The term dyscalculia refers to an acquired disorder of

number processing and calculation skills following brain

damage. Henschen was the first to identify this syndrome

in 1919.1 However, for a long time dyscalculia was treated

as one of the subcomponents of the Gerstmann syndrome

or as an impairment due to more generalised cognitive

deficits such as visuospatial and language disorders. It is

now well established that impairments in number processing

and calculation are independent from deficits in

general intelligence, language, reading, writing, semantic

memory and short-term memory.2

Acquired deficits in number processing and calculation

are rather frequent after brain lesions and may result

from both acute and neurodegenerative conditions. The

incidence of dyscalculia in patients with either left hemisphere

lesions or at the early stage of Alzheimer’s disease

is high

Cipolotti, L

Van Harskamp, N

UCL Discovery

14 I ACNR • VOLUME 5 NUMBER 4 • SEPTEMBER/OCTOBER 2005DyscalculiaIntroductionThe term dyscalculia refers to an acquired disorder ofnumber processing and calculation skills following braindamage. Henschen was the first to identify this syndromein 1919.1 However, for a long time dyscalculia was treatedas one of the subcomponents of the Gerstmann syndromeor as an impairment due to more generalised cognitivedeficits such as visuospatial and language disorders. It isnow well established that impairments in number pro-cessing and calculation are independent from deficits ingeneral intelligence, language, reading, writing, semanticmemory and short-term memory.2Acquired deficits in number processing and calculationare rather frequent after brain lesions and may resultfrom both acute and neurodegenerative conditions. Theincidence of dyscalculia in patients with either left hemi-sphere lesions or at the early stage of Alzheimer’s diseaseis high.ClassificationDyscalculia is not a unitary disorder and can take a vari-ety of different forms. Patients may present with specificimpairments in processing numbers, in calculation or inboth. Table 1 provides an overview of the types ofimpairment.Disorders of number processingPatients with number processing impairments may showselective deficits in either producing (e.g. reading, writingor repeating) or in comprehending (e.g. knowing that 5 isbigger than 4) Arabic numerals (e.g. 5), verbal numerals(e.g. five) or both.Patients with number production impairments mayshow selective deficits affecting syntactical or lexicalprocessing.• Syntactical errors involve the incorrect selection of thenumber class to which a number belongs (e.g. theyread 600 as “sixty”).• Lexical errors involve the incorrect selection of theposition of the individual elements within the correct-ly identified class (e.g. they read 29 as “forty-nine”).Patients with a deficit in number comprehension may nolonger be able to point to the larger of two Arabic numer-als (e.g. 345 and 785 or 265 and 2307) or match spokennumerals to the corresponding Arabic numeral. Deficitsin number comprehension may selectively affect a subcat-egory of numbers or different aspects of number mean-ing. For example, Cipolotti et al3 described a patient wholost the meaning of numbers above 4. Patients with aselective impairment in “cardinal” (which depicts thenumerosity of a set of entities and describe the manynessof the set) number meaning or “sequence” (which depictsthe position of a number word in the number sequenceand do not refer to numerosities) number meaning havebeen described.4Disorders of calculationIn order to carry out a complex calculation such as346+475, several independent calculation subprocessesare required. These include:• The identification of arithmetic symbols (e.g. +, x, -, :)• The retrieval of arithmetic facts. These are defined as avocabulary of “number combinations”, such as 5+3=8,3x4=12, 10-4=6. These facts are directly retrieved frommemory and the solution to these problems does notrequire further computational processes or strategies.• The execution of calculation procedures. These proce-dures allow access to specific algorithms required tosolve multi-digit calculation. Specific examples are thecarrying and borrowing procedures.Cognitive PrimerNatasja van Harskamp studiedNeuropsychology and Develop-mental Psychology at the Uni-versity of Utrecht. Since 1996 shehas been Clinical Neuropsych-ologist in the NeuropsychologyDepartment of the NationalHospital for Neurology andNeurosurgery. She has a specialinterest in acquired disorders ofcalculation and social cognition.Lisa Cipolotti studied Experi-mental Psychology at the Uni-versity of Padua and completed aPhD on Dyscalculia in 1993 atUCL. Since 1996 she has beenHead of the NeuropsychologyDepartment of the NationalHospital for Neurology andNeurosurgery. Her main researchis on the acquired disorders ofmemory, language and calculation.Correspondence to:Natasja J van Harskamp,Department of Neuropsychology,National Hospital for Neurologyand Neurosurgery,2nd Floor, Queen Mary Wing,Queen Square,London WC1N 3BG.Box No. 37Tel: 0207 829 8793Email: n.harskamp@ion.ucl.ac.ukl.cipolotti@ion.ucl.ac.ukDisorders of number processing Error exampleDisorders of number production • Disorders of syntactical processing (Syntactical errors) 5 read as “fifty”• Disorders of lexical processing (Lexical errors) 5 read as “seven”Disorders of number comprehension • Disorders of cardinal number meaning:Problems with identifying the quantity of a set 5 is larger than 6• Disorders of sequence number meaning: Problems with identifying the position of a number in a sequence 5 comes after 6Table 1 Overview of types of deficits occurring in dyscalculiaDisorders of calculation 47 49Disorders of arithmetic symbol processing: x 67 x 29e.g. adding when there is a multiplication sign 114 78Disorders of arithmetic fact retrieval:e.g. patients failing to retrieve automatically arithmetical facts such as 6x5• Operand errors, if the incorrect answer is the correct answer to a problem that shares one of the operands 6x5=25• Operation errors, if the incorrect answer is the correct answer to another problem involving the same 6x5=11operands, but a different operation • Table errors, if the incorrect answer is an answer that is a product of two other single digit numbers 6x5=32• Non table errors, if the incorrect answer is not an operand, table or operation error 6x5=41Disorders of calculation procedures: e.g. patients failing to apply specific calculation procedures such as:• Systematic smaller from larger subtraction errors• Inappropriate carry over procedures in the calculation of partial products and in the sum of partial productsDisorders of conceptual knowledge:e.g. the solution of an addition problem is smaller than the addends 8+7=6823 171- 644 - 58221 12735 723x 97 x 57245 151275   3852995 3100116 I ACNR • VOLUME 5 NUMBER 4 • SEPTEMBER/OCTOBER 2005• The retrieval of conceptual knowledge. Thisallows the understanding of the principlesunderlying both arithmetic facts and proce-dures (e.g. the solution of an addition prob-lem is greater than the addends).Each of these different cognitive processesappears to be functionally independent and dif-ferentially susceptible to brain damage.For example, patients with a selective impair-ment in the processing of arithmetic symbolsmisname and misidentify the arithmetical signsand perform the calculation according to theirmisidentification.5Several patients have been documented withselective arithmetic fact retrieval impairment.Typically, these patients are impaired in solvingvery simple single digit addition, subtraction,multiplication and division problems. Theyproduce many errors (e.g. 5+7= “13 roughly”)and their response times are abnormally slow(e.g. >3 sec.). Deficits in arithmetic factretrieval may be specific for type of operation.Thus, patients have been described with selec-tive impairment or preservation in subtraction,multiplication, addition and division problems(see table 2).Patients with selective deficits in calculationprocedures may have problems in executingthose procedures that specify the sequence ofsteps necessary to solve multi-digit problems,for example, the carrying and borrowing pro-cedures. Patients with a multi-digit subtrac-tion problem may systematically subtract thesmaller digit from the larger one regardless oftheir location in the top or bottom numbers.11Interestingly, this deficit can be specific fortype of procedure. Thus, patients may onlyhave difficulties with the borrowing procedurein complex subtraction, while still being ableto carry out complex addition, involving thecarry procedure.Few patients with deficits in conceptualknowledge have been reported. These patientsmay show a poor understanding of the con-ceptual aspects of calculation. For example,they produce highly implausible errors (e.g.the solution of subtraction is greater than theminuend) or may not apply very basic arith-metic principles such as order irrelevant-prin-ciple in multiplication (4x12=12x4) or repeat-ed addition of the second operand(4x12=12+12+12+12).8 Selective preservationof conceptual knowledge in the context ofsevere dyscalculia is also on record.12 Intact con-ceptual knowledge may be critical in develop-ing rehabilitation procedures for arithmeticdeficits (see below).AssessmentA variety of tasks may be used to assess numberprocessing and calculation (see table 3).Recently, a battery of Number Processing andCalculation (NPC) has been standardised byDelazer et al.13 Error analyses are also very use-ful additions to the assessment. They provideinvaluable information regarding the type offunctional impairment the patient presents.LocalisationNumerical skills have a discrete and independentbrain substrate. The majority of evidence basedon lesion studies has indicated the involvement ofthe left posterior areas. The reports available donot allow for a conclusive localisation of lesionswithin the left posterior quadrant. However, itappears that the left parietal lobe plays a crucialrole. Recent neuroimaging studies have investi-gated the neuronal correlates underpinningnumber processing and calculation. They oftenreport large neuronal networks of parietal, pre-frontal and cingulate areas. In particular, the hor-izontal segment of intraparietal sulcus bilaterally(HIPS) and the inferior frontal gyrus and theprecentral sulcus are mostly implicated.14Treatment and recovery Data on recovery from dyscalculia is rare, how-ever partial recovery occurs in most patientswith dyscalculia following vascular lesions.15According to the specific type of dyscalculia,different kinds of rehabilitative interventionmay be required. Principles of treatment inrehabilitation of dyscalculia mainly consist of:• reteaching lost arithmetic fact knowledgevia extensive training • errorless learning • the use of back-up strategies based on theprinciples underlying arithmetic facts suchas the order-irrelevant principle like8x6=6x8=48, decomposition strategies like4x8=2x8+2x8=32 or repeated addition ofthe second operand (3x5=5+5+5=15).Case reports show improvement followingintense rehabilitation of arithmetic facts andnumber transcoding deficits.16Cognitive PrimerMultiplication Subtraction Addition DivisionSubtraction preservation BB (Pesenti et al., 1994)6 x ✓ x -Subtraction impairment SS (van Harskamp et al, 2002)7 ✓ x ✓ -Multiplication preservation JG (Delazer and Benke, 1997)8 ✓ x x xMultiplication impairment VP (van Harskamp and Cipolotti, 2001)9 x ✓ ✓ -Addition impairment FS (van Harskamp and Cipolotti, 2001)9 ✓ ✓ x -Division impairment CB (Cipolotti and de Lacy Costello, 1995)10 ✓ ✓ ✓ x✓ =  spared; x = impaired; - = not testedTable 2: Examples of patients with selective impairments/preservations of arithmetic factsNumber processing tasksNumber production • Reading• Writing to dictation• Repetition • Transcoding between Arabic numerals and verbal numerals (5→five)Number comprehension • Magnitude comparison of Arabic numerals or written/spoken number names (e.g. 12-21)• Analogue number scale task (scale from 0-100, marked at 25; 50; 75, show me 25)• Parity judgment (odd or even number?)Calculation tasksArithmetic symbol processing • read• point• write the arithmetical signsArithmetic fact retrieval Simple arithmetic problems across the four basic operations such as 4+2, 3x4 or 5-2, 6:3.Procedural knowledge Multi-digit calculation, such as 294+12=306Conceptual knowledge Arithmetical principles such as commutativity (a+b=b+a); repeated addition; 10a x 10b; multiplication/division inversion; a-1xb; a+1; a-1; addition/subtraction inversion Table 3: Assessment of numerical processing and calculationReferences1. Henschen SE. Uber Sprach-Musik-und Rechenmechanismen und ihre Lokalisationen imGrosshirn. Zeitschrift fur die gesamte Neurologie und Psychiatrie 1919;52:273-98.2. Cipolotti L and van Harskamp NJ. Disturbances of number processing and calculation. In F.Boller & J. Grafman (Eds) Handbook of Neuropsychology: 305-31 Amsterdam-New York:Elsevier 2001.3. Cipolotti L, Butterworth B, Denes G. A specific deficit for numbers in case of dense acalculia.Brain: 1991;114:2619-37.4. Delazer M, Butterworth B. A dissociation of number meanings. Cognitive Neuropsychology1997;14:613-36.5. Laiacona M, Lunghi A: A case of concomitant impairment of operational signs and punctua-tion marks. Neuropsychologia 1997;35:325-32.6. Pesenti M, Seron X, Van Der Linden M. Selective impairment as evidence for mental organi-sation of arithmetical facts: BB, a case of preserved subtraction? Cortex 1994;30:661-71.7. Van Harskamp NJ, Rudge P, Cipolotti L. Are multiplication facts implemented by the leftsupramarginal and angular gyri? Neuropsychologia 2002;40:1786-93.8. Delazer M and Benke T. Arithmetic facts without meaning. Cortex 1997;33:697-710.9. Van Harskamp NJ, Cipolotti L. Selective impairments in addition, subtraction and multiplica-tion. Implications for the organistion of arithmetical facts. Cortex 2001;38:363-68.10. Cipolotti L and De Lacy Costello A. Selective impairment for simple division. Cortex1995;31:433-49.11. Sandrini M, Miozzo A, Cotelli M, Cappa F. The residual calculation abilities of a patient withsevere aphasia: evidence for a selective deficit of subtraction procedures. Cortex 2003;39:85-96.12. Hittmair-Delazer M, Sailer U and Benke T. Impaired arithmetic facts but intact conceptualknowledge - A single case study of dyscalculia. Cortex 1995;31:139-48.13. Delazer M, Girelli L, Grana A, Domahs F. Number processing and calculation-NormativeData from Healthy Adults. The Clinical Neuropsychologist 2003; 17(3):331-50.14. Dehaene S, Molko N, Cohen L, Wilson AJ. Arithmetic and the brain. Current opinion inneurobiology. 2004; 14:218-24.15. Basso A, Caporali A, Faglioni P. Spontaneous recovery from acalculia. J of the Int NeuropsychSoc 2005;11:99-107.16. Girelli L. and Seron X. Rehabilitation of number processing and calculation skills.Aphasiology, 2001;15(7): 695-12.

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