Learning and Individual Differences 20 (2010) 123–129

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Learning and Individual Differences j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / l i n d i f

Neurocognitive approaches to developmental disorders of numerical and mathematical cognition: The perils of neglecting the role of development Daniel Ansari ⁎ Department of Psychology and Graduate Program in Neuroscience, University of Western Ontario, Westminster Hall, London, ON, Canada N6G 2K3

a r t i c l e

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Article history: Received 12 December 2008 Received in revised form 29 May 2009 Accepted 3 June 2009 Keywords: Adult neuropsychology Developmental disorders Numerical cognition Atypical development Developmental cognitive neuroscience

a b s t r a c t The present paper provides a critical overview of how adult neuropsychological models have been applied to the study of the atypical development of numerical cognition. Specifically, the following three assumptions are challenged: 1. Profiles of strength and weaknesses do not change over developmental time. 2. Similar neuronal structures are activated in children and adults, as well as the notion that 3. Similarities in behavioral performance imply equivalence in underlying neurocognitive mechanisms. Data from behavioral and neuroimaging studies with both typically and atypically developing children is reviewed to illustrate the pitfalls of these assumptions. The present review proposes that, instead of resting on adult neuropsychological models, the use of both cross-sectional and longitudinal methods is required to elucidate the agerelated changes in brain and behavior that give rise to the breakdown of numeracy and mathematics. Empirical data derived from such studies will generate explanatory models of the development of atypical numerical cognition. © 2009 Elsevier Inc. All rights reserved.

1. Introduction While progress in research on the atypical development of numerical competencies continues to lag behind advances in the understanding of reading disorders, there now exist a growing body of studies reporting investigations into atypical number development (Berch & Mazzocco, 2007; Butterworth, 2005). It is, therefore, important to consider what theoretical models may be most appropriate to provide explanatory frameworks for this emergent body of findings. Fueled by advances in non-invasive functional neuroimaging methods, such as functional Magnetic Resonance Imaging (fMRI), there has been a rapidly accumulating body of studies uncovering the neural circuitry that underlies basic numerical competencies, such as the representation and processing of numerical magnitude (for reviews see: Brannon, 2006; Cohen Kadosh, Lammertyn, & Izard, 2008), as well as higher-level processes such as mental arithmetic algebra and calculus (Andoh et al., 1990; Ansari, 2008; Delazer et al., 2003; Ischebeck, Zamarian, Egger, Schocke, & Delazer, 2007; Krueger et al., 2008). This work has primarily been conducted with adult participants and has led to an impressive body of literature that has allowed researchers to map out the brain structures underlying different aspects of mature numerical and mathematical cognition. When combined with a long tradition of the study of numerical cognition in neuropsychological patients (Butterworth, 1999; Dehaene & Cohen, 1995; McCloskey, Caramazza, & Basili, 1985), these data have ⁎ Tel.: +1 519 661 2111x80548; fax: +1 519 850 2554. E-mail address: [email protected]. 1041-6080/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.lindif.2009.06.001

generated several models of numerical and mathematical processing that posit functional and structural dissociations between different aspects of numerical cognition. The present paper will first provide a brief review of some the existing neurocognitive models of numerical and mathematical cognition that have been frequently applied to interpret the performance profiles of children with mathematical difficulties and disorders. Subsequently, a critical overview of how these adult models have been used to interpret data from atypically developing children who present with mathematical difficulties will be provided. Against this background, it will be argued that much of the current application of neurocognitive data to interpret the behavior of children with mathematical difficulties and disorders ignores the crucial role that developmental processes play in these disorders. The limitations of the application of neuropsychological approaches to the study of developmental disorders will be highlighted and pathways for the integration of the role of developmental processes into neurocognitive models of mathematical difficulties and disorders will be proposed. 2. Numerical and mathematical processes in the adult brain For almost a century, neuropsychologists have associated areas of the parietal cortex with numerical and mathematical processing. This link between the parietal cortex and numerical cognition was initially uncovered through the study of adult neuropsychological patients (Henschen, 1919, 1925). Specifically, patients with damage to the parietal cortex were found to suffer from calculation deficits. Perhaps most prominent among these investigations, are those of Josef Gerstmann. He described patients with lesions to the angular gyrus

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presenting with calculation deficits in addition to agraphia, finger agnosia and left-right disorientation. This constellation of symptoms has become known as ‘Gerstmann syndrome’ (Gerstmann, 1940, 1957). With the advent of functional neuroimaging methods it became possible to study the neural correlates of numerical and mathematical processing in the healthy human brain. In the first functional neuroimaging study of calculation, Roland and Friberg (1985) used Xe intra-cartoid methodology to measure cerebral blood flow difference in the brain while participants calculated. Their results revealed increased blood flow during a subtraction task in bilateral regions of the angular gyrus as well as the prefrontal cortex. Subsequent studies using positron emission tomography (PET) and later functional Magnetic Resonance Imaging (fMRI) also revealed an association between parietal regions and calculation tasks (Burbaud et al., 1995; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Gruber, Indefrey, Steinmetz, & Kleinschmidt, 2001; Rueckert et al., 1996). Moreover, through the use of both data from neuropsychological patients and functional neuroimaging studies of the healthy adult human brain, investigators have started to differentiate between a variety of numerical and mathematical tasks and their underlying processes. Specifically, researchers differentiate between, on the one hand, the basic processing of numerical quantity, which is typically measured by tasks such as number comparison and estimation and processes related to arithmetic problem solving and calculation on the other. Against the background of a large body of data from neuropsychological patients, whose performance on a variety of numerical tasks had been tested, Dehaene and Cohen (1995) put forward a highly influential functional-structural model of numerical and mathematical processing in the brain: ‘The Triple Code Model’. In light of the data from neuropsychological patients, Dehaene and Cohen contend that there are three principle codes (‘quantity’, ‘verbal’ and ‘visual’) for numerical and mathematical processing that are functionally and structurally dissociated from one another. In other words, these codes are associated with the activation of anatomatically distinct brain regions and can function independently of one another. More recently, studies with healthy adults have explored the predictions of the ‘Triple Code Model’ through the use of functional neuroimaging methods. In a seminal study, Dehaene et al. (1999) used fMRI to show that, consistent with the predictions of the Triple Code Model, exact verbal calculation and approximate non-verbal calculation led to distinct patterns of brain activation. In a recent metaanalysis of functional neuroimaging studies, Dehaene, Piazza, Pinel, and Cohen (2003) put forward a model of the involvement of the parietal cortex in numerical and mathematical processing that predicted the existence of three different parietal circuits that subserve different aspects of processing and representation. Consistent with the ‘quantity code’ of the Triple Code model, bilateral regions of the IPS are thought to subserve the internal representation of quantity. Concurrent with early neuropsychological models and a large body of recent neuroimaging research, the model also predicts that areas of the left temporo-parietal cortex comprising the angular and supramarginal gyri are involved in the verbal processing of numerical information, such as mental arithmetic. Finally, the review posits a third circuit in bilateral regions of the superior parietal lobules (SPL) that subserve attentional processes required during number processing, such as the greater attentional resources that are required during multi-digit compared with single-digit calculation. Furthermore these regions are thought to subserve the frequently observed linkages between spatial and numerical processing (Hubbard, Piazza, Pinel, & Dehaene, 2005). In addition to the growing body of research showing differences between the neural correlates of verbal and non-verbal processing during numerical and mathematical tasks, researchers have started to map out differences and commonalities in the neural mechanisms

subserving numerical and non-numerical magnitude processing (Cohen Kadosh et al., 2005; Pinel, Piazza, Le Bihan, & Dehaene, 2004), the effects of different stimulus formats (for example, whether numerical problems are presented using Arabic numerals or number words) on brain activation (Cohen Kadosh, Cohen Kadosh, Kaas, Henik, & Goebel, 2007; Piazza, Pinel, Le Bihan, & Dehaene, 2007) and the relationship between individual differences in mathematical competence and brain activation during numerical and mathematical processing (Grabner et al., 2007). This impressive body of neuropsychological and neuroimaging data has profoundly influenced researchers interested in a variety of aspects of numerical and mathematical processing. This has been particularly so in the study of atypical numerical development, where the type of data reviewed above has influenced the interpretation of data from children with mathematical difficulties and disorders.

3. The case for the application of adult neuropsychology The data reviewed above from adult neuropsychological and neuroimaging studies demonstrates that different aspects of numerical and mathematical processing can be functionally separated from one another in the adult brain and appear to be subserved by different neural regions that can function relatively independently from one another. In view of this, it is very tempting to predict that mathematical difficulties and disorders and their subtypes can be understood as impairments to these dissociated aspects of processing. Indeed many researchers have advocated this approach to the interpretation of developmental disorders of numerical and mathematical cognition. In order to be able to evaluate this approach, the following section will review these theoretical proposals and their application to the empirical study of atypical number development. Against the background of a meta-analysis of adult neuroimaging studies of numerical and mathematical processing, Dehaene et al. (2003) state that: “Our hypothesis is that the cultural construction of arithmetic is made possible by pre-existing cerebral circuits that are biologically determined and are adequate to support specific subcomponents of number processing (Dehaene, 1997). This hypothesis supposes an initial prespecialization of the brain circuits that will ultimately support high-level arithmetic in adults.” (p.499). The authors go on to hypothesize that: “As in adult acalculia, at least two subtypes of developmental dyscalculia should be traceable to differential impairment of quantity vs. language processing circuits” (p.500). Thus the clear prediction put forward by these authors is that the brain circuits that are found to underlie different aspects of numerical and mathematical processing in adults are also present in children and that they can be differentially impaired, leading to different subtypes of developmental disorders of numerical and mathematical processing. A similar hypothesis was put forward by Butterworth (1999) who contends that developmental dyscalculia is the result of an impaired ‘number module’ in the brain that is innately specified. The neuropsychological notion of functionally dissociable processes has led many to apply the logic and terminology of adult neurospsychology to the study of atypical development of numerical and mathematical competencies in children. For example, in his review of research on mathematical disabilities, Geary (1993) proposes the existence of three subtypes of mathematical disabilities of a developmental origin: 1. Semantic memory subtype; 2. Procedural subtype, and 3. Visuo-spatial subtype. Against the background of data from studies of adult brain-damaged patients, Geary proposes ‘Neuropsychological Features’ for each of the subtypes of developmental cases of mathematical disabilities. The direct application of data from adult neuropsychological case studies to explain the subtypes of mathematical disabilities is justified by Geary in view of the fact that there are similarities in the behavioral performance of

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children with certain subtypes of mathematical disabilities and adult neuropsychological patients. As Geary (2004), in his discussion of origins of procedural deficits in children with mathematical disabilities states: “Based on the similarity between the deficits associated with MLD and those associated with acquired dyscalculia, neuropsychological studies of dyscalculia provide insights into the potential systems contributing to the procedural deficits of children with MLD” (p. 11). Thus, according to this position, the observed similarities in the behavioral performance profiles of adult neuropsychological patients and children with mathematical difficulties make adult neuropsychology a useful explanatory framework for developmental disorders of numerical and mathematical processing. Perhaps the most direct application of methodological and theoretical features of adult neuropsychological studies to developmental difficulties and disorders of numerical cognition are the singlecase studies of children with mathematical difficulties and disorders (Temple, 1992, 1997a,b). In these studies, the logic of single and double dissociations is directly applied to explore the strengths and weaknesses of children with mathematical difficulties and disorders. In her book entitled ‘Developmental Neuropsychology’ (Temple, 1997a,b), for example, presents single cases of children who show a procedural but not a semantic retrieval deficit and vice versa. The origins of the single-case profiles of performance on measures of different domains of numerical and mathematical skills are then interpreted with recourse to the adult neuropsychological literature. Several genetic developmental disorders are marked by impairments of both numerical and mathematical processing. One of these genetic developmental disorders is Williams syndrome (WS), which is caused by a microdeletion one copy of chromosome 7 of at least 28 genes (Donnai & Karmiloff-Smith, 2000). Williams syndrome is characterized by a profile of relative strengths and weaknesses. This uneven cognitive profile has lead to a significant amount of interest among cognitive psychologists and neuroscientists, as it might provide a window onto the functional independence of different cognitive processes. On the one hand, children with Williams syndrome have a striking impairment in visuo-spatial cognition and, on the other hand, present with relatively strong language skills. This has lead some researchers to argue that WS is a perfect example of innate modularity, since language is ‘intact’ while visuo-spatial cognition is ‘impaired’ (Bellugi, Wang, & Jernigan, 1994). In a recent study, O'Hearn and Landau (2007) apply this logic to the study of mathematical skills in WS and hypothesize that:“ One possibility is that some but not all components of mathematical reasoning are impaired in individuals with WS. This possibility reflects evidence that different components of mathematics are functionally distinct in adults” (p.239). Thus, these authors motivate their hypotheses concerning the performance of individuals with WS on the basis of adult neuropsychological studies. Specifically, against the background of the Triple Code model and subsequent neuroimaging studies, O'Hearn and Landau suggest that individuals with WS may present with relatively impaired non-verbal magnitude abilities that are thought to be dependent on visuo-spatial cognition, while having, at the same time, strong verbal mathematical skills. On the basis of the analysis of items on a standardized test of mathematics (Test of Early Mathematics 2; TEMA-2), the authors conclude that, while children with WS perform, on average, at the level of a mental age matched control group on the TEMA-2, participants with WS have particular difficulties with measures of numerical magnitude understanding, such as judging which of two numbers is closer to a reference digit. At the same time participants with WS were found to perform at a higher level than their MA-matched controls on tasks that required the reading of 3-digit Arabic numerals. In view of these data, O'Hearn and Landau argue that children with WS have a specific difficulty with tasks that tap the ‘mental number line’ and that such difficulties may be related to parietal impairments that have been observed in WS

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(Meyer-Lindenberg, Mervis, & Berman, 2006; Reiss et al., 2000). At the same time, in view of the relatively strong performance on tests of number reading, the authors argue that verbal skills are relatively good in children with WS. Such data then are interpreted as providing evidence for a parietally-mediated impairment of the ‘mental number line’ in combination with intact verbal number skills. Another developmental disorder of genetic origins where deficits in numerical and mathematical skills have been observed is velocardio-facial syndrome (VCSF). This genetic developmental disorder is caused by a microdeletion on chromosome 22q11. In a recent study of VCSF, De Smedt et al. (2007) used Dehaene et al.'s models based on data from adult neuropsychological patients and functional neuroimaging studies to interpret the performance of children with and without VCSF. Against the background of these models De Smedt et al. predicted that different subtypes of MD should be discernable according to differential impairment of the systems revealed in adult neuropsychological and neuroimaging studies. In their study, the authors found that children with VCSF performed at the level of their mental age matched controls on tests of number reading and the retrieval of arithmetic facts, but were impaired on tests that required the semantic manipulation of numerical quantities, such as number comparison. De Smedt et al. interpret this pattern of data as being consistent with the presence in VCSF of: “… a preserved verbal system but an impaired quantity system of number processing” (p. 893). Furthermore, in view of the strong association between the IPS and the representation and processing of numerical quantities, the authors predict that children with VCSF should present with functional abnormalities of the IPS during number processing. 4. Do the assumptions of adult neuropsychology hold for developmental disorders of numerical and mathematical cognition? The above review of studies and theoretical positions on atypical number development represents only a cross-section of published studies on mathematical disabilities and disorders that have relied on neuropsychological and adult neuroimaging studies to explain their data. What all of these studies have in common, to varying degrees are three assumptions: 1. Profiles of strength and weaknesses do not change over developmental time, 2. Similar neuronal structures are activated in children and adults, and 3. Similarities in behavioral performance imply equivalence in underlying neurocognitive mechanisms. In what follows below each of these assumptions is critically reviewed before alternative models and methodological approaches are discussed. 4.1. Profiles of strength and weaknesses do not change over developmental time The notion that adult data can be used to interpret findings from developmental populations presupposes that the relative pattern of strengths and weaknesses in a group of children with disorders of numerical and mathematical processing should be stable across developmental time. This notion of developmental stability is found, for example, in Dehaene et al.'s (2003) hypothesis that brain circuits subserving numerical and mathematical cognition are prespecified or in the notion of a ‘number module’ put forward by Butterworth (1999). Only if children and adults with developmental disorders of numerical cognition have similar profiles of strengths and weaknesses can the same theoretical frameworks be applied to understand the cognitive and neural architecture underlying their deficits. If, however, the starting point of development differs from the endstate then an atypical developmental trajectory must be assumed and it is a developmental process that leads to adult performance profiles, rather than the unfolding of a prespecified pattern of strengths and difficulties.

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An explicit evaluation of the view point that early profiles of strength and weaknesses should resemble those found among adults was provided by Paterson, Brown, Gsodl, Johnson, and KarmiloffSmith (1999). It is well known that adults with both Williams and Down Syndrome (DS) have difficulties with numerical and mathematical problems (Ansari, Donlan, & Karmiloff-Smith, 2007; Paterson, Girelli, Butterworth, & Karmiloff-Smith, 2006; Udwin, Davies, & Howlin, 1996). The application of neuropsychological models to these developmental disorders would therefore predict that infants with these two disorders should have difficulties in processing numerical magnitudes. In order to test this, Paterson et al. investigated basic numerosity discrimination abilities in infants with both WS and DS as well as in a control group of typically developing infants matched for chronological (CA) and mental age (MA). Infants were familiarized with a particular number of stimuli and then looking time was measured during the presentation of both familiar (same numerosity) or novel (different numerosity) and looking time differences between these conditions were calculated to establish whether infants were sensitive to the numerical changes. The data revealed that while infants with WS as well as their MA and CA matched controls looked significantly longer at the novel compared to the familiar numerosity, infants with DS did not. If the maturational assumption that cognitive modules simply unfold over developmental time were correct than both infants with WS and DS should have exhibited an impaired performance in their number discrimination abilities. This evidence suggests that for WS early sensitivity to numerosity is not a predictor of the breakdown of numerical and mathematical competencies observed in adulthood, while for DS it might be. The data reported by Paterson et al. (1999) may imply that while infants with DS appear to lack foundational skills (the ability to discriminate between visually presented numerosities), individuals with WS do not. These findings suggest that the causes of numerical difficulties and the developmental time point at which they can be observed differ between developmental disorders. This also demonstrates that children with mathematical disorders need to be tested either cross-sectionally at different time points of their developmental trajectory or, ideally, be followed longitudinally to understand the origins and ontogenetic causal chain resulting in the kinds of numerical and mathematical difficulties observed in later child and adulthood. Furthermore, these data justify calls for greater attention to be paid to cross-syndrome differences in the origins of mathematical disorders. It should be noted that even when children present with similar behavioral profiles as those reported in adult neuropsychological patients, this may not imply that these patterns are stable over developmental time and may simply represent a transitory pattern of performance at a particular point in development. 4.2. Similar neuronal structures are activated in children and adults The application of adult neuropsychological models to individuals with developmental deficits carries the assumption that a similar neuronal architecture supports numerical and mathematical processing in children and adults. This notion is implicit, for example, in the interpretation that impaired magnitude processing in individuals with WS implies impaired functioning of the parietal cortex or that: “As in adult acalculia, at least two subtypes of developmental dyscalculia should be traceable to differential impairment of quantity vs. language processing circuits (Dehaene et al., 2003; p.500)”. Only if a similar neuronal architecture is present in children and adults can models generated from adult cases of brain damage be extrapolated to explain the patterns of performance present in individuals with difficulties of a developmental origin. It is well known from both adult neuropsychological patients and functional neuroimaging that the intraparietal sulcus (IPS) plays a crucial role in the processing and representation of numerical magnitude, while

regions of the left temporo-parietal cortex such as the angular and supramarginal gyri have been consistently associated with mental arithmetic (Ansari, 2008; Dehaene, Molko, Cohen, & Wilson, 2004; Dehaene et al., 2003). Recent developmental neuroimaging data have revealed that activation of both these neuronal circuits is the outcome of a gradual process of developmental specialization. Specifically, in studies of numerical magnitude processing, adults have been found to activate regions of the IPS to a greater extent than children (Ansari & Dhital, 2006; Ansari, Garcia, Lucas, Hamon, & Dhital, 2005; Kaufmann et al., 2006; Kucian, von Aster, Loenneker, Dietrich, & Martin, 2008). Similarly, Rivera, Reiss, Eckert, and Menon (2005), using a cross-sectional design with children and adolescents between the ages of 8 and 19, revealed an agerelated increase in the recruitment of the left temporo-parietal regions during mental arithmetic tasks. Thus these studies clearly show that activation in the regions thought to subserve numerical processing and mental arithmetic in adult participants increase in their relative engagement as a function of developmental time. The observed age-related increase in parietal brain activation is not entirely inconsistent with a maturational model of development as such a model predicts the unfolding of prespecified processing within one region. However, developmental neuroimaging studies of both numerical magnitude processing and mental arithmetic revealed that in younger participants a network of other brain regions is activated to a greater extent than in older children. In both numerical magnitude processing tasks (such as number comparison) and mental arithmetic (addition and subtraction), children exhibited greater levels of activation in prefrontal regions then did the older children and adults. Thus it is not simply the case that brain regions are recruited by children to a lesser degree, than they are by adults, but instead that the network of activated regions differs between children and adults. The data from functional neuroimaging studies reviewed above suggest that the developmental increases in the brain circuits that are known to subserve numerical and mathematical processing in adults are correlated with the activation decreases in other brain regions, such as the prefrontal cortex as well as subcortical regions (Rivera et al., 2005). These regions appear to scaffold the ontogenetic process of specialization. Even when adults and children have been found to both exhibit similar levels of parietal activation during numerical processing tasks, hemispheric differences were found. Specifically, Cantlon, Brannon, Carter, and Pelphrey (2006) found that while adults activated both left and right intraparietal regions during a numerosity processing task, the activity in children was restricted to the right hemisphere. Thus development of numerical and mathematical competencies involves multiple brain regions and age-related differences in lateralization of activation patterns. It is therefore possible that mathematical difficulties and disorders of developmental origin result from dysfunction of brain regions other than those associated with numerical and mathematical processing in adulthood and that such impairments prevent the typical specialization of the neural regions that come to subserve typical adult numerical and mathematical processing. Put differently, in the case of brain-damaged patients, difficulties result from damage to brain circuits that have developed following a typical developmental trajectory and the support of other, intact, brain regions. In atypical development, however, the same brain region may function atypically, not because of focal lesions, but because their developmental trajectory was disrupted due to the atypical functioning of other brain regions that might have played a crucial role in their functional specialization. Indeed, while there is only sparse neuroimaging research on developmental disorders of numerical and mathematical processing, there is already some evidence to suggest that diffuse brain networks are structurally and functionally atypical in these children. In the first functional neuroimaging study of Developmental Dyscalculia (DD; a specific deficit in calculation in the presence of otherwise normal intellectual functioning), Kucian et al. (2006) found that children with

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DD showed atypical activation during approximate calculation in a wide area of cortical regions, including the prefrontal and inferior frontal cortex as well as the intraparietal sulcus. In a structural imaging study of individual with DD, Rotzer et al. (2007) found that children with DD exhibited (relative to typically developing controls) lower gray matter volumes in the right parietal cortex, but also in regions of the frontal cortex and the parahippocampal gyrus. Moreover, while Price, Holloway, Räsänen, Vesterinen, and Ansari (2007) found atypical activation of the right parietal cortex in children with DD in a functional neuroimaging study of non symbolic (arrays of squares) magnitude processing, disordered profiles of activation were also found in the right occipitotemporal cortex as well as in regions of the medial frontal cortex. These functional and structural data of children with specific difficulties in numerical and mathematical processing indicate that deficits cannot be localized to single brain regions, but instead suggest abnormalities in multiple neural circuits. A limitation of existing functional neuroimaging studies of disorders of numerical cognition is that children are tested within a certain age range and no comparisons of groups of children with mathematical disorders at different time points and/or longitudinal studies have been performed. It is thus unclear whether observed abnormalities are the product of atypical maturation of prespecified regions or whether atypical structure and function of these regions is the outcome of an atypical developmental trajectory. Even when a single brain region thought to subserve processing of a particular cognitive domain in adulthood shows increases in activation over developmental time or is shown to function atypically in children with difficulties in that cognitive domain, this may not necessarily be consistent with a maturational perspective on brain functioning. As (Johnson, 2001, 2003) has pointed out, there are at least three ways of conceptualizing developmental changes in brain function. One of them is the maturational perspective where a particular brain region's increasing structural and functional development enables a particular form of cognition and behavior to emerge. This perspective assumes that the functioning of cortical regions is innately prespecified and can thus be selectively damaged. Another perspective is what Johnson (2001, 2003) refers to as the skill learning perspective where a given area comes to subserve a particular cognitive function as a consequence of increasing practice and the acquisition of new skills through processes of development and enculturation. The third possibility for functional brain development put forward by Johnson is called the interactive specialization view. According to this model, increasing engagement of a neural region in a cognitive process is the product of interactions between brain regions. According to this view, age-related increases in regions known to subserve cognitive functions in adulthood are the product of initial biases, interaction and competition between cortical and subcortical regions. The greater engagement of frontal regions in numerical magnitude processing and mental arithmetic tasks in children compared to the activation patterns observed in older children and adults may suggest that these brain regions interact with parietal brain regions in the process of age-related specialization. According to this perspective then, the causes of disorders of cognition, such as numerical or mathematical difficulties, may not be reduced to the dysfunction of innately specified cortical modules of cognition that can be selective damaged, but instead need to be traced down to atypical interaction and competition between brain circuits that are involved in the developmental construction of cognitive competencies. Taken together, models other than those assuming the maturation of genetically prespecified brain circuits needs to be considered and empirically tested. This will require a developmental focus, where children with developmental difficulties and their typically developing peers are examined at various stages of development. Furthermore, such research will need to investigate not only differences in

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brain function and structure between age groups, but also how the interaction between brain regions changes over developmental time. One such approach is to look at age-related functional connectivity. In the study of reading development, for example, such approaches are starting to reveal dramatic effects of development on the functional interconnectivity of brain regions that subserve reading and the evidence also suggests different patterns of connectivity between children with and without reading difficulties (Bitan et al., 2007). Recent data looking at task-independent functional connectivity has revealed dramatic age-related changes in the functional correlations between brain regions (Fair et al., 2008). Such analysis could be applied to the study of the atypical development of numerical and mathematical processing. 4.3. Similarities in behavioral performance imply equivalence in underlying neurocognitive mechanisms A third assumption implicit in the application of adult neuropsychological and neuroimaging data to developmental disorders of cognition is that if similar behavioral profiles of performance are established in atypically developing children and brain-damaged patients, then there should be equivalent causes. This assumption is, for example, inherent in the studies reviewed above by Temple (1992); Geary (1993); O'Hearn and Landau (2007) and De Smedt et al. (2007). In other words, behavioral equivalence is taken to suggest a similar nature of disordered processes. This notion has been challenged by both behavioral and neuroimaging studies. While similar behavior may be revealed between two groups of participants, individual differences with the two groups may be explained by different factors. An example of this is a study comparing the counting abilities of children with WS with a group of mental age matched controls (Ansari et al., 2003). While the level of understanding of the meaning of counting among children with WS was at the level of their mental age matched controls, different factors correlated with individual differences in measures of the understanding of the meaning of counting in the two groups. Specifically, while individual differences in verbal but not visuo-spatial abilities predicted variability in the WS children's understanding of the meaning of counting, the opposite was true for the group of typically developing children where visuo-spatial but not verbal competencies predicted individual differences in the measure of children's understanding of the meaning of counting. Data like these illustrate that equivalent behavioral performance may not mean equivalent cognitive processes and, by extension, the involvement of similar brain regions. Instead, different mechanisms may give rise to similar behavioral profiles. In view of this, researchers should be going beyond comparing measures of performance between groups towards a deeper understanding of subtle differences in the factors that account for individual differences in performance and differences in strategies that might superficially produce comparable behavioral profiles. In addition to using multiple behavioral measures, assessing strategy variability (to evaluate, for example, compensatory strategies) and predictors of behavioral outcomes, neuroimaging can also provide a useful additional analysis to assess the degree to which equivalent behavioral performance between groups implies similar representation and processing. Several neuroimaging studies have revealed differences in brain activation in the presence of equivalent behavioral performance profiles between groups. For example, in a study of illusory contour perception (Kanizsa Figures), Grice et al. (2001) found that while both individuals with WS and a group of typically developing participants matched on chronological age could perceive the illusory controls equally well, the underlying eventrelated brain potentials associated with the presentation of the visual stimuli differed dramatically between the groups. Moreover, studies of typical development have shown that age-related changes in

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functional neuroanatomy can be found even when behavioral performance does not differ between different age groups (Schlaggar et al., 2002; Turkeltaub, Gareau, Flowers, Zeffiro, & Eden, 2003). Conversely, there is research suggesting that task related brain responses do not differ significantly across age groups while, at the same time, dramatic differences in behavioral performance between the groups of children of different ages are observed (Coch, Grossi, Coffey-Corina, Holcomb, & Neville, 2002). Taken together, these findings illustrate that a.) behavioral equivalence between age group can exist in the presence of age-related neuronal changes and b.) behavioral differences between groups can be detected in the absence of age-related changes in the neural correlates. These and other studies illustrate that equivalent behavioral performance between groups does not mean similar neuronal function or dysfunction. On the other hand, similar neuronal responses do not necessarily mean that equivalent behavioral performance is observed. In view of this, the extrapolation from similarities in behavioral profiles of children with mathematical difficulties and neuropsychological patients to similar underlying neural mechanisms needs to be examined with great caution. Furthermore, the inference of a similarity in processing is often reliant on a null result, such as a non-significant difference between a group of children with mathematical disorders and a control sample of typically developing children. Such null results may imply that the groups function at comparable levels, but may equally well indicate that the measurement used is not sensitive enough to detect differences in processing between the two groups. 5. Conclusions and future directions The study of adult neuropsychological patients and, more recently, the use of functional neuroimaging tools have provided unprecedented insights into the brain regions and processes that underlie numerical and mathematical abilities. Therefore, in concluding, it should also be clearly stated here that while the present review is critical of the use of adult neuropsychological and functional neuroimaging data to interpret developmental disorders, it is clear that such data can play a very useful role in informing the study of development. Specifically these models inform developmental researchers about the organization of mature cognitive processes, such as numerical and mathematical cognition, which can guide predictions concerning the developmental processes that may lead to adult-levels of cognitive and brain processing. However, it is these developmental processes that need to be better understood. As reviewed above, these compelling models have frequently been applied to provide explanatory frameworks for developmental difficulties and disorders of numerical and mathematical processing. The above discussion shows that many of the predictions derived from the study and modeling of data from adult neuropsychological patients and functional neuroimaging studies cannot be directly applied to development. Profiles of strength and weaknesses result from atypical developmental trajectories, which involve interactions between cognitive and neuronal processes, leading to an atypically organized brain. This process may be fundamentally different from those underlying cognitive deficits among brain-damaged patients where brain regions that developed along a normal developmental trajectory and their associated cognitive processes can be selectively damaged. Furthermore, even when similar behavioral profiles are found, this may not indicate equivalent cognitive and neuronal processes and could be reduced to lack of measurement sensitivity. In light of these considerations, the application of adult neuropsychological models and functional neuroimaging data may lead to false conclusions about the causes of these disorders. It is important to note that many of the limitations of applying adult neuropsychological models to the interpretation of developmental disorders outlined above are not necessarily specific to the

domain of numerical and mathematical cognition but are applicable to a wide variety of domains. Indeed, there are a number of theoretical papers that present critiques of the extrapolation of adult lesion studies to the study of development in domains other than the one discussed here (Goswami, 2003; Karmiloff-Smith, 1998). Since there are considerably more data on the neurocognitive basis of numerical and mathematical cognition from adults than children, how should researchers go about constraining their interpretation of the performance of children with mathematical difficulties and disorders without directly applying models of adult neurocognitive functioning to developmental cases? A starting point is to adopt a developmental perspective. This means going beyond the study of mathematically challenged children at one age, towards either a crosssectional analysis of children with the same difficulties at different ages or, ideally, the design of longitudinal studies. Studying a single age group does not equate to taking a development perspective. Adopting such an approach will enable researchers to go beyond the comparison of typically and atypically developing groups at one age to a study of differences in developmental trajectories (for examples, see: Ansari et al., 2007; Karmiloff-Smith et al., 2004; Thomas et al., 2009). Furthermore, more study of the brain circuits underlying task performance in atypically developing children should be sought. Most of the above-reviewed studies that adopt an adult neuropsychological perspective of developmental disorders on numerical and mathematical processing do so without reference to any neurophysiological data from these children. As discussed above, drawing inferences about neuronal processes on the basis of behavioral performance is, at best, speculative. Such neuroimaging studies should also seek to sample atypically developing children at different ages, or attempt longitudinal designs. Furthermore, models of functional brain development other than the maturational perspective, such as Johnson's (2001, 2003) ‘interactive specialization model’ should be considered. This will require that neuroimaging studies need go beyond localizing task-specific regions to an analysis of the interactions between regions and how these change differently, over developmental time, in typically and atypically developing children. Furthermore, children should be assessed using multiple measures that tap the same construct. Such an approach will increase measurement sensitivity and may reveal subtle deficits in aspects of processing that may be interpreted as looking normal, if only a single measurement tool is used. Finally, domains of competence other than numerical and mathematical processing should be measured at different developmental time points. It is possible that the interaction between numerical and mathematical cognition and processes related to other domains such as reading, language and visuo-spatial competence is atypical in children who develop impairments of their ability to deal with numbers and mathematical problems (Simon, 2008). Such an approach will also move the study of atypical development away from the current focus on specific disorders towards a greater understanding of the frequent co-occurrence or comorbidity of mathematical disorders with disorders of reading, attention and spatial processing.

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