Mental calculation is a complex cognitive operation that iscomposed of a set of distinct functional processes. Using functionalmagnetic resonance imaging (fMRI), we mapped brain activity inhealthy subjects performing arithmetical tasks and control tasksevoking a comparable load on visuo-constructive, linguistic,attentional and mnemonic functions. During calculation, as well asnon-mathematical tasks, similar cortical networks consisting ofbilateral prefrontal, premotor and parietal regions were activated,suggesting that most of these cortical areas do not exclusivelyrepresent modules for calculation but support more generalcognitive operations that are instrumental but not specific to mentalarithmetic. Significant differences between calculation and thenon-mathematical tasks were found in parietal sub-regions, wherenon-arithmetic number or letter substitution tasks preferentiallyactivated the superior parietal lobules whereas calculationpredominantly elicited activation of the left dorsal angular gyrus andthe medial parietal cortices. We interpret the latter activations toreflect sub-processes of mental calculation that are related to theprocessing of numerical representations during exact calculationand to arithmetical fact retrieval. Finally, we found that morecomplex calculation tasks involving the application of calculationrules increased activity in left inferior frontal areas that are known tosubserve linguistic and working memory functions. Taken together,these findings help to embed the specific cognitive operation ofcalculation into a neural framework that provides the required setof instrumental components. This result may further inform thecognitive modeling of calculation and adds to the understanding ofneuropsychological deficit patterns in patients.

IntroductionSince Henschen coined the term ‘acalculia’ in his classical

overview of calculation impairments in brain-damaged patients

(Henschen, 1919), there has been a large amount of work on

both characterizing the deficits in performing calculations and

on relating these to sites of lesions in neurological patients.

While disturbances of calculation may occasionally be found in

relative isolation from other cognitive impairments (Warrington,

1982), they are usually associated with deficits in wider

functional contexts than the performance of arithmetical

operations alone (Collignon et al., 1977). Neuropsychological

assessment has guided researchers to define different types of

acalculia. In particular, specific disturbances of arithmetical

skills, called pure anarithmetia, were distinguished from other

functional impairments (alexia, agraphia, visuospatial disorders

and constructional apraxia) that affect calculation without a

genuine deficit in number processing and calculation (Levin et

al., 1993). In other words, linguistic and visuo-spatial

malfunctions have inevitable repercussions on calculation, but

are not specific to it.

Mirroring the diversity of clinical manifestations, the

conceptual models developed for calculation assume a

composite multi-level process. In their abstract–modular model

of number processing McCloskey et al. distinguish input

mechanisms subserving the recognition of numbers and

operation symbols from output mechanisms for writing or

uttering numbers (McCloskey et al., 1985). In the case of written

numerical material these mechanisms involve visuo-spatial and

visuo-constructive computations and linguistic processes. To

perform arithmetical tasks a third component is postulated that

stores basic number facts and rules, as well as procedures

for more complex arithmetical operations. A different cognit-

ive model of number processing is the ‘triple code theory’

(Dehaene, 1992), which proposes two representational forms

for input and output devoid of semantic information (visual

arabic and auditory verbal code) and a third, analogical

‘magnitude’ or ‘quantity’ representation. While an asemantic

route is sufficient for reading or writing arabic numerals, mental

calculation requires access to numerical meaning via a semantic

route. Both models share in common the notion that calculation

relies on components of a more general nature that are also

required for other cognitive functions. These include sensory

feature input processing, visuo-constructive capacity, symbolic

operational and particularly linguistic processing, as well as

memory and attention.

Parallelling the diversity of functions recruited for the

execution of mental arithmetic, deficits in calculation have been

associated with a variety of lesion locations in patients. From the

early work of Henschen onwards, a key role has repeatedly, but

not exclusively, been assigned to the left angular gyrus (Boller

and Grafman, 1985). However, instead of pinpointing a specific

area as a ‘calculation center’, an overview of lesion studies allows

merely a vague conclusion, namely that left hemispheric lesions

are more likely than right hemispheric lesions to affect calcu-

lation and posterior damage more likely than anterior damage

(Dahmen et al., 1982; Grafman et al., 1982). These findings

suggest not only that several distinct and spatially segregated

areas subserve calculation, but also that the neuronal processes

performed in these areas support a broad set of cognitive oper-

ations and are not confined to calculation.

So far, only a few functional magnetic resonance imaging

(fMRI) studies have addressed the neural correlates of mental

arithmetic (Burbaud et al., 1995, 1999; Rueckert et al., 1996;

Chochon et al., 1999; Dehaene et al., 1999; Rickard et al., 2000).

Together, they as well as mostly earlier studies using positron

emission tomography (Roland and Friberg, 1985; Dehaene et

al., 1996; Sakurai et al., 1996; Pesenti et al., 2000) describe a

distributed network that subserves the performance of calcu-

lation tasks, including prefrontal, premotor and parietal cortices.

However, the exact interpretation of the observed activations

appears difficult for at least two reasons. Firstly, the experi-

mental conditions and statistical comparisons in most of these

studies non-selectively highlighted structures partaking in

calculation as well as lower levels of numerical processing.

Dissociating Neural Correlates of CognitiveComponents in Mental Calculation

O. Gruber1,2, P. Indefrey3, H. Steinmetz4 and A. Kleinschmidt4

1Max-Planck-Institute of Cognitive Neuroscience, Leipzig,

Germany, 2Institute of Medicine, Research Center Jülich,

Jülich, Germany, 3Max-Planck-Institute for Psycholinguistics,

Nijmegen, The Netherlands and 4Department of Neurology,

University of Frankfurt, Frankfurt, Germany

Cerebral Cortex Apr 2001;11:350–359; 1047–3211/01/$4.00© Oxford University Press 2001. All rights reserved.

Therefore, the different functional contributions of these various

brain sites could not be dissociated. Only recently, Dehaene

and co-workers attempted to dissociate different arithmetical

processes through direct comparisons of exact and approximate

calculation tasks (Dehaene et al., 1999). They observed relative

differences in brain activation produced by these tasks, in

particular an increase in activation in the left inferior frontal lobe

during exact calculations and bilaterally along the intraparietal

sulci during approximation. In another fMRI study from the

same laboratory, Chochon and co-workers found evidence for

a partial functional–neuroanatomical dissociation of different

numerical operations (digit naming, comparison, multiplication

and subtraction) (Chochon et al., 1999), which is in line with

neuropsychological observations in brain-damaged patients. The

limitations of both studies arise from the fact that while they do

compare different arithmetical processing of numerical material,

they do not (and cannot within this framework) account for

concomitant changes in load on underlying cognitive processes

recruited for task execution. In other words, the differences

observed between different more or less arithmetical tasks are

confounded with task-related differences in tapping, for

instance, working memory or covert verbalisation.

In the present study we therefore used fMRI to investigate

more closely the determinants of activity in the brain areas

cooperatively mediating mathematical performance. We chose

an alternative approach to those described above in comparing

brain activity during one type of mental arithmetic, i.e. exact

calculation, with activity evoked by construed control tasks that

were non-mathematical but imposed a similar load on the

instrumental cognitive processes. Firstly, we included a result

matching task to follow each series of calculations. This control

situation maintained a functional load on anarithmetical num-

erical processing while not necessitating any actual calculation.

Furthermore, to investigate a possible task-related specificity of

cerebral activation, we introduced several experimental tasks

at high cognitive levels that controlled for some of the other

instrumental components required for calculation (Figs 1 and 2).

Materials and Methods

Subjects and Imaging

We studied in depth six healthy male native German speakers (mean age

25.8 ± 2.9 years) who had given their written informed consent. All were

consistent right-handers as assessed with a modified German version of

the Edinburgh Inventory (Oldfield, 1971). All subjects were students or

employees at the Research Center Jülich. We did not formally assess

educational level but all had finished high school and had never

experienced manifest problems with arithmetic at school. Imaging was

performed on a 1.5 T MRI system (Siemens Vision, Erlangen, Germany)

using a standard, circularly polarized head coil. We obtained structural

images with a MPRAGE sequence (repetition time TR = 11.4 ms, echo time

TE = 4.4 ms, f lip angle = 15°, field of view FOV = 230 × 230 mm2, matrix

size = 256 × 256, voxel size = 0.9 × 0.9 × 1.25 mm3). For functional studies

we dynamically acquired series of gradient echo planar image volumes

oriented in parallel with the AC–PC plane and spanning almost the entire

cerebrum (TR per slice = 90 ms, TE = 66 ms, f lip angle = 90°). An image

volume was recorded every 3 s and comprised 16 contiguous slices (FOV

= 200 × 200 mm2, matrix size = 64 × 64, voxel size = 3.1 × 3.1 × 5.0 mm3

with 0.5 mm gaps between adjacent sections). Five functional series with

96 volumes each (112 volumes in the compound calculation condition)

were measured per subject, every series covering six repetitions of the

respective condition. The order of conditions was randomized across

subjects, each condition being announced prior to the onset of the

imaging series.

Experimental Conditions

Calculation Tasks

We used two calculation conditions. Simple calculation involved

multiplication and division tasks that did not go beyond stored table facts

(e.g. 3 × 8, 30/6). Compound calculation involved multiplication and

division tasks with larger operands (e.g. 3 × 18, 85/17) but remained in

the same number domain (1–99).

The assignment of a given calculation task to mere fact retrieval or to

a more complex computational process depends in part on the subject’s

proficiency. To verify our subjects’ arithmetical performance level we

carried out off-line tests by presenting 32 number multiplication and

division tasks on a computer screen (e.g. 4 × 6). The subjects were

instructed to respond both correctly and rapidly and the experimenter

pushed a response button as soon as the oral response was initiated. Half

of the tasks we assumed to require only fact retrieval, i.e. simple number

multiplication or division. Identical calculations also occurred during the

scanning period. Since numerical fact retrieval is highly overlearned, the

preceding off-line presentation should not pose a problem for the results

obtained during imaging. Carry-over of stimulus–response associations

from off-line testing to the scanning period was excluded by the use of

different stimulus presentation formats (whole task versus sequential, see

below). The other half were compound number multiplication or division

tasks, which were merely similar to those used during the subsequent

scanning period. This procedure was chosen to minimize the possibility

of result memorization but nonetheless to obtain valid performance time

differences for the conditions tested during imaging. The order of tasks

from the two classes was randomized.

During functional imaging, stimuli (see Fig. 1) were serially presented

on a screen which was positioned outside the magnet coil and could be

seen via a mirror placed above the subject’s eyes. The instruction was to

perform the calculations silently, i.e. without speaking and avoiding lip

or tongue movements. Since calculation with large multiplication factors

required more time than with small operands in prior testing, the rate of

item presentation in compound number calculation was slowed. This

resulted in fewer calculation steps in compound number calculation

(three operations of 9 s each) than in the other conditions (five operations

of 3 s each). Because the relative time that subjects are performing a task

inf luences the signal intensity changes in fMRI, we preferred to engage

subjects continuously on each task instead of choosing rate matching

with confounding long waiting periods in all tasks except for the

compound calculation task.

For similar reasons we did not employ an event-related design, as this

would mean modeling each response as a function of the time required

for a given computation by a given subject. Such a procedure is prone to

confound intensity and duration of a neural activity change due to the

intrinsic low-pass properties of hemodynamic signals.

Control Tasks

In order to selectively address those cognitive components of calculation

tasks that were not related to calculation proper, we chose to create novel

tasks by replacing numbers with letters and operation symbols with

difference of magnitude signs (> and <). When using letters instead of

numbers the resulting task was termed letter pseudo-multiplication and

pseudo-division, the instruction being to covertly verbalize the required

operation but to maintain the initial item as the result of every

computational step (DT divided by F equals DT; see Fig. 1E). When using

magnitude symbols instead of multiplication and division symbols the

result was a substitution task in which the tip of the symbol indicated

whether the left or the right element in the processed item was to be

replaced (Fig. 1C). While letter pseudo-multiplication and pseudo-division

is a nonsense task that matches for linguistic input and output processing,

the number substitution task not only evokes anarithmetical number

processing but also involves meaningful transformations that impose a

high load on visuo-constructive capacity. Finally, a combination of both

manipulations resulted in a letter substitution task analogous to the

number substitution task (Fig. 1D).

Baseline Condition

Both calculation and substitution tasks were followed by a result

Cerebral Cortex Apr 2001, V 11 N 4 351

matching period that was considered as the baseline condition. During

this period the correct result from a series of sequentially presented

possible results was to be indicated by a button press (Fig. 1). The

position of the correct item within the series was varied across trials to

prevent predictability. This procedure removed any ambiguity between

activations related to the result matching period and activations related to

the preceding experimental task.

In addition to the brief motor activation related to pushing the button,

the result matching period maintained visual input processing and a

working memory load. Although not required for correct performance,

access to semantic representations of letters and numbers, including a

representation of magnitude in the case of numbers, most probably also

took place during this period. In short, the result matching period

controlled in a meaningful context for all aspects of the activation

conditions that did not involve the calculation or substitution procedures

as such. The hypothesized processing components of all conditions are

given in Figure 2.

Data Analysis

Data processing used software for statistical parametric mapping

(SPM v.96b; Wellcome Department of Cognitive Neurology, London,

http://www.fil.ion.ucl.ac.uk/spm/). The first six images of each time

series were discarded to allow establishment of steady-state magnet-

ization. The image volumes were realigned, corrected for motion, global

signal intensity variation (proportional scaling) and low frequency

f luctuations (high pass filters of 60 and 90 s cut-off), co-registered,

normalized onto standard stereotactic space (using a template provided

courtesy of the Montreal Neurological Institute) and smoothed with an

8 mm full-width at half-maximum Gaussian kernel (Friston et al., 1995).

Image volumes obtained during the visual presentation of items after

the correct response (‘baseline’) and images of series with an incorrect

or missing response were modeled as separate conditions. These were

not included in the contrasts tested by the statistical analyses. Further-

more, one subject had to be discarded from statistical analysis due to

image artifacts which we were unable to explain or correct for.

Statistical tests were based on a fixed effects model and performed for

single experimental conditions by contrasting them with the individually

modeled ensuing result matching periods of variable length. Across

conditions we tested for component-specific effects by direct comparison

of the condition-specific effects (thus formally corresponding to inter-

actions). For condition-specific effects we report results for areas that

achieved a significance threshold of P < 0.05 (corrected) in either of the

two calculation conditions. To allow comparison we also list findings

in these areas for the other conditions, provided these foci passed a

significance level of P < 0.001 (uncorrected). For the same reason we list

responses exceeding the significance level of P < 0.001 (uncorrected) in

cases where the contralateral homologous area was activated at P < 0.05

(corrected). For comparisons between conditions we report responses

at the level of P < 0.05 (corrected). Areas for which we had a prior

hypothesis, e.g. the left angular gyrus, are reported if they were activated

at P < 0.001 (uncorrected). For visualization, activations are displayedFigure 1. Experimental design (black arrow indicates right hand key press).

Figure 2. Hypothesized processing components of the experimental conditions: result-matching (RM), letter pesudo-calculation (LPC), letter substitution (LS), number substitution(NS), simple number calculation (SNC), compound number calculation (CNC). The darker shade of cells representing visuospatial processing in substitution tasks indicates a relativelystronger recruitment of this processing component compared to calculation tasks.

352 Neural Correlates of Mental Calculation • Gruber et al.

at P < 0.001 (uncorrected) to enhance sensitivity in delineating spatial

extent.

Results

Off-line Testing of Performance

After removal of 17 trials with incorrect solutions and five

outliers (> mean calculation time + 3 SD) the grand mean

calculation time of the remaining 138 trials was 2.50 ± 1.20 s.

Specifically, simple calculation took 1.89 ± 0.86 s for

multiplication and 1.93 ± 0.80 s for division, whereas compound

calculation took 3.05 ± 1.21 s for multiplication and 3.03 ±

1.38 s for division. Despite the small sample size, there was a

significant effect when comparing simple with compound

number calculation (ANOVA, subjects F = 21.07, P < 0.01, items

F = 19.72, P < 0.001), simple calculations being faster for

multiplication and division. We found no significant effect of

calculation type (multiplication or division) and no interaction

between the two factors. Thus, the behavioral data give evid-

ence for additional costs in compound calculation compared

with simple calculation (presumably ref lecting additional

cognitive processes), whereas there was no such difference

between multiplication and division tasks in the subjects of this

study.

Functional Imaging

During functional imaging we could only obtain condition-

related error rates (and not reaction times) because our

experimental design avoided overt responses during calculation

as a potential source of artifacts. Analyzing these error rates we

found no significant differences between conditions (P < 0.05)

at the behavioral level, indicating that after compensation for

the additional time requirements of compound calculation our

experimental conditions were balanced for task difficulty.

Statistical comparisons of condition-specific brain activity levels

were performed in several steps. We first identified cortical

regions that were activated by each condition relative to the

respective result matching period. All five conditions showed a

similar left-dominant bilateral prefrontal, premotor and parietal

response pattern (Fig. 3 and Table 1).

In more detail, the commonly activated brain regions

comprised the prefrontal cortex lining the anterior inferior

frontal sulcus, the opercular part of the inferior frontal gyrus (BA

44), the premotor cortex (BA 6), the intraparietal sulcus and the

adjacent inferior and superior parietal cortices and the pre-

supplementary motor area, partially extending into the anterior

cingulate. The lateralization of prefrontal and parietal foci to

the left hemisphere during calculation shifted to a more sym-

metrical distribution in the substitution conditions. Letter

pseudo-calculation yielded the smallest extent of significant

activation. Our further analyses were aimed at revealing

differences between conditions. First, we contrasted simple

number calculation with letter pseudo-calculation and number

substitution. Each condition was considered with regard to the

activation it produced compared to its control period. Letter

pseudo-calculation matches number calculation for covert

linguistic processing of the presented items and operation

symbols and for maintenance of a result over a succession of

items.

The comparison between simple number calculation and

letter pseudo-calculation revealed left-sided activations at P <

0.001 (uncorrected) in dorsal parts of the inferior frontal gyrus

(BA 44/45), in the middle third of the inferior frontal sulcus, in

the ascending and descending parts of the intraparietal sulcus as

well as in the posterior cingulate cortex (BA 23/31) and the

adjacent precuneus (BA 31/7). A similar response pattern was

seen when contrasting compound number calculation with

letter pseudo-calculation (Fig. 4A), with responses in the left

frontal areas and the ascending intraparietal sulcus (P < 0.05,

corrected). Given that the item presentation rate was lower in

compound calculation than in pseudo-calculation, these areas

seem to be actively engaged in the computations performed and

not merely in processing input and output.

In further comparisons we tried to differentiate activations

related to visuo-spatial symbol manipulation from activations

related to calculation proper. In number substitution the

adequate item category is maintained but the operation symbols

changed. Although < and > were explicitly chosen as in the realm

of mathematically meaningful symbols, in the substitution task

they did not indicate arithmetical operations but reordering

of symbols engaging visuo-spatial and visuo-constructive cap-

acities. Testing for greater activity during number substitution

than during number calculation, we found bilateral activation of

the superior parietal lobule (Fig. 4B), suggesting that number

substitution had an even greater demand on such capacities than

number calculation.

Comparing the parietal activation foci in Figure 4A,B, it is

apparent that the parietal activation observed across the

conditions shown in Figure 3A–D spans the superior and inferior

parietal lobules separated by the intraparietal sulcus, i.e. several

functionally distinct brain areas. The opposite comparison,

i.e. probing for greater activity in number calculation than

substitution (Fig. 4C), revealed a focus in the left dorsal angular

gyrus (P < 0.001, uncorrected). Together with the similarly

localized activation focus shown in Figure 4A, these data

underline the key role of the left angular gyrus in mental

calculation. Moreover, this comparison confirmed the finding of

strong posterior cingulate cortex activation extending into the

precuneus (P < 0.05, corrected) that had already been observed

when contrasting number calculation with letter pseudo-

calculation.

We also tested whether our experiment could identify

correlates of automatic access to a number-specific magnitude

representation in an anarithmetic context. However, we

observed no significant activity differences when directly

contrasting number with letter substitution. This indicates that

when embedded in tasks as devised here, number (as opposed to

letter) processing does not detectably engage different neuronal

structures.

In a final step of the analysis we attempted to differentiate

between the two types of calculation tasks applied, i.e. simple

and compound number calculation. When testing for greater

activity in the simple compared to the compound number

calculation task we found a response in the aforementioned

medial parietal zone covering the precuneus and posterior

cingulate cortex (P < 0.001, uncorrected). Since item presenta-

tion rate was slowed for compound number calculation, this

finding may simply ref lect a higher rate of mathematical fact

retrieval in the more rapid succession of calculations with

simple numbers. Conversely, the opposite comparison probed

for activation differences due to the more complex calculation

processes required in compound than in simple calculation.

We found left-sided activity changes at P < 0.001 (uncorrected)

in dorsal parts of the inferior frontal gyrus (BA 44/45), in

ventrolateral prefrontal cortex adjacent to the anterior inferior

frontal sulcus and in the anterior cingulate cortex (at the border

of BA 8, 24 and 32; see Fig. 4D). Interestingly, the focus in the

Cerebral Cortex Apr 2001, V 11 N 4 353

354 Neural Correlates of Mental Calculation • Gruber et al.

anterior cingulate cortex was identical to that we observed when

comparing number substitution with simple number calculation,

i.e. it was sensitive to non-arithmetical challenge.

DiscussionThe present study aimed at a better understanding of functional

contributions from the diverse activation foci detected for

mental calculation in more general comparisons (Rueckert et al.,

1996; Sakurai et al., 1996; Burbaud et al., 1999). We therefore

construed experimental control conditions that were matched

for relevant components in the set of operations required for

mental arithmetic (see Fig. 2). In a first step the different

activation conditions, including the calculation tasks, were

contrasted with result matching to reveal activation areas related

to common task components. Subsequently, direct comparisons

across conditions were made to detect significant differences

between the tasks.

The common activations across arithmetical tasks and non-

arithmetical control tasks cover a distributed cortical network

that is in accordance with results from previous studies cited

above and comprises mostly left-sided prefrontal, premotor and

parietal regions (Table 1 and Fig. 3). These brain regions are well

known to contribute to linguistic and visuo-spatial processing

and to related aspects of working memory function, i.e. the

phonological loop and the visuo-spatial sketchpad [see for

example Baddeley, Paulesu and Swartz (Baddeley, 1992; Paulesu

et al., 1993; Swartz et al., 1995)]. Hence, these results support

the notion that most of the cortical areas involved in mathemat-

ical tasks represent neural substrates not only for calculation

proper but also for other, related cognitive operations.

Correspondingly, it has been suggested that most calculation

disturbances subsequent to brain lesions arise secondary to

one or several of three so-called instrumental problems: spatial

deficit, visuo-constructive impairment and aphasia (Collignon et

al., 1977).

Besides the processes already controlled for in the baseline

condition of result matching, letter pseudo-calculation shared

inner speech and lexical processing of operation symbols with

the other conditions. It was devised as an intermediate level

control task to identify those activations in the other four

conditions that were related to execution of a meaningful

operation, i.e. active processing of item representations held

in working memory (see Fig. 2). Comparing calculation with

pseudo-calculation, increased activity was found to be left-sided

in the middle third and dorsal posterior parts of the inferior

frontal gyrus and medial and inferior lateral parietal regions

(Fig. 4A). Similar frontal activation patterns were seen when

substitution tasks were compared with letter pseudo-calculation.

In sum, additional activation in the left inferior frontal cortex

was induced by all tasks involving meaningful operations on

working memory representations of letters or numbers. The

known involvement of the dorsal inferior frontal gyrus in

phonological [for an extensive review see Indefrey and Levelt

(Indefrey and Levelt, 2000)], syntactic (Caplan et al., 1998,

1999) and semantic (Poldrack et al., 1998, Friederici et al., 2000)

processing of language suggests that the operations required

by the tasks also evoked linguistic processes. Activations in

the middle third of the left inferior frontal sulcus, on the other

hand, were within the variation range of Brodmann areas 9 and

46, as previously reported (Rajkowska and Goldman-Rakic,

1995). Considering that both areas are known regions of

working memory functions [for reviews see Ungerleider and

Goldman-Rakic (Ungerleider, 1995; Goldman-Rakic, 1996] and

that, in contrast to pseudo-calculation, the meaningful tasks

required frequent exchanges and manipulations of items to be

held in working memory, it seems plausible to assume that the

observed activations were due to additional working memory

recruitment.

In the substitution tasks the operations to be performed

preferentially relied on visuo-spatial processing resources (i.e.

replacing the left or right digit), whereas the calculation tasks

required access to number meaning and magnitude repres-

entation, arithmetical processing of the memorized items and

retrieval of arithmetical facts. We further assessed activations

related to these different operations through direct comparisons

of the respective tasks. The greater activity in the superior

Table 1Brain regions commonly activated across conditions (compared to result matching) as determined by statistical parametric mapping (see Materials and Methods)

Area Compound no. calculation Simple no. calculation No. substitution Letter substitution Letter pseudo-calculation

Anterior inferior frontal sulcusLeft –48,44,8; 6.92b –44,44,12; 4.58b –44,44,8; 5.86b –44,48,8; 3.75a –56,44,0; 5.55b

Right 48,40,12; 4.31a 52,40,8; 3.37a 48,44,4; 5.82b 48,40,4; 3.97a nsPremotor cortex

Left –48,–8,40; 6.62b –48,0,32; 6.28b –48,0,36; 6.04b –48,4,32; 4.31a –48,–8,40; 5.30b

Right ns 44,–8,32; 4.85b 52,–8,36; 3.55a 52,–12,48; 3.71a 52,–8,40; 3.60a

Inferior frontal gyrus (opercular part)Left –52,8,20; 5.78b –52,8,16; 4.76b –52,8,16; 7.27b –56,8,12; 6.16b –64,12,12; 5.87b

Right ns ns ns ns nsIntraparietal sulcus and adjacent parietal lobules

Left –28,–68,48; 7.02b –36,–68,48; 5.80b –32,–64,52; 7.81b –24,–68,52; 7.50b –20,–72,48; 3.85a

Right 12,–76,52; 4.40a 24,–64,40; 5.72b 24,–68,52; 7.74b 24,–68,52; 7.49b 32,–60,60; 4.00a

Presupplementary motor area –4,16,48; 3.97a –4,8,56; 3.60a –4,8,52; 5.46b –4,8,52; 3.98a –12,12,52; 4.08a

The values given are the spatial coordinates of activation maxima and the Z scores obtained in these areas in the various conditions (cf. Fig. 1).aP < 0.001, uncorrected; bP < 0.05, corrected; ns, not significant.

Figure 3. Activations during mental calculation and related control tasks. The results from statistical parametric mapping in five subjects were rendered onto the lateral and medialsurfaces of a standard anatomical (T1-weighted) reference brain template. The activations show brain areas more active during experimental conditions A–E (cf. Fig. 1) than duringensuing result matching. For spatial coordinates see Table 1.

Cerebral Cortex Apr 2001, V 11 N 4 355

parietal lobule during substitution of numbers than during

calculation (Fig. 4B) indicated that the substitution conditions

successfully controlled for the demands on visuo-spatial and/or

visuo-constructive processing. In the reverse comparison we

found enhanced activity during calculation in the left dorsal

angular gyrus as well as in the posterior cingulate cortex and

Figure 4. Activations related to cognitive components of mental calculation. These activations were determined by contrasting the results from the calculation and control tasksshown in Figure 3 and are displayed on brain surface renderings. Calculation refers to compound number calculation in (A) and to simple number calculation in (B) and (C), substitutionto number substitution (cf. Fig. 1).

356 Neural Correlates of Mental Calculation • Gruber et al.

precuneus (Fig. 4C). Ever since the publications of Peritz and

Henschen (Peritz, 1918; Henschen, 1919) there have been

attempts to ascertain a possible role of the (left) angular gyrus as

a specific calculation center. Converging evidence from neuro-

psychology, electrophysiology and functional neuroimaging

points to an involvement of inferior parietal regions in math-

ematical tasks, although their exact functional contributions

remain ill defined. In three patients with left parietal ischemic

infarctions overlapping along the intraparietal sulcus calculation

disturbances were apparent, especially in more complex

arithmetical problems, and were ascribed to a disruption of

working memory (Takayama et al., 1994). A working memory

deficit was also proposed in a more recent case report of a

parietal lesion including the angular gyrus, where number

transcoding but not calculation was disturbed (Markowitsch

et al., 1999). In support of this view, certain components of

event-related potentials (ERP) that were recorded at parietal sites

during mental arithmetic have been associated with task diffi-

culty and were thought to ref lect the load imposed on working

memory (Pauli et al., 1996). A different functional description

was given for a patient with a left parietal intra-cerebral

hematoma (Warrington, 1982), attributing his acalculia to faulty

access to semantic entries of arithmetical facts. Correspondingly,

a recent study using cortical electrostimulation in the left

parietal lobe demonstrated a disruption of arithmetical fact

retrieval (Whalen et al., 1997). Finally, Dehaene and Cohen

assessed two patients with a double dissociation of lesion sites

and behavioral effects (Dehaene and Cohen, 1997). They con-

cluded that the inferior parietal lobe might contain an abstract

semantic representation of numbers that is necessary to guide

arithmetical fact retrieval and to perform calculations (Dehaene

et al., 1998). A different interpretation of the cause of acalculia

was reached in a recent very detailed case report of pure

Gerstmann syndrome subsequent to a lesion of left parietal

white matter (Mayer et al., 1999). Following the concept of

a ‘Grundstörung’, the common cognitive denominator of the

disturbed functions was hypothesized to be an impairment in

mental manipulation of images. Together, these studies indicate

that several operational levels of the processes required for

mental calculation may depend on functional integrity of lateral

inferior parietal cortices and the underlying white matter.

The most recent neuroimaging studies on number processing

and mental calculation have also examined determinants of

parietal activation in greater detail. ERP and fMRI data suggest

that the intraparietal sulci are preferentially involved in approxi-

mation and the angular gyri in exact calculation (Dehaene et

al., 1999). While Rickard et al. found activations restricted to

the intraparietal sulci alone (Rickard et al., 2000), a study by

Chochon et al. supports the assumption that both the intra-

parietal sulci and the inferior parietal lobules, i.e. dorsal angular

and/or supramarginal gyri, subserve the execution of number

processing tasks (Chochon et al., 1999).

Several of our findings further the understanding of the role(s)

that different parietal brain areas play in the processing of

numbers and in mental calculation. First of all, we failed to

corroborate the hypothesis of Dehaene’s triple code theory

that the bilateral intraparietal sulcus corresponds to a specific

semantic representation of numerical quantities (Dehaene et

al., 1998). In contrast, this brain region was non-specifically

activated under all experimental conditions when compared

with result matching, irrespective of the material (numbers

or letters) and the actual nature of the task performed. This

suggests that the intraparietal sulcus participates in processes

that are instrumental but not specific to mental calculation.

On the other hand, our results extend the previous findings of

(left) dorsal angular gyrus activation during number processing

by showing that there is more activation in this region when

the brain processes numbers in an arithmetical context than in a

different cognitive task. This indicates that the angular gyrus

is sensitive to the arithmetical nature of the cognitive operation

performed and supports the assumption of a role of the angular

gyrus in excact calculation (Dehaene et al., 1999). The most

recent, albeit indirect, evidence for the role of the angular gyrus

comes from another fMRI study that, when comparing activation

patterns in perfect and imperfect calculation performers, only

found differences in the left angular gyrus, with less activation in

perfect performers, and interpreted this finding in the context

of proficiency (Menon et al., 2000).

Another important novel finding in our study relates to the

functional contribution of the medial parietal structures

activated during number calculation, i.e. the posterior cingulate

cortex and the precuneus. Both neuropsychological investi-

gations and functional neuroimaging studies have established

the importance of these areas in memory and, in particular,

retrieval functions. In semantically cued word retrieval tasks the

precuneus has been shown to be sensitive to item imageability

(Fletcher et al., 1996). On the other hand, a more recent study by

Krause and co-workers indicates that the precuneus may be

involved in episodic associative memory retrieval independent

of presentation modality and imagery content of the presented

material (Krause et al., 1999). Our results provide experimental

support for a role of this area in arithmetic fact retrieval because

the number calculation and substitution tasks differed with

respect to the necessity of this process. This finding does not

exclude the involvement of additional brain areas during

arithmetical fact retrieval in exact calculation tasks. Along these

lines, our results are in accordance with the notion that exact

calculation involves structures subserving language functions

(Dehaene et al., 1999) (see Fig. 3B and 4A). However, activations

in left inferior frontal areas which presumably underlie these

functions were canceled out when compared with tasks without

calculation but with similar demands on language functions

(Fig. 4C). Not surprisingly, the recruitment of language func-

tions, even in number processing contexts, is not restricted to

arithmetic fact retrieval. This view receives additional support

from data presented by Chochon and co-workers (Chochon et

al., 1999).

In sum, our findings provide evidence that the angular gyrus

and the medial parietal areas play essential roles in a functional

circuit ensuring mathematical performance. They support a

role of the angular gyrus in exact calculation by showing that

this structure is particularly sensitive to number processing in an

arithmetical context. They furthermore suggest a role of medial

parietal structures in the retrieval of arithmetical facts. Although

these two processes inevitably interact during calculation, the

possibility of a selective impairment has been shown in a patient

who, subsequent to operation on a left parietal tumor, presented

acalculia, agraphia, finger agnosia, right–left disorientation and

apraxia but was found to display preserved arithmetical fact

retrieval (Delazer and Benke, 1997).

A well-known behavioral feature of mental calculation that

was also observed in our subjects is the so-called problem size

effect, meaning that responses are slower on problems with

larger operands (Campbell, 1987; Siegler, 1988). More difficult

calculations require planned sequential usage of stored math-

Cerebral Cortex Apr 2001, V 11 N 4 357

ematical rules. These processes take additional time compared

to direct retrieval of arithmetical facts from memory. In other

words, whenever direct retrieval of simple table facts is in-

sufficient to solve an arithmetical problem, it will be necessary

to decompose the problem, by the use of mathematical rules,

into sub-steps for which single table facts can be successfully

retrieved. For example, 6 × 12 may be solved through trans-

formation, decomposition and recomposition in the following

way: 6 × 12 = 6 × (10 + 2) = 6 × 10 + 6 × 2 = 60 + 12 = 72.

When comparing the respective activations in our study,

compound calculation elicited higher activity in dorsal parts of

the left inferior frontal gyrus, in the anterior inferior frontal

sulcus and in the anterior cingulate cortex (see Fig. 4D).

The anterior cingulate focus was also activated in the number

substitution task when compared with simple number calcu-

lation and can therefore not be specifically related to calculation.

Its activity might indicate factors such as attentional effort or

task difficulty in monitoring result generation. Activations in

the inferior frontal cortex were also detected in more basic

comparisons (cf. Figs 4A and 3A–E, respectively) and attributed

to linguistic processing (BA 44/45) as well as to the on-line

maintenance of information in the presence of additional,

possibly interfering, cognitive processes (anterior inferior

frontal sulcus). Activity in these regions once again increased

during compound calculation as compared with simple calcu-

lation, suggesting a further involvement of language and working

memory functions during the use of rule-based decomposition

and recomposition strategies.

In conclusion, this study has confirmed that an extended

bilateral prefrontal–premotor–parietal network subserves

mental calculation. However, to a large extent these cortical

areas do not seem to be exclusively involved in arithmetical

procedures, but also in other cognitive contexts relying on

similar instrumental components, such as working memory,

processing symbolic information, mental image transformations

and inner speech. In contrast, we have demonstrated specific

functional contributions of lateral (angular gyrus) and medial

(posterior cingulate and precuneus) parietal cortices to the

processing of numerical representations during exact calcula-

tion, including arithmetical fact retrieval. Finally, our findings

suggest that the use of decomposition and recomposition

strategies in more complex calculation problems does not rely

on neuronal resources specific to mental arithmetic, but recruits

left inferior frontal areas subserving language and working

memory functions.

NotesWe would like to thank the Research Centre Jülich for providing the

facilities that made this work possible and especially Dr Stefan Posse, Dr

Jon Shah and Maisa Grosse-Ruyken for help in data acquisition. A.K. is

supported by the Volkswagen-Stiftung.

Address correspondence to Dr Oliver Gruber, Max-Planck-Institute of

Cognitive Neuroscience, PO Box 500 355, D-04303 Leipzig, Germany.

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