Chapter 21

The neuropsychology of acquired calculation disorders

Marinella Cappelletti and Lisa Cipolotti

1 Acalculia Acalculia is an acquired disorder of number processing and calculation skills following cerebral damage (Henschen, 1919 ). The inability to use numbers can be very incapacitat-ing as it interferes with several everyday activities such as shopping, using bank accounts and telephones (Butterworth, 1999 ; Dehaene, 1997 ). Acalculia is not a unitary disorder and can take a variety of different forms: patients may present with impairments in number processing, in calculation or both. The incidence of acalculia in patients with left hemisphere lesions has been estimated between 16% and 28%, and 90% of patients at the early stage of Alzheimer disease present with acalculia (Carlomagno et al ., 1999 ). Knowing the incidence of acalculia and its impact on everyday life has helped to improve assess-ment techniques and it has recently promoted the development of rehabilitation programs. In this chapter, we will:

1 present the basic components of the number and calculation system;

2 discuss cases of selective impairment of number processing and calculation;

3 discuss cases of selective preservation of number processing and calculation;

4 briefly outline’ localization in number and calculation disorders;

5 propose some guidelines for the assessment and rehabilitation of acalculia in neurological patients.

2 The basic components of number and calculation processing

2.1 Number comprehension and production Number comprehension usually refers to the ability to generate a semantic representa-tion of numbers. This commonly refers to the quantity associated with numbers, for instance ‘21’ indicates the numerosity of 21. The representation of quantity is thought to be abstract in nature (Dehaene and Cohen, 1997 ; McCloskey et al ., 1990 ). The quantity expressed by numbers is often processed when comparing numbers, for instance when deciding which of two products is the most expensive. In doing so, we are usually faster and more accurate when two numbers (or two prices) are further apart, e.g. £1.20 and

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£2.55, relative to when they are close together, e.g. £1.20 and £1.15. This phenomenon is referred to as the ‘distance effect’ (Moyer and Landauer, 1967 ). Impairments in process-ing numerical quantity often result in a ‘reverse distance effect’; this indicates that the time needed to process numbers increases as the numerical distance between them increases. This is because the quantity associated with numbers can not be accessed auto-matically and other strategies, for instance counting, have to be used instead. Besides indicating quantities, numbers can also have a ‘nominal’ function, for instance ‘21’ may also indicate a bus number or the age of consent (Butterworth, 1999 ; Dehaene, 1997 ).

Number production refers to the process of converting a numeral’s semantic represen-tation into an output format. The most commonly used formats are the Arabic, e.g. ‘21’ and the verbal, e.g. ‘twenty-one’. The production of a number in one of these formats is based on input and output processes. For instance, reading Arabic numerals is based first on visually identifying the string of digits, second on translating this string of digits into a sequence of words according to a set of rules, and third on producing an oral or written output. This transformation of numbers from one format to another is usually referred to as ‘ transcoding’ (Dehaene and Cohen, 1995 ; Deloche and Seron, 1982 ; McCloskey et al . , 1986 ).

2.2 Calculation Oral and written arithmetical operations require a set of specific and independent processes. These include:

(i) The processing of arithmetical symbols (i.e. +, ×, –, ÷)

(ii) The retrieval of arithmetical facts. These are defined as a vocabulary of ‘number combinations’, such as ‘3 + 3 = 6’ or ‘3 × 3 = 9’.

(iii) The execution of calculation procedures. These procedures consist of the specific algorithms required to solve multi-digit calculations. Specific examples are the carrying (e.g. in ‘234 + 159’) and borrowing (e.g. in ‘234 – 159’) procedures.

(iv) The retrieval of conceptual knowledge. This allows the understanding of the principles underlying arithmetical facts and procedures (e.g. the principle of com-mutativity, such as a + b = b + a).

3 Selective impairment of number processing and calculation Patients may present with specific impairments in the processing of numbers, in calcula-tion, or in both. Table 21.1 provides an overview of the range of potential deficits which can be observed in acalculic patients.

Here we will describe

1 impairments of number transcoding;

2 impairments of quantity processing;

3 impairments of calculation, which in turn may consist of impairments in processing arithmetical signs, simple facts, procedures, or arithmetical conceptual knowledge.

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3.1 Impairments of number transcoding Several studies have described patients with disorders in reading and writing numbers (e.g. Cipolotti et al., 1995 ; Deloche and Seron, 1982 ; McCloskey et al., 1986 ; 1990 ; Sokol and McCloskey, 1988 ). Some of these studies have focused on errors made by patients when reading or writing numbers. This permitted the identification of two major cognitive mechanisms within number transcoding: the syntactic, and the lexical number processes.

Syntactic processes involve the specification of the relationship among the elements of the number (e.g. number class, i.e. to read aloud 600, one needs to retrieve the correct number class {hundred}). Syntactic errors are errors which occur when the wrong number class is selected (e.g. stimulus 5, response ‘fifty’ or stimulus two thousand and thirty eight, response ‘2000308’). For example, patient SF’s syntactic errors occurred in reading aloud Arabic numerals and could best be classified as ‘quantity-shift’ errors (e.g. 207 → ‘two thousand and seven’; 80 → ‘eight’) (Cipolotti, 1995 ).

The lexical processes involve the processing of individual elements in the numeral (i.e. to read aloud 600, once the correct class {hundred} has been retrieved, it is necessary to retrieve the correct element, {6}). Lexical errors involve the incorrect production of one or more of the individual elements in a numeral (stimulus 29, response ‘forty-nine’). For example, patient HY’s errors in reading aloud Arabic numerals were mainly lexical (Sokol et al ., 1991 ). This patient could access the correct number lexical class (ones, teens, tens). However, he had difficulty in accessing the correct position within that class (e.g. stimu-lus 17, response ‘thirteen’ or stimulus 902, response ‘nine hundred six’).

Other studies have demonstrated that transcoding skills can be independent from other numerical abilities. For example, patient HY was impaired in reading aloud Arabic numerals (McCloskey et al ., 1986 ). When attempting to read aloud an Arabic numeral, he often made mistakes such as reading number ‘5’ as ‘seven’, and ‘29’ as ‘forty-nine’. In contrast, results from a series of number comprehension tasks showed that patient HY was unimpaired in comprehending Arabic numerals. Thus, his impaired performance in

Table 21.1 Overview of the potential disorders of (A) number processing and (B) calculation (right column)

A. Disorders of number processing

Disorders of number production Disorders of lexical processing

Disorders of syntactical processing

Disorders of number comprehension Disorders of cardinal number meaning

Disorders of sequence number meaning

B. Disorders of calculation

Disorders of arithmetical symbol processing

Disorders of arithmetical fact retrieval

Disorders of calculation procedures

Disorders of conceptual knowledge

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reading aloud Arabic numerals reflected his impairment in number transcoding rather than a deficit in comprehending the Arabic stimuli.

This pattern of performance has also been shown by other patients (e.g. Cipolotti and Butterworth, 1995 ), and the opposite performance, (i.e. poor performance in calculation and preserved transcoding has also been documented) (e.g. Cipolotti and Butterworth, 1995 ; Sokol et al ., 1991 ). This suggests that transcoding skills are distinct from other numerical abilities.

A frequent type of error in writing numbers to dictation or from a written input has been referred to as ‘intrusion errors’. These consist of reproducing part of the input code when formulating the output one. This type of error has been described in patients with dementia of the Alzheimer type as well as in those with focal brain lesions (Della Sala et al ., 2000 ; Thioux et al ., 1999 ). According to Thioux and colleagues ( 1999 ), three differ-ent types of intrusion errors can be distinguished.

◆ The first which consists of producing a numerical output in the same format as the source one, has been called ‘complete lexical shift error’ or ‘perseveration’. For exam-ple when the Arabic numeral ‘75’ is written as ‘75’ instead of as the number word ‘seventy five’, (or conversely the ‘seventy five’ written as the number word ‘seventy five’ instead of as the numeral ‘75’).

◆ The second type of intrusion error is called ‘partial lexical shifting’, which occurs when only one part of the input is transposed onto the output, (e.g. the Arabic numeral ‘75’ written as ‘seventy5’).

◆ Thirdly, ‘intra lexical shift’ errors consist of writing a numeral in a mixture of verbal and Arabic formats, for example, the Arabic numeral ‘70’ written as ‘7ty’.

Some authors explain these errors in terms of the combination of impaired transcoding mechanisms on the one hand, and impaired inhibitory processes on the other (Thioux et al ., 1999 ). Others have suggested that perseveration and intrusion errors are attentional in nature and originate from a unique impairment in selective attentional capacities (Macoir et al. , 2002 ).

3.2 Impairments of quantity processing Other studies have reported patients with a basic failure of understanding and processing the quantity indicated by numbers (Cipolotti et al ., 1991 ; Dehaene and Cohen, 1997 ; Delazer and Benke, 1997 ; Delazer and Butterworth 1997 ; Lemer et al., 2003 ; Noel and Seron, 1993 ; Polk et al ., 2001 ). For example, patient NR showed impairment in under-standing Arabic numerals (Noël and Seron, 1993 ). Thus, she could no longer point to the larger of two Arabic numerals (e.g. 345 vs 785 or 265 vs 237). Moreover, she lost the ability to match spoken number names to the corresponding Arabic numeral. A pro-foundly acalculic patients (CG) presented with an unusual pattern of performance (Cipolotti et al., 1991 ). Although she had retained the ability to process abstract quanti-ties, she lost the meaning of all numbers above 4. For example, she was unable to say how many days were in a week or which was larger, 5 or 10. Her deficit was so pervasive that it seriously limited her activities of daily living. For example, CG was unable to do her own

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shopping, as she could no longer deal with money (or check her change). She was no longer able to make phone calls, use a calendar, or read the time, although her intellectual skills, language, memory and visuospatial abilities were largely preserved.

Recent studies have focused on quantity processing in neglect patients. Two of these have shown that neglect patients can correctly indicate whether a number is larger or smaller than the reference number ‘5’ (Cappelletti and Cipolotti, 2006 ; Vuilleumier et al., 2004 ). The patients’ performance in number comparison indicated that even the numer-ical stimuli appearing in the neglected hemifield could be processed even when not reported (Cappelletti and Cipolotti, 2006 ). Although neglect patients could still correctly perform quantity tasks, their performance indicated that they were slower to process numbers which are located to the left of ‘5’, (i.e. ‘4’) relative to numbers located to its right (i.e. ‘6’). Their performance in numerical tasks therefore, seems consistent with their impairment in processing within the left side of space (Vuilleumier et al ., 2004 ).

Thus, the idea that the spatial impairment in neglect may affect numerical processing has recently been explored in studies, specifically, which tested whether neglect may affect the mental ‘number line’; a metaphor used to represent numbers as oriented from smaller to larger (Dehaene et al., 1993 ). One way to test the integrity of a mental number line is with the bisection task. This consists of orally presenting two numbers, (e.g. ‘1’ and ‘5’) and asking which number falls in the middle. Neglect patients typically select a number which is larger than the middle one, for instance they say that ‘4’ and not ‘3’ is mid-way between ‘1’ and ‘5’ (Cappelletti et al ., 2007a ; Doricchi et al ., 2005 ; Zorzi et al ., 2002 ). This pattern of performance mirrors the classical leftward bias which neglect patients show when bisecting physical lines, although their performance also seems to depend on the line orientation (physical and numerical) (Cappelletti et al ., 2007 ).

3.3 Impairments of calculation Each of the different cognitive processes involved in calculation (i.e. processing arithme-tic symbols, arithmetic facts, arithmetic procedures and arithmetic conceptual knowl-edge), appear to be functionally independent as well as differentially susceptible to brain damage. Examples of selective impairments in such components of calculation are described below.

3.3.1 Disorders of arithmetic symbol processing

Very few patients with selective impairment in the comprehension of written arithmetic symbols have been reported (e.g. Ferro and Botelho, 1980 ; Laiacona and Lunghi, 1997 ). For example, Laiacona and Lunghi ( 1997 ) investigated a patient who misnamed and misidentified the arithmetic signs and performed written arithmetic operations accord-ing to their misidentification. For example, the patient systematically plussed (ie. added), the times (ie. the multiplications) (See Fig. 21.1a ). According to the authors, this deficit appeared to have little impact on the patient’s everyday life. He continued to be com-pletely autonomous. However, it is of interest to note, that this deficit was not specific to arithmetic symbols, but was part of a more general processing deficit for relational symbols. He also presented with an impairment in the use of punctuation marks.

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3.3.2 Disorders of arithmetic fact retrieval

Several patients have been documented with selective impairments of arithmetic fact retrieval. Typically, these patients have severe difficulties in performing very simple single-digit addition, subtraction, multiplication and division. They produce many errors (e.g. 5 + 7 = ‘13 roughly’, e.g. Dehaene and Cohen, 1991 ), and their response times are abnormally slow (e.g. >2 seconds). An error classification has been proposed by McCloskey et al ., ( 1991c ). Errors can be classed as operand errors , if the incorrect answer is the correct answer to a computation that shares one of the operands (e.g. 6 × 5 = 25); as operation errors , if the incorrect answer constitutes a correct answer to another computation involv-ing the same operands, but a different operation (e.g. 3 × 4 = 12); as table errors , if the incorrect answer represents the product of two different single digit numbers (e.g. 4 × 4 = 25); and as non table errors , if the incorrect answer is neither an operand, a table or an operation error (e.g. 9 x 8 = 52). Patients who make these errors often show intact knowl-edge of arithmetical principles and procedures. Typically, they are able to retrieve and apply the appropriate arithmetic steps required to solve complex arithmetic problems and they are able to define arithmetic operations (e.g. Delazer and Benke, 1997 ; Sokol et al ., 1991 ; Warrington, 1982 ). For this type of patient everyday activities such as check-ing their change or their bank statements pose great difficulties.

Impairment of arithmetic facts can also manifest in a highly selective manner. Selectively preserved and impaired arithmetic facts have been reported according to specific types of operation. For example, patients have been documented with selective impairments or selective preservations of multiplication (e.g. Dehaene and Cohen, 1997 ; McCloskey et al ., 1991b ; Grafman et al ., 1989 ; Delazer and Benke, 1997 ), and subtraction facts (e.g. Dagenbach and McCloskey, 1992 ; Pesenti et al., 1994 ; McNeil and Warrington, 1994 ; Lampl et al ., 1994 ; Dehaene and Cohen, 1997 ). Selective impairments of addition and division have also been reported. Cipolotti and de Lacy Costello ( 1995 ) described a patient who could no longer solve ‘4 ÷ 2’, while being able to solve ‘27 × 26’. Van Harskamp and Cipolotti ( 2001 ) described a patient with selective addition impairment. The patient could no longer add 2 + 3, but was still able to solve 13 – 6 and 8 × 9.

3.3.3 Disorders of calculation procedures

There have only been a few reports of patients with selective impairment of calculation procedures. Girelli and Delazer ( 1996 ) described patient MT, who systematically subtracted the smaller number from the larger one, irrespective of whether the larger was at the top or the bottom line (see Fig. 21.1b ). His impairment in solving multi-digit

Fig 21.1a Disorders of arithmetric symbol processing.

a) Performance of patient EB, with a selective impairment in arithmetical symbol processing (Laiacona and Lunghi, 1997). Specifically, the patient systematically plussed the times.

48× 67

115

59× 29

88

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3 SELECTIVE IMPAIRMENT OF NUMBER PROCESSING AND CALCULATION 407

subtractions was attributed to a defective knowledge of calculation procedures. This deficit was selective for subtraction, as the patient was able to correctly carry out multi-digit addition problems involving the carry over procedure (such as 78 + 26).

Other impairments in the use of calculation procedures may consist of misalligning the digit in multi-digit operations, of errors using carrying procedures in addition problems, or of not applying the problem-specific procedure in the correct order. For example, in multi-digit multiplications this consists of multiply digits in the upper number starting from the rightmost, by digits in the bottom number starting from the rightmost etc. (e.g. Benson and Weir, 1972 ; Grana et al ., 2006 ; Sokol et al ., 1991 ; Sokol and McCloskey, 1991 ).

3.3.4 Disorders of arithmetic conceptual knowledge

Delazer and Benke ( 1997 ) described a patient (JG) who had lost arithmetical conceptual knowledge, which is defined as ‘An understanding of arithmetical operations and laws pertaining to these operations’ (Hittmair-Delazer et al ., 1994 , p. 117). JG could retrieve multiplication tables from memory despite severe problems in all tasks requiring concep-tual knowledge. The patient could correctly calculate ‘9 x 9,’ but did not know that ‘9 x 9’ equals ‘3 + 3 + 3’. The patient was encouraged to use any ‘back-up strategies’, such as counting on fingers for simple addition problems. Usually this strategy is adopted spon-taneously by patients with impairments in arithmetic fact retrieval. However, JG never succeeded in representing problems with her fingers (Delazer and Benke, 1997 , p.705). She did not apply even very basic principles such as commutativity in multiplication (i.e. 4 × 12 = 12 × 4), and she was unable to recognise that multiplication can be transformed into repeated addition (4 × 12 = 12 + 12 + 12 + 12).

The opposite side of the dissociation, namely intact conceptual knowledge with impaired fact retrieval, has been reported in two single-case studies (Hittmair-Delazer et al ., 1994 , 1995 ). The first study described a patient who demonstrated excellent con-ceptual knowledge despite a severe acalculia. For example, when presented with arithme-tic calculations such as 8x6, he adopted the following strategy 8 × 10 = 80; 80:2 = 40; 40+8=48 (see Fig. 21.1c ). Another patient who was no longer able to solve simple elemen-tary arithmetic such as 2 + 3, showed excellent understanding and use of abstract equations such as (b × a): (a × b)=1 and (cd + ed):d= c + e (Hittmair-Delazer et al ., 1994 ). Similarly, a semantic dementia patient demonstrated well preserved understanding of arithmetic concepts despite the loss of arithmetic facts (Cappelletti et al ., 2005 ). He spon-taneously developed strategies which allowed him to perform computations when he could not access simple facts or use canonical procedures (e.g. using multiple additions

Fig. 21.1b Disorders of arithmetric symbol processing.

923− 644

321

171− 48

127

b) Performance of patient MT with a calculation procedure impairment. He showed ‘Smaller from Larger’ subtraction bugs (Girelli and Delazer, 1996).

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to solve a multiplication problem, see Fig. 21.1d ). These observations have been interpreted as demonstrating that conceptual knowledge is a functionally independent component of calculation.

4 Selective preservation of number processing and calculation It is now well established that numerical and calculation abilities can be largely independ-ent of other cognitive abilities, such as general intellectual skills (e.g. Remond-Besuchet et al., 1999 ), language (e.g. Rossor et al ., 1995 ; Thioux et al ., 1998 ), short-term memory (e.g. Butterworth et al ., 1995 ), and semantic knowledge (e.g. Cappelletti et al., 2001 ; Crutch and Warrington, 2004 ). For example, a patient with semantic dementia (IH) presented with severe impairment in understanding the meaning of words and the use of objects. Despite this severe impairment, the patient remained able to understand numerical con-cepts and to perform arithmetical operations (Cappelletti et al ., 2001 ), even in the later stages of illness (Cappelletti et al ., 2005 ). This suggests that a patient’s understanding of arithmetical concepts may remain preserved despite severe inability to process the meanings of words and objects. A similar pattern of performance on numerical and

Fig. 21.1c Disorders of arithmetric symbol processing.

c) Performance of patient BE with impaired arithmetical fact retrieval but preserved conceptual knowledge. When asked to perform arithmetical problems such as 8x6, he adopted the strategy below (Hittmair-Delazer et al, 1994)

8 × 10 = 80; 80 : 2 = 40; 40 + 8 = 48

Fig. 21.1d Disorders of arithmetric symbol processing.

d) An example of IH’s performance in solving a multidigit multiplication operation (Cappelletti et al., 2005). Number 531 was decomposed into the subparts 1, 30, and 500. Numbers in squared brackets indicate the main steps in performing the operation. Step [1] is 27×1. Steps [2] to [4] show how 327×30 was obtained. Step [2] is 327×5; [3] is 327×10 and [4] is 327×30; [5] is 327×100 obtained through multiple additions; [6] is 327×500, again obtained through multiple additions. Finally, [7] is (327×500)+(327×30)+(327×1). The final result is correct.

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calculation tasks has been reported in other semantic dementia patients (e.g. Crutch and Warrington, 2002 ).

5 Localisation of brain lesions in number and calculation disorders An overview of group and single case studies has indicated that numerical skills may have discrete and independent brain substrates (Cipolotti and van Harskamp, 2001 ). The majority of patients with number production and/or number comprehension impair-ments have left posterior lesions, invariably involving the parietal lobe. Similarly, the majority of patients with arithmetical fact retrieval impairments had lesions mainly implicating the left parietal lobe. This is also the case for the majority of patients with impairment of arithmetic fact retrieval for specific types of operations. In particular, patients with impairments of multiplication or subtraction typically have lesions encroaching on the left parietal lobule.

The neuropsychological evidence indicating the involvement of the parietal lobe in numerical processing has recently been corroborated by neuroimaging studies (e.g. Dehaene et al ., 1999, 2003; Pesenti et al ., 2000 ), including transcranial magnetic stimula-tion (TMS, e.g. Cappelletti et al ., 2007b ; Andres et al ., 2005 ). However, it is important to note that not all parietal lesions result in numerical impairments. For instance, only about 20% of patients with left parietal lesions show numerical deficits (e.g. Jackson and Warrington, 1986 ). Moreover, most patients with right parietal lesions show preserved comprehension of core numerical concepts (e.g. Cappelletti and Cipolotti, 2006 ; Cappelletti et al ., 2007 ; Cohen et al ., 2007 ; but c.f. Rosselli and Ardilla, 1989 ; Langdon and Warrington, 1997 for a different pattern of results). Similarly, not all numerical impair-ments are due to parietal lesions. For instance, deficits in arithmetical procedures have been reported in patients with more anterior localisation sites (e.g. Grana et al ., 2006 ). Future research will help to clarify the extent to which parietal regions are critical for number and calculation processing, and elucidate the role of the left and the right parietal lobes in these processes.

6 Assessment and rehabilitation of acalculia

6.1 Diagnosis and assessment tools The diagnosis of acalculia relies on the establishment of appropriate tools to detect the presence of number processing and/or calculation impairment in subjects with normal premorbid calculation skills. The diagnosis of acalculia can be made only when deficits in numeracy skills are not a secondary consequence of other cognitive deficits. In other words, one needs to exclude generalised impairment in language, attention, visuospatial functions or other cognitive skills underpinning the failure of number processing and calculation.

Formal assessment of the patient’s numeracy skills requires detailed evaluation of number processing and calculation abilities. Number processing can be assessed using

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number reading, writing to dictation, and repetition tasks. Numerical stimuli can be pre-sented as either Arabic numerals (e.g. 8), written ( eight ) or spoken number names (‘eight’). Patients can be simply asked to read, write or repeat the numeral presented. Comprehension of number quantity is usually assessed with magnitude comparison tasks, whereby patients are asked to indicate the larger of two numbers (e.g. which is big-ger: 6 or 9?).

Another test of number comprehension is the number composition task in which sub-jects have to compose the value of a given number using poker chips which range in value from 1 to 500. The ability to subitize is another way to test comprehension of number quantity. Subitizing is the ability to rapidly indicate the number of items contained in a small set (usually up to 4 or 5). Larger sets of objects require additional processing, such as counting. Patients’ subitizing can be tested by presenting them with displays contain-ing items which vary in number and position. Patients can simply be asked to report the number of items contained in each display. Intact quantity processing is expected to cor-respond to the ability to rapidly and correctly indicate the numerosity of the set; the time required to produce an answer is expected to increase with the number of items presented with a sharp increase after number 4 or 5. This is because enumeration over four items is slow, more error-prone, and thought to require counting (e.g. Trick and Pylyshyn, 1993 , Sathian et al ., 1999 ).

Calculation skills can be assessed with tasks exploring knowledge of arithmetical signs, arithmetical facts, and arithmetical procedures. To evaluate the ability to process arith-metical signs, patients are usually required to read, point to and write, the arithmetical signs. To test the ability to retrieve simple arithmetical facts, patients are asked to com-pute orally presented single-digit arithmetical calculations (e.g. 4 + 2, 3 × 4 or 5 – 2). The access to arithmetical procedures is usually assessed by performing multi-digit calcula-tion, such as 294 + 12=306, either in the written or in the oral modality.

A detailed functional assessment of the patient’s number processing and calculation skills is needed to establish which specific components within the number processing or calculation system are impaired. Several testing batteries which evaluate numerical skills in detail are available. One of these is the EC301 composite battery or its shorter form (EC301R), which permits a detailed evaluation of both calculation and number process-ing abilities in brain damaged adults (Deloche et al., 1994, 1995, 1996). A more recent set of numerical and calculation tasks have been developed for which normative data have also been provided (Delazer et al ., 2003 ). A standardised test which evaluates mental cal-culation is the Graded Difficulty Arithmetic Test, which is comprised of 12 multi-digit addition and subtraction problems (GDA, Jackson and Warrington, 1986 ).

Error analysis, although usually not included in standardised tests, is a very useful addition to the assessment, in so far as it can provide information about the nature of the impairment. Thus, when patients have problems in reading and writing numerals, a com-mon analysis of errors is the distinction between lexical or syntactical errors. When patients have problems in the retrieval of arithmetical facts, a common analysis is the error classification proposed by McCloskey et al . ( 1991c ) which distinguishes between operand errors, operation errors, table errors, and non-table errors.

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6.2 Natural recovery from acalculia Although acalculia is a frequent disorder in left-brain damaged patients, relatively little is known about its prognosis. Only one study has investigated the natural evolution of acal-culia in patients with left hemisphere vascular lesions (Caporali et al ., 2000 ). This study indicates that some patients recover completely in the first months post-stroke even with-out specific rehabilitation. Apparently, this improvement can also present in severely acalculic patients. This would suggest that initial severity of acalculia may not signifi-cantly influence recovery.

The authors suggest that recovery might be due to the resolution of the diachises in the first months after stroke (3-6 months post-onset). However, besides this more general and ‘passive’ recovery, the authors suggest that a more specialised functional reorganisa-tion might play a role several months post onset (>7 months). This suggestion would account for the fact that some patients continued to improve even several months after stroke. One patient, for example, first seen 5 months post onset, improved significantly on an acalculia test over the following 7 months. However, it is not clear whether the recovery was specific for acalculia. In fact, most patients also improve in other cognitive domains, such as language.

6.3 Rehabilitation of number processing and calculation skills The high frequency and disabling consequences of acalculia underline the importance of developing remediation techniques and intervention for arithmetical disorders. A com-plete and detailed assessment of the patient’s numerical processing and calculation skills is essential to understanding the nature of numeracy impairment (Girelli and Seron, 2001 ). According to the specific type of acalculia, different kinds of rehabilitative inter-vention may be required. Thus, a detailed assessment of a patient’s number processing and calculation skills will constitute the basis for designing suitable rehabilitation programmes. For example, a patient may show a highly selective impairment in the retrieval of simple multiplication facts. However, he/she may still be able to perform simple addition and subtraction. The re-acquisition of multiplication facts may then be facilitated by the use of back-up strategies based on addition (e.g. counting-on procedure: 3 × 6 = 6 + 6 + 6).

Recent reviews (e.g. Girelli and Seron, 2001 ; Lochy et al ., in press) have shown that very few studies have been devoted to the rehabilitation of numerical skills. These few studies concentrated on the rehabilitation of number transcoding deficits (e.g. Deloche et al ., 1989 ; Sullivan et al ., 1996 ), arithmetical fact retrieval (e.g. Domahs, 2003 , 2007 ; Girelli et al ., 1996 , 2002; Hittmair-Delazer et al., 1994 ; Whetstone, 1998 ), and the development of strategies for solving problems (Fasotti et al ., 1992 ).

6.3.1 Rehabilitation of transcoding skills

The rehabilitation of transcoding skills has concentrated on the ability to translate numer-ical stimuli into different codes (e.g. ‘four’ → 4). As an example we will discuss an inter-esting rehabilitation programme implemented by Deloche et al . ( 1989 ). They treated a patient with selective difficulties in the production of written verbal numerals

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from Arabic numerals (i.e. 7001 → seven thousand zero one ). This deficit was very severe. Indeed, the patient showed a 45% error rate. His errors were mostly syntactic (e.g. 114 → one hundred ten four). Deloche et al. ’s rehabilitation programme consisted of re-teaching a set of explicit transcoding rules using a step-by-step procedure. They used highly specific exercises which targeted a single transcoding rule. There were several facilitation procedures (e.g. colour cues, vocabulary panels) to help the patient with his learning process. For example, they rehabilitated the patient’s ability to transcode a 2-digit Arabic numeral such as 73 into the corresponding written number word seventy-three , by explicitly stating the rule ‘transcode the left digit by a ten name and the right digit by a unit name’. During training a vocabulary panel with two coloured columns was placed in front of him. The red part of the vocabulary panel contained the TENS names (ten, twenty, thirty etc up to ninety), and the blue part contained the UNIT names (one, two, etc up to nine). After 25 training sessions of 30 to 60 minutes each, the patient’s performance was close to ceiling. Interestingly, the long-term effects of the treatment were reassessed 7 months post-training. Only a very small increase in error rate relative to post-training evaluation was reported (6%). Overall, the patient continued to perform better than before training.

6.3.2 Rehabilitation of arithmetical facts

Rehabilitation of arithmetical fact retrieval consists primarily of attempts to re-teach ‘lost’ arithmetical knowledge via extensive practice. The underlying assumption is that practice may ‘re-create’ or ‘re-strengthen’ lost associations between problems and answers. Thus, for example, two patients (TL and ZA) with specific impairment of mul-tiplication fact retrieval underwent twice-weekly training sessions over a period of eight weeks (Girelli et al ., 1996 ). During the training session the arithmetical problems were presented in written form and simultaneously read aloud by the examiner. The patients were asked to answer, either verbally whilst pointing to the number on a table, or by writ-ing the Arabic numeral. Errors were always corrected immediately. Patient TL relearned the answers as ‘labels’, by reciting one operand’s table (e.g. 4 × 3 = 4, 8, 12,). Patient ZA relearned the answers as serial additions (e.g.4 × 3 = 3 + 3 + 3 = 12).

At the end of the session, both patients improved considerably and showed stable recovery at one month follow-up. Strikingly, after treatment, their overall error rate dropped to 10% (from pre-treatment error rates of 91% (TL), and 81% (ZA)). Not only did the error rates decrease dramatically over the course of remediation in both patients, but the nature of the errors also changed. This change in error patterns was interpreted as being due to different strategies used by the two patients in relearning their facts. As described above, patient TL relied on the recitation of Nx1 forward until she could access the solution to the problem. Patient ZA relied on the strategy of repeated addition of the second operand. Thus, back-up strategies may certainly play a facilitative role in the re-acquisition of simple arithmetic. Similar training programs for the rehabilitation of arithmetical fact impairment have been adopted in other cases as well (e.g. Hittmair-Delazer et al ., 1994 ; Whetstone, 1998 ).

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The conventional ‘drill’ approach to rehabilitation of arithmetical facts can be light-ened by simultaneous presentation of visual or auditory cues (Domahs et al ., 2004, 2007). Another approach has been to rehabilitate arithmetic facts based on conceptual knowl-edge rather than extensive training. This method is based on inducing a meaningful re-organisation of arithmetical facts according to the principles underlying them, for instance the commutative law or decomposition using known facts, (i.e. 10s, ties and 2s) (Girelli et al ., 2002 ).

Girelli and Seron ( 2001 ) also stressed the importance of minimising the opportunity to make mistakes, since repeated errors may strengthen the wrong associations between problems and answers. Moreover, the use of back-up strategies based on the principles underlying arithmetical facts such as the order-irrelevant principle (e.g. 8 × 6 = 6 × 8 = 48), decomposition strategies (e.g. 4 × 8 = 2 × 8 + 2 × 8 = 32) or repeated addition of the second operand (e.g. 3 × 5 = 5 + 5 + 5 = 15) seem important in rehabilitation of arithmetical facts. Overall, research has indicated that patients benefit significantly from training, and that improvement is retained over time (and in some cases, spontaneously generalises).

7 Conclusions This review described acalculia as a heterogeneous disorder consisting of impairments in processing numbers, in calculation, or in both. Specific components of number and cal-culation have been presented in the context of both normal functioning and impaired performance following cerebral damage. This included disorders in transcoding process-ing, quantity processing, calculation, and their sub-components. Cases of selective pres-ervation of number processing have also been analysed. This review also offered a short outline of lesions’ localization with respect to number and calculation disorders, and indicated the parietal areas most relevant for numeracy and its impairments.

Finally, we have proposed some guidelines for the assessment of acalculia and discussed some of the rehabilitation programmes for acalculia in neurological patients. These pro-grammes have indicated the importance of systematic intervention in acalculia, but have also shown that there are several important aspects of numerical skills for which no spe-cific interventions have been developed. Future studies are needed to create treatments for different numeracy impairments and to test whether these treatments generalise to real-life situations; the purpose of any rehabilitation programme.

Acknowledgments This work was supported by a grant from the Wellcome Trust (MC).

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