Calculation disturbances in adults with focal hemispheric damage

The role of the intraparietal sulcus in numeracy: A review of parietal lesion cases

Prominent theories of numeracy link the intraparietal sulcus (IPS) to approximate representations of quantity that undergird basic math abilities. The goal of this review is to better understand the neural basis of mathematical cognition through the lens of acalculia, by identifying numeracy-focused single case studies of patients with parietal lesions and testing for causal relationships between numeracy impairments and the locus of parietal damage. A systematic literature review identified 27 single case studies with left parietal lesions and categorized administered tasks across four numeracy domains: Approximation, Calculation, Ordinality/Cardinality, and Transcoding. We compared published lesion images by drawing a sphere at the inferred center-of-mass and assigning each case to an anatomical group (IPS or Other Parietal damage) based on overlap with left IPS and original anatomical description. We performed Fisher's Exact Test to compare behavioral performance on each numeracy domain between the two groups. As an exploratory follow-up, we used Activation Likelihood Estimation (ALE) to identify sites of damage within parietal cortex preferentially associated with impairments in each domain. We found that Approximation impairments were significantly more frequent in the IPS group (p = .003). The exploratory ALE analysis revealed that only Approximation impairment cases significantly overlapped with the IPS, while impairments in other domains were localized to different regions of the parietal lobe. Based on the pattern of impairments shown across these cases, we conclude that damage to the left IPS is linked to impairments in approximation ability specifically. Our findings support theoretical claims linking IPS to magnitude representation, but do not provide evidence that IPS critically underpins performance across all numeracy tasks. Instead, our findings are more compatible with models of dissociable circuits of numerical processing within the parietal lobe.

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Brain laterality of numbers and calculation: Complex networks and their development

This chapter reviews notions about the lateralization of numbers and calculation in the brain, including its developmental pattern. Such notions have changed dramatically in recent decades. What was once considered a function almost exclusively located in the left hemisphere has been found to be sustained by complex brain networks encompassing both hemispheres. Depending on the specific task, however, each hemisphere has its own role. Much of this progress was determined by the convergency of investigations conducted with different methods. Contrary to traditional wisdom, the right hemisphere is not involved in arithmetic just as far as generic spatial aspects are concerned. Very specific arithmetic functions like remembering the spatial templates for complex operations, or processing of zero in complex numbers, are indeed sustained in specific right-sided areas. The system used in the typical adult appears to be the result of a complex pattern of development. The numerical brain clearly evolved from less mature to more advanced brain networks because of growth and education. Children seem to be equipped with the ability to represent the number nonverbally from a very early age. The bilateral processing of number-related tasks is however a late acquisition.

The brain under pressure: Exploring neurophysiological responses to cognitive stress

Stress is an increasingly dominating part of our daily lives and higher performance requirements at work or to ourselves influence the physiological reaction of our body. Elevated stress levels can be reliably identified through electroencephalogram (EEG) and heart rate (HR) measurements. In this study, we examined how an arithmetic stress-inducing task impacted EEG and HR, establishing meaningful correlations between behavioral data and physiological recordings. Thirty-one healthy participants (15 females, 16 males, aged 20 to 37) willingly participated. Under time pressure, participants completed arithmetic calculations and filled out stress questionnaires before and after the task. Linear mixed effects (LME) allowed us to generate topographical association maps showing significant relations between EEG features (delta, theta, alpha, beta, and gamma power) and factors such as task difficulty, error rate, response time, stress scores, and HR. With task difficulty, we observed left centroparietal and parieto-occipital theta power decreases, and alpha power increases. Furthermore, frontal alpha, delta and theta activity increased with error rate and relative response time, while parieto-temporo-occipital alpha power decreased. Practice effects on EEG power included increases in temporal, parietal, and parieto-occipital theta and alpha activity. HR was positively associated with frontal delta, theta and alpha power whereas frontal gamma power decreases. Significant alpha laterality scores were observed for all factors except task difficulty and relative response time, showing overall increases in left parietal regions. Significant frontal alpha asymmetries emerged with increases in error rate, sex, run number, and HR and occipital alpha asymmetries were also found with run number and HR. Additionally we explored practice effects and noted sex-related differences in EEG features, HR, and questionnaire scores. Overall, our study enhances the understanding of EEG/ECG-based mental stress detection, crucial for early interventions, personalized treatment and objective stress assessment towards the development of a neuroadaptive system.

The brain lateralization and development of math functions: progress since Sperry, 1974

In 1974, Roger Sperry, based on his seminal studies on the split-brain condition, concluded that math was almost exclusively sustained by the language dominant left hemisphere. The right hemisphere could perform additions up to sums less than 20, the only exception to a complete left hemisphere dominance. Studies on lateralized focal lesions came to a similar conclusion, except for written complex calculation, where spatial abilities are needed to display digits in the right location according to the specific requirements of calculation procedures. Fifty years later, the contribution of new theoretical and instrumental tools lead to a much more complex picture, whereby, while left hemisphere dominance for math in the right-handed is confirmed for most functions, several math related tasks seem to be carried out in the right hemisphere. The developmental trajectory in the lateralization of math functions has also been clarified. This corpus of knowledge is reviewed here. The right hemisphere does not simply offer its support when calculation requires generic space processing, but its role can be very specific. For example, the right parietal lobe seems to store the operation-specific spatial layout required for complex arithmetical procedures and areas like the right insula are necessary in parsing complex numbers containing zero. Evidence is found for a complex orchestration between the two hemispheres even for simple tasks: each hemisphere has its specific role, concurring to the correct result. As for development, data point to right dominance for basic numerical processes. The picture that emerges at school age is a bilateral pattern with a significantly greater involvement of the right-hemisphere, particularly in non-symbolic tasks. The intraparietal sulcus shows a left hemisphere preponderance in response to symbolic stimuli at this age.

Aphasia and Math: Deficits with Basic Number Comprehension and in Numerical Activities of Daily Living

OBJECTIVE

In the present study, we explored numerical problems in individuals with aphasia. We investigate whether numerical deficits, usually accompanying aphasia, can be observed on number comprehension tasks that do not necessarily require an oral response.

METHOD

Individuals with aphasia were classified into anterior, posterior, and global subgroups according to the lesion type. To investigate numerical cognition, we used a relatively recent tool, the Numerical Activities of Daily Living (NADL).

RESULTS

The results showed that individuals with aphasia have problems with tasks of basic number comprehension as well as in most NADL. In the formal part of the NADL, anterior aphasic patients made comparatively more errors than the posterior aphasic patients. Global aphasic patients presented an invariably poor performance on almost all tasks.

CONCLUSION

The results provide insight into how numerical deficits may impair an individual with aphasia in activities of daily living. This study is a preliminary attempt to start the validation process of the NADL for the Greek population.

EEG correlation during the solving of simple and complex logical-mathematical problems

Solving logical-mathematical word problems is a complex task that requires numerous cognitive operations, including comprehension, reasoning, and calculation. These abilities have been associated with activation of the parietal, temporal, and prefrontal cortices. It has been suggested that the reasoning involved in solving logical-mathematical problems requires the coordinated functionality of all these cortical areas. In this study was evaluated the activation and electroencephalographic (EEG) correlation of the prefrontal, temporal, and parietal regions in young men while solving logical-mathematical word problems with two degrees of difficulty: simple and complex. During the solving of complex problems, higher absolute power and EEG correlation of the alpha and fast bands between the left frontal and parietal cortices were observed. A temporal deactivation and functional decoupling of the right parietal-temporal cortices also were obtained. Solving complex problems probably require activation of a left prefrontal-parietal circuit to maintain and manipulate multiple pieces of information. The temporal deactivation and decreased parietal-temporal correlation could be associated to text processing and suppression of the content-dependent reasoning to focus cognitive resources on the mathematical reasoning. Together, these findings support a pivotal role for the left prefrontal and parietal cortices in mathematical reasoning and of the temporal regions in text processing required to understand and solve written mathematical problems.

Reassessing lateralization in calculation

The role of the left hemisphere in calculation has been unequivocally demonstrated in numerous studies in the last decades. The right hemisphere, on the other hand, had been traditionally considered subsidiary to the left hemisphere functions, although its role was less clearly defined. Recent clinical studies as well as investigations conducted with other methodologies (e.g. neuroimaging, transcranial magnetic stimulation and direct cortical electro-stimulation) leave several unanswered questions about the contribution of the right hemisphere in calculation. In particular, novel clinical studies show that right hemisphere acalculia encompasses a wide variety of symptoms, affecting even simple calculation, which cannot be easily attributed to spatial disorders or to a generic difficulty effect as previously believed. The studies reported here also show how the right hemisphere has its own specific role and that only a bilateral orchestration between the respective functions of each hemisphere guarantees, in fact, precise calculation. Vis-à-vis these data, the traditional wisdom that attributes to the right hemisphere a role mostly confined to spatial aspects of calculation needs to be significantly reshaped. The question for the future is whether it is possible to precisely define the specific contribution of the right hemisphere in several aspects of calculation while highlighting the nature of the cross-talk between the two hemispheres.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Anodal transcranial direct current stimulation of the right anterior temporal lobe did not significantly affect verbal insight

Humans often utilize past experience to solve difficult problems. However, if past experience is insufficient to solve a problem, solvers may reach an impasse. Insight can be valuable for breaking an impasse, enabling the reinterpretation or re-representation of a problem. Previous studies using between-subjects designs have revealed a causal relationship between the anterior temporal lobes (ATLs) and non-verbal insight, by enhancing the right ATL while inhibiting the left ATL using transcranial direct current stimulation (tDCS). In addition, neuroimaging studies have reported a correlation between right ATL activity and verbal insight. Based on these findings, we hypothesized that the right ATL is causally related to both non-verbal and verbal insight. To test this hypothesis, we conducted an experiment with 66 subjects using a within-subjects design, which typically has greater statistical power than a between-subjects design. Subjects participated in tDCS experiments across 2 days, in which they solved both non-verbal and verbal insight problems under active or sham stimulation conditions. To dissociate the effects of right ATL stimulation from those of left ATL stimulation, we used two montage types; anodal tDCS of the right ATL together with cathodal tDCS of the left ATL (stimulating both ATLs) and anodal tDCS of the right ATL with cathodal tDCS of the left cheek (stimulating only the right ATL). The montage used was counterbalanced across subjects. Statistical analyses revealed that, regardless of the montage type, there were no significant differences between the active and sham conditions for either verbal or non-verbal insight, although the finding for non-verbal insight was inconclusive because of a lack of statistical power. These results failed to support previous findings suggesting that the right ATL is the central locus of insight.

The zero effect: voxel-based lesion symptom mapping of number transcoding errors following stroke

Zero represents a special case in our numerical system because it is not represented on a semantic level. Former research has shown that this can lead to specific impairments when transcoding numerals from dictation to written digits. Even though, number processing is considered to be dominated by the left hemisphere, studies have indicated that both left as well as right hemispheric stroke patients commit errors when transcoding numerals including zeros. Here, for the first time, a large sample of subacute stroke patients (N = 667) was assessed without being preselected based on the location of their lesion, or a specific impairment in transcoding zero. The results show that specific errors in transcoding zeros were common (prevalence = 14.2%) and a voxel-based lesion symptom mapping analysis (n = 153) revealed these to be related to lesions in and around the right putamen. In line with former research, the present study argues that the widespread brain network for number processing also includes subcortical regions, like the putamen with connections to the insular cortex. These play a crucial role in auditory perception as well as attention. If these areas are lesioned, number processing tasks with higher attentional and working memory loads, like transcoding zeros, can be impaired.

Cortical dynamics during simple calculation processes: A magnetoencephalography study

OBJECTIVE

We elucidated active cortical areas and their time courses during simple calculation by using whole-scalp magnetoencephalography.

METHODS

Twelve healthy volunteers were asked to view meaningless figures (figure viewing) or digits (digit viewing) and add single digits (calculation). The magnetic signals of the brain were measured using a helmet-shaped 122-channel neuromagnetometer during the three tasks.

RESULTS

The occipital, inferior posterior temporal, and middle temporal areas of each hemisphere and the left superior temporal area (STA) were activated during all tasks (approximately 250 ms after the stimulus onset). The calculation-related sources were located in the left inferior parietal area (IPA) in 8 subjects, right IPA in 5, left STA in 3, right STA in 5, right inferior frontal area in 2, and left inferior frontal area in 1. The IPA and STA of the left hemisphere were activated more strongly and significantly earlier than those of the right hemisphere: the left IPA was activated first (mean activation timing: 301 ms), followed by activations of the left STA (369 ms), right IPA (419 ms), and right STA (483 ms).

CONCLUSIONS

Simple digit addition is executed mainly in the left IPA and left STA, followed by the recognition processes of results in the right IPA and right STA.

SIGNIFICANCE

This study clarified the cortical process during simple calculation, with excellent temporal and spatial resolution; the IPA and STA of the left hemisphere were activated more strongly and earlier than the corresponding areas of the right hemisphere.

Re-assessing acalculia: Distinguishing spatial and purely arithmetical deficits in right-hemisphere damaged patients

Arithmetical deficits in right-hemisphere damaged patients have been traditionally considered secondary to visuo-spatial impairments, although the exact relationship between the two deficits has rarely been assessed. The present study implemented a voxelwise lesion analysis among 30 right-hemisphere damaged patients and a controlled, matched-sample, cross-sectional analysis with 35 cognitively normal controls regressing three composite cognitive measures on standardized numerical measures. The results showed that patients and controls significantly differ in Number comprehension, Transcoding, and Written operations, particularly subtractions and multiplications. The percentage of patients performing below the cutoffs ranged between 27% and 47% across these tasks. Spatial errors were associated with extensive lesions in fronto-temporo-parietal regions -which frequently lead to neglect- whereas pure arithmetical errors appeared related to more confined lesions in the right angular gyrus and its proximity. Stepwise regression models consistently revealed that spatial errors were primarily predicted by composite measures of visuo-spatial attention/neglect and representational abilities. Conversely, specific errors of arithmetic nature linked to representational abilities only. Crucially, the proportion of arithmetical errors (ranging from 65% to 100% across tasks) was higher than that of spatial ones. These findings thus suggest that unilateral right hemisphere lesions can directly affect core numerical/arithmetical processes, and that right-hemisphere acalculia is not only ascribable to visuo-spatial deficits as traditionally thought.

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.

Right-hemisphere (spatial?) acalculia and the influence of neglect

The present study aimed at exploring basic number and calculation abilities in right-hemisphere damaged patients (RHD), focusing primarily on one-digit orally presented tasks, which do not require explicit visuo-spatial abilities. Twenty-four non mentally-deteriorated RHD patients [12 with clinical neglect (RHDN+), 12 without clinical neglect (RHDN-)], and 12 healthy controls were included in the study. Participants were administered an ad hoc numerical battery assessing abilities such as counting, number magnitude comparison, writing and reading Arabic numerals and mental calculation, among others. Significant differences emerged among healthy controls and both the RHDN+ group and the RHDN- group, suggesting that the mathematical impairment of RHD patients does not necessarily correspond to the presence of left-neglect. A detailed analysis of the sub-tests of the battery evidenced expected differences among RHDN+ patients, RHDN- patients, and controls in writing and reading Arabic numerals. Crucially, differences between RHDN+ patients and controls were also found in tasks such as mental subtraction and mental multiplication, which do not require written visuo-spatial abilities. The present findings thus suggest that unilateral right hemisphere lesions may produce specific representational deficits that affect simple mental calculation, and not only the spatial arrangement of multi-digit written numbers as previously thought.

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.

Mapping mental calculation systems with electrocorticography

OBJECTIVE

We investigated intracranially-recorded gamma activity during calculation tasks to better understand the cortical dynamics of calculation.

METHODS

We studied 11 patients with focal epilepsy (age range: 9-28years) who underwent measurement of calculation- and naming-related gamma-augmentation during extraoperative electrocorticography (ECoG). The patients were instructed to overtly verbalize a one-word answer in response to auditorily-delivered calculation and naming questions. The assigned calculation tasks were addition and subtraction involving integers between 1 and 17.

RESULTS

Out of the 1001 analyzed cortical electrode sites, 63 showed gamma-augmentation at 50-120Hz elicited by both tasks, 88 specifically during naming, and 7 specifically during calculation. Common gamma-augmentation mainly took place in the Rolandic regions. Calculation-specific gamma-augmentation, involving the period between the question-offset and response-onset, was noted in the middle-temporal, inferior-parietal, inferior post-central, middle-frontal, and premotor regions of the left hemisphere. Calculation-specific gamma-augmentation in the middle-temporal, inferior-parietal, and inferior post-central regions peaked around the question offset, while that in the frontal lobe peaked after the question offset and before the response onset. This study failed to detect a significant difference in calculation-specific gamma amplitude between easy trials and difficult ones requiring multi-digit operations.

CONCLUSIONS

Auditorily-delivered stimuli can elicit calculation-specific gamma-augmentation in multiple regions of the left hemisphere including the parietal region. However, the additive diagnostic value of measurement of gamma-augmentation related to a simple calculation task appears modest.

SIGNIFICANCE

Further studies are warranted to determine the functional significance of calculation-specific gamma-augmentation in each site, and to establish the optimal protocol for mapping mental calculation.

Effects of mental rotation on acalculia: differences in the direction of mental rotation account for the differing characteristics of acalculia induced by right and left hemispheric brain injury

We observed a 59-year-old right-handed man with an infarction in his right-middle cerebral artery that included the parietal lobe, who abnormally manipulated mental images in the horizontal direction, resulting in calculation disturbances. Three years later, the patient suffered an infarction in the left parietal lobe and displayed abnormalities during the creation of mental images; i.e., he rotated them in the vertical direction, which again resulted in calculation disturbances. These mental imagery disturbances might indicate that a common acalculia mechanism exists between the right and left hemispheres.

Comparison of the choice effect and the distance effect in a number-comparison task by FMRI

Behavioral and neurophysiological studies of numerical comparisons have shown a "distance effect," whereby smaller numerical distances between two digits are associated with longer response times and higher activity in the parietal region. In this experiment, we introduced a two-choice condition (between either the smaller/lower or the larger/higher of two digits) and examined its effect on brain activity by fMRI. We observed longer response times and greater activity with the choice of smaller numbers ("choice effect") in several brain regions including the right temporo-parietal region, (pre)cuneus, superior temporal sulcus, precentral gyrus, superior frontal gyrus, bilateral insula, and anterior cingulate cortex. These regions correspond to areas that have been suggested to play a role in attentional shift and response conflict. However, brain activity associated with the distance effect disappeared even though the behavioral distance effect remained. Despite the absence of the distance effect on brain activity, several areas changed activity in relation to response time, including regions that were reported to change activity in both a distance effect and a reaction-time-related manner. The result suggested that the level of task load may change the activity of regions that are responsible for magnitude detection.

Neuroscience of learning arithmetic--evidence from brain imaging studies

It is widely accepted that the human brain is remarkably adaptive not only in child development, but also during adulthood. Aim of this work is to offer an overview and a systematic analysis of neuroimaging studies on the acquisition of arithmetic expertise. In normally developing children and adults, the gain of arithmetic competence is reflected by a shift of activation from frontal brain areas to parietal areas relevant for arithmetic processing. A shift of activation is also observed within the parietal lobe from the intraparietal sulci to the left angular gyrus. Increases in angular gyrus activation with gaining of expertise have also been documented in other cognitive domains. It appears that the left angular gyrus activation is modulated by inter-individual differences in arithmetic performance. The comparison of normal individuals with exceptionally performing individuals (e.g., calculating prodigies) suggests that the experts' arithmetic proficiency relies on a more extended activation network than the network found in non-experts. In expert individuals with long-lasting, extensive mathematical training, specific structural brain modifications are also evident.

Behavioral and near-infrared spectroscopy study of the effects of distance and choice in a number comparison task

Extensive behavioral and neurophysiological numerical comparison studies have shown that response times are longer and parietal activities are stronger when the numerical distance between two digits is smaller (the distance effect). However, only a few behavioral studies have considered the effect of the choice of larger or smaller numerals in numerical comparisons. Using near-infrared spectroscopy (NIRS), we investigated the neural basis of choosing larger/smaller numerals in number comparison tasks in which subjects were required to choose a larger or smaller digit. Our results showed that choosing a smaller digit induced significantly longer response times (the choice effect) and stronger parietal activities. We also obtained significantly longer response times as the distance effect in accordance with previous works. However, NIRS data did not show any significant difference corresponding to distance effect. Our results and previous studies suggest that the parietal cortex is involved not only in measuring numerical quantities, but also in choosing a numerically larger/smaller quantity among the categories of choice. Potentials and limitations of NIRS were discussed.

Common brain regions underlying different arithmetic operations as revealed by conjunct fMRI-BOLD activation

The issue of how and where arithmetic operations are represented in the brain has been addressed in numerous studies. Lesion studies suggest that a network of different brain areas are involved in mental calculation. Neuroimaging studies have reported inferior parietal and lateral frontal activations during mental arithmetic using tasks of different complexities and using different operators (addition, subtraction, etc.). Indeed, it has been difficult to compare brain activation across studies because of the variety of different operators and different presentation modalities used. The present experiment examined fMRI-BOLD activity in participants during calculation tasks entailing different arithmetic operations -- addition, subtraction, multiplication and division -- of different complexities. Functional imaging data revealed a common activation pattern comprising right precuneus, left and right middle and superior frontal regions during all arithmetic operations. All other regional activations were operation specific and distributed in prominently frontal, parietal and central regions when contrasting complex and simple calculation tasks. The present results largely confirm former studies suggesting that activation patterns due to mental arithmetic appear to reflect a basic anatomical substrate of working memory, numerical knowledge and processing based on finger counting, and derived from a network originally related to finger movement. We emphasize that in mental arithmetic research different arithmetic operations should always be examined and discussed independently of each other in order to avoid invalid generalizations on arithmetics and involved brain areas.

Effects of age and mild cognitive impairment on direct and indirect access to arithmetic knowledge

The present study aimed at investigating age-related changes and mild cognitive impairment (MCI) related effects in simple arithmetic. To pursue this goal, MCI patients, healthy old adults and young adults performed three computerised tasks. The production (e.g., 3 x 4=?) and the verification task (3 x 4 12?) evaluated direct access to multiplication knowledge, the number-matching task (3 x 4 34?, 'do 3 x 4 and 34 have the same digits?') tested indirect access. In verification and number-matching, interference from related distractors (e.g., 3 x 4 followed by 16) relative to unrelated distractors (17) reflects access to stored fact representations as well as efficiency of inhibition processes. Results indicated that, compared to young adults, MCI and healthy old adults were slower in responding across tasks. In production and verification, analyses of individual latency regression slopes and intercepts suggested that these age effects were related to differences at peripheral processing stages (e.g., encoding) rather than at the central (arithmetic retrieval) stage. Differences between MCI and healthy elderly emerged only in the number-matching task. While in verification effects were comparable between groups, in number-matching MCI patients were more susceptible to interference from irrelevant information than healthy old participants. Overall, the present findings indicate that aging has a general effect on peripheral processing speed, but not on arithmetic memory retrieval. Parietal cortico-subcortical circuits mediating arithmetic fact retrieval (Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83-120; Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219-250) thus seem to be preserved in normal aging and MCI. In contrast, MCI patients show enhanced interference in number-matching. This task-specific lack of inhibition may point to dysfunctional frontal cortico-subcortical networks in MCI.

Acalculia from a right hemisphere lesion dealing with "where" in multiplication procedures

The present study describes in detail, for the first time, a case of failure with multiplication procedures in a right hemisphere damaged patient (PN). A careful, step-by-step, error analysis made possible to show that an important portion of PN's errors could be better explained as spatial in nature and specifically related to the demands of a multi-digit multiplication. These errors can be distinguished from other types of errors, including those, expected after a right hemisphere lesion, determined by a generic inability to deal with spatial material, or from other deficits, like neglect, affecting cognitive capacities across the board. The best explanation for PN's problems is that he might have difficulties in relying on a visuo-spatial store containing a layout representation specific to multiplication. As a consequence, while knowing what, when and how to carry out the various steps, PN does not know where. What he may thus lack is a spatial schema of multiplication. Such schema is thought to help normal calculators in overcoming working memory demands of complex calculation by representing the information of where exactly each sub-step should be placed.

Parietal rTMS distorts the mental number line: Simulating ‘spatial’ neglect in healthy subjects

Patients with left-sided visuospatial neglect, typically after damage to the right parietal lobe, show a systematic bias towards larger numbers when asked to bisect a numerical interval. This has been taken as further evidence for a spatial representation of numbers, perhaps akin to a mental number line with smaller numbers represented to the left and larger numbers to the right. Previously, contralateral neglect-like symptoms in physical line bisection have been induced in healthy subjects with repetitive transcranial magnetic stimulation (rTMS) over right posterior parietal lobe. Here we used rTMS over parietal and occipital sites in healthy subjects to investigate spatial representations in a number bisection task. Subjects were asked to name the midpoint of numerical intervals without calculating. On control trials subjects' behaviour was similar to performance reported in physical line bisection experiments. Subjects underestimated the midpoint of the numerical interval. Repetitive transcranial magnetic stimulation produced representational neglect-like symptoms in number bisection when applied over right posterior parietal cortex (right PPC). Repetitive TMS over right PPC shifted the perceived midpoint of the numerical interval significantly to the right while occipital TMS had no effect on bisection performance. Our study therefore provides further evidence that subjects use spatial representations, perhaps akin to a mental number line, in basic numerical processing tasks. Furthermore, we showed that the right posterior parietal cortex is crucially involved in spatial representation of numbers.

Inferior Parietal RTMS Affects Performance in an Addition Task

Neuropsychological and neuroimaging studies strongly suggest that the inferior parietal cortex is important for calculation. However, the evidence from neuroimaging experiments for a left hemispheric dominance in calculation is not as clear as one would expect from the studies of patients. Often a concomitant activation of the homologous inferior parietal region of the right hemisphere is reported in the same tasks. The objective of this study was to replicate basic findings of acalculic patients and to investigate discrepancies between data from patients and results from neuroimaging studies in an addition task. Repetitive transcranial magnetic stimulation (rTMS) was applied over inferior parietal areas and the adjacent intraparietal sulcus (IPS) while subjects solved double-digit addition tasks. From studies of acalculic patients it was hypothesised that left hemispheric rTMS stimulation should result in longer reaction times (RTs) in the addition task. On addition trials without TMS subjects showed the classical problem size effect with longer RTs the larger the sum of the two operands. Magnetic stimulation over left inferior parietal areas disrupted performance significantly. The effect was specific to the left hemisphere stimulation. There was no increase in RTs for rTMS stimulation over the right hemisphere.

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.

Electrophysiological evidence for differential processing of numerical quantity and order in humans

It is yet unclear whether the processing of number magnitude and order rely on common or different functional processes and neural substrates. On the one hand, recent neuroimaging studies show that quantity and order coding activate the same areas in the parietal and prefrontal cortices. On the other hand, evidence from developmental and neuropsychological studies suggest dissociated mechanisms for processing quantity and order information. To clarify this issue, the present study investigated the spatio-temporal course of quantity and order coding operations using event-related potentials (ERPs). Twenty-four subjects performed a quantity task (classifying numbers as smaller or larger than 15) and an order task on the same material (classifying numbers as coming before or after 15), as well as a control order task on letters (classifying letters as coming before or after M). Behavioral results showed a classical distance effect (decreasing reaction times [RTs] with increasing distance from the standard) for all tasks. In agreement with previous electrophysiological evidence, this effect was significant on a P2 parietal component for numerical material. However, the difference between processing numbers close or far from the target appeared earlier and was larger on the left hemisphere for quantity processing, while it was delayed and bilateral for order processing. There was also a significant distance effect in all tasks on parietal sites for the following P3 component elicited by numbers, but this effect was larger on prefrontal areas for the order judgment. In conclusion, both quantity and order show similar behavioral effects, but they are associated with different spatio-temporal courses in parietal and prefrontal cortices.

Verbal mediation of number knowledge: Evidence from semantic dementia and corticobasal degeneration

Patients with corticobasal degeneration (CBD) appear to have impaired number knowledge. We examined the nature of their number deficit while we tested the hypothesis that comprehension of larger numbers depends in part on verbal mediation. We evaluated magnitude judgments and performance on number conservation measures rooted in Piagetian theory in nonaphasic patients with CBD (n=13) and patients with a fluent form of progressive aphasia known as semantic dementia (SD; n=15). We manipulated the numbers of the arrays and the visual-spatial properties of the stimuli being compared during magnitude judgments and Piagetian conservation measures. CBD patients were consistently impaired judging the magnitudes of larger numbers (4-9), while they had minimal difficulty with smaller numbers (magnitudes < or = 3). By comparison, SD patients performed all measures of number knowledge at a ceiling level regardless of number magnitude. Neither patient group was significantly impacted by manipulations of the spatial properties of the stimuli. CBD patients' impairment with larger numbers despite minimal aphasia, and SD patients' intact performance despite an aphasia, challenge the proposal that understanding larger numbers depends on verbal mediation.

Manipulation of frontal brain asymmetry by cognitive tasks

The purpose of the present study was to evaluate whether verbal fluency tasks may specifically induce relatively greater left than right hemispheric activation in the dorsolateral prefrontal cortex. The effectiveness of the manipulation was evaluated by EEG, which was recorded during performance of the verbal fluency task and during two control conditions, i.e., a baseline condition without cognitive demands, and a mental arithmetic task, respectively. The results demonstrate that the desired effect can only be achieved in individuals with good performance on the verbal fluency task. Good and poor performers do not only differ in lateral asymmetry, but also in the most affected region within the prefrontal cortex. Whereas good performers show relatively increased activation in the cortical region and hemisphere putatively most specialized for this kind of task (i.e., the left dorsolateral frontal cortex), poor performers show a marked shift of frontopolar asymmetry to the right.

A functional MRI study of simple arithmetic—a comparison between children and adults

The purpose of this study was to examine brain areas involved in simple arithmetic, and to compare these areas between adults and children. Eight children (four girls and four boys; age, 9-14 years) and eight adults (four women and four men; age, 40-49 years) were subjected to this study. Functional magnetic resonance imaging (fMRI) was performed during mental calculation of addition, subtraction, and multiplication of single digits. In each group, the left middle frontal, bilateral inferior temporal and bilateral lateral occipital cortices were activated during each task. The adult group showed activation of the right frontal cortex during addition and multiplication tasks, but the children group did not. Activation of the intraparietal cortex was observed in the adult group during each task. Although, activation patterns were slightly different among tasks, as well as between groups, only a small number of areas showed statistically significant differences. The results indicate that cortical networks involved in simple arithmetic are similar among arithmetic operations, and may not show significant changes in the structure during the second decade of life.

The Number Space and Neglect

Recent cognitive models of numerical abilities have postulated that number processing may in part rely on a representation of quantities where magnitude is organized by spatial proximity, along a "mental number line" extending from left to right. We describe four experiments that examined whether such a spatial representation of number would be affected by the presence of unilateral neglect after right hemisphere damage. When asked to judge whether a single number shown at fixation was smaller or larger than "5", patients with neglect were selectively slower to respond to "4", but when asked to compare numbers to "7" they were selectively slower to respond to "6". This is consistent with a representational deficit for numbers located to the left of a reference point along the mental number line and was not found in other right brain-damage patients without neglect. No effect of represented number position was found in a non-numerical task requiring judgements of the physical size of single digit characters. Finally, when asked to classify numbers as indicating hours earlier or later than six o'clock, neglect patients showed a reverse pattern with slower responses to numbers larger than "6', consistent with a representational deficit for hour numbers located on the left side of an imagined clock-face. Our findings demonstrate that unilateral spatial neglect may produce specific representational deficits in number processing that implicate different spatial representations according to the task demands.

The differential involvement of inferior parietal lobule in number comparison: a rTMS study

Number processing is known to involve several sites within the posterior regions of parietal cortex. Here, we investigated whether neural activity in the inferior parietal lobule (IPL) is essential for number processing, by observing the effects of interfering with its activity during the execution of a standard number comparison task. Subjects performance on the task was significantly slowed down when we delivered trains of repetitive transcranial magnetic stimuli (rTMS) to the posterior parietal scalp site overlying the left IPL, while rTMS did not affect the number comparison task if delivered to homologous, contralateral (right) IPL. In conclusion, the present findings add support to a growing body of evidence from neuropsychology and neuroimaging studies that the left inferior parietal lobule is an important component of the networks subserving the representation of quantity.

Learning complex arithmetic—an fMRI study

Aim of the present functional magnet resonance imaging (fMRI) study was to detect modifications of cerebral activation patterns related to learning arithmetic. Thirteen right-handed subjects were extensively trained on a set of 18 complex multiplication problems. In the following fMRI session, trained and untrained problems (closely matched for difficulty) were presented in blocked order alternating with a number matching task and a fact retrieval task. Importantly, left hemispheric activations were dominant in the two contrasts between untrained and trained condition, suggesting that learning processes in arithmetic are predominantly supported by the left hemisphere. Contrasting untrained versus trained condition, the left intraparietal sulcus showed significant activations, as well as the inferior parietal lobule. A further significant activation was found in the left inferior frontal gyrus. This activation may be accounted for by higher working memory demands in the untrained as compared to the trained condition. Contrasting trained versus untrained condition a significant focus of activation was found in the left angular gyrus. Following the triple-code model [Science 284 (1999) 970], the shift of activation within the parietal lobe from the intraparietal sulcus to the left angular gyrus suggests a modification from quantity-based processing to more automatic retrieval. The present study shows that the left angular gyrus is not only involved in arithmetic tasks requiring simple fact retrieval, but may show significant activations as a result of relatively short training of complex calculation.

Calculation impairment in neurodegenerative diseases

We examined oral calculation in patients with corticobasal degeneration (CBD; N=17), frontotemporal dementia (FTD; N=17), and Alzheimer's disease (AD; N=20), as well as 17 healthy seniors matched for age and education. Our calculation model involves at least three components: numerosity, combinatorial processes, and executive resources such as working memory. We assessed addition, subtraction, multiplication, and division involving small numbers (small, single-digit answers) and large numbers (larger, single- and double-digit answers). We also assessed dot counting for small numbers (2-5) and large numbers (6-9), as well as a measure of working memory. All patient groups differed from healthy seniors in oral calculation. CBD (36% correct) and FTD (65% correct) demonstrated a significant overall impairment in oral calculation relative to AD (76% correct). CBD (66% correct) had more difficulty counting dots overall relative to AD (94% correct) and FTD (86% correct), consistent with our hypothesis that the calculation deficit in CBD is due in large part to a numerosity deficit. FTD had more difficulty relative to AD in their performance of reverse digit span, consistent with our hypothesis that FTD patients' executive resource limitation contributes to their pattern of calculation impairment.

Intraoperative mapping of the cortical areas involved in multiplication and subtraction: an electrostimulation study in a patient with a left parietal glioma

OBJECTIVES

Advances in neuroimaging studies have recently improved the understanding of the functional anatomy of the calculation processes, having in particular underlined the central role of the angular gyrus (AG). In this study, the authors applied this knowledge to the surgical resection of a glioma invading the left AG, by localising and sparing the cortical areas involved in two different components of calculation (multiplication and subtraction), using direct electrical stimulations.

METHODS

A calculation mapping was performed in a patient without deficit except a slightly impaired performance for serial arithmetic subtraction, during the resection under local anaesthesia of a left parieto-occipital glioma invading the dominant AG. After somatosensory and language mappings, cortical areas involved in single digit multiplications and subtractions of seven were mapped using the method of electrostimulation, before glioma removal.

RESULTS

Distinct sites specifically involved in multiplication or subtraction were detected within the left AG, with a precise spatial distribution and overlapping. All the eloquent (somatosensory, language, and calculation) areas were surgically spared. Postoperatively, the patient had a transient complete deficit for arithmetic subtraction, without either multiplication or language disturbance. The tumour removal was complete.

CONCLUSIONS

These findings suggest: firstly, the usefulness of an intraoperative calculation mapping during the removal of a lesion involving the left dominant AG, to avoid permanent postoperative deficit of arithmetic processes while optimising the quality of tumour resection; secondly, the possible existence of a well ordered and dynamic anatomo-functional organisation for different components of calculation within the left AG.

Acalculia and dyscalculia

Even though it is generally recognized that calculation ability represents a most important type of cognition, there is a significant paucity in the study of acalculia. In this paper the historical evolution of calculation abilities in humankind and the appearance of numerical concepts in child development are reviewed. Developmental calculation disturbances (developmental dyscalculia) are analyzed. It is proposed that calculation ability represents a multifactor skill, including verbal, spatial, memory, body knowledge, and executive function abilities. A general distinction between primary and secondary acalculias is presented, and different types of acquired calculation disturbances are analyzed. The association between acalculia and aphasia, apraxia and dementia is further considered, and special mention to the so-called Gerstmann syndrome is made. A model for the neuropsychological assessment of numerical abilities is proposed, and some general guidelines for the rehabilitation of calculation disturbances are presented.

The selective impairment of arithmetical procedures

The theoretical distinction between arithmetic facts and procedures was first made by Groen and Parkman (1972). This was confirmed with a neuropsychological single case described by Warrington (1982) who had impaired arithmetical facts but well preserved arithmetical procedures. Since this time there have been several patients described who showed a selective impairment of arithmetic facts. There have also been reports of cases with impaired arithmetical procedures. However, there has not yet been a case reported with the selective impairment of procedures in the context of intact arithmetic facts. This paper describes a patient, SR, with probable Alzheimer's dementia who had well preserved addition, multiplication and subtraction facts but who nevertheless had severe difficulties with a range of arithmetical procedures such as multidigit sums, decimals and fractions. The implications of this case for current theoretical models are discussed.

The mental number line and the human angular gyrus

To investigate the hemispheric organization of a language-independent spatial representation of number magnitude in the human brain we applied focal repetitive transcranial magnetic stimulation (rTMS) to the right or left angular gyrus while subjects performed a number comparison task with numbers between 31 and 99. Repetitive TMS over the angular gyrus disrupted performance of a visuospatial search task, and rTMS at the same site disrupted organization of the putative "number line." In some cases the pattern of disruption caused by angular gyrus rTMS suggested that this area normally mediates a spatial representation of number. The effect of angular gyrus rTMS on the number line task was specific. rTMS had no disruptive effect when delivered over another parietal region, the supramarginal gyrus, in either the left or the right hemisphere.

Some quick arithmetic

Mathematical abilities for the four simple arithmetic operations were studied on a sample of 53 female right-handers. True and false statements were presented auditorily and a manual response was required as to the trueness of the statements. Response times, accuracy, and laterality index showed no significant ear advantage, but the responses to true statements were significantly faster than to the false ones. The shortest response times were found for addition problems, and the longest for subtractions. The correlation between the size of the difference between the two operands and response time was not conclusive but the trend was unexpectedly positive.

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.

Language and calculation within the parietal lobe: a combined cognitive, anatomical and fMRI study

We report the case of a patient (ATH) who suffered from aphasia, deep dyslexia, and acalculia, following a lesion in her left perisylvian area. She showed a severe impairment in all tasks involving numbers in a verbal format, such as reading aloud, writing to dictation, or responding verbally to questions of numerical knowledge. In contrast, her ability to manipulate non-verbal representations of numbers, i.e., Arabic numerals and quantities, was comparatively well preserved, as evidenced for instance in number comparison or number bisection tasks. This dissociated impairment of verbal and non-verbal numerical abilities entailed a differential impairment of the four arithmetic operations. ATH performed much better with subtraction and addition, that can be solved on the basis of quantity manipulation, than with multiplication and division problems, that are commonly solved by retrieving stored verbal sequences. The brain lesion affected the classical language areas, but spared a subset of the left inferior parietal lobule that was active during calculation tasks, as demonstrated with functional MRI. Finally, the relative preservation of subtraction versus multiplication may be related to the fact that subtraction activated the intact right parietal lobe, while multiplication activated predominantly left-sided areas.

Neuroanatomical substrates of arabic number processing, numerical comparison, and simple addition: a PET study

Positron emission tomography was used to localize the cerebral networks specifically involved in three basic numerical processes: arabic numeral processing, numerical magnitude comparison, and retrieval of simple addition facts. Relative cerebral blood flow changes were measured while normal volunteers were resting with eyes closed, making physical judgment on nonnumerical characters or arabic digits, comparing, or adding the same digits. Processing arabic digits bilaterally produced a large nonspecific activation of occipito-parietal areas, as well as a specific activation of the right anterior insula. Comparison and simple addition fact retrieval revealed a fronto-parietal network involving mainly the left intraparietal sulcus, the superior parietal lobule and the precentral gyrus. Comparison also activated, but to a lesser extent, the right superior parietal lobe, whereas addition also activated the orbito-frontal areas and the anterior insula in the right hemisphere. Implications for current anatomo-functional models of numerical cognition are drawn.

The calculating brain: an fMRI study

To explore brain areas involved in basic numerical computation, functional magnetic imaging (fMRI) scanning was performed on college students during performance of three tasks; simple arithmetic, numerical magnitude judgment, and a perceptual-motor control task. For the arithmetic relative to the other tasks, results for all eight subjects revealed bilateral activation in Brodmann's area 44, in dorsolateral prefrontal cortex (areas 9 and 10), in inferior and superior parietal areas, and in lingual and fusiform gyri. Activation was stronger on the left for all subjects, but only at Brodmann's area 44 and the parietal cortices. No activation was observed in the arithmetic task in several other areas previously implicated for arithmetic, including the angular and supramarginal gyri and the basal ganglia. In fact, angular and supramarginal gyri were significantly deactivated by the verification task relative to both the magnitude judgment and control tasks for every subject. Areas activated by the magnitude task relative to the control were more variable, but in five subjects included bilateral inferior parietal cortex. These results confirm some existing hypotheses regarding the neural basis of numerical processes, invite revision of others, and suggest productive lines for future investigation.

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.