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Saul R. Korey Department of Neurology, Albert Einstein College of Medicine, Bronx, New YorkDepartment of Pediatrics, Albert Einstein College of Medicine, Bronx, New YorkRose F. Kennedy Center for Research on Intellectual and Developmental Disabilities, Albert Einstein College of Medicine, Bronx, New York
Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to “many” with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.
The goal of this review is to update child neurologists and other professionals concerned with developmental disorders about the brain basis of much neglected calculation and dyscalculia. Parents of children with isolated academic problems without physical abnormality, epilepsy, overt language or cognitive deficiency, or unacceptable behaviors are likely to have heard or read of spectacular advances in genetics and neuroscience. They may consult child neurologists or other physicians seeking an up-to-date explanation for their child's unexpected developmental problem, asking whether there is a test or medication that might help. Most physicians see few school-age children with “pure” dyscalculia because it is considered the responsibility of educators and school psychologists. Neurologists need to be able to convey to parents the important biologic implications of advances in neuroscience research, stressing that they are not clinical diagnostic tools or likely to help in the management of the individual child with dyscalculia or other pure developmental disorders. Although neurologists will likely be asked about genetic contributions to dyscalculia,
up-to-date knowledge of the genetics of brain development exceeds the competence of most practitioners and the scope of this review.
Definition and prevalence
Dyscalculia applies to any otherwise competent child who fails to learn at the expected age that number names refer, one-to-one, exclusively and sequentially, to each item in any set of objects, including their own ten fingers, and, later, to Arabic symbols, and to estimates of the approximate number (numerosity) of items in the set (approximate number sense—ANS). Failure to grasp the abstract concept of number (cardinality), let alone the place principle (ordinality), means calculation eludes such children. Calculation, like reading, is a culturally derived, specifically taught, slowly learned skill. It calls for the ongoing interaction of environmental experiences with multigenetically influenced development of visual, spatial, and language proficiency, working and long-term memory, focused attention, motivation, and other intellectual and executive competences. Dysfunction of any one or more of these abilities can contribute to difficulty or failure to acquire an understanding of numbers. This means that dyscalculia, like dyslexia, is not a single disorder attributable to a single gene abnormality, unique cognitive deficiency, or maldevelopment or damage affecting one specific brain locus or pathway.
reported a 6% prevalence of dyscalculia, but only 1.8% (a third of those affected) with pure dyscalculia, compared with 4.2% in whom dyscalculia was associated with another prevalent developmental disorder like dyslexia, attention-deficit disorder,
Similarly, among 50 11- to 13-year-old Swedish children with developmental dyscalculia, 16 (32%) were considered to have a specific deficit accessing number concepts and the 34 others (68%) more general problems with calculation.
There are some standardized tests of various calculation skills, but none that is universally accepted; therefore prevalence estimates rest on school reports. The prevalence of dyscalculia approximates that of dyslexia, some 5% to 6% of elementary school children.
Yet an October 24, 2015, PubMed search (by S.R. Snodgrass) retrieved 615 papers on epidemiology of dyslexia and 15 (2.4%) on dyscalculia; there were 96 on brain networks in dyslexia and 15 on networks in dyscalculia. This disparity shows how much less research attention dyscalculia has attracted despite comparable prevalences
describe four stages in children's acquisition of number competence: (1) In neonates, toddlers, and persistently throughout life, subitizing (from subito, quick, sudden) is the innate nonverbal ability to detect at a glance a difference in number of items between two small arrays of up to four or so randomly placed items.
(2) Preschoolers learn by rote the verbal labels for counting, usually at first as an undifferentiated sequence of single digit names without referents. Only by age three to four years does the average preschooler acquire gradually, one-by-one, the name of the numeral that applies specifically to one object, then to two, then three, then four, and so on, progressively developing a more abstract sense of exact number and learning that each name symbolizes a specific number, irrespective of the size or appearance of its referent.
The child needs time to understand the uniqueness (cardinality) of each number, that they occur always in the same counting order (ordinality), and, eventually, that the last digit name provides the exact total number of items in any serially counted set. By kindergarten most children will count reliably five to ten randomly spaced objects without repetition or skipping, indexing their gradual grasp of both cardinality and ordinality of numbers. (3) By first grade they will have learned to link number names to the visually coded Arabic digit symbols and to understand that adding one or subtracting one from a set yields the next bigger or smaller number of items in the set. They become progressively able to give fast memorized verbal answers to single-digit addition and subtraction of numbers up to five, and will have discovered the utility of their fingers for calculation.
(4) Only at this stage will many elementary school children find out that you can count items serially across several sets to yield the total of their union.
Children will typically require two or three more years to memorize rote answers to addition and subtraction of larger single digits and to visualize the spatial representation of consecutively larger numbers on the linear number line as scaffold for ordinality and mastery of the decimal system. Mental arithmetic with small numbers is subject to much practice in the first grades of school, although probably somewhat less today thanks to ubiquitous cell phones and other calculators. Automatic answers to overlearned single digit calculations and, after several years, to multiplication tables up to five, by tens and, mostly less securely, to the six to nine times tables, generally become coded verbally for long-term storage, thus bypassing the need for repeated mental computation.
Understanding dissociations in dyscalculia: a brain imaging study of the impact of number size on the cerebral networks for exact and approximate calculation.
In contrast to single-digit addition and multiplication, the reverse operations, subtraction and division, are less likely to be automatized and continue to call for more cognitive resources and more widespread brain circuitry
Figure 1Comparison of functional magnetic resonance imaging brain activations and deactivations between the four arithmetic operations in 20 normal adults solving single one- to nine-digit problems compared with identifying digit 5 in a string of meaningless symbols and digits presented visually in a choice reaction time paradigm (correct versus incorrect).
Rosenberg-Lee et al. Neuropsychologia. 2011; 49:2592-2608).
with a precipitous drop for crowded uncountable numbers, i.e., “many.” High-strength functional magnetic resonance imaging (fMRI) in normal adults shows the clearest topographic number tuning in the right posterior superior parietal lobe, with small numbers of dots (one to seven) in displays more narrowly tuned and allocated more cortical space than larger ones whose tunings are progressively broader and overlapping.
Thus cortical representation of the nonverbal visual ANS is analogous to that of other sensory estimates like sound pitch, weight, temperature, and pain,
which all share the characteristic of logarithmically decreasing precision with increasing magnitude. Thus unlike the uniformly spaced numbers on the linear ordinal number line, spaces between consecutive numbers in our brain and mental number line narrow as number grows (distance effect). The combined magnitude and distance effects mean that ANS discrimination is slower and less accurate for seven versus nine, than two versus four (magnitude effect), and for eight versus ten than eight versus twelve (distance effect). It is the ratio between the numbers in the sets that determines ANS discriminability, with the requirement of a ratio of at least 2:1 between all but very small sets.
How the exact linear number system requisite for calculation evolves from nonsymbolic subitizing and the ANS is not fully understood, but at least in early childhood, they are postulated to share ventral occipitotemporoparietal and lateral prefrontal number networks.
Magnetoencephalography mapping in ten competent adults whose task was fast number identification in sets of two to eight random dots revealed that subitized arrays of two to four dots activated an early bilateral occipital response followed by a bilateral 250-400 ms temporoparietal response insensitive to the number of dots.
Larger sets requiring counting induced a delayed bilateral potential whose latency increased linearly by 300 ms per additional dot over five to a mean of 1690 ms for eight dots; this late potential spread over parietal and frontal areas, suggesting that working memory, attention, and verbal number processing circuitry had come on line. A near-infrared spectroscopy study of blood flow in 11 university students naming 200 ms dot or corresponding Arabic numeral displays revealed that accuracy dropped from 100% for two to four dots to 40% for six dots, supporting dissociation of subitizing and ANS.
trained two macaques to choose, after a 1-s delay, which of two random dot displays matched the number of dots in the previous display. The choice activated, after 100 ms, “primary number neurons” deep in the posterior IPS then, after some 60 ms, lateral prefrontal number neurons whose sustained activation during the reaction time interval indexed prefrontal attentional and working memory networks in preparation for programming the appropriate pointing command in the motor cortex.
In the IPS, small clusters of “primary number neurons” were aligned in order of their tuning thresholds, with narrower more specific tuning to small than larger numbers (number effect), progressively losing sharpness toward the blurred “many” estimate.
Tuning overlap thus increased as distance between numbers decreased (distance effect), mirroring frequency tuning of neurons in the primary auditory cortex.
In monkeys the visual “number neurons” of the IPS and adjacent parietal areas are interleaved with others activated by other sensory estimate tasks like length, surface, and duration.
The convergence in the IPS of exact number and magnitude estimate networks may explain some of the disparities in localization and lateralization among functional studies comparing counting with ANS. Whether the root basis of dyscalculia is inadequacy of the visual ANS or the exact number system circuitries remains debated.
The cellular studies in monkeys suggest this dichotomy is overly simplistic.
Brain networks supporting number skills in man
It is Gerstmann's 1924 influential description of an adult with a left parietal stroke responsible for dyscalculia plus right-left confusion, finger agnosia, and dyslexia that pinpointed the parietal lobe as the cortical nexus for calculation. Kleinschmidt and Rusconi
provides a simplified schematic of major reciprocal tracts of the left and right posterior parietal magnitude pathways centered on the IPS and adjacent gyri (Fig 2). Both hemispheres receive continuous preprocessed visual information punctuated by intermittent other inputs, via direct superior occipitoparietal projections—not shown in Dehaene's diagram—and via asymmetric indirect inferior projections from the ventral occipitotemporal sulcus and adjacent fusiform gyrus. Also not shown in Dehaene's diagram is the superior longitudinal fasciculus, the massive reciprocal right and left posteroanterior tracts that link parietal to prefrontal and limbic cortices. Complex behaviors like calculation require focused attention, responsible for on-line choice of optimal behavior by integrating incoming bottom-up sensory inputs with current top-down organismal priorities and memorized past experiences.
Figure 2Schematic of the major number pathways in lateral views of the left and right hemispheres, including their massive posterior callosal interhemispheric connections. Not shown are the superior occipitoparietal (spatial, where) and inferior occipitoventrotemporal (identification, what) afferents. For clarity, Dehaene emphasizes frontal parts of the verbal system and omits strong links between the visual number form area and posterior perisylvian phonemic processing cortices and their interconnections with widespread other parts of the verbal system.
The dorsal occipitoparietal—so-called where—visual stream informs the posterior superior parietal lobules, predominantly the right, about location, size, movement, and other spatial characteristics of visual stimuli. Spatial skills are reportedly predictive of later arithmetic ability.
understanding the decimal place principle, and for mastering geometry and other mathematical skills not considered here. The ventral temporo-occipital—so-called what—visual stream is largely indirect on the left, linking IPS and angular gyrus to widespread language processing cortices, including speech sound retrieval (posterior superior temporal and supramarginal gyri—Wernicke area,
semantic processing in more anterior temporal cortices, and, through the arcuate fasciculus, speech output programming in inferior frontal—Broca area).
Gerstmann syndrome's right-left confusion reflects both impaired attention and loss of somatosensory information from the damaged left parietal lobe. Deficient connection of the anterior IPS to the abutting hand and finger representations of somatosensory cortex explains finger agnosia and may contribute to the dyscalculia in view of the role of the fingers in contemporaneous learning to count and write, not to mention to the invention of the decimal system.
report that in Indo-European languages written from left to right, counting tends to start on the left hand and universally proceeds from left to right on our mental number line. He describes the opposite of Arabic in which both writing and counting proceed from right to left, whereas directionality tends to be random in native speakers of Hebrew who read and write from right to left but calculate from left to right.
The dyslexia of Gerstman syndrome has to do with the ventral visual stream, implicated in categorization of familiar classes of objects, faces, and colors. On the left written symbols like letters, Arabic numbers, and, in specifically educated individuals, printed words, mathematical or chemical formulas, musical scores, and such, which working memory retrieves on-line from long-term stores, can be transmitted to the supramarginal and angular gyri directly, thus efficiently bypassing the need for further processing. Dehaene et al.
dub the left ventral occipitotemporal cortex the “visual word form area” or “letter box” in view of its critical role in fluent reading. He hypothesizes that the abstract concept of exact number and the invention of arithmetic from subitizing and the ANS had to await the evolutionarily recent development of written language, which only goes back some 8000 years but rapidly spread to all but isolated illiterate societies.
Bilingual persons tend to calculate faster lifelong in their native language than in their second—perhaps now better—language. This testifies to the bonding of numbers to language and also to the drill used to inculcate elementary arithmetic facts.
Fast answers to easy problems like adding single digits and multiplying numbers two to four activated not only the IPS of both hemispheres but also the left angular gyrus, recipient of overlearned verbal material, while deactivating the right. As the subjects had to muster additional cognitive resources and more effortful strategies for multiplying larger numbers (five to nine), for subtraction, and, especially, for division, answers tended to be slower, less accurate, and associated with left angular deactivation and activation of the right, together with widespread mostly bilateral cortical, limbic, and subcortical networks and their connecting white matter tracts. Further proof of the contribution of widespread circuitry to calculation is that inactivation of either left or right IPS with transcranial magnetic stimulation interfered with both multiplication (retrieval from memory) and subtraction (requiring mental calculation).
Further, fine-grained fMRI of the IPS, angular, and superior parietal lobule suggests some cortical specializations across the four basic arithmetic operations previously illustrated earlier in Fig 2.
In short, the hub of the number-processing network consists of the callosally interconnected IPS sulci and adjacent posterior parietal gyri of both hemispheres, on which somatosensory, visual, auditory, and language inputs, as well as attention and other executive commands, converge. Particular number skills activate partially dissociable lateralized circuitry, depending in part on the degree to which they engage language and spatial abilities, with lateralization of number skills less strong than that of language. The interacting frontoparietal circuitry that supports exact number skills and ANS is massively and reciprocally connected via the superior longitudinal fasciculus with prefrontal, motor, limbic, and mesotemporal cortices and with multiple subcortical areas.
Development of pathways supporting calculation in children
Whether a rare developmental Gerstmann syndrome occurs in childhood
is questionable because, by adolescence, the footprint of congenital or early left or right focal hemispheric damage was not dyscalculia but at most marginal deficit on standardized tests of mathematical ability.
Focal lesions in adults typically compromise a subroutine of complex skills like language, causing varied subtypes of aphasia exceptional in childhood. Children with very early unilateral lesions may appropriate alternate, uncommitted circuitry to develop skills crucial for survival like walking
Privileged age-related resiliency of language and calculation development in the face of early focal lesions suggests they too have high evolutionarily value. Their sparing following focal lesions in an otherwise normal immature brain contrasts starkly with genetically influenced enduring developmental disorders, which usually improve to a variable degree but with persistent very specific underlying skill deficits.
In young children, calculating increases not only parietal oxygen consumption but also in cortical and subcortical circuitry supporting attention, working and long-term explicit and implicit memory, and other cognitive resources. There are exceptions to decreasing extraparietal activation with increasing age and arithmetic competence: skilled professional abacus users and calculation wizards who devote much of their time to figuring out answers to complex mental calculations recruit brain areas ordinary adult calculators no longer activate.
and other experts who exemplify the environmental effects of learning on brain plasticity.
To summarize, imaging studies indicate number processing activates interconnected number networks centered in both right and left parietal lobes, in particular the IPS, with the left angular gyrus somewhat more involved in language-dependent manipulation of exact numbers and the right in continuous magnitude estimation, including ANS, although this remains disputed.
The relevance of brain imaging and electrophysiologic research studies to the otherwise healthy individual child with dyscalculia is tenuous because, at present, neither provides specific diagnostically helpful evidence regarding its cause. Technical progress in both fields is dazzling, and there is no dispute regarding their value for research, although their anatomic and functional brain maps remain coarse and not fully overlapping. Understanding the brain basis of dyscalculia is a young field in which, typically, small numbers of subjects studied weaken the power of conclusions, diversity of tasks confounds results, and thus far mostly cross-sectional design limits full appreciation of maturational changes.
Among tentative results, MRI morphometry disclosed in 23 seven- to nine-year old dyscalculic children smaller volumes of many gray matter areas of the bihemispheric number network than in well-matched competent controls, and fractional anisotropy revealed especially marked axonal disorganization of posterior callosal interparietal connecting fibers.
In a reaction time and visual event-related potential (ERP) study, seven adolescent girls with severe pure dyscalculia and matched unaffected controls had to report whether a flashed Arabic digit was smaller or larger than five.
Presence in both groups of the expected distance effect—discrimination easier the more distant the target from five—and normal early automatic ERPs suggested that deficient visual perception of digits or their positions on the number line did not explain the dyscalculia, whereas absence of a right parietal 400-450 poststimulus ERP in the dyscalculic group pointed to inadequate complex processing. Notably, the subjects, who had no other deficit in spatial skills, had severe difficulty identifying fingers in rotated pictures of hands. fMRIs in 18 9- to 12-year-old children with pure dyscalculia were compared with those of 20 matched controls on each of three tasks: choosing which of two answers was the exact or the approximate sum of two previously flashed single digits, or which of two displays of nine or fewer randomly dispersed small drawings of vegetables or fruits was the larger.
fMRIs of the two groups did not differ when answering exact additions or estimations of number magnitude problems, but did in judging approximate answers to additions (Fig 3). Although the groups activated the same brain networks, the dyscalculic children's allocation for approximate assessment of number magnitude was weaker, more variable, and less widespread. Lack of fMRI differences from controls in retrieval of exact calculation facts and nonsymbolic magnitude estimation suggested neither was critical to dyscalculia, leading the investigators to postulate a specific deficit in approximate number appraisal.
Figure 3Comparison of functional magnetic resonance imaging brain activations by three quantitative visual choice reaction time tasks in 18 11-year-old school agers with dyscalculia compared with 20 matched unaffected children. Top boxes: closest number estimate to the correct exact sum of two displayed single digits; middle boxes: correct exact sum; bottom boxes: which of the two displays had the larger number of objects (number magnitude). Box C displays the same data as Box A but with weaker significance criteria. Kucian et al. Behavioral and Brain Functions. 2006;2:31.
Reproduced from open access paper. DD, developmental dyscalculia; FWE, family-wise error correction; FDR, false discovery rate correction: statistical tools used to evaluate the significance of each task's selective fMRI activation of brain regions of interest.
carried out four meta-analyses of 19 suitable fMRI studies relevant to numerical skill. The number of subjects per study ranged from eight to 27; only five studies included dyscalculic children. Meta-analysis of eight studies indicated that in unaffected children Arabic digit magnitude comparison activates parietal, limbic, and frontal areas mostly bilaterally, whereas ANS comparison of nonsymbolic items like dot displays favors right activation. Another analysis showed that competent children activate more frontal and broader networks than adults during nonsymbolic ANS comparisons and that simple calculation is associated with stronger biparietal and also much more widespread activations, including of basal ganglia and cerebellum, taken as evidence of their less mature number skills. Finally, dyscalculic children activate less left IPS and inferior parietal language cortex and more frontoparietal and widespread areas than unaffected children, supporting mobilization of compensatory strategies to mitigate numerical incompetence. A recent review
of the contentious question of whether dyscalculia is the result of deficient exact numeracy or ANS makes clear that the issue is by no means settled; it is counterintuitive that a skill as complex as calculation would have a single neurological pathogenesis, an argument that applies equally to other developmental disorders like language.
Intervention-induced alterations in the calculating brain
As virtually all children in whom brain imaging studies were performed had been exposed to teaching of arithmetic in school, long-term changes in the brain correlates of arithmetic operations are inevitably a combined function of genetically orchestrated brain maturation and environmentally diverse experience and training. For example, accuracy and speed of responses to simple (n + 1) and complex (n + n) single-digit addition problems were faster and more accurate in 45 intelligence quotient–matched typical third graders than 45 second graders.
What difference does a year of schooling make? Maturation of brain response and connectivity between 2nd and 3rd grades during arithmetic problem solving.
At both ages the correlate of complex addition was fMRI activation of a bilaterally widespread network that included the left IPS and language network, as well as prefrontal cortices and other areas implicated in general cognitive functions, notably attention and retrieval from long-term memory. Differences in connectivity between second and third graders and between simpler and more demanding tasks illustrate functional differences related to competence and, presumably, to acquired dedicated cognitive strategies linked to both maturation and learning.
recorded specific electroencephalography (EEG) changes attributable to skill acquisition by comparing trained with untrained tasks. Twenty normal adults were trained by computer on two consecutive days to respond as fast as possible to ten difficult multiplications and 10 equally difficult tridimensional visual-spatial tasks. On day three the participants were presented with the ten trained and ten novel calculation and visual tasks. They responded faster and more accurately to trained than untrained stimuli of both types, and improvement was associated with increased parietal and parieto-occipital theta (3-6 Hz) EEG oscillations, interpreted as evidence for localized practice-enhanced learning of both numerical and spatial skills and for the relevance of theta EEG rhythms to fact retrieval from long-term memory stores.
In normally competent children and adults during or immediately following acquisition of a variety of arithmetic skills, increased expertise was correlated with decreased frontal and increased parietal fMRI activation.
A study of 17 typical college students showed that 30 minutes of training in mental multiplication and subtraction problems on each of five consecutive days resulted in bilaterally decreased activation in frontoparietal areas and caudate nucleus and increased activation in temporoparietal areas, findings attributed to a switch from calculation to retrieval of learned answers from cortical memory stores.
Trained multiplication—but not subtraction—increased left angular gyrus activation, supporting the role of language in learning the times tables. Using this paradigm to study changes in brain activation during training, these investigators
compared increasingly long trains of repeated with novel multiplication problems. There were detectable differences in activation between novel and repeated problems within 15 minutes of the start of training; over time, repeated problems progressively decreased activation in frontoparietal areas and increased activation in the left angular area. Interpretation was again that subjects were moving from calculation requiring working memory and other executive skills to more automatic fast retrieval of learned results from neocortical verbal semantic memory. Angular gyrus activation was correlated with problems for which 28 healthy male students stated they knew rather than calculated their answers.
That more competent calculators activate the left angular gyrus more than do less competent individuals corroborates progressive reliance on retrieval of learned responses rather than calculation. However, angular activation does not seem specific for calculation as it also occurred when counting the number of faces in familiar versus unfamiliar complex tridimensional objects.
recorded behavioral responses and fMRIs before and after longer training in 16 dyscalculic eight- to ten-year-old children and 20 age-matched controls. They developed a clever computer game to enhance learning of the spatial representation of numbers: children had to write on paper number lines marked only with 0 and 100 the approximate location of 20 Arabic digits, answers to 20 additions and 20 subtractions, and estimates of the number of dots in 20 sets of random dots. The problems appeared on a “space ship” at position 50 above the number line. Correct answers were rewarded. Training consisted of playing the game at home for 15 minutes five days/week for five weeks. fMRIs were recorded before and after the five training weeks, and again five weeks later with no further practice. The children enjoyed the game. The children with dyscalculia had less bilateral parietal activation than unaffected controls before training. Training decreased widespread—especially frontal, but surprisingly also parietal—activations in both groups, more so in the dyscalculic children. This effect persisted after the five weeks of no-training and was associated with better performance on the problems and on standardized math tests as well.
and others have developed other engaging computer games designed to mitigate dyscalculia. Callaway's game focuses on numerosity coding, for example, learning that an array includes a specific number of items and that adding or subtracting one or more items changes that number to a specific new number. His game and others emphasize the link between dimensionality and number. It is too soon for reliable evaluation of the long-term effectiveness of these remediations and the brain correlates of intervention in dyscalculia.
Testing and intervention for dyscalculia
Physicians' first responsibility when consulted for dyscalculia is to consider whether there is any evidence for a biologically treatable neurological disorder such as epilepsy or a genetic, metabolic, or structural brain disorder that might mandate EEG, brain imaging, or other investigations unrelated to the focus of this article. Physicians concerned with development need to discuss with the parents how far to go in probing for the biologic etiology of the problem, making clear which tests are for the child's or family's benefit and which to answer research questions. Their more pressing responsibility is to probe for the highly prevalent associated developmental problems such as attention-deficit disorder, anxiety, or depression amenable to counseling and pharmacologic interventions, and for unfavorable environmental circumstances to be addressed. Dealing with these nonspecific issues together with cognitive counseling may determine the success or failure of remedial educational interventions.
The main clinical function of neuropsychological testing is to identify specific roots of the problem by teasing out cognitive dysfunctions educators need to address or circumvent. Probing neuropsychological testing is of course essential to the interpretation of atypical brain circuitry suggested by electrophysiology, imaging, or other research tools. The major tool available to the school and clinic to evaluate the effectiveness of a particular intervention is comparison of pretraining and post-training standardized test scores, and comparison of proficiency with similarly educated peers who did or did not experience an intervention. Convincing data comparing the effectiveness of different conventional educational approaches may be in the offing,
but whether brain-based treatment approaches have a realistic future remains to be seen.
There is currently a great deal of interest in the potential benefits of noninvasive transcranial magnetic, electrical, and random noise stimulation aimed at altering intrinsic oscillations of underlying brain circuitry in individuals with developmental and psychiatric disorders.
For example, repeated transcranial direct current electrical stimulation of areas identified as relevant to particular cognitive abilities was reported to have long-term beneficial effects in children and adults with learning disabilities.
Lateralization of activity in the parietal cortex predicts the effectiveness of bilateral transcranial direct current stimulation on performance of a mental calculation task.
At this stage this is still preliminary research requiring much more work before it can be recommended ethically in the clinic, irrespective of exploding numbers of lay publications touting beneficial results.
Summary
Like reading, calculation is an acquired skill that engages complex brain circuitry and individualized strategies. It involves two major abilities, vision and, to a variable degree, language. The first is innate visual sensitivity to small (two to four) numbers—subitizing—which later blends into the ANS, the ability to detect magnitude differences in numerosity between item displays and estimate approximate number of items, with precipitously declining precision as number grows. This innate spatial ability is analogous to gauging stimulus magnitude and intensity in other sensory modalities. The second, an exact cardinal and ordinal abstract number system, enables calculation and takes years to develop. Learning the visual symbols and algorithms for number competence requires extensive exposure and specific teaching and depends on other cognitive abilities like language and executive skills. The accelerating advances in science and technology that feed advances in society's convenience, well-being, and, unfortunately, weapons depend on an educated workforce comfortable with quantitative concepts and logical reasoning. Greater attention to math competence and developmental dyscalculia is no accident.
Calculation activates domain-specific networks centered on the interconnected IPS and adjacent gyri of the left and right hemispheres. The IPS receives preprocessed visual inputs from occipital cortices, either directly (so-called where circuit) or after further preprocessing in ventral occipitotemporal and language cortices (so called what circuit). In addition the IPS and adjacent multimodal cortical areas are reciprocally connected with prefrontal cortices activated by focused attention and working memory, and hippocampal circuitry involved in learning and retrieval from long-term memory stores, as well as with limbic and multiple subcortical nodes. To what degree impaired processing of specific or nonspecific, exact or approximate, symbolic and nonsymbolic stimuli contributes to subtypes of dyscalculia remains under investigation.
The assistance of S.R. Snodgrass, MD, and J.N. Constantino, MD, who reviewed an earlier draft of this review, and the suggestions of the Editor and two anonymous reviewers are gratefully acknowledged.
Financial support: None.
Conflict of interest: None.
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Developmental dyscalculia is a familial learning disability.
Understanding dissociations in dyscalculia: a brain imaging study of the impact of number size on the cerebral networks for exact and approximate calculation.
What difference does a year of schooling make? Maturation of brain response and connectivity between 2nd and 3rd grades during arithmetic problem solving.
Lateralization of activity in the parietal cortex predicts the effectiveness of bilateral transcranial direct current stimulation on performance of a mental calculation task.
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