| Hein, Jakob : The Specific disorder of arithmetical skills. Prevalence study in an urban population sample and its clinico-neuropsychological validation. Including a data comparison with a rural population sample study. |
Power today is mainly power through and by the command of figures Brüning, 1996 . Mathematical power means having the experience and understanding to participate constructively in society Romberg, 1993 . Consequently, individuals who have deficits acquiring these important powers deserve attention and efforts to facilitate their access to mathematical knowledge. At the core of this help must be the understanding of their insufficiencies.
The need for the institution of an universally applicable quantification first arose with the emergence of agriculture and animal breeding, because then the reaction to nature itself became insufficient. The interaction with and the planning of the environment originated the first mathematical terminology Institut, 1993 .
This process continued with the development of trade and crafts. Fingers and toes obviously played a significant role in constituting numerical phraseology. In certain cultures, numerical systems were based on the values five (e.g. the language of the Khmer), ten (e.g. Arabic numbers) or twenty (e.g. language of the Mayas). Despite their different cultures, developmental stages and religions, the basic structures of the number systems of all these peoples are very similar. The first form of recording numbers was to carve a corresponding number of notches in a wooden stick. This evolved in the Roman and Egyptian cultures into a symbolic system with which any natural number could be symbolized. In these systems, a new symbol had to be used for every new decimal power. In other cultures, such as the Chinese, Sumeric and Indian, another method developed, in which the place of a figure determined its decimal value. India is also where the symbol and the figure for ’Zero‘ originated.
With the emergence of number systems the first arithmetical rules came to pass. But
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it was only the specialization of professions and slave labor in the Greek city-states that made scientific mathematics possible with well-known mathematic prodigies such as Archimedes or Pythagoras. The first examples of mathematical demonstrations are known from that time.Mathematics remained a science in the hands of a few for many centuries. M. Alchwarismi, an Arabic mathematician, published around 800 A.D. a compendium of mathematics for merchants called ’Algebra et Almucabala‘, creating the term ’algebra‘, Arabic for ’carrying over‘. His own name later evolved to become ’algorithm‘. The premise for a general distribution of mathematic science was created by the German A. Riese in the sixteenth century by his conception of principles for mental arithmetic. With the rise of industrialization and the introduction of mandatory schooling such an distribution of mathematical knowledge took place to an extensive scale. At the same time this process elucidates the ever increasing mathematical understanding in general: in the Middle Ages, multiplication was exclusively taught on universities.
In our time, electronic calculators and computers have come to dominate certain areas of mathematics. This also opened new areas of and possibilities for mathematical research. But it makes knowledge of mathematics far from unnecessary. As the United States Conference Board of the Mathematical Sciences pointed out: ’a strong mathematics education is at the basis of the nation‘s need for a competent workforce and an informed society‘ 1995 . The scope of this statement can be extended to include all developed, if not all nations (e.g. Boissiere, Knight, &Sabot, 1985 ; Rivera-Batiz, 1992 ).
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In the present manuscript we are using the term ’Specific disorder of arithmetical skills‘ as defined by the World Health Organization in its ’International Classification of Diseases, 10th Edition. Classification of mental and behavioural disorders: clinical descriptions and diagnostic guidelines‘ (ICD-10). In its chapter F81: ’Specific developmental disorders of scholastic skills‘ the ’Specific disorder of arithmetical skills‘ is coded as F81.2 and defined as follows:
’This disorder involves a specific impairment in arithmetical skills, which is not solely explicable on the basis of general mental retardation or of grossly inadequate schooling. The deficit concerns mastery of basic computational skills of addition, subtraction, multiplication, and division (rather than of the more abstract mathematical skills involved in algebra, trigonometry, geometry, or calculus).‘
The ICD-10 then goes on to lay out the diagnostic guidelines for the Specific disorder of arithmetical skills as follows: ’The child‘s arithmetical performance should be significantly below the level expected on the basis of his or her age, general intelligence, and school placement, and is best assessed by means of an individually administered, standardized test of arithmetic. Reading and spelling skills should be within the normal range expected for the child‘s mental age, preferably as assessed on individually administered, appropriately standardized test. The difficulties in arithmetic should not be mainly due to grossly inadequate teaching or to the direct effects of defects of visual, hearing, or neurological function, and should not have been acquired as a result of any neurological, psychiatric or other disorder.‘ The disorder is distinguished from the ’Acquired arithmetical disorder‘, or Acalculia, coded as R48.8 in the ICD-10. Latter diagnosis stands for a loss of previously present arithmetical skills as opposed to the failure to acquire them in the former WHO, 1992 .
Unless otherwise noted, we will use the term ’Specific disorder of arithmetical skills‘ in the following if the criteria above apply, even if the quoted authors have used another terminology. This is meant to facilitate readability and understanding. For a discussion of the terminology ’Specific disorder of arithmetical skills‘ see chapter 6.1.
The other major classification system of mental and behavioural disorders which is used chiefly in Northern America, the Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition (DSM-IV), has a parallel diagnosis with quite similar diagnostic criteria. In the chapter ’Learning disorders‘ the ’Mathematics disorder‘ is coded as 315.1. The DSM-IV then lays out the diagnostic criteria for the condition: ’The essential feature of Mathematics Disorder is mathematical ability (as measured by individually administered standardized tests of mathematical calculation or reasoning) that falls substantially below that expected for the individual‘s chronological age, measured intelligence, and age-appropriate education (Criterion A). The disturbance in mathematics significantly interferes with academic achievement or with activities of daily living that require mathematical skills (Criterion B). If a sensory deficit is present, the difficulties in mathematical ability are in excess of those usually associated with it (Criterion C)‘ Association, 1994 . The similarities in the definitions of both classification systems are evident, certainly in their content, but in principal
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passages even in their wording.
It is noted by many researchers, as well as both in the DSM IV Association, 1994 and the ICD-10 WHO, 1992 that there is a lack of research on that field of study. At the same time, other disorders, such as dyslexia, another learning disorder on the field of language acquirement, received considerably more attention of the scientific community.
As an illustration of that circumstance we conducted a data-search in ’Medline‘, a database containing information on medical publications. A search for the keyword ’dyscalculia‘, a term often used synonymously for the Specific disorder of arithmetical skills, in the data for the years 1966-1996 produced a list of 81 publications. An analogous search for ’dyslexia‘, a synonym for the Specific reading disorder (F81.1, ICD-10), in ’Medline‘-data for the years 1966-1996 found 3273 publications containing that keyword.
The research focus on the field of learning disorders has not changed significantly in the last years, either. A ’Medline‘-search found 1 publication containing the keyword ’dyscalculia‘ and 81 publications containing ’dyslexia‘ in the database for 1997. This means, that from 1966-1996 as many publications concerning dyscalculia have been published as 1997 on dyslexia alone.
Alkmaeon, a student of Pythagoras, physician and philosopher, who probably also participated in bisections of the eye, claimed around 500 BC the brain to be the central organ of the senses. Galen went around 200 AD even further and assumed there to be a specialization inside the cerebral cortex as well as the white matter. In the third century Christianity became the official religion of the Roman Empire and Christian doctrine was to dominate much of science. Its most popular philosopher St. Augustine (354-430 AD) moved away from the corporeal view of thought and claimed the existence of an immortal soul that can be influenced by but is independent of the body.
This view endured 1400 years and the brain as a site of human thought only regained interest when Descartes in the eighteenth century localized the ’animal spirits‘, responsible for muscle motion, blood circulation, respiration, sensory impressions, appetites, passions and memory in the cerebrospinal fluid. He agreed with the Christian concept of the soul, but thought it had to interact with the animal spirits in some corporeal site. This he proposed to be the pineal gland since it was the only singular part of the brain.
By developing the phrenological theory, F.J. Gall moved in the beginning of the
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nineteenth century beyond Cartesian principles. He was convinced that the superior human intelligence is owed to the greater development of the human cerebral cortex and that there is a specialization of the cerebral cortex into ’organs‘. Gall thought that one could draw conclusions about the different regions of the brain by examining the outside features of the skull. Together with J.C. Spurzheim they popularized this view as ’organoscopy‘ or ’cranioscopy‘ as they themselves called it. It later became known as phrenology. A convexity of the skull was assumed to reflect a well-developed underlying cortical gyrus and vice versa with regard to a concavity Hunt, 1993 . By those comparisons Gall and Spurzheim tried to localize a number of cerebral functions. Because they thought mathematicians to have a protrusion in the temporal area of the skull, just behind and above the eye (see Fig. 1), they concluded that the calculation abilities are located in ’a convolution on the most lateral portion of the external, orbital surface of the anterior lobes‘ (cited in Levin, Goldstein, &Spiers, 1993 ). By the middle of the nineteenth century, the phrenological view had became so popular that there were for example 29 phrenological societies in Great Britain alone.Fig.1 Contemporary illustration of phrenology. The proposed site for underlying calculation abilities is marked ’X‘ (from Hunt, 1993 ).
Although it is now known that the outer surface of the skull does not reflect its inner surface, let alone structures of the brain, it should be pointed out that almost 200 years later, a positron emission tomography study reported the involvement of the prefrontal cortex of the dominant hemisphere in visual calculation, a localization comparable to that of Gall and Spurzheim Sakurai, Momose, Iwata, Sasaki, &Kanazawa, 1996 .
To disprove the phrenologist theories which he thought lacked scientific methodology, the French physiologist P. Flourens developed around 1840 the experimental technique of ablation. In 1861 P. Broca and a few years later, in 1874,
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C. Wernicke introduced the clinico-anatomical correlations into neurophysiology and thus truly scientific methods were established in the field. These developments were greatly facilitated by new staining methods for neuronal tissue. Another important factor was the discovery of the X-rays by W.C. Röntgen in 1895 and their rapid utilization for imaging. It was then, that the first articles on arithmetical disorders and their site of localization in the brain were published.Lewandowsky and Stadelmann described in 1908 a patient with deficits in addition, subtraction and the decimal system, who was found to have a hematoma localized at the left occipital side of the cortex Lewandowsky &Stadelmann, 1908 . They concluded that the calculation center of the brain is localized at that site. This meant two significant achievements: on one hand it was the first attempt to make a clinico-anatomical correlation in a disorder of arithmetic and Lewandowsky and Stadelmann were the first to describe arithmetical abilities independent from language abilities. They pointed out that in order to make a correct anatomic correlation, only cases of calculation disorders without aphasia should be considered. Seven years later, Poppelreuter described patients with shooting injuries to the head. He found arithmetical disorders in 12 of his patients. These also had a left- or double-sided hemianopsy. Concluding from the localization of the optic center, he considered the site of arithmetical abilities to be located in the cortex of the occipital lobes Poppelreuter, 1915 . In 1917 Sittig examined the calculation abilities of aphasic patients and proposed an influence of the left retrolrolandic area on number writing Sittig, 1917 . Peritz postulated 1918 the calculation center to be in the left angular gyrus (cited in Rüdiger, 1994 ).
The first systematic analysis of arithmetical disorders was conducted by Henschen, who coined the term acalculia (’Akalkulie‘). He reviewed 305 cases with calculation disorders in the literature in addition to 67 cases of his own. Henschen stated that calculation abilities are to be considered separately from language abilities and their disorders. He concluded the existence of ’separate centers for letters (words) and figures (numbers)‘. But he was also convinced that these centers would ’only with stimulation by adequate stimuli become active and conscious‘, thus postulating a cooperation of multiple regions of the nervous system Henschen, 1919 . Analogous to language production, Henschen assumed a motoric calculation center in the third convolution of the left frontal lobe and a sensory calculation center in the left angular gyrus Henschen, 1925 . But he also determined the need for further research and, in accordance with Lewandowsky &Stadelmann, 1908 , that only very circumscribed cases of acalculia should be analyzed in order to gain further information on the brain mechanisms involved in calculation Henschen, 1925 .
Soon after Henschen, Berger was the first to make a distinction between primary and secondary forms of calculation disorders. While primary calculation disorders develop independent from other cerebral disorders, secondary calculation disorders are those, that evolve as ’a consequence of damage or the loss of other cerebral abilities.‘ Berger, 1926 . Berger thought secondary
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calculation disorders to be more common. As causes for these impairments of arithmetical abilities he found attention deficits, memory and language disorders. In 18 of his own cases with the initial presentation of a calculation disorder he diagnosed after precise clinical examination only 3 cases of primary calculation disorder. In his publication he was the first to describe isolated deficits for certain arithmetical operations, namely division and multiplication. After an anatomical examination of his cases he concluded lesions of the temporal and the occipital lobe of the dominant hemisphere to be chiefly involved in cases of primary calculation disorders. Berger deduced a collaboration of these areas and possibly the frontal cortex of the dominant hemisphere in the acquisition of calculation abilities. A publication by Head, in which he proposed a terminology linking a calculation disorder to each form of aphasia, is significant as the first publication on the matter by an English-speaking author Head, 1926 .Gerstmann published an article in 1927 in which he described a symptom cluster devised of bilateral finger agnosia, right-left confusion, agraphia and acalculia Gerstmann, 1927 . This syndrome later came to be named after Gerstmann. No specific type of calculation disorder has been ascribed to the syndrome and the syndrome itself has been the subject of considerable controversy. Gerstmann later described the syndrome to be a consequence of lesions of angular gyrus of the dominant hemisphere Gerstmann, 1940 .
Leonhard described the role of spatial conceptions for arithmetical in a subgroup of 21 in 91 high-achieving individuals. He thought these people to have an internal representation of numbers and numerosity in a spatial fashion which they use for calculation procedures. The author makes a distinction between these ’figurative calculators‘ on one hand and the ’number-picture calculators‘ and ’number-word calculators‘ on the other. The ’figurative calculators‘ either calculate by counting (4 of 21 subjects) or by ’measuring‘ inside their internal numerical representation. In his paper, Leonhard pointed out that small multiplication tasks are solved by the use of rote verbal knowledge and not mental calculation Leonhard, 1938 . In 1948 Goldstein attempted to summarize the knowledge on calculation disorders in a book on language disturbances Goldstein, 1948 .
The neurologist R. Cohn was the first to attempt to develop a comprehensive model of calculation disorders. He chose multiplication tasks as the basis for his examinations Cohn, 1961 , and reasoned that five basic abilities are necessary for this mathematical operation: the recognition of numbers and operand, number ordering, a static memory for multiplication tables, a dynamic memory for carrying over results and addition. From case studies of dyscalculia he assumed 3 main causes of dyscalculia: disturbances in number ordering, memory problems and perserverations.
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Hécaen, Angelergues and Houllier strove for a more systematic model, formed on neurophysiological knowledge. Their model sought to segregate the calculation process in its components, then specify the errors resulting from malfunction of any of these components and finally ascribe these types of errors to lesions in particular regions of the brain Hécaen, Angelergues, &Houillier, 1961 ). Hécaen et al. arrived at three different types of acalculia:
Type 1: Acalculia of the spatial type, in which the patients have problems to align digits correctly or maintain the decimal place of them or make errors such as inversion, reversal or neglecting of numbers. The authors thought this form to be linked with a ratio of 12:1 to right-sided lesions.
Type 2: Acalculia resulting from alexia and agraphia for numbers, in which the patient is unable to read or write numbers. This form can occur independent of an inability to read or write linguistic material. Hécaen et al. thought this form to be correlated mostly with posterior, mainly left-sided cerebral lesions.
Type 3: Anarithmetria as the inability to calculate. This Hécaen et al. thought to be mostly associated with posterior, dominantly left-sided lesions.
Groen and Parkman developed a model for calculation based on the measurements of reaction time for simple addition tasks. From their results with First-graders they deduced that the reaction time for a calculation task depends in a linear fashion on the complexity of the task itself Groen &Parkman, 1972 . After reaction time studies in adults they modified their model. They now proposed that the reaction time was composed of fact retrieval and counting. While results of certain calculation tasks are stored in long-term-memory and are simply retrieved in a constant time, the counting time for unknown tasks would vary depending on their problem size Parkman, 1972 . In addition to those results, Ashcraft and Battaglia found that the retrieval time for arithmetical facts stored in the long-term-memory also depends on their magnitude Ashcraft &Battaglia, 1978 .
Besides the clinico-anatomical correlations for acalculia, attempts were made to obtain functional imaging of the calculation process itself. Sokoloff et al. attempted such a task with the nitrous oxygen technique and could not find any changes of global cerebral blood flow or oxygen consumption in mental arithmetic compared to baseline Sokoloff, Mangold, Wechsler, Kennedy, &Kety, 1955 .
More telling data only came forward with the advancement of functional imaging
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techniques. Roland and Friberg used radionucleotide imaging. They measured an increased cerebral blood flow in the prefrontal, inferior frontal and angular cortices of both hemispheres when they confronted subjects with a subtraction task Roland &Friberg, 1985 . In a positron emission tomography study (PET), Sakurai et al. demonstrated an increased activation of the left prefrontal and the left posterior superior temporal gyrus in 9 subjects when presented with multiple calculation tasks Sakurai, Momose, 1996 . Dehaene et al. made a more differentiated attempt in their PET study. They found that both in multiplication and comparison tasks, the occipital cortices are activated bilaterally as well as the left precentral gyrus and the supplementary motor area. In addition, multiplication only increased activation in both inferior parietal gyri, the left fusiform and lingual gyri as well as the right cuneus when compared to baseline whereas comparison yielded increased activation in the right superior temporal, the right middle temporal as well as the right superior and inferior frontal gyri. They deduced an at least partial distinction of networks for both mathematical tasks Dehaeneet.al, 1996 .The most recently published functional imaging study was again carried out by Dehaene and his colleagues. They assessed functional magnetic resonance images (fMRI) during approximation and exact calculation tasks in three male and four female probands. The fMRI showed greater activation in the parietal lobes for approximation than exact calculation. Specifically, the inferior parietal lobe showed increased activation in areas previously shown to be involved in tasks such as visually guided hand and eye movements, mental rotation and attention orienting. In exact calculation however, there was greater activation of mostly the left inferior frontal lobe, an area shown to be involved in verbal association tasks. The authors reach the conclusion that different cerebral networks are used in both tasks. Since fMRI has a low temporal resolution there is also an alternative interpretation of the shown data, namely that the activation differences are due to a secondary stage of mathematical reasoning rather than the primary information processing. Dehaene et al. rule this interpretation out with the employment of data from event related potential (ERP) in the same tasks. ERP have a high temporal resolution and the authors were able to show significant differences corresponding to the fMRI data in these tasks during the first 400 ms of a trial, when the probands were only presented with the mathematical tasks but not yet the choice stimuli (exact calculation vs. addition) Dehaene, Spelke, Pinel, Stanescu, &Tsivkin, 1999 .
The increase of data on mental calculation created a demand for neurophysiologic models. These have the advantage of yielding testable hypothesis and thus allow for a more detailed research. It is important though, that these models are modified once certain aspects of them have been found not to be true.
McCloskey and his colleagues analyzed a number of cases of ’acquired dyscalculia‘, or Acalculia. They specified the model of Hécaen and colleagues in an attempt to make more adequate neurophysiologic predictions. McCloskey and colleagues divided the process of calculation into two major modules, the Number Processing System and the Calculation System McCloskey, Caramazza, &Basili, 1985 .
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The Number Processing System, in their model responsible for the perception of the necessary information, is subdivided into comprehension and formation of numbers, both composed of a verbal and a digit component. The Calculation System, responsible for the processing of the information is subdivided into a module for recognizing the operand, another for access to basic arithmetical knowledge and a third for the operation itself.
McCloskey et al. claimed that with this model they could not only describe their cases of acalculia but also make predictions about future cases and eventually find anatomical correlates for the different components McCloskey, 1992 , McCloskey, Aliminosa, &Macaruso, 1991 .
Campbell and Clark countered the model of McCloskey et al. with their own Encoding Complex Theory of calculation. They argued that McCloskey et al. had failed to produce anatomical correlates of their proposed models and that several of the deficits described by McCloskey et al. may also occur in probands without deficits in calculation. They thus advised that the model should be abandoned Campbell &Clark, 1988 . Campbell and Clark suggested that mental calculation is a collaborative process of associated networks that relate through inhibitory synaptic transmission Campbell, 1990 . Consequently the authors concluded that disorders of calculation seldom occur solitarily, but most commonly in connection with other disorders Clark &Campbell, 1991 .
A more specific model of neuronal networks was devised by Dehaene and Cohen. They reviewed the literature on acalculia and its anatomical correlates as well as their own cases of acalculia and proposed a ’Triple-code model‘ for calculation Dehaene &Cohen, 1995 . Their model postulates three main representations of numbers in the brain:
1) a visual arabic code located to the left and right inferior occipito-temporal areas of the brain. It represents numbers as strings of digits. The function of this representation is multidigit operation and the evaluation of numbers are even or uneven.
2) a magnitude code, located in the left and right parietal areas of the brain. Here, numbers are represented as distributions along an oriented number line. This serves to evaluate of the quantity and proximity of numbers as well as their comparison.
3) a verbal code, located in the perisylvian areas of the dominant hemisphere. It represents numbers as sequences of words. By this representation arithmetical rote memory is accessed and the plausibility of results can be controlled.
With the triple-code model, Dehaene and Cohen were not only able to fit in the results of many previously described cases but could also make adequate predictions about the forms of acalculia connected with lesions of specific areas of the brain. Their model of verbal code and its use in arithmetical rote memory retrieval, first postulated by Leonhard Leonhard, 1938 ,
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would also explain the strong interference between multiplication and object naming tasks, as described by Campbell Campbell, 1994 and why multilinguistics subjects retrieve to the language in which they first learned arithmetic when confronted with simple calculation tasks Shannon, 1984 . Dehaene and Cohen showed in detailed case descriptions that there is evidence for asemantic routes for transcoding numerals, i.e. patients with impaired calculation abilities whose ability to read and write down arabic numbers is preserved Dehaene &Cohen, 1997 . This contradicts the model of McCloskey et al. which claims the existence of a central quantitative representation, independent of its form. They further argued, that the model of McCloskey et al. fails to explain the dissociation of the sparing of certain types of operations (e.g. subtraction) when others (e.g. addition and multiplication) are impaired. Additional dissension against unique internal numerical representation comes from a case study of McNeil and Warrington. Their patient was able to perform simple oral addition and subtraction but his ability for written addition was severely impaired McNeil &Warrington, 1994 .Further evidence for the triple-code model comes from a study Dehaene et al. carried out in bilingual probands. Eight students, fluent in both Russian and English, were trained on a set of approximate and exact additions in either language. When these trained tasks were subsequently tested the probands performed significantly faster on the exact addition tasks in the language they were previously trained in, regardless of the language itself. For approximation tasks the performance was equivalent in the two languages. The probands were then presented with new tasks in a similar numerical magnitude. The authors found that the subjects performed faster only on the previously trained exact calculations. Dehaene et al. draw the conclusion that these data are further proof for at least two independent arithmetical representations. While exact arithmetic relies on a language based representation, approximation tasks and advanced mathematical understanding relies more on a language independent conceptualization Dehaene, Spelke, 1999 .
The Specific disorder of arithmetical skills, first described by Cohn Cohn, 1968 , differs in at least two important aspects from Acalculia, summarized by Rourke &Conway, 1997 :
First, the tasks involved in executing learned arithmetical skills differ from those involved in learning arithmetic. While in the former, information is to a large extend retrieved, in the latter, processes such as maturation of concept formation and adaptive reasoning skills are involved.
Second, there is evidence that the tasks involved in learning arithmetic are associated with right-hemispherical processes. It is only after the successful learning that arithmetical facts can be retrieved and executed by left-hemispherical systems.
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As a consequence, in the clinical cases involving adults with acalculia there will be a dominance of left-sided lesions, whereas in children also right-hemispheric lesions or dysfunctions will interfere with arithmetical learning.
The facts about the mathematical development of children have recently been summarized in a comprehensive book Geary, 1994 . Here, only the facts most pertinent to our study are referred to.
There is now considerable evidence that human infants already possess an impressive number of basic skills necessary for mathematics. Wynn demonstrated that infants can most likely even perform simple calculations. She showed a small number of objects to 32 infants with a mean age of 5 months. Then a screen was raised and the babies saw a hand either adding or taking one object away. After that, the screen was dropped, with either a correct or an incorrect number of objects on display. Wynn found that the infants looked significantly longer at the incorrect display, looking time being now a standard procedure in the testing of infant cognition Wynn, 1992 . In the light of these results Bryant Bryant, 1992 raised the question, whether Piaget Piaget, 1952 was right, when he claimed that infants do not understand the concept of calculation.
Starkey and his colleagues found that infants can recognize the number of a small set of objects and can tell whether there has been a change in number. They can make a correct connection between the number of acoustic stimuli (i.e. drumbeats) and the number of simultaneously displayed objects. Infants were even able to correlate the number of acoustic stimuli with the number of displayed objects when the rate and duration of the acoustic stimuli were varied randomly. This led Starkey et al. to assume that infants possess a sense of numerosity Starkey, Spelke, &Gelman, 1990 . Mix et al. were not able to fully reproduce the findings of Starkey et al. They could only find an significant correlation to the infants‘ looking time when the acoustic stimuli were constant in rate and duration. They cast a doubt on the thesis that infants are already able to genuinely represent the numerosity of a set of objects Mix, Levine, &Huttenlocher, 1997 . In a recent review article, Geary gives an overview of ’Potential Biologically Primary Mathematical Abilities‘, citing not only studies in humans, but also in mammals and other animal species as evidence. The author concludes that the ability of numerosity, ordinality, counting, and simple arithmetic are likely inherent mathematical abilities, as they can be found independent from cultures and even in nonhuman primates Geary, 1995 .
The fact that infants already possess certain mathematics skills, raises the question which factors contribute to these abilities. Gillis et al. found in a study of 264 reading-disabled and 182 matched control twin pairs a significant genetic influence on mathematics performance score, with the genetic factors predicting the mathematical performance even slightly stronger than environmental factors Gillis, DeFries, &Fulker, 1992 .
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Benbow and Stanley published data on the sex differences for mathematical reasoning ability. In a first publication, they found a significant majority of boys among the mathematically precocious, which even increases through the high school years Benbow &Stanley, 1980 . They then extended their research to include all levels of mathematical giftedness. The results, however, were similar. By the age of 13 years Benbow and Stanley found in 39820 high school students a highly significant sex difference in mathematical reasoning ability in favor of boys across all levels of capability. This difference was most pronounced at the high end of the distribution: here boys outnumbered girls by a ratio of 13:1, even though girls and boys were matched by intellectual ability, age, grade and voluntary participation. No such differences for verbal abilities were found Benbow &Stanley, 1983 .
The reasons for this difference remain unclear. In their sample, Benbow and Stanley could not find gender differences in formal training in mathematics. Neither did their data support the theory of divergent gender socialization playing a significant role Benbow &Stanley, 1983 . The authors suspect that greater spatial ability in males might play a role Benbow &Stanley, 1980 . Another contributing factor might be a specific ’mathematics anxiety‘. Hembree found this to have a negative effect on mathematics performance and to be more pronounced in girls Hembree, 1990 .
Schwank studied the problem-solving algorithms in girls from an educational perspective. The author comes to the conclusion, that girls approach problems in a conceptual way when boys use a sequential manner. Thus boys find their solutions ’in dialogue with the material‘ whereas girls will try to solve the problem as a whole Schwank, 1990 . Educational techniques are usually poorly adapted to develop conceptual thinking. Von Aster suggests that consequently, if a girl has a problem with a mathematics task, she should not be encouraged to ’try again‘ but to ’think it through again.‘ von Aster, 1999 . A review of the findings regarding gender differences was conducted by Geary. The author suggests that the sex differences are limited to biologically secondary mathematical domains. Geary concludes from the literature that there might be two main reasons why the male mathematical performance is consistently found to be better: more elaborate neurocognitive systems that support spatial naviagation as a result of sexual selection, and different social preferences and styles. These advantages lead to a positive feedback mechanism and further interest in mathematics, increasing the male advantage in certain mathematical domains Geary, 1996 .
There is sufficient evidence that there is a specialization between the hemispheres of the brain in their contribution to certain cognitive performance tasks Geschwind &Galaburda, 1987 . There are also reports of distinct patterns in hemisphere specialization, such that left-handers might have a
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less pronounced lateralization than right-handers, whereas males have a more lateralized pattern of cerebral organization than females. Also, intellectually gifted adolescents were found to rely more on right-hemisphere functioning in basic information processing O'Boyle, Gill, Benbow, &Alexander, 1994 .Annett and Manning investigated the relation between arithmetical ability, hand preference and hand skill. In a general population sample of school children they found arithmetical ability to be positively associated with left-handedness in both sexes Annett &Manning, 1990 . Accessory to these results are the data of Peters, who found strong right-handedness to be significantly associated with a lack of mathematical giftedness Peters, 1991 .
O‘Boyle et al. observed an enhanced activation of the right hemisphere in the EEG of mathematically gifted, left-handed males compared to normal controls when confronted with a basic processing task O'Boyle, Alexander, &Benbow, 1991 . They later corroborated these findings, when they showed that the finger tapping rate of both hands was reduced in 24 male, left-handed, mathematically precocious subjects while confronted with a verbal task. In 16 controls of average ability the verbal task reduced the tapping rate of the right but not of the left hand. The authors suggest that enhanced right hemisphere involvement is the physiological correlate of mathematical precocity in males O'Boyle, Gill, 1994 .
Piaget devised a theory on the ’Child‘s conception of number‘ Piaget, 1952 . That theory was evolved by Aebli into a more comprehensive didactic concept Aebli, 1973 . He emphasized the process of learning mental calculation as an advancing abstraction from reality in four steps. Other didactic models of mathematics learning are analogous (e.g. Grissemann, 1996 ).
The first step is an action including real objects. Already there is an abstraction, but it can be compared with experience. (e.g. ’If I have five apples and take three away, how many are there left?‘)
The second step is a symbolic illustration of the arithmetical operation. The realistic representation is modified into a more abstract form. (e.g. ’If I erase three of the five circles on the blackboard, how many are there left?‘)
The third step is the transformation of symbols into numbers. These have the advantage of universal applicability. (e.g. ’How much is 5-3?‘)
As the fourth and final step Aebli identifies the automatization of known results through repetition.
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Cohn described in 1968 a group of 12 children with deficits in mathematics skills and coined the term ’Developmental dyscalculia‘ for their condition Cohn, 1968 . Slade and Russell described four children with a Specific disorder of arithmetical skills. They maintained it to be a primarily cognitive deficit Slade &Russell, 1971 .
The first systematic assessment of the Specific disorder of arithmetical skills was undertaken by Kosc. He defined it as a ’structural disorder of mathematical abilities which has its origin in a genetic or congenital disorder of those parts of the brain that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities adequate to age, without a simultaneous disorder of general mental functions.‘ Kosc studied a population of 375 Slovak fifth-graders and found 24 (6.4%) of them to have a Specific disorder of arithmetical skills Kosc, 1974 . Four years later, Spellacy and Peter laid foundation to the criterion of discrepancy between arithmetical and general achievement to define the disorder in a publication about a group of 14 children with a Specific disorder of arithmetical skills Spellacy &Peter, 1978 .
Badian attempted to identify the prevalence of learning disorders in a study of 1476 schoolchildren of grades 1 through 8. He defined poor achievement as a score below the 20th percentile of the Stanford Achievement Test. The author found low reading achievement in 2.2%, low mathematics achievement in 3.6% and low achievement in both reading and mathematics in 2.7% of his sample Badian, 1983 . In total he identified 6.3% of schoolchildren to have a below-average achievement in mathematics. Häußer examined 181 schoolchildren in a rural area and detected 12 children with a Specific disorder of arithmetical skills, equaling a prevalence rate of 6.6% Häußer, 1995 . The most comprehensive investigation of the demographic features of the Specific disorder of arithmetical skills was conducted by Gross-Tsur et al. They tested 3029 (75%) of the fourth-graders of the city of Jerusalem. Children were diagnosed with a Specific disorder of arithmetical skills if their arithmetic achievement score was equal to or below the mean score of children being two grades younger. Gross-Tsur et al. thus found a prevalence of the Specific disorder of arithmetical skills of 6.5% Gross-Tsur, Manor, &Shalev, 1996 .
Klauer used a slightly different approach when examining a sample of 546 third graders. He defined the Specific disorder of arithmetical skills as a discrepancy of two standard deviations between actual mathematics achievement and expected mathematics achievements as predicted by performance in other academic areas. The expected performance was determined by a regression analysis. Klauer thus arrived at a prevalence rate of 4.4% for the Specific disorder of arithmetical skills. In his sample, the condition was more prevalent in girls Klauer, 1992 . Lewis et al. found a prevalence rate of 1.3% for a Specific disorder of arithmetical skills and a prevalence of 3.6% of schoolchildren with combined mathematics and reading underachievement Lewis, Hitch, &Walker, 1994 .
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A number of publications has been concerned with the way children with a Specific disorder of arithmetical skills access arithmetic problems. Bull and Johnston examined the short-term memory, processing speed, sequencing ability and retrieval ability of seven-year-old children. They found that processing speed was the most reliable predictor for the arithmetical abilities of the children. The authors conclude that arithmetic difficulties in children are mostly due to their inability to automate facts Bull &Johnston, 1997 . A study by Barrouillet and his colleagues of seventh-graders with learning disabilities demonstrated that these children did not only make more errors and had problems with fact retrieval but that they also had difficulties with inhibiting the retrieval of associations irrelevant to the given task Barrouillet, Fayol, &Lathulière, 1997 . Jordan and Montani observed that third graders with specific difficulties in mathematics did make more errors than controls under timed conditions. When there was no time limit for the resolution of the given tasks however, there were no significant differences between both groups, but the group with difficulties in mathematics did rely more than the control group on primitive back-up strategies such as finger-use or counting. A third group with general learning difficulties made more errors than controls under any conditions Jordan &Montani, 1997 . The use of back-up strategies, immature arithmetical concepts and a low rate of instant fact retrieval in children with underachievement in mathematics was also found by other researchers Geary, 1990 ; Geary, Bow-Thomas, &Yao, 1992 ; Hitch &McAuley, 1991 ; Swanson, 1993 .
A transversal study of students with a Specific disorder of arithmetical skills was conducted by Ostad. He examined students with a Specific disorder of arithmetical skills in grade 1 (n=32), grade 3 (n=33) and grade 5 (n=36) and compared them to corresponding numbers of students without difficulties in mathematics. The author was not only able to confirm the finding that students with a Specific disorder of arithmetical skills relied more on a small variety of rather primary back-up strategies, but also that the use of these strategies did not change significantly from year to year Ostad, 1997 . Similar to those results is a longitudinal 3-year prospective follow-up study of children with a Specific disorder of arithmetical skills by Shalev et al. Their results indicate that in almost half of the affected children the Specific disorder of arithmetical skills persists independent of gender, socioeconomic status, and educational intervention. This persistence is most likely in children with severe arithmetic disabilities and those with a sibling who also suffers from a Specific disorder of arithmetical skills Shalev, Manor, Auerbach, &Gross-Tsur, 1998 .
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In his careful examination of the subject, Kosc also proposed six subtypes of the Specific disorder of arithmetical skills, four of which are closely resembling those of Hécaen et al. Hécaen, Angelergues, 1961 , described above. In addition, he also proposed two additional forms of the Specific disorder of arithmetical skills: the ’practognostic‘ form with the inability to manipulate objects for mathematical purposes and the ’ideognostic‘ form, standing for the inability to understand mathematical concepts per se Kosc, 1974 .
Rourke and his colleagues examined in a series of publications from 1978-1993 the Specific disorder of arithmetical skills from a neuropsychological perspective. They divided children with specific learning disabilities into two groups: children with deficiencies mainly in graphomotor and spelling performance (subtype A) and children with an impaired mechanical arithmetics and spelling performance but normal reading and mathematics comprehension (subtype R-S). They then went on to distinguish the two groups as follows:
The subtype R-S children are characterized by deficiencies in the rote aspects of psycholinguistic skills and pronounced deficiencies in the more complex semantic-acoustic aspects of psycholinguistic skills. These deficiencies are more pronounced in children of younger age inside the group. Older children of the group have outstanding deficiencies in the semantic-acoustic aspects of psycholinguistic skills. Rourke et al. thought the R-S-group children to have a higher performance IQ (pIQ) than verbal IQ (vIQ). Their performance on visual-spatial-organizational, psychomotor, tactile-perceptile and nonverbal learning tasks is normal. Rourke et al. thought these children to be well adapted socio-emotionally.
The subtype A children were characterized by deficient performances on visual-spatial-organizational, psychomotor, tactile-perceptual as well semantic-acoustic tasks of novel material. These children have profound problems on nonverbal learning tasks and show no benefit from informational feedback or continued experience. They have a higher vIQ than pIQ. Socio-emotionally they exhibit deficiencies in the adaptation to novelty, social competence, emotional stability and activity level.
Rourke et al. assumed the R-S-group to be an expression of right-hemispheric functional disorders and the group A, or Non-verbal-learning-disorder (NLD) children, as being impaired by left-hemispheric disorders. Both subtypes can show arithmetical disabilities, but the type of arithmetical problems of the groups will be different: while the R-S group will make more mechanical arithmetical errors the subgroup A children will show more severe arithmetical errors as well as difficulties with the spatial organization of numbers. Therefore the focus of assessment and therapy for both groups has to be substantially different Rourke, 1993 .
Von Aster tested the hypotheses of Rourke and colleagues in a sample of 41 psychiatrically referred schoolchildren. 20 of those children had a specific and 21 a combined disorder of arithmetical skills. He found no IQ differences for 21 of those children. 6 of 9 boys but only 1 of 11 girls with a specific arithmetical disorder were
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diagnosed with a Gerstmann syndrome. Von Aster found the highest rate of neurological soft signs among boys with a combined disorder of scholastic skills and the lowest among girls with a specific arithmetical disorder. Internalizing disorders were most frequent in children without an IQ-discrepancy or a Gerstmann syndrome. Girls with specific arithmetical disorders manifested almost exclusively internalizing disorders and boys mostly externalizing disorders. In conclusion, von Aster is only partially able to support the findings of Rourke et al. and argues for a gender difference in learning disabilities von Aster, 1994 .Similar data to those of von Aster were published by Shalev et al. in 1995. 25 children with a Specific disorder of arithmetical skills who showed evidence for either right or left-hemispheric dysfunctions were thoroughly assessed medically and with both an arithmetical and a psychological test battery. Children with an abnormal perinatal history, neurologic disorder or signs of bilateral dysfunction were not included. The authors were unable to identify patterns of arithmetical errors that were specific to left or right hemisphere dysfunction. They concluded that there is no specific neuropsychological profile for children with a Specific disorder of arithmetical skills, but that input from both hemispheres is necessary for the development of arithmetical skills. In contrast to the suggestions of Rourke et al., the authors found that left hemisphere dysfunction had a more severe impact on arithmetical abilities Shalev, Manor, Amir, Wertman-Elad, &Gross-Tsur, 1995 .
Von Aster proposed a neuropsychologic model of arithmetical learning based on the triple-code-model by Dehaene and Cohen (see chapter 1.4.2.1.5.) In his model, arithmetical facts first pass through a basal processing mechanism. If processed correctly, the facts then meet a modular structure for numerical representation, the triple-code-model as outlined by Dehaene and Cohen. Based on the findings in very young children (see chapter 1.4.2.2.1.), von Aster proposes the magnitude code to be a congenital ability. The verbal code is then developed in early childhood and the visual-arabic number representation in school von Aster, 1999 (see Figure 2).
Figure 2: Triple code model for arithmetical learning
From that model, von Aster derives four different types of the Specific disorder of arithmetical skills:
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In the first type, the basic processing mechanism is disturbed, resulting in global deficits in arithmetical abilities.In the second type, the verbal representation of arithmetical abilities is deficient. Here especially the arithmetical abilities of counting and adding are impaired.
In the third type, the visual-arabic code is impaired. In this type of the Specific disorder of arithmetical skills especially reading, writing and comparison of written numbers are affected.
In the fourth type, due to a disturbed magnitude code, almost all arithmetical abilities are insufficient in spite of an intact basic processing capacity.
Most data on the socio-emotional characteristics of children with learning disabilities are not linked to a specific form of learning disorder. Waldie and Spreen examined a group of 65 learning disabled subjects who reported police contact at a mean age of 19 years. Six years later, they interviewed the same subjects again and then separated them into groups according to whether repeated police contact had occurred (n=40) or not (n=25). The authors found that of all the examined factors, only poor social judgement and impulsivity were reliable predictors, as subjects with high scores on these domains were much more likely to report repeated police contact Waldie &Spreen, 1993 . Huntington and Bender point out, that children with learning disorders attribute failure and success more to internal, personal factors and have a poorer self-concept Huntington &Bender, 1993 . Wright-Strawderman and Watson assessed 53 children with learning disabilities with the Children‘s Depression Inventory and found a prevalence of 19 children (35.8%) who scored in the depressed range Wright-Strawderman &Watson, 1992 . This is comparable to the rate of 26% found by Goldstein et al. Goldstein, Paul, &Sanfilippo-Cohn, 1985 and the unpublished results of Wong (40%) and Chaskelson (29%) (cited in Wright-Strawderman &Watson, 1992 ). There are also indications that children with learning disorders are at an increased risk for suicide Hayes &Sloat, 1988 .
It seems that the characteristics, which constitute risk factors for the socio-emotional well-being of learning-disabled children, are prevalent in children and adolescents with a Specific disorder of arithmetical skills at a rate which at least equals that of all learning-disabled subjects. In addition to the results of Rourke et al. (see chapter 1.4.2.2.7.) other publications have been concerned with the socio-emotional characteristics of children with a Specific disorder of arithmetical skills. Johnson and Myklebust described a group of 14 children with a Specific disorder of arithmetical skills. They found them to be not well adapted socially, poor at estimating distance and time and lacking self-help skills Johnson &Myklebust, 1967 . Badian and Ghublikian examined 360 students and found that 16 boys with underachievement in mathematics relative to reading were rated significantly
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lower on a personal-social behavior skill by their teachers than two comparison groups. These ratings were very similar from both English and Mathematics teachers. They did not find significant personal-social characteristics for either girls with underachievement in mathematics or in any of the other groups Badian &Ghublikian, 1983 .
Comorbidity is the co-occurence of at least two different disorders in the same individual. The best-documented comorbidity for the Specific disorder of arithmetical skills is perhaps the co-occurence of dyslexia. Ackerman et al. proposed that the majority of children with a reading disability evident early in childhood will go on to develop a deficiency in arithmetical skills in the higher grades Ackerman, Anhalt, &Dykman, 1986 . Badian found that almost half of the students with underachievement in mathematics have also a reading deficiency Badian, 1983 . Lewis found an almost threefold higher prevalence for a combined deficiency of scholastic skills than for underachievement in mathematics alone Lewis, Hitch, 1994 (see also chapter 1.4.2.2.5.) Gross-Tsur et al. found a prevalence of 17% for dyslexia in children with a Specific disorder of arithmetical skills Gross-Tsur, Manor, 1996 . The authors cited above hypothesize about the reasons for this comorbidity, but data for the analysis of the problem are scarce.
There is now considerable evidence of a genetic contribution to reading disabilities DeFries, Fulker, &LaBuda, 1987 as well as mathematic performance Gillis, DeFries, 1992 . Light and DeFries attempted to find the factors contributing to this comorbidity. They examined 259 twin pairs (149 monozygotic, 111 same-sex dizygotic) in which at least one twin had a history of reading problems and compared them to a sample of 134 monozygotic (MZ) and 93 same-sex dizygotic (DZ) twins without a history of learning problems. They found that 49% of the MZ and 32% of the DZ twins of reading disabled probands were mathematics disabled. Light and DeFries were able to show that approximately 26% of proband‘s reading deficits are due to genetic factors which also influence mathematics performance. The authors conclude that genetic and environmental factors contribute almost equally to the observed covariance in reading and mathematics scores Light &DeFries, 1995 .
Another disorder that is often observed in co-occurrence with the Specific disorder of arithmetical skills is the Attention Deficit Hyperactivity Disorder (ADHD). Shaywitz and Shaywitz noticed that children with ADHD often show problems in arithmetical achievement Shaywitz &Shaywitz, 1984 . In his aforementioned publication Badian found that as many as 42% of children with low mathematics achievement showed evidence of attentional-sequential deficits Badian, 1983 . In a publication by Gross-Tsur et al. 26% of 143 children with a Specific disorder of arithmetical skills showed evidence for ADHD in their parent‘s or teacher‘s rating scales Gross-Tsur, Manor, 1996 .
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Stanford and Hynd found that learning disabled children and those with ADHD were rated equally by their teachers and parents on subjects such as daydreaming and inactivity. Only on closer questioning the observers perceived other forms of inattention and withdrawal in the two groups Stanford &Hynd, 1994 .Evidence for further comorbid disorders is rare. But a publication by Shalev and Gross-Tsur indicates that this is more likely due to negligent observation than the absence of other comorbidities. The authors assessed seven third-graders with a Specific disorder of arithmetical skills who had not made significant academic progress despite their schooling in a special arithmetic class. They found neurological conditions with a direct influence on the children‘s cognitive abilities in all children. Four of these children had Attention Deficit Disorder without hyperactivity (ADD), one suffered petit mal-seizures, one ADHD and a developmental form of Gerstmann syndrome and one had severe dyslexia for numbers. Shalev and Gross-Tsur conclude that a thorough medical and neuropsychological assessment is necessary when the diagnosis of a Specific disorder of wal skills is made Shalev &Gross-Tsur, 1993 , a view shared by others (i.e. O'Hare, Brown, &Aitken, 1991 ).
Although the Specific disorder of arithmetical skills itself has not received as much attention as other learning disabilities (see chapter 1.4.) there is a considerable number of studies on the psychoeducational characteristics of other conditions indicating that a Specific disorder of arithmetical skills might be associated with these predicaments. In a recent review paper by Gross-Tsur et al. the authors conclude that: ’In fact, (the Specific disorder of arithmetical skills) is the most frequently encountered learning disability in children with epilepsy, fragile X carriers, Turner‘s syndrome and phenylketonuria.‘ Gross-Tsur, Manor, &Shalev, 1993 .
Other conditions should be added to that list. Klebanov et al. examined the school achievement of children with very low birthweight. They found those children with a birth weight below 1,000 grams to be most severely affected by risks such as grade failure and placement in special classes, even when they controlled for maternal education and neonatal stay. The only persistent academic difference of significance however, was in the mathematics score of the Woodcock-Johnson Achievement Battery. The authors controlled the group of normal birth-weight children to those of extremely low birth weight for intelligence. While the differences between both groups in reading achievement then abided, the extremely low birth-weight children still scored significantly lower than controls on arithmetical tasks Klebanov, Brooks-Gunn, &McCormick, 1994 . A study from Vohr and her colleagues suggests that mainly those low birth-weight infants might be affected that develop broncho-pulmonary dysplasia (BPD). While the full-scale IQ-scores of 15 BPD children were similar to those of weight-matched children the BPD children had a comparable reading but significantly lower arithmetic score on the Wechsler Intelligence Scale for Children - Revised edition (WISC-R)
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Vohret.al, 1991 .Prenatal alcohol exposure also seems to affect the arithmetical abilities of children. Aronson and Hagberg made a follow-up of 24 children of alcoholic mothers. The authors discerned difficulties in mathematics, logical conclusion, visual perception, short-term memory and spatial relation abilities in these children Aronson &Hagberg, 1998 . Analogous to these results, Streissguth et al. found in a series of studies a strong correlation between maternal alcohol consumption during pregnancy on one hand and learning deficits as well as poorer spatial ability on the other. The level of impairment was dependent on the consumed amount of alcohol and the most prominent learning deficits were those in arithmetic Streissguth, Bookstein, Sampson, &Barr, 1989 , Streissguthet.al, 1994 , Streissguth, Barr, Sampson, &Bookstein, 1994 . In a later study, the mathematics deficits of these individuals were examined more profoundly. Koperafrye, Dehaene and Streissguth observed particular difficulties in calculation and estimation relative to controls, but intact number reading and writing abilities Koperafrye, Dehaene, &Streissguth, 1996 .
Segalowitz and Brown questioned 616 high-school students about mild head injuries. They found that, contrary to a hospital-reported rate of 2-3%, there was a prevalence of head injuries of 31.2% among these students. They did not find an increased incidence of dyslexia or speech impairment in the group with head injuries, but they could show an association to depression, hyperactivity and mixed handedness. Although they did not assess academic achievement, the authors showed a correlation of head injuries and dislike of mathematics as a subject. This dislike was more pronounced the earlier the injury had occurred Segalowitz &Brown, 1991 . In the light of these data a case study by Levin et al. is interesting. They describe a 17-year-old boy who had sustained a right parietal skull fracture and right temporal hemorrhage at the age of 7 months. He showed a disproportionate impairment on arithmetical approximation and transcoding tasks, naming tasks as well as reading and spelling. In a fMRI study the authors were able to show that in this patient an early interhemispheric transfer of visuospatial skills, usually located in the right parietal area, to left parietal region had taken place Levinet.al, 1996 .
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