Acalculia: an historical review of localization

Impaired Arithmetic Fact Retrieval in an Adult with Developmental Dyscalculia: Evidence from Behavioral and Functional Brain Imaging Data

Developmental dyscalculia (DD) is a developmental disorder characterized by arithmetic difficulties. Recently, it has been suggested that the neural networks supporting procedure-based calculation (e.g., in subtraction) and left-hemispheric verbal arithmetic fact retrieval (e.g., in multiplication) are partially distinct. Here we compared the neurofunctional correlates of subtraction and multiplication in a 19-year-old student (RM) with DD to 18 age-matched controls. Behaviorally, RM performed significantly worse than controls in multiplication, while subtraction was unaffected. Neurofunctional differences were most pronounced regarding multiplication: RM showed significantly stronger activation than controls not only in left angular gyrus but also in a fronto-parietal network (including left intraparietal sulcus and inferior frontal gyrus) typically activated during procedure-based calculation. Region-of-interest analyses indicated group differences in multiplication only, which, however, did not survive correction for multiple comparisons. Our results are consistent with dissociable and processing-specific, but not operation-specific neurofunctional networks. Procedure-based calculation is not only associated with subtraction but also with (untrained) multiplication facts. Only after rote learning, facts can be retrieved quasi automatically from memory. We suggest that this learning process and the associated shift in activation patterns has not fully occurred in RM, as reflected in her need to resort to procedure-based strategies to solve multiplication facts.

Comprehension of computer code relies primarily on domain-general executive brain regions

Computer programming is a novel cognitive tool that has transformed modern society. What cognitive and neural mechanisms support this skill? Here, we used functional magnetic resonance imaging to investigate two candidate brain systems: the multiple demand (MD) system, typically recruited during math, logic, problem solving, and executive tasks, and the language system, typically recruited during linguistic processing. We examined MD and language system responses to code written in Python, a text-based programming language (Experiment 1) and in ScratchJr, a graphical programming language (Experiment 2); for both, we contrasted responses to code problems with responses to content-matched sentence problems. We found that the MD system exhibited strong bilateral responses to code in both experiments, whereas the language system responded strongly to sentence problems, but weakly or not at all to code problems. Thus, the MD system supports the use of novel cognitive tools even when the input is structurally similar to natural language.

Neurological Aspects of Foreign Accent Syndrome in Stroke Patients

Foreign Accent Syndrome (FAS) is an intriguing motor speech disorder which has captured the interest of the scientific community and media for decades. At the moment, there is no comprehensive model which can account for the pathophysiology of this disorder. This paper presents a review of 112 FAS cases published between 1907 and October 2016: these were analyzed with respect to demographic characteristics, lesion location, associated neurocognitive symptoms, and comorbid speech and language disorders. The analysis revealed that organic-neurogenic FAS is more frequent in women than in men. In organic-neurogenic FAS over half of the patients acquired the foreign accent after a stroke. Their lesions are typically located in the left supratentorial regions of the brain, and generally involve the primary motor cortex and premotor cortex (BA 4 and 6), and/or the basal ganglia. Although neurocognitive data are not consistently reported, vascular FAS patients regularly suffer frontal executive dysfunctions. On the basis of a careful comparison of the cognitive and theoretical accounts of FAS, AoS and ataxic dysarthria, it is concluded that FAS should be regarded a dual component motor speech disorder in which both planning and motor execution of speech may be affected.

Predicting Math Achievement Through Neuropsychological Interpretation of WISC-III Variance Components

Although prevalence estimates suggest that mathematics learning disorders (MLD) are as common as reading disorders, there has been comparatively little research conducted that examines the psychological processes involved in math competency for typical children, and the characteristics, etiology, and treatment of children with MLD. Previous research in disabled populations has implicated dysfunctional right hemisphere cognitive processes as a cause of MLD and suggested that impaired visual-spatial skills lead to specific MLD error patterns. In this study of 587 unidentified children with variable intellectual test profiles, the cognitive predictors necessary for math competency were interpreted from a neuropsychological orientation. Results revealed that complex interactions between WISC-III Verbal and Performance subtests were predictive of Albert Einstein College of Medicine math word problems and computation skills, suggesting that semantic/mathematics knowledge, working memory, executive function, novel problem solving, and visual-perceptual-motor processes are necessary for mathematics performance. Contrary to the right hemisphere hypothesis of math competency, results suggest that left hemisphere crystallized abilities and frontal executive functions are most predictive of mathematics achievement for children with variable test profiles. Consistent with theoretical and empirical advances regarding lateralization of function, the numerous predictor commonalities found support a reconceptualization of the left-verbal/right-nonverbal dichotomy of the cognitive processes underlying mathematics competency.

Spectrum of Mathematical Weaknesses: Related Neuropsychological Correlates

Math disorders have been recognized for as long as language disorders yet have received far less research. Mathematics is a complex construct and its development may be dependent on multiple cognitive abilities. Several studies have shown that short-term memory, working memory, visuospatial skills, processing speed, and various language skills relate to and may facilitate math development and performance. The hypotheses explored in this research were that children who performed worse on math achievement than on Full-Scale IQ would exhibit weaknesses in executive functions, memory, and visuoperceptual skills. Participants included 436 children (27% girls, 73% boys; age range = 5-17 years, M(age) = 9.45 years) who were referred for neuropsychological evaluations due to academic and/or behavioral problems. This article specifically focuses on the spectrum of math weakness rather than clinical disability, which has yet to be investigated in the literature. Results suggest that children with relative weakness to impairments in math were significantly more likely to have cognitive weaknesses to impairments on neuropsychological variables, as compared with children without math weaknesses. Specifically, the math-weak children exhibit a weakness to impairment on measures involving attention, language, visuoperceptual skills, memory, reading, and spelling. Overall, our results suggest that math development is multifaceted.

The Accountant Who Lost Arithmetic: A Case Report of Acalculia With a Left Thalamic Lesion

Acalculia is usually from dysfunction of the dominant parietal cortex. We report a case of acalculia associated with a lesion of the left thalamus.

Developmental cognitive neuroscience of arithmetic: implications for learning and education

In this article, we review the brain and cognitive processes underlying the development of arithmetic skills. This review focuses primarily on the development of arithmetic skills in children, but it also summarizes relevant findings from adults for which a larger body of research currently exists. We integrate relevant findings and theories from experimental psychology and cognitive neuroscience. We describe the functional neuroanatomy of cognitive processes that influence and facilitate arithmetic skill development, including calculation, retrieval, strategy use, decision making, as well as working memory and attention. Building on recent findings from functional brain imaging studies, we describe the role of distributed brain regions in the development of mathematical skills. We highlight neurodevelopmental models that go beyond the parietal cortex role in basic number processing, in favor of multiple neural systems and pathways involved in mathematical information processing. From this viewpoint, we outline areas for future study that may help to bridge the gap between the cognitive neuroscience of arithmetic skill development and educational practice.

The neural correlates of calculation ability in children: an fMRI study

Most studies investigating mental numerical processing involve adult participants and little is known about the functioning of these systems in children. The current study used functional magnetic resonance imaging (fMRI) to investigate the neural correlates of numeracy and the influence of age on these correlates with a group of adults and a group of third graders who had average to above average mathematical ability. Participants performed simple and complex versions of exact and approximate calculation tasks while in the magnet. Like adults, children activated a network of brain regions in the frontal and parietal lobes during the calculation tasks, and they recruited additional brain regions for the more complex versions of the tasks. However, direct comparisons between adults and children revealed significant differences in level of activation across all tasks. In particular, patterns of activation in the parietal lobe were significantly different as a function of age. Findings support previous claims that the parietal lobe becomes more specialized for arithmetic tasks with age.

Aberrant functional activation in school age children at-risk for mathematical disability: a functional imaging study of simple arithmetic skill

We used functional magnetic resonance imaging (fMRI) to explore the patterns of brain activation associated with different levels of performance in exact and approximate calculation tasks in well-defined cohorts of children with mathematical calculation difficulties (MD) and typically developing controls. Both groups of children activated the same network of brain regions; however, children in the MD group had significantly increased activation in parietal, frontal, and cingulate cortices during both calculation tasks. A majority of the differences occurred in anatomical brain regions associated with cognitive resources such as executive functioning and working memory that are known to support higher level arithmetic skill but are not specific to mathematical processing. We propose that these findings are evidence that children with MD use the same types of problem solving strategies as TD children, but their weak mathematical processing system causes them to employ a more developmentally immature and less efficient form of the strategies.

Developmental Changes in Mental Arithmetic: Evidence for Increased Functional Specialization in the Left Inferior Parietal Cortex

Arithmetic reasoning is arguably one of the most important cognitive skills a child must master. Here we examine neurodevelopmental changes in mental arithmetic. Subjects (ages 8-19 years) viewed arithmetic equations and were asked to judge whether the results were correct or incorrect. During two-operand addition or subtraction trials, for which accuracy was comparable across age, older subjects showed greater activation in the left parietal cortex, along the supramarginal gyrus and adjoining anterior intra-parietal sulcus as well as the left lateral occipital temporal cortex. These age-related changes were not associated with alterations in gray matter density, and provide novel evidence for increased functional maturation with age. By contrast, younger subjects showed greater activation in the prefrontal cortex, including the dorsolateral and ventrolateral prefrontal cortex and the anterior cingulate cortex, suggesting that they require comparatively more working memory and attentional resources to achieve similar levels of mental arithmetic performance. Younger subjects also showed greater activation of the hippocampus and dorsal basal ganglia, reflecting the greater demands placed on both declarative and procedural memory systems. Our findings provide evidence for a process of increased functional specialization of the left inferior parietal cortex in mental arithmetic, a process that is accompanied by decreased dependence on memory and attentional resources with development.

Neural correlates underlying mental calculation in abacus experts: a functional magnetic resonance imaging study

Experts of abacus operation demonstrate extraordinary ability in mental calculation. There is psychological evidence that abacus experts utilize a mental image of an abacus to remember and manipulate large numbers in solving problems; however, the neural correlates underlying this expertise are unknown. Using functional magnetic resonance imaging, we compared the neural correlates associated with three mental-operation tasks (numeral, spatial, verbal) among six experts in abacus operations and eight nonexperts. In general, there was more involvement of neural correlates for visuospatial processing (e.g., right premotor and parietal areas) for abacus experts during the numeral mental-operation task. Activity of these areas and the fusiform cortex was correlated with the size of numerals used in the numeral mental-operation task. Particularly, the posterior superior parietal cortex revealed significantly enhanced activity for experts compared with controls during the numeral mental-operation task. Comparison with the other mental-operation tasks indicated that activity in the posterior superior parietal cortex was relatively specific to computation in 2-dimensional space. In conclusion, mental calculation of abacus experts is likely associated with enhanced involvement of the neural resources for visuospatial information processing in 2-dimensional space.

Prefrontal cortex involvement in processing incorrect arithmetic equations: evidence from event-related fMRI

The main aim of this study was to investigate the differential processing of correct and incorrect equations to gain further insight into the neural processes involved in arithmetic reasoning. Electrophysiological studies in humans have demonstrated that processing incorrect arithmetic equations (e.g., 2 + 2 = 5) elicits a prominent event-related potential (ERP) compared to processing correct equations (e.g., 2 + 2 = 4). In the present study, we investigated the neural substrates of this process using event-related functional magnetic resonance imaging (fMRI). Subjects were presented with arithmetic equations and asked to indicate whether the solution displayed was correct or incorrect. We found greater activation to incorrect, compared to correct equations, in the left dorsolateral prefrontal cortex (DLPFC, BA 46) and the left ventrolateral prefrontal cortex (VLPFC, BA 47). Our results provide the first brain imaging evidence for differential processing of incorrect vs. correct equations. The prefrontal cortex activation observed in processing incorrect equations overlaps with brain areas known to be involved in working memory and interference processing. The DLPFC region differentially activated by incorrect equations was also involved in overall arithmetic processing, whereas the VLPFC was activated only during the differential processing of incorrect equations. Differential response to correct and incorrect arithmetic equations was not observed in parietal cortex regions such as the angular gyrus and intra-parietal sulcus, which are known to play a specific role in performing arithmetic computations. The pattern of brain response observed is consistent with the hypothesis that processing incorrect equations involves detection of an incorrect answer and resolution of the interference between the internally computed and externally presented incorrect answer. More specifically, greater activation during processing of incorrect equations appears to reflect additional operations involved in maintaining the results in working memory, while subjects attempt to resolve the conflict and select a response. These findings allow us to further delineate and dissociate the contributions of prefrontal and parietal cortices to arithmetic reasoning.

Neural substrates of mathematical reasoning: a functional magnetic resonance imaging study of neocortical activation during performance of the necessary arithmetic operations test

Brain activation was examined using functional magnetic resonance imaging during mathematical problem solving in 7 young healthy participants. Problems were selected from the Necessary Arithmetic Operations Test (NAOT; R. B. Ekstrom, J. W. French, H. H. Harman, & D. Dermen, 1976). Participants solved 3 types of problems: 2-operation problems requiring mathematical reasoning and text processing, 1-operation problems requiring text processing but minimal mathematical reasoning, and 0-operation problems requiring minimal text processing and controlling sensorimotor demands of the NAOT problems. Two-operation problems yielded major activations in bilateral frontal regions similar to those found in other problem-solving tasks, indicating that the processes mediated by these regions subserve many forms of reasoning. Findings suggest a dissociation in mathematical problem solving between reasoning, mediated by frontal cortex, and text processing, mediated by temporal cortex.

The functional neuroanatomy of simple calculation and number repetition: A parametric PET activation study

We examined cerebral activation patterns with positron emission tomography (PET) in 12 right-handed normal volunteers while they were completing simple calculation tasks or merely repeating numbers. Using a parametric experimental design, during calculation we found activation in the medial frontal/cingulate gyri, left dorsolateral prefrontal cortex, left anterior insular cortex and right anterior insular cortex/putamen, left lateral parietal cortex, and the medial thalamus. Number repetition engaged bilateral inferior sensorimotor cortex, bilateral temporal areas, and left inferior frontal cortex. These results suggest a functional anatomical network for simple calculation, which includes aspects of attention, auditory, and motor processing and the phonological store and articulatory loop components of working memory; they add some support for a special role of the parietal cortex in calculation tasks.

Dissociating prefrontal and parietal cortex activation during arithmetic processing

Lesion and brain-imaging studies have implicated the prefrontal and parietal cortices in arithmetic processing, but do not exclude the possibility that these brain areas are also involved in nonarithmetic operations. In the present study, we used functional magnetic resonance imaging to explore which brain areas contribute uniquely to numeric computation. Task difficulty was manipulated in a factorial design by varying the number of operands and the rate of stimulus presentation. Both manipulations increased the number of operations to be performed in unit time. Manipulating the number of operands allowed us to investigate the specific effect of calculation, while manipulating the rate of presentation allowed us to increase task difficulty independent of calculation. We found quantitative changes in activation patterns in the prefrontal and parietal cortices as well as the recruitment of additional brain regions, including the caudate and midcerebellar cortex, with increasing task difficulty. More importantly, the main effect of arithmetic complexity was observed in the left and right angular gyrus, while the main effect of rate of stimulus presentation was observed in the left insular/orbitofrontal cortex. Our findings indicate a dissociation in prefrontal and parietal cortex function during arithmetic processing and further provide the first evidence for a specific role for the angular gyrus in arithmetic computation independent of other processing demands.

Cortical activation during retrieval of arithmetical facts and actual calculation: a functional magnetic resonance imaging study

By using functional magnetic resonance imaging (fMRI), the neural substrates involved in mental recitation of the single-digit multiplication table and serial subtraction were studied. The former depends mostly on well-learned arithmetical facts, while the latter requires arithmetic processing. Activation during each task was compared with that in a number counting control. During the recitation of single-digit multiplication, the activated regions included the area lying along the left intraparietal sulcus, the premotor and supplementary motor areas, and the posterior portion of the left inferior frontal gyrus. The areas activated during serial subtraction included these areas as well as the bilateral prefrontal and right parietal areas. From the results obtained during retrieval of the multiplication table in this study and previous studies, it was concluded that semantic memory of the multiplication table is stored in the area lying along the intraparietal sulcus and that the frontal areas play an executive role in utilizing the semantic memory of arithmetical facts. It was assumed that the arithmetical facts requiring actual calculation are also stored in the same region. The additional activation during serial subtraction compared with the activation during retrieval of the multiplication table is probably due to the processes of actual calculation. These processes include proper alignment of digits, which may have caused the right parietal activation, and maintaining digits needed for the mental serial subtractions, which may have caused the bilateral prefrontal activation.

Functional optimization of arithmetic processing in perfect performers

Lesion and imaging studies to date have not clarified which sub-regions of the parietal lobe are specialized for arithmetic processing, and which perform supporting functions. We used functional magnetic resonance imaging to investigate parietal lobe function during arithmetic processing. Functional optimization was examined by analyzing regional differences in brain activation between perfect (100% accuracy) and imperfect performers. Perfect performers had significantly less activation only in the left angular gyrus, a finding that may be associated with skill mastery and long-term practice effects. The present results provide the first direct evidence of localized functional optimization for arithmetic processing in the human brain.

Psychophysiology by the end of the 20th century

The first real breakthrough in the research of brain organization and thinking in the 20th century was made in neurophysiological investigations performed in direct contact with different sites of the brain, which became possible in diagnosis and treatment. The second breakthrough is happening at present. It is based on the opportunities provided by the non-invasive technique. The theory of the unique character of the brain system consisting of rigid and flexible elements maintaining thinking was created as well as concepts on the reliability in the system, of the error detector and intrinsic protective mechanisms of the brain. In the clinic these data enabled us to help patients who had lost various functions due to stroke. In confirmation with the above theory it was revealed that the same task could be solved in the brain by systems consisting of different elements due to environmental changes or even direction of attention. Data on the functional properties or every zone of the cortex and subcortex as well as cerebellum are rapidly increasing in number. The first priority lies in neurophysiologically penetrating into the physiological character and micromosaic of the activation sites of PET. The main aim of future brain research lies in the investigation of the fine physiological rearrangements which underlie thinking, i.e. deciphering its brain code. This is going to be the basis for the third, extremely valid breakthrough in the research on brain organization of thinking.

Sequential information processing during a mental arithmetic is reflected in the time course of event-related brain potentials

OBJECTIVE

We hypothesized that mental arithmetic is a complex of sequential information processing. In order to test the hypothesis, event-related potentials (ERPs) were measured during 3 mental tasks.

METHODS

Fifteen normal human subjects performed the following tasks; (1) subjects added up every digit delivered successively on a computer display, (2) subjects counted the number of presented digits, or (3) subjects counted the number of meaningless patterns. Spatiotemporal differences between ERP waveforms under the 3 tasks were studied within the period of 1200 ms post-stimulus.

RESULTS

During the adding task, N120-P180-N220 complex advanced in latency in the left frontal, central and parietal regions. P300 increased in amplitude in the frontal and temporal regions during adding and counting digits, which was specific to the digit presentation. A positive slow potential depended on the adding task and showed two spatiotemporal distributions; one was a widespread brain activity over the frontal, central, temporal and parietal regions observed within 400-820 ms, and another was a brain activity restricted to the frontal region lasting up to 1150 ms.

CONCLUSIONS

The results suggest that the early portion of ERPs reflects identification of physical attributes of stimuli and numeric meaning of digits, and that the positive slow potential reflects processes associated with calculation.

Assessment of calculation and number processing by adults: cognitive and neuropsychological issues

Calculation and number processing abilities were assessed in normal (n = 138) and traumatic brain-injured (n = 15) Brazilian literature subjects. The study aimed (i) to analyse the effects of demographic factors and to provide tentative norms adjusted for the relevant variables, (ii) to examine the factorial structure of the battery and to evaluate its clinical validity for diagnosis purposes, and (iii) to question the power of current models to account for effects and dissociations found for these groups. Analysis indicated a main effect of education on most subtests and of sex on three, but none for age. Cut-off scores for normality were defined at Percentile 10 with reference to education. The sensitivity of the battery to the presence of arithmetical impairments was considered satisfactory since 11 out of the 15 patients showed pathological scores. A principal component analysis indicated that the different sub-tests were grouped into three factors, which were tentatively interpreted with reference to current information-processing models. The multiple single-case analysis of dissociations in patients' performance suggested some limits with respect to anatomo-functional models of calculation and number processing.

A functional magnetic resonance imaging study of mental subtraction in human subjects

The neuronal network involved in a precise type of calculation procedure, mental subtraction, was investigated by means of functional magnetic resonance imaging. Two tasks were used requiring covert production of numbers: (1) with calculation; (2) without calculation. During the first task, activation was observed in the left dorsolateral prefrontal and premotor cortices, in Broca's area and bilaterally in the inferior parietal cortex. During the second task, activation was mainly observed in Broca's area and to a less extent in the left prefrontal and premotor cortices. Statistical comparison of data in the two situations revealed that the procedure of mental subtraction is mediated by a distributed system which includes predominantly the left dorsolateral prefrontal cortex and the inferior parietal cortex bilaterally.

Medial prefrontal cortex generates frontal midline theta rhythm

Frontal midline theta rhythm (Fm theta) is a distinct theta activity of EEG in the frontal midline area that appears during concentrated performance of mental tasks in normal subjects and reflects focused attentional processing. To tomographically visualize the source current density distributions of Fm theta, we recorded Fm theta by using a 64-channel whole-head MEG system from four healthy subjects, and applied a new analysis method, synthetic aperture magnetometry (SAM), an adaptive beam forming method. Fm theta was observed in the MEG signals over the bilateral frontal regions. SAM analysis showed bilateral medial prefrontal cortices, including anterior cingulate cortex, as the source of Fm theta. This result suggests that focused attention is mainly related to medial prefrontal cortex.

Priming arithmetic facts in amnesic patients

In this study, amnesic patients showed significant repetition priming effects in arithmetic fact retrieval tasks. The results indicate that repetition priming effects in arithmetic depend not on explicit recognition, but on the activation of specific long-term representations of arithmetic facts. Processing dissociations between easy and difficult items suggest that the priming effects results from the stage of fact retrieval and not from peripheral activation. This claim is also supported by encoding and naming tasks, which showed only slight priming effects as compared to the priming found in calculation tasks. Significant priming was found for identical (5 x 6 and 5 x 6) and complement problems (5 x 6 and 6 x 5), the latter showing a smaller magnitude of priming.

Mathematical cognitive style and arithmetic sign comprehension: a study of EEG alpha and theta activity

The localization of arithmetic sign comprehension was investigated using EEG spectral parameters as indicators of cortical engagement. Right-handed male subjects were selected on the basis of scores on the Mathematics Cognitive Style Survey and assigned to 2 groups, a 'left hemisphere oriented (LHO)' (N = 9) and 'right hemisphere oriented (RHO)' (N = 9) group. Subjects were presented with 4 conditions, a motoric baseline condition, two arithmetic fact retrieval tasks employing either a sign operator or verbal operator and a sign comprehension task which required subjects to fill in a missing sign (e.g. 6 ? 4 = 24). Both across subject correlational analysis of EEG alpha 1 asymmetry and performance as well as within subject analysis of condition means indicated a somewhat unique contribution of the right hemisphere to sign comprehension. LHO subjects exhibited greater relative left mid-temporal lobe activation than RHO subjects but less relative left frontal activation (theta band) than RHO subjects during the verbal operator task. It was tentatively concluded that this frontal lobe asymmetry difference was due to a mismatch in strategy preference and coding requirements among RHO subjects.

The isolation of calculation skills

The unusual preservation of calculation skills in a patient with severe global aphasia is described. The implications for the relationship between numerical and language abilities are discussed.

Dynamic activities of the frontal association cortex in calculating and thinking

We found 5-7 Hz magnetic theta waves in the frontal association cortex of adult human subjects during calculation and musical imagination by using 37-channel SQUID gradiometers. Simultaneous recording from the left and right cerebral hemispheres with two sets of 37-channel gradiometers revealed that the theta activity appeared in a waxing and waning manner in the frontal cortices of both hemispheres during the mental exercises. Electrical current dipoles for the theta waves were estimated to occur repeatedly and scatteringly in various parts of the frontal lobes of both hemispheres during continuous and intense mental exercises for 2 min. The results suggest a dynamic mode of activities in the frontal association cortex during mental effort such as calculating and thinking.

Cognitive mechanisms in numerical processing: evidence from acquired dyscalculia

This article discusses cognitive neuropsychological research on acquired dyscalculia (i.e., impaired numerical processing resulting from brain damage), surveying issues of current interest, and illustrating the ways in which analyses of acquired deficits can contribute to an understanding of normal processing. I first review the logic whereby inferences concerning normal cognition are drawn from patterns of impaired performance. I then consider research exploring the general functional architecture of the cognitive numerical processing mechanisms, and finally turn to studies aimed at probing the internal structure and functioning of individual processing components.

Theory-based assessment of acquired dyscalculia

This article describes a theory-based approach to assessment of acquired dyscalculia. A model of the normal cognitive number-processing/calculation system is presented, and methods are discussed for characterizing number-processing/calculation deficits in terms of functional damage to the mechanisms specified in the model.