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Chapter 6. General discussion

6.1 The terminology of the Specific disorder of arithmetical skills

Many different names have been employed to label the inability of children to acquire basic arithmetical skills. In English publications the term ’developmental dyscalculia‘ Cohn, 1968 , Kosc, 1974 , Gross-Tsur, Manor, 1996 , von Aster, 1994 , has been the original name for the disorder and is most frequently used, but also the singular ’dyscalculia‘ has been used synonymously Spellacy &Peter, 1978 , O'Hare, Brown, 1991 . Other authors employ the names ’specific arithmetical difficulties‘ Lewis, Hitch, 1994 , Hitch &McAuley, 1991 , ’specific mathematical difficulties‘ Jordan &Montani, 1997 , ’arithmetic learning disabilities‘ Rourke, 1993 , Trembley, Caponigro, &Gaffney, 1980 , Siegel &Feldman, 1983 , Shekim &Dekirmenjian, 1978 , ’mathematic disabilities‘ Garnett &Fleischner, 1987 , Geary, 1993 , Ginsburg, 1997 or ’arithmetic disorders‘ Batchelor, 1989 . Further authors, especially those embracing the neuropsychologic model of Rourke and colleagues, consider the inability to acquire arithmetic knowledge as cardinal symptom of the ’nonverbal learning disorder‘ Rourke, 1993 , Brookshire, Butler, Ewing-Cobbs, &Fletcher, 1994 , Rourke, 1989 or a symptom of the ’developmental right hemisphere syndrome‘ Gross-Tsur, Shalev, Manor, &Amir, 1995 . In the German literature, the ICD-10 translation for the diagnosis is ’Rechenstörung‘ (calculation disorder) Dilling, Mombour, &Schmidt, 1993 , but the terms ’Umschriebene Rechenschwäche‘ (circumscribed weakness in calculation) von Aster &Göbel, 1990 and ’Dyskalkulie‘ Grissemann, 1996 are also used.

This cacophony could be considered unimportant. Unfortunately it is of significance for several reasons. Without a common terminology it is markedly harder to make the condition better known to psychiatrists, psychologists, teachers and educators, let alone the general public, which leads to a low level of understanding for the children with the condition. This stands in striking contrast to dyslexia, where the affected children can often benefit from an extensive network of public health resources, special education and therapeutic interventions. This problem was already noted more than 20 years ago Kosc, 1974 . Since then, the importance of the problem has grown rather than diminished. The resources for special education are limited and families can only gain access to them on the basis of acknowledged diagnoses.

Secondly, each of the aforementioned terms are each defined slightly different by its authors. This makes it harder to compare the sparse data on the field and to find scientifically the most effective ways to treat the condition. Consequently, it would be highly desirable to have a common name and the same diagnostic criteria for the

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As described in chapter 1.3. we used in the present publication the name and the diagnostic criteria of the International classification system of the World Health Organization (ICD-10), as it came closest to the criteria of uniformity we desired. But, as we could show in chapter 4, and as it is argued by many of the leading researchers on the field (e.g. von Aster, 1994 ; Gross-Tsur, Manor, 1996 ), at the core of the condition is an insufficiency to acquire arithmetic knowledge, caused by different factors. This can eventually lead to a ’Specific disorder of arithmetical skills‘. However, the loss of previously present arithmetical skills also leads to the clinical picture of a ’Specific disorder of arithmetical skills‘. We therefore think that the term ’Developmental dyscalculia‘, chosen by the earliest authors and used by many of the leading contemporary authors of the field should be universally used, which beyond the fact that it is a historically and scientifically valid designation has the potential of wide international use.

6.2 The validity of the diagnosis of a Specific disorder of arithmetical skills

As mentioned in chapter 1.4. and noted by many authors, research on the field has been rather parsimonious for indeterminate reasons. Levine and his colleagues suspect, that since arithmetical knowledge is often equaled with intelligence, individuals with the disorder are simply considered less intelligent and consequently given less consideration. Furthermore, the authors argue, the overwhelming majority of people, including medical doctors, psychologists and teachers, did experience a level of mathematics, above which they did not fully comprehend the underlying concepts. This leads to the statement: ’I was never any good at mathematics, either.‘, underestimating the grave deficits of the affected children Levine, Lindsay, &Reed, 1992 .

In our sample we found more girls than boys with a suspected Specific disorder of arithmetical skills. As shown in chapter 1.4.2.2.2. there is much evidence that boys have an advantage over girls in mathematical reasoning ability. As opposed to dyslexia, it could be shown that the Specific disorder of arithmetical skills affects both genders at least equally Gross-Tsur, Manor, 1996 , if it is not more prevalent in girls Klauer, 1992 . In their socialization process girls are taught to rely more on quiet, internalizing problem-solving strategies, whereas boys are taught a more aggressive, externalizing approach. This could be an explanation why a learning disorder mainly affecting boys requires more attention in classrooms and subsequently in the medical and psychological communities.

The problems cited above have led some to question the existence of a Specific disorder of arithmetical skills altogether. Grissemann for example advises with regard to the problem ’to overcome the medical paradigm‘ Grissemann, 1996 .

We do not think that everybody who is slow in learning mathematics should be treated by specialists. The diagnostic criteria of the ICD-10 clearly demand that ’the

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These findings are supported by our own data. In chapter 4 we could show that there are children with a distinct inability to acquire mathematical skills. As other investigators ( Kosc, 1974 ; Shalev &Gross-Tsur, 1993 ), we found accompanying medical conditions in some of our probands, could prove functional lesions and even found certain structural abnormalities with modern imaging techniques in others. We did not find a high correlation of the teachers' evaluations with their students' results on standardized academic achievement tests, or with the results of our thorough clinical and neuropsychological validation process.

We therefore conclude from our own findings, as laid out in chapter 4.4., as well as from the literature mentioned in chapter 1.4.2.2.9. that a careful clinico-neurological and neuropsychological evaluation process should be obligatory before the establishment of the diagnosis of a Specific disorder of arithmetical skills. We find this of particular importance, since the symptomatology has been shown to be persistent in longitudinal studies ( Ostad, 1997 ; Shalev, Manor, 1998 ), and the socio-emotional well-being of the affected individuals is at risk (see chapter 1.4.2.2.5.). We do not mean to deprecate the intense and extensive need of these children for special education, which clearly falls into the educational realm. We only argue that at the beginning of the therapeutic process should stand an exact and thorough examination of the individual's resources, and that treatable medical conditions contributing to the diagnosis should be attended to.

Furthermore, as mentioned in chapter 6.1. we would find it desirable if the condition became better known among all professions concerned with the care of children, as a problems to acquire the ’intrinsically cumulative‘ Levine, Lindsay, 1992 mathematical knowledge should be detected and appropriately approached as early as possible.

6.3 Neuropsychological models

Upon our clinical and neuropsychological examination of probands with a suspected

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Instead, our data corroborate the neuropsychological model of von Aster, 1999 for the Specific disorder of arithmetical skills (see chapter 1.4.2.2.7. and Figure 2). While one proband (T.H.) displayed the first type of deficit according to this model (a deficient basic processing mechanism), another (M.W.) could be described with the second type (deficient verbal code), and two probands (S.P. and D.B.) suited the fourth type (a disturbed magnitude representation.) The fact that none of the selected probands fitted the third type (deficient visual-arabic code) of von Aster's model might be due to the fact that these individuals not only have problems with reading and writing numbers, but also words. Our screening process would consequently either not have detected such probands or they are within the group of probands with a combined learning disorder.

Especially as the underlying triple-model of Dehaene et al. is increasingly substantiated by newer publications (see chapter 1.4.2.1.5.), and since our findings could be well predicted with the conception of von Aster, we endorse it as the currently most sufficient neuropsychological model.

6.4 Diagnostic criteria for the Specific disorder of arithmetical skills

The central diagnostic criterion for the diagnosis of a Specific disorder of arithmetical skills in the most common classification systems, the ICD-10 and the DSM-IV, is a significant discrepancy between the individual‘s general intelligence and his or her mathematical performance. From our findings we do not consider this to be a very good criterion. Although we did find distinct problems in solving arithmetical tasks in four of five examined probands, only one of them met the diagnostic criteria of the ICD-10.

One point of criticism with the criterion of discrepancy is that it is ill-defined. Most authors use our criteria, a performance of one standard deviation within the normal range in a standardized, individually administered intelligence test in contrast to a mathematical performance below in a standardized, individually administered academic achievement test. This procedure is advised in the ICD-10, but not obligatory. In other publications a mathematical achievement two grades below the actual grade of the proband has been used as a diagnostic criterion Gross-Tsur, Manor, 1996 , or all probands with an IQ below 90 were excluded Lewis, Hitch, 1994 .

Even if the guidelines for the use of the standardized test would be specified, we still

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In a recent review, Rispens and van Yperen raise the same critical points against the discrepancy criteria for Specific Developmental Disorders of the common classification systems. The authors reach the conclusion, that the discrepancy should be abandoned and propose ’...to develop diagnostic criteria that refer to behavioral characteristics and underlying psychological and biological processes‘ of the Specific Developmental Disorders Rispens &van Yperen, 1997 . In another publication, Kulak analyzes the relatively larger body of research on the field of reading disorders and tries to draw conclusions for directions of research on the Specific disorder of arithmetical skills. She suggests that individuals with a severe form of either disorder do not only differ quantitatively in their acquisition of knowledge but in their quality. She further suggests a careful componential analysis of the skills involved in the acquisition of mathematical knowledge, as this has been a useful approach to reading disorders Kulak, 1993 .

Considering our data, we agree with the above authors that diagnostic criteria which are descriptions of qualitative symptoms are superior to a mere quantitative discrepancy criterion between ill-defined standards. Until these qualitative criteria are established, and when it is impractical to apply then, such as in the screening of larger groups of people, we favor the discrepancy approach of Kamphaus et al. They propose a regression analysis to define Learning disabilities, in which the individuals IQ performance is related with a regression approach to his or her performance in another test. The authors found that this approach generates a more homogeneous distribution of Specific learning disabilities in a study population Kamphaus, Frick, 1991 .

6.5 Conclusions

We conclude that there is a discrete group of individuals with severe problems in the acquisition of mathematical skills. These problems go beyond a quantitative range, but differ qualitatively from the ways unaffected individuals gain mathematical knowledge. These differences can be defined and predicted within a well-defined neuropsychological model supported by scientifically reliable data. We think that these qualitative differences should be best employed as diagnostic criteria rather

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