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## 1. Introduction

Mathematical and logical abilities are fundamental to our ability to not only calculate but also perform basic skills such as telling time, and dialing a phone number. This research paper considers both the cognitive processes which provide fundamental numerical skills such as number comprehension and production, number comparison, and arithmetic, and their neural substrates.

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## 2. Fundamental Numerical Abilities

### 2.1 Numeral Comprehension, Production, and Transcoding

An incredible advance for humanity was the ability to represent numerical quantities symbolically using numerals. We are capable of representing numerical quantities in a myriad of ways including arabic numerals (12), written verbal numerals (twelve), spoken numerals (‘twelve’), and a variety of concrete representations (e.g., twelve hatch marks on a stick). Each of these symbolic representations must be converted into a meaningful common form (tens: 1, ones: 2) in order to allow comparison of quantities and calculation. The translation rules from numerals to quantity, and quantity to numerals are varied and surprisingly complex. Arabic numerals have important spatial ordering (relative to the rightmost numeral) with some symbols representing quantities (123456789) and others representing a placeholder such as the ‘0’ in ‘102.’ Verbal numerals have diﬀerent syntactic and quantity related components (one thousand twenty three) which unlike arabic numerals, only provide information about non-zero entries. Critical to success with numerals are: conversion of numerals into their meaning (numeral comprehension), production of numerals which correspond to the quantity we intend (numeral production), and translation of numerals from one form (e.g., 6) into another (six, numeral transcoding).

### 2.2 Representation of Quantity, Counting and Number Comparison

Our most fundamental numerical abilities include the representation of quantity information, the ability to apprehend quantity by counting, and the ability to compare two numerical quantities to determine (for example) the larger of two quantities.

### 2.3 Arithmetic

Solving arithmetic problems requires a variety of cognitive processes, including comprehension and production of numerals, retrieval of arithmetic table facts (such as 3 x 7 = 21), and the execution of procedures specifying the sequence of steps to be carried out (e.g., multiply the digits in the right-most column, write the one digit of the product, and so forth).

## 3. Components of Numerical Cognition

In this section a purely functional question is considered: What are the diﬀerent cognitive processes in the brain which allow for our fundamental numerical abilities? A particularly successful line of research in answering this question has come from the study of impairments in numerical abilities after brain injury. This cogniti e neuropsychological approach seeks to answer the question: How must the cognitive system be constructed such that damage to that system results in the observed pattern of impaired performance? The existence of multiple numerical cognition components has been based in part on evidence that some numerical components are impaired after brain damage, while other numerical abilities are spared. While the next sections will single out individual patients, there are in each case several documented cases with similar patterns of performance. These remarkable case studies are described below.

### 3.1 Numeral Comprehension and Production Processes

A distinction between calculation and numeral comprehension and production processes has been drawn in part on the ﬁndings from brain-damaged patients such as ‘DRC,’ who reveal striking impairments in remembering simple arithmetic facts (e.g., 6 4 24), while other abilities such as comprehending and producing written and spoken numerals are unimpaired. This evidence implies separate functional systems for comprehending and producing numerals from those for calculation. Several cases with similar ﬁndings, and the opposite pattern of impairment have been reported.

Comprehension and production processes are also clearly composed of several functional components. For example, some patients reveal highly speciﬁc impairments only in producing arabic numerals, such as producing ‘100206’ in response to ‘one hundred twenty six.’ This pattern of performance reveals several separable components of numeral processing, including distinctions between: numeral comprehension and production, verbal and arabic numeral processes, and lexical and syntactic processing within one component (given the correct nonzero numerals in the response, despite impairment in the ability to represent the correct pattern of zeros for the arabic numeral).

An ongoing question in the numerical cognition literature revolves around the question of asemantic transcoding procedures. This notion suggests that it may be possible to translate numerals directly into other forms of numerals (e.g., 4 four) without having an intermediary step in which the meaning of the numbers is represented. Note that normal transcoding requires the comprehension of the initial problem representation, the representation of that numerical quantity in an abstract, asemantic form, and the subsequent production of the quantity in the format. Are there asemantic transcoding algorithms? The available evidence is mixed, with some studies suggesting there may be asemantic transcoding algorithms, yet there is still some question as to whether or not there is suﬃcient evidence to support this notion (Dehaene 1997).

### 3.2 Numerical Magnitude Representations

Response latencies for determining the larger of two numerals (from healthy adults) suggests that we represent numerical quantities in terms of a magnitude representation comparable to that used for light brightness, sound intensity and time duration, and use these magnitude representations to compare two or more numbers. Moyer and Landauer (1967) found that humans are faster at comparing numerals with a large diﬀerence (e.g., 1 vs. 9) than they are when they compare two numerals with a small diﬀerence (e.g., 4 vs. 5). Further, when the diﬀerences between the numerals are equated (e.g., comparison 1: 2 vs. 3, comparison 2: 8 vs. 9) the comparison between numbers which are smaller (e.g., 2 vs. 3) is performed more quickly than the comparison of larger numbers (e.g., 8 vs. 9). These ﬁndings led Moyer and Landauer to conclude that the arabic numerals are being translated into a numerical magnitude representation along a mental number line with increasing imprecision the larger the quantity being represented (a psychophysical representation which conforms to Weber’s Law). This system appears to be central to the ability to represent numerical quantities, estimate, and compare numerical quantities meaningfully.

These magnitude representations are found in human adults, infants and children, and in animals. They appear to be related to our representation of other magnitudes, such as time duration and distance (Whalen et al. 1999). A crucial challenge for the development of numerical literacy is the formation of mappings between the exact number symbol representations which are learned in school, and the approximate numerical quantity representations which are present very early in life.

### 3.3 Arithmetic

Evidence from brain-damaged patients, as well as those with normal and impaired development have revealed that there are several functional systems required in order to be able to perform calculations, such as 274 x 59. Several distinctions can be drawn based on the study of brain-damaged patients and their impairments. First, there is a distinction between the ability to remember simple arithmetic facts (such as 6 x 4 = 24), and the ability to perform calculation procedures (such as those required for carrying numbers and placing zeroes in multidigit multiplication). Several brain-damaged patients reveal impairments which are speciﬁc to either fact retrieval, or multidigit calculation procedures, indicating that the ability to retrieve arithmetic facts and perform multidigit calculations are represented by diﬀerent neural substrates.

Within the simple process of remembering arithmetic facts such as 6 x 4 = 24, there are multiple processes involved. Brain-damaged patients have also revealed selective impairment of the ability to recognize the arithmetic operator ( + , ‒ , x , / ), despite unimpaired ability to retrieve facts from memory. Others illustrate that it is possible to fully comprehend the process of multiplication or addition, but yet be unable to retrieve multiplication or addition facts (Warrington 1982).

There is currently some debate as to the form in which arithmetic facts are stored. Three major theories are under discussion. One possibility is that arithmetic facts are stored and retrieved in a sound-based representation (consistent with the idea of rote verbal learning, like a nursery rhyme). This notion is popular in part because multilingual individuals often report that they believe they are remembering arithmetic facts in the language in which they acquired arithmetic, even if they have subsequently used a diﬀerent language in which they are ﬂuent.

However, an alternative view is that arithmetic facts are stored in an abstract, meaning-based form which is not tied to a speciﬁc modality (e.g., spoken or written). This proposal was ﬁrst suggested in part because we perform arithmetic with a variety of numeral forms (primarily arabic and spoken numerals), and so perhaps there is a central store of arithmetic facts which are independent of any numeral form. Several brain-damaged patients have been reported who reveal impairment which is independent of the form in which the problems were presented or answered, consistent with an amodal representation.

Supporters of the sound-based representation of arithmetic facts suggest that representations of numeral magnitude (one likely candidate for a meaningbased representation of arithmetic facts) might not be related to arithmetic fact retrieval. Some braindamaged patients reveal the inability to determine exact arithmetic responses (e.g., 2 2 3) but nevertheless can reject a highly implausible and answer such as 2 2 9, indicating that there is an approximate number representation which provides the meaning of numbers that may be separate from the process of retrieving exact arithmetic facts (Dehaene 1997).

## 4. Localization of Number Processes in the Brain

Neuropsychological evidence has played a crucial role in informing functional theories of numerical cognition and its components. However, most studies have until now been framed within the context of cognitive neuropsychology, which examines the behavior of brain damaged patients at a purely functional level, without concern for brain localization or lesioned site. Accordingly, models of number processing have been framed exclusively in terms of cognitive processes without reference to brain structures. For this reason, much less is known about the relation between these cognitive processes these involved in the processing and their neural substrates.

Interpreting the available evidence is not entirely straightforward, in part because most reports sought to relate brain areas to arithmetic tasks, and not to speciﬁc cognitive processes such as arithmetic fact retrieval. For example, studies exploring lesioneddeﬁcit correlations have focused typically on identifying lesion loci associated with impaired performance of some calculation task or tasks. However, an association between lesion site and impairment of a calculation task does not in itself constitute strong evidence for the damaged brain area to be implicated in arithmetic processing, because impaired performance could have resulted from disruption to nonarithmetic processes required by the task (such as attention, working memory, numeral comprehension, and numeral production).

These points also apply to research involving brain recording and cortical stimulation methods. In several recent studies, subjects performed a serial subtractions task (i.e., count backwards by sevens from 1,000) while brain activity was recorded, or cortical simulation was applied. A number of brain areas were found to be associated with performance of the task, including the left parietal lobe, the frontal lobes, and thalamic regions. Although intriguing, these ﬁndings do not necessarily imply that these brain areas are implicated in arithmetic processing; the reported data did not exclude the possibility that some or all of the brain areas are involved in nonarithmetic processes required by the serial subtractions task (e.g., comprehending the initial stimulus number, producing responses).

This series of studies left little insight into the localization of speciﬁc cognitive processes, and resulted in a report by Kahn and Whitaker (1991), who echo Critchley’s statement that ‘disorders of calculation may follow lesions in interior or posterior brain, left or right hemisphere, and both cortical and sub cortical structures.’ While this may suggest the task of localizing arithmetic processes is daunting, it reveals that more recent data which focus on the localization of separate number processes (e.g., arithmetic fact retrieval), rather than data which seeks to determine which areas are active for a speciﬁc task (e.g., serial subtractions) may reveal signiﬁcant insights into the relation between the mind and brain for numerical cognition.

### 4.1 Dehaene’s Anatomical and Functional Theory of Number Processing

Stanislas Dehaene and colleagues were the ﬁrst researchers to provide a theory of number processing which includes both the diﬀerent functional components and their localization in the brain. According to the Triple Code Model there are three separate number codes in the brain: verbal, arabic, and magnitude. Verbal codes are located in the left hemisphere language areas (e.g., Broca’s and Wernicke’s areas), and are responsible for holding numbers in memory, arithmetic fact retrieval, and comprehending and producing spoken numerals. Written numerals may also recruit temporal areas involved in visual word recognition. Arabic numerals are thought to be representing in temporal areas which are distinct from the visual word recognition area, and which are thought to be present in both hemispheres. This center is responsible for the recognition and production of arabic numerals. The ability to estimate and compare numbers involves quantity representations found in parietal areas of both hemispheres.

According the Triple Code Model, arithmetic table facts are stored in a sound-based form in language processing centers such as Broca’s area. There are four fundamental components involved in calculation, which are: rote verbal memory, semantic elaboration, working memory, and strategy use. Dehaene proposes that retrieval of rote verbal arithmetic facts may be retrieved from a corticostriatal loop through the left basal ganglia, which is thought to store other linguistic material such as rhymes. In some cases, solving simple arithmetic facts may also involve semantic collaboration (such as determining that 9 7 10 6, and retrieving the answer to 10 6). If this semantic collaboration is involved, then the Triple Code Model predicts that parietal centers which represent numerical quantity will be involved. The next sections consider the available evidence regarding the localization of diﬀerent arithmetic processes.

### 4.2 Numeral Comprehension and Production Processes

There is a large body of evidence suggesting that the comprehension and production of written and spoken verbal numerals is largely subsumed by the lexical and syntactic processing centers used for other linguistic material. For example, brain imaging studies have revealed that left-hemisphere language centers such as Broca’s area are active during the production of spoken numerals, while Wernicke’s area is active during the comprehension of spoken numerals, consistent with other language-based material (Hochon et al. 1999). Evidence from patients with disconnected brain hemispheres provides converging evidence. The left hemisphere shows a strong advantage over the right hemisphere in both the production and comprehension of verbal numerals. The right hemisphere can eventually produce spoken numerals, though the process is extremely laborious, and there is evidence which suggests that this hemisphere uses the counting string to produce the correct spoken numeral (Dehaene 1997).

There do, however, appear to be diﬀerences in the localization of visual numeral processes. Evidence from brain imaging experiments, and from cases in which the two brain hemispheres are disconnected, suggest that both hemispheres represent the ability to recognize arabic numerals. This region, located in the posterior temporal lobe, appears to represent arabic numerals exclusively and not other types of written symbols. In contrast, passive recording of numerical tasks involving written verbal numerals (e.g., twentyeight) have provided evidence for a single, left hemisphere representation for written verbal numerals, which appears to be a subset of written word representations. In fact, there are striking impairments in which patients are unable to produce any letter symbol, or even their signature, yet are completely able to produce arabic numerals (Butterworth 1999).

### 4.3 Representations of Numerical Quantity

Perhaps the strongest single ﬁnding in the localization of numerical processes is that bilateral parietal regions represent numerical magnitude. Patients with disconnected hemispheres have revealed the ability to perform number comparison in either hemisphere (Dehaene 1997). Studies of the electrical brain signature during number comparison using a technique called e ent related potentials (ERP), has revealed several electrical waveforms during the comparison of two numerals. Speciﬁcally, approximately one-tenth of a second after presenting two numbers, there is bilateral activation of parietal lobes, which varies according to the diﬃculty of the comparison. Further, this study also looked at the other waveforms and determined that they were involved in either the comprehension of the visual or spoken numerals, or in the action of responding.

As was presented earlier, responses are faster to comparisons involving large diﬀerences (e.g., which is more: 1 or 9) relative to comparisons involving smaller diﬀerences (e.g., which is more: 4 or 5). The bilateral parietal signal, called N1 (ﬁrst negative activation) varies in amplitude according to the diﬃculty of the problem. The larger the diﬀerence between the numerals, the larger the N1 brain signal (Dehaene 1996). This suggests that the parietal activation was related to the operation of representing the magnitude of the numbers, and comparing them to decide the larger of the two numbers.

Brain imaging experiments involving functional magnetic resonance imaging (f MRI) have also provided evidence of parietal involvement in representing magnitude. In these experiments, the activation from a number comparison task has been compared with the activation in either a number reading task, or a simple calculation task. Each of these tasks presented a single numeral, which was compared with one in memory and required a spoken response, so that the brain activation for numeral comprehension, numeral production and working memory could be approximately equated, leaving the activation from the cognitive processes of interest to vary. In each case, relative to the control conditions (such as number reading) the inferior parietal lobe of both hemispheres was more active during number comparison than in the control conditions. This suggests that this area is likely to be involved in both the representing of numerical magnitudes, and in the comparison process itself (Pinel et al. 1999).

One question which has not been addressed in the reports of localization is: How do we represent numerical magnitude in the brain? There are a few studies which provide some relevant insights into the brain’s representation of magnitude. Studies of parietal cortex in cats have found neurons which are tuned to speciﬁc magnitudes. For example, one neuron ﬁred maximally when ﬁve items were presented to the cat, and responded in a weaker fashion to related quantities such as 4 and 6). Comparable patterns of performance have also been reported in human subjects (Dehaene 1997).

### 4.4 Arithmetic

A number of ﬁndings from studies of groups of impaired-brain damaged patients suggest that posterior cortical regions, particularly parietal regions, may play a role in arithmetic fact retrieval (Butterworth 1999). Impairment in calculation after brain damage, termed acalculia, occurs much more frequently after injury of the parietal lobes than to damage in other centers. Damage to left posterior brain regions impair numerical tasks, including those involving arithmetic, more so than damage elsewhere. However, these studies generally do not distinguish between the multiple components of complex calculation, including arithmetic fact retrieval, calculation procedures, and numeral comprehension and production.

Several single case studies have also implicated left parietal regions as a center for arithmetic fact retrieval, including the previously described patient ‘DRC,’ and multiple cases studied by Takayama et al. (1994). The application of electrical stimulation to the left parietal lobe (prior to neurosurgery) has also resulted in transient impairment to arithmetic fact retrieval during stimulation when the subject is otherwise completely capable of recalling arithmetic facts. Thus it appears that the left parietal lobe, and perhaps both parietal lobes, play a major role in the retrieval of arithmetic facts from memory.

Parietal cortex may not be the only region involved in retrieving arithmetic facts from memory. Calculation impairments have also been found after damage to frontal lobes and subcortical structures including the basal ganglia (Dehaene 1997). Some of these areas may be involved in processes other than arithmetic fact retrieval. For example, evidence from single case studies suggest that damage to the frontal lobes may produce impairment and an inability to produce multidigit calculation procedures (rather than impairing arithmetic fact retrieval). The hypothesis that frontal lobes play a role in calculation procedures is consistent with the ﬁnding from multiple brain imaging studies that complex calculation such as repeated subtractions activate not only parietal centers (thought to be involved in arithmetic fact retrieval) but also other centers such as the frontal area, which maybe involved in holding answers in memory, and performing multidigit calculation procedures.

Patients who have little or no communication between their cerebral hemispheres also provide some evidence as to the localization of arithmetic fact retrieval. When each hemisphere is given a calculation task, only the left hemisphere can retrieve arithmetic facts, and the right hemisphere produces very high error rates (80 percent) (Dehaene 1997). The right hemisphere’s inability to perform the arithmetic task cannot be attributed to numeral comprehension or response production impairments. As was discussed earlier, these patients reveal the ability in each hemisphere to perform number comparison, suggesting that both hemispheres can both represent numerical magnitudes and comprehend arabic numerals.

In summary, current evidence indicates that the parietal lobe plays a major role in simple arithmetic fact retrieval. Other subcortical regions such as the basal ganglia may also be involved. It is currently thought that frontal areas are involved in multidigit calculation procedures, and the working memory demands of complex calculation. These conclusions are somewhat at odds with the assumptions made by the Triple Code Model presented above. Dehaene suggests that the most frequent lesion sites which result in acalculia are in the left inferior parietal region, because this area provides semantic relations between numbers, and can inform fact retrieval. Thus lesions in this area might aﬀect access to arithmetic memory without destroying the rote facts themselves. However, several cases do report speciﬁc fact retrieval deﬁcits as a result of parietal lesions.

### 4.5 Recent Functional Brain Imaging (fMRI) Studies

The advent of f MRI, an excellent tool for measuring the localization of brain activity during cognitive tasks, has reinvigorated the study of the relation between human processing abilities and their corresponding neural substrates. For example, Dehaene and colleagues have performed several comparisons between activation levels during simple numerical tasks such as multiplication and number comparison (deciding the larger of two numbers). In one such f MRI study, Dehaene and colleagues revealed signiﬁcant activation in bilateral inferior parietal regions in a comparison task relative to the multiplication task. This could be for one of two reasons. First, parietal regions could be active primarily during the comparison and not during arithmetic fact retrieval. However, another possibility is that parietal brain areas are involved in both multiplication and the comparison, but number comparison activates this region to a greater extent than does multiplication.

Additional evidence from comparisons between subtraction, multiplication, number comparison and number naming suggest that parietal brain regions are involved in both arithmetic and number comparison. Each of the arithmetic tasks and number comparison revealed signiﬁcant parietal activation relative to the number naming control (all of which had comparable number comprehension and production requirements).

## 5. Summary

Numerical processes in the brain have several subsystems, including those for numeral comprehension and production, representations of number magnitude, remembering speciﬁc arithmetic facts, and performing more complex mathematical procedures. Verbal representations of number appear to be represented largely in the left hemisphere, and are considered to be a subset of non-numerical language representations. Arithmetic fact retrieval appears to involve more than one center, but certainly involves parietal and subcortical centers.

Unlike verbal numbers and words, arabic numeral representations have highly bilateral representations, as do representations of number magnitude. Magnitude representations appear to be responsible for number comparison, estimation, and play a crucial role in arithmetic. The representations of number appear to mirror other psychophysical representations such as time duration and light intensity. Finally, complex arithmetic procedures such as those for multidigit calculation appear to recruit frontal brain regions for holding partial facts in memory, and following a stepwise procedure.

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