Selective acalculia with sparing of the subtraction process in a patient with left parietotemporal hemorrhage

Probing Our Built-in Calculator: A Systematic Narrative Review of Noninvasive Brain Stimulation Studies on Arithmetic Operation-Related Brain Areas

This systematic review presented a comprehensive survey of studies that applied transcranial magnetic stimulation and transcranial electrical stimulation to parietal and nonparietal areas to examine the neural basis of symbolic arithmetic processing. All findings were compiled with regard to the three assumptions of the triple-code model (TCM) of number processing. Thirty-seven eligible manuscripts were identified for review (33 with healthy participants and 4 with patients). Their results are broadly consistent with the first assumption of the TCM that intraparietal sulcus both hold a magnitude code and engage in operations requiring numerical manipulations such as subtraction. However, largely heterogeneous results conflicted with the second assumption of the TCM that the left angular gyrus subserves arithmetic fact retrieval, such as the retrieval of rote-learned multiplication results. Support is also limited for the third assumption of the TCM, namely, that the posterior superior parietal lobule engages in spatial operations on the mental number line. Furthermore, results from the stimulation of brain areas outside of those postulated by the TCM show that the bilateral supramarginal gyrus is involved in online calculation and retrieval, the left temporal cortex in retrieval, and the bilateral dorsolateral prefrontal cortex and cerebellum in online calculation of cognitively demanding arithmetic problems. The overall results indicate that multiple cortical areas subserve arithmetic skills.

Dissociating Arithmetic Operations in the Parietal Cortex Using 1 Hz Repetitive Transcranial Magnetic Stimulation: The Importance of Strategy Use

The triple-code model (TCM) of number processing suggests the involvement of distinct parietal cortex areas in arithmetic operations: the bilateral horizontal segment of the intraparietal sulcus (hIPS) for arithmetic operations that require the manipulation of numerical quantities (e.g., subtraction) and the left angular gyrus (AG) for arithmetic operations that require the retrieval of answers from long-term memory (e.g., multiplication). Although neuropsychological, neuroimaging, and brain stimulation studies suggest the dissociation of these operations into distinct parietal cortex areas, the role of strategy (online calculation vs. retrieval) is not yet fully established. In the present study, we further explored the causal involvement of the left AG for multiplication and left hIPS for subtraction using a neuronavigated repetitive transcranial magnetic stimulation (rTMS) paradigm. Stimulation sites were determined based on an fMRI experiment using the same tasks. To account for the effect of strategy, participants were asked whether they used retrieval or calculation for each individual problem. We predicted that the stimulation of the left AG would selectively disrupt the retrieval of the solution to multiplication problems. On the other hand, stimulation of the left hIPS should selectively disrupt subtraction. Our results revealed that left AG stimulation was detrimental to the retrieval and online calculation of solutions for multiplication problems, as well as, the retrieval (but not online calculation) of the solutions to subtraction problems. In contrast, left hIPS stimulation had no detrimental effect on both operations regardless of strategy.

Language-specific numerical estimation in bilingual children

We tested 5- to 7-year-old bilingual learners of French and English (N = 91) to investigate how language-specific knowledge of verbal numerals affects numerical estimation. Participants made verbal estimates for rapidly presented random dot arrays in each of their two languages. Estimation accuracy differed across children's two languages, an effect that remained when controlling for children's familiarity with number words across their two languages. In addition, children's estimates were equivalently well ordered in their two languages, suggesting that differences in accuracy were due to how children represented the relative distance between number words in each language. Overall, these results suggest that bilingual children have different mappings between their verbal and nonverbal counting systems across their two languages and that those differences in mappings are likely driven by an asymmetry in their knowledge of the structure of the count list across their languages. Implications for bilingual math education are discussed.

Non-invasive mapping of calculation function by repetitive navigated transcranial magnetic stimulation

Concerning calculation function, studies have already reported on localizing computational function in patients and volunteers by functional magnetic resonance imaging and transcranial magnetic stimulation. However, the development of accurate repetitive navigated TMS (rTMS) with a considerably higher spatial resolution opens a new field in cognitive neuroscience. This study was therefore designed to evaluate the feasibility of rTMS for locating cortical calculation function in healthy volunteers, and to establish this technique for future scientific applications as well as preoperative mapping in brain tumor patients. Twenty healthy subjects underwent rTMS calculation mapping using 5 Hz/10 pulses. Fifty-two previously determined cortical spots of the whole hemispheres were stimulated on both sides. The subjects were instructed to perform the calculation task composed of 80 simple arithmetic operations while rTMS pulses were applied. The highest error rate (80 %) for all errors of all subjects was observed in the right ventral precentral gyrus. Concerning division task, a 45 % error rate was achieved in the left middle frontal gyrus. The subtraction task showed its highest error rate (40 %) in the right angular gyrus (anG). In the addition task a 35 % error rate was observed in the left anterior superior temporal gyrus. Lastly, the multiplication task induced a maximum error rate of 30 % in the left anG. rTMS seems feasible as a way to locate cortical calculation function. Besides language function, the cortical localizations are well in accordance with the current literature for other modalities or lesion studies.

Task-Specific Facilitation of Cognition by Anodal Transcranial Direct Current Stimulation of the Prefrontal Cortex

We previously speculated that depression of cerebellar excitability using cathodal transcranial direct current stimulation (tDCS) might release extra cognitive resources via the disinhibition of activity in prefrontal cortex. The objective of the present study was to investigate whether anodal tDCS over the prefrontal cortex could similarly improve performance when cognitive demands are high. Sixty-three right-handed participants in 3 separate groups performed the Paced Auditory Serial Addition Task (PASAT) and the more difficult Paced Auditory Serial Subtraction Task (PASST), before and after 20 min of anodal, cathodal, or sham stimulation over the left dorsolateral prefrontal cortex (DLPFC). Performance was assessed in terms of the accuracy, latency, and variability of correct verbal responses. All behavioral measures significantly improved for the PASST after anodal DLPFC stimulation, but not the PASAT. There were smaller practice effects after cathodal and sham stimulation. Subjective ratings of attention and mental fatigue were unchanged by tDCS over time. We conclude that anodal stimulation over the left DLPFC can selectively improve performance on a difficult cognitive task involving arithmetic processing, verbal working memory, and attention. This result might be achieved by focally improving executive functions and/or cognitive capacity when tasks are difficult, rather than by improving levels of arousal/alertness.

The neural bases of the multiplication problem-size effect across countries

Multiplication problems involving large numbers (e.g., 9 × 8) are more difficult to solve than problems involving small numbers (e.g., 2 × 3). Behavioral research indicates that this problem-size effect might be due to different factors across countries and educational systems. However, there is no neuroimaging evidence supporting this hypothesis. Here, we compared the neural correlates of the multiplication problem-size effect in adults educated in China and the United States. We found a greater neural problem-size effect in Chinese than American participants in bilateral superior temporal regions associated with phonological processing. However, we found a greater neural problem-size effect in American than Chinese participants in right intra-parietal sulcus (IPS) associated with calculation procedures. Therefore, while the multiplication problem-size effect might be a verbal retrieval effect in Chinese as compared to American participants, it may instead stem from the use of calculation procedures in American as compared to Chinese participants. Our results indicate that differences in educational practices might affect the neural bases of symbolic arithmetic.

Non-symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

When solving 22-7 is much more difficult than 99-12

We describe the case of a 69-year-old professor of mathematics (GV) who was examined 2 years after left-hemispheric capsular-thalamic haemorrhage. GV showed disproportionate impairment in subtractions requiring borrowing (22 - 7). For large subtraction problems without borrowing (99 - 12) performance was almost flawless. Subtractions with borrowing mostly relied on inadequate attempts to invert subtractions into the corresponding additions (22 - 7 = x as 7 + x = 22). The hypothesis is advanced that difficulty in the inhibitory components of attention tasks (Stroop test, go-no-go task) might be the responsible factor of his calculation impairment. A deficit in subtractions with borrowing might be related to left-hemispheric damage involving thalamo-cortical connections.

Dissociations in numerical abilities revealed by progressive cognitive decline in a patient with semantic dementia

This study describes a 3-year follow-up investigation of the deterioration of number abilities in a semantic dementia patient (IH). A few studies have previously reported the decline of number knowledge in patients with degenerative disorders, although almost never in semantic dementia (Diesfeldt, 1993; Girelli, Luzzatti, Annoni, & Vecchi, 1999; Grafman, Kempen, Rosenberg, Salazar, & Boller, 1989). These studies described the change of the patients' performance mainly in terms of increased errors in number tasks. On the other hand, dissociations between different types of number abilities, or different arithmetical operations, have been reported in patients with focal lesions. In the present investigation, the cognitive basis of number processing was revealed throughout the patient's cognitive decline. Two major results emerged from a longitudinal study: First, the patient's conceptual knowledge of arithmetic was well preserved despite severe impairment of nonarithmetic conceptual knowledge. Second, the patient's progressive decline revealed patterns of dissociations between different number abilities. These were between (1) multiplication and other arithmetical operations, which particularly emerged in the use of algorithms; (2) impaired knowledge of number facts and procedures on one hand, and conceptual knowledge of arithmetic on the other; and (3) different types of transcoding skills. The implications of these dissociations for the cognitive architecture of number processing are discussed.

Three parietal circuits for number processing

Did evolution endow the human brain with a predisposition to represent and acquire knowledge about numbers? Although the parietal lobe has been suggested as a potential substrate for a domain-specific representation of quantities, it is also engaged in verbal, spatial, and attentional functions that may contribute to calculation. To clarify the organisation of number-related processes in the parietal lobe, we examine the three-dimensional intersection of fMRI activations during various numerical tasks, and also review the corresponding neuropsychological evidence. On this basis, we propose a tentative tripartite organisation. The horizontal segment of the intraparietal sulcus (HIPS) appears as a plausible candidate for domain specificity: It is systematically activated whenever numbers are manipulated, independently of number notation, and with increasing activation as the task puts greater emphasis on quantity processing. Depending on task demands, we speculate that this core quantity system, analogous to an internal "number line," can be supplemented by two other circuits. A left angular gyrus area, in connection with other left-hemispheric perisylvian areas, supports the manipulation of numbers in verbal form. Finally, a bilateral posterior superior parietal system supports attentional orientation on the mental number line, just like on any other spatial dimension.

Common substrate for mental arithmetic and finger representation in the parietal cortex

The history of mathematics provides several examples of the use of fingers to count or calculate. These observations converge with developmental data showing that fingers play a critical role in the acquisition of arithmetic knowledge. Further studies evidenced specific interference of finger movements with arithmetic problem solving in adults, raising the question of whether or not finger and number manipulations rely on common brain areas. In the present study, functional magnetic resonance imaging (fMRI) was used to investigate the possible overlap between the brain areas involved in mental arithmetic and those involved in finger discrimination. Solving subtraction and multiplication problems was found to increase cerebral activation bilaterally in the horizontal part of the intraparietal sulcus (hIPS) and in the posterior part of the superior parietal lobule (PSPL). Finger discrimination was associated with increased activity in a bilateral occipito-parieto-precentral network extending from the extrastriate body area to the primary somatosensory and motor cortices. A conjunction analysis showed common areas for mental arithmetic and finger representation in the hIPS and PSPL bilaterally. Voxelwise correlations further showed that finger discrimination and mental arithmetic induced a similar pattern of activity within the parietal areas only. Pattern similarity was more important for the left than for the right hIPS and for subtraction than for multiplication. These findings provide the first evidence that the brain circuits involved in finger representation also underlie arithmetic operations in adults.

Early Numerical Competencies of Students with Different Forms of Mathematics Difficulty

The purpose of the study was to examine differences in early numerical competencies, as well as subtraction skill, as a function of children's mathematics difficulty (MD) status: computational difficulty (CD), word problem-solving difficulty (PD), concurrent difficulty (CDPD), or neither difficulty (i.e., typically developing; TYP). Based on measures of addition and word-problem skill, second grade students (N = 332) were classified in terms of MD status. Then, students were assessed on three early numerical competency measures (Number Line Estimation, Number Sets Test, and Counting Knowledge) as well as a measure of subtraction. On Number Line Estimation and Number Sets Test, students with CD and those with PD scored comparably, but both outperformed students with CDPD. On Counting Knowledge-Double First, students with CDPD scored lower than the three contrasting groups. On subtraction, students with CD outperformed those with PD, and students with PD and those with CDPD performed comparably. Findings are discussed in terms of differential performance as a function of difficulty status and implications for understanding and teaching subtypes of MD.

Role of distinct parietal areas in arithmetic: an fMRI-guided TMS study

Although several parietal areas are known to be involved in number processing, their possible role in arithmetic operations remains debated. It has been hypothesized that the horizontal segment of the intraparietal sulcus (hIPS) and the posterior superior parietal lobule (PSPL) contribute to operations solved by calculation procedures, such as subtraction, but whether these areas are also involved in operations solved by memory retrieval, such as multiplication, is controversial. In the present study, we first identified the parietal areas involved in subtraction and multiplication by means of functional magnetic resonance imaging (fMRI) and we found an increased activation, bilaterally, in the hIPS and PSPL during both arithmetic operations. In order to test whether these areas are causally involved in subtraction and multiplication, we used transcranial magnetic stimulation (TMS) to create, in each participant, a virtual lesion of either the hIPS or PSPL, over the sites corresponding to the peaks of activation gathered in fMRI. When compared to a control site, we found an increase in response latencies in both operations after a virtual lesion of either the left or right hIPS, but not of the PSPL. Moreover, TMS over the hIPS increased the error rate in the multiplication task. The present results indicate that even operations solved by memory retrieval, such as multiplication, rely on the hIPS. In contrast, the PSPL seems to underlie processes that are nonessential to solve basic subtraction and multiplication problems.

Core multiplication in childhood

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.

The counting function and its representation in the parietal cortex in humans and animals

Current data provide evidence that the ability to assess numbers is present not only in adult humans, but also in animals and children of preverbal age. Studies of behavior in infants and animals have demonstrated that the perception of number, the discrimination of quantities, and elementary addition and subtraction appear during onto- and phylogenesis before the appearance of speech. Number perception in humans and animals has common features: the greater the difference between numbers, the easier they are to discriminate; for a given difference between numbers, increases in size lead to increased difficulty in discrimination. Clinical data on counting impairments in patients and functional tomography studies of number operations in healthy subjects have shown that the key structures involved in number perception in humans are located in the parietal cortex. As demonstrated by experiments on monkeys and dogs, recognition of number in these species is also associated with the parietal area of the cortex. The similarity of the morphofunctional bases of "counting behavior" in humans and animals suggests that counting can be regarded as a functional mechanism of adaptive behavior which formed during evolution.

Cortical calculation localization using electrostimulation

Object

A naming task has been used to spare cortical areas involved in language. In the present study, a calculation task was combined with electrostimulation mapping (awake surgery) to spare cortical areas involved in calculation in patients undergoing surgery for brain lesions. The organization of language and calculation areas was analyzed in relation to these surgical data.

Methods

Twenty patients with brain lesions close to areas possibly involved in calculation (dominant parietal lobe and F2) were prospectively studied over a 4-year period. Four patients had preoperative symptoms of acalculia and therefore were not included in the brain mapping procedure.

Results

In 16 patients, direct electrostimulation caused calculation interferences in localized small cortical areas (< 2 cm2). Of the 53 calculation interferences found, 23 were independent of language areas, especially those in the inferior left parietal lobule. Various patterns of interference were observed (11 complete acalculia, 5 acalculia with wrong answers, 2 hesitations, and 5 mixed responses), although error patterns were fairly similar across angular, parietal, and frontal stimulation sites. Calculation areas in 4 patients could not be spared for oncological reasons; postoperatively, 3 of these patients showed significant acalculia symptoms. In contrast, none of the patients whose calculation areas were spared had arithmetic difficulties 1 month after surgery. Improvements in acalculia symptoms after surgery were also found in 3 of the 4 patients with preoperative calculation difficulties.

Conclusions

To limit the risk of personal and professional disturbances caused by acquired anarithmetia in patients undergoing surgery for brain tumors or epilepsy, the authors think it is necessary to use a calculation task during brain mapping, especially when operating in the dominant parietal lobe.

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.

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.

Safe Removal of Glioblastoma Near the Angular Gyrus by Awake Surgery Preserving Calculation Ability-Case Report-

A 67-year-old patient presented with progressive agraphia, alexia, and impaired ability to calculate persisting for 4 weeks. He showed preserved ability to do single-digit addition and subtraction. Magnetic resonance imaging demonstrated a tumor in the left parietal lobe. A malignant glioma was suspected, and awake craniotomy was performed to remove the tumor with functional cortical mapping to determine the cortices involved in calculation and language. His calculation ability was mapped on the angular gyrus, and partial resection of the tumor was achieved without deterioration of that ability. The histological diagnosis was glioblastoma multiforme. The patient's calculation ability improved dramatically after the operation.

Learning by strategies and learning by drill—evidence from an fMRI study

The present fMRI study investigates, first, whether learning new arithmetic operations is reflected by changing cerebral activation patterns, and second, whether different learning methods lead to differential modifications of brain activation. In a controlled design, subjects were trained over a week on two new complex arithmetic operations, one operation trained by the application of back-up strategies, i.e., a sequence of arithmetic operations, the other by drill, i.e., by learning the association between the operands and the result. In the following fMRI session, new untrained items, items trained by strategy and items trained by drill, were assessed using an event-related design. Untrained items as compared to trained showed large bilateral parietal activations, with the focus of activation along the right intraparietal sulcus. Further foci of activation were found in both inferior frontal gyri. The reverse contrast, trained vs. untrained, showed a more focused activation pattern with activation in both angular gyri. As suggested by the specific activation patterns, newly acquired expertise was implemented in previously existing networks of arithmetic processing and memory. Comparisons between drill and strategy conditions suggest that successful retrieval was associated with different brain activation patterns reflecting the underlying learning methods. While the drill condition more strongly activated medial parietal regions extending to the left angular gyrus, the strategy condition was associated to the activation of the precuneus which may be accounted for by visual imagery in memory retrieval.

Effet de la grandeur décimale sur la lecture de chiffres dans les gliomes du carrefour gauche

An increasing number of studies are focusing on the anatomo-functional organisation of number processing and some cognitive models have been recently developed. Nevertheless, relationships between areas implicated in number processing, and language areas and circuits remain unclear. Recently, Dehaene and Cohen, in their "triple-code model of number processing", (Dehaene and Cohen, 1995) distinguished two alternative number representation and processing systems: one depending on verbal processes, the other representing a quantity manipulation. According to this model, the retrieval of "arithmetical facts" (AF), learned by rote at school and memorised in a verbal form (such as the multiplication table or simple addition problems) can be considered as a verbal automatism; conversely, subtraction problems, which require mental manipulation of the quantities, represent an abstract, semantic elaboration: "Actual Calculation" (AC). The anatomical correlate of the retrieval of AF (depending on automatic verbal associations) seems to correspond to the left-hemispheric perisylvian areas, while impairment of the actual calculation (AC) depends on the intraparietal region, particularly in the left dominant hemisphere. The present study describes the neuropsychological assessment of three patients, tested after surgery for left parieto-occipital tumors. Two of them were affected by an anaplasic glioma, the third by a low-grade glioma. The cognitive evaluation included: words of Rey, numeral (directed and reversed) span, reading of "simple" numbers (from 1 to 10) and of "complex" numbers (many decimals), writing (dictation) and reading a standard text, finger denomination and right-left distinction. All patients showed language disturbances, dysgraphia and severe dyslexia. In reading numbers, we identified two types of errors: lexical and syntactic. "Lexical errors" consisted in a wrong choice among words in the number's lexicon. For instance, all patients made errors in reading "complex" numbers composed by many decimals, switching single numbers but respecting the decimal size and the structure of the whole number (such as 69107 instead of 68107). On the other hand, only one patient committed syntactic errors, misunderstanding the decimal size and the structure of the number. We considered lexical errors as verbal errors, and syntactic errors as semantic errors, affecting the notion of quantity. We tried to explain verbal disturbances as well as lexical errors as a consequence of lesion of the left-hemispheric perisylvian areas, while syntactic errors as a consequence of impairment of the intraparietal region.

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.

Does the Left Inferior Parietal Lobule Contribute to Multiplication Facts?

We report a single case, who presents with a selective and severe impairment for multiplication and division facts. His ability to retrieve subtraction and addition facts was entirely normal. His brain lesion affected the left superior temporal and to lesser extent in the left middle temporal gyri and the left precentral gyrus extending inferiorly to the pars opercularis of the left frontal lobe. Interestingly, the left supramarginal and angular gyri (SMG/AG) were spared. This finding realised a double dissociation with a previously reported patient, who despite lesions in the SMG/AG did not have a multiplication impairment (van Harskamp et al., 2002). The previously suggested crucial role of the SMG/AG in the retrieval of simple multiplication facts is therefore poorly supported (Cohen et al., 2000; Lee, 2000).

Spontaneous recovery from acalculia

A topic much considered in research on acalculia was its relationship with aphasia. Far less attention has been given to the natural course of acalculia. In this retrospective study, we examined the relationship between aphasia and acalculia in an unselected series of 98 left-brain-damaged patients and the spontaneous recovery from acalculia in 92 acalculic patients with follow-up. There was a significant association between aphasia and acalculia although 19 participants exhibited aphasia with no acalculia and six acalculia with no aphasia. We observed significant improvement between a first examination carried out between 1 and 5 months post-onset and a second examination carried out between 3 and 11 months later (mean: 5 months). The mechanisms of spontaneous recovery are discussed.

Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group Independent Component Analysis of the mental addition and subtraction of fractions

The suitability of a previously hypothesized triple-code model of numerical processing, involving analog magnitude, auditory verbal, and visual Arabic codes of representation, was investigated for the complex mathematical task of the mental addition and subtraction of fractions. Functional magnetic resonance imaging (fMRI) data from 15 normal adult subjects were processed using exploratory group Independent Component Analysis (ICA). Separate task-related components were found with activation in bilateral inferior parietal, left perisylvian, and ventral occipitotemporal areas. These results support the hypothesized triple-code model corresponding to the activated regions found in the individual components and indicate that the triple-code model may be a suitable framework for analyzing the neuropsychological bases of the performance of complex mathematical tasks.

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 residual calculation abilities of a patient with severe aphasia: evidence for a selective deficit of subtraction procedures

We report the case study of a severe fluent aphasic patient, who showed relatively preserved numerical abilities. A detailed investigation of number processing indicated good numerical comprehension and a relative sparing of addition and subtraction abilities; on the other hand, multiplication and division were severely impaired. A further study of multi-digit operations showed that the patient's performance was characterized by a selective impairment of the borrowing procedure, in which she applied the so-called Smaller-from-Larger bug, typically observed in children learning to calculate. The present case provides further evidence for the dissociation between operations based on verbal sequences and on quantity manipulation, respectively impaired and preserved in patients with severe aphasia. Moreover, it provides evidence indicating that procedures may be dissociated from conceptual knowledge within a single arithmetical operation.

Approximate quantities and exact number words: dissociable systems

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.

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.

Spared numerical abilities in a case of semantic dementia

We report a case study of a patient (IH) with a progressive impairment of semantic memory affecting all categories of knowledge apart from numbers. Pictorial material was better understood than words, but was still severely impaired. The selective preservation of nearly all aspects of numerical knowledge suggested that this domain might have different neuropsychological status from other aspects of semantic memory.

Selective impairments for addition, subtraction and multiplication. implications for the organisation of arithmetical facts

This study reports for the first time a selective impairment for simple addition in patient FS. Moreover, patient VP presented with a selective impairment for simple multiplication and patient DT with a selective impairment for simple subtraction. These findings are discussed in the context of two of the most influential models for the organisation of arithmetical facts in memory (Dehaene and Cohen, 1995, 1997, and Dagenbach and McCloskey, 1992). Dehaene and Cohen (1995, 1997) have proposed that dissociation between arithmetical facts result from a selective impairment to two different types of processing: rote verbal memory for multiplication and simple addition vs. quantity processing for subtraction and division. Dagenbach and McCloskey (1992) suggest dissociation between arithmetical facts result from a selective damage to segregated memory networks specific for each operation. We will argue that our findings are problematic for Dehaene's model and in good accord with McCloskey's view.

Understanding dissociations in dyscalculia: a brain imaging study of the impact of number size on the cerebral networks for exact and approximate calculation

Neuropsychological studies have revealed different subtypes of dyscalculia, including dissociations between exact calculation and approximation abilities, and an impact of number size on performance. To understand the origins of these effects, we measured cerebral activity with functional MRI at 3 Tesla and event-related potentials while healthy volunteers performed exact and approximate calculation tasks with small and large numbers. Bilateral intraparietal, precentral, dorsolateral and superior prefrontal regions showed greater activation during approximation, while the left inferior prefrontal cortex and the bilateral angular regions were more activated during exact calculation. Increasing number size during exact calculation led to increased activation in the same bilateral intraparietal regions as during approximation, as well the left inferior and superior frontal gyri. Event-related potentials gave access to the temporal dynamics of calculation processes, showing that effects of task and of number size could be found as early as 200-300 ms following problem presentation. Altogether, the results reveal two cerebral networks for number processing. Rote arithmetic operations with small numbers have a greater reliance on left-lateralized regions, presumably encoding numbers in verbal format. Approximation and exact calculation with large numbers, however, put heavier emphasis on the left and right parietal cortices, which may encode numbers in a non-verbal quantity format. Subtypes of dyscalculia can be explained by lesions disproportionately affecting only one of these networks.

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.

Noncommutability of the N + 0 arithmetical rule: a case study of dissociated impairment

It has been shown that some arithmetical problems are stored in the form of individual facts representations (e.g., 3 x 4 = 12) whereas others are solved by general stored rules (e.g., 0 x N = 0). We describe the performance of a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Although her performance was almost perfect for rule-based problems in all arithmetical operations, she was severely impaired for 0 + n problems in contrast with her relatively good performance for the corresponding n + 0 problems. This dissociation extends to addition the noncommutability of arithmetical rules described in multiplication.

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.

Differential contributions of the left and right inferior parietal lobules to number processing

We measured cerebral activation with functional magnetic resonance imaging at 3 Tesla while eight healthy volunteers performed various number processing tasks known to be dissociable in brain-lesioned patients: naming, comparing, multiplying, or subtracting single digits. The results revealed the activation of a circuit comprising bilateral intraparietal, prefrontal, and anterior cingulate components. The extension and lateralization of this circuit was modulated by task demands. The intraparietal and prefrontal activation was more important in the right hemisphere during the comparison task and in the left hemisphere during the multiplication task and was intensely bilateral during the subtraction task. Thus, partially distinct cerebral circuits with the dorsal parietal pathway underlie distinct arithmetic operations.

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.

Arithmetic facts without meaning

The paper presents a single-case study of patient J.G. showing severe calculation problems (and also agraphia, finger agnosia, right-left disorientation and apraxia) after the surgery of a left parietal tumor. Although the patient completely lost conceptual knowledge of arithmetic, she preserved part of memorised fact knowledge (multiplications and some additions and subtractions). The study indicates that arithmetic facts can be represented at a superficial level without understanding of the operation performed. The study completes a valid double dissociation between two types of knowledge involved in calculation (memorised facts and conceptual knowledge).

A deficit for arithmetical procedures: lack of knowledge or lack of monitoring?

A patient is described with a specific deficit for arithmetical procedures. Unlike in previously described cases, where the observed problems could be attributed to the systematic application of disturbed algorithms, this patient's difficulty seems to stem from an inability to monitor the sequence of operations that calculation procedures specify. Criteria are provided for distinguishing impairments in written calculation due to the application of defective knowledge of the procedures from those determined by lack of monitoring. The role of monitoring and control processes in different calculation components is also discussed.

Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic

We describe two acalculic patients, one with a left subcortical lesion and the other with a right inferior parietal lesion and Gerstmann's syndrome. Both suffered from "pure anarithmetia": they could read arabic numerals and write them to dictation, but experienced a pronounced calculation deficit. On closer analysis, however, distinct deficits were found. The subcortical case suffered from a selective deficit of rote verbal knowledge, including but not limited to arithmetic tables, while her semantic knowledge of numerical quantities was intact. Conversely the inferior parietal case suffered from a category-specific impairment of quantitative numerical knowledge, particularly salient in subtraction and number bissection tasks, with preserved knowledge of rote arithmetic facts. This double dissociation suggests that numerical knowledge is processed in different formats within distinct cerebral pathways. We suggest that a left subcortical network contributes to the storage and retrieval of rote verbal arithmetic facts, while a bilateral inferior parietal network is dedicated to the mental manipulation of numerical quantities.

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.

Subtraction bugs in an acalculic patient

We report a patient, MT, who presented a specific, though not isolated, deficit in written calculation. Despite a preserved knowledge of simple arithmetic - single-digit addition and subtraction - he failed systematically in multi-digit subtraction. The nature of errors was consistent across problems and reflected the application of a disturbed underlying algorithm. Moreover, the pattern of error observed mimies a very common finding in developmental studies on arithmetical procedure acquisition (Fuson, 1990, 1992, Young and O'Shea, 1981; VanLehn, 1986, 1990). The data suggest that, within calculation skills, syntax may exist as a system of stable, but inappropriate, rules which are independent of any underlying conceptual knowledge.

Selective impairment for simple division

We report the case of a patient, C.B., with a calculation impairment. His performance on simple division problems was severely impaired while his performance on simple and complex addition, subtraction and multiplication problems was largely spared. The patient was also found to be impaired in solving simple arithmetical problems such as Nx? = M. The implications of these findings for the organization of arithmetical facts are discussed.