Exploring working memory in children with low arithmetical achievement

Learning and Individual Differences 15 (2005) 189 – 202 www.elsevier.com/locate/lindif

Exploring working memory in children with low arithmetical achievement Antonella D’Amicoa,T, Maria Guarnerab a

Dipartimento di Psicologia, Universita` di Palermo, Via delle Scienze, Parco d’Orleans, Edificio 15, 90128 Palermo, Italy b Dipartimento dei Processi Formativi, Universita` di Catania, Italy Received 1 June 2004; received in revised form 21 January 2005; accepted 24 January 2005

Abstract This research aimed at exploring the working memory functions in children with low arithmetical achievement and normal reading, compared to age matched controls (mean age 9 years). All the children completed a series of working memory tasks, involving the central executive functions (using both linguistic and numerical material), the phonological loop (using words, pseudo-words and digits) and the visual sketchpad (using both static visualspatial patterns and visual-spatial sequences). Results indicated that poor arithmeticians performed worse than agematched controls in all the visual sketchpad tasks and in all the central executive tasks, whether they used linguistic or numerical material. On the contrary, the only phonological loop measure on which poor arithmeticians underachieved was the digit span forward. This selective impairment in the short-term recall of numerical material is explained as a difficulty in the access and retrieval of information from the numerical lexicon. D 2005 Elsevier Inc. All rights reserved. Keywords: Arithmetic difficulties; Dyscalculia; Working-memory

1. Introduction The study of individual differences in mathematical cognition is a recent issue of interest to many psychologists (i.e. Geary, 1993, 2003; Jordan, Hanich, & Kaplan, 2003; Swanson & Sachse-Lee, 2001). A good strategy for understanding these individual differences is to investigate the underlying factors T Corresponding author. Tel.: +39 91 7028420; fax: +39 91 7028430. E-mail address: [email protected] (A. D’Amico). 1041-6080/$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.lindif.2005.01.002

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that influence the development of normal mathematical skills in children since these may be considered as the causal factors of arithmetical learning difficulties. Among these cognitive factors, a significant role has been attributed to working memory (Baddeley, 1986), considered as a measure of the capacity to manipulate and transform material while remembering information. The working memory model consists of three main components: the central executive, the phonological loop and the visual-spatial sketchpad. The central executive is responsible for performing a series of high order functions, such as the inhibition of irrelevant information, switching between retrieval plans or between different strategies and the temporary activation of long-term memory information (Baddeley, 1996; Lehto, Juuja¨rvi, Kooistra, & Pulkkinen, 2003; Miyake et al., 2000); the phonological loop and the visual-spatial sketchpad are assumed to be slave systems coordinated by the central executive and are devoted respectively to storing phonological (Vallar & Baddeley, 1984) and visual-spatial information (i.e. Della Sala, Gray, Baddeley, Allamano, & Wilson, 1999; Logie, 1991; Logie & Pearson, 1997; Quinn, 1988; Quinn & McConnell, 1996). Thus, while the central executive component is considered to work both with verbal-phonological and visual-spatial information, the slave mechanisms are highly specialized and domain dependent. Different studies have reported a linear relationship between performances in digit span (Gathercole, Pickering, Knight, & Stegmann, 2004; Hoosain & Salili, 1987) or in central executive tasks (Bull & Scerif, 2001; Gathercole et al., 2004) and mathematical abilities as evaluated using standardized achievement batteries. Other studies have demonstrated that conditions that overload the working memory capacity, e.g. solving mental calculations rather than written calculations, result in poorer performances both in adults (Hitch, 1978) and children (Adams & Hitch, 1997). Finally, a series of experimental studies has shown that concurrent tasks involving the central executive component (Furst & Hitch, 2000; Lemaire, Abdi, & Fayol, 1996; Logie, Gilhooly, & Wynn, 1994; Seitz & SchumannHengsteler, 2000), the phonological loop (Furst & Hitch, 2000; Haughey, in Hitch, Cundick, Haughey, Pugh, & Wrigth, 1987; Lau & Hoosain, 1999; Lee & Kang, 2002; Lemaire et al., 1996; Logie et al., 1994; Seitz & Schumann-Hengsteler, 2000) or the visual-spatial sketchpad (Heathcote, 1994; Lee & Kang, 2002) have a deleterious effect on the execution of calculation processes. The role played by the different components of working memory in arithmetical performance has been further explored in a series of studies involving children with mathematical difficulties and controls (e.g. Bull & Johnston, 1997; Bull, Johnston, & Roy, 1999; McLean & Hitch, 1999; Passolunghi & Siegel, 2001). The literature examined is quite consistent in highlighting an impairment of the central executive process in children with arithmetical difficulties. Indeed, these subjects perform worse than controls in more general executive tasks such as the Wisconsin Sorting Card Tasks (Bull et al., 1999) and in memory tasks with a high executive demand such as Case, Kurland and Goldberg’s (1982) counting span (Siegel & Ryan, 1989) or the digit span backward (Geary, Hamson, & Hoard, 2000; Geary, Hoard, & Hamson, 1999). Siegel and Ryan (1989), however, have shown that children with arithmetical difficulties showed impairment only in the central executive tasks that involved numerical information, while no differences between groups were found in the central executive tasks involving linguistic information, such as the Daneman and Carpenter Listening span task (1980). This result led the authors to retain that poor arithmetical performers did not have a general executive disorder, but rather that their failure in the central executive tasks derived from their difficulty in manipulating and maintaining numerical information. Furthermore, McLean and Hitch (1999) demonstrated that children with poor arithmetical skills showed impairment in central executive tasks that require a switch between numerical/ linguistic retrieval strategies and are assumed to measure the ability to inhibit a pre-activated retrieval

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strategy. More precise results have come from other recent research experiments (Gathercole et al., 2004; Passolunghi & Siegel, 2001; Swanson & Sachse-Lee, 2001) that have demonstrated how poor mathematicians fail in different central executive tasks that involve uniquely verbal information; these results support the hypothesis of domain independent central executive impairment in these children. The numerical/linguistic nature of the material to be recalled appears, on the contrary, to be an important constraint in the phonological loop functions of children with mathematical difficulties; indeed, while converging evidence has demonstrated that these subjects perform normally in non-word repetition (McLean & Hitch, 1999) and word span (Bull & Johnston, 1997; Passolunghi & Siegel, 2001), their results in the short-term recall of digits are often inconsistent. In a number of studies low arithmetical achievers performed normally in digit span tasks (Bull & Johnston, 1997; Geary et al., 1999, 2000), whereas in others they performed at a significantly lower level than controls (Passolunghi & Siegel, 2001; Swanson & Sachse-Lee, 2001; see also McLean & Hitch, 1999, who found group differences very close to the statistical significance). Passolunghi and Siegel (2001), however, argue that the difficulties found in digit span tasks, rather than being the result of a phonological loop impairment, are more likely due to slower access to number representation in long-term memory. The results of previous research by Dark and Persson Benbow (1991) are consistent with this interpretation; they reported that children gifted in the mathematical area showed particular ability in the short-term recall of digits. In this case, too, the authors’ interpretation is that the memory span of these children is enhanced by their skill and speed in identifying numerical items. In other words, the stronger long-term memory representation of numbers in the arithmetically gifted children may facilitate their item identification, and this in turn may enhance their digit span performance. With regard to the functioning of the visual-spatial sketchpad, Bull et al. (1999) found some differences between low-ability and high-ability mathematicians in the Corsi Block task (1972), that requires the reproduction of sequences of positions presented serially and is assumed to measure the spatial span. These differences, however, disappeared when IQ and reading ability were statistically controlled for, suggesting that the visual-spatial problems of low ability mathematicians were only indirectly related to their arithmetical skills. McLean and Hitch (1999), on the contrary, reported that children with arithmetical difficulties and normal reading abilities showed impairment in the Corsi Block task, even though they performed normally in the short-term recall of random visual-spatial patterns (Matrix task, adapted from Wilson, Scott, & Power, 1987), a result more recently replicated by Swanson and Sachse-Lee (2001), using a very similar task. In conclusion, the literature on the working memory abilities of children with arithmetical difficulties has produced some inconsistent results. For this reason, in the present study we try to draw a complete and extensive picture of all the working memory functions of children with arithmetical difficulties but normal reading abilities, focusing particularly on the differences between tasks that use linguistic and numerical material. For this purpose, we selected a group of Italian children with low arithmetical achievement and normal reading skills (mean age: 9 years and 3 months) and a group of age-matched controls with normal arithmetical and reading achievement. The two groups were presented with a series of working memory tasks, chosen according to the most recent literature on the assessment of working memory in children (Gathercole & Pickering, 2000; Pickering & Gathercole, 2001). In particular, the experimental working memory battery used in the present study consisted of three typologies of tasks: the Listening Span Task (D’Amico, 2002a; adapted from Daneman & Carpenter, 1980), the Digit Span Backward (Wechsler, 1974), the Making Color Trails Task (adapted from D’Elia & Satz, 1989) and the Making Verbal Trails (Reitan, 1958) were used to measure different aspects of the

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central executive component, and involved both linguistic and numerical information; the Non-word Repetition task (D’Amico, 2002b), the Bisyllabic Word Span (Sechi, D’Amico, Longoni, & Levi, 1997) and the Digit Span Forward (Wechsler, 1974) were used to measure the phonological loop functions, again using an equal amount of linguistic and numerical information; the Corsi Span Task (Corsi, 1972) and the Matrix Task (Cornoldi et al., 1997) were used to assess the functioning of the visual sketchpad, short-term memory for spatial sequential information and short-term memory for visual static patterns respectively.

2. Method 2.1. Participants A group of teachers at an Italian primary school were asked to identify all the pupils attending their classes (fourth and fifth grades) who showed scholastic difficulties in the arithmetical area. The teachers were then asked to pair each child selected with a child of the same gender and similar age, with normal scholastic achievement. The teachers identified 25 children with low arithmetical achievement and 25 controls. The next step in the group selection relied upon the use of standardized tests as described below. The numerical comprehension and calculation processes of the whole group were assessed using 10 subtests drawn from the ABCA battery (Lucangeli, Tressoldi, & Fiore, 1998). The 10 subtests included were: mental calculation, written calculation, retrieval of combinations and numerical facts, seven-table completion forward, four-table completion backward, number dictation, denomination of arithmetic symbols, insertion of symbols bbQ and bNQ between two numbers, increasing arrangements of numbers and decreasing arrangements of numbers. These 10 subtests were scored following the author’s recommendations as follows: (1) the raw score obtained in each subtest was compared to the ABCA normative data; (2) if the raw score was below the 10th percentile a standard score of b0Q was given, otherwise if the raw score was above the 10th percentile a standard score of b1Q was given (Lucangeli, Fiore, & Tressoldi, 1998). Thus, the total arithmetical standard score could range from 0 to 10. Children who obtained a total standard score below 5 were then considered as bpoor arithmetical achieversQ. Furthermore, two reading tasks, considered to test different aspects of reading abilities, were administered to exclude all children with below average reading abilities from the poor arithmetic group. The first was a reading comprehension task (Prova MT, Cornoldi & Colpo, 1981) requiring pupils to read a passage to themselves and to answer 10 questions about its content. Following the norms of the MT test (Cornoldi & Colpo, 1981), the children that answered at least six questions correctly were considered to be normal readers. The second task, drawn from the Sartori, Job, and Tressoldi (1995) battery, required children to read a list of 48 non-words aloud. For the particular error distribution reported in normative data from the test, Sartori et al. (1995) suggest considering normal readers to be those children that perform at least 36 non-words correctly (N58 percentile). Only 14 children (6 males, 8 females, mean age in months=113.1, S.D.=4.8), out of the initial 25, matched the criterion for inclusion in the Poor Arithmetic group. They showed normal reading ability (Reading Comprehension M=7.6, S.D.=1.6, Non-words Reading M=40, S.D.=5.4) but low achievement in the arithmetical area (total standard score M=2.2, S.D.=1.5). The children in the Control group were 14 age/gender matched controls (6 males, 8 females, mean age in months=111, S.D.=6),

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who showed normal achievement both in Arithmetic (total standard score M=7.9, S.D.=1.5) and Reading (Reading Comprehension M=8.3, S.D.=1.1, Non-words Reading M=43.3, S.D.=4.1). A series of T tests confirmed that children with poor and normal arithmetical skills did not differ in chronological age, t(26)=1, pN0.25, in reading comprehension, t(26)= 1.2, pN0.10, or in non-word reading, t(26)= 1.8, pN0.05, while their Arithmetic standard scores were significantly different, t(26)= 9.8, pb0.0001. A multivariate analysis of variance (MANOVA) was also carried out on the raw scores of the two groups in the 10 arithmetic subtests. Results again revealed a significant main effect of Group, F(10,17)=5.3, pb0.001, and an effect size in the large range [eta]2=0.76 (Cohen, 1969), indicating that poor arithmetical achievers had lower scores than controls. Univariate analyses of variance indicated that poor arithmeticians achieved significantly lower scores in mental calculation, F(1,26)=17.2, pb0.0001, [eta]2=0.40, written calculation, F(1,26)=21.8, pb0.0001, [eta]2=0.46, retrieval of combinations and numerical facts, F(1,26)=43.3, pb0.0001, [eta]2=0.62, seven-table completion forward, F(1,26)=7.4, pb0.01, [eta]2=0.22, four-table completion backward, F(1,26)=11.6, pb0.005, [eta]2=0.31, number dictation, F(1,26)=6.1, pb0.05, [eta]2=0.19, denomination of arithmetic symbols, F(1,26)=6.3, pb0.05, [eta]2=0.19, insertion of symbols bbQ and bNQ between two numbers, F(1,26)=19.2, pb0.0001, [eta]2=0.42, increasing arrangements of numbers, F(1,26)=14.6, pb0.001, [eta]2=0.36, and decreasing arrangements of numbers F(1,26)=15.9, pb0.0001 [eta]2=0.38. Table 1 shows the descriptive statistic for each dependent variable, the univariate tests, the corresponding degrees of freedom and effect sizes (eta2). The effect size are in the large range for all the considered dependent variables, except for the seven-table completion forward, the number dictation and the denomination of arithmetic symbols, for which the effect size is in the medium range. A more accurate picture of the results of the poor arithmetical and control groups in the 10 arithmetic subtests is shown in Table 2, which also reports the percentage of subjects who obtained a raw score Table 1 Raw scores of poor arithmeticians and controls at the ABCA battery (Lucangeli et al., 1998), followed by the univariate tests for each dependent variable included in the MANOVA analysis, with the corresponding degrees of freedom and effect sizes (eta2) Arithmetical tasks drawn from the ABCA battery (1) Mental calculation (2) Written calculation (3) Retrieval of combinations and numerical facts (4) Seven-table completion forward (5) Four-table completion backward (6) Numbers dictation (7) Denomination of arithmetic symbols (+  : b N) (8) Insertion of symbols bbQ and bNQ among two numbers (9) Increasing arrangement of numbers (10) Decreasing arrangement of numbers

Poor arithmeticians (N=14)

Normal arithmeticians (N=14)

Univariate test

M

S.D.

M

S.D.

df

F

p

eta2

1.8 3.9 3.4

1.2 1.7 2.4

5.4 7.4 9.2

3 2.2 2.3

1,26 1,26 1,26

17.2 21.8 43.3

b0.0001 b0.0001 b0.0001

0.40 0.46 0.62

2.6 2.3 7.6 4.8

1.4 1.7 0.6 1.3

3.7 3.8 8 6

0.6 0.4 0 1.1

1,26 1,26 1,26 1,26

7.4 11.6 6.1 6.3

b0.01 b0.005 b0.05 b0.05

0.22 0.31 0.19 0.19

3.1

2.1

5.8

0.8

1,26

19.2

b0.0001

0.42

2.7 2.3

1.8 1.7

4.7 4.3

0.6 0.6

1,26 1,26

14.6 15.9

b0.001 b0.0001

0.36 0.38

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Table 2 Percentage of poor arithmeticians and controls falling below the 10th percentile at the ABCA battery (Lucangeli et al., 1998) Arithmetical tasks drawn from the ABCA battery

Poor arithmeticians

Controls

(1) Mental calculation (2) Written calculation (3) Retrieval of combinations and numerical facts (4) Seven-table completion forward (5) Four-table completion backward (6) Numbers dictation (7) Denomination of arithmetic symbols (+  : b N) (8) Insertion of symbols bbQ and bNQ among two numbers (9) Increasing arrangement of numbers (10) Decreasing arrangement of numbers

100 100 85.7 57.1 64.3 35.7 85.7 92.9 85.7 71.4

50 35.7 14.3 28.6 7.1 0 42.9 7.1 21.4 7.1

below the 10th percentile in each subtest. It is clear that even if the differences between groups are statistically significant for all the tasks, a lot of interindividual and intraindividual differences may be observed in both the poor arithmetic group and in the control one. However, many authors have argued that one impediment to the systematic study of arithmetical processes is the large number and complexity of the specific domains involved in this area (Geary, 1994). All normal children show uneven patterns of competencies in mathematics, and the intraindividual differences are much more apparent in children with arithmetical difficulties and developmental dyscalculia (Geary, 1993; Geary & Hoard, 2001; Sokol, Macaruso, & Gollan, 1994; Temple, 1992) who are, however, quite often examined as a single group. The poor arithmeticians involved in the present research had particular difficulty in written and mental calculation (100% failed), tasks that also presented some difficulties for children in the control group (respectively, 50% failed in mental calculation and 35.7% in written calculation). On the contrary, only 35.7% of the poor arithmeticians and none of the controls underscored in the number dictation subtest. In the three subtests dedicated to examining long-term knowledge of arithmetical facts (tasks 3, 4 and 5), poor arithmeticians encountered particular difficulties in the retrieval of combination and numerical facts (85.7% failed in this task). Most poor arithmeticians (85.7%), as well as a significant number of controls (42.9%) underscored in the denomination of the arithmetical symbols subtest. It must be noted, however, that only two of the poor arithmeticians failed in the denomination of the operational symbols such as b+Q, b Q, bQ, and b:Q, while most errors involved the denomination of symbols bbQ and bNQ. The particular difficulty in the use of these symbols may also have affected the performance of the poor arithmeticians in inserting the symbols bbQ and bNQ between two numbers, while it seemed to have little significant influence on the performance of controls. It is more likely, however, that poor arithmeticians failed in these tasks because of their general difficulty in judging number magnitude, and this also affected their performance in the increasing and decreasing arrangement of numbers. 2.2. Materials and procedures The working memory abilities of poor arithmeticians and controls were individually measured at school using the test battery described below. Each child performed the tests in two sessions, in random order of presentation.

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2.2.1. Central executive tasks 2.2.1.1. Listening span task. The Daneman and Carpenter (1980) listening span task is one of the most famous tasks used for the assessment of the central executive component, as it requires the simultaneous storage and manipulation of verbal information. In the Italian version by D’Amico (2002a), the experimenter reads a set of sentences aloud, asking the child to express a true/false judgment on the content of each sentence and to recall, at the end of the set, the last word of each sentence. Examples of true and false sentences are respectively bThe child has a red umbrellaQ or bThe ship floats on the skyQ. bSkyQ and bumbrellaQ are, in these examples, the words to be recalled at the end of the set presentation. The test is composed of 12 sets of increasing length from 2 to 5 sentences (three sets for each length); following the Daneman and Carpenter (1980) procedure, the listening span score corresponds to the longest set of sentences of which the last words have been correctly recalled. For assigning the score, the true/false judgment is irrelevant. 2.2.1.2. Digit span backward. Recalling digits in the inverse order of presentation is generally considered to involve both the phonological loop and the central executive (Morra, 1994; Pickering & Gathercole, 2001), as the sequence spoken by the experimenter must be stored and reversed to produce the correct answer. However, another interpretation of the processes involved in these tasks has been proposed by Li and Lewandowsky (1995), with a series of converging evidence which suggests that backward recall relies on visual-spatial representation of the input material. The administration procedure of the digit span backward (drawn from the Wechsler Scale for Children—Revised, 1974) is as follows: the experimenter reads a series of digits aloud (starting from sets of 2 digits up to sets of 8 digits), asking children to recall the digit set in the reverse order (i.e.: 3, 4, 8 becomes 8, 4, 3). The score corresponds to the maximum length of the digit set recalled in the reverse order of presentation. 2.2.1.3. Making verbal/color trails. The switching of retrieval strategies is one of the processes considered to involve the central executive component (Baddeley, 1996; Bull & Scerif, 2001). Indeed, the switching process requires the simultaneous inhibition of pre-activated irrelevant information. The making trails tasks (Reitan, 1958) are a good way of measuring the ability to switch between two different retrieval strategies. In particular, for the purpose of the present research, we used two making trails tasks for their high correlation with arithmetical abilities as demonstrated in the study by McLean and Hitch (1999). To perform the Verbal Trails task, children were asked to shift between the numerical and the alphabetic code, starting to recite the numerical sequence (from 1 to 11) and the alphabetical sequence (from A to M), alternating each number with the corresponding letter (i.e.: 1, A, 2, B, 3, C, etc.). The Verbal Trails score corresponds to the total time taken to perform the task. The making Color Trails task (adapted by McLean & Hitch, 1999, from an adult version by D’Elia & Satz, 1989) consisted in the presentation of an A4 sheet on which 22 circles were printed. Half the circles were pink and half yellow. Each series of colored circles contained a number from 1 to 11 and each number appeared once in the pink circles and once in the yellow circles. The children were asked to make a written trail, alternating numbers and colors, starting with byellow 1Q up to bpink 11Q. The Color Trails score corresponds to the total time taken to complete the task.

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2.2.2. Phonological loop tasks 2.2.2.1. Digit span forward task. The Digit Span Forward Task, drawn from the WISC-R Scale (Wechsler, 1974) is one of the tasks most often used to measure verbal short-term memory. The administration procedure is as follows: the experimenter reads a series of digits aloud (starting from sets of 2 items up to sets of 8 items), asking children to recall the item set in the same order as the presentation. The Digit Span Forward score corresponds to the maximum length of the item set recalled in the correct order of presentation. 2.2.2.2. Bisyllabic word span task. For the purpose of measuring children’s short-term memory for words instead of numbers, a word span test was used (Sechi et al., 1997) consisting of a series of frequently occurring Italian bisyllabic words like bsoleQ (sun) and bgattoQ (cat). The task administration and the scoring method were similar to those used for the digit span tasks: the experimenter read a series of words aloud (starting from sets of 2 up to sets of 8 items), asking children to recall the words in the same order as the presentation. The Bisyllabic Word Span score corresponds to the maximum length of the word set recalled in the same serial order. 2.2.2.3. Non-word repetition task. Repetition of non-words is considered in different studies (Gathercole & Baddeley, 1996; Gathercole & Pickering, 2000) to be a bpureQ measure of phonological short-term memory since, by using an unfamiliar phonological sequence, it limits the long-term retrieval process, and involves the rehearsal process to a very limited extent (Baddeley, Gathercole, & Papagno, 1998). The non-word repetition task used in this research (D’Amico, 2002b), replicating the procedures of the research mentioned above, requires children to repeat, one at a time, a series of non-words pronounced by the experimenter. The task consists of 18 non-words from 5 to 9 phonemes in length. The Non-word Repetition score corresponds to the total number of phonemes correctly repeated. 2.2.3. Visual-spatial sketchpad tasks 2.2.3.1. Corsi’s blocks span task (Corsi, 1972). This is a well-known task, very frequently used to measure short-term memory for spatial sequential information. In addition, the task is considered to involve the encoding of visual stimuli, the retention of information over time and response selection, prior to overt response execution (Fischer, 2001; Orsini, 1994; Smyth & Scholey, 1994). The testing set is composed of nine 4.5 cm square blocks arranged in random order on a board. The experimenter taps from two up to nine blocks following a series of fixed sequences, and the child is required to reproduce the tapping sequences in the same serial order. Blocks are tapped at a rate of one per second and each block is tapped only once in each sequence. According to the procedure described by Spinnler and Tognoni (1987), the test stops when a child fails to reproduce two out of three sequences of the same length. The Corsi Blocks span score corresponds to the maximum length of the sequence reproduced in the correct serial order. 2.2.3.2. Matrix task (Cornoldi et al., 1997). The Matrix Task was used to measure memory for positions. Children were presented with a total of six 55 cm matrixes, with three, four or five colored squares in various positions (two matrixes for each of the three, four or five square conditions). Each matrix was presented for 10 s; after the matrix had been removed the child was asked to remember the positions of

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the colored squares and to point them out on an identical blank matrix. The Matrix Task score corresponds to the total number of positions correctly remembered. Cornoldi et al. (1997) consider these tasks to be a measure of passive visual-spatial memory employing only a small amount of the central executive resources, as the subjects may recall the presented positions without performing any active spatial operation. Because of the static nature of the visual patterns, similar versions of the task have been used in different studies (Della Sala et al., 1999; McLean & Hitch, 1999; Wilson, Scott, & Power, 1987) to measure visual short-term memory. 2.3. Results A series of T tests for independent samples was performed to examine the differences in working memory ability between the poor arithmetic and normal arithmetic groups. Results are reported in Table 3, with effect size estimates and confidence intervals (Cohen’s d statistic, 1969). Results revealed significant differences between the groups in almost all the tasks considered to involve the central executive component. The children with poor arithmetical skills obtained lower scores in the Listening Span task, t(26)= 3.28, pb0.005, and in the Digit Span Backward, t(26)= 3.04, pb0.005, and took longer than the controls to complete the Making Verbal Trails, t(26)=2.19, pb0.05, and the Making Color Trails, t(26)=1.99, pb0.05. The effect size is in the large range for all the considered dependent variables, except for the Making Color Trails, for which the effect size is in the medium range. As far as the tasks considered to involve the phonological loop are concerned, the poor arithmetical group performed worse than controls only in the Digit Span Forward task, t(26)= 3.88, pb0.001, while no differences emerged in the Bisyllabic Word Span, t(23)= 0.96, pN0.1, or in the Non-word Repetition test, t(26)= 0.98, pN0.1 (effect size in the small range for both the variables). On the contrary, differences between groups were Table 3 Working memory scores of poor arithmetic and normal arithmetic groups Working memory tasks Central executive tasks Listening span task Making verbal trails (s) Making colors trails (s) Digit span backward

Poor arithmeticians (N=14)

Normal arithmeticians (N=14)

Difference

M

M

S.D.

M

S.D.

0.86TT 15.5T 17.28T 0.93TT

0.26 7.07 8.65 0.3

1.35 0.83 0.76 1.12

2.13 0.04 0.03 1.88

0.49 1.58 1.5 0.29

S.D.

d

CI upper CI lower

2 59.5 70.5 3

0.5 14.3 27.4 0.7

2.9 44 53.2 3.9

0.8 22.2 17.3 0.9

Phonological loop tasks Digit span forward 4.6 Bisyllabic word span 4.7 Non-words repetition 120.7

0.5 0.7 1.7

5.8 5 121.2

1 0.8 0.9

1.21TTT 0.31 0.31 0.32 0.50 0.51

1.52 0.40 0.37

2.31 1.18 1.1

0.64 0.4 0.39

Visual sketchpad tasks Matrix task Corsi span task

3.8 0.9

16.7 5.5

2.7 1

4.93TTT 1.25 1.14TT 0.37

1.7 1.26

2.51 2.03

0.8 0.42

11.8 4.3

The difference statistic p values are based on independent sample t-tests. Confidence interval for effect size were calculated using Cohen’s d statistic (1969). Tpb0.05, TTpb0.005, TTTpb0.001, two-tailed.

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found in both the tasks considered to involve the visual-spatial sketchpad. Poor arithmeticians obtained lower scores than controls in the Matrix task, t(26)= 3.95, pb0.001, and in the Corsi Span task, t(26)= 3.1, pb0.005 (effect size in the large range for both the variables).

3. Discussion Results indicated low performances in children with poor arithmetical skills, when compared to controls, in all the central executive tasks. The results obtained by poor arithmeticians in the Listening Span task and Digit Span Backward demonstrate their difficulties in retrieving information from temporarily activated components of long-term memory, while new information has to be encoded and retrieved from the phonological loop. This impairment could explain many of the difficulties of the poor arithmeticians involved in the present research (see Table 2) in tasks such as the mental and written calculations that require pupils to simultaneously retain information (the amount carried, the loan, etc.) and to transform the new incoming items (i.e. the new addends). Furthermore, poor arithmeticians performed worse than controls in the Making Verbal Trails and Making Color Trails, tasks that require shifting between numerical/linguistic retrieval plans and that are assumed to measure the ability to inhibit a pre-activated retrieval strategy. Many authors (i.e. Barrouillet, Fayol, & Lathulie`re, 1997; Bull & Scerif, 2001; Passolunghi & Siegel, 2001; Swanson & Sachse-Lee, 2001) argue that this inability to control and inhibit irrelevant information affects different arithmetical abilities, from the retrieval of numerical facts, to the selection of addends in simple addition problems. Finally, it should be noted that the central executive impairment of poor arithmeticians is apparent both in tasks involving linguistic information (Listening Span task), and numerical information (Digit Span Backward) or both linguistic and numerical information (Making Verbal Trails). These results are consistent with the evidence from recent literature (Gathercole et al., 2004; Passolunghi & Siegel, 2001; Swanson & Sachse-Lee, 2001), and are therefore considered to confirm the hypothesis of a domain independent central executive disorder in these subjects. However, the results of the present research lead us to hypothesize that the phonological loop is not the major factor in explaining arithmetical difficulties. Indeed, we have demonstrated that poor arithmeticians have normal word span abilities, as already claimed by Bull and Johnston (1997) and Passolunghi and Siegel (2001), as well as normal non-word repetition abilities, as reported in McLean and Hitch’s study (1999). Their selective impairment in the short-term recall of digits may not therefore be due to a phonological loop impairment. A lot of converging evidence has demonstrated that verbal short-term memory performance relies upon a number of different factors: of these a major role is played by the strength of long-term memory representation of the material used in the short-term memory tasks (Case et al., 1982; D’Amico, 2002b; Dempster, 1981, 1985; Gathercole, 1995; Gathercole & Adams, 1994; Hulme, Maughan, & Brown, 1991; Roodenrys, Hulme, & Brown, 1993). Children with arithmetical difficulties are considered to have particularly weak long-term memory representation of the purely numerical–and nonlinguistic–material, as indicated by their slowness in digit naming tasks (Bull & Johnston, 1997; Geary, 1993; Geary et al., 2000, Hitch & McAuley, 1991) but not in letter naming (Bull & Johnston, 1997). Taken together, this evidence suggests that the digit span impairment shown by poor arithmeticians could be considered as only one of the symptoms of a more general difficulty in acquiring long-term representation of numerical material. This assumption could also explain the difficulty in numerical fact retrieval of poor arithmeticians. This is a very frequent disorder, affecting 85.7% of the poor arithmetical group included in

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the present study (see Table 2), and is considered to be a process of activation and retrieval of information from long-term semantic memory (e.g. Ashcraft, 1982; Geary, 1993; Geary, Brown, & Samaranayake, 1991; Jordan et al., 2003; McCloskey, Caramazza, & Basili, 1985). The other partially novel finding of the present research regards the impairment of both visual-spatial passive short-term memory (Matrix Task) and visual-spatial sequential memory (Corsi Block Task) in poor arithmeticians. As indicated above, the literature regarding this issue is not really consistent (Bull et al., 1999; McLean & Hitch, 1999; Swanson & Sachse-Lee, 2001); we are, however, inclined to believe that many of the difficulties of poor arithmeticians could be explained with reference to their visualspatial short-term memory impairment. There is disagreement about the nature of the relationship between visual-spatial abilities and calculation processes; Rourke (1993; see also Rourke and Conway, 1997; Sechi, Giordani, and Becciu, 1995) has hypothesized that visual-spatial abilities support the alignment process in performing arithmetical operations; but, like McLean and Hitch (1999) who obtained similar results, an analysis of the type of errors made by the poor arithmetic group in the present study provided little support for this hypothesis, as only 2 children out of 14 made misalignment errors. Another interesting explanation of the use of the visual sketchpad in performing arithmetical tasks is provided by Heathcote (1994). The author compares the visual sketchpad to a bmental blackboardQ where individuals store information while performing an operation. If we accept this view, the visualspatial sketchpad impairment of the poor arithmeticians involved in the present research could explain their low scores in mental calculation. However, we are more inclined to adopt Campbell’s (1994) and Dehaene’s (1997) perspective to explain the visual sketchpad impairment of our poor arithmeticians. These authors hypothesize that visual-spatial abilities are of great importance in performing arithmetical tasks involving comparisons between quantities that are based on an analogical magnitude code (see also Girelli, Lucangeli, & Butterworth, 2000) and require subjects to generate and use the mental representation of the number line. From this perspective, the visual sketchpad impairments of the poor arithmeticians involved in the present research could be the underlying factor affecting their performance in tasks that require judgement of number magnitudes, such as the insertion of symbols bbQ and bNQ between two numbers and the increasing and decreasing arrangements of numbers, tasks in which almost all poor arithmeticians scored badly while controls generally performed well (see Table 2). In conclusion, the results of the present research support the view that the central executive and the visual sketchpad play an important role in arithmetical abilities, while the involvement of the phonological loop in arithmetical achievement seems to have been overestimated in the literature that based its observation only on the digit span performance of low arithmeticians (i.e. Gathercole et al., 2004; Hoosain & Salili, 1987). This latter conclusion, however, needs to be confirmed in later studies, that should also seek to identify the role of each of the working memory components in explaining both the individual differences in arithmetical development and the different impairments observed in children with arithmetical disorders.

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