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Calculating without reading?Comments on Cohen and Dehaene(2000)Agnesa Pillon a & Mauro Pesenti aa Université catholique de Louvain, BelgiumPublished online: 10 Sep 2010.
To cite this article: Agnesa Pillon & Mauro Pesenti (2001) Calculating without reading?Comments on Cohen and Dehaene (2000), Cognitive Neuropsychology, 18:3, 275-284, DOI:10.1080/02643290042000143
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CALCULATING WITHOUT READING? COMMENTSON COHEN AND DEHAENE (2000)
Agnesa Pillon and Mauro PesentiUniversité catholique de Louvain, Belgium
Cohen and Dehaene (Cognitive Neuropsychology, 2000, Vol. 17, pp. 563–583) reported the case of a purealexic patient who preserved some calculation abilities despite severely impaired Arabic numeral read-ing. They argued that these residual abilities and the general pattern of performance of the patient canbe fully explained within their anatomo-functional triple-code model. Here, we show how the lack ofspecification of the assumed architecture, the failure to provide sufficiently detailed data, and theabsence of adequate refutation of alternative accounts make this study unsuitable for constrainingtheories of numerical cognition.
INTRODUCTION
In Calculating without reading, Cohen andDehaene (2000; hereafter, C&D) report the case ofa pure alexic patient (VOL) whose performancepattern suggests complex relations between therepresentations and processes involved in mappingArabic onto verbal numerals and those involved inretrieving various aspects of numerical knowledgefrom Arabic numerals. The case is used to addressspecific aspects of numerical knowledge involved invarious tasks, and how they are related within a gen-eral architecture of number processing andcalculation.
In this comment we argue that although C&Doffer plausible theoretical propositions related tothis issue, their study does not meet two basicrequirements for a single-case study to constraintheories of normal cognition. The first unfulfilledrequirement is that it must be possible to derive a
patient’s pattern of performance from thecharacteristics of the assumed architecture (i.e., thefunctional components and their relations) and thenature of representations and computational prin-ciples of its components, especially those assumedto be impaired (Caramazza, 1986, 1992). We thinkthat, in C&D’s study, the nature of representationsand computations in the hypothetical numericalarchitecture is not sufficiently detailed to allowVOL’s pattern of performance to be interpretedreliably. The second unfulfilled requirement bearsupon a more general principle of scientific argu-mentation. Data support a theory not only if they fitthis theory, but if they compel us to prefer it to otherplausible ones. In a single-case study, this entailsnot only accounting for a patient’s performancewithin one’s own set of assumptions, but also reject-ing, with adequate evidence, accounts based onother plausible postulates. C&D never consideralternative accounts for their data and, accordingly,
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Ó 2001 Psychology Press Ltdhttp://www.tandf.co.uk/journals/pp/02643294.html DOI:10.1080/02643290042000143
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Requests for reprints should be addressed to Agnesa Pillon or Mauro Pesenti, Unité de Neuropsychologie cognitive, Faculté dePsychologie, Université catholique de Louvain, place Cardinal Mercier 10, B-1348 Louvain-la-Neuve, Belgium (Email:[email protected]; [email protected]).
AP and MP are Research Associates of the National Fund for Scientific Research (Belgium). We wish to thank Xavier Seron,Dana Samson, and an anonymous reviewer for helpful comments on an earlier draft of this article, and the PAI/IUAP Program fromthe Belgian Government for financial support.
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neither seek nor report evidence to evaluate otherpossible accounts. We will show how the failure tomeet these two basic requirements makes C&D’sstudy unable to constrain models of numerical cog-nition. We first summarise C&D’s observationsand the derived theoretical claims. Then we exam-ine each of these claims and ask whether they arecommensurate to the level of detail of the theoreti-cal framework and to the amount and nature of evi-dence provided.
C&D’S ACCOUNT OF VOL’SPATTERN OF PERFORMANCE
Although her comprehension of Arabic numeralsand her production of verbal numerals were correctper se, VOL was impaired at reading aloud single-digit Arabic numerals. When solving arithmeticalproblems, she correctly read aloud the operands inabout 20% of the cases. Yet, she gave the result ofthe target problem with a relatively good level ofaccuracy in subtraction, addition, and division(72%, 61%, and 66% correct, respectively), but notin multiplication (only 13% correct). Interestingly,she gave the correct result for the problem errone-ously read in 77% of the cases (e.g., 5×9 was read asfour times six and answered twenty-four). This con-trast between impaired naming and multiplicationand relatively spared comparison and other arith-metic operations led C&D (2000, p. 580) to pro-pose two distinct routes for Arabic numeralprocessing, “one that is the mandatory input path-way to naming and multiplication processes, andthe other that is able to supply comparison, addi-tion, subtraction, and simple division routines.” Inpatient VOL, only the direct Arabic-to-verbalmapping route would be impaired, whereas theindirect semantic route (i.e., through access to andmanipulation of the quantities represented bynumerals) would be spared. Multiplication prob-lems could only be solved by retrieval of the corre-sponding facts stored as verbally coded associationslinking operands and products. Arabic-to-verbalmapping would thus be the prerequisite for multipli-cations, which explains answers stemming fromerroneous reading of operands through this route.
C&D also tentatively link these functional pro-posals with an anatomical organisation of their “tri-ple-code” model (Dehaene & Cohen, 1995),according to which, digit structural representationand quantity information are processed by bothhemispheres, whereas only the left hemisphere pro-cesses verbal numerals. VOL’s lesion damaged theleft-hemispheric digit recognition processes, pre-cluding the transcoding of Arabic numerals intowords (hence, the retrieval of stored multiplicationfacts) and the access to the left-hemispheric quan-tity representation. Her residual numerical abilitieswould reflect intact right-hemispheric processes.
C&D’S THEORETICAL CLAIMS ANDTHE SIGNIFICANCE OF THE DATA
Do Arabic numeral reading errors reflectfunctional Arabic-to-verbal mappingimpairments?
C&D hypothesise that VOL’s naming difficultiesarise from the verbal system being deprived of itsnormal input, but without specifying whether thedamage was to the Arabic recognition system or tothe Arabic-to-verbal mapping system. Here, wefocus on the functional aspects of this claim, anddiscuss its anatomo-functional aspects in the lastpart of this section.
At a functional level, structural processing ofdigits seemed relatively spared. VOL could dis-criminate digits from letters, and her errors did notreflect visual confusions. Therefore, the deficitcausing VOL’s naming difficulties would lie in theasemantic Arabic-to-verbal mapping system.However, this conclusion stems from C&D’s priorassumption that Arabic numerals are named with-out semantic access, which is quite controversial(for a discussion, see McCloskey, 1992; Seron &Noël, 1995). It might turn out to be right but, to bestated as a motivated account of VOL’s naming dif-ficulties, alternative assumptions must be empiri-cally ruled out, the more straightforward beingreading digits through semantic access. Then,VOL’s naming errors would stem from an inabilityto access semantics from the structural representa-tion of Arabic digits. Unfortunately, C&D do not
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specify the characteristics of their asemantictranscoding system, and they do not derive nor testspecific predictions about the pattern of namingerrors generated when the system is not properlyaddressed. For instance, in case of damage to theasemantic route, should one expect substitutionsbetween phonologically similar verbal numerals, orbetween semantically similar (i.e., close in magni-tude) numerals as expected in case of damage to thesemantic route? In the reading tasks, VOL was pre-sented with one- to four-digit Arabic numerals,and she made almost exclusively digit substitutionerrors. If the visual representation computed on thebasis of Arabic numerals cannot properly addressthe Arabic-to-verbal mapping system, one shouldexpect VOL, besides substitutions, to make syntac-tic errors due to loss or exchange of digit positionsduring transfer. Such errors were not observed,which shows that she at least retained the ability toanalyse the positions of multi-digit numerals, tomaintain and transfer these positions to thetranscoding system, and to map them into theappropriate multi-word numeral structure. Thisseems rather inconsistent with a “destruction ordeafferentation” (C&D, 2000, p. 573) of theprocessing component whose output is used by theArabic-to-verbal mapping system. C&D should atleast explain why only the lexical part (identity ofthe digit) of the structural representation was notcorrectly transferred and processed, while its syn-tactic part (positional structure) was.
Thus, the data reported do not allow the readerto decide whether Arabic naming errors stem fromimpairment to the asemantic or to the semanticroute. C&D argue that their hypothesis naturallyfollows from observations made with other tasks.Since VOL could access quantity information fromArabic numerals but could not read them aloud,then naming necessarily relies on an asemantic pro-cessing route. In the next two sections, we discussthe relevance of the evidence supporting the
premise of this argument and explain why it isinadequate.
Distinct processing routes for naming andcomparing Arabic numerals?
VOL could accurately compare two Arabic numer-als though she could not read them aloud. Thiswould support the idea of two processing routes if itis shown that a single route cannot yield, whendamaged, different levels of performance reflectinginherently different levels of difficulty of the tasks.
If naming and comparing rely on the same pro-cessing route, the structural representation ofArabic numerals would give access to their meaningand, thereafter, to the verbal numerals retrievedfrom the phonological lexicon. Assuming thatVOL can compute structural and magnitude repre-sentations, the functional damage would then belocated in the structural-to-magnitude representa-tion addressing procedures, causing weak activationof the numeral meaning, and resulting in approxi-mate/degraded quantity information. This couldnevertheless allow VOL to give accurate responsesin a magnitude comparison task since it is logicallypossible to compare numerals without accessingtheir exact magnitude (as demonstrated byDehaene & Cohen, 1991). This is especially truefor the comparison task used in C&D’s study,where the numerical distance between the numbersin the pairs ranged from 2 to 26 (mean distance:8.8). Accessing only approximate quantity infor-mation might thus suffice in most cases. Moreover,accurate responses might be given even when thequantity information accessed is totally wrong, pro-vided that the relative size of the two numbers is notreversed1. This account would not hold if it hadbeen shown that VOL could access exact quantities(e.g., verifying whether there is a correct matchbetween Arabic numerals and arrangements ofpoker chips of various numerical values, compari-
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1 C&D rejected in advance this objection by arguing that VOL made very few reversing errors when reading aloud pairs of two-digit numerals. In our view, this only suggests that if VOL had based her responses in comparison on the quantity information sheaccessed in reading, the probability of errors in comparison would be very low. Note that data in comparison and in naming werecollected with different stimuli on different occasions. It would have been more convincing if VOL had had to read aloud andimmediately after to compare the pairs.
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sons with splits one or two, etc.). In this study,nothing compels us to accept that VOL “showed abetter access to quantity information than could beexpected on the basis of her number reading perfor-mance” (C&D, 2000, p. 567). The data are com-patible with the hypothesis of two distinctprocessing routes, but they do not need to make thishypothesis.
We showed that, in VOL’s performance, noth-ing allows one to characterise the naming route as anon-quantity-based one. In our opinion, there isno strong basis for characterising the comparisonprocess as mandatorily involving quantity informa-tion either. That numerical comparison normallyrelies on quantity information is a widely sharedassumption. However, aspects of VOL’s perfor-mance question whether quantity information isabsolutely required to compare numerals. Indeed,VOL correctly accessed the knowledge of the rankof Arabic numerals and the corresponding rank ofverbal numerals (she could retrieve the correct ver-bal numeral by reciting the conventional sequenceof number words up to the rank retrieved from thepresented digit). This raises the question of the sta-tus of rank information in number semantics. Rankand quantity information might be two logicallydissociable aspects of number semantics whoseintegration forms the core meaning of a numericalconcept (for a dissociation between these twoaspects, see Delazer & Butterworth, 1997; see alsoFuson, 1988; Gallistel & Gelman, 1992; Wynn,1992, 1995). Thus, that VOL performed the com-parison task on the basis of rank informationbecause of weak activation of quantity informationcannot be dismissed on the basis of the resultsreported for the comparison task. There is evidencethat VOL did not compare Arabic numerals by(covert) counting, but she might directly accessinformation about their rank value (e.g., some rep-resentation of their “sequence meaning”; cf. Fuson,1988, 1992). In such conditions, short latenciesand distance effects might also be expected, as isfound when comparing the order of letters (Hamil-ton & Sanford, 1978; Taylor, Kim, & Sudevan,1984). A possible account for the dissociationobserved between naming and comparing is thatdigit naming (without verbal sequence recitation)
is dependent on (impaired) quantity informationwhereas comparing (and naming after verbal reci-tation) might be based on (spared) rank informa-tion only. Such an account does not entail that rankand quantity are represented and processed by twodistinct systems. Rank information associated withnumerals is acquired earlier than the quantity theyrepresent (Wynn, 1990, 1992) and might just bemore easily accessible within the number semanticsystem or/and less vulnerable to brain damage.
A single processing route for comparing,adding, subtracting, and dividing Arabicnumerals?
Because VOL performed almost perfectly the com-parison task and, with reasonable accuracy, simplesubtractions, divisions, and additions, C&D pro-pose that the same processing route (access to andmanipulation of quantities) was used in all cases.However, observing a good performance in twotasks does not suffice to infer that both rely on thesame processing route or functional system. Oneneeds independent evidence that the same sort ofknowledge—here, quantities—is used, with spe-cific additional knowledge or processing for eachtask.
As we understand it, C&D propose that, inVOL as well as in healthy subjects, simple subtrac-tions could rely only on quantity manipulations.However, they do not specify the kind of computa-tion involved in these manipulations (i.e., howquantities encoded as distributions of activation onan internal number line are manipulated). There-fore, it is not clear how errors arising from faultymanipulations could be distinguished from errorsarising from faulty access to stored facts. For exam-ple, the fact that “most erroneous responses to sub-tractions were false by only 1 unit” (C&D, 2000,p. 578) might reflect a faulty access to the quantityof one of the operands rather than errors in quantitymanipulation during calculation.
C&D (2000, p. 576) propose “two parallel cir-cuits” for additions: “rote verbal retrieval andstrategical manipulations” or “back-up strategies”
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such as counting or “referring to 10 (6 +5 = 6 +4 +1= 10 + 1 = 11)”2. However, it is unclear whetherthese circuits are two equally efficient means toreach correct results or if they must be recruited in acomplementary manner. In the above-mentionedexample, is 6 +4 retrieved from memory or com-puted? The same ambiguity holds for divisions.C&D (2000, p. 577) propose that, for large prob-lems, the corresponding multiplication fact isretrieved, but “the simplest division problems, suchas those presented to VOL, can also be solved usingsemantic procedures. For instance, halving a num-ber can be performed by referring to the corre-sponding tie addition fact (6 ÷2 =3 because 3 +3 =6).” In this example, how is the result of 3 + 3retrieved? These ambiguities prevent precise pre-dictions in case of damage to the verbal retrieval cir-cuit resulting in compensatory reliance on quantitymanipulations. If verbal retrieval and quantity ma-nipulation are equally efficient, then subtractions,divisions, and additions should yield the same levelof performance in VOL—as was actually observed.If rote verbal retrieval and quantity manipulationare complementary, then subtractions should yieldbetter performance than additions and divisions inVOL—which was not observed. If VOL solvedadditions by strategical counting and/or decompo-sition, her response times should be longer for addi-tions than for subtractions and divisions—whichwas not reported.
Predictions are even more obscure concerningthe problems that VOL should solve accurately/erroneously. If quantity manipulation is the rule,poorer performance should be observed for largethan for small problems, whatever the operation. Ifquantity manipulation and verbal retrieval are bothrequired for additions and divisions, the size of theproblems should affect error rates for subtractionsonly. If additions require both procedures, VOLshould resort either to quantity manipulations forsome problems and rote retrieval for some others, orto quantity manipulations and rote retrieval formost if not all problems (e.g., 3 + 4 = 3 + 3 +1).Finally, regarding divisions, VOL should be able to
cope either with problems with 2 as the divisor,whatever the size of the first operand and result(e.g., better performance both for 6 ÷ 2 and 10 ÷ 2than for 6 ÷3 and 15 ÷3), or with problems with 2 asthe result, whatever the size of the operands (6 ÷ 3or 18 ÷9), because these problems can be solved bycounting.
Given the lack of specifications on VOL’sassumed back-up strategies, and the lack of demon-stration that her pattern of performance can bederived from these specifications, it is not possibleto evaluate the claim that she solved subtractions,additions, and divisions by manipulating quanti-ties. For the same reasons, other accounts cannot bedismissed. For instance, one can assume that arith-metical facts are stored in memory as abstractdeclarative (instead of verbal) knowledge directlyaccessed from the structural representation ofArabic numerals. Then, VOL would solve subtrac-tions, additions, and divisions by directly retrievingthe answers, and her errors would result from rela-tively impaired access to this stored knowledge.Otherwise, VOL could solve these problems byrelying on both direct memory retrieval and quan-tity manipulation, and her errors would result fromimpaired access to quantities.
Mandatory verbal retrieval for multiplicationfacts?
Multiplication errors were generally the correctanswers for the operands (erroneously) read aloud,which clearly shows that multiplication was moredependent on the verbal coding of the operandsthan other operations. However, it does not implythat the products were retrieved from a verbalstore—only that they were retrieved after or on thebasis of the verbal coding of the operands. Further-more, even if VOL actually retrieved the resultsfrom verbal memory, it might not be the only avail-able routine. Indeed, healthy subjects use back-upstrategies based on calculation when memoryretrieval fails to provide an answer (LeFevre,Sadeski, & Bisanz, 1996). Why does VOL not
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2 It is unclear whether “quantity manipulation”, “semantic routines”, “strategical quantity manipulations”, and “back-up strategies”have the same status in C&D’s proposals.
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resort to such strategies, as she does for other opera-tions? C&D argue that these back-up strategiescannot be distinguished from retrieval because theyinvolve other stored facts (e.g., computing 7×8 as 8× 8 – 8). However, the set of multiplications pre-sented to VOL also included problems that do notresort to fact retrieval (N×1 problems, N×12 prob-lems potentially solvable by addition), and virtuallyall problems could be solved by counting by 2, by 3,etc3. Yet, different rates of errors were not reportedfor these problems. On the contrary, problemsmatched in result magnitude (mainly N ×1 prob-lems) gave rise to as many errors (C&D, Table 2).No explanation is given for this result: Why wasVOL able to access quantities and strategicallymanipulate them to retrieve the results for addi-tions and divisions, and not for at least some multi-plications? A possible interpretation is that, in fact,quantity-based and compensatory strategies wereno longer available to VOL, because access to quan-tity information from Arabic numerals was dam-aged. This conclusion is reasonable since no strongevidence was provided that VOL resorted to quan-tity-based or compensatory strategies for subtrac-tions, additions, and divisions. Thus, let us againassume that arithmetical facts (for all operations)are stored as declarative knowledge, access to whichcan proceed directly from the structural representa-tion of the operands, and that solving arithmeticproblems requires both retrieval and quantitymanipulations, the latter being specially recruitedfor less familiar problems. If access to quantitiesfrom Arabic numerals was impaired while access tostored facts was spared, VOL would be moreimpaired at multiplication, because the set of multi-plication problems probably comprised less familiarproblems. To compensate for these difficulties,VOL could choose systematically to first transcodethe operands verbally so as to access the quantitysystem and the stored multiplication facts from averbal input (i.e., by the same route as when solvingorally presented multiplications). Then, errors forother operations should reflect the impaired accessto quantity information as well—which is notinconsistent with the data.
Because they are learned as verbal associations,multiplication facts might be represented in a verbalas well as in an abstract knowledge store. Then,VOL would retrieve not only the operands, but alsothe results of multiplications through the impairedverbal route (the direct verbal route or the quantity-to-verbal route). However, this additional point isnot required by the observations. C&D (2000, p.570) note that, in the occasional errors (as a func-tion of the read operands) made by VOL in multi-plications, “the results fell close to within thecorrect row or column of the spoken problemwithin the multiplication table.” Are such tableerrors predicted by the hypothesis of rote verbalretrieval for multiplications? How are multiplica-tion facts organised within the verbal store? Shoulderrors showing phonologically based confusionsnot be expected? Here again, without more detailedspecifications about the representation and pro-cessing principles of the rote verbal memory formultiplication facts, it is not possible to distinguishequally reasonable and compatible accounts.
Intact abilities of the right hemisphere orresidual abilities of the partially impaired lefthemisphere?
C&D proposed that VOL’s residual abilities inprocessing quantities reflect the normal function-ing of intact numerical processes in the right hemi-sphere (RH) since her left occipito-temporaldamage was assumed to alter processing of Arabicnumerals, depriving the left-hemispheric (LH)Arabic-to-verbal mapping and quantity systemsfrom their normal input.
As previously shown, the study provides no evi-dence likely to constrain this argument. C&Dassume that the functional implication of VOL’sLH damage is to prevent processing of Arabicnumerals but without specifying whether the dam-age was to the left Arabic recognition system or tothe Arabic-to-verbal connection pathway. Withintheir theoretical framework, this has distinct impli-cations. Damage to the LH recognition system
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3 Unfortunately, it is not known whether the patient is able to use such counting sequences.
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itself would result in incorrect input to both the LHverbal and semantic systems. Damage altering theconnection between the recognition and verbal sys-tems would spare the access to the left semanticcomponent. Hence, VOL’s residual abilities couldbe accounted for without assuming RH routes, andher errors might simply reflect the residual abilitiesof only partially damaged left components. Dam-age to the left Arabic recognition system predictsthat (1) the digit substitution errors made by VOLin naming should reflect confusions between simi-larly shaped digits (i.e., errors reflecting visual prox-imity between target and response or/and othervisuospatial features), and (2) if presented withArabic numerals to her right hemifield, VOLshould be unable to access semantics, or should alsoprovide responses reflecting visually based confu-sions. Such predictions were not tested, althoughthe main hypothesis underlying the anatomo-func-tional proposal was damage to the visual recogni-tion system.
Even if we accept C&D’s assumptions about thefunctional locus of impairment in VOL, and even ifthere was independent evidence for RH numericalcapacities in other patients, this case would notdemonstrate RH numerical capacities. This issueincludes three main aspects. First, up to whichpoint is the RH able to process Arabic numeralsfrom perception? We agree that the RH seems ableto build structural representations of Arabicnumerals. RH processing capacities might endthere, with the output of this process being trans-ferred to the LH, as C&D state in their conclusion.However, they also claim that “both the left- andthe right-hemispheric visual systems are able tobuild a structural representation of Arabic numerals... and to access their meaning, i.e., the quantity towhich they refer” (C&D, 2000, p. 573) or that“quantity manipulation abilities are distributed overthe two hemispheres” (C&D, 2000, p. 576). Thiscannot be inferred either from VOL’s performance
or from other pure alexic patients (Cohen &Dehaene, 1995; McNeil & Warrington, 1994;Miozzo & Caramazza, 1998) who “showed sparedcomparison abilities, whereas sparing of arithmeticproblem solving seems at first sight to be more vari-able across patients and operation types” (C&D,2000, p. 573). Even leaving aside this inter-patientvariability, it remains to show that the residual abil-ities reflected RH rather than LH capacities aftertransfer of structural representations computed inthe RH4. Second, are RH processing componentsdifferent or a mere duplication of the left ones?C&D only speculate that the right and left process-ing routes involve the same kind of knowledge andcomputations, but the right route does it less accu-rately: There might be a LH advantage for visualidentification of digits, the right quantity systemmight be inherently less precise than the left, orboth hemispheres might be equally accurate, butthere could be some loss of information due tocallosal transfer. Finally, if the left and right com-ponents are different, what are their respectivefunctions during normal number processing?Unfortunately, the nature of computation andrespective contributions of RH and LH compo-nents during normal processing, and the means todistinguish RH intact processing of approximatequantity from LH impaired processing of (moreprecise) quantity information are not clearlyformulated.
ANOTHER POSSIBLE ACCOUNT OFVOL’S PERFORMANCE
It might be objected that C&D’s explanation ofVOL’s entire pattern is more elegant and coherentthan our multiple alternative accounts. However,putting together these alternatives would not nec-essarily yield a less coherent explanation. Let usassume a number processing system composed of
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4 Admittedly, data from split-brain patients (e.g., Cohen & Dehaene, 1996) speak against the view that the RH is only able tocompute digit structural representations. With Arabic numerals presented to their left hemifield, these patients are able to performcomparison tasks. However, to the best of our knowledge, performance is generally not normal.
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one functional component5, building a structuralrepresentation of Arabic numerals, one processingnumber semantics, and one representing storedarithmetic knowledge (Figure 1). The “Numbersemantics” system represents and processes the coremeaning of numerals integrating their rank andquantity properties. The “Stored arithmetic knowl-edge” represents, in an abstract format, simple,highly familiar, and frequently used arithmeticalfacts (for all operations), as well as usual arithmeticproperties (like parity, square roots, prime num-
bers, etc.; it is not relevant here whether these vari-ous aspects of stored arithmetic knowledge are rep-resented in partly independent networks). In thiscontext, VOL’s pattern of performance could beaccounted for as follows. The Arabic-to-semanticprocessing route was impaired, so that only rankand approximate quantities information wereaccessed. From this partial meaning, VOL couldretrieve the correct verbal numeral by reciting theverbal sequence up to its rank, and perform com-parisons but not tasks requiring exact number
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Figure 1. A possible alternative account of VOL’s performance (plain lines: spared routes; dashed lines: impaired route).
5 By “functional” system, we mean a component within a functional architecture that is assumed to be involved in specificoperations. It does not matter, within this definition, whether this processing component is located within one hemisphere ordistributed over both hemispheres. In case of a functional component distributed over the two hemispheres, if the normal functioningof the assumed operations required intact processing of both, then only one functional component is assumed.
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meaning such as reading aloud Arabic numerals.Access to stored arithmetic knowledge would bespared. If only familiar arithmetic knowledge isstored and/or reliably retrieved, then less familiararithmetic problems (and arithmetic properties)would require access to both stored arithmeticalknowledge and quantity manipulations. Therefore,solving simple problems and accessing parity infor-mation, though not perfect, would yield a betterperformance than naming. The need to rely onquantity manipulations for less familiar facts wouldaccount for the different levels of accuracy acrossthe four operations. In order to compensate for herdifficulties in multiplication, VOL would strategi-cally attempt to retrieve the results on the basis ofthe verbal coding of the operands so that she couldaccess the number knowledge system (numbersemantics and stored arithmetical knowledge)through the spared verbal-to-semantic route.
Let us stress, in order to avoid any misunder-standing, that we are neither advocating here thisparticular architecture of number processing norclaiming that it could account for a larger range ofobservations than just this study. Providing finespecifications and discussing the whole range ofevidence to date would fall beyond the scope of thiscommentary. We are not even defending this spe-cific account of VOL’s performance. Instead, wewant to show that, with the evidence reported inthis study and other reasonable assumptions about apossible architecture, it is possible to draw a conclu-sion exactly opposite to that of C&D.
CONCLUSION
Deriving claims about normal processing systemsfrom impaired performance requires clear andmotivated hypotheses about the nature and func-tional locus of the deficits. Although C&D do putforward a hypothesis about the locus of VOL’sdamage, this hypothesis is ambiguous and notmotivated by adequate evidence. The data reportedgenerally support the proposed theory, but they donot compel us to prefer it to other accounts. Wehave shown that, due to a lack of specification ofsome aspects of the model and the absence of criti-
cal tests of alternative assumptions, it is possible toaccount for VOL’s performance in exactly theopposite way as C&D did. If so, their study clearlydoes not contribute to constraining theories aboutnumber processing. We also hope to have shownthat single-case studies in neuropsychology, if theyaim at constraining theories of normal cognition,should not neglect the old precept: “To give supportto your theory, assume just the opposite of what youwant to demonstrate, and then show that this oppo-site view cannot work.”
Manuscript received 15 November 1999Revised manuscript received 28 June 2000
Revised manuscript accepted 6 July 2000
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