This article includes discussion of acalculia, acquired dyscalculia, anarithmetia, agraphic acalculia, alexic acalculia, aphasic acalculia, and spatial acalculia. The foregoing terms may include synonyms, similar disorders, variations in usage, and abbreviations.
Calculation ability represents a complex cognitive process. It has been understood to represent a multifactorial skill, including verbal, spatial, memory, and executive function abilities. Calculation ability is frequently impaired in cases of focal brain pathology, especially posterior left parietal damage, and dementia. Acalculia is also common in posterior cortical atrophy. Contemporary neuroimaging studies suggest that arithmetic is associated with activation of specific brain areas, specifically the intraparietal sulcus; language and calculation areas are partially overlapped and partially independent. Acalculia recovery is variable and depends on different factors, such as the extension and etiology of the brain pathology.
• Calculation ability represents a multifactorial skill including verbal, spatial, memory, and executive function abilities.
• Calculation disturbances are usually observed in cases of posterior left parietal damage.
• The intraparietal sulcus has been proposed to represent the most crucial brain region in the understanding and the use of quantities.
• A major distinction can be established between primary and secondary acalculia.
• Acalculia is most often caused by a stroke, tumor, or trauma and it is usually present in dementia.
Historical note and terminology
Henschen introduced the term "acalculia" to refer to the impairments in mathematical abilities in patients with brain damage (Henschen 1925). Berger distinguished 2 different types of acalculia: primary and secondary acalculia. Secondary acalculia refers to calculation defects resulting from a different cognitive deficit: memory disorders, attention impairments, language defects, spatial deficits, etc. (Berger 1926). Gerstmann proposed that acalculia is observed together with agraphia, disorders in right-left orientation, and finger agnosia, representing the basic brain syndrome usually known as "Gerstmann syndrome" (Gerstmann 1940). Gerstmann syndrome has been associated with left angular gyrus damage (Vallar 2007). It has been proposed that Gerstmann syndrome represents a disorder in the spatial representation of the body-scheme and mental rotations occurring after left parietal damage (Gold et al 1995).
Hecaen and colleagues distinguished 3 major types of calculation disorders: (1) alexia and agraphia for numbers, (2) spatial acalculia, and (3) anarithmetia (Hecaen et al 1961). Alexia and agraphia for numbers represent calculation disturbances resulting from difficulties in reading and writing quantities. Spatial acalculia represents a disorder of spatial organization where the rules for setting written digits in their proper order and position are not followed; spatial neglect and number inversions are frequently found in this disorder. Anarithmetia corresponds to primary acalculia. It implies a basic defect in computational ability.
Boller and Grafman considered that calculation abilities can be disrupted as a result of: (1) inability to appreciate the meaning of the number names, (2) visuospatial defects that interfere with the spatial arrangement of numbers and the mechanical aspects of mathematical operations, (3) inability to recall mathematical facts and appropriately use them, and (4) defects in mathematical thinking and in understanding underlying operations (Boller and Grafman 1985).
A general cognitive model of number processing and calculation has been proposed by McCloskey and colleagues (McCloskey and Caramazza 1987; McCloskey et al 1991). A distinction is drawn between the number processing system, which comprises the mechanisms for comprehending and producing numbers, and the calculation system, which encompasses processing components required specifically for carrying out calculation. In the case of brain pathology, these components can be dissociated. Facts (eg, the multiplication tables), rules (eg, n x 0 = 0), and procedures (eg, the multiplication process goes from left to right) are included as elements of the calculation system. Errors in calculation observed in brain-damaged and normal subjects can result from inappropriate fact retrieval, misuse of arithmetical rules, and procedural errors.
Ardila and Rosselli have proposed a classification of acalculias (Ardila and Rosselli 2002). A basic distinction between anarithmetia (primary acalculia) and acalculia resulting from other cognitive defects (secondary acalculias) is included. Secondary acalculias may result from linguistic defects (oral or written), spatial deficits, and frontal-type disturbances, particularly perseveration, memory, and attention impairments. However, there is a certain degree of overlap among the acalculia subtypes proposed by Ardila and Rosselli. Thus, in anarithmetia some spatial deficits can be observed. Spatial acalculia associated with right hemisphere pathology is also partially an alexic acalculia (eg, some inability to read complex numbers as a result of left hemineglect can be observed). Even though arithmetical calculation is a rather complex cognitive activity requiring the participation of many elements, brain damage may result in a relatively restricted disorder. Rosca reported a patient with preserved arithmetic facts but impaired procedural knowledge (Rosca 2009b). Klessinger and colleagues studied a 56-year-old retired university professor suffering a vascular lesion in the left middle cerebral artery territory with extensive damage to perisylvian temporal, parietal, and frontal cortices, resulting in right hemiplegia, severe aphasia, and apraxia of speech. Regardless of the severe language disturbance and his difficulties with processing both phonological and orthographic number words, he demonstrated largely intact algebraic fact knowledge, procedures, and conceptual knowledge; this case illustrates that some aspects of mathematical processing can be preserved despite severe disruption to the language ability (Klessinger et al 2007).
Hittmair-Delazer and colleagues described a patient affected by an inability to recall and use "arithmetical facts" of single-digit multiplication and division (Hittmair-Delazer et al 1994). This impairment contrasted with the preservation of a wide range of complex notions (cardinality judgments, recognition of arithmetical signs, written calculations, solving arithmetical problems, additions and subtractions). This patient's difficulty stemmed from an inability to monitor the sequence of operations that calculation procedures specified. Dehaene and Cohen described 2 patients with pure anarithmetia, 1 with a left subcortical lesion and the other with a right inferior parietal lesion and Gerstmann syndrome (Dehaene and Cohen 1997). The patient with the subcortical lesion suffered from a selective deficit of rote verbal knowledge (eg, arithmetical tables), whereas the semantic knowledge of numerical quantities was intact. The patient with the inferior parietal lesion suffered from a category-specific impairment of quantitative numerical knowledge, with preserved knowledge of rote arithmetical facts. Domahs and colleagues reported a patient with left frontal lesions due to a cerebrovascular disorder presenting defects in simple multiplication when problems were given in a mixed operations list (multiplication, addition, and subtraction) but not when the same multiplications were presented in isolation (a kind of “task-switching acalculia”) (Domahs et al 2011). Tanaka and colleagues described an unusual case of “abacus-based acalculia” (Tanaka et al 2012). Abacus users typically manipulate a mental representation of an abacus. The authors evaluated a patient who was a good abacus user and transiently lost her "mental abacus" after a right hemispheric stroke involving the dorsal premotor cortex and inferior parietal lobule.
Troiani and colleagues proposed that numerical quantifier understanding (which requires magnitude processing, for example, "at least three") and logical quantifier understanding (which can be comprehended using a simple form of perceptual logic, such as “"some") depend on different brain neural networks; whereas numerical quantifier understanding depends on a lateral parietal-dorsolateral prefrontal network, logical quantifier understanding depends on a rostral medial prefrontal-posterior cingulate network (Troiani et al 2009). It has been further proposed that the numerical processing and calculation system includes 4 independent elements: quantitative numerical knowledge, numerical recodification, qualitative numerical knowledge, and calculation. In cases of brain pathology, each component may be impaired selectively without affecting the others, indicating that each one has different brain representation.
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