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Neuropsychologia 45 (2007) 1642–1648

Algebra in a man with severe aphasia

Nicolai Klessinger ∗, Marcin Szczerbinski, Rosemary Varley ∗∗Department of Human Communication Sciences, University of Sheffield, Sheffield S10 2TA, UK

Received 9 March 2006; received in revised form 5 January 2007; accepted 7 January 2007Available online 14 January 2007

bstract

We report a dissociation between higher order mathematical ability and language in the case of a man (SO) with severe aphasia. Despite severelympaired abilities in the language domain and difficulties with processing both phonological and orthographic number words, he was able to judgehe equivalence of and to transform and simplify mathematical expressions in algebraic notation. SO was sensitive to structure-dependent properties

f algebraic expressions and displayed considerable capacity to retrieve algebraic facts, rules and principles, and to apply them to novel problems.e demonstrated similar capacity in solving expressions containing either solely numeric or abstract algebraic symbols (e.g., 8 − (3 − 5) + 3 versus− (a − c) + a). The results show the retention of elementary algebra despite severe aphasia and provide evidence for the preservation of symbolicapacity in one modality and hence against the notion of aphasia as asymbolia.

2007 Elsevier Ltd. All rights reserved.

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eywords: Language; Number; Mathematics; Aphasia; Algebra; Asymbolia

. Introduction

The role of language in mathematics is a hotly debatedssue in the cognitive sciences (Brannon, 2005; Gelman &utterworth, 2005; Gelman & Gallistel, 2004) and several linesf enquiry contribute to the debate. Studies with animals and pre-erbal human infants have demonstrated basic numerical andalculation abilities (Dehaene, Dehaene-Lambertz, & Cohen,998; Flombaum, Junge, & Hauser, 2005; Wynn, 1992), indi-ating the autonomy of basic number capacity from language.owever, recent cross-linguistic studies demonstrated language

ffects on the development of basic calculation skills (HoudeTzourio-Mazoyer, 2003) and on number representation and

omparison (Nuerk, Weger, & Willmes, 2005). Studies investi-ating numerical abilities in Amazonian tribes, who possess onlyrudimentary lexicon for number words, point to possible lim-

tations for the acquisition of mathematics without supportinginguistic lexical resources (Gordon, 2004; Pica, Lemer, Izard,

Dehaene, 2004). Similarly, experiments with bilingual adultsave shown a language training effect when performing exact aspposed to approximate calculations (Spelke & Tsivkin, 2001).

∗ Corresponding author. Tel.: +44 114 22 22418; fax: +44 114 273 0547.∗∗ Co-corresponding author. Tel.: +44 114 22 22449; fax: +44 114 273 0547.

E-mail addresses: Nicolai.Klessinger@gmx.net (N. Klessinger),.a.varley@sheffield.ac.uk (R. Varley).

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028-3932/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.neuropsychologia.2007.01.005

Further evidence for a possible role of language in calcula-ion comes from functional imaging studies. Some studies haveound evidence for the recruitment of left hemisphere perisyl-ian language areas during exact calculation (Cohen, Dehaene,hochon, Lehericy, & Naccache, 2000; Dehaene, Spelke, Pinel,tanescu, & Tsivkin, 1999). However, others have implicated

he involvement of a bilateral parietofrontal network and theilateral inferior temporal gyri, including areas associated withisuospatial working memory and mental imagery, during men-al arithmetic (Houde & Tzourio-Mazoyer, 2003; Venkatraman,nsari, & Chee, 2005; Zago et al., 2001).In the field of neuropsychology, although aphasia is

ften associated with impaired number and calculation abil-ty (Delazer & Bartha, 2001; Delazer, Girelli, Semenza, &enes, 1999), dissociations between language and mathemat-

cs have also been demonstrated. Double dissociations betweenphasia and acalculia have been described with reports of pre-erved language skills despite impaired mathematical abilitiesButterworth, 1999), or retained mathematical skills despiteeverely impaired language (Rossor, Warrington, & Cipolotti,995; Varley, Klessinger, Romanowski, & Siegal, 2005). Thesetudies indicate considerable autonomy in the neurocognitive

echanisms underpinning language and calculation.Compared to the rich body of evidence on number processing

nd elementary arithmetic, relatively little is known about theognitive processes involved in more complex and advanced

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athematical abilities such as solving fractions or algebraicxpressions (Butterworth, 1999; Dehaene, 1997). Similarly, theole language plays in these processes is still to be determinedCampbell & Epp, 2004). There are few neuropsychologicaltudies which report on higher mathematical functions in brainamaged patients.

With regard to algebra, one study reported the case of DA,ho despite severe disruptions in basic calculation abilities

ould still solve abstract algebraic equations (Hittmair-Delazer,ailer, & Benke, 1995). DA was impaired in solving sim-le addition, subtraction, division or multiplication problems,ut could correctly simplify abstract expressions such asb × a) ÷ (a × b) or (a + b) + (b + a) and make correct judge-ents whether abstract algebraic equations like b − a = a − b or

d ÷ c) + a = (d + a) ÷ (c + a) were true or false. In the study, alge-raic expressions were used to test for conceptual knowledge ofrithmetic. Conceptual knowledge entails deeper understand-ng of arithmetical operations and laws pertaining to theseperations (Semenza, 2002), such as commutativity for addi-ion (i.e., 4 + 7 = 7 + 4), or non-commutativity for subtractioni.e., 7 − 4 �= 4 − 7). Similarly, transforming and judging equiv-lences of expressions in algebra requires knowledge of suchathematical principles (Resnick, Cauzinille-Marmeche, &athieu, 1987). DA showed a dissociation between impaired

rithmetical fact knowledge (such as 2 + 3) and intact knowl-dge of arithmetical procedures (such as sequences of necessaryteps to perform multi-digit calculations) and conceptual knowl-dge (e.g., a + b = b + a). The findings from DA have beennterpreted as evidence for two parallel, independent levelsf mathematical processing, one formal-algebraic and onerithmetical–numerical (Hittmair-Delazer et al., 1995), and fur-her, as preliminary evidence for the existence of independenteuronal circuits for algebraic knowledge and mental calculationDehaene, 1997).

Despite the claim of autonomy between algebraic and cal-ulation processes, some functional imaging studies revealommon activations across these behaviours. Anderson, Qin,ohn, Stenger, and Carter (2003) found that brain regionsctive during algebra equation solving in adults were also areasnvolved in calculation and mental arithmetic (e.g., Dehaene,iazza, Pinel, & Cohen, 2003; Dehaene et al., 1999; Gruber,ndefrey, Steinmetz, & Kleinschmidt, 2001; Menon, Rivera,

hite, Glover, & Reiss, 2000; Zago et al., 2001). These includedhe left intraparietal sulcus, left precuneus, left inferior frontalyrus, left pre-frontal regions and left and right supramarginalyri.

However, the question regarding the relation of language andlementary algebra has not been directly addressed in a neu-opsychological case study to date. As DA’s language abilitiesere intact, his case did not show whether algebraic reason-

ng can be sustained in the face of severe language impairment.ith regard to the relation between natural language and alge-

ra, there are similarities between the two systems. In algebra,

here is a ‘syntactic’ level of mathematical processing which isecessary to determine whether an algebraic equation is well orll-formed (e.g., a ÷ (b − (c + a)) versus a (÷ b − (c + a)), sim-lar to syntactic processing in natural language (e.g., the man

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logia 45 (2007) 1642–1648 1643

its the ball versus the man the ball hits). Similarly, an algebraicxpression contains operator signs, abstract variables (denoteds letters, e.g., a or b) and numerals (such as 3 or 5 express-ng specific quantities) which constitute a symbolic system andre analogous to the language lexicon (with comparable abstractnd concrete words, e.g., creature versus dog). However, it hastill to be determined whether syntactic and symbolic capacitiesn algebra are related to natural language or whether they areargely independent.

We address this question in the current investigation, andeport the case of a man (SO) with severe aphasia who retainedcapacity for solving algebraic expressions. SO also retained

alculation ability and his performance in number processingnd basic calculation tests has been reported elsewhere (Varleyt al., 2005). In summary, SO displayed good understandingnd production of Arabic numerals, but had mild impairmentn processing phonological number words, and marked disrup-ion of processing orthographic number words. He was able toranscode numbers across different formats, with intact perfor-

ance in matching Arabic numerals to orthographic numberords, and only mild impairment in other forms of transcod-

ng. SO was accurate on an analogue estimation task, and wasble to perform all basic single-digit arithmetical operations, andddition and subtraction of two-digit operands. He had some dif-culty in multiplication and division of two-digit multipliers andivisors. He was able to add and subtract fractions and was ableo calculate the result of mathematical expressions containingrackets. SO was the only participant in the previously reportedase series who pre-morbidly had a high level competence inathematics. Thus, with regard to arithmetic, SO demonstrated

argely intact knowledge of arithmetical facts and proceduresnd intact conceptual knowledge in face of severe aphasia.

In this report, we examine whether higher order mathematicsn the form of elementary algebra was retained despite severeanguage impairment and whether language is necessary to gainccess to the algebraic-formal level of mathematical processing.n particular, SO’s capacity to comprehend and solve mathemat-cal expressions including terms with not only numbers but alsobstract variables (represented as letters, e.g., a, b or y) wasxplored.

. Method

.1. Participants

SO and five healthy male volunteers participated in the study. All participantsave informed consent prior to participation in the study, and the protocol waspproved by the North Sheffield Research Ethics Committee (NS200291449).O was a 56-year-old retired university professor. He had been a professor inaculty of Science and, although he was not a professor of mathematics, theesearch in which he was engaged required an advanced competence in theubject. He was pre-morbidly right-handed and, at the time of the investigation,as 3.5 years post-onset of a left hemisphere vascular lesion. A CT scan revealedlarge lesion resulting from a proximal occlusion of the left middle cerebral

rtery (Fig. 1). There was extensive peri-rolandic damage, with lesion of theosterior aspect of the middle and inferior frontal gyri. Large sections of the leftemporal lobe were lesioned, and damage extended to the left amygdala, coronaadiata, lenticular nucleus and operculum. Within the left parietal lobe, damagextended to the anterior portion of the superior and inferior parietal lobules,

1644 N. Klessinger et al. / Neuropsycho

Fig. 1. Structural brain scan (CT) for patient SO showing a vascular lesion in leftmpA

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iddle cerebral artery territory with extensive damage to perisylvian temporal,arietal and frontal cortices. (From Varley et al., 2005; copyright (2005) Nationalcademy of Sciences, U.S.A.)

ncluding the supramarginal gyrus. However, some inferior parietal structures,ncluding the angular gyrus were spared.

Consistent with this lesion, SO showed right hemiplegia, severe aphasiand apraxia of speech. His language impairment crossed lexical and syntacticomains and impacted both on comprehension and expression. His spontaneousanguage production was severely limited in both speech and writing. His outputas minimal and restricted to occasional social word forms (‘bye’, ‘hello’).

The healthy control participants were matched to SO on age (median = 55,ange from 52 to 58 years) and education (all had a university degree). In addition,ll the control participants possessed advanced competence in mathematics. Allve had higher degrees in either a mathematical, science or engineering subjecthich included a substantial mathematical component. Two were universityrofessors, one was an IT systems engineer, one a computer programmer, andne a senior accountant.

.2. Language evaluation

SO’s language performance was assessed at lexical and grammatical lev-ls. Tests were taken from the Action for Dysphasic Adults (ADA) Auditoryomprehension Battery (Franklin, Turner, & Ellis, 1992), the Psycholinguis-

ic Assessments of Language Processing in Aphasia (PALPA, Kay, Lesser, &oltheart, 1992), or were devised for the study. Comprehension of both spokennd written concrete nouns was relatively preserved (ADA spoken word–pictureatching: 61/66; ADA written word–picture matching: 57/66). Synonym judge-ent tests that allow evaluation of both high and low imageability words revealedgreater degree of lexical impairment across spoken and written modalities

ADA spoken synonym matching: 145/160; ADA written synonym matching:12/160). Errors were predominantly on low imageability items (24/80 com-ared to 4/80 on high imageability ones). Lexical retrieval in speech and writingas severely impaired (PALPA 54), with SO being unable to name any items in

ither modality.Grammatical processing was severely impaired. SO performed at or below

hance level on a reversible sentence comprehension test performed in bothpoken and written modalities (comprehension of spoken reversible sentences:2/100; written reversible sentences: 43/100). The reversible sentence testequired matching spoken or written sentences to a corresponding picture in theresence of a distractor picture showing the reversed roles of the protagonistse.g., the man killed the lion/the lion killed the man). The test included 50 activend 50 passive sentences. However, SO’s performance on written grammati-

ality judgements indicated that sentence parsing mechanisms were relativelyntact, and he obtained an above chance score (35/40; p < 0.01). The grammat-cality test was not administered in the spoken modality to avoid prosodic cuesnfluencing judgements. SO showed a digit span of 5 items, when tested viaecognition paradigm (PALPA 13).

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logia 45 (2007) 1642–1648

.3. Processing algebraic expressions

SO’s capacity to solve algebraic expressions was examined across a series ofasks, which were also administered to the five matched control subjects in ordero determine the range of normative scores. Sets of algebraic expressions weredministered in paper and pencil format. Participants were not allowed to use aalculator, but were allowed to write down intermediate steps in the solution ofny given problem. There was no time restriction on tasks.

.3.1. Judging the equivalence of algebraic expressionsParticipants had to make a judgement as to whether the algebraic expression

iven on the right of an equation was the correct simplification or transfor-ation of the expression on the left (e.g., 2b × 5b = 7b2). The set of 70 itemsas balanced with regard to whether the presented equations were correct or

ncorrect and included all basic mathematical operations (addition, subtraction,ultiplication, division), and some bracketed and squared expressions.

.3.2. Transforming and simplifying algebraic expressionsSO and control subjects had to transform and simplify algebraic expressions.

roblems were grouped according to the main mathematical operation involvedaddition/subtraction, multiplication, division, fractions, bracket expressions).

.3.2.1. Addition and subtraction of algebraic terms. Participants simplifiednd transformed 50 algebraic expressions, 25 involving addition, and 25 sub-raction. Each subset consisted of two-term (e.g., 6c + 3 = ?), three-term (e.g.,3b − 2b − 5b = ?) or four-term expressions (e.g., 2a + 5a + 11 + 3 = ?), whichould include any mixture of numeric, abstract variables or combined terms andp to two different variables. A further set of 20 expressions involved mixedperators. Items included up to three different variables in three or more termse.g., 4z − 7y + 2y = ?, or, e.g., 4b − a − 2c − 3b + 2c + 6a = ?).

.3.2.2. Multiplication and division of algebraic terms. The multiplication setonsisted of 24 two-term expressions, in which an algebraic term was eitherultiplied by a number (e.g., 3 × 2a = ?), an algebraic term with another alge-

raic term containing a different variable (e.g., 3y × 5a = ?), or by a term with theame variable (“squared terms”, e.g., 4a × 2a = ?). The whole set was balancedith regard to whether terms consisted of numbers, abstract variables or both.he division set included 20 two-term division problems, involving 10 pairedxpressions, where for the second member of the pair the position of dividendnd divisor were reversed (e.g., y ÷ 2y and 2y ÷ y). In addition to division, theseroblems examined structure-dependent properties of algebraic expressions andere constructed to parallel reversible sentence comprehension stimuli (e.g., thean killed the lion/the lion killed the man). Again the set was balanced for typesf terms. Problems appeared in pseudo-random order with a minimum of threentervening items between the members of each pair.

.3.2.3. Algebraic fractions and mixed expressions. Participants transformed0 algebraic expressions involving two-term fraction expressions with abstractumerator or denominator (10 problems, e.g., 3

a− 2

a=?, or 4

2c+ 1

c=?) and

hree-term expressions with mixed operators (10 problems, e.g., 3cb+2cbc

=?, ora2 × 4

b+ a

b=?).

.3.2.4. Algebraic expressions involving brackets. Participants solved a total of20 algebraic expressions with brackets. The set included expressions with singlerackets in addition and subtraction of alphanumeric terms (40 problems, e.g.,y − (4y − x) = ?, or 9c − 9c − (5c − 2) = ?), single brackets in multiplication (40roblems, e.g., (7a + 4) × 2c = ?), double brackets in addition and subtraction (16roblems, e.g., 9c − (3b − (6c + b)) = ?, or 2b + (3b + c) − (4c + 5b) = ?) and sin-le brackets in addition and subtraction of abstract terms without numbers (24roblems, e.g., b − (b − b) + b = ?). Expressions could involve up to two differ-nt variables and were balanced with regard to the position of brackets (initial or

nal, or, in the case of double brackets embedded or serial). All subsets were pre-ented pseudo-randomly, allowing for a maximum of four successive problemsf the same kind. The set included a total of 76 “syntactic” bracket expressions,n which the use of a serial order strategy to solve the expression would lead ton incorrect result (e.g., 2a − (c − 3c) + 7a = ?, or a − ((12a − 5a) + 6) = ?).

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.3.3. Simplifying abstract algebraic expressions versus numericxpressions

Performance on expressions containing only numeric versus abstract alge-raic terms was compared. SO and control subjects completed a set of 80roblems, consisting of 40 abstract algebraic expressions, which contained onlybstract non-numeric terms (e.g., b − (a − c) + a = ?), and 40 matched numericxpressions, which only contained numbers and were parallel to the structure ofhe abstract expressions (e.g., 8 − (3 − 5) + 3 = ?). Half of the paired expressionsnvolved only addition and subtraction, and half involved multiplication or divi-ion (e.g., (cb − ab) ÷ b = ? versus (5 × 2 − 3 × 2) ÷ 2 = ?). Out of the 40 pairedxpressions, 30 contained brackets or double brackets. Numeric and abstractxpressions were presented in separate blocks.

.3.4. Scoring and error analysisSolutions to algebraic expressions were scored for accuracy. A strict scoring

riterion was used: answers to complex expressions were scored as correct onlyf they included the most parsimonious simplification possible, and, in the casef answers to bracket expressions, if all brackets were removed and transformedorrectly (e.g., only c − a rather than cb−ab

bwas counted as a correct answer to

he expression (cb − ab) ÷ b = ?).To examine SO’s sensitivity to structure-dependent properties of algebraic

xpressions, a qualitative error analysis was performed on the 20 reversible divi-ion problems, and on all mixed algebraic expressions with syntactic brackets76 out of 120). For division problems, an error was considered to be syntac-ic when the answer indicated that positions of divisor and dividend had beeneversed (e.g., 2a ÷ a = 1

2 ). In bracket expressions, errors were scored as syn-actic when SO had applied a serial order strategy, thus omitting the bracket andot taking into account its embedded structure (e.g., 8x − (y + 3y) = 8x + 2y).

An additional error analysis was performed on the matched set of abstracton-numeric versus numeric expressions, focussing on different types of errors:i) syntactic errors, (ii) rule errors, which involved either the false applicationf a correct rule (such as operator errors, e.g., 3a + 2b = 6ab), or the correctpplication of a false rule (e.g., a ÷ a = 0, or a × a = 2a), and (iii) calculationrrors (e.g., a − b + (a − b) = a − 2b).

. Results

Across all tasks SO made only very occasional use of writtenntermediate steps during calculation, even when solving longxpressions. Most often, he mentally performed all the necessary

ransformations and produced the solution directly.

Despite his severe aphasia SO displayed considerable com-etence in elementary algebra (Table 1). In general, he wasble to verify, transform and simplify algebraic expressions

able 1cores (accuracy in percent) on algebraic expressions

ask SO Controls (range)

udging equivalence of algebraicexpressions (out of 70)

54 (77) 57–70 (81–100)

ransforming and simplifying algebraic expressionsAddition or subtraction only (out of

50)48 (96) 46–50 (92–100)

Mixed addition/subtraction (out of 20) 18 (90) 15–20 (75–100)Multiplication (out of 24) 12 (50) 22–24 (92–100)Division (out of 20) 16 (80) 18–20 (90–100)Fractions and mixed operators (out of

20)10 (50) 15–19 (75–95)

Bracket expressions (out of 120) 94 (78) 102–119 (85–99)

implifying abstract algebraic vs. numeric expressionsAbstract expressions (out of 40) 25 (63) 26–38 (65–95)Numeric expressions (out of 40) 27 (68) 34–40 (85–100)

alues in parentheses are in %.

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logia 45 (2007) 1642–1648 1645

nvolving abstract and numeric terms across all four basic math-matical operations. This capacity extended to bracket andraction expressions. SO’s performance lay within the con-rol range on two tasks (addition or subtraction and mixedddition/subtraction), although his performance fell outside theange of those of mathematically highly competent controls inhe remaining tasks (Table 1).

SO could judge the equivalence of algebraic expression cor-ectly (54/70) and his overall performance on this set waslose to the range of the healthy control subjects (57–70).O was accurate in addition and subtraction of algebraic

erms and could simplify and transform expressions with upo six alphanumeric terms and three different variables (e.g.,a − 3b + 3a − 2c + 3b + 4c = 10a − 2c). With regard to multi-lication, SO’s score lay well outside the normal range and hehowed specific difficulty with problems which resulted in aquared term (e.g., 3y × 2y), or required the multiplication ofwo mixed terms (e.g., 3a × 4b). These expressions accountedor all his 12 errors. His errors were not random in that he alwayspplied a false rule by adding the numbers and only multiplyinghe variables (e.g., 3y × 2y = 5y2, or 3a × 4b = 7ab). However,n subsequent testing sessions he applied the correct rule inracketed and some mixed algebraic expressions involving mul-iplication. To examine this context-dependency of performance,n additional set of 40 two-term multiplication problems wasdministered at the end of the investigation. In this set SO’score was perfect, showing that he had either relearned the ruleuring the course of the study, or that he could not retrieve theule at the start of the investigation, but could do so at a latertage.

Division performance was relatively intact, and SO solved6 out of 20 problems correctly. A subsequent error analysisevealed a specific difficulties with division problems of theype na ÷ na = 1 and na ÷ ma = n/m. SO consistently applied thealse rules “na ÷ na = 0” and “na ÷ ma = n/ma” to such prob-ems. This accounted for all four of his errors in the divisionet.

SO showed some capacity to simplify algebraic fractions andore complex problems with mixed operators, demonstrating

hat he understood how to combine diverse mathematical opera-ions and rules. However, he solved only 10 out of 20 expressionsorrectly and his score in these tasks was well below the normalange (Table 1). A closer look at his errors revealed that in fourases he solved the first step correctly, but then could not findhe correct most parsimonious simplification possible (e.g., heolved a

2 × 4b

+ ab

=? with a2 × 4

b+ a

b= 4a

2b+ a

b= 5a

3b).

SO could successfully transform algebraic expressionsnvolving brackets, and could correctly identify and apply the

athematical properties and rules which are necessary to clearhe brackets before simplifying. However, again his performanceell short of the control range (78% compared to 85–99%).

With regard to awareness of structural principles in expres-ions involving mixed alphanumeric terms, SO was sensitive

o reversibility in the set of division problems, i.e., whether anlgebraic term took the role of divisor or dividend in such anxpression, and to the embedded structure in 76 syntactic mixedlgebraic bracket expressions (Table 2). Although he produced

1646 N. Klessinger et al. / Neuropsycho

Table 2SO’s scores (accuracy in percent) and errors on ‘syntactic’ algebraic expressions

Task SO Number ofsyntactic errorsa

Reversibility in division (out of 20) 16 (80) 0“Syntactic” algebraic bracket

expressions (out of 76)52 (68) 11

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racket expressions, errors were scored as syntactic when the participant hadrongly applied a serial order strategy, thus omitting the bracket and not taking

nto account its embedded structure.

1 syntactic errors, he still displayed some capacity to solve suchroblems correctly.

Despite his score lying outside the range of the healthyontrols, SO performed well on the set of abstract algebraicxpressions, achieving an accuracy of 63% (Table 1); thus hiserformance was only slightly below his score on the matchedumeric expressions (68%, McNemar Test, p = 0.804). The errornalysis on the two sets revealed that he produced only onesyntactic” error in each set, and that the most common typef error in transforming abstract expressions was rule errorsTable 3). SO applied false rules to novel expressions, e.g.,e solved the problem b × b = ? with 2b through applying thealse rule of adding (instead of multiplying) the two factors.

hen given the problem a × a × a = ?, he extended the false rulend gave the answer 3a. He also falsely applied correct rules,uch as transforming fractions through simplifying numeratornd denominator by the same factor, e.g., solving the expres-ion a ÷ (ba − ac) = ? as a

ba−ac= 1

b−ac, instead of the correct

nswer 1b−c

. Thus, SO displayed some capacity to generateovel (although false) rules, and to apply them to novel abstractlgebraic material.

In summary, despite his scores often lying outside the rangef the five healthy matched controls, SO displayed good overallerformance on the algebraic tasks. His performance was largelyntact with regard to judging the equivalence of algebraic equa-ions and transforming algebraic expressions involving addition,

ubtraction, divisions and brackets (as demonstrated by scoresithin or close to the control range). In contrast, he showed aore impaired performance on algebraic expressions involvingultiplication, fractions and combinations of these operations

able 3umber and type of errors produced by SO in sets of abstract algebraic vs.umeric expressions

ype of error Abstractexpressions(n = 40)

Numericexpressions(n = 40)

ule errorsFalse application of correct rule 5 6Correct application of false rule 5 2

alculation errors 1 3o response/error not specified 3 1yntactic errors (out of possible 23) 1 1

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logia 45 (2007) 1642–1648

ith both scores lying well outside the range of the healthyontrols. SO could judge the equivalence of and simplify expres-ions by applying the relevant transformation rules. He displayedensitivity to the embedded structure of equations through hisapacity to solve algebraic expressions with up to two brackettructures. Furthermore, SO’s performance was almost the samen structurally equivalent expressions whether they containedbstract algebraic variables or numbers.

. Discussion

SO’s performance revealed considerable residual ability forlgebraic processing in face of severe aphasia. He demonstratedargely intact algebraic fact knowledge (such as a + a = 2a),rocedures (such as the sign-change rule in clearing negativerackets), and conceptual knowledge (e.g., order relevance inubtracting, a − b �= b − a). SO’s performance in addition andubtraction of algebraic terms was well within the normal range.owever, he was considerably impaired on some algebraic

xpressions involving multiplication, fractions or mixed opera-ors, mirroring the selective difficulties he had shown previouslyn two-symbol multiplication and division in arithmetic tasksVarley et al., 2005).

With regard to the hypothetical two parallel indepen-ent levels of mathematical processing (formal-algebraic andrithmetical–numerical) two conclusions can be drawn. First,O demonstrates that access to the formal-algebraic level ofathematical processing is possible despite severely impaired

anguage. However, in comparison to DA (Hittmair-Delazer etl., 1995), who despite having intact language did not haveccess to the arithmetical–numerical level (through impairedrithmetical fact knowledge), SO had access to both levels ofathematical processing, as demonstrated by his ability to solve

umeric, mixed alphanumeric and abstract algebraic expres-ions. It seems that either intact conceptual knowledge and intactanguage processing (DA), or intact conceptual knowledge andargely intact arithmetical processing (SO) are sufficient for ele-entary algebraic computations. Second, with regard to the

uggested autonomy between the two levels (Hittmair-Delazer etl., 1995), the evidence of a common, analogue impairment inulti-symbol multiplication in arithmetic and algebraic tasks

n SO suggests either that the two systems are not entirelyutonomous, or they are both dependent upon common perfor-ance mechanisms (e.g., working memory).In terms of imperfect performance, some of the errors pro-

uced by SO were systematic. At the level of surface behaviour,hey could be described as the result of the application and gen-ration of rules. Given SO’s inability to generate phrase andlause structures in language, it is remarkable that he displaysehaviour that appears to indicate structure-dependent trans-ormations in algebra. We remain neutral as to whether thisenerativity demonstrated in algebra should be accounted fory invoking the concept of rule manipulation (Pinker, 1994), or

hether it is more adequately described as template manipula-

ion (Langacker, 1987).In some instances SO consistently applied a false rule.

or example, in algebraic fraction expressions with different

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enominators, SO would falsely add up both numerators andenominators, rather than finding the common denominator anddding up the numerators only. Some errors were inconsistentnd possibly the result of random variation in performance (e.g.,alculation errors, such as 17a − 9a = 6a, or sign errors, such asb × 2b = 9b), and were also observed in the matched controlubjects. SO’s other errors indicated that relearning might haveaken place during the investigation (e.g., problems of the typea ÷ na = ? were falsely answered with “0” in the beginning, butorrectly answered with “1” later on).

There are a number of possible explanations for imperfecterformance. These include issues specifically of mathematicalompetence, but also of the interaction of calculation with othereneral processing resources, i.e., attention, working memory,xecutive functions and language (Houde & Tzourio-Mazoyer,003). With respect to language, inner speech may be importantn solving complex problems. Although inner speech may note intrinsic to algebraic computations, it may provide a scaffoldhat permits offloading of information in working memory andhus assists in solution of problems involving multiple sequen-ial steps. Alternatively, although the computation of specificalculation rules may be autonomous from language, their stor-ge may be supported by a linguistic code (Campbell & Epp,004), and this might explain SO’s inconsistency in retrievinghose facts correctly.

The case of SO demonstrates that an individual with severephasia can still sustain algebraic reasoning. The history ofphasiology contains debates regarding the relationship betweenphasia and symbolic abilities more generally (Finkelnburg,870; Head, 1926; Hughlings-Jackson, 1878; Saygin, Dick,ronkers, & Bates, 2003). SO’s performance provides evidence

gainst the notion of aphasia as asymbolia. In algebra, let-ers are used to denote variables, i.e., abstract symbols, whichepresent non-specific numerical quantities and therefore cane substituted by any number or numeric term. SO’s similarerformance across sets of algebraic expressions containingither solely numeric or solely abstract variables indicated aophisticated ability to manipulate highly abstract symbolicepresentations.

At a neuroanatomical level, the extent of SO’s brain dam-ge involves structures that have shown to be active duringolving of algebra equations in healthy participants. In thesetudies, active regions included the left inferior frontal gyrusnd pre-frontal regions (BA 45/46). Activation was bilateral inhe parietal lobes, and included the supramarginal gyri and theanks of the intraparietal sulcus (Anderson et al., 2003; Qin etl., 2003). SO demonstrates that the inferior zones of the leftiddle and inferior frontal gyri and the anterior portion of the

eft intraparietal gyrus can be damaged, but the capacity to solvebstract symbolic expressions can be preserved. However, SO’sesion did not extend to posterior parts of the parietal lobe andight hemisphere fronto-parietal systems were intact. Activationn these areas were also observed during algebra equation solv-

ng in the studies mentioned, and have also been associated with

ental calculation (e.g., Dehaene et al., 1999; Zago et al., 2001).hese regions and undamaged parietal zones appear sufficient

o support algebraic function in face of unilateral left lesion.

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With regard to the relation between algebra and language ourndings show a clear dissociation. First, if his residual lexicalomprehension abilities in language are differentiated into highnd low imageability words, SO’s performance is more impairedn abstract words, which is in contrast to his ability to manipu-ate abstract symbols in algebra. Second, SO showed differentialensitivity to ‘syntactic’ features in algebraic expressions com-ared to sentences. Although structural parsing mechanismsere intact as reflected by his score on a grammaticality judge-ent task, he was insensitive to structure-dependent features

f sentences such as Subject/Object status of a noun phrase.y contrast, he showed considerable sensitivity to structure-ependent features of algebraic expressions such as embeddedracket structure.

The case of SO adds further empirical evidence to the viewhat some aspects of mathematical processing can be sustainedespite severe disruption to the language faculty. This has noween demonstrated not only with respect to arithmetic (GelmanButterworth, 2005; Varley et al., 2005), but also to elementary

lgebra. The capacity to compute algebraic expressions contain-ng only abstract variables demonstrates the relative autonomyf symbolic manipulations in the face of severe aphasia, androvides evidence for the considerable functional independencef some linguistic and higher mathematical functions in thedult cognitive system. Whether these findings extend to evenore complex algebraic expressions and equations remains

ndetermined, and is open to further investigation. However, ifatural language is not available, preservation of other forms ofymbolic coding might act together with language-independentepresentations of number to produce successful performancen solving and judging at least elementary forms of algebraicxpressions.

cknowledgements

We want to thank SO and all healthy participants for theirillingness to participate in this study, and Charles Romanowski

or his assistance in obtaining the CT scans.

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