The ability to appreciate quantities, understand numbers, and calculate is a cognitive domain in its own right. This domain has a left inferior parietal hub involving the horizontal portion of the intraparietal sulcus, an area that appears dedicated to processing symbolic numerical information. However, many other areas and their cognitive functions can affect calculation abilities. This chapter discusses the evaluation for acalculias and includes assessment of left parietal neighborhood signs and symptoms included in the Gerstmann syndrome.
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.
Primary
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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.
The Examination for Acalculias
Calculation ability implies use of numerical concepts. Small quantities from 1 to 3 do not need to be counted, a process known as subitization and present in small children and in other animals; however, larger quantities involve the representation of quantity implicit in numbers, the numerical position within other numerical symbols, the relationships between a number and other numbers, and the relationships between numerical symbols and their verbal representations. Much of the testing for the acalculias focuses on these numerical properties. Before actual testing, however, the examiner must determine if the patient has symptoms of acalculia and whether they have the premorbid ability and education to perform calculation tests. The examiner must be aware of the many types of symptoms that can reflect difficulty with calculation abilities ( Table 12.2 ). A good neurobehavioral history is helpful in detecting these symptoms (see Chapter 6 ). Some patients never learned or developed their innate mathematical abilities, and others have poor skills because of increasing reliance on calculators and other electronic devices. The examiner can start by assessing if the patient has basic counting ability by asking them to count objects in the room or dots on a page ( Fig. 12.2 ).
| Eighteen Patients with Acalculia Variant Alzheimer Disease | N |
| Difficulty with calculations, simple math, and numbers | 10 |
| Difficulty writing checks and managing checkbooks | 6 |
| Difficulty reading numbers and words | 5 |
| Word-finding problems | 8 |
| Difficulty writing and spelling | 6 |
| Inability to do finances and pay bills | 4 |
| Difficulty with depth perception AND dressing apraxia | 2 |
| Difficulty calculating the tip at restaurants | 2 |
| Inability to managing clients’ bills | 2 |
| Difficulty with motor manipulation of objects or tasks | 1 |
| Difficulty counting money | 1 |
| Having trouble reading a tape measure | 1 |
| Trouble with tape and visual scales and plans | 1 |
| Difficulty with the numbers in dialing a telephone | 1 |
| Difficulty setting a timer with numbers | 1 |
| Difficulty using a measuring cup | 1 |
| Difficulty reading the time on a clock | 1 |
| Problems judging time and distance driving | 1 |
| Unable to remember how many zeros are in 10,000 | 1 |
| Complaint of confusing 8 million with 8 thousand | 1 |
Examiners most frequently rely on serial subtraction tests to briefly screen for calculation difficulty. Patients must count backward from 100 by 7s or count backward from 20 by 3s, as are used for testing attention and mental control (see Chapter 7 ). In the first, the examiner asks the patient to subtract by 7 beginning with the number 100, for example, 93, 86, 79, 72, 65, et cetera. The number of errors are the number of incorrect subtractions; if the patient makes an incorrect subtraction at one level, the examiner corrects the patient and instructs the patient to continue subtracting from the corrected number. Alternatively, the subsequent “correct” subtractions are determined from the incorrect number, that is, if the patient subtracts 7 from 100 as 94, then the subsequent correct subtraction is 87 and not 86. Patients should be able to get three or more subtractions in a row. Counting backward from 20 by 3s is a similar, but easier, version of serial subtractions.
Some clinicians briefly screen for calculation difficulty with an oral word problem. Although able to clarify the presence or absence of calculation difficulty, these word problems do not specify the type of acalculia. One example is to ask the patient to calculate the number of books on two shelves, if there are 18 books with twice as many on one shelf as the other. Another example is to ask the patient to report the number of nickels in $1.35, or the number of quarters in $6.75. Tell the patient: “The girl went to the store with $5 to buy 2 sodas for $1.85 each. What was her change?” Individuals of limited educational background may fail formal mathematical tasks but demonstrate competence when asked to handle money.
For routine screening, it is not necessary to test all four major arithmetic operations, and problems in addition and subtraction are sufficient. The patient can perform two simple double-digit calculations such as 47 + 18 = ___ and 82 – 16 = ___. During these calculations, have the patient specifically read the arithmetic signs and describe their operation. In addition, ask the patient for items reflective of numeracy (the term is used here as the ability to apply mathematics to the real world). Start by asking the number of days in a week and weeks in a half a year, and then follow this up with a simple word problem.
The examiner can more extensively evaluate for multiple aspects of calculation ability with a screening battery of tests ( Table 12.3 ). The examiner can determine if there is a primary acalculia by eliminating secondary acalculias based on their mistakes and any accompanying deficits. First, ask the patient to read and transpose verbal numbers to Arabic numerals and vice versa. This assesses for a language (“lexical-symbolic”) secondary acalculia. Second, ask the patient to write numbers of one, two, or more digits. This assesses for apraxia or motor deficits in writing. Third, ask them to write a dictated addition and subtraction problem with double or triple digits without performing the problem. Spatial errors in alignment may emerge. Fourth, the examiner asks a series of simple mental calculations. To test mental calculation, ask the patient to do 4 + 13, 21 – 3, 6 x 5, and 25/5 in their head. Alternatively, give the continuous additions task in which the patient must add the successive overlapping pairs as rapidly as possible: 5, 2, 7, 3, 4, et cetera (adds last to next to last for 7, 9, 10, 7). Finally, follow this up with actual performance on written calculations.
| 1. Transcoding | ||||
| A. Spoken verbally ⇒ Written Arabic numeral responses | ||||
| 381, 495, 1405, 7643, 10280, 132406 | ||||
| B. Written Arabic numerals ⇒ Read aloud responses | ||||
| 279, 756, 2450, 7658, 12038, 134906 | ||||
| C. Written Arabic numerals | ⇒ Written number word responses | |||
| 316, 572, 2364, 4125, 28054, 303008 | ||||
| D. Written number words | ⇒ Written Arabic numeral responses | |||
| Three hundred five | Four hundred six thousand four hundred fifty three | |||
| Five thousand eight hundred seventy six | One thousand five hundred eighty one | |||
| Twelve thousand three hundred ninety | Two hundred eighty seven | |||
| 2. Writing numbers | ||||
| Instruct to write a series of numbers to dictation: 21 84 2,304 84 730 17,241 3 107 | ||||
| 3. Spatial alignment | ||||
| Align without calculating; one item for each operation (each with three- and two-digit numbers): | ||||
| A. 376 + 25 B. 621 – 72 | ||||
| C. 73 x 308 D. 869 ÷ 69 | ||||
| 4. Mental calculations and estimates (“perform in your head”) | ||||
| 16 + 5 27 + 14 29 – 7 15 – 6 | ||||
| 7 x 8 11 x 3 12 ÷ 4 28 ÷ 7 | ||||
| Estimate of 39 x 28.7: a. 600, b. 1200, c. 2300, d. 1.5 | ||||
| 5. Written calculation performance (examples; see also Table 12.4 ) | ||||
| A. Addition 376 + 27 495 + 51 | ||||
| B. Subtraction 129 – 32 621 – 73 | ||||
| C. Multiplication 824 x 16 417 x 325 | ||||
| 6. Arithmetic sign recognition | ||||
| Reading aloud: 1 + 2 5 x 7 9 – 3 16 ÷ 4 | ||||
| Writing from dictation: 6 + 2 13 x 4 25 ÷ 5 42 – 20 | ||||
| 7. Analog-ordinal processing | ||||
| A. Place numbers 57, 32, 78, 12, 93 on the following analog line: | ||||
| 0_______________________________________________________________________________________________________100 | ||||
| B. Which number in each pair is larger? | ||||
| 6 or 5 14 or 12 24 or 27 133 or 143 | ||||
| 782 or 872 2,142 or 1,242 6,580 or 8,650 71,142 or 72,300 | ||||
| 8. Numerosity and arithmetic reasoning (see also Table 12.5 ) | ||||
| How many days in a week? How many seasons in a year? How many months in a year? How many hours in a day? How many days in a year? | ||||
| Estimate either weights, lengths, or number of elements: e.g., truck (weight), players on a basketball team, chair height, number of schoolchildren in a classroom, frog length. | ||||
Related posts:
Common Clinical Disorders Affecting Mental Status Testing
Essential Behavioral Neuroanatomy
Praxis and Related Cortical Movement Abnormalities
Constructional, Perceptual, and Spatial Abilities
General Mental Status Scales, Rating Instruments, and Behavior Inventories
Overview of Neuropsychological Testing
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