Dtsch Arztebl Int 2012; 109(45): 767-78 ; DOI: 10.3238/arztebl.2012.0767

__Background:__ Dyscalculia is defined as difficulty acquiring basic arithmetic skills that is not explained by low intelligence or inadequate schooling. About 5% of children in primary schools are affected. Dyscalculia does not improve without treatment.

__Methods:__ In this article, we selectively review publications on dyscalculia from multiple disciplines (medicine, psychology, neuroscience, education/special education).

__Results:__ Many children and adolescents with dyscalculia have associated cognitive dysfunction (e.g., impairment of working memory and visuospatial skills), and 20% to 60% of those affected have comorbid disorders such as dyslexia or attention deficit disorder. The few interventional studies that have been published to date document the efficacy of pedagogic-therapeutic interventions directed toward specific problem areas. The treatment is tailored to the individual patient’s cognitive functional profile and severity of manifestations. Psychotherapy and/or medication are sometimes necessary as well.

__Conclusion:__ The early identification and treatment of dyscalculia are very important in view of its frequent association with mental disorders. Sufferers need a thorough, neuropsychologically oriented diagnostic evaluation that takes account of the complexity of dyscalculia and its multiple phenotypes and can thus provide a basis for the planning of effective treatment.

Difficulty in learning arithmetic, like difficulty in learning to read and write, is a common learning disorder in childhood. The prevalence of each of these disorders among primary-school pupils is about 5%, a relatively constant figure across the countries in which these disorders have been studied (1, e1). Dyscalculia is often associated with mental disorders (2, 3, e2). Many affected children acquire a negative attitude to counting and arithmetic, which, in turn, often develops into a specific mathematics anxiety or even a generalized school phobia (4). Unless specifically treated, dyscalculia persists into adulthood (5–7); it can lastingly impair personality development, schooling, and occupational training. Dyscalculia is also an economic issue, as adults with poor arithmetic skills suffer a major disadvantage on the job market. About 22% of young adults fall into this category (e3, e4). The mental disorders commonly associated with dyscalculia are expensive to treat (7). Thus, the early recognition and differential diagnosis of learning disorders are an important matter not just for child psychiatrists, who must often deal with the secondary conditions that arise from these disorders, but also for general practitioners and pediatricians, as the delayed acquisition of pre-scholastic skills in the nursery and kindergarten years may already be an early sign of a problem (3). In this article, we selectively review the literature on dyscalculia.

Learning objectives

This article is intended to acquaint the reader with

- the typical development of numerical and arithmetical skills,
- the instruments used for the evaluation and differential diagnosis of dyscalculia, and
- current methods of treating dyscalculia.

The typical development of numerical and arithmetical skills

Neuroscientific models let us view the development of numerical processing in the brain as a neuroplastic maturation process that leads, over the course of childhood and adolescence, to the establishment of a complex, specialized neural network (2, 8). This development begins with the very simple (basic) numerical skills that give even babies of a few months of age an elementary grasp of quantity and number. Infants can clearly distinguish one item from two items, and they are capable of a rough estimation of number for three or more items (e5). When language begins to appear, children become able to symbolize numbers linguistically with number-words: They learn to count out loud and perform simple, verbal arithmetical manipulation of quantities and numbers. Next, a second way of symbolizing numbers is learned in the preschool and primary school years. This is the Arabic numeral system, with its own “grammar” that differs markedly from the representation of numbers in spoken language. The Arabic place-value system is a highly economical way of symbolizing numbers that simplifies numerical calculations. For example, the number 1 768 329 is written with a mere seven Arabic numerals, while its English expression, “one million seven hundred sixty-eight thousand three hundred twenty-nine,” is 62 letters long, not counting dashes and spaces. Finally, in parallel with the development of the spoken and written (Arabic-numeral) symbolization of numbers and the associated operative capacities, children develop a numerospatial conceptual ability (a “mental number line”) enabling them to operate with numerical symbols. The mental number line seems to be of fundamental importance for arithmetical thinking and for calculating in one’s head (e6). The early, basic numerical skills thus give an initial meaning to the processes of symbolization (number words, Arabic numerals), while the mental line extends the semantic range of the concept of number to a higher, more abstract level. An overview of the relevant aspects of numerical and arithmetical skills is provided in Box 1.

Various aspects of numerical-arithmetical functions and relevant skills in each
functional area

Functional imaging studies have shown that, with increasing practice and expertise, these components coalesce in a neural network in multiple brain areas that is activated when the performance of the corresponding tasks is required (8). Number-words are processed in the speech areas of the left perisylvian region, Arabic numerals in the occipital lobes. Basic numerical representations and the numerical-spatial representations that develop later on are subserved by parietal areas on both sides of the brain that become increasingly functionally specialized as the child grows older and acquires more education (e7, e8). The development and maturation of these domain-specific brain functions depends on the maturation of numerous domain-specific or multi-domain functions, including attention and working memory (mainly subserved by the frontal lobes), language, sensorimotor function (e.g., finger counting), and visuospatial ideation; they also depend on experience (e.g., practice and stimulation in everyday life and the type of teaching methods used). Clearly, at any time in a child’s development, many different factors could disturb or delay the maturation of these neural networks, causing clinically evident manifestations of various kinds (Figure 1).

Potentially
causative factors
and different levels
on which developmental
dyscalculia
can manifest itself

Definition and etiology

Definition

Each of the two main classification systems for mental disorders that are currently in use, the ICD-10 (10) and the DSM-IV (11), classifies disorders of the acquisition of written language (ICD-10: F81.0 and F81.1), isolated dyscalculia (F81.2), and dyscalculia combined with dyslexia (F81.3) as independent categories of specific disorders of scholastic skills. Dyscalculia is defined as a serious impairment of the learning of basic numerical-arithmetical skills in a child whose intellectual capacity and schooling are otherwise adequate. It is supposed to be demonstrable by standardized psychometric testing that reveals poor calculating ability despite normal intelligence.

In the DSM-V, due to be published in 2013, the specific developmental disorders of scholastic skills will be grouped together as dimensions within a single category, in each of which there can be a greater or lesser degree of impairment; there will no longer be a strict requirement for the psychometric demonstration of a discrepancy with otherwise normal intelligence. This change is being introduced to take better account of the heterogeneity of these disorders with respect to performance profiles and comorbidities, and thus to improve the clinical utility of the DSM diagnoses. In situations where proof of a discrepancy is required (as is still the case for medicolegal evaluations in Germany and Austria), we recommend the use of multi-component intelligence tests, consisting of several subtests, which can provide a detailed picture of the child’s deficiencies—for example, the current (fourth) version of the Hamburg-Wechsler Intelligence Test for Children (e9).

The literature also reflects a distinction between “disorders” and “disabilities,” with the latter term being more commonly used by education professionals. For example, a child is said to have a mathematical learning disability (MLD) when his or her performance on a test of mathematical skills falls below an arbitrarily chosen percentile in a study population, without reference to the child’s general intelligence level (e10). Variations in defining criteria account for the wide variation among the study populations and testing procedures encountered in published studies, and this renders comparisons across studies difficult. The distinction between dyscalculia, considered as a disorder, and MLD is mainly important for research. In clinical treatment planning, the child’s actual performance profile is of central importance, regardless of whether dyscalculia or MLD is said to be present.

Etiology

Dyscalculia has many contributory causes. For the ones that will be discussed below, no firm conclusions can yet be drawn about the relations and interactions between them, as too little evidence is available. There is, however, a consensus in the current literature supporting the multifactorial origin of developmental dyscalculia and other learning disorders (2). Dyscalculia is also often present in children suffering from neurological diseases (e.g., epilepsy, premature birth, metabolic disorders) and genetic syndromes (e.g., fragile X syndrome, Williams-Beuren syndrome, velocardiofacial syndrome).

Developmental dyscalculia tends to run in families (e11), possibly because of a genetic predisposition (e12, e13).

There may be a primary genetic vulnerability to the impaired development of basic numerical functions or of linguistic, visuospatial, and executive ones. As we now know, the maturation of these functions can also be affected by environmentally determined, epigenetically mediated influences, such as stress (Figure 1). This explains the marked association of dyscalculia with attention deficit hyperactivity disorder (ADHD) on the one hand, and with dyslexia on the other. Dyscalculia is not a single, uniform entity; rather, its subtypes can be classified systematically and in detail on the basis of their varying etiologies, underlying neural bases, cognitive representations, and skills levels (Figure 1).

Comorbidities and commonly associated accompanying phenomena

According to von Aster and Shalev (2), 20% to 60% of all persons with dyscalculia also have learning difficulties of other types, e.g., dyslexia (12) or ADHD (13, 14, e2). A combination of dyslexia and dyscalculia was found in 7.6% of 788 randomly sampled Dutch fourth- and fifth-graders (e14). Even among ninth-graders, 52% of the variance in calculating ability was accounted for by reading skills (e15). It even seems to be the case that deficient phonological skills (considered the most important precursor skill for written language) in preschool children are associated with poorer performance on formal calculating tasks in the early years of primary school (e16). A disorder of linguistic development in nursery-school and kindergarten age seems to be a risk factor for poor calculating ability (e17).

Only scant empirical data are available to date on the comorbidity of dyscalculia with attention deficit disorders. Rubinsten and colleagues (13) studied the effects of methylphenidate, a stimulant, on calculating ability in three groups of children with ADHD: The first group had no learning disorder, the second had a mathematical disability, and the third had dyscalculia. In all groups, the stimulant was found to improve those aspects of calculating ability that depend on working memory (e.g., “carrying” while calculating with multi-digit numbers), but it had no effect on basic numerical skills. The authors concluded that children who have both ADHD and either mathematical disability or dyscalculia need not only pharmacotherapy, but also specific learning therapy.

Accompanying manifestations

As is pointed out in a current review, mental illness is much more common among children with learning disorders than among age-matched children without learning disorders (30–50% versus 8–18%, [e2]). Children with dyscalculia often have severe emotional distress because they do poorly in school; this can lead, in turn, to a negative attitude about mathematical tasks or even to mathematics anxiety and school phobia (4). Mathematics anxiety tends to become chronic and to persistently impair skill development. Its effects can be seen on multiple levels—physiological (palpitations, diaphoresis), cognitive (feelings of helplessness, impaired working memory [e18]) and behavioral (avoidance). Severe mathematics anxiety also seems to impair basic numerical processing: The mental representation of number is less precise in persons with severe mathematics anxiety than in persons with mild or no mathematics anxiety (e19).

The diagnostic assessment and differential diagnosis of dyscalculia

Practical experience shows that there is a wide variety of performance profiles for numerical and calculating skills; this fact suggests that there are different subtypes of dyscalculia (15, 16, e20). Multiple cognitive factors have been discussed in the literature as potential contributory causes of dyscalculia (16, e10). Among these, a deficient understanding of quantity has been the most extensively studied by empirical means.

When dyscalculia is suspected, a detailed diagnostic evaluation is needed in order to take proper account of the complexity of this learning disorder and to produce an accurate picture of the affected child’s particular strengths and weaknesses in the area of numbers and calculations. The diagnostic instruments used for this purpose are of two main types, the curricular and the neuropsychological (Table). As the affected children often perform far below grade level on numerical and calculating tasks, the use of curricular tests alone may not yield a complete picture of the actual performance deficit; this can, in turn, lead to inappropriate interventions with little promise of efficacy, because the child’s performance is not really at the level for which the intervention was designed.

In children of kindergarten age, specific precursor skills can be identified (Box 1) that have been found to be reliable predictors of later calculating ability (e33, e34). Working memory has also been found to be such a predictor (17–19, e34, e35). Most standardized tests of calculating ability are designed for primary-school children and are not normed for any higher level than the sixth grade (e30). The BASIS-MATH 4–8 test, which was published in 2010, is the only German-language test that can be used up to the eighth grade (e36). It provides no normed data, but rather a single threshold value for the fourth through eighth grades, which can be interpreted as a rough indication of the mastery of basic mathematical concepts. BASIS-MATH has proved its usefulness in practice, as it yields not just a quantitative test score, but also information about the child’s calculating strategies, which can be of service in the planning of therapeutic interventions.

Differential-diagnostic considerations

Children with mathematical learning disability, with dyscalculia, and with combined dyscalculia and dyslexia have markedly different cognitive-neuropsychological performance profiles, as shown in Figure 2 (14, 20). In view of the multiplicity of the functional components participating in these disturbances and the wide range of potential mental and neuropediatric comorbidities, it is clear that the diagnostic evaluation must go beyond the strictly mathematical components to include a thorough personal, familial, and scholastic developmental history. The evaluation of the child’s state of general cognitive development must also take account of non-numerical cognitive domains, social and emotional well-being, and findings from the fields of developmental neurology and, where appropriate, neurophysiology (EEG, neuroimaging) (2, 15, 21, 22) (Box 4).

Differential-diagnostic considerations for the identification and characterization of
learning difficulties in the numerical-arithmetical area, taking special account of the
distinction between dyscalculia and mathematical learning disability (MLD)

Methods for the treatment of dyscalculia

The treatment of children and adolescents with dyscalculia is a complex matter because of the heterogeneity of the disorder and the comorbid disorders often associated with it. For the best chance of a lasting therapeutic benefit, the treatment should be individually tailored to the findings of the diagnostic evaluation. It should be adapted to the patient’s individual cognitive functional profile and may incorporate medications and psychotherapy as well, if these are warrented by severe accompanying psychopathological manifestations such as anxiety, depression, or ADHD (23, e37, e38).

Even though the disorder is heterogeneous, certain aspects of numerical calculating skills do seem to be especially common in, and characteristic of, patients with dyscalculia (Box 2).

Numerical-arithmetical skills that present particular
difficulty to children with dyscalculia and mathematical
learning disability

The following skills are particularly important for the domain-specific aspects of treatment:

- basic numerical skills,
- the establishment and consolidation of numero-spatial representations,
- the development of arithmetical reasoning,
- procedural knowledge, and
- the automatization of factual knowledge.

All of these skills fundamentally depend on successively acquired layers of understanding, in constant interaction with the practicing of skills for their consolidation. For example, when a child acquires arithmetical facts, attention should be paid to the basic underlying arithmetical reasoning, as well as to frequent practice. As can be seen from the examples in Box 3, the various solving strategies differ both with respect to the complexity of the solution procedures involved and with respect to the underlying arithmetical reasoning. Counting up from the first summand to the answer is an immature solving strategy compared to breaking down the problem into its several components, but it is, at least, more mature and reflects a better comprehension of arithmetic than simply counting everything that is to be summed. The latter immature strategy demands more from working memory, which must store both interim results, 6 and 4. Repeated practice fosters the automatic recall of learned content, thereby lessening the demand on working memory. This demand is especially high in multi-digit calculations with carrying, where interim results must be retained and manipulated (e.g., adding 87 and 45).

Illustration of various solution pathways and strategies for single-digit addition
and multiplication tasks (also called “arithmetical facts”)

Most of the empirical studies on interventions to improve calculating ability have been carried out in English-speaking countries by special-education professionals. Their findings are, to some extent, inconsistent. A meta-analysis by Kroesbergen and van Luit (24) of 58 interventional studies among primary-school pupils revealed that

- most interventions dealt with basic numerical skills;
- interventions to promote basic numerical skills were more effective than interventions to promote precursor skills and/or problem-solving strategies; the sample size varied from 3 to 376, and the Cohen’s d values (effect sizes) from –0.44 to above 3 (24);
- shorter interventions were more effective than longer ones (the duration of interventions ranged from one week to one year across studies);
- interventions performed in person by teaching staff (“direct instruction”) were more effective than computerized ones (“mediated instruction”).

A further meta-analysis of studies of individualized intervention in the English-speaking countries revealed that individualized intervention is highly effective at improving calculating ability (d = 0.91 [e40]), that the particular method of intervention that is used largely determines the degree of effectiveness, and that interventions conveying an understanding of strategy are much more effective than those in which the subject matter is passively communicated (“strategy instruction” versus “direct instruction”).

The following methods of intervention were found to be especially effective:

- repeated practice;
- segmentation of subject matter;
- small, interactive groups;
- the use of cues in strategy-learning (e40).

A recently published meta-analysis by Ise and colleagues (25) evaluated eight studies from the German-speaking countries. The overall effect strength was 0.50, which is considered an intermediate value (neither strong nor weak). No difference in effectiveness was found between curricular and non-curricular treatment approaches, but all interventions tended to be more effective the longer and the more intensively they were carried out. Most of the currently available German-language programs for promoting numerical calculating skills have not been empirically tested for efficacy, and many seem to lack an adequate theoretical basis.

The scientifically studied learning programs in the German-speaking countries include “Mengen, Zählen, Zahlen” (MZZ; “Quantities, Counting, and Numbers”), which was designed for preschool children (e41); the computerized and largely curricular intervention programs “Elfe und Mathis” (“Elfe and Mathis”) (e42); and the learning program “Calcularis” (e43), which is based on neuroscientific models of the development of number processing and arithmetic. The effect strength of these intervention programs has not been studied. MZZ is intended to promote the establishment of basic numerical and scholastic precursor skills (the concepts of quantity and number, counting order, and various types of notation). “Calcularis” begins with basic numerical skills and addresses the automatization of various number representations in increasingly large numerical domains, the understanding of arithmetical operations, and the establishment of factual arithmetical knowledge. “Calcularis” has an adaptive structure, i.e., it can be adapted to the individual child’s learning difficulties. In a study of a “Calcularis” prototype with functional neuroimaging (fMRI), the improvement in numerical-spatial and arithmetical skills was found to be associated with altered patterns of neural activation in the frontal and parietal lobes (e44).

Overview: the role of the treating physician

Dyscalculia, if untreated, persists into adulthood (5, 6, e3, e4). The affected persons often suffer from secondary, associated disturbances and are at a disadvantage on the job market (6, e2).

When dyscalculia is suspected, a detailed diagnostic evaluation should be performed. The primary task of specialists in child and adolescent psychiatry, and of the school health services, is to determine whether any comorbid (associated) disturbances are present. The differential evaluation of performance on numerical-arithmetical and non-numerical skills lies in the area of competence of school psychologists and neuropsychologists. Ideally, the performance profile that is generated by the diagnostic evaluation should serve as a point of departure for intervention planning.

The effective treatment of dyscalculia demands special expertise, which is most likely to be found among graduates of specialized training and continuing-education programs that have been certified by recognized professional associations. In Germany, for example, the relevant organizations are the Bundesverband Legasthenie und Dyskalkulie (BVL, National Dyslexia and Dyscalculia Association) and the Fachverband Integrative Lerntherapie (FIL, Association for Integrative Learning Therapy). Recently, bachelor’s and master’s degree programs for specific training in learning therapy have been initiated at universities and professional training institutes.

Learning therapy can be carried out either in school, in conjunction with school, or outside school. In Germany, the cost of learning therapy is borne by the Youth Services Department after confirmation that the legally mandated condition of an impending mental impairment is fulfilled. As a rule, interventions can succeed only when they are ecologically valid, i.e., when they can take effect in the setting of the child’s everyday life.

A further role for the treating physician or psychologist may be to point out that an established legal framework exists for giving the affected persons special means to compensate for their learning difficulty in situations calling for high performance, including situations where their performance will be evaluated (tests). In Germany, Switzerland, and Austria, the availability of methods to acknowledge an individual's learning difficulty within the school setting (e.g., by allowing more time to complete written examinations etc.) is far from uniform, and they may be entirely lacking. Whatever opportunities of this kind are available should be tried out in the individual case and made use of where appropriate.

In summary, the main role of the treating pediatrician or family physician centers on the early recognition of dyscalculia (and other learning disorders) and on counseling of the child’s parents and other carers about the further diagnostic and therapeutic measures that are indicated. Early recognition largely depends on information provided by the child’s parents or other carers. Depending on the age of the child, specific questions should be asked about his/her understanding of quantity, counting skills, and mathematical performance in school to date. The history should also include questions about any secondary disturbances that might be present, e.g., learning disorders in other areas and/or psychopathological manifestations, dislike of school, mathematics anxiety, and/or school phobia.

Conflict of interest statement

PD Dr. Kaufmann receives license fees for test procedures such as TEDI-MATH and BVN5-11, for the textbook “Dyskalkulie,” and for the edited text “Entwicklungsneuropsychologie.”

Prof. von Aster receives royalties from Pearson-Verlag for the ZAREKI-R/K test and from the publishing house Vandenhoeck & Ruprecht for the book “Rechenstörungen bei Kindern.” The Lilly company has paid him honoraria for the preparation of scientific continuing education events and has reimbursed his travel and accommodation costs.

Manuscript submitted on 13 June 2012, revised version accepted on 5 October 2012.

Translated from the original German by Ethan Taub, M.D.

Corresponding authors

PD. Dr. rer. nat. Liane Kaufmann

Landeskrankenhaus Hall

Abteilung für Psychiatrie und Psychotherapie A

Milserstr. 10

6060 Hall in Tirol, Austria

liane.kaufmann@ymail.com

Prof. Dr. med. Michael von Aster

Klinik für Kinder- und Jugendpsychiatrie

Psychotherapie und Psychosomatik

DRK-Kliniken Berlin Westend

Spandauer Damm 130

14050 Berlin, Germany

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Kaufmann L, Lochy A, Drexler A, Semenza C: Deficient arithmetic fact retrieval – storage or access problem? A case study. Neuropsychologia 2004; 42: 482–96. MEDLINE

e36.

Moser Opitz E, Reusser L, Moeri Müller M, Anliker B, Wittich C, Weesemann O, Ramseier E: BASIS-MATH 4–8. Basisdiagnostik Mathematik für die Klassen 4–8. Göttingen: Hogrefe 2010.

e37.

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e38.

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e39.

e40.

e41.

Krajewski K, Nieding G, Schneider W: „Mengen, zählen, Zahlen“. Die Welt der Mathematik entdecken. Berlin: Cornelsen 2007.

e42.

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e43.

Von Aster MG, Käser T, Kucian K, Gross M: Calcularis – Rechenschwäche mit dem Computer begegnen. Schweizerische Zeitschrift für Heilpädagogik 2012; 18: 32–6.

e44.

Kucian K, Grond U, Rotzer S, Henzi B, Schönemann C, Plangger F, Gälli M, Martin M, von Aster MG: Mental number line training in children with developmental dyscalculia. NeuroImage 2011; 57: 782–95. MEDLINE

UMIT—Private University of Health Sciences, Medical Informatics and Technology, Institute for Applied Psychology Hall in Tyrol, Austria: Ao. Prof. Dr. rer. nat. Kaufmann

Department of Child and Adolescent Psychiatry, DRK-Kliniken Berlin Westend and MR-Center, University Children’s Hospital of Zurich, Switzerland: Prof. Dr. med. von Aster

Department of Child and Adolescent Psychiatry, DRK-Kliniken Berlin Westend and MR-Center, University Children’s Hospital of Zurich, Switzerland: Prof. Dr. med. von Aster

Various aspects of numerical-arithmetical functions and relevant skills in each
functional area

Numerical-arithmetical skills that present particular
difficulty to children with dyscalculia and mathematical
learning disability

Illustration of various solution pathways and strategies for single-digit addition
and multiplication tasks (also called “arithmetical facts”)

Potentially
causative factors
and different levels
on which developmental
dyscalculia
can manifest itself

Differential-diagnostic considerations for the identification and characterization of
learning difficulties in the numerical-arithmetical area, taking special account of the
distinction between dyscalculia and mathematical learning disability (MLD)

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