BOOK EXCERPT
Vision, Perception, and Cognition:
A Manual for the Evaluation and Treatment of the Adult With Acquired Brain Injury, Fourth Edition
Barbara Zoltan MA, OTR/L

Chapter 11
Acalculia

Acalculia

The ability to perform calculations is crucial to many areas of occupational performance. The skill is used for tasks such as reading price tags, paying bills, counting out money for a purchase, addressing letters, measuring for a recipe, or writing checks.^1,2 Apart from language, calculations are perhaps the only culturally determined semantic system that the majority of the population is expected to acquire and master.³ Our “number sense” is useful for survival and helps us make sense of a world of discrete objects that form sets and whose combinations follow the rules of arithmetic.⁴

An individual’s internal representation of numerical quantities develops rapidly in the first year of life.⁴ This ability underlies our ability, later in life, to learn symbols for numbers and perform simple calculations.^4,5 Our adult numerical skills not only include these simple calculations but also include the ability to translate among Arabic numerals, written number names, and spoken number names.⁶ Dehaene outlines the following number processing skills of the normal adult⁴:

1. Read, write, produce, or comprehend numerals in both Arabic and verbal forms
2. Convert numbers in these formats to internal quantities and vice-versa
3. Compute single-digit addition, subtraction, multiplication, and division operations
4. Coordinate several such elementary operations to solve a complex, multidigit arithmetic problem

In order to understand and interpret any problems the client may have with calculations, it is important to understand the theoretical underlying mechanisms required for successful performance. This successful performance depends on different types of knowledge, including arithmetic facts (ie, 4 x 5 = 20), knowledge of procedures (ie, use of carrying over), and conceptual knowledge (ie, understanding the principles underlying the facts and the procedures).⁷

It is generally assumed that the cognitive numerical processing mechanisms include numeral comprehension, numeral production, and cognitive processes specific to arithmetic.^8,9 They include components of comprehension of operation symbols (eg, =) and words (eg, plus), retrieval of arithmetic facts, and execution of calculation procedures.^9,10 The number processing system is generally distinguished from the calculation system. The number processing system includes the number comprehension and number production subsystems.¹⁰ The calculation system, on the other hand, includes the comprehension of operation symbols, the retrieval of arithmetic facts, and the execution of arithmetic procedures.^9-11 Within each of these subsystems, a further distinction is made between components for processing Arabic numbers (ie, numbers in digit form such as 362) and components for processing verbal numbers.²

The number comprehension subsystem translates Arabic or verbal number inputs into internal semantic representation for use in subsequent cognitive processing.² Within the Arabic and verbal number comprehension components, a distinction is made between lexical and syntactic processing. Lexical processing is the comprehension of the individual elements in a number. For example, the digit 3 or the word three. Syntactic processing involves the analysis of the relations among elements. This skill refers to word order or the ability to produce an internal representation of the number as a whole.²

As previously described, number production components serve to translate internal semantic representations of numbers into sequences of digit or word representations for output.^2,11 Performing calculations requires the three elements of cognitive processes for number comprehension and production and cognitive processes that are specific to calculation procedures. These processes include comprehension of operation symbols or words, retrieval of number facts, and execution of the procedures themselves. Retrieval of number facts is central to almost any form of arithmetic problem solving.¹¹

The number processing and calculation system model is summarized in Figure 11-1 and Figure 11-2. Although the number processing and calculation system model is generally accepted, some theorists believe it to be somewhat oversimplified.¹² These theorists describe an alternative theory of encoding complex. They hypothesize that excitatory and inhibitory associative processes contribute collectively to the performance of numerical tasks. Rather than assuming that brain damage would selectively impair comprehension, calculation, or production, this model suggests that partial dysfunction might by associated with selective damage to inhibitory processes.¹² A nonmodular system, in which multiple numerical codes activate one another in the course of numeral processing and arithmetic tasks, is envisioned.⁹ These numerical codes are assumed to be interconnected in an associative network.

McClosky presents a model of number processing for which “…all transcoding, including reading aloud both Arabic numerals and written number names, occurs via a single route.”⁹ Still others propose a multiroute model, which includes both semantic and nonsemantic routes. These theorists hypothesize that the activation of one of the two routes will inhibit the function of the other route.⁴ Others hypothesize that internal representations of numbers are not abstract but rather are format specific.⁶ According to this model, “…number transcoding and calculations are based on a series of modality-specific codes (eg, verbal, visual, and visuospatial) that can be directly interconnected without the mediation of an abstract code.”⁶ Dehaene and Cohen support this concept by proposing a “Triple-Code Model,” which includes the three main representations of numbers as follows: visual Arabic code, an analogical quantity or magnitude code, and a verbal code.¹³

Figure 11-1. Examples of test items from the number-processing section of the dyscalculia test battery.2 (Reprinted with permission from Macaruso P, Harley W, McCloskey M. Assessment of acquired dyscalculia. In: Margolin DM, ed. Cognitive Neuropsychology in Clinical Practice. New York, NY: Oxford University Press; 1992:4055.)

Hittman-Delazer et al describe three components of the arithmetic system.⁵ These components are 1) arithmetic facts, which are thought to be stored in a specific semantic network system from where they can be retrieved as “labels” without a calculation process, 2) arithmetic procedures, defined as sequences of steps necessary to perform multi-digit operations, and 3) a recognition system for arithmetic signs.⁵ Other theorists add a fourth component of conceptual knowledge. This involves the understanding and use of arithmetic principles.

Recent neuroimaging studies have lent support to the concept of multiple network involvement in numerical processing abilities. Both cortical and subcortical networks have been found to underlie the ability to understand, produce and mentally manipulate numbers in various formats.¹⁴ Research has shown even extremely simple calculations, such as 5 – 2, involve multiple brain areas.⁴ One functional magnetic resonance imaging (fMRI) study indicated brain regions including parietal, frontal, and anterior cingulate areas were activated during number processing.¹⁵ A study by Ruechert et al also revealed that activation during calculations involved the coordinated effort of several different cortical areas.¹⁶ Another fMRI study of eight college students demonstrated that the parietal areas as well as the prefrontal cortex and possibly the thalamus are all activated during numerical processing.¹⁷ A study conducted by Chochon et al confirmed the involvement of the left and right parietal areas in calculations but state they may not be functionally equivalent.¹⁵ In addition, this study demonstrated that there exists partially distinct cerebral networks or circuits that underlie distinct arithmetic operations.¹⁵ A study conducted by Dehaene et al supports the notion of distinct networks when their study results pointed to partially distinct networks for multiplication and number comparisons.¹⁸

Figure 11-2. Examples of test items from the calculation section of the dyscalculia test battery.2 (Reprinted with permission from Macaruso P, Harley W, McCloskey M. Assessment of acquired dyscalculia. In: Margolin DM, ed. Cognitive Neuropsychology in Clinical Practice. New York, NY: Oxford University Press; 1992:4055.)

Indeed, deficits in acalculia have been reported as the result of numerous brain regions.¹ It can occur with a number of other deficits such as aphasia or as part of Gerstmann’s syndrome^13,19 or as a primary deficit involving any or all four arithmetic operations.¹⁰ Denburg and Tranel have described three general categories of acalculia, which are summarized in Table 11-1.

Even two seemingly similar operations such as subtraction and multiplication can be dissociated (ie, the client is able to perform one operation but not the other).⁶ Some studies, in fact, have found that subtraction appears to be better preserved than multiplication and addition.²⁰ One study has also described a client who exhibited acalculia for addition, subtraction, and division, but had intact multiplication.¹⁰ This client also had intact ability to distinguish math signs. These results point to the possibility of different processing systems responsible for each of the basic arithmetic operations. Another study of clients with major left hemisphere lesions showed their inability to name, add, subtract, or multiply digits accompanied with an intact ability to state which of two numbers is larger.⁶

Tohgi et al support these findings in a description of a client with a left frontal lobe infarct.²¹ The client was able to add and subtract numbers but could not multiply or divide.²¹ The deficit was attributed to a difficulty in retrieving facts from the multiplication table and the calculation procedures themselves.

Adapted from reference 1: Denburg N, Tranel D. Acalculia and disturbances of the body schema. In: Heilmann, Valenstein E, eds. Clinical Neuropsychology. 4th ed. New York, NY: Oxford University Press; 2003.

Corbett et al describe still another clinical picture related to acalculia.²² A 60-year-old male with a subcortical infarct had difficulties in numerical syntax, a loss of ability to manipulate math concepts, and impaired working memory.²² This study is one of several that have demonstrated that localization for number processing and calculations is not limited to cortical structures.²³

Sokol et al describe a client with ABI with impaired retrieval of arithmetic facts but intact abilities in execution of calculation procedures.²⁴ These authors point to two separate functionally distinct components in the cognitive calculation system.

Rosselli and Ardila studied 41 clients with left hemisphere damage and 21 clients with right hemisphere damage.²⁵ All groups presented difficulties with calculations tasks. Left hemisphere clients showed a significantly higher number of errors in reading numbers and arithmetic signs, counting backwards, and performing successive operations. These clients, however, showed a better comprehension of written numbers. Lucchelli and DeRenzi also describe a client with good comprehension of operation symbols but impaired retrieval of math facts and execution of calculation procedures.²⁶

Takayama et al describe clients who could read, repeat, and accurately verbalize numbers; had normal counting ability; understood the basic processes of calculation, and showed little difficulty in the retrieval of table values.²⁷ Their errors were made in the process where a number of steps were carried out simultaneously. These steps included retrieval of the number fact or table value, appropriate spatial alignment of the digits, appropriate procedural access, and retention and use of any integers remaining from the previous product.²⁷ These authors concluded that a working memory deficit could have a strong effect on multi-digit arithmetic problems and that the left parietal lobe may be a specialized area for working memory for calculation.²⁷

Delazer and Benke describe a 56-year-old female with a left parietal tumor who demonstrated that arithmetic facts could be represented at a superficial level without really understanding the operation she performed.²⁰ A client studied by Hittman-Delazer et al demonstrated the opposite.⁵ Despite this client’s inability to perform arithmetic fact problems such as 2 + 3, he was able to process algebraic expressions and had an excellent understanding of complex arithmetic text problems. He understood arithmetic principles and applied them in a variety of tasks. The client’s clinical picture lead these authors to hypothesize that conceptual knowledge (in addition to arithmetic fact knowledge and arithmetic procedures) is a functionally independent component of the calculation system.⁵

The previously described theoretical models, neuroimaging, and clinical research studies indicate that acalculia can be manifest in a number of different ways. Calculation ability is likely mediated by several cognitive processes including, language, memory, visual spatial, and attentional abilities.^1,7,20 In addition, conceptual knowledge or abstraction is considered to be an important part of the calculation or number processing system.⁵ Semenza et al associate a calculations deficit with either a deficit in the knowledge or memory of the procedure or a monitoring deficit.⁷ They build on this concept by outlining the characteristics of the client’s performance that would indicate the type of calculation deficit. These characteristics and associated deficits are summarized in Table 11-2.

Adapted from reference 7: Semenza C, Miceli L, Girelli L. A deficit for arithmetical procedures: lack of knowledge or lack of monitoring? Cortex. 1997;33(3):483-498.

Acalculia can have broad implications relating to the client’s occupational performance and, therefore, warrants a complete evaluation. Assessment of calculation skills should be varied and include written and oral calculation, comprehension and use of operations, and tasks involving the spatial components of arithmetic.¹ The assessment should parse out a true primary calculation deficit versus one that is secondary to deficits in areas such as language, attention, memory, or executive skills.¹ It is also important initially to find out, perhaps through family interviews, the client’s premorbid calculation abilities.

Evaluation of Acalculia

Test 1 – Clinical Observations

Description
Observe the client during occupation-based activities involving number processing abilities (ie, measuring during kitchen tasks, calculating money during a purchase on a community outing, balancing his or her checkbook, etc).

Test 2 – Dyscalculia Battery²

Description
This test battery has items representing both the number processing system and calculations system. Within each of these major sections are items covering all the component skills. Please refer to Figure 11-1 and Figure 11-2 for examples of test items.

Test 3 – Functional Calculations Evaluation

Description
An evaluation can be developed that contains test items for recognition of numbers, simple mathematical operations, and complex mathematical operations, including concepts associated with these operations.^28,29 Test items that are functionally oriented should be incorporated into the evaluation (eg, coin recognition, calculating change, check writing, or budgeting).

Scoring
Nonstandardized
The clinician measures the client’s level of performance on each category of tasks described.

Validity
To improve validity, rule out poor visual attentiveness and oculomotor skills, decreased attention, problem solving, mental inflexibility, and aphasia as causes of poor performance.

References

1. Denburg N, Tranel D. Acalculia and disturbances of the body schema. In: Heilmann, Valenstein E, eds. Clinical Neuropsychology. 4th ed. New York, NY: Oxford University Press; 2003.
2. Macaruso P, Harley W, McCloskey M. Assessment of acquired dyscalculia. In: Margolin DM, ed. Cognitive Neuropsychology in Clinical Practice. New York, NY: Oxford University Press; 1992:4055.
3. Spiers PA. Acalculia revised: current issues. In: Deloche G, Seren X, eds. Mathematical Disabilities: A Cognitive Neuropsychological Perspective. Hillsdale, NJ: Lawrence Erlbaum Assoc; 1987.
4. Dehaene S. Cerebral basis of number counting and calculation. In: Gazzaniga, Michael S, ed. The New Cognitive Neuroscience. 2nd ed. Cambridge, Mass: MIT Press; 2000.
5. Hittman-Delazer M, Sailer U, Benke T. Impaired arithmetic facts but intact conceptual knowledge—a single-case study of dyscalculia. Cortex. 1995;31(1):139-147.
6. Cipolotti L, Butterworth B. Toward a multiroute model of number processing: impaired number transcoding with preserved calculations skills. J Exp Psychol. 1995;124(4):375-390.
7. Semenza C, Miceli L, Girelli L. A deficit for arithmetical procedures: lack of knowledge or lack of monitoring? Cortex. 1997;33(3):483-498.
8. Basso A, Burgio F, Caporali A. Acalculia, aphasia and spatial disorders in left and right brain-damaged patients. Cortex. 2000;36(2):265-280.
9. McCloskey M. Cognitive mechanisms in numerical processing: evidence from acquired dyscalculia. Cognition. 1992;44:107-157.
10. Lample Y, Eshel Y, Gilad R, Sarova-Pinhas I. Selective acalculia with sparing of the subtraction process in a patient with left parietotemporal hemorrhage. Neurology. 1994;44(9):1759-1761.
11. McCloskey M, Harley W, Sokol SM. Models of arithmetic fact retrieval: and evaluation in light of findings from normal and brain-damaged subjects. J Exp P Psych Learning Mem Cogn. 1991;17(3):377-397.
12. Clark JM, Campbell JID. Integrated versus modular theories of number skills and acalculia. Brain Cogn. 1991;17:204-3-239.
13. Dehaene S, Cohen L. Cerebral pathways of calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex. 1997;33(2):210-250.
14. Delazer M, Girelli L, Semenza C, Denes G. Numerical skills and aphasia. J Int Neuropsychol Soc. 1999;5(3):213-221.
15. Chochon F, Cohen L, van de Moortele PF, Dehaene S. Differential contributions of the left and right inferior parietal lobules to number processing. J Cogn Neurosci. 1999;11(6):617-630.
16. Ruechert L, Lange N, Partiot A, et al. Visualizing cortical activation during mental calculation with functional MRI. Neuroimage. 1996;3(2):97-103.
17. Rickard TC, Romero SG, Basso G, et al. The calculating brain: an fMRI study. Neuropsychologia. 2000;38(3):325-335.
18. Dehaene S, Tzourio N, Frak V, et al. Cerebral activations during number multiplication and comparison: a PET study. Neurolpsychologia. 1996;34(11):1097-1106.
19. Grafman J, Kampen D, Rosenberg J, Salazar AM, Boller F. The progressive breakdown of number processing and calculation ability: a case study. Cortex. 1989;25:121-133.
20. Delazer M, Benke T. Arithmetic facts without meaning. Cortex. 1997;33(4):697-710.
21. Tohgi H, Saitoh K, Takahashi S, et al. Agraphia and acalculia after a left prefrontal (F1, F2) infarction. J Neurol Neurosurg Psychiatry. 1995;58:629-632.
22. Corbett AJ, McCusker EA, Davidson OR. Acalculia following a dominant-hemisphere subcortical infarct. Arch Neurol. 1986;43(9):964-966.
23. Kahn HJ, Whitaker HA. Acalculia: an historical review of localization. Brain Cogn. 1991;17:102-1151.
24. Sokol SM, McCloskey M, Cohen NJ, Aliminosa D. Cognitive representations and processes in arithmetic: inferences from the performance of brain-damaged subjects. J Exp Psych Learning Mem Cogn. 1991;17(3):355-376.
25. Rosselli M, Ardila A. Calculation deficits in patients with right and left hemisphere damage. Neuropsychologlia. 1989;27(5):607-617.
26. Lucchelli F, DeRenzi E. Primary dyscalculia after a medical frontal lesion of the left hemisphere. J Neurol Neurosurg Psychiatry. 1993;56:304-307.
27. Takayama Y, Sugishita M, Akiguchi I, Kimura J. Isolated acalculia due to left parietal lesion. Arch Neurol. 1994;1:2860-291.
28. Lezak M. Neuropsychological Assessment. New York, NY: Oxford University Press; 1983.
29. Luria AR. Higher Cortical Functions in Man. 2nd ed. New York, NY: Basic Books Inc; 1980.
30. Parente R, Herrmann D. External aids to cognition. In: Parente R, Herrmann D, eds. Retraining Cognition: Techniques and Applications. 2nd ed. Austin, Tex: PRO-ED Inc; 2003.

Resources

Benson DF. Disorders of visual gnosis. In: Brown JW, ed. Neuropsychology of Visual Perception. Hillsdale, NJ: Lawrence Erbaum Assoc; 1989.

Wheatley C. Evaluation of cognitive dysfunction. In: Early MB, ed. Physical Dysfunction: Practice Skills for the Occupational Therapy Assistant. St. Louis, Mo: Mosby; 1998.