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The British Psychological Society
British Journal of Educational Psychology (2008), 78, 181–203 q 2008 The British Psychological Society
www.bpsjournals.co.uk
Working memory as a predictor of written arithmetical skills in children: The importance of central executive functions Ulf Andersson* Department of Behavioural Sciences, Linko¨ping University, Sweden Background. The study was conducted in an attempt to further our understanding of how working memory contributes to written arithmetical skills in children. Aim. The aim 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. Sample and method. A total of 141 third- and fourth-graders were administered arithmetical tasks and measures of working memory, fluid IQ and reading. Regression analysis was used to examine the relationship between working memory and written arithmetical skills. Results. Three central executive measures (counting span, trail making and verbal fluency) and one phonological loop measure (Digit Span) were significant and predictors of arithmetical performance when the influence of reading, age and IQ was controlled 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 was hypothesized that monitoring and coordinating multiple processes, and accessing arithmetical knowledge from long-term memory, are important central executive functions during arithmetical performance. The contribution of the phonological loop and the central executive (concurrent processing and storage of numerical information) indicates that children aged 9–10 years primarily utilize verbal coding strategies during written arithmetical performance.
Empirical studies show that working memory is an important factor in children’s mathematical 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, Linko¨ping University, SE-581 83 Linko¨ping, Sweden (e-mail:
[email protected]). DOI:10.1348/000709907X209854
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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 memory system. 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 that the 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 of the 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 threecomponent 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 9to 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 Moperators, which is the maximum number of schemes that can be concurrently activated 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,
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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 monitoring simultaneous 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 (trailmaking task), and the ability to hold and manipulate information accessed from longterm memory, in a sample of 33 third- and fourth-graders. One-third of the children had specific 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 are central 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 examined and 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;
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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 Kytta¨la¨, 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 these studies 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 with working 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 coordinating simultaneous 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)
(2)
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. It was also predicted that the phonological loop and the visuospatial sketchpad should contribute to written arithmetical calculation, independent of the contribution of the central executive, IQ, reading and age (cf. Henry & MacLean, 2003; Kytta¨la¨ et al., 2003; Maybery & Do, 2003).
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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 trailmaking 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) and the 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 Participants A 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 thirdgraders, 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 procedure The 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
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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 this order: 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 task The 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 performance of 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, scoring procedure) 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 7 minutes to complete that task. The same test procedure (i.e. instructions, paper-andpencil, 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
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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 retrieval The 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 to the 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 task The 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 task The 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. The children’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 task This 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 and B. 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 (1-2-34-5-6-7-8) and instructed to solve it before the actual trail-making tasks commenced.
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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 A conditions (i.e. B 2 A ¼ difference) was used the dependent measure.
Colour Stroop task The 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 test procedure 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 time for 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 task This 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 size employed 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
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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 task This 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 the child 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 matrices were 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 task The 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 eight digits. 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 span The 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
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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, testing stopped 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 correlations ranging 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 tapping different 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 and the 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 arithmetical skills In 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 regression analysis 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.
18.31 6.06 5.69 6.55 8.13 16.33 22.42 4.26 7.44 29.61 73.77 22.65 5.06 5.68
M 8.20 3.28 2.94 2.94 3.89 6.41 7.27 0.85 2.04 8.34 42.86 11.63 0.75 1.07
SD .87 –
2 .90 .65 –
3 .90 .66 .76 –
4 .78 .67 .76 .66 –
5 .60 .55 .55 .51 .58 –
6
†
N ¼ 141, df ¼ 138, correlation coefficients greater than .16 are significant at the 5% level. Time-based test, resulting in negative correlations.
1. Arithmetical composite measure 2. Arithmetical calculation standard 3. Arithmetical equations 4. Arithmetical combinations 5. Arithmetic fact retrieval 6. Reading task 7. Raven’s matrices 8. Counting span task 9. Visual-matrix span 10. Verbal fluency 11. Trail-making† 12. Stroop task† 13. Digit Span task 14. Corsi-block span
Tasks .58 .45 .56 .52 .45 .35 –
7 .55 .45 .54 .50 .37 .28 .36 –
8 .50 .41 .52 .42 .44 .28 .41 .42 –
9 .40 .39 .41 .28 .43 .29 .27 .11 .18 –
10
Table 1. Descriptive statistics and partial correlations among the tasks used in the study controlling for chronological age 12 2.28 2.23 2.24 2.29 2.29 2.27 2.18 2.09 2.25 2.10 .29 –
11 2 .61 2 .52 2 .58 2 .55 2 .55 2 .42 2 .61 2 .36 2 .44 2 .31 – .46 .40 .40 .43 .34 .21 .23 .38 .32 .20 2.26 2.21 –
13 .33 .26 .34 .28 .38 .18 .29 .28 .39 .31 2.38 2.24 .26 –
14
Working memory and arithmetic 191
192 Ulf Andersson Table 2. Regression analysis of written arithmetical performance: The contribution of IQ, reading, age and different working memory tasks B
SE
b
t
Model 1 Arithmetical composite measure Block 1 Fð3; 137Þ ¼ 55:57, p , :001, R 2 ¼ :55 Age 0.23 .07 0.20 3.41** Raven’s 0.45 .07 0.40 6.53** Reading 0.57 .08 0.44 7.16** Block 2 Fð10; 130Þ ¼ 32:78, p , :001, R 2 ¼ :72, DR 2 ¼ :17 Age 0.16 .06 0.13 2.74** Raven’s 0.18 .07 0.16 2.67* Reading 0.36 .07 0.28 5.20** Counting span 2.16 .55 0.22 3.95** Visual-matrix span 0.43 .24 0.11 1.83 Verbal fluency 0.14 .05 0.15 2.81** Trail making 0.00 .01 20.17 22.51** Stroop task 0.00 .04 20.03 20.65 Digit Span 1.71 .58 0.16 2.96** Corsi-block span 2 0.31 .42 20.04 20.74 Model 2 Arithmetical composite measure Block 1 Fð4; 136Þ ¼ 49:60, p , :001, R 2 ¼ :59 Counting span 3.12 .60 0.32 5.19** Verbal fluency 0.21 .06 0.22 3.74** Trail making 0.00 .01 20.40 26.37** Digit Span 1.82 .66 0.17 2.78** Block 2 Fð7; 133Þ ¼ 45:87, p ¼ :000, R 2 ¼ :71, DR 2 ¼ :11 Counting span 2.31 .53 0.24 4.35** Verbal fluency 0.13 .05 0.14 2.68** Trail making 0.00 .01 20.19 22.93** Digit Span 1.88 .57 0.17 3.31** Age 0.18 .06 0.15 3.18** Raven’s 0.20 .07 0.18 2.91** Reading 0.38 .07 0.30 5.51** Model 3 Arithmetical composite measure Fð8; 132Þ ¼ 61:19, p , :001, R 2 ¼ :79 Arithmetical fact retrieval 0.86 .12 0.41 7.07** Age 0.21 .05 0.18 4.31** Raven’s 0.16 .06 0.14 2.78** Reading 0.20 .06 0.16 3.09** Counting span 1.92 .46 0.20 4.22** Verbal fluency 0.00 .04 0.05 1.12 Trail making 0.00 .01 20.10 21.70 Digit Span 1.35 .49 0.12 2.76**
pr2†
.04 .14 .17 .02 .01 .06 .03 .00 .02 .01 .00 .02 .00
.08 .04 .12 .02 .04 .01 .02 .02 .02 .02 .07
.08 .03 .01 .02 .03 .00 .00 .01
† 2 pr ¼ squared part correlations, represents the unique contribution for each variable. *p , .05, **p , .01.
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;
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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 (DR 2 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 written arithmetical 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 much variance 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 (DR 2 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 additional support 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 (R 2 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-making task 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 to check 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
194 Ulf Andersson Table 3. Regression analysis of the different tasks of written arithmetical skills: The contribution of IQ, reading, age and different working memory tasks
Model 4
Model 5
Model 6
Arithmetical calculation standard Fð10; 130Þ ¼ 16:71, p , :001, R 2 ¼ :56 Age Raven’s Reading Counting span Visual-matrix span Verbal fluency Trail making Stroop task Digit Span Corsi-block span Arithmetical equations Fð10; 130Þ ¼ 23:48, p , :001, R 2 ¼ :64 Age Raven’s Reading Counting span Visual-matrix span Verbal fluency Trail making Stroop task Digit Span Corsi-block span Arithmetical combinations Fð10; 130Þ ¼ 16:32, p , :001, R 2 ¼ :56 Age Raven’s Reading Counting span Visual-matrix span Verbal fluency Trail making Stroop task Digit Span Corsi-block span
B
SE
b
t
pr2†
0.08 0.04 0.15 0.65 0.14 0.07 2 0.01 0.00 0.65 2 0.17
.03 .03 .04 .27 .12 .03 .01 .02 .29 .21
0.18 0.08 0.29 0.17 0.09 0.18 2 0.15 2 0.01 0.15 2 0.06
2.91** 1.08 4.24** 2.41* 1.20 2.71** 2 1.83 2 0.14 2.26* 2 0.84
.03 .00 .06 .02 .00 .02 .01 .00 .02 .00
0.03 0.07 0.11 0.78 0.23 0.06 2 0.01 0.00 0.38 2 0.17
.02 .03 .03 .22 .10 .02 .01 .02 .23 .21
0.06 0.18 0.24 0.23 0.16 0.17 2 0.13 2 0.01 0.10 2 0.06
1.07 2.63** 4.00** 3.58** 2.46* 2.91** 2 1.75 2 0.16 1.62 2 0.84
.00 .02 .04 .03 .02 .02 .00 .00 .00 .00
0.05 0.07 0.10 0.72 0.06 0.02 2 0.01 2 0.02 0.69 2 0.09
.03 .03 .03 .24 .11 .02 .01 .02 .26 .19
0.12 0.18 0.23 0.21 0.04 0.04 2 0.17 2 0.08 0.18 2 0.03
1.95 2.41* 3.33** 2.93** 0.54 0.66 2 2.01* 2 1.15 2.66** 2 0.49
.01 .02 .04 .03 .00 .00 .01 .00 .02 .00
† 2 pr ¼ squared part correlations, represents the unique contribution for each variable. *p , .05, **p , .01.
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 significant predictors. 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,
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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 the working 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. Table 4. Regression analysis of arithmetical fact retrieval: the contribution of IQ, reading, age and different working memory tasks
Model 7
Fð10; 130Þ ¼ 15:44, p , :001, R 2 ¼ :54 Age Raven’s Reading Counting span Visual-matrix span Verbal fluency Trail making Stroop task Digit Span Corsi-block span
B
SE
b
t
pr2†
0.00 0.00 0.20 0.30 0.22 0.00 0.00 0.00 0.44 0.32
0.04 0.04 0.04 0.33 0.14 0.03 0.01 0.02 0.35 0.25
2 0.08 0.06 0.34 0.07 0.11 0.19 2 0.17 2 0.05 0.08 0.09
2 1.21 0.76 4.85** 0.92 1.51 2.85** 2 2.01* 2 0.76 1.25 1.26
.00 .00 .08 .00 .00 .03 .01 .00 .00 .00
†
pr2 ¼ squared partial correlation, represents the unique contribution for each variable. *p , .05, **p , .01.
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 central executive 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 & BeebeFrankenberger, 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.
196 Ulf Andersson
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 accounted for 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 & BeebeFrankenberger, 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 executive functions 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 the phonological 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 from long-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 significant predictors 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 the visual-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
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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 (Fu ¨ rst & 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 most probably 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; Fu ¨ rst & 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 the regression model (model 3). In contrast to Bull and Scerif (2001), but in line with Swanson and BeebeFrankenberger (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 of information 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,
198 Ulf Andersson
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 eliminating the 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 function is 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 central executive 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 their working 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
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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. An alternative 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 of information 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 phonological loop 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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