure that the children

understood the task. All instructions regarding the tasks were presented orally. The

group test session and the individual test session took approximately 90 minutes

each. The sessions were divided into two 40–45 minute sessions with a 15-minute

pause between them.

Tasks and procedure

Written arithmetical taskThe task was a paper-and-pencil test and consisted of three subtests. In subtest 1,

arithmetical calculation standard, the child was asked to solve six addition problems

and six subtraction problems (67 þ 42; 78 2 43; 568 þ 421; 658 2 437; 56 þ 47;

65 2 29; 545 þ 96; 384 þ 278; 824 2 488; 4,203 þ 825; 8,010 2 914;11,305 2 5,786) in 10 minutes. Thus, the task was designed so that the test

items became successively more difficult. The problems were presented horizontally,

because this is the primary form of presentation when starting to teach children

multi-digit arithmetic in Sweden. The children responded in Arabic form (e.g. 103).

Half the problems involved regrouping (i.e. carrying or borrowing). The children

were instructed that they could solve the problems in any way they wanted, and

that they should not struggle and spend to much time on a single problem but

instead try the next problem. Paper-and-pencil were allowed during performanceof the task. The number of correctly solved problems was used as dependent

measure.

In subtest 2, arithmetical equations, consisted of 12 arithmetic equations presented

horizontally (61 þ ___ ¼ 73; ___ £ 4 ¼ 16; ___ £ 5 ¼ 40; ___ þ 25 ¼ 500;

1; 000 2 ___ ¼ 550; ___ 2 8 ¼ 6; 8 £ ___ ¼ 24; ___ 2 50 ¼ 50; ___ 2 445 ¼ 55;

13 ¼ 6 þ ___; 136 ¼ ___ þ 27; 360 ¼ ___ 2 610). The task was to fill in the right

number so the equation was correct. The children were allowed 7 minutes to complete

that task. The same test procedure (i.e. instructions, paper-and-pencil, scoringprocedure) as in subtest 1 was used. In subtest 3, arithmetical combinations, the child

was presented with an answer and two to four numbers that had to be combined with

one to three arithmetic operations (addition, subtraction and multiplication) in order to

attain the predetermined answer. For example, if the answer was 12 and the three

numbers were 5, 8 and 9, a correct combination would be 9 þ 8 2 5. Thirteen problems

were included in subtest 3 (6, 17 ¼ 23; 24, 8 ¼ 16; 27, 113 ¼ 140; 9, 1 ¼ 9; 11,

26 ¼ 15; 5, 8, 9 ¼ 12; 10, 50, 90 ¼ 30; 11, 19, 25 ¼ 33; 4, 16, 4 ¼ 0; 25, 19, 11 ¼ 5;

4, 2, 5, 9 ¼ 9; 2, 5, 30, 60 ¼ 100; 1, 3, 8, 25 ¼ 0). The children were allowed 7minutes to complete that task. The same test procedure (i.e. instructions, paper-and-

pencil, scoring procedure) as in subtests 1 and 2 was used. The intercorrelations among

the three subtests of the arithmetical task were significant, and ranged from r ¼ :67

(p , :05) to r ¼ :77 (p , :05). Thus, the shared variance among the subtests ranged

186 Ulf Andersson


from r 2 ¼ :45 to r 2 ¼ :59, suggesting that the different subtests both taxed a common

component and different components of written arithmetical skill. As a consequence,

analyses were performed on a composite measure of the three subtests as well as on

each separate subtest.

Arithmetic fact retrievalThe child had to solve simple addition problems (e.g. 7 þ 6), by direct retrieval from

long-term memory. The test material consisted of 14 addition problems presented

horizontally. The child was instructed to provide an answer right away and encouraged

to guess if the answer was not available right away. One problem at a time was presented

on the computer screen and each problem was preceded by the word ‘READY’. When

the child announced that he/she was ready, the experimenter pressed the mouse button

and a problem was displayed on the computer screen until the child had responded tothe problem. A timer controlled by SuperLAB PRO 1.74 software started at the onset

of the problem and was stopped when the experimenter pressed the mouse button after

the child had given an oral response to the problem. During performance, the

experimenter also continually checked the child’s answers and registered each error.

The number of correctly solved problems with response times within 3 seconds was

used as the dependent measure (cf. Russell & Ginsburg, 1984).

Reading taskThe task was to read 12 short stories, 15–135 words in length, as fast and accurately as

possible (e.g. Lena and John are siblings. Lena plays with her doll. John plays with his

dog) and then answer a number of multiple-choice comprehension questions (question

1: John has a: brother; doll; car; sister, question 2: Lena plays with her: dog; doll; ball;

cycle) in relation to each story (Malmquist, 1977). The total number of questions was 33

and the child had 5 minutes to complete the test. Total number of correctly answered

questions constituted the dependent measure.

Central executive tasks

Semantic verbal fluency taskThe task was included to tap controlled retrieval of information from long-term memory

(Baddeley, 1996; Ratcliff et al., 1998; Riva et al., 2000). The child was instructed to

generate as many words as possible from two semantic categories (animals and food).

Sixty seconds were allowed for each category, and the child was encouraged to keep

trying to generate words, even if it was difficult, until the experimenter said stop. Thechildren’s responses were recorded by means of an Apple iPod MP3 player. The total

number of words correctly retrieved within the allowed time interval was used as the

dependent measure.

Trail-making taskThis paper-and-pencil test was used to assess the ability to switch between operations,

or retrieval strategies (Lee, Cheung, Chan, & Chan, 2000; Lehto et al., 2003; McLean &

Hitch, 1999; Miyake et al., 2000). The task included two different test conditions, A andB. In the A condition, the material consisted of 25 encircled numbers on a sheet of

paper. The task was to connect the 25 circles in numerical order as fast and accurately as

possible. Each child was presented with a practice trial consisting of eight circles (1-2-3-

4-5-6-7-8) and instructed to solve it before the actual trail-making tasks commenced.



In the B condition, half of the circles had a number in the centre (1–13), and half had a

letter (A–L). Children were asked to start at number 1 and make a trail with a pencil so

that each number alternated with its corresponding letter (i.e. 1-A-2-B-3-C : : : 12-L-13).

Each child was presented with a practice trial consisting of eight circles (1-A-2-B-3-C-4-D)

and instructed to solve it before the actual trail-making tasks commenced. In order to

obtain a ‘purer’ measure of shifting, the difference in solution time for the B and Aconditions (i.e. B 2 A ¼ difference) was used the dependent measure.

Colour Stroop taskThe Stroop (1935) task was used to measure inhibition control, and was administered

on three separate sheets of paper, one for each test condition. On the congruent

and incongruent conditions, the colour words, red, green, blue and yellow, were

presented in two columns with 12 words in each column. In the first test condition,colour-naming condition, the child’s task was to name aloud as quickly as possible

the colour of 24 colour patches. In the congruent condition, the colour words named

the ink colour in which they were printed (e.g. the word ‘RED’ printed in red ink).

The task was to name aloud as quickly as possible the ink colour in which each word

was printed (i.e. blue, red, yellow, green). In the incongruent condition, the words

named a colour incongruent with the ink colour in which they were printed (e.g. the

word ‘RED’ in green ink). The task was to name aloud as quickly as possible the ink

colour in which each word was printed while ignoring the word’s identity. The testprocedure included a short pause between each paper, and the order of test

conditions was the same for all participants; that is, colour naming, congruent and

incongruent. Prior to each test condition, the child was presented with a practice

trial to ensure that the children understood the task. The experimenter used a

stopwatch to measure the total time it took to name the colours, read the 24 words

or name the ink colour of the 24 words. During performance, the experimenter also

continually checked the participant’s answers and registered each error. In order to

obtain a ‘purer’ measure of inhibition control, the difference in total response timefor the incongruent condition and the first test condition, that is, the colour-naming

condition (i.e. incongruent-colour naming ¼ difference), was used as the measure of

inhibition control.

Counting-span taskThis task tapped the ability to coordinate the performance of two separate operations

(e.g. simultaneous storage and processing of numeric information; Baddeley, 1996;Engle et al., 1999; Gathercole et al., 2004; Towse & Hitch, 1995). The task was

administered using the SuperLAB PRO 1.74 software program, which was run on an

Apple Power Mac G4 laptop computer. The stimuli consisted of black and red dots, each

with a diameter of 1 cm, arranged in a random pattern. All patterns included four black

dots, whereas the number of red dots varied from two to seven dots. The black dots

were included to prevent the child from using a subitizing strategy when counting the

red dots. The task was to count the red dots in each pattern and then recall the number

of red dots from each pattern in the sequence in correct serial order. The first span sizeemployed was two, the next was three and so on up to six patterns. Two sequences

were presented for each span size. After a practice trial, the child was tested on span size

two according to the following procedure. The first dot pattern was presented on

the computer screen and after the child had counted the red dots the child pressed

188 Ulf Andersson


the space bar and the next pattern was displayed on the screen. After counting the dots

in the second pattern, the child pressed the space bar again, a question mark appeared

on the screen and the child had to recall the number of red dots in the two patterns.

Testing stopped when the participant failed to repeat both trials at any particular span

length. Span size was determined as the longest sequence perfectly recalled, plus .5

points for each subsequent sequence of the same length recalled correctly (cf. Hulme,Roodenrys, Brown, & Mercer, 1995).

Visual-matrix span taskThis task tapped the ability to coordinate performance of two separate operations (e.g.simultaneous storage and processing of visual information; Baddeley, 1996; Engle et al.,

1999; Gathercole et al., 2004; Towse & Hitch, 1995). The same computer and software

program was used as for the counting-span task (see above). The child was presented a

number of dots in a matrix. The task was to remember the location of the dots in the

matrix. A total of 32 matrices constituted the test material. The matrices were made up of

different numbers of squares. Each square was 2 cm, and was drawn on a white

background. The dots were black with a diameter of 1 cm. One matrix at a time was

displayed on the computer screen for 5 seconds. Then the matrix was removed, and thechild was asked a process question: ‘Were there any dots in the first column?’ After

answering the process question, the child was required to draw dots in the correct

squares in an identical matrix. The first matrix had six squares (2 £ 3) and included two

dots. The matrices in the next span size had nine squares (3 £ 3) and also included two

dots. The third span size also had nine squares but included three dots. The complexity

of the matrices increased for each new span size by either increasing the size of the

matrix or increasing the number of dots. The complexity ranged from a matrix of six

squares and two dots to a matrix of 56 squares and nine dots. Two different matriceswere presented for each span size. Testing stopped when the child had failed twice to

repeat both trials at any particular span length. Thus, testing proceeded as long as the

child succeeded to reproduce one of the two trials of the same span length. Visual-matrix

span was measured as the most complex matrix remembered correctly, plus .5 points if

the participant managed to replicate correctly both trials in the same span length.

Test of the phonological loop

Digit Span taskThe experimenter read series of digits, 1–9, to the participant from the computer screen

(Apple Power Mac G4 laptop computer) at a rate of one digit per second. The Microsoft

PowerPoint software program (version 10.1.5) administered the presentation of the

digits. The first span size employed was three, the next was four, and so on up to eightdigits. Two sequences were presented for each span size. The task was to remember

the digits and recall them in correct serial order (e.g. 7-2-8-6: recall!). Testing stopped

when the child made a mistake in both trials of the same span length. The same scoring

procedure as in the counting-span task was used.

Test of the visuospatial sketchpad

Corsi-block spanThe apparatus consisted of 10 square wooden blocks each of 3 cm which were painted

blue and glued in random positions on a white board (28 £ 21.5 cm). To aid the



experimenter, the numbers 1–10 were painted on the side of each block facing away

from the child. The child was instructed to observe the experimenter tap a sequence of

blocks and then attempt to repeat the sequence in the same order. Blocks were tapped

at a rate of one per second and the first trial was a sequence of length 2. Two different

sequences were presented for each span size. Testing proceeded as long as the child

succeeded in reproducing one of the two trials of the same span length. Thus, testingstopped when the child failed to repeat both trials at any particular span length. Span

size was determined as the longest sequence perfectly repeated, plus .5 points for each

subsequent sequence of the same span length repeated correctly.

Results

As a first step in order to examine the present hypotheses, partial correlations were

calculated between the tasks used in the study when controlling for age (in months).

The results of these analyses and mean and standard deviations are displayed in Table 1.

Performance on the written arithmetical task and the arithmetical fact retrieval task

was significantly correlated with all measures used in the study, the partial correlationsranging in size from 2 .23 to .90. The partial correlation patterns for the three different

subtests of the arithmetical task were very similar. As expected, the partial correlations

between the arithmetical tasks, and IQ (Raven’s), and reading were among the strongest

(from .45 to .60; Geary, 1993; Gersten, Jordan, & Flojo, 2005; Lee et al., 2004; Swanson

& Beebe-Frankenberger, 2004). Furthermore, scores on the reading task and Raven’s

matrices were significantly correlated with all working memory tasks, ranging from .18

to .61. The partial correlations among the seven working memory tasks ranged from .09

(non-significant) to .44, suggesting that the different tasks employed are indeed tappingdifferent aspects of the working memory system. It is interesting to note that the mean

partial correlation among the five central executive tasks is .26. These modest to weak

partial correlations converge with results from previous studies, and imply that the

central executive can be divided into separate but related functions (Lehto et al., 2003;

Miyake et al., 2000). As such, the present finding corroborates the current view of both

unity and diversity among central executive functions (Duncan, Johnston, Swales, &

Freer, 1997; Miyake et al., 2000; Zoelch et al., 2005). Furthermore, the partial

correlations between performance on the written arithmetical composite measure andthe five central executive tasks were 2 .28 or stronger. In sum, the present patterns of

partial correlations among the working memory tasks and the written arithmetical tasks

indicate the possibility that the different central executive functions contribute variance

to arithmetical skill.

The contribution of different working memory components to written arithmeticalskillsIn order to examine that the four central executive functions contribute to written

arithmetical skill independent of the contribution of the two slave components, IQ,

reading and age, and that the two slave components also contribute independent of the

contribution of the central executive, IQ, reading and age a hierarchical regressionanalysis was calculated on the composite measure. The first block included measures of

reading, IQ and age, whereas the second block included all seven working memory

tasks. All variables in the blocks were entered simultaneously. The results of the analysis

are presented in Table 2.

190 Ulf Andersson


Table

1.

Des

crip

tive

stat

istics

and

par

tial

corr

elat

ions

among

the

task

suse

din

the

study

contr

olli

ng

for

chro

nolo

gica

lag

e

Task

sM

SD2

34

56

78

910

11

12

13

14

1.A

rith

met

ical

com

posi

tem

easu

re18.3

18.2

0.8

7.9

0.9

0.7

8.6

0.5

8.5

5.5

0.4

02

.61

2.2

8.4

6.3

32.A

rith

met

ical

calc

ula

tion

stan

dar

d6.0

63.2

8–

.65

.66

.67

.55

.45

.45

.41

.39

2.5

22

.23

.40

.26

3.A

rith

met

ical

equat

ions

5.6

92.9

4–

.76

.76

.55

.56

.54

.52

.41

2.5

82

.24

.40

.34

4.A

rith

met

ical

com

bin

atio

ns

6.5

52.9

4–

.66

.51

.52

.50

.42

.28

2.5

52

.29

.43

.28

5.A

rith

met

icfa

ctre

trie

val

8.1

33.8

9–

.58

.45

.37

.44

.43

2.5

52

.29

.34

.38

6.R

eadin

gta

sk16.3

36.4

1–

.35

.28

.28

.29

2.4

22

.27

.21

.18

7.R

aven

’sm

atri

ces

22.4

27.2

7–

.36

.41

.27

2.6

12

.18

.23

.29

8.C

ounting

span

task

4.2

60.8

5–

.42

.11

2.3

62

.09

.38

.28

9.V

isual

-mat

rix

span

7.4

42.0

4–

.18

2.4

42

.25

.32

.39

10.Ver

bal

fluen

cy29.6

18.3

4–

2.3

12

.10

.20

.31

11.Tr

ail-m

akin

g†73.7

742.8

6–

.29

2.2

62

.38

12.St

roop

task

†22.6

511.6

3–

2.2

12

.24

13.D

igit

Span

task

5.0

60.7

5–

.26

14.C

ors

i-blo

cksp

an5.6

81.0

7–

N¼

141,df

¼138,co

rrel

atio

nco

effici

ents

grea

ter

than

.16

are

sign

ifica

nt

atth

e5%

leve

l.†T

ime-

bas

edte

st,re

sultin

gin

neg

ativ

eco

rrel

atio

ns.



The three control variables included in the first block together accounted for 55% of

the variation in arithmetical performance. Thus, consistent with previous studies,

individual differences in reading, IQ and age contribute to children’s arithmetical skills

(Bull & Scerif, 2001; Keeler & Swanson, 2001; Swanson & Beebe-Frankenberger, 2004;

Table 2. Regression analysis of written arithmetical performance: The contribution of IQ, reading, age

and different working memory tasks

B SE b t pr2†

Model 1 Arithmetical composite measureBlock 1 Fð3; 137Þ ¼ 55:57, p , :001, R 2 ¼ :55

Age 0.23 .07 0.20 3.41** .04Raven’s 0.45 .07 0.40 6.53** .14Reading 0.57 .08 0.44 7.16** .17

Block 2 Fð10; 130Þ ¼ 32:78, p , :001, R 2 ¼ :72, DR 2 ¼ :17Age 0.16 .06 0.13 2.74** .02Raven’s 0.18 .07 0.16 2.67* .01Reading 0.36 .07 0.28 5.20** .06Counting span 2.16 .55 0.22 3.95** .03Visual-matrix span 0.43 .24 0.11 1.83 .00Verbal fluency 0.14 .05 0.15 2.81** .02Trail making 0.00 .01 20.17 22.51** .01Stroop task 0.00 .04 20.03 20.65 .00Digit Span 1.71 .58 0.16 2.96** .02Corsi-block span 20.31 .42 20.04 20.74 .00

Model 2 Arithmetical composite measureBlock 1 Fð4; 136Þ ¼ 49:60, p , :001, R 2 ¼ :59

Counting span 3.12 .60 0.32 5.19** .08Verbal fluency 0.21 .06 0.22 3.74** .04Trail making 0.00 .01 20.40 26.37** .12Digit Span 1.82 .66 0.17 2.78** .02

Block 2 Fð7; 133Þ ¼ 45:87, p ¼ :000, R 2 ¼ :71, DR 2 ¼ :11Counting span 2.31 .53 0.24 4.35** .04Verbal fluency 0.13 .05 0.14 2.68** .01Trail making 0.00 .01 20.19 22.93** .02Digit Span 1.88 .57 0.17 3.31** .02Age 0.18 .06 0.15 3.18** .02Raven’s 0.20 .07 0.18 2.91** .02Reading 0.38 .07 0.30 5.51** .07

Model 3 Arithmetical composite measureFð8; 132Þ ¼ 61:19, p , :001, R 2 ¼ :79Arithmetical fact retrieval 0.86 .12 0.41 7.07** .08Age 0.21 .05 0.18 4.31** .03Raven’s 0.16 .06 0.14 2.78** .01Reading 0.20 .06 0.16 3.09** .02Counting span 1.92 .46 0.20 4.22** .03Verbal fluency 0.00 .04 0.05 1.12 .00Trail making 0.00 .01 20.10 21.70 .00Digit Span 1.35 .49 0.12 2.76** .01

†pr2¼ squared part correlations, represents the unique contribution for each variable.*p , .05, **p , .01.

192 Ulf Andersson


Wilson & Swanson, 2001). When the seven working memory tasks in block 2 were

entered into the regression equation, the model accounted for a total of 72% of the

individual differences in written arithmetical performance. Out of this 72%, different

working memory tasks accounted for 17% variance (DR2 ¼ :17). In addition to age, fluid

IQ (Raven’s) and reading, four working memory tasks (counting span, verbal fluency,

trail making and Digit Span) turned out to be significant predictors of writtenarithmetical performance. The squared partial correlations displayed in Table 3 show

that the counting span contributed 3% variance, the trail-making tasks 1%, whereas the

verbal fluency task and the Digit Span task accounted for 2% variance each. Thus, the

four working memory tasks account for equal amounts of variance. In addition, age,

Raven’s and reading accounted for 2, 1 and 6% variance, respectively. The hierarchical

regression analysis clearly demonstrates that the different measures of working memory

contribute to children’s arithmetical performance; however, the regression analysis in

combination with the partial correlations displayed in Table 2 also show that muchvariance is shared with fluid IQ, reading and age. These three measures account for 55%

of the variance when entered first into the regression, whereas working memory

accounts for an additional 17% variance. The question is how much additional variance

is accounted for by measures of fluid IQ, reading and age? To address this question and to

establish the total amount of variance accounted for by the four significant working

memory tasks when entered first in the regression equation (i.e. block 1), a second

hierarchical regression analysis was computed. As can be seen in the middle part of

Table 3, the four working memory measures together accounted for 59% of the variance.The three control variables, age, Raven’s and reading, contributed an additional 11%

variance (DR2 ¼ :11). Hence, when entered first in the regression, the four working

memory measures accounted for 4% extra variance in children’s written arithmetical

skills compared with reading ability, fluid IQ and age.

To master accurate and automatic retrieval of basic arithmetic facts is an important

prerequisite for efficient arithmetical skill (Geary, 1993; Gersten et al., 2005; McCloskey,

Caramazza, & Basili, 1985). The very strong partial correlation obtained between

arithmetic fact retrieval and arithmetical performance (see Table 1) provides additionalsupport to that connection. Therefore, a third regression analysis was computed to

investigate whether the seven significant predictors in model 1 will continue to

contribute variance to arithmetical performance, when arithmetic fact retrieval is

entered into the model. All eight variables were entered simultaneously in one block.

The outcome of the analysis is displayed in the lower part of Table 3. Model 3 accounted

for 79% of the variance, an increase in R2 of 7% compared with model 1 (R2 ¼ :72) that

can be attributed to arithmetic fact retrieval. Another important finding related to

model 3 is that the significant contribution of the verbal fluency task and trail-makingtask were eliminated, whereas the other five tasks (age, reading, Raven’s, counting span

and Digit Span) remained significant predictors of written arithmetical performance.

The present study sought to address the limitations in previous research regarding

the use of general measures of written arithmetical skill, by employing a relatively pure

measure of written arithmetical skill. However, the correlations among the three

subtests of written arithmetical skill were not perfect (range from r ¼ :67 to r ¼ :77),

suggesting that the different subtests, to some extent, tax different written arithmetical

skill components. Therefore, three additional regression analyses were performed tocheck out the possibility that the three different subtests are differentially predicted by

the different working memory functions (cf. Rasmussen & Bisanz, 2005; see also Jordan

& Hanich, 2000; Trbovich & LeFevre, 2003). As can be seen in Table 3, the three analyses



generated quite similar results, but a few differences also emerged. Model 4 accounted

for 56% variance in performance on the arithmetical calculation standard subtest, and

age, reading skill, counting span, verbal fluency and Digit Span emerged as significantpredictors. In model 5, Raven’s, reading, counting span, visual-matrix span and verbal

fluency were significant predictors of performance on the arithmetical equations

subtest. The complete model predicted 64% of the variability. Performance on the

arithmetical combinations subtest was significantly predicted by Raven’s, reading,

Table 3. Regression analysis of the different tasks of written arithmetical skills: The contribution of IQ,

reading, age and different working memory tasks

B SE b t pr2†

Model 4 Arithmetical calculation standardFð10; 130Þ ¼ 16:71, p , :001, R 2 ¼ :56Age 0.08 .03 0.18 2.91** .03Raven’s 0.04 .03 0.08 1.08 .00Reading 0.15 .04 0.29 4.24** .06Counting span 0.65 .27 0.17 2.41* .02Visual-matrix span 0.14 .12 0.09 1.20 .00Verbal fluency 0.07 .03 0.18 2.71** .02Trail making 20.01 .01 20.15 21.83 .01Stroop task 0.00 .02 20.01 20.14 .00Digit Span 0.65 .29 0.15 2.26* .02Corsi-block span 20.17 .21 20.06 20.84 .00

Model 5 Arithmetical equationsFð10; 130Þ ¼ 23:48, p , :001, R 2 ¼ :64Age 0.03 .02 0.06 1.07 .00Raven’s 0.07 .03 0.18 2.63** .02Reading 0.11 .03 0.24 4.00** .04Counting span 0.78 .22 0.23 3.58** .03Visual-matrix span 0.23 .10 0.16 2.46* .02Verbal fluency 0.06 .02 0.17 2.91** .02Trail making 20.01 .01 20.13 21.75 .00Stroop task 0.00 .02 20.01 20.16 .00Digit Span 0.38 .23 0.10 1.62 .00Corsi-block span 20.17 .21 20.06 20.84 .00

Model 6 Arithmetical combinationsFð10; 130Þ ¼ 16:32, p , :001, R 2 ¼ :56Age 0.05 .03 0.12 1.95 .01Raven’s 0.07 .03 0.18 2.41* .02Reading 0.10 .03 0.23 3.33** .04Counting span 0.72 .24 0.21 2.93** .03Visual-matrix span 0.06 .11 0.04 0.54 .00Verbal fluency 0.02 .02 0.04 0.66 .00Trail making 20.01 .01 20.17 22.01* .01Stroop task 20.02 .02 20.08 21.15 .00Digit Span 0.69 .26 0.18 2.66** .02Corsi-block span 20.09 .19 20.03 20.49 .00

†pr2¼ squared part correlations, represents the unique contribution for each variable.*p , .05, **p , .01.

194 Ulf Andersson


counting span, trail making and the Digit Span task. This model (model 6) accounted for

a total of 56% variance.

Since arithmetical fact retrieval is an important aspect of arithmetical skill, and

including the arithmetical fact retrieval task in model 3 changed the contribution of

different working memory functions, it is interesting to examine what working memory

functions predict it. Furthermore, previous studies showed that the contribution of theworking memory system is different depending on the type of arithmetical task at hand

(Fuchs et al., 2005; Rasmussen & Bisanz, 2005). For that reason, a seventh regression

model was computed, and the results of the analysis are displayed in Table 4.

Model 7 captured 54% of the variation in arithmetical fact retrieval. The important

result was that only three variables accounted for unique variance in arithmetical fact

retrieval. Reading was the strongest predictor, accounting for 8% variance, whereas the

verbal fluency task and the trail-making task accounted for 3% and 1% variance,respectively.

Discussion

The aim of the present study was to examine the contribution of different centralexecutive functions, the phonological loop and the visuospatial sketchpad to children’s

written arithmetical skills. Overall, the present findings give further weight to previous

evidence that working memory in general and the central executive in particular

contribute to children’s arithmetical skills beyond the contribution of fluid IQ, reading

and age (cf. Bull & Scerif, 2001; Keeler & Swanson, 2001; Lehto, 1995; Swanson & Beebe-

Frankenberger, 2004; Wilson & Swanson, 2001). More specifically, three tasks tapping the

central executive (i.e. counting span, verbal fluency, trail making) and one task tapping

the phonological loop (i.e. Digit Span) accounted for 59% variance in written arithmetic,which is 4% more variance than captured by the three control variables (age, reading and

Raven’s). A substantial amount (54%) of variance in arithmetic fact retrieval was

accounted for by the 10 tasks employed in the study, but only the reading task, and two

central executive tasks (i.e. verbal fluency, trail making) emerged as significant predictors.

Table 4. Regression analysis of arithmetical fact retrieval: the contribution of IQ, reading, age

and different working memory tasks

B SE b t pr2†

Model 7 Fð10; 130Þ ¼ 15:44, p , :001, R 2 ¼ :54Age 0.00 0.04 20.08 21.21 .00Raven’s 0.00 0.04 0.06 0.76 .00Reading 0.20 0.04 0.34 4.85** .08Counting span 0.30 0.33 0.07 0.92 .00Visual-matrix span 0.22 0.14 0.11 1.51 .00Verbal fluency 0.00 0.03 0.19 2.85** .03Trail making 0.00 0.01 20.17 22.01* .01Stroop task 0.00 0.02 20.05 20.76 .00Digit Span 0.44 0.35 0.08 1.25 .00Corsi-block span 0.32 0.25 0.09 1.26 .00

†pr2¼ squared partial correlation, represents the unique contribution for each variable.*p , .05, **p , .01.



The novel and important findings in this study concern the contribution of different

central executive functions to written arithmetical skill. In accordance with the

prediction, three out of four central executive functions (concurrent processing and

store of information, shifting, retrieval of information from long-term memory,

inhibition control) contributed to performance in written arithmetic. Similar to a

number of studies, the ability to concurrently process and store information accountedfor individual differences in children’s arithmetical skills (Bull & Scerif, 2001; Fuchs

et al., 2005; Gathercole & Pickering, 2000; Gathercole et al., 2004; Keeler & Swanson,

2001; Lee et al., 2004; Lehto, 1995; Noel et al., 2004; Swanson, 1994; Swanson & Beebe-

Frankenberger, 2004; Wilson & Swanson, 2001). The significant contribution of shifting

between operations constitutes a replication of previous studies (Bull et al., 1999; Bull &

Scerif, 2001; McLean & Hitch, 1999), whereas the contribution of retrieval of

information from long-term memory to children’s arithmetical performance is a novel

finding not previously reported. Thus, the present results regarding central executivefunctions converge with results reported by Bull and Scerif (2001) showing that

different central executive functions contribute to written arithmetical performance in

children.

Consistent with the second prediction, the capacity of the phonological loop to store

and process numerical information, tapped by the Digit Span task, accounted for

variance in children’s arithmetical performance independent of the contribution of

reading, age and IQ (cf. Maybery & Do, 2003). It is important to note that the measure of

concurrent processing and storage of numerical information (i.e. counting span) and thephonological loop measure (i.e. Digit Span) remained significant predictors even when

the contribution of arithmetical fact retrieval in addition to reading, fluid IQ and age was

controlled. This finding indicates that these two working memory functions seem to

operate independently of the long-term memory system. In contrast, the trail-making

and verbal fluency tasks did not capture variance in arithmetical performance when fact

retrieval was entered into the analysis. The fact that the contribution of the verbal

fluency task was eliminated when arithmetic fact retrieval was included in the

regression was expected, as this task taps the retrieval of semantic information fromlong-term memory. The non-significant contribution of the trail-making task was less

expected and indicates that this measure of shifting also depends on information from

long-term memory. Thus, the contribution of these two central executive functions

during arithmetical performance seems to be related to retrieval of task-relevant

information (e.g. arithmetic rules, arithmetic facts) and shifting between sets of

arithmetic knowledge stored in long-term memory (Swanson & Sachse-Lee, 2001). This

interpretation is supported by the finding that the trail-making task and the verbal

fluency task were the only working memory tasks which emerged as significantpredictors of arithmetical retrieval performance. Thus, as might be expected, processes

of activating and retrieving information from long-term memory ascribed to the

central executive are the main working memory contributors to direct retrieval of

basic arithmetic facts. Other working memory processes, such as temporary storage of

information and coordinating different sub-processes, are not required when

performing this highly automated task.

The present study shows that the counting-span task predicted individual differences

in the composite measure of written arithmetical calculation (models 1–3), but thevisual-matrix span task, a task assumed to tap similar processes as the counting-span task

(i.e. concurrent processing and storage of (visual) information), did not. Neither was the

Corsi-block span task associated with written arithmetical performance, unlike the Digit

196 Ulf Andersson


Span task. However, the visual-matrix span task, as well as the counting-span task,

emerged as significant predictors of the arithmetical equations subtest (model 5), and

the trail-making task was only a significant predictor of the combinations subtest

(model 6). Thus, an important finding is that the contribution of the different working

memory functions to written arithmetical calculation differs to some extent depending

on how the calculation task is designed (cf. Rasmussen & Bisanz, 2005; see also Jordan &Hanich, 2000; Trbovich & LeFevre, 2003). The results from the regression models

indicate that children at this age (mean age ¼ 10 years and 4 months) primarily rely on

verbal strategies that draw upon the phonological loop and the central executive

function of concurrent processing and storage of numerical information when solving

written arithmetical problems. The contribution of the phonological loop indicates that

written arithmetical performance in children requires a verbal storage system which can

represent visually presented numbers, and retain interim results, for example, carry and

borrow information by means of a phonological code (Furst & Hitch, 2000; Logie et al.,1994). Still, when children at the present age solve arithmetical equations they seem to

employ both visual (the visual-matrix span) and numerical-verbal strategies (counting

span). The employment of visual coding strategies when solving arithmetical equations

is possibly a reflection of the fact that they entail a (mental) rearrangement of the

numbers in the problem (e.g. 61 þ ___ ¼ 73; 73 2 61 ¼ 13). This rearrangement

process requires concurrent storage and processing of visual information and thus

engages the visuospatial sketchpad in addition to the central executive. The significant

contribution of the trail-making task to the arithmetical combinations subtest is mostprobably due to the fact that the child had to combine two to four numbers with one to

three arithmetic operations (addition, subtraction and multiplication) in order to attain

the predetermined answer. That is, switching between operations constituted an

essential requirement when solving this type of arithmetical problem.

Arithmetical performance, especially multi-digit tasks, involves a variety of

processes: retrieval of arithmetic rules and arithmetic facts from long-term memory,

calculating and storing interim results, and performing carrying or borrowing

operations (Ashcraft, 1992, 1995; Furst & Hitch, 2000; Geary, 1993; McCloskey et al.,1985; Seitz & Schumann-Hengsteler, 2002). Hence, a theoretically straightforward

account of the contribution of the counting-span task may be that it reflects individual

differences in the ability to monitor and coordinate the different sub-processes (e.g.

simultaneous demands of storing and processing numeric information) involved in

arithmetical calculation (cf. Swanson & Beebe-Frankenberger, 2004). These crucial

central executive processes seem to operate independently of the long-term memory

system, as indicated by the finding that the counting-span task remained to predict

arithmetical performance when arithmetical fact retrieval was included in theregression model (model 3).

In contrast to Bull and Scerif (2001), but in line with Swanson and Beebe-

Frankenberger (2004), inhibition control did not provide a significant contribution to

arithmetical performance. One possible explanation to this non-significant association is

that inhibition is a very fundamental process in working memory which is involved in

almost all working-memory-controlled functions (Cantor & Engle, 1993; Conway &

Engle, 1994; Pennington, 1994). For example, the function of shifting requires

inhibition of the ongoing operation in order to start a new operation. Retrieval ofinformation from long-term memory probably involves inhibition as well, because when

information is retrieved it is important to prevent task-irrelevant information from

entering the working memory system along with the target information. Furthermore,



demands on inhibition processes are most probably critical in all working memory

functions that involve continuous temporary storage of information. That is, prior

information held in working memory must be inhibited from interfering with the

present information (i.e. intrusion errors; Passolunghi & Siegel, 2001). Thus, processes

of inhibition control involved in arithmetical performance were probably captured by

all other tasks that were associated with arithmetical performance, thereby eliminatingthe significant correlation between the Stroop task and written arithmetic.

From a developmental point of view, the present findings demonstrate that there are

reasons to assume that an efficient and flexible working memory system is important for

children’s learning of arithmetic (cf. Gathercole & Alloway, 2004; Gathercole &

Pickering, 2000, 2001; Swanson, 2006). The present results suggest that almost all

components and functions of working memory are important for children’s skill

development in arithmetic. However, the central executive function of coordinating and

monitoring two separate operations seems to be particularly important, as this functionis responsible for handling many different processes involved in performing and

learning arithmetic. For example, working capacity is required in many classroom

learning activities, such as comprehending and following complex instructions or taking

notes whilst listening to the teacher (Gathercole & Alloway, 2004). Furthermore,

learning entails the integration of new information with already existing knowledge, a

process that is assumed to require the capacity to simultaneously process and store

information and a function that is provided by the central executive component

(e.g. Baddeley, 2000; Swanson & Beebe-Frankenberger, 2004). This specific centralexecutive function in combination with the phonological loop, and to some extent the

visuospatial sketchpad, appear to be critical for the child’s ability to develop a mixture of

solution strategies (i.e. verbal and visual strategies) and to use the most efficient strategy

when solving different forms of arithmetical problems.

The fact that the verbal fluency task and the trail-making task emerged as significant

predictors of automatic fact retrieval suggests that accessing arithmetical knowledge

from long-term memory and shifting between sets of arithmetic knowledge are

important central executive functions for children’s arithmetical skill development.Thus, it seems that a child’s capability to develop a high skill level in written arithmetical

calculation, and particularly in automatic fact retrieval, is constrained by the central

executive functions responsible for interaction with the long-term memory system.

The independent contribution of the different working memory components and

functions suggests that the relationship between individual differences in working

memory and arithmetic is mediated by a number of resources, not only processing

efficiency but also storage capacity and (central) executive ability (Bayliss, Jarrold,

Baddeley, Gunn, & Leigh, 2005; Bayliss, Jarrold, Gunn, & Baddeley, 2003; Engle et al.,1999). This, in turn, speaks in favour of a multi-resource view of working memory,

instead of a (single) resource-sharing view, which states that individual differences in

working memory capacity is determined by the efficiency of separate resource pools

for processing and storage (Baddeley, 1986; Baddeley & Hitch, 1974; Case, Kurland, &

Goldberg, 1982; Daneman & Carpenter, 1980; Engle et al., 1999; Shah & Miyake, 1996).

The above discussion demonstrates that there are a number of reasons why children

with poor working memory might have problems with learning arithmetic. One way to

help these children improve their learning might be to reduce the demands on theirworking memory while performing learning activities (see Gathercole & Alloway, 2004).

This can be accomplished by providing external memory aids and giving short and

simple instructions (possibly in writing; see Gathercole & Alloway, 2004; Gathercole

198 Ulf Andersson


et al., 2006). As the different arithmetical subtests used in the present study seem to

draw upon different working memory components, another way to help children with

low working memory capacity is to present arithmetical problems in the most

favourable format or in multiple formats. That is, employing the presentation format that

imposes the least demand on the aspect of working memory which is less efficient, and

instead utilizes the efficient aspects of working memory for that particular child. Analternative method to reducing the working memory demand while learning arithmetic

might be to increase the child’s working memory capacity through training. Klingberg

and colleagues have presented behavioural and neurophysiological evidence in

children and adults that the working memory capacity can be enhanced by systematic

training, and that the training effect also generalizes to non-trained tasks requiring

working memory capacity (Klingberg et al., 2005; Klingberg, Forssberg, & Westerberg,

2002; Olesen, Westerberg, & Klingberg, 2004). Finally, the present findings suggest that

a few working memory tasks (i.e. counting span, verbal fluency, trail making, Digit Span)can be used to predict children’s future mathematical skills in arithmetic and, as such,

they can be used as a complement to traditional arithmetical screening tasks when

screening for possible future learning difficulties in arithmetic (cf. Gathercole &

Pickering, 2000, 2001; Swanson, 2006; see also Swanson, Saez, & Gerber, 2006).

In conclusion, the present study demonstrates that children’s written arithmetical

skills are constrained by their working memory capacity. A key finding is that the

phonological loop and three different central executive functions (i.e. coordination of

concurrent processing and storage of numerical information, shifting, retrieval ofinformation from long-term memory) contribute to written arithmetical performance

in children. These findings demonstrate that performing arithmetic tasks involves a

number of processes that must be handled by a flexible and efficient working memory

system. More specifically, one crucial central executive function is to monitor and

coordinate the multiple processes during arithmetical performance. Another key

process performed by the central executive is to access arithmetical knowledge (e.g.

arithmetic rules, arithmetic facts) from long-term memory, which also involves shifting

between sets of arithmetic knowledge. Moreover, the contribution of the phonologicalloop and the central executive function of concurrent processing and storage of

numerical information indicate that children aged 9–10 years utilize verbal coding

strategies during arithmetical performance and that temporary storage capacity is

important even when performing written arithmetical tasks. Another important finding

is that the contribution of the different working memory functions to written

arithmetical calculation differs to some extent depending on how the calculation task is

designed.

Acknowledgements

This research was supported by a grant from The Bank of Sweden Tercentenary Foundation

(J2002-0210: 2).

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Received 25 October 2006; revised version received 18 April 2007


working memory as a predictor of written arithmetical skills

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central executive measures

longterm memory

societyworking memory

mclean hitch

hitch mcauley

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