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Approximate quantities

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Approximate quantities and exact number words: dissociable systemsIntroductionClinical description of BRIInitial evaluation of number processingClinical description of LECExperimental investigationElementary arithmetic problemsMethodControlsPatient BRIMultiplicationSubtractionAdditionDivisionDissociations between operationsPatient LECMultiplicationSubtractionAdditionDivisionDissociations between operationsDiscussionExact versus approximate calculationMethodControlsPatient BRIPatient LECDiscussionSubitizing and countingMethodControlsPatient BRIPatient LECDiscussionComparison and addition of quantitiesMethodControlsPatient BRIPatient LECDiscussionGeneral discussionSummary of BRI and LEC's resultsTwo distinct patterns of acalculiaAnatomical correlationsConclusionAcknowledgementsReferencesNeuropsychologia 41 (2003) 1942–1958Approximate quantities and exact number words: dissociable systemsCathy Lemera,∗, Stanislas Dehaenea, Elizabeth Spelkeb, Laurent Cohena,caINSERM U562, “Neuroimagerie Cognitive” Service Hospitalier Frédéric Joliot, CEA/DRM/DSV,4 Place du General Leclerc, 91401 Orsay Cedex, FrancebDepartment of Psychology, Harvard University, Cambridge, MA, USAcService de Neurologie 1, Clinique Paul Castaigne, Hˆopital de la Salpˆetrière, Paris, FranceReceived 7 February 2003; received in revised form 28 April 2003; accepted 19 May 2003AbstractNumerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolicrepresentation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending onwhether the deficit affectstheverbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularlythe presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion ofthe left parietal lobe and Gerstmann’s syndrome and another with semantic dementia with predominantly left temporal hypometabolism.As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slownessin approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and witharrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intactapproximation abilities, and showed preserved processing of non-symbolic numerosities.© 2003 Elsevier Ltd. All rights reserved.Keywords: Acalculia; Aphasia; Gerstmann’s syndrome; Semantic dementia1. IntroductionIn recent years, several studies have confirmed that anevolutionary precursor of human arithmetic abilities existsin animals. Behavioral studies have revealed that mon-keys can performed simple operations such as ordering oftwo sets based on their numerosity (Brannon & Terrace,1998), that this ability is cross-modal (Hauser, Dehaene,Dehaene-Lambertz, & Patalano, 2002), and that it is foundeven in untrained animals (Hauser, Carey, & Hauser, 2000).Three arguments suggest a genuine homology between suchanimal abilities and the human number sense. First, theneural bases of these abilities have begun to be explored atthe single-cell level, revealing neurons tuned to number infrontal and parietal areas that are plausible homologs of thecorresponding areas observed by functional neuroimagingduring arithmetic tasks in humans (Dehaene, 2002; Nieder,Freedman, & Miller, 2002; Nieder & Miller, 2003;Sawamura, Shima, & Tanji, 2002; Simon, Mangin, Cohen,Le Bihan, & Dehaene, 2002). Second, human infants exhibitsimilar approximate number discrimination and compari-∗Corresponding author. Tel.: +33-1-42-16-27-45.E-mail address: [email protected] (C. Lemer).son abilities in the first year of life, before the acquisitionof number words (Brannon, 2002; Brannon, Wusthoff,Gallistel, & Gibbon, 2001; Wynn, Bloom, & Chiang, 2002;Xu & Spelke, 2000).Third, adults who have learned number words and Ara-bic symbols, show approximation and non-symbolic dis-tance effects parallel to those observed in animals andinfants (Dehaene & Cohen, 1997; Dehaene, Spelke, Pinel,Stanescu, & Tsivkin, 1999). Of course, language compe-tence enables humans to go beyond other species in arith-metic, and to develop symbol systems that support exact cal-culation and higher mathematics. These observations haveled to the view that human arithmetic rests on the integrationof two distinct systems, a verbal system of number words anda non-symbolic representation of approximate quantities.In the present paper, we explore the consequences of thisview for neuropsychology, by exhibiting two new cases withmultiple double dissociations and within-subject associa-tions of deficits. The verbal versus quantity distinction hasalready been used to account for several puzzling features ofacalculia, including the dissociations between multiplicationand subtraction, and between exact and approximate abili-ties. We now show that this distinction predicts whether apatient will show impairments in the processing of numbers0028-3932/$ – see front matter © 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0028-3932(03)00123-4C. Lemer et al./Neuropsychologia 41 (2003) 1942–1958 1943presented non-symbolically as sets of dots, in basic tasksthat are more similar to those used in young children and inanimals.The first dissociation that we examine is between differ-ent arithmetic operations with Arabic numerals. Accordingto the triple-code model (Dehaene & Cohen, 1995), distinctarithmetic operations put a differential burden on verbaland quantity representations. Multiplication problems arethought to be solved by accessing a table of memorizedfacts (Ashcraft, 1992) stored in the form of verbal associ-ations without reference to quantity. In contrast, subtrac-tion facts are not learned by rote at school, and may beresolved through the mental manipulation of quantities.This view can explain the frequent observation of a dou-ble dissociation between those two operations (Cohen &Dehaene, 2000; Dagenbach & McCloskey, 1992; Dehaene& Cohen, 1997; Delazer & Benke, 1997; Lampl Eshel,Gilad, & Sarova-Pinhas, 1994; Lee, 2000; McNeil &Warrington, 1994; Pesenti, Seron, & Van der Linden, 1994;van Harskamp & Cipolotti, 2001; van Harskamp, Rudge,& Cipolotti, 2002). It also accounts for the frequent as-sociation of


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