Calculations and Related Functions

Acalculia

Acalculia is an acquired disturbance of the ability to calculate with numbers. The term “acalculia” dates to 1925 when Henschen studied impairment of computational skills from brain injury and noted the significance of the angular gyrus and intraparietal fissure for mathematical ability. The ability to calculate can be disturbed from a primary disorder of numerical ability in this region or from the secondary effects of other cognitive deficits ( Table 12.1 ). Primary acalculia involves the loss of numerical concepts and the inability to understand or execute basic arithmetical operations, whereas secondary acalculia involves the loss of the ability to calculate from language, spatial, motor praxis, or executive impairments. Primary acalculia is often associated with the neighborhood signs of digit agnosia, right-left discrimination difficulty, and transitional agraphia in Gerstmann syndrome, discussed further in this chapter.

There are different classifications of acalculia, but one clinically useful classification presented here divides them into four primary and four secondary acalculias. The main primary acalculia, often referred to as anarithmetia, is an impairment of a “digital magnitude code” in the left inferior parietal area with decreased digital processing of quantity and operations. Anarithmetia is a specific disturbance of global quantification abilities and the use of syntactic rules and symbols for calculation. On testing, they fail in quantitatively comparing numbers and in doing procedures such as borrowing, carrying over, or solving successive mathematical tasks. Patients with anarithmetia have calculation defects in both oral and written operations and usually other “neighborhood” cognitive difficulties, such as Gerstmann syndrome. These patients may still be able to count aloud, perform rote numerical learning such as the multiplication tables, and conserve some numerical knowledge. Anarithmetia occurs in patients when damage affects the horizontal segment of the intraparietal sulcus, which borders the superior aspect of the inferior parietal lobule ( Fig. 12.1 ).

Other primary acalculias include an impairment of an analog magnitude code in right and left parietal areas with decreased analog processing of position and relationships; semantic impairment of mathematical symbols (“mathematical asymbolia”) in the left temporo-occipital junction with decreased knowledge of mathematical signs; and impairment of the retrieval of mathematical facts out of proportion to other executive or memory deficits. These other primary acalculias are much less common than anarithmetia. Primary acalculia from impairment of the analog code requires biparietal injury and may affect the ability to tell basic ordinal differences, for example, ordering by relative size. These patients are often very impaired and cannot be tested for acalculia in any more depth. Numerical asymbolia is a rare semantic deficit, and primary acalculia from disproportionate difficulty retrieving mathematical information, such as multiplication tables, may be a frequently missed frontal-executive deficit. In addition, there are rare and debated reports of primary acalculias that just involve difficulty with specific numbers such as 7, and other reports of acalculia limited to the verbal modality but not involving Arabic numerals and not consequent to greater language impairment.

The most common secondary acalculia is due to language impairments in appreciating the verbal names and Arabic numeral notations for numbers. This type of acalculia may accompany Broca aphasia, with impaired calculation syntax or numerical agrammatism including difficulty with number sequences. Examples of acalculia with Broca aphasia include inability to complete mathematical operations, transcoding (verbal to numerical) errors (for example, “twenty-three” to 23), and “stack” errors (for example, 27 is read as 7). Acalculia also accompanies Wernicke aphasia with lexical and semantic errors in saying, reading, and writing numbers. This is often manifest as paraphasic errors: for example, on hearing or reading 45 they may repeat 37 and write 51. In addition, they may make lexicalization errors (for example, on hearing three thousand two hundred ten they may write 300020010) and decomposition errors (for example, on reading 3210 they may say “thirty-two and ten”). Acalculia can accompany the different forms of alexia, both central and peripheral, with similar deficits for numbers as for words, including inability to read number signs with central alexias and number-by-number reading in the syndrome of alexia without agraphia (see Chapter 8).

The other secondary acalculias result from spatial, motor praxis, or executive deficits (see Chapters 10, 11, and 13). Spatial acalculia often accompanies occipitoparietal damage with perceptual impairments and right parietal damage with spatial deficits, including hemispatial neglect. Patients with spatial acalculia can count, perform successive operations, and may do somewhat better on mental calculations than on written calculations. Their main problem is in spatially organizing and placing numbers in the correct location due to visuospatial difficulty, including hemispatial neglect. Secondary acalculia may result from an inability to write numbers secondary to an ideomotor or limb-kinetic apraxia of the hands with difficulty mechanically writing numbers. Another secondary acalculia arises from prefrontal injuries with executive and working memory dysfunction in mental operations that require holding information and manipulating it, for example, multistep numerical problems and serial reversal tasks, such as counting backward from 100 by 7s. These patients are impaired in executive functions necessary for maintaining and manipulating numbers. In contrast to spatial and motor apraxic acalculia, patients with frontal acalculia do better on written calculations than on mental calculations.