Developmental Science 13:3 (2010), pp 508–520

DOI: 10.1111/j.1467-7687.2009.00897.x

PAPER How is phonological processing related to individual differences in children’s arithmetic skills? Bert De Smedt,1,2 Jessica Taylor,1 Lisa Archibald3 and Daniel Ansari1 1. Numerical Cognition Laboratory, University of Western Ontario, Canada 2. Centre for Parenting, Child Welfare & Disabilities, Katholieke Universiteit Leuven, Belgium 3. Language, Reading and Cognitive Neuroscience Laboratory, University of Western Ontario, Canada

Abstract While there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single-digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem-solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short-term memory. The present findings indicate that the quality of children’s long-term phonological representations mediates individual differences in singledigit arithmetic, suggesting that more distinct long-term phonological representations are related to more efficient arithmetic fact retrieval.

Introduction When performance on tests of reading and arithmetic competence is measured among typically developing children, strong correlations are obtained (e.g. Bull, Espy & Wiebe, 2008; Bull & Johnston, 1997; Hecht, Torgesen, Wagner & Rashotte, 2001). While such correlations may reasonably be explained by recourse to general cognitive competencies, which account for significant variance of individual differences in both domains of competence, there are a number of reasons to suggest that reading and arithmetic may be specifically related to one another, independent of general intellectual ability. First of all, a high percentage of children with developmental dyscalculia, who show deficits in arithmetic in spite of normal intellectual ability, also present reading difficulties (Shalev, 2007). Children with comorbid developmental dyscalculia and dyslexia have been found to differ in characteristic ways from their peers who are impaired in only one of the two domains (e.g. Jordan, Hanich & Kaplan, 2003). Second, cognitive neuropsychological case studies and cognitive neuroimaging studies also point to an overlap between

reading and arithmetic (see also Simmons & Singleton, 2008). More specifically, both reading (e.g. Pugh, Mencl, Jenner, Katz, Frost, Lee, Shaywitz & Shaywitz, 2001; Schlaggar & McCandliss, 2007) and arithmetic (e.g. Dehaene, Piazza, Pinel & Cohen, 2003) have been found to be strongly associated with regions of the left temporo-parietal cortex, such as the left angular and supramarginal gyri. In particular, the left temporoparietal cortex is more active when reading nonwords versus real words, and this dorsal system is thought to underlie phonological decoding or mapping graphemes onto phonemes (e.g. Pugh et al., 2001; Schlaggar & McCandliss, 2007), which is crucial for learning to read in alphabetic scripts (e.g. Hulme, 2002). Most interestingly, there is a consistent association between reading ability and left angular gyrus activation in children (Hoeft, Hernandez, Mcmillon, Taylor-Hill, Martindale, Meyler, Keller, Siok, Deutsch, Just, Whitfield-Gabrieli & Gabrieli, 2006; Meyler, Keller, Cherkassky, Lee, Hoeft, Whitfield-Gabrieli, Gabrieli & Just, 2007; Shaywitz, Shaywitz, Pugh, Fulbright, Constable, Mencl, Shankweiler, Liberman, Skudlarski, Fletcher, Katz, Marchione, Lacadie, Gatenby & Gore, 1998; Temple,

Address for correspondence: Daniel Ansari, Department of Psychology and Graduate Program in Neuroscience, University of Western Ontario, Westminster College, 361 Windermere Road, London, ON, N6G 2K3, Canada; e-mail: [email protected]  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

Phonological processing and arithmetic 509

Poldrack, Salidis, Deutsch, Tallal, Merzenich & Gabrieli, 2001; Turkeltaub, Gareau, Flowers, Zeffiro & Eden, 2003). In the domain of arithmetic, neuroimaging studies have shown that the left angular gyrus is particularly related to operations like multiplication, which are solved by means of fact retrieval from long-term memory, a process that is assumed to rely on verbal codes (Dehaene et al., 2003). A recent developmental neuroimaging study (Rivera, Reiss, Eckert & Menon, 2005) revealed that the left supramarginal gyrus exhibits age-related increments in activation during arithmetic problem solving, thus suggesting an increasing recruitment of phonologically represented arithmetic facts during arithmetic problem solving, coupled with a decreasing reliance on effortful procedural strategies. In addition to functional neuroimaging data, left temporoparietal white matter has been found to be associated with both reading (Niogi & McCandliss, 2006) and arithmetic (Van Eimeren, Niogi, McCandliss, Holloway & Ansari, 2008). This recruitment of similar brain regions and structural networks may suggest that reading and arithmetic share some core neurocognitive processes. In light of this, it is important to understand the factors that explain the link between reading and arithmetic and their developmental trajectories. Both abilities rely on the processing of the phonological code of the language. For example, arithmetic requires the retrieval of phonological forms corresponding to number words and arithmetic facts are assumed to be stored as phonological codes in long-term memory (see also Simmons & Singleton, 2008). Surprisingly, only a limited number of empirical studies have examined the relationship between phonological processing abilities and general arithmetical abilities in typically developing children (Durand, Hulme, Larkin & Snowling, 2005; Fuchs, Compton, Fuchs, Paulsen, Bryant & Hamlett, 2005; Fuchs, Fuchs, Compton, Powell, Seethaler, Capizzi, Schatschneider & Fletcher, 2006; Hecht et al., 2001; Leather & Henry, 1994; Nol, Seron & Trovarelli, 2004; Rasmussen & Bisanz, 2005; Simmons, Singleton & Horne, 2008; Swanson, 2004; but see Passolunghi, Mammarella & Altoe, 2008). Nonetheless, the existing body of data reveals a strong relationship between phonological processing and arithmetic. Importantly, these associations remained significant when general reading ability was controlled for (Fuchs et al., 2005; Fuchs et al., 2006; Hecht et al., 2001; Simmons et al., 2008), suggesting that the relationship between reading and arithmetic is mediated by a common reliance on phonological processing. One of the most comprehensive studies on the relationship between phonological processing and math competence was undertaken by Hecht et al. (2001) who examined the associations between various phonological processing skills and general math achievement in a fouryear longitudinal study of primary school children from second to fifth grade. Specifically, Hecht et al. examined the effects of phonological awareness, rate of access, and  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

phonological working memory on individual differences in mathematical ability. Their longitudinal findings showed that only phonological awareness uniquely predicted growth in math skills over and above the influence of the other phonological skills. Against the background of their data, Hecht et al. (2001) argued that since it has been repeatedly shown that phonological awareness, or the conscious sensitivity to the phonological structure of language, plays a pivotal role in the development of reading skills (e.g. Bradley & Bryant, 1983; Hulme, Hatcher, Nation, Brown, Adams & Stuart, 2002; Wagner, Torgesen, Rashotte, Hecht, Barker, Burgess, Donahue & Garon, 1997), it may be a factor constraining the development of both reading and math. Data from cognitive neuroscience studies also suggest that phonological awareness might be a specific locus of the association between phonological processing and arithmetic, as both abilities lead to consistent activation of the left temporo-parietal cortex. Indeed, developmental neuroimaging studies of reading showed that the left angular gyrus is typically activated during rhyme judgment tasks (Shaywitz et al., 1998; Temple et al., 2001) and that left angular gyrus activation during reading relates to individual differences in phonological awareness tasks (Turkeltaub et al., 2003). In view of this convergent body of evidence showing both behavioral and neuronal connections between arithmetic and phonological awareness, the present study will particularly focus on the influence of phonological awareness on children’s arithmetical skills. While there is evidence for a strong relationship between phonological awareness and arithmetic processing, an important limitation of the existing behavioral studies is that they do not reveal which specific aspects of arithmetical processing are particularly affected by phonological awareness. This is because these studies assessed arithmetic using general standardized achievement tests. Such tests yield a total score which only reflects performance averaged across different domains of mathematical cognition (including counting, calculation, word problem solving and geometry). Moreover, these total scores often express performance only in terms of accuracy, which, compared to a fine-grained analysis of both accuracy and response time, is a less sensitive measure for investigating very specific cognitive processes. Although Hecht et al. (2001) also examined the association between the time to solve simple arithmetic problems and phonological awareness, these authors failed to find unique associations between growth in arithmetic solution time and phonological awareness from fourth to fifth grade. This null result may be explained by the focus on arithmetic speed but not accuracy, and by the lack of an investigation of the effect of phonological awareness on different arithmetic operations or on different problem sizes. In view of this, it remains to be understood which components

510 Bert De Smedt et al.

of arithmetic processing phonological awareness is related to. As reviewed above, data from cognitive neuroscience suggest an overlap between reading and arithmetic at the neural level in the left angular gyrus. Neuroimaging studies of arithmetic indicate that the left angular gyrus is modulated by specific types of arithmetic problems, i.e. problems of a relatively small problem size (Grabner, Ansari, Reishofer, Stern, Ebner & Neuper, 2007). Because the solutions to small problems are retrieved significantly more often than those of relatively larger problem sizes (Barrouillet, Mignot & Thevenot, 2008; Campbell & Xue, 2001; Imbo & Vandierendonck, 2008), these data suggest that the left angular gyrus is particularly engaged during the retrieval of arithmetic facts. Given the overlap between phonological awareness and arithmetic in the left angular gyrus, these neuroimaging data predict that phonological awareness should be particularly related to small problems. This may be particularly so because a crucial factor determining the performance on phonological awareness tasks involves the quality of children’s longterm phonological representations (e.g. Fowler, 1991). In reading, it is well demonstrated that structured phonological representations are necessary in order to learn to read an alphabetic script (e.g. Hulme, 2002). In arithmetic, the quality of long-term phonological representations might be particularly important for the retrieval of existing arithmetic facts, which are stored in a phonological code in long-term memory. Indeed, it can be hypothesized that more distinct longterm phonological representations of such facts will be easier, i.e. more accurate and faster, to retrieve. In this way a relationship between long-term phonological representations in reading and arithmetic can be predicted. In light of this, the aim of the present study was to move beyond general standardized mathematics achievement measures and to examine the relationship between individual differences in phonological awareness and single-digit arithmetic. By means of an experimental task in which we manipulated arithmetic problem size, we tried to discern the precise locus of the relationship between phonological awareness and arithmetic. In view of the above, we predicted that performance on the phonological awareness task should be particularly related to small problems, which are assumed to be retrieved from long-term memory in a verbal code and hence would rely on the integrity of phonological longterm representations. Moreover, we wanted to go beyond broad measures of arithmetic competence and their relationship to phonological awareness and investigated whether this relation differs across different arithmetical operations. Dehaene et al. (2003) hypothesized that distinct singledigit operations might rely on different codes, and proposed that multiplication relies on a verbal code, whereas subtraction relies more on a quantity code.  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

Against this background, we should expect the relationship between phonological awareness and arithmetic to be the most prominent in multiplication. As we hypothesized that phonological awareness would be particularly related to fact retrieval, we focused on children who had already acquired sufficient basic arithmetic facts in all operations, but at the same time exhibited sufficient variability in their performance to meaningfully capture individual differences. This led to the selection of fourth and fifth grade children. All children were presented with a computerized single-digit arithmetic verification task, comprising addition, subtraction, multiplication and division. They also completed a Phoneme Elision task, which is a widely accepted measure of phonological awareness, to examine their quality of phonological representations in long-term memory. Two control measures were included to evaluate alternative explanations for any associations between Phoneme Elision and arithmetic verification. First, both tasks involve the temporary maintenance of a phonological code while processing the task. In Phoneme Elision, participants hold a word in mind while determining the phonological information to be deleted, and in the arithmetic verification tasks, participants may rehearse the solution or the problem while deciding on the correct response. Therefore, we included a measure of phonological short-term memory (STM) to examine the possibility that associations between both tasks are explained by shared task demands. It should be noted that measures of phonological STM may also rely on the quality of long-term phonological representations (e.g. Bowey, 1996; Snowling, 2000). In view of the present study, it is therefore crucial to use an STM task as a control that requires the short-term storage of phonological information but draws only minimally on the quality of long-term phonological representations. One task specifically designed for this is nonword repetition, which involves the immediate repetition of novel phonological forms (e.g. Gathercole, 1995; Gathercole, Willis & Baddeley, 1991). In particular, nonword repetition for nonwords of low wordlikeness is known to tap mainly the temporary storage of short-term rather than long-term phonological representations (Gathercole, 1995). Therefore, we administered the Nonword Repetition Test (NRT) designed by Dollaghan and Campbell (1998), which incorporates only nonwords of low wordlikeness. Different from the NRT, our measure of phonological awareness, Phoneme Elision, includes familiar words. Therefore, it may be assumed that our measure of phonological awareness drawing on familiar vocabulary for manipulation will rely to a greater extent on the quality of long-term phonological representations than the NRT. Second, the relation between phonological awareness and arithmetic might be explained by the covariation of both measures with reading ability (see Bull & Johnston,

Phonological processing and arithmetic 511

1997) and therefore, a standardized reading test was also administered.

Method Participants Participants included 37 children (25 girls, 12 boys) between 9 and 11 years, with a mean age of 10 years (SD = 8 months). For all of them, written informed parental consent was obtained. Children were recruited through the University of Western Ontario’s Developmental Psychology Participant Pool and the Thames Valley District School Board, in order to have participants from a variety of socioeconomic backgrounds. All were native English speakers without a history of learning disorders and were enrolled in the fourth (n = 23) or fifth grade (n = 14) of primary school. Materials Arithmetic verification task The problems of this task were selected from the so-called ‘standard set’ of single-digit arithmetic problems (Lefevre, Sadesky & Bisanz, 1996), which excludes tie problems (e.g. 5 + 5) and problems containing a 0 or 1 as operand or answer. This set comprises 56 problems per operation. From this set, 20 small and 20 large problems were selected for each operation, yielding a total set of 160 problems. Problem size was defined as described in Campbell and Xue (2001) (see also Barrouillet et al., 2008; Imbo & Vandierendonck, 2008). For addition and multiplication, problems were defined as small when the product of the operands was smaller than or equal to 25 (e.g. 6 + 3; 7 · 3) and as large if the product of their operands was larger than 25 (e.g. 6 + 7; 7 · 8). Subtraction and division problems were defined on the basis of their inverse relation with addition and multiplication, respectively. The position of the largest operand was counterbalanced in addition and multiplication. Likewise, the size of the subtrahend ⁄ divisor and the difference ⁄ quotient was counterbalanced. All stimuli were presented in white on a black background in Courier New (size 60) on a 17-inch laptop monitor, using the E-prime software (Version 1.1, Psychology Software Tools). Each problem was horizontally presented in Arabic format for 3000 ms upon which an equal sign appeared for 500 ms, followed by two response alternatives, one correct and one incorrect. Children had to indicate the position of the correct alternative by pressing with the index finger the key on the same side as the correct answer. The left response key was the ‘A’ key, and the right response key was the ‘L’ key. Both keys were marked with yellow stickers and children were encouraged to keep their index  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

fingers on both keys during the task. The position of the correct answer was balanced across problems. In addition and subtraction, incorrect answers were created by adding or subtracting 1 or 2 to the solution; in multiplication, incorrect answers were table related and were created by adding ⁄ subtracting 1 multiplicand from ⁄ to the product; in division, incorrect answers were 1 more or less than the correct answer. Children were asked to respond as accurately and as quickly as possible. Ten practice problems were administered to familiarize children with the task requirements and key assignments. The 160 test problems were randomly divided into four blocks of 40 problems. Children were given breaks between blocks. Phonological awareness The standardized Phoneme Elision subtest from the Comprehensive Test of Phonological Processing (Wagner, Torgesen & Rashotte, 1999) was used. This test is a well-accepted measure of phonological awareness (e.g. Hecht et al., 2001; Schatschneider, Francis, Foorman, Fletcher & Mehta, 1999). In this test, children were asked to repeat an existing word and then to say what the word would be if a specified phoneme in the word were to be deleted. In the first three test items, larger chunks of phonemes were required to be omitted by the child. For example, children were instructed to say popcorn and then were asked what would be left if they were to say popcorn without saying pop. The remaining 17 test items required the omission of single phonemes, like in saying cup without the phoneme ⁄ k ⁄ . All phonemes were consonants and the position of the omitted phoneme varied randomly. A real word remained after omitting the requested phoneme. Items were one- and two-syllable words consisting of three to five phonemes. Each part of the test was preceded by three practice items to make the child familiar with the task requirements. For each child, a standardized score (M = 10, SD = 3) was derived using age-appropriate norms. Control measures Phonological short-term memory. The Nonword Repetition Test (NRT) (Dollaghan & Campbell, 1998) was administered as a measure of phonological STM. This test consisted of 16 nonwords, four stimuli each containing one, two, three, and four syllables following an alternating consonant-vowel structure, yielding a total of 90 phonemes. All nonwords incorporated novel phonological forms with low English wordlikeness (e.g. equal stress across syllables, tense vowels only), to minimize reliance on long-term phonological representations. A detailed description of the development and administration of the NRT is provided in Dollaghan and Campbell (1998). The nonwords were presented auditorily via a

512 Bert De Smedt et al.

digital audio recording of an adult female speaker following the phonetic transcription and pronunciation guidelines described by Dollaghan and Campbell (1998). The child was asked to immediately repeat the presented nonword. Verbal repetition responses were recorded for subsequent analyses. The percentage of correctly recalled phonemes was calculated for each child. Reading. The Sight Word Efficiency subtest of the Test of Word Reading Efficiency (Torgesen, Wagner & Rashotte, 1999) was administered. In this test, children are asked to read as many individually printed words as possible in 45 seconds. A standardized score (M = 100; SD = 15) was calculated using age-appropriate norms. Calculation. The Calculation subtest of the WoodcockJohnson III Tests of Achievement (Woodcock, McGrew & Mather, 2001) was administered to ensure that all participants had normal mathematical competency. This standardized paper-and-pencil test consists of a broad range of calculation problems. Unlike the experimental, timed arithmetic verification task, this test is an untimed measure of individual differences in children’s mathematical competence. As such, a comparison of this test and the experimental, speeded arithmetic verification measure will enable the evaluation of speeded versus unspeeded measures of mathematical abilities and their associations with phonological awareness. A standardized score (M = 100; SD = 15) using age-appropriate norms was derived. Procedure All measures were individually administered in a quiet room. The different tasks were presented in a random order in one session, which took about one hour.

Results Descriptive analyses On the arithmetic verification task, all children performed above chance level (50%) in addition, subtraction and multiplication. However, a large number of children performed below chance on the division problems, probably owing to a lack of instruction in division. Therefore, the data on the division trials were excluded. The mean percentages and response times for the arithmetic verification task are shown in Table 1. There was no evidence of a speed–accuracy trade-off, as shown by non-significant correlations that were either negative or close to zero (Addition: r(37) = ).01, p = .94; Subtraction: r(37) = .07, p = .69; Multiplication: r(37) = ).29, p = .08). A 2 (problem sizes) · 3 (operations) · 2 (grades) repeated measures ANOVA, with problem size and operation  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

Table 1 Descriptive statistics for the arithmetic verification task broken down by operation Addition Accuracy Small 92.97 Large 85.14 Reaction time Small 1025.65 Large 1713.06

(7.50) (12.66) (454.74) (1003.33)

Subtraction

Multiplication

90.81 (8.54) 77.57 (15.57)

84.86 (10.77) 72.43 (19.95)

1384.18 (691.99) 2182.22 (1289.05)

1803.43 (1020.74) 2353.98 (1628.15)

Note: Accuracy is expressed in percentage correct. Reaction time is expressed in milliseconds.

as within-subject factors and grade as a between-subject factor, was conducted on children’s accuracies and reaction times to evaluate the main effects and interactions of problem size and operation on children’s accuracy and reaction time. With regard to accuracy, there was a main effect of problem size (F(1, 35) = 68.69, p < .01) and of operation (F(2, 70) = 14.85, p < .01), while the main effect of grade was not significant (F(1, 35) = 3.26, p = .08). There was a significant interaction between grade and problem size (F(1, 35) = 11.17, p < .01), indicating that fifth graders were more accurate than fourth graders for large (t(70) = )2.89, p < .01) but not for small problems (t(70) = )0.44, p = .66). There was also a significant grade · operation interaction (F(2, 70) = 5.40, p = .01), indicating that fifth graders were more accurate in multiplication (t(70) = )3.27, p < .01), whereas no grade differences occurred in addition (t(70) = )0.49, p = .62) and subtraction (t(70) = )0.83, p = .41). The size · operation (F(2, 70) = 2.58, p = .08) and size · operation · grade (F(2, 70) = 0.14, p = .87) interactions were not significant. The analysis of the reaction times showed a main effect of problem size (F(1, 35) = 24.67, p < .01), operation (F(2, 70) = 10.99, p < .01) and grade (F(1, 35) = 5.52, p = .02) but no significant problem size · operation (F(2, 70) = 1.07, p = .35), size · grade (F(1, 35) = 2.90, p = .10), operation · grade (F(2, 70) = 2.58, p = .08) and size · operation · grade (F(2, 70) = 0.14, p = .87) interactions, indicating that fifth graders had overall faster responses than fourth graders. These data show the presence of the problem size effect in all operations for speed and accuracy. For the purpose of correlational and regression analyses, the accuracy and speed data for small and large problems were initially averaged across operations, and their descriptive statistics are shown in Table 2. The means, standard deviations, and range of the scores on Phoneme Elision, NRT, and the general achievement tests are displayed in Table 2. Unfortunately, due to failure of the microphone to record responses, NRT data were available for only 32 children. The minima and maxima of the various measures showed that all data were well distributed without any ceiling or floor effects. The means of the standardized measures were close to the expected population average, indicating that the selected sample

Phonological processing and arithmetic 513

Table 2 Descriptive statistics for the collected measures Variable Arithmetic Accuracy (% correct) Reaction time (ms) Phonological Awareness Phonological Memory Achievement Reading Calculation

Small Large Small Large Phoneme Elision NRT Word Reading Efficiency Woodcock-Johnson

M

SD

Minimum

Maximum

89.55 78.38 1404.42 2083.09 10.65 83.32

7.42 12.49 526.39 1109.47 2.23 8.31

68.33 51.67 557.21 578.08 6 64.58

100.00 100.00 2567.29 6340.05 14 95.83

102.76 98.57

11.69 7.90

83 81

126 116

Note:  n = 32.

indicated that only accuracy on the large problems was related to WJ Calculation, whereas no other significant associations between the arithmetic verification task and general calculation ability emerged. It was somewhat striking to find low correlations between the experimental arithmetic measure and the standardized test of math achievement. This might be explained by the limited variability in standardized test scores. We therefore calculated associations between the raw scores on the WJ Calculation and the arithmetic verification task. High correlations were found with accuracy (Small: r(37) = .37, p = .02; Large: r(37) = .62, p < .01) but not with speed (Small: r(37) = ).28, p = .09; Large: r(37) = ).21, p = .20) on the arithmetic verification task. This might be explained by the fact that the latter task is an online measure, whereas the WJ Calculation is an untimed measure of arithmetical ability, highlighting the importance of the inclusion of both timed and untimed measures to fully assess individual differences in mathematical competence. Phoneme Elision was significantly related to accuracy on the arithmetic verification task. This relationship appeared to be somewhat specific to the small problem sizes, as significant relationships were found between phonological awareness and reaction time for the small problems, while a similar relationship was not found for the large problems. Thus, in line with our expectations, the associations between phonological awareness and arithmetic were particularly prominent on the small problems. Scatterplots displaying these associations are presented in Figure 1.

is representative for the general population. The standard scores on the reading test and WoodcockJohnson (WJ) Calculation were both within 2 standard deviations of the mean, that is, between 70 and 130. This shows that none of the children in this sample showed exceptionally low or high performance. Correlational analyses In order to assess the extent to which individual differences in performance on the measures collected were related to one another, Pearson correlation coefficients were calculated (Table 3). Because the children from the present study were selected from two different grades, we examined the effect of grade on performance. Grade was significantly related to reaction times on the arithmetic verification task for both problem sizes and to accuracy on the large problems. Consistent with the ANOVA reported above, an inspection of the means for each grade showed that fourth graders were slower for small (1563.01 vs. 1143.88) and large (2404.26 vs. 1555.45) problems, and less accurate for large problems (74.71 vs. 84.40) than fifth graders. There were no significant associations between grade and the standardized tests, as the scores on these tests were calculated using age-appropriate norms. As can be expected, accuracy and reaction times of both problem sizes on the arithmetic verification task were highly related to each other. An examination of the relationship between individual differences on WJ Calculation and the arithmetic verification task

Table 3 Correlations between the administered measures

1 2 3 4 5 6 7 8 9

Grade Accuracy small Accuracy large Reaction time small Reaction time large Phoneme Elision NRT Reading WJ Calculation

1

2

3

4

5

6

7

8

.09 .38* ).35* ).36* .25 .15 ).09 .16

.78** ).13 .06 .49** .02 .40* .21

).26 ).08 .44** .13 .31 .34*

.82** ).44** ).09 ).15 .08

).25 ).06 ).16 .08

.22 .38* .04

.28 ).22

).33*

Note: Grade was coded as a dummy variable: fourth graders were coded as )1 and fifth graders were coded as +1.  n = 32. * p < .05; ** p < .01.

 2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

514 Bert De Smedt et al.

100 95

Accuracy

90 85 80 75 70 65 60 5

6

7

8

9 10 11 12 Phoneme Elision

13

14

15

3000

Regression analyses

Reaction time

2500

2000

1500

1000

500

0

.48, p < .01) and speed (Addition: r(37) = ).34, p = .04; Subtraction: r(37) = ).46, p < .01; Multiplication: r(37) = ).29, p = .08). For the large problems, there was only a trend for associations between phoneme elision and accuracy (Addition: r(37) = .32, p = .05; Subtraction: r(37) = .26, p = .13; Multiplication: r(37) = .42, p < .01), but no associations with speed on the large problems were found (Addition: r(37) = ).24, p = .15; Subtraction: r(37) = ).27, p = .11; Multiplication: r(37) = ).19, p = .27). These associations showed that the relationship between phonological awareness and small problems appears to be similar across the administered operations. Still, it should be acknowledged that unlike addition and subtraction, the association between phonological awareness and multiplication occurred for both problem sizes, at least for correlations between phonological awareness and multiplication accuracy.

5

6

7

8

9 10 11 12 Phoneme Elision

13

14

15

Figure 1 Scatterplots showing the significant associations between Phoneme Elision and accuracy (top panel) and reaction time (bottom panel) on the small problems. The solid line represents the linear regression for this relationship.

Turning to the control measures, performance on the NRT was neither related to the arithmetic verification task nor to Phoneme Elision. Reading ability was found to be significantly correlated with accuracy on the large problems and with Phoneme Elision. Although Phoneme Elision was related to the arithmetic verification task, it was not related to performance on the WJ Calculation test, suggesting that the relationship between phonological awareness and individual differences in mathematical competence is specific to online, speeded measures of arithmetic. We also examined the correlations between Phoneme Elision and arithmetic verification for each operation separately. These associations were found to be similar to those reported in Table 3. Phoneme Elision was associated with performance on the small problems in both accuracy (Addition: r(37) = .42, p < .01; Subtraction: r(37) = .31, p = .06; Multiplication: r(37) =  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

The correlations between phonological awareness and arithmetic reported above could be said to be confounded by individual differences in general reading competence. In other words, individuals with higher reading scores might also perform better on the tests of arithmetic verification, revealing a general relationship between reading and arithmetic rather than one that is specific to phonological awareness. To examine the extent to which phonological awareness explained unique variance in arithmetic over and above the association between general reading ability and arithmetic, we conducted a series of hierarchical regression analyses. Two steps were sequentially included to examine whether phonological awareness is related to arithmetic performance over and above the influence of grade (Step 1) and reading ability (Step 2). The results of these analyses (Table 4) showed that after controlling for these potentially confounding factors, phonological awareness still explained significant unique amounts of variance in accuracy and speed on the small problems. These analyses also indicate that phonological awareness does not add significant variance to children’s accuracy and reaction time on the large problems after controlling for grade and reading ability. While phonological awareness predicted significant variance in children’s performance on arithmetic problems with a small size, this association may be explained by similar task demands rather than common underlying cognitive processes, as both the Phoneme Elision and the arithmetic verification task require subjects to hold a temporary phonological code in STM. We therefore calculated the correlation between NRT, which measures phonological STM, and arithmetic to examine this issue. Table 3 showed no associations between NRT and arithmetic. There was also no significant relationship between NRT and phonological

Phonological processing and arithmetic 515

Table 4 Hierarchical regression analyses predicting small and large problems Small problems Step Accuracy Reaction time

1. 2. 3. 1. 2. 3.

Large problems

Predictor

b

t

R2

DR2

b

t

R2

DR2

Grade Reading Phoneme Elision Grade Reading Phoneme Elision

.099 .413 .389 ).351 ).185 ).353

0.59 2.65* 2.34* )2.22* )1.17 )2.06*

.010 .179 .296 .123 .157 .253

.010 .169* .117* .123* .034 .096*

.382 .356 .257 ).356 ).198 ).108

2.44* 2.42* 1.57 )2.25* )1.25 )0.60

.146 .272 .322 .127 .167 .174

.146* .126* .051 .127* .040 .010

Note: * p < .05.

additions and large subtractions were grouped into a procedural category. Correlational analyses indicated that the association between Phoneme Elision and arithmetic appeared even more specific. Phoneme Elision was significantly related to retrieval accuracy (r(37) = .49, p < .01) and speed (r(37) = ).41, p = .01) but not to procedural accuracy (r(37) = .31, p = .06) and speed (r(37) = ).26, p = .12). We again conducted hierarchical regression analyses with the same sequential steps as those reported in Table 4, to examine whether phonological awareness predicted variance in arithmetic over and above the influence of grade (Step 1) and reading ability (Step 2). These analyses (Table 5) demonstrated that after accounting for these potential confounds, phonological awareness still significantly explained unique variance in accuracy of retrieval problems and a marginally significant amount of variance in speed on these problems. By contrast, phonological awareness did not significantly explain variance in the accuracy and reaction times of the procedural problems. Moreover, there were no associations between NRT and these new arithmetic measures (retrieval accuracy: r(32) = .05, p = .78; retrieval speed: r(32) = ).08, p = .67; procedural accuracy: r(32) = .05, p = .79; procedural speed: r(32) = ).08, p = .65), indicating that the specific association between phonological awareness and retrieval problems cannot be explained by common task demands.

awareness in the present sample. Therefore, phonological STM was not a mediating factor between phonological awareness and arithmetic. Additional analyses Based on previous research on single-digit arithmetic (e.g. Barrouillet et al., 2008; Campbell & Xue, 2001; Imbo & Vandierendonck, 2008), it can be assumed that small problems are associated with a high probability of retrieval from long-term memory and large problems are more procedurally based. However, we did not directly examine the strategies that children applied during the solution of both problem types. Interestingly, two recent studies analysed the frequencies of retrieval use as a function of operation and problem size, with problem size defined in the same way as in the present study. Imbo and Vandierendonck (2008) examined single-digit addition and multiplication. As shown in the Appendix, the problems used by these authors largely overlap with the problems administered in the current study. The mean percentages of fourth graders for retrieval use reported by Imbo and Vandierendonck (2008) were 88% in small additions and 37% in large additions. For multiplication, the mean percentages were 85% in small problems and 76% in large problems. Barrouillet et al. (2008) examined strategy use in singledigit subtraction. They reported 64% retrieval use in small subtractions and 27% retrieval use in large subtractions. These frequencies in subtraction were obtained in third graders and one could expect these percentages to increase in higher grades, as in the present study. Against this background, we re-analysed our data by grouping small additions, small subtractions and multiplications in a so-called retrieval category. The large

Discussion What is the precise locus of the overlap between reading and arithmetic and their associated disorders? Behavioral data have shown that phonological awareness tasks

Table 5 Hierarchical regression analyses predicting retrieval and procedural problems Retrieval problems Step Accuracy Reaction time

1. 2. 3. 1. 2. 3.

2

Procedural problems 2

Predictor

b

t

R

DR

Grade Reading Phoneme Elision Grade Reading Phoneme Elision

.190 .381 .375 ).376 ).176 ).316

1.14 2.44* 2.24* )2.40* )1.12 )1.84

.036 .180 .289 .141 .172 .249

.036 .144* .108* .141* .031 .077

Note: * p < .05;  p = .07.

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b

t

R2

DR2

.225 .274 .189 ).229 ).273 ).100

1.36 1.70 1.04 )1.86 )1.73 )0.56

.051 .125 .153 .090 .163 .171

.051 .075 .028 .090 .074 .008

516 Bert De Smedt et al.

uniquely predict growth in mathematical skills over and above the influence of other phonological skills (Hecht et al., 2001). As phonological awareness is a crucial factor for successful reading development (e.g. Bradley & Bryant, 1983; Hulme et al., 2002; Wagner et al., 1997), it might account for variability in children’s development of both reading and arithmetic. Despite the welldocumented association between phonological awareness and both reading and arithmetic, the theoretical interpretation of these relationships remains unclear. One major drawback of the studies reported so far is that all of them have looked at arithmetic as measured through general standardized tests, which typically assess a variety of arithmetical abilities without a time constraint, thereby failing to capture specific aspects of online cognitive processing. The use of such heterogeneous measures of arithmetic competence have thus far made it impossible to assess specifically which aspects of arithmetic are particularly affected by the quality of long-term phonological representations. Against the background of findings from cognitive neuroscience research that suggest a specific neural overlap of phonological processing and arithmetic, the present study extends the above-mentioned behavioral work and provides the first behavioral evidence for a highly specific association between phonological processing and arithmetic. Developmental neuroimaging studies of reading have revealed that the left angular gyrus is consistently activated during phonological awareness tasks (Shaywitz et al., 1998; Temple et al., 2001), and that this activation is related to performance differences in phonological awareness tasks and reading ability (Turkeltaub et al., 2003). At the same time, studies in arithmetic have shown that the left angular gyrus is particularly active whenever the answer to a problem is directly retrieved from long-term memory, which is the case in problems of small problem size (Grabner et al., 2007), multiplication (Dehaene et al., 2003), and problems that have been highly practiced (Delazer, Domahs, Bartha, Brenneis, Lochy, Trieb & Benke, 2003). These overlapping neural processes may therefore indicate that the association between phonological processing and arithmetic should be specifically prominent during arithmetic fact retrieval and that the quality of long-term phonological representations might be a key factor accounting for the overlap in the neural resources recruited in both reading and arithmetic. Specifically, such data led us to hypothesize that more distinct long-term phonological representations should predict more efficient retrieval. The present study explicitly tested this proposal by examining the associations between phonological awareness and different arithmetic problem types. The regression analyses showed that phonological awareness is specifically related to small but not to large problem sizes. Additional analyses showed that phonological  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

awareness is uniquely related to problems that, in children, are likely to be retrieved from long-term memory, i.e. small additions, small subtractions and multiplications (Barouillet et al., 2008; Imbo & Vandierendonck, 2008), but not to problems that are likely to be solved by means of procedural strategies, such as large additions and large subtractions. These associations remain after controlling for general effects of reading ability. Moreover, phonological STM was not related to arithmetic performance or to phonological awareness in this age range, which indicates that these associations are not merely due to shared task demands or to individual differences in phonological STM. While previous studies have shown that phonological awareness is related to arithmetic, the present data provide a more fine-grained picture by revealing that it is the quality of long-term phonological representations rather than the short-term storage of phonological representations that is related to individual differences in arithmetic. Our data showed that the associations between phonological awareness and small problem sizes occur for each of the administered operations separately. This suggests that the effect of phonological awareness on small problem sizes is not operation-specific, at least not in children. This finding is in contrast with adult neuropsychological models of arithmetic that have shown that different arithmetical operations rely on distinct codes, indicating that multiplication relies on a verbal code whereas subtraction relies more on a quantity code (e.g. Dehaene et al., 2003). One possible explanation is that operation-specificity emerges throughout development and that the reliance on different codes might only become apparent in adults. Another explanation could be that the suggested distinction between operations is not absolute and that arithmetic performance might be better characterized in terms of strategies that people apply during problem solving. For example, it is widely agreed that, regardless of operation, single-digit arithmetic problems are either solved by retrieval or by using a procedural strategy, like counting or transformation (e.g. Siegler, 1996). Retrieval strategies are assumed to rely on a verbal code, and hence to be related to phonological processing, whereas procedural strategies might rely more on a quantitybased code. Retrieval strategies might occur not only in multiplication (and addition), but also in subtraction, in particular on the small problems (Barrouillet et al., 2008; Campbell & Xue, 2001). This retrieval strategy effect might explain the absence of operation-specificity in the association between phonological awareness and small problem sizes. Furthermore, Imbo and Vandierendonck (2008) showed that frequency of retrieval is high for both small and large multiplications, which echoes our findings with regard to multiplication. Indeed, phonological awareness was correlated with accuracy on both small and large multiplications; the same unique associations with phonological awareness were found

Phonological processing and arithmetic 517

when large multiplications were additionally included in a set of small problem sizes. Although our findings are consistent with previous reports that revealed associations between phonological processing and arithmetic (Durand et al., 2005; Hecht et al., 2001; Leather & Henry, 1994; Nol et al., 2004; Rasmussen & Bisanz, 2005; Simmons et al., 2008; Swanson, 2004), we did not find a relationship between phonological awareness and an untimed measure of general math achievement. By contrast, we showed that the association between phonological awareness and arithmetic is highly specific and can be best captured by online timed measures of arithmetic. This might explain why not all studies have found a relationship between phonological processing and arithmetic (Passolunghi et al., 2008) as such associations might be dependent on the amount of fact retrieval necessary to complete arithmetic achievement tests. One aspect of phonological processing, which also might explain the correlation between reading and arithmetic, not tested in the present study is phonological working memory. We used an STM task requiring the brief retention of phonological information (nonwords) only. Working memory tasks, on the other hand, require not only storage but also processing of information and draw upon executive information processing resources. For example, a classical phonological working memory task is reading span, in which people have to judge the veracity of a sentence while memorizing the final words of the presented sentences and recalling them at the end of a trial (Daneman & Carpenter, 1980). Several studies have demonstrated the influence of phonological working memory in both reading (e.g. De Jong, 1998) and mental arithmetic (for a review, see DeStefano & Lefevre, 2004). The phonological awareness task used in the present study closely resembled working memory tasks, because children were asked to retain phonological information while completing some processing task, i.e. phonological segmentation, on the input material. In line with this, Hecht et al. (2001) argued that the association between phonological awareness and math achievement is explained by the fact that both skills require central executive resources from working memory. One aspect of our results renders this suggestion unlikely. If the central executive accounted for the associations between phonological awareness and arithmetic, phonological awareness should be particularly related to the most difficult, i.e. large, problems, as these problems are typically shown to put the largest load on the central executive (Lemaire, Abdi & Fayol, 1996). The present findings indicate that this is not the case. Instead, we found strong evidence for a specific association with the problems that require less working memory (those with a small problem size), which makes it unlikely that our findings could be explained by individual differences in central executive resources. It might be worthwhile for future studies to include both measures of phonological  2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

awareness and phonological working memory when examining the associations between phonological processing and arithmetic, as phonological working memory might also underlie the development of number facts (e.g. Barrouillet et al., 2008). The absence of an association between Phoneme Elision and NRT merits further comment. It has been argued that measures of phonological awareness and phonological STM should correlate as they both draw upon phonological representations (e.g. Bowey, 1996; Snowling, 2000). However, our measure of phonological STM was specifically designed to minimize the reliance on long-term representations (Dollaghan & Campbell, 1998) and required the repetition of novel phonological material of low wordlikeness, involving short-term rather than long-term phonological representations (e.g. Gathercole, 1995). By contrast, our measure of phonological awareness involved familiar words and drew upon long-term phonological representations. This task difference might explain the absence of a correlation between NRT and Phoneme Elision. It should be noted that we used only one task of phonological awareness, which might limit the findings of the present study. However, we employed a classic standardized measure of phonological awareness which has been widely used in different areas of behavioral (e.g. Hecht et al., 2001; Schatschneider et al., 1999) and neuroimaging (e.g. Dougherty, Ben-Shachar, Deutsch, Hernandez, Fox & Wandell, 2007) research. Future studies should take into account the different strategies children apply during arithmetic problem solving. Although our findings point to an association between phonological awareness and the use of retrieval strategies, it is not clear from the present findings whether this association is explained by the frequency of retrieval use, or by the efficiency (in terms of accuracy and speed) of retrieval use, or to both. Therefore, it might be worthwhile to investigate how phonological awareness is related to strategy use in arithmetic. Although we did not find any associations between phonological STM and arithmetic, the role of phonological memory in performing arithmetic should not be underestimated. Phonological memory might be of critical importance in nonretrieval strategies when intermediate results need to be maintained during calculation (e.g. Imbo & Vandierendonck, 2007). Although one might expect a relation between phonological memory and large problems, the present study failed to find any evidence for that hypothesis. This might be explained by the fact that, in the present study, the memory load on these problems was limited. The association between phonological memory and arithmetic might only become apparent when larger problems, like multidigit calculation, are examined, or when arithmetic data are analysed in terms of strategies children apply (e.g. Imbo & Vandierendonck, 2007). The results of the present study also have important implications for subsequent developmental neuroimaging

518 Bert De Smedt et al.

studies of arithmetic. Although cognitive neuroscience research provided the basis for examining a specific association between phonological awareness and arithmetic, none of the reported imaging studies examined phonology and arithmetic in one and the same sample (of children). Our findings provide the first direct behavioral evidence for this association in one sample of children. These findings suggest an interesting avenue for future developmental neuroimaging research, hypothesizing that similar activation patterns should be observed during a phonological awareness task and during the solution of small problems of an arithmetic task. To conclude, the present study examined associations between phonological awareness and arithmetic and provided some of the first evidence for the specific locus of this relationship, namely the importance of the quality of long-term phonological representations for the retrieval of arithmetic facts.

Acknowledgements The authors wish to thank all children, parents and schools that participated in this study. Bert De Smedt is a Postdoctoral Fellow of the Research Foundation Flanders, Belgium. This research was supported by the National Science and Engineering Research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI), the Ontario Ministry of Research and Innovation and the Canada Research Chair Program. This research was also partially supported by Grant GOA 2006 ⁄ 01 from the Research Fund K.U.Leuven, Belgium.

References Barrouillet, P., Mignon, M., & Thevenot, C. (2008). Strategies in subtraction problem solving in children. Journal of Experimental Child Psychology, 99, 233–251. Bowey, J.A. (1996). On the association between phonological memory and receptive vocabulary in five-year-olds. Journal of Experimental Child Psychology, 63, 44–78. Bradley, L., & Bryant, P.E. (1983). Categorizing sounds and learning to read – a causal connection. Nature, 301, 419–421. Bull, R., Espy, K.A., & Wiebe, S.A. (2008). Short-term memory, working memory, and executive functioning in preschoolers: longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33, 205–228. Bull, R., & Johnston, R.S. (1997). Children’s arithmetical difficulties: contributions from processing speed, item identification, and short-term memory. Journal of Experimental Child Psychology, 65, 1–24. Campbell, J.I.D., & Xue, Q.L. (2001). Cognitive arithmetic across cultures. Journal of Experimental Psychology: General, 130, 299–315. Daneman, M., & Carpenter, P.A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning and Verbal Behavior, 19, 450–466.

 2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506. de Jong, P.F. (1998). Working memory deficits of reading disabled children. Journal of Experimental Child Psychology, 70, 75–96. Delazer, M., Domahs, F., Bartha, L., Brenneis, C., Lochy, A., Trieb, T., & Benke, T. (2003). Learning complex arithmetic – an fMRI study. Cognitive Brain Research, 18, 76–88. DeStefano, D., & Lefevre, J.A. (2004). The role of working memory in mental arithmetic. European Journal of Cognitive Psychology, 16, 353–386. Dollaghan, C., & Campbell, T.F. (1998). Nonword repetition and child language impairment. Journal of Speech, Language and Hearing Research, 41, 1136–1146. Dougherty, R.F., Ben-Shachar, M., Deutsch, G.K., Hernandez, A., Fox, G.R., & Wandell, B.A. (2007). Temporalcallosal pathway diffusivity predicts phonological skills in children. Proceedings of the National Academy of Sciences of the United States of America, 104, 8556–8561. Durand, M., Hulme, C., Larkin, R., & Snowling, M. (2005). The cognitive foundations of reading and arithmetic skills in 7- to 10-year-olds. Journal of Experimental Child Psychology, 91, 113–136. Fowler, A.E. (1991). How early phonological development might set the stage for phoneme awareness. In S.A. Brady & D. Shankweiler (Eds.), Phonological processes in literacy: A tribute to Isabelle Y. Liberman (pp. 97–117). Hillsdale, NJ: Erlbaum. Fuchs, L.S., Compton, D.L., Fuchs, D., Paulsen, K., Bryant, J.D., & Hamlett, C.L. (2005). The prevention, identification, and cognitive determinants of math difficulty. Journal of Educational Psychology, 97, 493–513. Fuchs, L.S., Fuchs, D., Compton, D.L., Powell, S.R., Seethaler, P.M., Capizzi, A.M., Schatschneider, C., & Fletcher, J.M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98, 29– 43. Gathercole, S.E. (1995). Is nonword repetition a test of phonological memory or long-term knowledge? It all depends on the nonwords. Memory and Cognition, 23, 83–94. Gathercole, S.E., Willis, C., & Baddeley, A.D. (1991). Differentiating phonological memory and awareness of rhyme: reading and vocabulary development in children. British Journal of Psychology, 82, 387–406. Grabner, R.H., Ansari, D., Reishofer, G., Stern, E., Ebner, F., & Neuper, C. (2007). Individual differences in mathematical competence predict parietal brain activation during mental calculation. NeuroImage, 38, 346–356. Hecht, S.A., Torgesen, J.K., Wagner, R.K., & Rashotte, C.A. (2001). The relations between phonological processing abilities and emerging individual differences in mathematical computation skills: a longitudinal study from second to fifth grades. Journal of Experimental Child Psychology, 79, 192– 227. Hoeft, F., Hernandez, A., Mcmillon, G., Taylor-Hill, H., Martindale, J.L., Meyler, A., Keller, T.A., Siok, W.T., Deutsch, G.K., Just, M.A., Whitfield-Gabrieli, S., & Gabrieli, J.D.E. (2006). Neural basis of dyslexia: a comparison between dyslexic and nondyslexic children equated for reading ability. Journal of Neuroscience, 26, 10700–10708.

Phonological processing and arithmetic 519

Hulme, C. (2002). Phonemes, rimes, and the mechanisms of early reading development. Journal of Experimental Child Psychology, 82, 58–64. Hulme, C., Hatcher, P.J., Nation, K., Brown, A., Adams, J., & Stuart, G. (2002). Phoneme awareness is a better predictor of early reading skill than onset-rime awareness. Journal of Experimental Child Psychology, 82, 2–28. Imbo, I., & Vandierendonck, A. (2007). The development of strategy use in elementary school children: working memory and individual differences. Journal of Experimental Child Psychology, 96, 284–309. Imbo, I., & Vandierendonck, A. (2008). Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults? Psychological Research ⁄ Psychologische Forschung, 72, 331–346. Jordan, N.C., Hanich, L.B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74, 834–850. Leather, C.V., & Henry, L.A. (1994). Working memory span and phonological awareness tasks as predictors of early reading ability. Journal of Experimental Child Psychology, 58, 88–111. Lefevre, J., Sadesky, G.S., & Bisanz, J. (1996). Selection of procedures in mental addition: reassessing the problem size effect in adults. Journal of Experimental Psychology: Learning, Memory and Cognition, 22, 216–230. Lemaire, P., Abdi, H., & Fayol, M. (1996). The role of working memory resources in simple cognitive arithmetic. European Journal of Cognitive Psychology, 8, 73–103. Meyler, A., Keller, T.A., Cherkassky, V.L., Lee, D.H., Hoeft, F., Whitfield-Gabrieli, S., Gabrieli, J.D.E., & Just, M.A. (2007). Brain activation during sentence comprehension among good and poor readers. Cerebral Cortex, 17, 2780– 2787. Niogi, S.N., & McCandliss, B.D. (2006). Left lateralized white matter microstructure accounts for individual differences in reading ability and disability. Neuropsychologia, 44, 2178– 2188. Nol, M.P., Seron, X., & Trovarelli, F. (2004). Working memory as a predictor of addition skills and addition strategies in children. Current Psychology of Cognition, 22, 3– 25. Passolunghi, M.C., Mammarella, I.C., & Altoe, G. (2008). Cognitive abilities as precursors of the early acquisition of mathematical skills during first through second grades. Developmental Neuropsychology, 33, 229–250. Pugh, K.R., Mencl, W.E., Jenner, A.R., Katz, L., Frost, S.J., Lee, J.R., Shaywitz, S.E., & Shaywitz, B.A. (2001). Neurobiological studies of reading and reading disability. Journal of Communication Disorders, 34, 479–492. Rasmussen, C., & Bisanz, J. (2005). Representation and working memory in early arithmetic. Journal of Experimental Child Psychology, 91, 137–157. Rivera, S.M., Reiss, A.L., Eckert, M.A., & Menon, V. (2005). Developmental changes in mental arithmetic: evidence for increased functional specialization in the left inferior parietal cortex. Cerebral Cortex, 15, 1779–1790. Schatschneider, C., Francis, D.J., Foorman, B.R., Fletcher, J.M., & Mehta, P. (1999). The dimensionality of phonolog-

 2009 The Authors. Journal compilation  2009 Blackwell Publishing Ltd.

ical awareness: an application of item response theory. Journal of Educational Psychology, 91, 439–449. Schlaggar, B.L., & McCandliss, B.D. (2007). Development of neural systems for reading. Annual Review of Neuroscience, 30, 475–503. Shalev, R. (2007). Prevalence of developmental dyscalculia. In D.B. Berch & M.M.M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 49–60). Baltimore, MD: Paul H. Brookes Publishing. Shaywitz, S.E., Shaywitz, B.A., Pugh, K.R., Fulbright, R.K., Constable, R.T., Mencl, W.E., Shankweiler, D.P., Liberman, A.M., Skudlarski, P., Fletcher, J.M., Katz, L., Marchione, K.E., Lacadie, C., Gatenby, C., & Gore, J.C. (1998). Functional disruption in the organization of the brain for reading in dyslexia. Proceedings of the National Academy of Sciences of the United States of America, 95, 2636–2641. Siegler, R.S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press. Simmons, F.R., & Singleton, C. (2008). Do weak phonological representations impact on arithmetic development? A review of research into arithmetic and dyslexia. Dyslexia, 14, 77–94. Simmons, F., Singleton, C., & Horne, J. (2008). Phonological awareness and visuo-spatial sketchpad functioning predict early arithmetic attainment: evidence from a longitudinal study. European Journal of Cognitive Psychology, 20, 711–722. Snowling, M.J. (2000). Dyslexia (2nd edn.). Malden, MA: Blackwell Publishers. Swanson, H.L. (2004). Working memory and phonological processing as predictors of children’s mathematical problem solving at different ages. Memory and Cognition, 32, 648–661. Temple, E., Poldrack, R.A., Salidis, J., Deutsch, G.K., Tallal, P., Merzenich, M.M., & Gabrieli, J.D.E. (2001). Disrupted neural responses to phonological and orthographic processing in dyslexic children: an fMRI study. NeuroReport, 12, 299–307. Torgesen, J.K., Wagner, R., & Rashotte, C. (1999). Test of Word Reading Efficiency (TOWRE). New York: Psychological Corporation. Turkeltaub, P.E., Gareau, L., Flowers, D.L., Zeffiro, T.A., & Eden, G.F. (2003). Development of neural mechanisms for reading. Nature Neuroscience, 6, 767–773. Van Eimeren, L., Niogi, S.N., McCandliss, B.D., Holloway, I.D., & Ansari, A. (2008). White matter microstructures underlying mathematical abilities in children. NeuroReport, 19, 1117–1122. Wagner, R.K., Torgesen, J.K., & Rashotte, C.A. (1999). Comprehensive Test of Phonological Processing. Austin, TX: Pro-Ed. Wagner, R.K., Torgesen, J.K., Rashotte, C.A., Hecht, S.A., Barker, T.A., Burgess, S.R., Donahue, J., & Garon, T. (1997). Changing relations between phonological processing abilities and word-level reading as children develop from beginning to skilled readers: a 5-year longitudinal study. Developmental Psychology, 33, 468–479. Woodcock, R.W., McGrew, K.S., & Mather, N. (2001). Woodcock-Johnson III Tests of Achievement. Itasca, IL: Riverside Publishing. Received: 16 July 2008 Accepted: 26 March 2009

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Appendix Arithmetic problems administered in the current study and in Imbo and Vandierendonck (2008)

Imbo & Vandierendonck (2008)

The current study Addition

Subtraction

2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9

5)2 5)3 6)2 6)4 7)2 7)3 7)4 8)3 8)5 8)6 9)2 9)4 9)5 9)6 9)7 10)2 10)3 10)4 10)6 10)8 11)4 11)5 11)6 11)7 12)3 12)4 12)5 12)7 12)8 13)4 13)6 13)7 13)8 14)6 14)9 15)8 15)9 16)7 17)8 17)9

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3 4 5 6 7 8 2 4 5 6 2 3 6 7 8 9 2 3 6 7 9 2 3 4 7 3 4 6 8 9 2 4 5 7 9 3 4 6 7 8

Multiplication 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 9 9 9

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

3 4 5 7 8 9 2 6 7 8 2 3 5 7 8 2 6 8 9 3 4 5 7 8 9 2 3 6 8 3 6 7 9 2 3 4 5 6 7 8

Addition 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3 4 5 6 7 8 9 2 4 5 6 7 8 9 2 3 5 6 7 8 9 2 3 4 6 7 8 9 2 3 4 5 7 8 9 2 3 4 5 6 8 9 2 3 4 5 6 7 9 2 3 4 5 6 7 8

Multiplication 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

3 4 5 6 7 8 9 2 4 5 6 7 8 9 2 3 5 6 7 8 9 2 3 4 6 7 8 9 2 3 4 5 7 8 9 2 3 4 5 6 8 9 2 3 4 5 6 7 9 2 3 4 5 6 7 8

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