Encyclopedia of Clinical Neuropsychology

Living Edition
| Editors: Jeffrey Kreutzer, John DeLuca, Bruce Caplan


  • Kelly BroxtermanEmail author
  • Natalie Wahmhoff
  • Elaine Clark
  • Alyssa Beukema
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-56782-2_1433-2


Mathematical Task Calculation Ability Primary Deficit Wide Range Achievement Number Knowledge 
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Acalculia is an acquired impairment in which people have difficulty performing mathematical tasks, such as adding, subtracting, multiplying, and dividing. Acalculia deficits can manifest in a wide variety of number processing and calculation abilities.


Generally, authors have agreed on two major distinctions: primary and secondary acalculia (Growth-Marnat 2000). These terms were first described by Berger in 1926 (Boller and Grafman 1983). Primary acalculia refers to a basic defect in computational abilities, not resulting from separate cognitive deficits. It is also known as anarithmetia. Deficits in primary acalculia include poor estimation, number comparison difficulties, and difficulty understanding procedural rules and numerical signs. In primary acalculia, these deficits will exist regardless of whether tasks are presented in an oral or written format (Ardila and Rosselli 2002).

Secondary acalculia refers to calculation defects due to primary deficits in other areas: memory disorders, attention impairments, language defects, spatial deficits, etc. (Berger 1926). The secondary acalculias include aphasic acalculia, alexic acalculia, agraphic acalculia, frontal acalculia, and spatial acalculia.

Aphasic acalculia occurs in patients with Broca’s and Wernicke’s aphasia. The deficits seen in patients with Broca’s aphasia are linguistic in nature, and these individuals often have impairments in the syntax of calculation. This includes problems when translating word representations of numbers (three hundred and forty-five) to their numeral form (345). They may also read numbers with morphological errors (15 is read as 50) (Ardila and Rosselli 2002; Basso et al. 2000). When the secondary acalculia stems from Wernicke’s aphasia, deficits are more severe. Reading and writing of numbers often have semantic errors, and poor verbal memory often impacts the calculation abilities of these patients (Grafman and Rickart 2000).

Alexic acalculia is the inability to read number and correlations with the inability to read text. People with this type of acalculia may focus only on beginning digits (538 is read as 53). For those with alexic acalculia, mental calculation abilities exceed written calculation abilities (Ardila and Rosselli 2002).

Agraphic acalculia is the inability to write numbers. Like aphasic acalculia, agraphic acalculia correlates with Broca’s and Wernicke’s aphasia. In Broca’s aphasia, acalculia deficits manifest as omissions, substitutions, and order reversal. In Wernicke’s aphasia, difficulties are especially evident when required to write quantities when they are orally dictated. Those with Wernicke’s aphasia also tend to make paralexias and paragraphias (Ardila and Rosselli 2002; Growth-Marnat 2000).

Frontal acalculia deficits occur in conjunction with attention difficulties, perseveration, and impairment of more complex math concepts (Dehaene et al. 1998). Patients with frontal acalculia often have significant damage to the prefrontal areas of their brain, which can cause trouble with mental operations, operations that need to be carried out successively, backward operations, and numerical problems that require multiple steps. In these individuals, written mathematical problem-solving is easier to do than mental math operations. While complex concepts are difficult for patients with frontal acalculia, more basic math concepts are usually maintained (Ardila and Rosselli 2002).

Spatial acalculia impacts written mathematical tasks more than mental math tasks. A difficulty with writing numbers is quite apparent in these cases and manifests in several ways. Writing on only one side of the page, inability to write numbers in a straight line, and general disorganization are some of the deficits that impact math performance (Basso et al. 2000). Patients with spatial acalculia often forget where to place remainders and carried numbers, despite understanding the basic division and multiplication functions. Math procedure signs are often undetected or switched (add instead of subtract).


Acalculia can result from stroke, tumors, and trauma. It is also seen in patients with degenerative dementia (Ardila and Rosselli 2002).

Prognostic Factors and Outcomes

There is noted variability in prognosis for acalculia, ranging from no recovery to full recovery. For primary acalculia, improvement is limited. In the case of secondary acalculias, when recovery from the primary deficit, such as aphasia, alexia, and agraphia, occur, the corresponding acalculia deficits tend to improve as well.

Neuropsychology and Psychology of Acalculia

Primary acalculia is associated with left posterior parietal lesions. More specifically, damage to the left angular and supramarginal gyri occurs with primary acalculia (Grafman and Rickart 2000). It is suggested that there are separate neuropathways for rote number knowledge and semantic number knowledge.

Neuroimaging techniques reveal that several brain areas are active when performing calculations and also that the pattern differs according to what type of calculation is done (Dehaene et al. 1998). This occurs to the many abilities that calculation often requires, including verbal, spatial, executive functioning, and memory. The areas most associated with calculation are the upper cortical surface and anterior aspect of the left middle frontal gyrus, the bilateral supramarginal and angular gyrus, the left dorsolateral prefrontal and premotor cortices, Broca’s area, inferior parietal and left parietal cortex, and the inferior occipitotemporal regions (Ardila and Rosselli 2002).

It is important to keep in mind that damage to the right hemisphere and the frontal lobes also impact the occurrence of acalculia, especially when it is a secondary acalculia.


The arithmetic section of the Wide Range Achievement Test (WRAT) has often been used to test operational skills. The Key Math, which is designed for children and adolescents, tests most targeted and specific abilities that are suggested for an acalculia assessment (Grafman and Rickart 2000). Many authors have suggested experimental batteries that target specific functions and include error analysis. These batteries often assess skills in the following areas: number recognition, number writing, number transcoding, quantification, magnitude estimation, basic arithmetic operations, calculation fact verification, multicolumn calculations, magnitude comparison, fractions, algebra, and numeric knowledge. When possible, these skills should be assessed in both written and oral forms (Ardila and Rosselli 2002; Grafman and Rickart 2000).


Some authors have suggested beginning rehabilitation with an error analysis if it was not completed during the assessment. This will provide explicit areas to target during rehabilitation (Grafman and Rickart 2000). Long-term rehabilitation programs should begin simply and progressively work toward more complex tasks. With secondary acalculia, focusing rehabilitation on the primary deficit may significantly improve the secondary acalculia deficits (Ardila and Rosselli 2002).


References and Readings

  1. Ardila, A., Matute, E., & Inozemtseva, O. (2003). Progressive agraphia, acalculia, and anomia: A single-case report. Applied Neuropsychology, 10, 205–214.CrossRefPubMedGoogle Scholar
  2. Ardila, A., & Rosselli, M. (2002). Acalculia and dyscalculia. Neuropsychology Review, 12, 179–231.CrossRefPubMedGoogle Scholar
  3. Basso, A., Burgio, F., & Caporali, A. (2000). Acalculia, aphasia, and spatial disorders in left and right brain-damaged patient. Cortex, 36, 265–280.CrossRefPubMedGoogle Scholar
  4. Berger, H. (1926). Uber Rechenstorunger bei Herderkraunkunger des Grosshirns. Arch Psychiatr Nervenkr, 78, 236–263Google Scholar
  5. Boller, F., & Grafman, J. (1983). Acalculia: Historical development and current significance. Brain and Cognition, 2 (3), 205–223.CrossRefPubMedGoogle Scholar
  6. Dehaene, S., Cohen, L., & Changeux, J. P. (1998). Neuronal network models of acalculia and prefrontal deficits. In R. W. Parks, D. S. Levine, & D. L. Long (Eds.), Fundamentals of neural network modeling: Neuropsychology and cognitive neuroscience (pp. 233–255). Cambridge, MA: MIT.Google Scholar
  7. Grafman, J., & Rickart, T. (2000). Acalculia. In M. J. Farah & T. E. Fienberg (Eds.), Patient based approaches to cognitive neurosciences: Issues in clinical and cognitive neuropsychology. Cambridge, MA: MIT.Google Scholar
  8. Growth-Marnat, G. (Ed.). (2000). Neuropsychological assessment in clinical practice. New York: Wiley.Google Scholar
  9. Scruggs, T. E., & Mastropieri, M. A. (2000). Acalculia. In Encyclopedia of special education (Vol. 1, 2nd ed., p. 27). New York: Wiley.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kelly Broxterman
    • 1
    Email author
  • Natalie Wahmhoff
    • 2
  • Elaine Clark
    • 2
  • Alyssa Beukema
    • 1
  1. 1.School PsychologyThe Chicago School of Professional PsychologyChicagoUSA
  2. 2. Department of Educational PsychologyUniversity of UtahSalt Lake CityUSA