## Abstract

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’.

## 1. Introduction

According to traditional wisdom, calculation is, in right-handers, a left hemisphere function, with a crucial role for the parietal lobe. ‘Primary acalculia’ was in fact, as described below, mostly observed in parietal lesions to the left hemisphere; ‘secondary acalculia’ may instead be a consequence of either language or visuo-spatial disorders following damage to the left or the right hemisphere, respectively. Maths was indeed believed to be represented and processed on the same side as language, even if the two functions are largely independent of each other (this specific issue has been addressed previously by clinical studies [1–3], and via neuroimaging [4]). Noticeably, when language is represented on the right, as shown in the very infrequent case of right-handed people with aphasia following a right hemisphere lesion (a condition known as ‘crossed aphasia’), calculation too seems to be represented on the right [5], although less prominently than in people with left hemispheric specialization for language [1]. The right hemisphere contribution to calculation was for a long time believed to be negligible, as concluded, for example, on the basis of observations on split-brain patients [6]. A role for the right hemisphere, if any, was recognized in supporting complex (i.e. multi-digit^{1}) calculation, whenever a spatial arrangement of a given written operation is required (see [7] for a review).

The dramatic increment in investigations about mathematical cognition in the past three decades has contributed to shaping a much more complicated picture than that held by traditional beliefs, however, by distinguishing several different aspects of mathematical functions. Moreover, the advent of neuroimaging and brain stimulation techniques (e.g. transcranial magnetic stimulation (TMS), intraoperative direct cortical electrostimulation (DCE)) has suggested that the neurological underpinnings of mathematical functions are much more extended than was previously believed. The implications of such findings with respect to lateralization have been so far only sparsely discussed. In the more restricted realm of calculation, it may be safe to say that the attention of investigators was focused on understanding where each operation is carried out rather than on how a calculation is performed in each hemisphere.

The present article does not intend to exhaustingly review the new literature on the neuroscience of calculation. It rather aims at reporting and discussing some critical findings in the attempt to answer a number of fundamental questions about the lateralization of calculation processes in the brain. These questions concern the specific roles played by each of the two hemispheres with a particular focus on the specific role of the right hemisphere.

Lesion and inhibition (i.e. TMS and DCE) studies, in fact, tell us that a unilateral lesion is sufficient to disrupt some aspects of calculation. Surprisingly, for example, even simple calculation involving arithmetical facts (i.e. like multiplication tables which are, in learned calculators, believed to be retrieved from long-term memory) may be disturbed by a unilateral lesion of either the left or the right hemisphere. Why is it that lesions to one of the two hemispheres produce apparently similar errors? One possibility is that the two hemispheres do the same thing. If that were the case, however, one should ask why is the spared hemisphere not sufficient to provide the solution? One answer that perhaps has been taken for granted and kept implicit, is that difficulties with arithmetical facts are always the result, in the case of lesions in the non-dominant hemisphere, of a generic defect to be expected after brain damage when the increment in processing demands exceeds the capacity of a suffering system. This explanation is plausible and, indeed, may account for some findings. Thus, if traditional wisdom were to be taken seriously, errors with arithmetical facts after lesions of the dominant hemisphere would be due to a disturbance to devoted retrieval mechanisms or stored representations [8]. By contrast, a lesion in the contralateral, non-dominant hemisphere could provoke in some cases a generic processing shortcoming that would affect any relatively demanding cognitive task, leading also to problems with simple (i.e. one-digit) calculation. This explanation lacks empirical support, however. In turn, a demonstration of the fact that the processes in each of the two hemispheres are indeed specific is also missing.

Neuroimaging studies, again in the instance of arithmetical facts, seem to invariably evidence bilateral activation, most often not entirely symmetrical and, crucially, in some cases superior on the right hemisphere [9]. By a logic analogous, although not entirely overlapping, to that mentioned above for lesion studies: why does the right hemisphere activate at all in a task (i.e. solving simple multiplications) that is believed to concern primarily the left hemisphere [10]? How can findings that show superior right hemisphere activation be interpreted? What does the right hemisphere do? More precisely, is the role of the right hemisphere specific? If this is the case, in what does the action of the right hemisphere consist?

These and germane questions stem from a survey of the literature on clinical studies and, in more recent years, from neuroimaging and reversible inactivation studies focusing on the specific role for the right hemisphere. Very recent findings will be specifically discussed that start providing a coherent answer. Suggestions for future research will be discussed.

## 2. The right hemisphere localization of calculation processes

One question that is poorly understood in traditional studies is what are the missing mathematical functions attributed to the right hemisphere, besides visuo-spatial ones. Even the role of visuo-spatial functions needs to be further specified: does the calculation process just recruit generic visuo-spatial abilities with the contribution of the right hemisphere or does the contribution of the right hemisphere consist in supporting abilities of spatial nature that are specific for calculation processes? Studies directly addressing the role of the right hemisphere have been only a very recent enterprise.

The following review will focus on describing the main findings obtained by employing three methods: clinical studies, neuroimaging and reversible inhibition. These methods were chosen because they have made significant contributions to better understanding the neurological underpinnings of calculation and the lateralization of arithmetical functions. Besides indications for localization, these studies seem to provide evidence about the nature of the representations and processes at play in relation to the specific way the right hemisphere may indeed support calculation. These studies have the merit of making clear that even simple calculation may require the right hemisphere and contributed to partially answering some of the questions posed in the introduction.

### (a) Clinical studies

Clinical studies are the most time-honoured methodology in investigating the brain–behaviour relation. Over other methods, clinical studies have the advantage to show effects that may not be expected by the observer/experimenter: counterintuitive findings may lead the researcher's attention towards new pathways of investigation. Many phenomena of unexpected dissociation (when one domain is preserved or affected) have thus shaped the history of neuropsychology. In addition, single case studies of calculation disorders have led the way to further investigations in cognitive neuroscience even though conclusions about the anatomy can only be drawn on the basis of group studies. In the following subsections, group and single case neuropsychological studies on calculation will be briefly reviewed.

#### (i) Groups studies

A number of clinical group studies investigated different aspects of number and calculation processing by focusing on lateralization, i.e. whether they related to functions or lesions of one of the two hemispheres, and on the comparison between left brain-damaged patients, right brain-damaged patients and a control group of healthy participants [2,11–15]. Most of these studies converged on the fact that patients with lesions to the left hemisphere have greater overall impairment than the right hemisphere lesion group and the controls. Importantly, they also reported that right hemisphere-damaged patients present deficits, although the interpretation and specification of such deficits varied considerably across studies.

The first author to suggest a possible contribution of the right hemisphere to calculation was Henschen [16]. He thought that the right hemisphere contributed to calculation only in a compensatory manner, in the case of large lesions in the left hemisphere. Likewise, decades later, Goldstein [17] still believed that there was no evidence for any role of the right hemisphere in calculation. As reported above, Sperry *et al*. [6] much later also argued that the capacity for calculation of the right hemisphere is ‘almost negligible’.

Later studies pointed out that the incidence of calculation disorders in right hemisphere lesions is not so negligible. Hécaen & Angelergues [12], who only considered retro-Rolandic lesions, reported the alexia/agraphia variety of acalculia in 8% of the cases; primary acalculia (anarithmethia) occurred in up to 15% of the cases, while calculation disorders of spatial type were present in 75% of patients with right retro-Rolandic lesions. Further data were provided in [13]: alexia/agraphia for numerals was found in 2.1%, anarithmethia in 20.2% and spatial acalculia in 31% of cases of right hemisphere lesions.

Grafman *et al.* [18], in an extensive investigation distinguishing patients along right/left and the anterior/posterior dimensions, concluded that left posterior patients performed exceedingly worse than right posterior patients, who, in turn, were much worse than both the right and the left anterior groups. The authors obtained similar results after correcting the acalculia scores with the results of other cognitive tests (Raven's progressive matrices, Token Test, Crosses Test and assessment of constructional apraxia). The authors could thus conclude that, even if different factors (like impairment of intelligence, visuo-constructive difficulties and aphasia) may contribute to calculation disorders, acalculia can still be partially independent from such disorders. Both right hemisphere patients and patients of the left posterior group affected by Wernicke's aphasia showed spatial disorders. However, the type and the nature of errors committed by right hemisphere patients were not even discussed.

Warrington & James [19] compared unilateral right and left cortical damaged patients using tachistoscopic number estimation, in which participants were required to estimate the number of dots (3–7) flashed in the left, right or centre of the visual field for 100 ms. Participants were also tested on dot counting: they were required to count the total number of dots or dashes on a card (maximum 15) with no time constraints. They reported that the right hemisphere group (particularly the right parietal one) was impaired on the number estimation task. Performance on the dot counting task was not related to laterality of lesion. The authors interpreted their findings suggesting that right hemisphere-damaged patients failed in these tasks owing to visuo-spatial agnosia and visual disorientation.

Ardila & Rosselli [20] tested 21 right hemisphere patients (6 pre-Rolandic and 15 retro-Rolandic). Their findings suggested that acalculia most frequently appeared in written calculation; mental calculation was better preserved. In reading and writing numbers, ‘spatial’ alexia and agraphia led to particular errors, e.g. inability to use the spaces to join and separate numbers, difficulty in conserving the written line in a horizontal position, increased left margins and unsteadiness in maintaining left margins, disrespect of spaces and spatial disorganization of written material. A loss of calculation automatisms and reasoning errors were found; impossible results were not rejected. These errors may not be specific to right brain damage, however.

Basso *et al.* [2] studied 26 right brain-damaged participants with an extensive battery number and calculation deficits, excluding patients affected by unilateral neglect. Only three out of 26 patients in this group were classified as acalculic, and their performance was not described in detail.

To summarize thus far: although acalculia following a right hemisphere lesion was consistently found in clinical group studies, it was very seldom reported in detail. The various proportions reported possibly reflected the sensitivity of different calculation tests employed across studies, and the fact that some works excluded patients affected by neglect (see a more detailed review of this issue in the next section). Moreover some studies considered only retro-Rolandic patients, which further complicated the exact quantification of the phenomenon.

A recent study of Benavides-Varela *et al*. [21] further examined this issue by testing 30 right hemisphere patients on a comprehensive battery of numerical abilities (numerical activities of daily living, NADL, [22]), and a variety of neuropsychological tests. Patients performed below the cut-offs in number comprehension, transcoding and written operations (particularly in subtraction and multiplication). Lesion analyses also showed that typical spatial errors (e.g. omissions of the left-most digits, misalignment, number inversions, etc.) were associated with extensive lesions in fronto-temporoparietal areas, frequently associated with neglect. By contrast, purely arithmetical errors (e.g. failure to recall basic number facts, to carry, to borrow, etc.) were related to more confined lesions in the lower parietal lobe, particularly in the right angular gyrus and its proximity. Regression models revealed that spatial errors were mainly predicted by composite measures of visuo-spatial attentional and representational abilities. By contrast, specific errors of arithmetical nature were not predicted by visuo-spatial abilities. Importantly, and in contrast with traditional views, the proportion of arithmetical errors in this group of patients was much higher than that of spatial errors across various calculation tasks. The reverse pattern was true for number comprehension tasks assessing numerical comparison, and number line estimation. This distribution of errors may of course depend on the specific nature of the tasks that were administered, but the same could be told of studies tapping more tasks of a spatial nature. Benavides-Varela *et al*. [21] concluded that right hemisphere lesions can directly affect core arithmetical processes and, therefore, that the contribution of the right hemisphere goes beyond a visuo-spatial support to calculation, as had been traditionally thought.

#### (ii) Single case studies

Dehaene & Cohen's [23] influential model of numerical cognition proposes a contrast between a retrieval system and an approximation system that was initially inspired by single case neuropsychological findings. Approximate semantic knowledge dissociates indeed from rote memory of calculation, as shown in a case of extended left hemisphere damage [24]: patient NAU could cope with approximation but could not perform even the simplest exact calculation. The authors tentatively proposed that approximation might be a function of the right hemisphere. In a Dehaene & Cohen's [25] later study, the dissociation between the two ways of calculating was supported by the observation of two patients with contrasting symptoms. In patient BOO, right handed, with a lesion in the left lenticular nucleus, head of the caudate, internal capsule and insula, a problem with simple multiplication was far worse than problems with addition and subtraction. Patient MAR was left handed, and after a lower parietal lesion to the right, dominant, hemisphere, was grossly impaired in subtraction with relative sparing of multiplication and addition. In interpreting these findings Dehaene & Cohen [25] suggested that simple single digit calculation can be solved via either a ‘direct route’ or an indirect ‘semantic route’. In the direct route operands, transcoded into verbal code, would eventually elicit the rote memory for the operation. Addition and multiplication would mostly rely on this route. Crucial structures sustaining the direct route would include a left-sided cortico-subcortical circuit including the basal ganglia and the thalamus. In the indirect ‘semantic route’, the operands represent quantities on which semantically meaningful manipulations can be performed. By this route subtraction would be preferentially performed. Crucially, this route is also assumed to monitor the plausibility of a result retrieved by the direct route by appealing to approximate magnitude knowledge. The route would be sustained bilaterally by the inferior parietal cortex and by the left perisylvian network. Given the handedness of patient MAR, however, one cannot make safe inferences about what would be the involvement of the typical right hemisphere in the given tasks. Subsequent work by the same [26] and other authors [10] showed in fact dissociations between different arithmetical operations, but within the left hemisphere.

Single case studies of acalculia in right brain-damaged patients are rare and mostly anecdotal (e.g. [27,28]). The only detailed case was reported by Granà *et al.* [7]. This patient had a posterior right hemisphere lesion, involving the polar planum, Heschl gyrus, the temporal planum, posterior insula, the supramarginal gyrus and the angular gyrus, with dilatation of the right ventricle. No left hemisphere lesion was detected. An important portion of this patient's errors could be demonstrated to be spatial in nature and specifically related to the demands of both mental and visual representation of multi-digit multiplication. According to Granà *et al.* these errors reflected difficulties in relying on a visuo-spatial layout representation specific to multiplication. Thus the patient, while knowing what, when and how to carry out the various steps, did not know where. Such a deficit was never reported again. It needs to be carefully looked for and convincingly distinguished from other types of deficits, however, and cannot be easily shown in a standard assessment.

#### (iii) The influence of neglect

One symptom that has been frequently called upon for explaining numerical errors in right hemisphere patients is unilateral spatial neglect [29,30]. Zorzi & colleagues [31] showed, for instance, that left-sided neglect patients following right hemisphere brain lesions are impaired in the mental bisection of numerical intervals, that is: when asked to indicate the midpoint of a numerical interval, they misplaced their answer towards larger numbers. This bias was replicated in several studies [32–39] and was interpreted as a representational deficit for numbers located to the left of a reference point along the mental number line (but see also [40] for a different interpretation).

The issue of how neglect directly influences mental calculation (besides the numerical mental space, as described above) has been explored in neuropsychological studies only recently. Benavides-Varela *et al*. [41] compared right hemisphere patients with and without left-sided neglect on a numerical battery, including counting, magnitude comparison, writing and reading Arabic numerals and mental calculation. Both patients with and without neglect differed significantly from otherwise matched healthy controls, suggesting that the maths impairment of right hemisphere patients does not necessarily correspond to the presence of neglect. Moreover, a difference between patients with neglect and healthy controls was found in tasks such as one-digit mental subtraction and multiplication. Still, the restricted number of items of the battery used in this study, which was appropriately designed for basic clinical screening, did not allow further correlations with specific error categories (spatial or core arithmetical ones, but see also [21]). Dormal *et al*. [42] also compared right hemisphere patients with and without neglect specifically on mental addition and subtraction of two-digit numbers. Just as in the study of Benavides-Varela *et al*. [41], neglect patients performed worse in subtraction (multiplication was not assessed). Interestingly neglect patients performed nearly perfectly on matched additions. The authors interpreted this result as demonstrating a causal relationship between attending to the left side of space and solving subtractions. Similarly, a recent case study [43] provided evidence for a reverse dissociation by describing a left brain-damaged patient exhibiting right unilateral visuo-spatial and representational neglect. The patient experienced difficulties in solving addition but not subtraction problems and also in judging the numerical magnitude of numbers located to the right of a reference point along the mental number line (i.e. larger than a standard). These findings are in keeping with other behavioural (e.g. [44–49]), and neuroimaging studies [50,51] showing that performing basic addition and subtractions induces spatial shifts to the right and left sides of the attentional space.

To summarize, impairments of the spatial representation of numbers have been repeatedly found in neuropsychological studies. The association between neglect and deficits in tasks involving numerical interval bisection, magnitude comparison and calculation are highly consistent across studies. What remains elusive is which (if any) other proper numerical functions (besides non-numerical ones such as attentional shifts) in the right hemisphere are necessary to carry out arithmetical operations.

One example of such functions was provided in a recent study [52] directly assessing the role of the right hemisphere in transcoding numbers (i.e. from Arabic to the verbal code and vice versa). The study clearly indicated that errors in transcoding complex numbers containing zero, a very common finding in right hemisphere patients, are not related to neglect, nor are they related to a generic depletion of cognitive resources. These errors, which may easily interfere with complex calculation, reflect a specific damage to mechanisms that are independent from those necessary to complete transcoding numbers without zeros. Such mechanisms for dealing with zero in complex numbers may require, according to Benavides-Varela *et al*. [52], setting-up appropriate empty-slot structures and parsing and mastering of categorical spatial relations between digits. The lesion study suggested a specific location of these mechanisms in the right insula and immediate surroundings. A recent study with a much larger sample of stroke patients further corroborated this finding [53].

In conclusion, clinical studies, whatever their limits, show, besides ‘spatial acalculia’ as in tradition, other patterns of acalculia that are hardly directly attributable or predicted by generic spatial deficits. Neglect does not account for a wide range of errors. This indicates that, if spatial functions in calculation are surely sustained by the right hemisphere, other arithmetical functions that are not clearly spatial in nature, as simple mental calculation, are involved as well. Specific, non-generic, spatial functions, like remembering the spatial layout for written operations were however revealed by careful investigations. A role in processing zero within complex numbers has also be shown. The contrast between a retrieval and an approximation system has also been explored: the tentative proposal that it may reveal something about the right hemisphere function is better supported by other methodologies than clinical observation.

### (b) Neuroimaging studies

Neuroimaging allows us with increasing precision to understand the relationship between activity in certain brain areas and specific mental functions. It has the advantage, over most clinical studies, of allowing strictly controlled, repeatable experiments. One major disadvantage, however, with respect to lesion studies and reversible inactivation, is that activation shown in a given area does not imply that the area is crucial for a specific function.

Consistent with neuropsychological literature on acalculia, the first neuroimaging investigations of mathematical functions [54–56] indicated a pivotal role of the left hemisphere in calculation, with little specification about the contribution of the right hemisphere. However, the anatomical correlation of functions soon made significant progress from the original idea that most maths skills are almost exclusively located in the left parietal lobe, with the exception of some spatial components located in the right hemisphere. The ‘triple code’ model [23], based on clinical and neuroimaging observations, gained popularity by drawing a more complex picture. The model, while assuming that Arabic and magnitude representations of numbers are available to both hemispheres in the vicinity of the parieto-occipito-temporal junction (but extending to the occipito-ventral on the left), assigns the verbal representation underlying the retrieval of arithmetical facts exclusively to the left hemisphere. Only the left hemisphere would possess the representation of the sequence of words corresponding to verbal numerals and lexical mechanisms for identifying and producing spoken numerals. These lexical mechanisms, which would not be specific for numbers, would be implemented in the classic language (Broca's and Wernicke's) areas, including the inferior frontal and the superior and middle temporal gyri, as well as basal ganglia and thalamic nuclei.

Subsequent studies identified, within the parietal lobe, three major circuits for number processing [57]. Each of these circuits would play a distinct functional role in arithmetic:

(1) A core system, called into action whenever numbers are manipulated and quantity processing is required, would be bilaterally located in the horizontal segment of the intraparietal sulcus (HIPS). This circuit would be systematically activated whenever numbers are processed, independently of the notation used for numbers, and the activation increases when the task requires the processing of quantities. According to the first studies, this area seems to be activated in calculation [54,56,58]. It is more active in approximate than in exact calculation [55], however, and more consistently in subtraction than in multiplication [57–60]. In the right hemisphere, activation increases when computing two addition or subtraction operations instead of one [61].

(2) A left angular gyrus area, in connection with other left hemisphere areas, would support the retrieval of arithmetical facts. In particular, the left angular gyrus would support the verbal aspects of numerical processing. It is thought to mediate the retrieval of facts stored in verbal memory, but not other numerical tasks, like subtraction, number comparison or complex calculation, which are related to quantity processing.

(3) A bilateral posterior superior parietal system would support visuo-spatial processes, attention shifts and spatial working memory related to numerical processing. These shifts are proposed to help in localizing the position of a number on a mental space continuum during both calculation (i.e. while mental addition shifts attention to the right, mental subtraction shifts attention to the left [50]) and relative numerical judgements (i.e. during numerical comparison, e.g. [62]). This mechanism is also active in approximation or counting.

A further step forward was made in [55] and [63]. Two techniques, functional magnetic resonance imaging (fMRI) and evoked potentials (ERPs) converged in obtaining the same results: both studies employed only addition in a verification task. The respective locations of retrieval (exclusively left-sided) and approximation (sustained bilaterally) confirmed earlier suggestions of lesion studies [24,25]. Importantly, in [55] and in [63], ERPs for exact and approximate addition differed within the first 400 ms of a trial, being more negative for exact calculation rather than approximation in an earlier phase, and vice versa later on in the epoch. Interestingly, an effect of number size was found in exact calculation: its increment led to increased activation of the same parietal regions as during approximation.

A quantitative meta-analysis of brain areas needed for numbers and calculation was subsequently performed by Arsalidou & Taylor [9] on fMRI-based available evidence. The authors calculated laterality indexes of activated voxels for addition, subtraction and multiplication. Obtained indexes revealed hemispheric asymmetries in the parietal cortex. While addition was left-lateralized and subtraction was bilateral or left-lateralized, the right hemisphere was found dominant for multiplication, thus highlighting an important difference from the triple code model. The role of the right hemisphere was further emphasized by Rosenberg-Lee *et al*. [64]. These authors examined functional overlap and dissociations within the parietal lobe across addition, subtraction, multiplication and division, by separately considering the intraparietal sulcus (IPS), the superior parietal lobule and the angular gyrus. Importantly, multiplication evoked significantly greater activation in the right posterior IPS.

Evidence for a very specific role of the right hemisphere in numerically related functions other than calculation, worth mentioning here, was recently provided in an fMRI study by Vetter and colleagues [65]. In particular, the authors studied the neural correlates of visual enumeration under different attentional loads in a dual-task paradigm. They found that relatively intact subitizing under low attentional load compared with impaired subitizing under high attentional load was associated with an increment in the response in the right temporoparietal junction. These data suggest that this brain region has a particular role in determining capacity limits of enumeration within the subitizing range.

In summary, recent neuroimaging studies show that, in contrast with tradition, the right hemisphere contributes to several aspects of number processing and may be even more important than the left one in the solution of single digit operations like multiplication. The dichotomy between retrieval (left-sided) and approximation (bilateral) receives further support with respect to lesion studies and the suggestion is put forward that the solution size of the problem should be considered, as an important factor in lateralization.

### (c) Reversible inactivation: transcranial magnetic stimulation and direct cortical electrostimulation

Two methods of reversible inactivation are considered here: TMS and DCE. These methods complement neuroimaging with causality. The fact that an fMRI study shows activation in a certain area does not imply that the area is crucial for a specific function. By contrast, this conclusion may be safely reached if the temporary disruption of a brain area by means of TMS or DCE impacts accuracy and/or reaction times in a task.

In TMS an electrical current produces a magnetic field that passes through the scalp and skull, inducing an electrical field sufficient to alter neuronal activity in the brain. TMS is employed to test whether or not a region in which change in neural activity is associated with a given task is also *necessary* for the performance of the task.

The purpose of intraoperative DCE is to gather precise information about the brain localization of functions that can be spared while removing pathological tissues [66]. Electrodes are applied directly to the cortex. Because the brain lacks pain receptors, the patient can be alert during the operation in order to interact with the operating team. Errors in tasks sustained by inhibited areas reveal the location of certain functions. These areas will be spared in the operation, when possible.

Using TMS, a few studies investigated whether the right hemisphere is related to specific calculation abilities (see [67] for a review). While Göbel *et al*. [68] found no effect of repetitive TMS (rTMS) on the participants' performance in additions, other studies found a bilateral involvement of the HIPS in exact arithmetic.

Andres *et al*. [69], using fMRI-guided rTMS, found that disruption to both the left and right HIPS leads to impaired multiplication. Interestingly, frequent errors of the retrieval type (table results other than the target) suggested that disruption of the HIPS impaired retrieval processes. No comparison of TMS effects over the left versus the right hemisphere was reported, however, in terms of type of errors.

Using repetitive navigated TMS systematically on 52 spots distributed over the two hemispheres and anatomically identified for every subject, Maurer *et al*. [70] further showed that single digit addition and multiplication-related areas are predominantly localized in the left hemisphere, most prominently in the superior temporal gyrus and in the angular gyrus, respectively, while single digit subtraction-related areas are in the right hemisphere, in particular in the angular gyrus.

Salillas & collaborators [71] used single pulse TMS with different stimulus-onset asynchronies (SOAs) (150, 200, 250, 300 ms) time-locked to the visual presentation of easy or hard addition and multiplication problems. Stimulation was delivered to right/left HIPS or right/left ventral portion of the intraparietal sulcus (VIPS) contralateral to lateralized problems; the interhemispheric fissure was used as a control site. In a second experiment, ipsilateral stimulation was directly contrasted with stimulation delivered contralateral to the problem. In both experiments addition and multiplication differed in hard but not easy problems. Addition involved the HIPS bilaterally. Multiplication seemed to rely instead on the left HIPS but involved the right VIPS, insofar as the right VIPS disruption predicted problem size effects. The contrast between ipsilateral and contralateral stimulation at different SOAs highlighted the temporal course for the involvement of the different sites: a left HIPS disruption appeared late in the 300 ms SOA, and it was preceded by an involvement of the right VIPS from 150 to 250 ms SOAs. Thus efficiency in simple multiplication seems to be partly dependent on the VIPS in the right hemisphere, considered to be critical for motion representation and automatization. Similar to Salillas *et al*. [72] the results suggested that more occipital areas may be involved in the automatic solution search and that verbal and visuo-spatial processes interact in exact calculation.

Findings with DCE also suggest a role of the right hemisphere in simple calculation. Yu *et al.* [73] found that inhibition of the right parietal lobe impaired performance on simple subtraction. The right parietal involvement in subtraction was believed to be related to quantity processing rather than to verbal numerical processing. However, Della Puppa *et al.* [74] further demonstrated with this technique a right hemisphere involvement also in multiplication (positive sites, namely those inducing errors in at least two-thirds of the trials, were reported for the angular gyrus, the supramarginal gyrus, the interparietal sulcus and the superior parietal lobule) and addition (in the supramarginal gyrus).

Semenza *et al*. [75] very recently analysed DCE data in patients undergoing operations in either the left or the right parietal lobe. A first important finding was that positive sites for both addition and multiplication were found after inhibition of each side (table 1). Crucially, no individual site was found positive for both addition and multiplication. The fact that positive sites were operation-specific thus ruled out the possibility that errors came from a generalized depletion of processing resources. For the first time an error analysis was performed. Commission (substitution) errors were found to be of a different nature after DCE in the two hemispheres, unveiling the function of the hemisphere contralateral to the inhibition. Multiplication errors, when inhibition was on the right, mostly consisted in replacing the correct solution with another table solution involving one of the operands. Such errors are interpretable as reflecting the search for a solution through retrieval, a mechanism believed to pertain to left hemisphere functioning. By contrast, when inhibition was on the left, a significant number of errors consisted of close approximation, reflecting approximation mechanisms proper of the right hemisphere. Furthermore, right hemisphere inhibition led to a larger deviation from the correct solution, consistent with the idea that approximation is a function more lateralized on the right. Errors in addition were also prevalently of the approximation type after left inhibition; for both hemispheres inhibition led to underestimation of the correct result, with a larger deviation from correct after right hemisphere inhibition.

In summary, reversible inactivation proved critical in supporting data obtained with neuroimaging, by demonstrating a causal role of the right hemisphere in single digit operations. Importantly, the specificity of inhibition sites speaks against unspecific difficulty effects. The retrieval/approximation dichotomy was specified and shown also for multiplication. The involvement of VIPS was shown to depend on problem size.

## 3. Discussion

Recent literature starts providing answers to the questions posed in the introduction. When put together, recent studies reveal, in fact, that not only do both hemispheres contribute to calculation and may take over functions of the contralateral hemisphere in some clinical situations [76], but also that each hemisphere seems to play its specific role in normal functioning.

The right hemisphere, in particular, does not just offer generic processing support when calculation becomes hard. The finding of operation-specific positive sites in the right parietal lobe speaks against such a hypothesis. If inhibiting a site provokes a generic shortage of processing capacity, the effect would hold irrespective of the operation and would not be as operation-consistent as found with DCE in Semenza *et al*. [75].

The idea that the right hemisphere solely contributes with spatial abilities is no longer tenable either, because it only provides a partial idea of what the right hemisphere really does. Spatial abilities are indeed a very important aspect within the competence of the right hemisphere and this is true also in calculation. This review suggests that other arithmetic functions which may not rely on spatial abilities are also carried out in the right hemisphere, however. Thus, the fact that a great proportion of errors in right hemisphere acalculia are not clearly of spatial origin shows that the difference between the two hemispheres in calculation cannot be captured entirely with the traditional verbal/spatial dichotomy. The right hemisphere may offer support to written calculation via memorizing the spatial layouts necessary for such complex tasks, but whether this type of function is exclusively a competence of the right hemisphere remains to be demonstrated. Surely one interesting function, which may intervene in calculation, concerns processing of zero in complex numbers: this function cannot be described as ‘spatial’, however, without a theory that clearly specifies why that would be the case.

The clearest indications for a specific, causal, role of the right hemisphere in calculation come from reversible inhibition. The studies conducted with this technique were on simple calculation, thus minimizing the possibility of an intervention of spatial functions. Thus TMS reveals that activation in simple calculation happens at times that are different and specific for each hemisphere and in different sites. In particular, the right VIPS seems to have a specific role, although not a necessary one (otherwise positive sites for VIPS would have been found with DCE). Crucially disruption to VIPS predicted problem size effects. The right VIPS may thus be called upon in efficiently performing harder problems. Further studies are needed to elucidate such VIPS-supported mechanisms that may indeed involve spatial exploration of internal representations, however.

Finally, results from analyses of errors in DCE seem to be the best indication so far of a bilateral orchestration, with a different contribution by each hemisphere, between two basic functions: retrieval and approximation, eventually leading to efficient simple calculation. Crucially, the studies showed that these functions might be playing a role irrespectively of the operation type. Interestingly, these findings on calculation seem to converge with other clinical studies and studies employing inactivation in tasks other than calculation. For instance, Warrington & James [19] also showed that right hemisphere-damaged patients showed impairments on number estimation tasks. TMS studies on number comparison [68,69,77] remarkably converge to the conclusion that the left hemisphere is able to perform precise number discrimination (e.g. impaired comparison of close numbers) whereas the right hemisphere can only approximate number magnitude (e.g. impaired comparison of distant numbers).

Findings on errors with DCE may be scarce in quantity, and need confirmation, but data from both addition and multiplication converge in supporting a crucial role of the right hemisphere. The fact that positive sites are invariably operation-specific is a very important demonstration that the right hemisphere is really specifically and crucially involved in addition and multiplication. These data seem also more clear with respect to previous literature about the distribution of functions, retrieval and approximation, between the left and the right hemisphere. While, consistently with neuroimaging studies, the role of the left hemisphere in rote retrieval is confirmed, approximation may still be carried out bilaterally but, indeed, with a crucial role for the right hemisphere. This is unlike in other bilateral tasks, such as number comparison, which remains undisturbed by (unilaterally applied) DCE.

Tapping the time course of each hemisphere's involvement, as also suggested by TMS, but perhaps more interestingly done with techniques such as ERP and magnetoencephalography, may provide in the near future further advancement in understanding these dynamics of lateralization in simple calculation.

We thus propose that each of the two hemispheres must enter the calculation process with its own role, the one played by the right hemisphere being complex and not at all an ancillary one. It may not be sufficient, though, to try to explain hemispheric differences in calculation in a dichotomous linguistic/spatial distinction as has been done in the past. Performance on tasks like dealing with mental layouts for complex operations, processing of zero in complex number transcoding, determining capacity limits of enumeration within the subitizing range, approximation, attention allocation in calculation, etc. cannot easily be accounted for by a single, more generic function. In some cases the right hemisphere might concur with the left hemisphere in the same task, while in others, the two hemispheres might work independently. Future research will highlight exactly how. For the time being, it is essential to start with the right questions.

## Data accessibility

This article has no additional data.

## Authors' contributions

Both authors drafted the article and approved the final version.

## Competing interests

We have no competing interests.

## Funding

The writing of this paper was made possible by the support of Progetto strategico ‘NEURAT’ (STPD11B8HM_004) from the University of Padova to C.S.

## Footnotes

One contribution of 19 to a discussion meeting issue ‘The origins of numerical abilities’.

↵1 Simple and complex arithmetic tasks respectively refer to one-digit (e.g. 2 × 3) and multi-digit problems (e.g. 36 × 45, except for those tasks that can be easily solved by retrieving principles or properties, such as 45 × 100). Arithmetic facts are those problems whose solution is directly retrieved from long-term memory.

- Accepted October 2, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.