NeuroImage 57 (2011) 771–781

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Effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency Bert De Smedt a,b,⁎, Ian D. Holloway a, Daniel Ansari a,⁎ a b

Numerical Cognition Laboratory, Department of Psychology, University of Western Ontario, London, ON, Canada Parenting and Special Education Research Group, Department of Education, Katholieke Universiteit Leuven, Belgium

a r t i c l e

i n f o

Article history: Received 22 June 2010 Revised 9 December 2010 Accepted 12 December 2010 Available online 21 December 2010 Keywords: Problem size effect Arithmetic fluency Intraparietal sulcus Hippocampus Fact retrieval Procedural strategies

a b s t r a c t Most studies on mathematics learning in the field of educational neuroscience have focused on the neural correlates of very elementary numerical processing skills in children. Little is known about more complex mathematical skills that are formally taught in school, such as arithmetic. Using functional magnetic resonance imaging, the present study investigated how brain activation during single-digit addition and subtraction is modulated by problem size and arithmetic operation in 28 children aged 10–12 years with different levels of arithmetical fluency. Commensurate with adult data, large problems and subtractions activated a fronto-parietal network, including the intraparietal sulci, the latter of which indicates the influence of quantity-based processes during procedural strategy execution. Different from adults, the present findings revealed that particularly the left hippocampus was active during the solution of those problems that are expected to be solved by means of fact retrieval (i.e. small problems and addition), suggesting a specific role of the hippocampus in the early stages of learning arithmetic facts. Children with low levels of arithmetical fluency showed higher activation in the right intraparietal sulcus during the solution of problems with a relatively small problem size, indicating that they continued to rely to a greater extent on quantity-based strategies on those problems that the children with relatively higher arithmetical fluency already retrieved from memory. This might represent a neural correlate of fact retrieval impairments in children with mathematical difficulties. © 2010 Elsevier Inc. All rights reserved.

Introduction The last five years have witnessed a growing amount of research on mathematics learning within the field of developmental cognitive neuroscience (for an overview see Ansari, 2008; Houde et al., 2010). In response to the need for improving mathematics education in many countries around the globe coupled with the importance of being numerate in a modern western society (Duncan et al., 2007), the field of mathematics learning has been put forward as an ideal workspace for interdisciplinary research in the field of “Educational Neuroscience” or “Mind, Brain, and Education” (De Smedt et al., 2010; Szucs and Goswami, 2007). The majority of studies in this field have focused on the neural correlates of very elementary numerical processing skills, such as magnitude comparison (see Houde et al., 2010 for a metaanalysis). Less is known about the neural correlates of more complex mathematical skills that are formally taught in school, such as arithmetic, and particularly about how individual differences in

⁎ Corresponding authors. De Smedt is to be contacted at Department of Educational Sciences, Katholieke Universiteit Leuven, Vesaliusstraat 2, P.O. Box 3765, 3000 Leuven, Belgium. Fax: +32 16 32 59 33. Ansari, Department of Psychology, University of Western Ontario, London, ON, Canada N6G 2K3. Fax: +1 519 661 3961. E-mail addresses: [email protected] (B. De Smedt), [email protected] (D. Ansari). 1053-8119/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2010.12.037

arithmetical competence modulate the neural processes recruited by solving arithmetic problems. Being fluent and efficient in performing basic calculations has been regarded as an important building block for the development of mathematical skills (Kilpatrick et al., 2001). Some children are known to have difficulties in mastering these basic number combinations (Jordan et al., 2003; Geary, 2004), but the neural correlates of these developmental impairments remain unknown. Against this background, the current study aimed to investigate how arithmetic problem solving affects brain activation in children who are in the process of becoming arithmetically fluent and how such neural processes differ between children with different levels of arithmetical fluency. Converging evidence has identified a fronto-parietal network during calculation in the adult brain, comprising the inferior parietal and prefrontal cortices (e.g., Dehaene et al., 2003; Nieder and Dehaene, 2009, for a review). Adult neuroimaging data have shown that the activation of this network is modulated by problem size (Stanescu–Cosson et al., 2000; Molko et al., 2003), operation (Dehaene et al., 2003; Lee, 2000) and individual differences in mathematical competence (Grabner et al., 2007, 2009b). Stanescu–Cosson et al. (2000) were the first to examine the neural correlates of the well-known problem size effect in arithmetic. This problem size effect is manifested in more accurate and faster responses on problems with relatively numerically small operands and answers

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(e.g., 3 + 2) than on problems with comparatively large operands and answers (e.g., 8 + 6) (Zbrodoff and Logan, 2005). Current explanations of the problem size effect in arithmetic consider it to be driven by differences in strategy use between small and large problems (Zbrodoff and Logan, 2005): Small problems are usually solved by means of fact retrieval, whereas large problems are more often solved by more errorprone and time-consuming quantity-based procedural strategies, such as counting or decomposing a problem into smaller problems (Campbell and Xue, 2001; Lefevre et al., 1996). Stanescu–Cosson et al. (2000) showed, by means of a parametric design, that the bilateral intraparietal sulcus and several frontal regions, including the left inferior and superior frontal gyri, the left precentral gyrus and the bilateral cingulate, showed significantly more activation during addition with large compared to small problem sizes. Interestingly, the same network was also more active during approximate (i.e., choosing which of two incorrect answers is closer to the correct solution, e.g. 4 + 5 = 8 or 3) than during exact (i.e., selecting which of two solutions is correct, e.g. 4 + 5 = 9 or 7) calculation. These findings likely reflect the greater involvement of quantity-based processes, such as estimation, that are necessary to solve an approximate calculation problem. The similarity between effects of problem size on brain activation and the networks activated during approximate calculation further supports the suggestion that the greater involvement of fronto-parietal regions during calculation of problems of a large compared to a small problem size reflects the larger involvement of quantity-based processes and the engagement of working memory resources. Stanescu-Cosson et al. also reported a region that showed the opposite effect. The left angular gyrus showed modulation during calculation of problems with a small problem size relative to large problems, possibly reflecting the greater involvement of phonologically represented rote arithmetic facts during the of small problems. There is also evidence to suggest that the four basic arithmetic operations are associated with the engagement of different neural networks. Specifically, the study of neuropsychological patients has demonstrated a double dissociation between subtraction and multiplication: Lesions in the left perisylvian cortex resulted in impairments in multiplication but not subtraction, while lesions to regions of the intraparietal cortex were associated with difficulties with subtraction but not multiplication (e.g., Cohen et al., 2000; Dehaene and Cohen, 1997). Functional neuroimaging data in healthy adults have revealed that the intraparietal sulcus and the posterior superior parietal lobe are more active during subtraction than during multiplication, whereas the left angular and supramarginal gyri are modulated to a stronger degree during multiplication relative to subtraction (Chochon et al., 1999; Lee, 2000). As appears to be the case for the problem size effect, the engagement of different brain regions as a function of arithmetic operation likely reflects the use of different strategies (Barrouillet et al., 2008; Campbell and Xue, 2001; Imbo and Vandierendonck, 2008). Multiplication is expected to be mainly solved by rote arithmetical fact retrieval, which recruits the left perisylvian language areas, while subtraction requires quantitybased procedural strategies that have been demonstrated to rely on the intraparietal sulci. Because there are large individual differences in learning arithmetic at the behavioral level (Dowker, 2005), neuroimaging studies have started to address the neural correlates of these individual differences in arithmetic. Grabner et al. (2007) found that individual differences in mathematical competence among adults are associated with the engagement of the left angular gyrus during multiplication, showing stronger activation in individuals of higher mathematical competence. Moreover, Grabner et al. (2009b) observed that these competencerelated differences in the left angular gyrus were absent after training, suggesting that competence-related differences can be attenuated by acquiring a high level of expertise in a particular set of problems. The degree to which the effect of problem size on brain activation differs as a function of mathematical competence has also been

addressed in patients with atypical mathematical development. Molko et al. (2003) investigated brain activation during arithmetic in adults with Turner Syndrome, a genetic disorder that is associated with a high prevalence of persistent deficits in mathematics. Interestingly, Molko et al. observed an effect of problem size on brain activation in the right intraparietal sulcus during exact addition in healthy controls but not in patients with Turner Syndrome. These data point to an abnormal recruitment of the right intraparietal sulcus as a function of problem size in individuals with lower mathematical competence and suggests a functional correlate of their arithmetic impairments. In sharp contrast to this large number of studies on arithmetic in healthy adults and in patients, little is known about the neural correlates of arithmetic in children. Moreover, the effects of problem size, operation and mathematical competence on the calculating brains of children have not been systematically investigated. Most of the existing investigations have examined brain activation during small addition. During the solution of these problems, primary school children seem to activate a network including the prefrontal and inferior parietal cortices (Davis et al., 2009a; Kucian et al., 2008; Meintjes et al., 2010). Rivera et al. (2005) investigated in children aged 8–19 years how the activation of the network underlying small addition and subtraction changes as a function of chronological age. They showed that, over developmental time, activation in the prefrontal cortex as well as areas in the basal ganglia and parahippocampal gyrus decreased with age, indicating that younger children recruit more working memory and attentional resources during arithmetic in order to engage in procedural problem–solving strategies. Age-related increases in brain activation were observed in the left inferior parietal cortex, i.e. the supramarginal gyrus, anterior angular gyrus and adjoining intraparietal sulcus, pointing to an increased functional specialization in the left parietal cortex for arithmetic with age. These brain activation data, however, were reported for the average of small addition and subtraction problems and no investigation of operation effects and any age-related changes in these were provided. In their seminal study, Stanescu–Cosson et al. (2000), highlighted that the cerebral basis of the problem size effect in arithmetic might be an important neural correlate for understanding typical and atypical mathematical development in children. Indeed, the problem size effect captures the use of different strategies – i.e. directly retrieving the answer from memory vs. using quantity-based procedures – to solve arithmetic problems in adults (Zbrodoff and Logan, 2005) and in children (Barrouillet et al., 2008; Imbo and Vandierendonck, 2008). Throughout development, children develop an increasing reliance on fact retrieval and a decreasing reliance on procedural strategies (Siegler, 1996), yet both strategies continue to exist into adulthood. There appears to be large inter-subject variability in this development and impairments in arithmetic strategy use have been highlighted as the hallmark of mathematics learning disabilities and subtypes of this learning disorder have been related to differential impairments in arithmetic strategies (Geary, 2004), pointing to the existence of a ‘retrieval’ or ‘semantic memory’ subtype (i.e. difficulties in arithmetic fact retrieval) and a ‘procedural’ subtype (i.e. impairments in understanding and executing procedural strategies). Against this background, the cerebral basis of the problem size effect might be a neural correlate for understanding individual differences in children's mathematical development and might help us to understand the various subtypes of learning disorders in mathematics. To the best of our knowledge, there are no studies available that examined the effect of problem size on brain activation in children. In the present functional magnetic resonance imaging (fMRI) study we examined the effect of problem size on children's brain activation during arithmetic. We predicted that small problems are more frequently solved by fact retrieval and will therefore show greater activation in the left-lateralized language areas. Large problems, on the other hand, put greater emphasis on quantity-based procedural strategies and, therefore, will show relatively larger activation in the

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intraparietal sulcus, compared to small problems. This problem size effect was tested in two arithmetic operations, i.e. addition and subtraction. We selected these two operations because they can be matched in terms of numerals in the problem. Because addition is more frequently solved by fact retrieval compared to subtraction (Barrouillet et al., 2008), we expected that addition would show greater activation than subtraction in the left perisylvian cortex. Conversely, we predicted that subtraction, which puts a greater emphasis on quantity-based procedural strategies than addition, would show relatively larger activation of the intraparietal sulcus than addition. We also examined whether the effects of problem size and operation on brain activation are affected by children's level of arithmetic proficiency. We therefore selected two groups of children with average reading skills but who differed in their mastery of basic calculation problems, as measured by a timed standardized test of arithmetical fluency, and examined whether the neural signature of the problem size effect differed between the two groups. In view of the above, the present study applied a 2 (Problem Size: small vs. large) × 2 (Operation: addition vs. subtraction) × 2 (Group: typical vs. low arithmetical fluency) full factorial ANOVA with Problem Size and Operation as within-subject factors and Group as between-subjects factor. Functional imaging data were collected by means of an event-related fMRI design, which allowed us to examine brain activation during only those problems that were solved correctly. We focused on children aged 10–12 years (grade 5 to 7) who were still in the process of becoming arithmetically fluent, but had received a considerable amount of instruction in basic number combinations. This is different from most existing fMRI studies, which have typically sampled very broad age ranges that are heterogeneous in terms of the amount of received mathematics instruction. Materials and methods Participants Participants were 28 healthy right-handed children aged 10– 12 years, with no neurological or psychiatric disorders. Children completed standardized tests that assessed their mathematical ability (Woodcock-Johnson (WJ) III Math Fluency and Calculation), reading ability (WJ III Letter–Word Identification and Reading Fluency), verbal IQ (WISC-IV Vocabulary), and nonverbal IQ (WISC-IV Block Design). We only included children who showed typical intellectual and reading development (standard score N 85). Performance on the WJ Math Fluency test was used to define two competence groups. Children were included in the low arithmetical fluency (LAF) group if their standardized score on the WJ Math Fluency subtest was below or at 85, which corresponds to 1 SD below the standardization sample mean. Children were included in the typical arithmetical fluency (TAF) group if their performance on the WJ Math Fluency was within the normal range, i.e. within 1 SD of the standardization sample mean. This study was approved by the Health Sciences Research Ethics board of the University of Western Ontario. Written informed consent was obtained from the children's guardians and written assent was obtained from the participating children. Only children who had at least two functional runs without excessive motion were included in the analyses (details of motion exclusion criteria are listed below). Data of seven children were discarded because of motion during the functional imaging and the data of one child were removed due to technical acquisition problems. Furthermore, due to their poor performance on the in-magnet arithmetic task (accuracy b 80%), two children were excluded. The final sample consisted of 18 children (6 males; age range: 10.08– 12.92 years; M = 11.77 years), who all attended grades 5 to 7. The descriptive statistics of both competence groups on age and the psychometric tests are given in Table 1. The groups did not differ in the number of boys and girls (Fisher's Exact test: p = 0.64), chronological

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age (t(16)= 0.62, p = 0.55, d = 0.31) or months of education (t(16) = 0.63, p = 0.54, d = 0.32). The groups differed in their performance on WJ Math Fluency (t(16)= 6.30, p b 0.01, d = 3.17), but not on the WJ Calculation (t(16) = 1.74, p = 0.10, d = 0.87). There were no group differences on the WJ letter–Word identification (t(16) = 1.60, p = 0.13, d = 0.80) but children from the LAF group performed more poorly than TAF children on the WJ reading fluency (t(16) = 2.13, p = 0.05, d = 1.07), even though we only selected children with typical reading development (standardized score N 85). There were no group differences in IQ (Vocabulary: t(16) = 0.68, p = 0.51, d = 0.34; Block design: t(16) = 1.34, p = 0.20, d = 0.68). Procedure Participation in this study involved two test sessions. Test Session 1 took place a few days before MR imaging. This session involved a training session in a mock-scanner, during which children were familiarized with the procedures associated with fMRI research participation. Children practiced keeping their head still and performing the experimental arithmetic task while lying in the mock-scanner. During this session, psychometric test data were also collected. Test Session 2 involved the acquisition of the MR imaging data. After the experimental arithmetic task, children also completed two other tasks in the scanner, which are not discussed here. Psychometric tests Mathematical competence was assessed by the Math Fluency subtests and the Calculation subtest of the Woodcock-Johnson III tests of Achievement (Woodcock et al., 2003). In the Math Fluency subtest, children had to answer as many single-digit addition, subtraction and multiplication problems as possible within a 3-minute period. The Calculation subtest is an untimed measure in which children had to solve increasingly difficult calculation problems. The Reading Fluency and Letter–Word Identification subtest of the Woodcock-Johnson III tests of Achievement (Woodcock et al., 2003) were administered as measures of reading ability. In Reading Fluency, children had to read a series of sentences as fast as possible and to indicate whether the sentence is true or false. In Letter–Word Identification, children needed to read out a list of words of increasing difficulty. Intellectual ability was tested by the WISC-IV Vocabulary and Block design subtests (Wechsler, 2003) as measures of verbal and non-verbal IQ, respectively. Experimental design and procedure All participants completed a single-digit arithmetic task during functional MR imaging, in which they had to select the correct answer to a presented problem. The problems of this task were selected from all possible pairwise combinations of the digits between 2 and 9, with Table 1 Participant characteristics. Variable

Age (years) Months of education WJ Math fluency WJ Calculation WJ Reading fluency WJ Letter word WISC Vocabulary WISC Block design

TAF-group (n = 10)

LAF-group (n = 8)

M

SD

M

SD

11.88 58.90 95.80 91.20 111.60 107.40 11.30 12.90

0.89 8.06 6.23 7.19 15.46 8.77 2.31 2.13

11.65 56.63 79.50 85.62 98.63 101.00 10.25 10.88

0.70 7.11 4.24 6.19 8.31 8.04 4.17 4.16

Note. TAF = Typical Arithmetical Fluency; LAF = Low Arithmetical Fluency.

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the exclusion of tie problems (e.g., 4 + 4) 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 80 problems. Problem size was defined as described in Campbell and Xue (2001) (see also Barrouillet et al., 2008; Imbo and Vandierendonck, 2008). For addition, problems were defined as small when the product of the operands was smaller than or equal to 25 (e.g., 5 + 2) and as large if the product of their operands was larger than 25 (e.g., 8 + 7). Subtraction problems were defined on the basis of their inverse relation with addition. The position of the largest operand was counterbalanced in addition; likewise, the size of the subtrahend and the difference was counterbalanced in subtraction. All stimuli were presented in white on a black background in Arial (font size 60) with the E-prime 2.0 software (Psychological Software Tools, Pittsburgh, PA). Problems were presented horizontally in Arabic format (e.g., 4 + 3), after which two response alternatives, one correct and one incorrect, were simultaneously shown on the left and right side of the screen. Participants were asked to indicate the correct answer by pressing the response button in their left hand if the correct answer was on the left side of the screen and on the response button in their right hand if the correct answer was on the right side of the screen. Participants were encouraged to answer as accurately and quickly as possible. The position of the correct answer was balanced. Incorrect answers were created by adding or subtracting 1 or 2 from the solution. The task was presented in an event-related fMRI design. Four functional runs were collected for each child. Functional runs started with 15 s of fixation, followed by 20 trials, comprising 5 trials of each problem size for each operation. A trial involved the presentation of a problem for 2000 ms after which an equal sign appeared for 750 ms in the centre of the screen followed by a 250 ms blank screen. Next, the response alternatives were presented for 2000 ms and participants were asked to press on the side of the correct answer. After the presentation of the response options, a jittered inter-trial interval between 5.5 s and 10.5 s (jittered in steps of 2.5 s, mean 8 s) was randomly introduced into the time series in order to enable the deconvolution of the hemodynamic response functions. Consequently, the duration of a trial could be 10.5 s, 13 s or 15.5 s. Each run took approximately 5 min. MRI data acquisition and analysis Functional and structural images were acquired in a 3 T Siemens Tim Trio whole-body MRI scanner, using a Siemens 32-channel head-coil (Siemens, Erlangen, Germany). Subjects' heads were stabilized with pillows to minimize head movement. A gradient echo-planar imaging T2*-sequence sensitive to the blood-oxygenation level-dependent (BOLD) contrast was used to acquire 45 functional images per volume, which were collected in an interleaved order (3 mm thickness, 80× 80 matrix, repetition time (TR): 2500 ms, echo time (TE): 30 ms, flip angle: 78°) and covered the whole brain. Between 109 and 117 volumes were acquired for each functional run. High-resolution anatomical images were acquired with a T1 weighted MPRAGE sequence (1× 1 × 1 mm. TI= 900 ms, TE= 4.25 ms, TR= 2300 ms, flip angle: 9°). All preprocessing steps and analyses were conducted with Brain Voyager QX, version 2.1 (Brain Innovation, Maastricht, The Netherlands). Only the functional runs without excessive movement artefacts – i.e. those runs with overall motion less than 3 mm across the run and with less than 2 mm motion between adjacent functional volumes – were included in the imaging analyses, yielding a total of 61/ 72 (85%) runs. The first 3 volumes of each run were discarded to allow for signal stabilization. Preprocessing of the data involved slice scantime correction, correction for 3D head motion and linear trend removal. After initial automatic alignment, the alignment of participant's functional images to their corresponding high-resolution anatomical image was manually fine-tuned. The co-registered functional dataset

was then transformed into Talairach space (Talairach and Tournoux, 1988), upon which it was spatially smoothed with a 6-mm full width at half maximum Gaussian smoothing kernel. For each subject, a general linear model that modelled the correctly solved trials and included the incorrectly solved trials as a regressor of no interest was calculated. A two-gamma hemodynamic response function was used to model the expected BOLD signal. A whole-brain random-effects full factorial 2 (Problem Size) × 2 (Operation)× 2 (Group) ANOVA was conducted on the fMRI data, with Problem size and Operation as within-subject factors and Group as between-subjects factor. Statistical F-maps were created for each main effect and for each interaction. Because these F-maps do not contain information about the direction of the main effects, tcontrasts were calculated to determine the direction of any significant main effects. Statistical maps were subsequently corrected for multiple comparisons using cluster-size thresholding (Goebel et al., 2006). This method requires first the setting of an initial voxel-level uncorrected threshold, in this study p b 0.001. After that, the thresholded maps are submitted to a whole-slab correction criterion based on the estimate of the map's spatial smoothness and on an iterative procedure (Monte Carlo simulation) for estimating cluster-level false positive rates. After 1000 iterations, the minimum cluster-size that yielded a false-positive rate (α) of 0.05 or less was used to threshold the statistical maps. In other words, the algorithm calculates the size that a cluster would need to be (cluster threshold) to survive correction for multiple comparisons at a given statistical level. Results Behavioral data The mean accuracy and reaction time for each problem type are displayed for each group in Table 2. A repeated measures ANOVA with Problem Size (small vs. large) and Operation (addition vs. subtraction) as within-subject factors and Group (TAF vs. LAF) as a betweensubjects factor was calculated on children's accuracy and reaction time. With regard to accuracy, there was a main effect of Problem Size (F(1,16) = 21.67, p b 0.01, ηp² = 0.58) showing more accurate performance on the small than on the large problems. There was a main effect of operation (F(1,16) = 8.11, p = 0.01, ηp² = 0.34), indicating that additions were solved more accurately than subtractions. There was a significant Problem Size × Operation interaction (F(1,16) = 4.58, p = 0.05, ηp² = 0.22), which showed that operation differences emerged on the large (t(17) = 2.97, p b 0.01) but not on the small problems (t(17) = 0.08, p = 0.93). Children from the LAF group performed less accurately than the TAF children (F(1,16) = 19.14, p b 0.01, ηp² = 0.54). Group did not interact with Problem Size (F(1,16) = 2.24, p = 0.15, ηp² = 0.12), Operation (F(1,16) = 4.00, p = 0.06, ηp² = 0.20) or the Problem Size× Operation interaction (F(1,16) =1.60, p =0.22, ηp² =0.09). Analysis of the reaction time data showed a highly similar pattern. There were main effects of Problem Size (F(1,16) = 44.86, p b 0.01,

Table 2 Arithmetic task performance. Condition

Small Addition Subtraction Large Addition Subtraction

Accuracy (% correct)

Reaction time (ms)

TAF

TAF

LAF

M

SD

M

97.33 97.66

2.85 3.87

93.33 92.50

93.16 90.50

4.04 7.41

88.13 75.63

SD

LAF

M

SD

M

7.97 7.77

629.07 653.48

109.84 113.81

833.39 854.24

168.64 162.88

8.79 10.80

743.55 951.78

170.65 366.67

1043.72 1232.36

206.13 220.08

Note. TAF = Typical Arithmetical Fluency; LAF = Low Arithmetical Fluency.

SD

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ηp² = 0.74) and Operation (F(1,16) = 13.34, p b 0.01, ηp² = 0.45), showing that small problems were solved faster than large problems and that additions were solved faster than subtractions. A significant Problem Size × Operation interaction (F(1,16) = 5.58, p = 0.03, ηp² = 0.26) indicated that additions were solved faster than subtractions on the large (t(17) = −3.28, p b 0.01) but not on the small problems (t(17) = −0.94, p = 0.36). TAF children were faster than the children from the LAF group (F(1,16) = 10.37, p b 0.01, ηp² = 0.39). Again, the effects of Problem Size (F(1,16) = 1.38, p = 0.26, ηp² = 0.08), Operation (F(1,16) = 0.04, p = 0.85, ηp² = 0.00) and the Problem Size × Operation interaction (F(1,16) = 0.01, p = 0.92, ηp² = 0.00) did not differ between groups. Imaging data Main effect of problem size The main effect of problem size on brain activation is displayed in Table 3 and Fig. 1. Small problems activated more strongly the left hippocampus and left anterior medial temporal lobe. These problems were also associated with stronger activations in the posterior part of the bilateral angular gyrus. Additional clusters that were more active in small compared to large problems were observed in the medial prefrontal cortex, the posterior cingulate, the right postcentral gyrus, the right posterior insula, the left lateral sulcus, right superior temporal sulcus and the right inferior temporal gyrus. These differences in activation between small and large problems were due to less deactivation (compared to baseline) during the small problems than during the large problems. Large problems more strongly activated the anterior cingulate cortex and the anterior insula, bilaterally, and the left middle and inferior frontal gyrus, the left precentral gyrus as well as the bilateral intraparietal sulcus. Main effect of operation Only the left hippocampus showed higher activation during addition than during subtraction (Table 4; Fig. 2). Subtraction activated more strongly the frontal cortex bilaterally, comprising the anterior cingulate cortex, the anterior insula, the middle frontal gyrus and the inferior frontal gyrus as well as the left precentral gyrus. Larger parietal activation was observed during subtraction in the bilateral superior parietal lobe, including the intraparietal sulcus. Subtraction also activated more the left middle temporal gyrus and the cerebellum, bilaterally.

Table 3 Main effect of problem size on brain activation. Contrast

Brain area

Small N large Bil Medial prefrontal cortex L Lateral sulcus L Anterior medial temporal lobe R Superior temporal sulcus R Angular gyrus Bil Posterior cingulate cortex R Postcentral gyrus L Angular gyrus R Posterior insula L Hippocampus R Inferior temporal gyrus Large N small L Intraparietal sulcus Bil Anterior cingulate cortex L Precentral gyrus L Middle frontal gyrus L Anterior insula R Intraparietal sulcus R Anterior insula L Inferior frontal gyrus

x

y

z

k

t

−6 −54 −42 51 54 −3 48 −51 36 −21 36

29 −16 8 −4 −61 −16 −16 −58 −16 −16 23

−2 16 −26 −11 13 37 25 13 22 −11 −23

11,311 5787 3861 3141 1747 1280 1236 904 512 375 301

6.13 6.91 7.40 8.41 5.71 6.26 6.39 5.28 5.15 5.59 5.55

−30 6 −48 −27 −30 27 30 −42

−58 23 8 8 17 −52 23 50

40 37 37 52 1 37 4 7

3707 3453 2755 2190 1872 1609 698 298

−5.98 −14.28 −6.46 −7.74 −6.17 −6.49 −5.33 −4.84

775

Problem size × operation interaction The Problem Size × Operation interaction revealed one significant activation cluster in the right posterior superior parietal lobe (peak voxel: 33, −70, 46; k = 679, t = 4.98, p b 0.0001; Fig. 3). The bar chart in Fig. 3, depicting the average beta weights (expressed in z-scores) of the different conditions, shows a large effect of problem size in subtraction, but not in addition. Main effect of group The main effect of Group revealed no significant activation clusters. Group × problem size interaction The Group × Problem Size interaction revealed two significant activation clusters in the right intraparietal sulcus (peak voxel: 29, −65, 25: k = 322, t = 5.74, p b 0.0001; and peak voxel: 28, −61, 42: k = 307, t = 5.41, p b 0.0001), as illustrated in Fig. 4. The beta weights (expressed in z-scores) for each problem size and group are shown for both clusters in Fig. 4. There was a problem size effect for both clusters in the TAF group, with less activation in the right intraparietal sulcus during small than during large problems. In the LAF group, a different effect of problem size on brain activation in the intraparietal sulcus was observed: in one cluster (28, −61, 42), no effect of problem size on brain activation was found, whereas in the other cluster (29, −65, 25) the problem size effect was reversed, showing larger activation in the right intraparietal sulcus during small than during large problems. This latter observation is counterintuitive and might be explained by the low accuracy in large subtractions in the LAF group compared to the TAF group. As a result, the large subtractions in the LAF group may involve the relatively easy large subtractions while in the TAF group it might contain both easy and more difficult large subtractions. We subsequently averaged the beta weights of both intraparietal sulcus clusters together for each subject and each problem size (Fig. 5). This clearly revealed a problem size effect in the TAF group, with less activation in the right intraparietal sulcus in small than in large problems, whereas no problem size effect was observed in the LAF group. Group × operation interaction There were no significant activation clusters in the Group × Operation interaction, indicating that the effect of operation on brain activation was the same in both groups. Group × problem size × operation interaction No significant activation clusters were observed in the Group × Problem Size × Operation interaction. This suggests that the observed Problem Size × Operation interaction was the same in both groups. It also indicates that the Group × Problem Size interaction was similar in addition and subtraction. Discussion The network observed during calculation in our study was comparable to the brain activation patterns that have been associated with the computation of small addition problems (compared to baseline) in primary school children (Davis et al., 2009b; Kucian et al., 2008; Meintjes et al., 2010). The current study, however, significantly extends these data by showing that this calculation network is specifically modulated by arithmetic problem size, operation and individual differences in arithmetical fluency. Commensurate with adult data, large problems and subtractions activated a fronto-parietal network, including the intraparietal sulci. In contrast to published data from adults, the present findings revealed that the left hippocampus was active during the solution of those problems that are expected to be solved by means of fact retrieval, i.e. small problem sizes and addition problems. In addition to these main effects, arithmetical fluency level was found to modulate the effect of

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Fig. 1. Statistical maps illustrating the effect of problem size on brain activation. Areas showing greater activation during small than during large problems are depicted in blue; areas showing more activation during large than during small problems are shown in orange.

problem size on brain activation in the right intraparietal sulcus, indicating that individual differences in children's arithmetical fluency modulate brain function during calculation.

Table 4 Main effect of operation. Contrast

Brain area

x

y

z

k

t

L Hippocampus

−18

−10

−14

676

6.64

L Intraparietal sulcus Bil Anterior cingulate cortex R Middle frontal gyrus L Anterior insula R Superior parietal lobe L Inferior frontal gyrus L Middle temporal gyrus Bil Cerebellum L Precentral gyrus L Middle frontal gyrus R Precuneus R Anterior insula R Inferior frontal gyrus L Cerebellum R Cerebellum

−18 3 24 −30 36 −39 −51 −3 −45 −24 12 33 39 −33 33

−58 26 2 17 −40 32 −61 −67 5 −1 −61 17 26 −67 −55

40 34 55 1 52 7 −2 −23 34 49 46 4 16 −23 −26

2666 2510 1208 1123 1029 940 821 776 758 718 691 626 468 421 351

−6.14 −6.37 −5.92 −7.05 −6.09 −5.94 −5.34 −5.93 −6.04 −5.89 −5.84 −5.26 −5.58 −5.67 −4.94

Add N sub Sub N add

The impact of arithmetical fluency level A key finding of the present study was that children with low levels of arithmetical fluency did not exhibit the same modulation of the right intraparietal sulcus as a function of problem size compared to children with typical levels of arithmetic fluency. Specifically, this latter group of children showed less activation of the right intraparietal sulcus during the calculation of problems with a relatively small problem size in comparison to activation correlated with solving arithmetic problems with a relatively large problem size. This is consistent with adults, who also show less activation in the intraparietal sulcus during the solving of small compared to large arithmetic problems (Stanescu Cosson et al., 2000). Such findings indicate that problems that are likely to be retrieved from long-term memory (e.g., problems with a relatively small problem size) require less activation of quantity representations in the parietal cortex than those problems that are typically solved by procedural strategies. Children with low arithmetical fluency, on the other hand, recruited the right intraparietal sulcus to the same extent during both large and small problems. This might indicate that these children continued to rely, to a greater extent than their typically developing peers, on quantity-based calculation strategies for the small problems, which the children with relatively higher arithmetical fluency might have retrieved from memory and thus required less engagement of the parietal cortex. Impairments in fact retrieval are known to be the

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Fig. 2. Statistical maps illustrating the effect of operation on brain activation. Areas showing greater activation during addition than during subtraction are depicted in blue; areas showing more activation during subtraction than during addition are shown in orange.

hallmark of children with mathematical difficulties (Geary, 2004, 2010; Jordan et al., 2003) and the current study provides the first evidence for a neural correlate of these impairments in children with mathematical difficulties. It should be noted that the current data are not consistent with the evidence presented in two previous fMRI studies that compared brain activation during calculation in children with and without mathematical difficulties (Davis et al., 2009a, 2009b; Kucian et al., 2006). Specifically, these studies failed to observe activation differences in the parietal cortex between children with and without mathematical difficulties. These studies did, however, not systematically manipulate problem size in their set of arithmetic problems and, consequently, they were not able to examine the effect of arithmetic problem size on brain activity in children with and without mathematical difficulties. Furthermore, Davis et al. (2009b) and Kucian et al. (2006) used a block design, which collapsed brain activation during both correct and incorrect trials, whereas we employed an event-related fMRI design and only analyzed the correctly solved trials. As we compared children of different mathematical ability levels, who logically differed in their performance on the arithmetic task they had to perform in the scanner, averaging brain activation over correct and incorrect responses would have introduced a confound in the estimation of the functional signal (Murphy and Garavan, 2004). These differences

in design may explain the difference between the current findings and those of Davis et al. (2009b) and Kucian et al. (2006). The fMRI data clearly indicate atypical functional activation in the right intraparietal sulcus in children with low arithmetical fluency. It has been suggested that only the right parietal cortex shows a process of developmental specialization for the (formatindependent) representation of magnitude, highlighting the crucial importance of the right intraparietal sulcus for arithmetical development (Holloway et al., 2010). Moreover, earlier developmental data have found that children with dyscalculia show structural (Molko et al., 2003; Rotzer et al., 2008) and functional (Mussolin et al., 2010; Price et al., 2007) alterations of the right intraparietal sulcus during number processing. This is consistent with adult data in patients with Turner syndrome, a genetically mediated form of mathematics disorder, who show an abnormal recruitment of the right intraparietal sulcus during arithmetic (Molko et al., 2003). It could be that children with mathematical difficulties rely on an impaired system of magnitude representation, which might prevent them from moving towards arithmetic fact retrieval. Alternatively, the results reflect a persistent reliance on the use of quantity-based strategies during arithmetic in children with low levels of arithmetic fluency, which could be the consequence of the atypical functioning or development of brain regions involved in arithmetic fact retrieval.

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Fig. 3. Problem size × Operation interaction in the right posterior superior parietal lobe (Peak voxel: 33, −70, 46: k = 679, t = 4.98, p b 0.0001). The bar chart represents the mean parameter estimates (averaged across all voxels of the activated cluster) across all participants in the right posterior superior parietal lobe. Y-axis depicts the BOLD signal represented in z-scores. Error bars depict 1 standard error of the mean.

Future developmental imaging studies should investigate this issue further. Effects of problem size and operation The brain areas that were more activated in small than in large problems were the medial prefrontal cortex, the left anterior medial temporal lobe and hippocampus, the posterior cingulate and the posterior part of the angular gyrus, which were all characterized by less deactivation (compared to baseline) during the small problems than during the large problems. These areas correspond to the socalled default-mode network in children (Supekar et al., 2010), which is the brain network that typically shows less reductions (relative to baseline) in brain activation during tasks that require less cognitive load (e.g., Raichle et al., 2001). As expected, large problems, compared to small problems, activated more strongly a fronto-parietal network. This network corresponds to the adult brain areas that have been previously found to be more active during large than during small additions (Molko et al., 2003; Stanescu– Cosson et al., 2000), those that are more active during the use of selfreported procedural strategies than during arithmetical fact retrieval (Grabner et al., 2009a) and those that are more active during untrained than during trained problems (e.g., Delazer et al., 2003; see Zamarian et al., 2009, for a review). Activation in the frontal cortex, including the

bilateral anterior cingulate, the anterior insula, the left middle and inferior frontal gyrus, reflects the increasing demands on working memory and attentional resources during more effortful tasks (Duncan and Owen, 2000), such as the use of procedural strategies, which are used by children when solving large problems (Barrouillet et al., 2008). Activation in the intraparietal sulcus suggests the larger involvement of quantity-based processes during the solution of problems with a large compared to a small problem size. It is noteworthy that activations in the intraparietal sulcus were observed in its more posterior part, a finding consistent with recent observations by Wu et al. (2009; see also Simon et al., 2002), who showed that particularly the posterior part of the intraparietal sulcus (hIP3), rather than its anterior part, is active during mental arithmetic. An analogous fronto-parietal network was more active during subtraction than during addition, which is not unexpected as subtraction is more frequently solved by procedural strategies than addition (Barrouillet et al., 2008). In contrast to our expectations, the (left) angular gyrus did not show larger activation during addition than during subtraction, as is typically observed in adults (Dehaene et al., 2003). However, the left hippocampus emerged as the only significant activation cluster that was more active during addition than during subtraction. We further corroborated this observation by performing a conjunction analysis of the problem size and operation effects ([small–large] ∩ [addition–subtraction]). This additional analysis revealed that the left hippocampus (−21 −13 −14)

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Fig. 4. Problem size × Group interaction revealing two significant activation clusters in the right intraparietal sulcus (peak voxel: 28, −61, 42: k = 307, t = 5.41, p b 0.0001; peak voxel: 29, −65, 25: k = 322, t = 5.74, p b 0.0001). Bar charts depict the mean parameter estimates (averaged across all voxels of the activated cluster) per group in the right intraparietal sulcus for each significant cluster (left: 28 −61 42; right: 29 −65 25). Y-axis depicts the BOLD signal represented in z-scores. Error bars depict 1 standard error of the mean.

is the only region that is more active during both small problems and additions than during large problems and subtractions. Thus, the left hippocampus is the only region of the default mode network which exhibits overlap in the effects of problem size and operation. This might suggest that this region is particularly important in arithmetic, especially for those problems that could be expected to be solved by fact retrieval. The hippocampus is known to play a crucial role in the general retrieval of facts from memory (e.g., Squire et al., 2004) and it has been

Fig. 5. Mean parameter estimates (averaged across all voxels of the activated cluster) averaged per subject per condition for the two right intraparietal sulcus clusters that showed a significant Problem size × Group interaction. Y-axis depicts the BOLD signal represented in z-scores. Error bars depict 1 standard error of the mean.

suggested that the medial temporal lobe plays a time-limited role in semantic memory, particularly in the early stages of learning and the retrieval of recently learned facts (Smith and Squire, 2009). Van Opstal et al. (2008) showed that the hippocampus plays a role in the initial learning of ordinal sequences, after which activation is transferred to the angular gyrus. Recent observations by Uddin et al. (2010; see also Rushworth et al., 2006) revealed that the posterior part of the angular gyrus shows strong structural and functional connectivity with hippocampal regions. One potential hypothesis is that the hippocampus plays an important role in the initial consolidation of arithmetic facts in long-term memory, whereas over development, the angular gyrus becomes more involved when highly automatized arithmetic facts are retrieved. Consistent with this hypothesis, Rivera et al. (2005) reported that the activation in the left hippocampus and parahippocampal gyrus during the solution of small arithmetic problems decreases with chronological age, while activation in the supramarginal and anterior angular gyri during arithmetic increases over developmental time. Their sample included children ranging in age from 8 to 19 years, and it is therefore not clear at what age the angular gyrus is starting to play a role during arithmetic, an issue that should be addressed in future longitudinal developmental imaging studies. The present study also indicated that the right posterior superior parietal lobe showed significantly greater modulation in response to large subtraction problems relative to the other problem types. This brain activation pattern parallels the behavioral results, which showed that the large subtractions comprise the most difficult problem type. The posterior superior parietal lobe has been associated with number processing and calculation, as outlined in a meta-analysis by Dehaene et al. (2003). This brain area is not thought to be specific to the number

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domain, but has been shown to play a central role in the orienting of visuospatial attention (e.g., Corbetta et al., 2008). Dehaene et al. (2003) suggested that the posterior superior parietal lobe supports the attentional orientation to quantities on a mental number line, which is needed during numerical tasks that require extensive quantity-based processing, such as number comparison, approximation, and subtraction. The current data show that more posterior superior parietal resources are needed when solving the more difficult problems, i.e. large subtractions. In future studies, this relationship should be further evaluated by presenting independent tasks of visuospatial attentional orientation and examining the overlap between these tasks and the problem-size effect. Implications for educational neuroscience The current findings highlight that predictions derived from adult neuroimaging studies may not always be adequate for characterizing functional brain organization in children (Ansari, 2010), by clearly illustrating that the brain regions in children that subserve the computations of problems with a high probability of the use of retrieval strategies are not the same as those reported in previous studies with adults. In view of this, it can be speculated that retrieval is a graded phenomenon, with different levels in the automaticity of arithmetic fact retrieval being associated with different brain regions — i.e. the hippocampus may be important in the initial encoding and retrieval of arithmetic facts, the angular gyrus subserves the retrieval of fully consolidated facts and the intraparietal sulcus is involved in modulating the relative contribution of quantitative information to arithmetic problem solving. Behavioral measures, such as verbal reports, might not always capture such different levels of arithmetic fact retrieval. Neuroimaging data might provide a level of analysis and measurement that cannot be accessed by behavioral studies alone, adding new insights to theories of arithmetic fact retrieval development in children. Related to this, we observed a group difference in the modulation of parietal activation as a function of problem size in the absence of such an interaction effect on either reaction time or accuracy data. This again suggests that brain imaging can provide an additional level of explanation that may uncover subtle processing differences between groups of children that may not be detected through the measurement of behavioral data alone. It should finally be emphasized that the type of received math instruction might have a tremendous effect on the brain activation patterns during calculation. Delazer et al. (2005) examined the effect of two methods to learn new arithmetic operations on brain activation during calculation. One approach involved learning by drill (i.e. learn the association between problem and answer) whereas the other approach comprised learning by strategy (i.e. apply a sequence of problem solving steps to calculate the solution). Findings revealed that the drill approach activated the angular gyrus more strongly than the strategy approach. The children of the current study all came from the same school district and all received a similar type of math instruction. Future longitudinal and cross-sectional studies should therefore compare brain activation of groups of children who differ in their type of math instruction (e.g. degree of emphasis on math fact retrieval) and investigate the extent to which this potential hippocampal–angular gyrus shift is modulated by the kind of math instruction children receive at school. Acknowledgments The authors wish to thank all children and parents who participated in this study. This research was supported by the National Science and Engineering Research Council (NSERC) of Canada, an operating grant from the Canadian Institutes of Health Research (CIHR), and grant GOA 2006/01 from the Research Fund KULeuven, Belgium. Special thanks are due to Bea Goffin for her help and support during the collection of the data.

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