A Feeling for Numbers: Shared Metric for Symbolic and Tactile Numerosities.

0.49 MB·English

by  Krause, Florian

ORIGINALRESEARCHARTICLE
published:25January2013
doi:10.3389/fpsyg.2013.00007
A feeling for numbers: shared metric for symbolic and
tactile numerosities
FlorianKrause1*,HaroldBekkering1 andOliverLindemann2
1DondersInstituteforBrain,CognitionandBehaviour,RadboudUniversityNijmegen,Nijmegen,Netherlands
2DivisionofCognitiveScience,UniversityofPotsdam,Potsdam,Germany
Editedby: Evidenceforanapproximateanalogsystemofnumbershasbeenprovidedbythefinding
LouiseConnell,Universityof that the comparison of two numerals takes longer and is more error-prone if the seman-
Manchester,UK
ticdistancebetweenthenumbersbecomessmaller(so-callednumericaldistanceeffect).
Reviewedby:
Recentembodiedtheoriessuggestthatanalognumberrepresentationsarebasedonpre-
CarmeloM.Vicario,Universityof
Queensland,Australia vioussensoryexperiencesandconstitutethereforeacommonmagnitudemetricsharedby
ZairaCattaneo,Universityof multipledomains.Herewedemonstratetheexistenceofacross-modalsemanticdistance
Milano-Bicocca,Italy effectbetweensymbolicandtactilenumerosities.Participantsreceivedtactilestimulations
*Correspondence: ofdifferentamountsoffingerswhilereadingArabicdigitsandindicatedverballywhether
FlorianKrause,DondersInstitutefor
the amount of stimulated fingers was different from the simultaneously presented digit
Brain,CognitionandBehaviour,
RadboudUniversityNijmegen,P.O. or not.The larger the semantic distance was between the two numerosities, the faster
Box9104,6500HENijmegen, and more accurate participants made their judgments.This cross-modal numerosity dis-
Netherlands. tanceeffectsuggestsadirectconnectionbetweentactilesensationsandtheconceptof
e-mail:[email protected]
numerical magnitude.A second experiment replicated the interaction between symbolic
andtactilenumerositiesandshowedthatthiseffectisnotmodulatedbytheparticipants’
fingercountinghabits.Takentogether,ourdataprovidenovelevidenceforasharedmetric
for symbolic and tactile numerosities as an instance of an embodied representation of
numbers.
Keywords:numbercognition,tactileperception,fingercounting
INTRODUCTION In modern psycholinguistic research,several authors empha-
It has been argued that numbers are cognitively represented in sized the idea of embodied cognition (Wilson, 2002), which
an approximate and analog manner (e.g.,Dehaene et al.,1993). basicallyholdsthateachsemanticrepresentationisgroundedin
Mainevidenceforthisnotioncomesfromtheso-callednumeri- previous sensorimotor experiences and therefore closely linked
caldistanceeffect(MoyerandLandauer,1967).Whenparticipants tolow-levelperceptualandmotorcodes(GlenbergandKaschak,
are asked to perform a magnitude judgment (i.e.,compare two 2002;Barsalou,2008;FischerandZwaan,2008).Interestingly,the
numbers by their semantic size) responses are slower,when the roleofembodiedrepresentationshasalsobeenrecentlydiscussed
semanticdistancebetweenthetwonumbersissmall(e.g.,2vs.3), inthecontextofnumberprocessing.Forinstance,recentresearch
comparedtowhenthesemanticdistanceislarge(e.g.,1vs.4;Gal- hasshownthattheperceptionof abstractnumericalstimulihas
listelandGelman,1992).Thiseffectofthenumericaldistancehas a direct influence on response selection (Daar and Pratt, 2008)
beenconsistentlyexplainedbyarepresentationaloverlapofneigh- as well as movement generation (Vicario, 2012), demonstrating
boring numbers on a hypothetical analog mental continuum of acloselinkbetweennumericalconceptsandaction.Ithasbeen
numericalmagnitudes(e.g.,Restle,1970;DehaeneandChangeux, speculatedthatnumericalmagnitudeinformationbecomesmean-
1993).Thatis,aparticularnumberdoesnotonlyactivatetherep- ingfulonlywhenitcanbesomehowmappedtoconcretebodily
resentationofexactlythisnumber,butalsotherepresentationof experiences with size and magnitude in everyday life (Andres
thenumbersnexttoit.Consequently,thefurtheraparttwonum- etal.,2008;Lindemannetal.,2009).Asimilarimportantroleof
bers are,the less do they activate each other and the easier it is size-related sensorimotor representations for numbers has been
todiscriminatebetweenthem.Supportforthisideaisalsopro- suggestedbyarecenttheoryonmagnituderepresentationspro-
videdbystudiesonhumanandnon-humancorticalactivationsin posedbyWalsh(2003),whichassumestheexistenceof ashared
responsetonumerosityinformationthatdemonstratedtheexis- generalized representation of magnitude. That is, numbers are
tenceofnumber-sensitiveneuronswithoverlappingtuningcurves thoughttobeprocessedbyasinglesystemwhichsimultaneously
in macaque (Nieder and Miller, 2003) as well as in the human codesforsize-relatedinformationfromothercognitivedomains
parietalcortex(Piazzaetal.,2004).Althoughtheexistenceofan like,forinstance,sensoryandmotorrepresentationsof physical
analogrepresentationinhumansandanimalsisveryestablished, size or temporal duration. Evidence for this notion comes from
theoriginandnatureof thisspecificsemanticrepresentationof behavioral studies showing interferences between numbers and
magnitudeinformationiscontroversiallydebated(seee.g.,Cohen othertypesofmagnitudeinformation,suchasthephysicalsizeof
KadoshandWalsh,2009). numbersymbols(Tzelgovetal.,1992),theperceivedtime(Oliveri
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Krauseetal. Afeelingfornumbers
etal.,2008),theperceivedsizeofanobject(Badetsetal.,2007),and perception in the tactical and in the visual modality share the
theaperturesizewhileobjectgrasping(Lindemannetal.,2007). sameprocesses.Thesefindingssuggestthatallsensorynumeros-
Another observation often interpreted as evidence for an ityinformationarerepresentedbythesamemodality-independent
embodiedrepresentationofnumbersistheexistenceofastrong magnitudesystem.
associationoffingersandnumbersinmostadults.Thisassociation Takingintoaccounttheembodiedviewoncognition(e.g.,Wil-
is probably resulting from the habit to use fingers while count- son, 2002; Barsalou, 2008) and the idea of a single generalized
ing (Lindemann et al., 2011; Bender and Beller, 2012; Moeller metric for magnitudes (Walsh, 2003), one might speculate that
et al., 2012). For instance, in Italian adults this association has tactile numerosity processing is based on the very same analog
beendemonstratedbyafacilitationtorespondtonumbers1to5 magnituderepresentationthatisactivatedwhenreadingsymbolic
withthefingersoftherighthand,andtonumbers6to10withthe numbers or solving arithmetic problems. Surprisingly,however,
fingersofthelefthand(amappingcongruenttotheprototypical verylittleisknownabouttherelationshipandthecommonalities
fingercountingstrategyoftheparticipants;DiLucaetal.,2006), between tactile and symbolic numbers. We assumed that tactile
aswellasbyafacilitationtojudgeifanumberissmallerorlarger numerosity judgments are based on the same analog represen-
thanfivewhenprimedwithafingerconfigurationcompatibleto tations as involved in symbolic number processing irrespective
theindividual’scountingstrategy(DiLucaandPesenti,2008).In of differences in format and modality. To examine this hypoth-
additiontothis,weknowthatthepreferencetostartcountingwith esis, we made use of the numerical distance effect (Moyer and
theleftorwiththerighthandvariesstronglybetweenindividu- Landauer,1967).Weconductedtwoexperiments,inwhichpar-
als and is independent of handedness (Lindemann et al.,2011). ticipants received tactile stimulations on their fingers of the left
Interestingly,manual counting habits,like the individual finger- or right hand while reading an Arabic digit. The participants’
numberassociationsandstartingpreferenceshavebeenshownto task was to indicate as fast as possible whether the visually pre-
affect symbolic number processing in adults, even if the use of sentednumbermatchedtheamountofstimulatedfingers.Ifboth
fingers is not required (Fischer,2008;Domahs et al.,2010). For tactile and symbolic numerosities are indeed mapped onto the
instance,Fischer(2008)showedthattheassociationof numbers same analog magnitude metric,we expected to observe a cross-
withaspatialresponse(SNARCeffect;Dehaeneetal.,1993)was modal numerosity distance effect reflected by an inverse linear
stronglyaffectedbywhetherparticipantsstartedtocountontheir relationbetweenthejudgmentlatenciesinthemagnitudecompar-
right or left hand. Only for participants who started counting isontaskandthesemanticdistancebetweentheto-becompared
ontheirlefthandaSNARCeffectcouldbeobserved.Inanother numerosities.Crucially,weusedasame-differenttask,andnota
study Domahs et al. (2010) investigated finger-based sub-base- magnitudecomparisontask.Thatis,ifalternatively,symbolicand
fiveeffectsinanArabicnumbercomparisontaskinthreedifferent tactilenumerositiesactivatedifferentanalogmagnituderepresen-
groups–Germandeafsigners,Germanhearingadults,andChi- tationsorsame-differentcomparisonsfindplaceonverbalcodes,a
nesehearingadults.Theirresultsrevealedthatsub-base-fiveeffects modulationoftheresponselatenciesasafunctionofthesemantic
werelargerinthetwoGermangroupswhichuseasub-base-five distanceisnotexpected(cf.VanOpstalandVerguts,2011;Defever
fingercountingsystem,comparedtotheChinesegroupwhichuses etal.,2012).
asub-base-10fingercountingsystem.Takentogether,thesestud- Moreover, if the acquired associations between fingers and
ies speak for an important role of finger representations for the number modulate adults’ processing of symbolic numerosity
processingofsymbolicnumericalinformation. information,onemightexpectthatcountinghabitsalsoaffectthe
Whileanincreasingamountofstudiesinvestigatedthecogni- enumeration or perception of numbers in the subitizing range.
tiveeffectsofthefinger-numberassociations,untiltodayonlyfew We therefore aimed to explore additionally the influence of fin-
studieshaveexaminedtactileorhapticnumerosityprocessingas gercountinghabitsonthetactileperceptionofnumerositiesand
such. As we know from recent experiments on tactile and hap- their comparison with symbolic numbers. To do so, we used
ticperception(Riggsetal.,2006;Plaisieretal.,2009;Plaisierand an adapted version of the finger counting questionnaire of Lin-
Smeets,2011),tactilenumerosityperceptionseemstobebasedon demann et al. (2011) to classify the starting preference of our
thesamedistinctcognitiveprocessesastheenumerationofvisual participants and tested whether detection times or cross-modal
items(Atkinsonetal.,1976).Forinstance,Riggsetal.(2006)stim- numerositydistanceeffectsaremodulatedbythesehabits.
ulatedthefingertipsoftheirparticipantsandaskedthemtoname
thenumberof stimulatedfingers.Theauthorsfoundthatjudg- EXPERIMENT1
mentswerebasedonserialcountingprocessesifmorethanthree The aim of the first experiment was to investigate if tactile
fingers were stimulated,since enumeration became more error- numerosities are mapped to the same analog representation of
prone and slower with increasing set-size. In contrast,however, numerical magnitude as symbolic numerosities, as expected by
forsmallnumerosities(i.e.,lessthanfourfingers)tactileenumer- thenotionofageneralizedmagnitudesystem(Walsh,2003).Par-
ation was fast,effortless,and highly accurate (Riggs et al.,2006; ticipantshadtoverballyindicateiftactilepresentednumerosities
PlaisierandSmeets,2011;butseealsoGallaceetal.,2008)–aphe- wereidenticalordifferenttovisuallypresentedArabicdigits.We
nomenonwellknownfromvisionresearchandcalled“subitizing” expected to find a cross-modal semantic distance effect in the
(Kaufman et al.,1949). Recently,support for subitizing has also numerosity judgments reflected by longer response times when
beendemonstratedforactivetouchandthehapticexplorationof comparing tactile and symbolic numerosities that are close in
theamountofobjectsinthehand(Plaisieretal.,2009;Plaisierand distance. Furthermore, if finger counting habits affect this ana-
Smeets,2011).Thatis,thereisincreasingevidencethatnumerosity log representation of numerical magnitude, both starting hand
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Krauseetal. Afeelingfornumbers
preferencesaswellasspecificfingerpreferencesshouldmodulate ofstimulatedfingerswasequalordifferenttothenumericalsize
across-modalnumerositydistanceeffect. of thevisuallypresenteddigit.Responsesweregivenverballyby
uttering“Tee”(when the numerosities were identical) or“Toh”
METHOD (when the numerosities were different). Since voice onset times
Participants servedasdecisiontimemeasures,wedecidedtouseverbalutter-
Twenty-fourstudents(fivemale,twoleft-handed)between17and ancesforwhichthefirsttransientisphonologicallythesame.The
33yearsofage(mean=21.33,SD=3.61)participatedinthestudy targetstimuli(tactileandvisual)disappearedassoonasaverbal
in return of C5 or credit points. All of them reported to have responsewasgivenandablankscreenwaspresentedforavari-
normalorcorrected-to-normalvision. abletimebetween1000and1500ms.Nofeedbackwasgivenfor
erroneousresponses.Thenexttrialstartedaftertheexperimenter
Setup classifiedgivenresponses.
Participantswereseatedinfrontofatablewithacomputerscreen
(viewing distance approximately 60cm) and two custom-made Design
tactile stimulation devices (one for each hand; see also van Ede Theexperimentconsistedoffourblocks.Eachblockcontained128
etal.,2010),eachconsistingoffivepiezoelectricBraillecells(Metec trials(tworepetitionsofallpossiblecombinationsofeightstim-
AG,Stuttgart,Germany).EachBraillecellhadeightpins,arranged ulationpatternsontwohandsandfourvisuallypresenteddigits).
intwogroupsoffour,whichcanberaisedandloweredforabout Alltrialswerepresentedinrandomizedorder.Thedurationofthe
1mm. The tactile stimulation devices were each placed into a experimentwasapproximately30min.
wooden, sound-shielded box on the table in front of the par-
ticipant, such that he or she could neither see the hands being RESULTS
stimulated,norhearmechanicalnoisefromthestimulation.The Fingercountingpreferences
orientationofthetactilestimulationdeviceswithintheboxeswas The analysis of the finger counting questionnaire yielded that
suchthatparticipantscouldplacetheirhandsinacomfortablehor- 58.3% of the participants preferred to start counting with their
izontallyorientedrestingposition.Adynamicmicrophoneanda left and 41.7% with their right hand. Twenty-one participants
custom-made voice-key device was used to record voice-onsets. reportedatypicalunimanualcountingpatternforWesternsub-
Theexperimentwascontrolledusingcustom-madesoftware.The jects and to start counting with the thumb. One participant
experimenter was seated out of the participants’vision at a sec- reported a counting pattern that could not be classified accord-
ondtableandusedakeyboardtoenterwhichverbalresponsewas ing to existing categories of starting hand and preferred finger
given. sequence (cf. Lindemann et al., 2011) and therefore had to be
excluded from the analysis. The other two participants started
Material countingwiththepinkieandcountedinsuccessiveordertothe
Visualtargetstimulicomprisedthedigits“1,”“2,”“3,”and“4”pre- thumb.Thereportedfingercountingpatternwasusedtoclassify
sentedinalightgraycolorinfrontofadarkbackground.Tactile thestimulatedsetof fingersintocountinghabitcompatibleand
targetstimuliconsistedofthesimultaneousstimulationofoneto incompatiblesetsforallparticipants.Thatis,for21participants
fourfingersofeithertheleftorrighthand.Toexaminetheimpact the medial fingers (thumb,index finger,middle finger,and ring
ofthecountinghabits,alwaysonetofoursuggestivefingerswere finger) were classified as counting habit compatible and the lat-
stimulatedstartingwitheitherthethumborpinkie.Thatis,there eralfingers(pinkie,ringfinger,middlefinger,andindexfinger)as
were in total eight patterns of stimulation for each hand: four countinghabitincompatible,whilefortwoparticipantsthelateral
medialfingersetsinwhichthenumberofstimulatedfingerswas fingerswereclassifiedascountinghabitcompatibleandthemedial
startedwiththethumb(1=[Thumb],2=[Thumb,IndexFinger], fingersascountinghabitincompatible.
3=[Thumb, Index Finger, Middle Finger], 4=[Thumb, Index
Finger,Middle Finger,Ring Finger]) and four lateral finger sets Numerositycomparisons
startingwiththepinkie.(1=[Pinkie],2=[Pinkie,RingFinger], Responses that deviated more than three SD from the mean
3=[Pinkie,RingFinger,MiddleFinger],4=[Pinkie,RingFinger, responsetimeofeachparticipant(anticipatoryresponses:0.04%;
MiddleFinger,IndexFinger]).Dependingonthereportedfinger slowresponses:1.63%)wereexcludedfromfurtheranalysis.Erro-
countingpreferenceseachstimulationpatterncouldbeclassified neousresponsesoccurredin7.37%ofallremainingtrialsandwere
aseitherfingercountingcompatibleorincompatible. excludedfromtheresponsetimeanalysis.
Individual finger counting habits and starting preference of Medianresponsetimesanderrorswereeachenteredinasepa-
eachparticipantweredeterminedbyafingercountingquestion- rate repeated measures ANOVA with the within-subject factors
naire(Lindemannetal.,2011). Semantic Distance (0, 1, 2, 3), Set of Fingers (counting habit
compatible,countinghabitincompatible),Hand(left,right),and
Procedure the between-subject factor Starting Hand (left,right). Reported
Each trial began with the presentation of a fixation cross for degreesoffreedomfortheFstatisticswereHuynh–Feldtcorrected,
500ms, followed by the simultaneous onset of the visual and whennecessary.
tactile target stimuli. Tactile stimulation consisted of a repeated In line with our hypothesis, the reaction time analysis
switchingbetweenraised(20ms)andlowered(30ms)statesofall revealed a significant main effect of Semantic Distance, F(1.84,
pins.Participantswereinstructedtodecidewhethertheamount 38.73)=25.03,p<0.001,η2 = 0.54, showing an interaction of
p
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Krauseetal. Afeelingfornumbers
tactile and symbolic numerosities. Responses were faster for a
numerical distance between tactile and symbolic numerosity of
3 compared to a distance of 2,t(22)=−4.21,p<0.001,as well
1400
asforadistanceof2comparedtoadistanceof1,t(22)=−6.06,
p<0.001.Therewasnosignificantdifferencebetweenadistance
of 1 and a distance of 0 (same numerosity in both modali- s)
m
ties), t(22)=1.52, p=0.14 (see Figure 1). There was a signif- (1200
icant main effect of Set of Fingers, F(1, 21)=11.55, p<0.01, RT
η2 = 0.36, reflecting shorter reaction times when stimulating n
p a
e
the counting habit compatible fingers (i.e., for most partici- M
pants starting from thumb to pinkie; 1131ms), compared to 1000
the counting habit incompatible fingers (i.e., for most partici-
pants starting from pinkie to thumb;1185ms). The main effect
of Hand did not reach significance, F(1, 21)=0.41, p=0.53,
800
η2 = 0.02. Interestingly, the factors Semantic Distance and 0 1 2 3
p Semantic Distance
Setof Fingersinteractedsignificantly,F(3,63)=8.80,p<0.001,
η2 = 0.30. Post hoc F-tests showed a stronger effect of Seman-
p FIGURE1|Thecross-modalsemanticdistanceeffectbetweentactile
tic Distance for the counting habit incompatible finger stimu-
andsymbolicnumerosities.Thelargerthesemanticdistancebetween
lations, F(2.20, 46.09)=26.25, p<0.001, η2p = 0.56, than for bothnumerosities,theshorterthemeanresponsetime.Errorbars
the compatible stimulations, F(2.01, 42.17)=15.09, p<0.001, represent95%within-subjectconfidenceintervals(cf.LoftusandMasson,
η2 = 0.42. No significant effects were observed for the interac- 1994).
p
tionsSemanticDistance×Hand,F(2.41,50.61)=2.41,p=0.09,
η2 = 0.10,SemanticDistance×StartingHand,F(3,63)=0.02,
p 63)=1.71,p=0.17,η2 =0.08,SetofFingers×Hand×Starting
p=1.0,η2 = 0.001,theSetof Fingers×Hand,F(1,21)=0.04, p
p Hand,F(1,21)=0.3.70,p=0.07,η2 = 0.15,andSemanticDis-
p=0.84, η2p = 0.002, Set of Fingers×Starting Hand, F(3, tance×Set of Fingers×Hand×Staprting Hand,F(3,63)=1.82,
63)=0.20, p=0.66, η2p = 0.01, Hand×Starting Hand, F(1, p=0.15,η2p =0.08.
21)=1.08,p=0.31,η2 = 0.05,SemanticDistance×SetofFin-
p
gers×Hand,F(3,63)=0.61,p=0.61,η2 = 0.03,SemanticDis-
p DISCUSSION
tance×SetofFingers×StartingHand,F(3,63)=0.32,p=0.81, As hypothesized, we found a cross-modal numerosity distance
η2p = 0.02, Semantic Distance×Hand×Starting Hand, F(3, effectinthemagnitudecomparisontaskwhenparticipantswere
63)=1.90,p=0.14,η2 =0.08,SetofFingers×Hand×Starting instructedtocomparetactilepresentednumerositieswithsymbol-
p
Hand,F(1,21)=0.06,p=0.81,η2 = 0.003,andSemanticDis- icallypresentednumerosities.Thatis,participantsbecamefaster
p
tance×Set of Fingers×Hand×Starting Hand,F(3,63)=0.06, and made fewer errors to judge the difference between tactile
p=0.98,η2 =0.003. andsymbolicnumerosities,whenthesemanticdistancebetween
p
The error analysis also revealed a significant main effect of both numerosities was increased. This finding suggests that tac-
SemanticDistance,F(2,42)=21.02,p<0.001,η2 =0.50.Thatis, tilenumerositiesaremappedtothesameanalogrepresentationof
p
participantsmadefewererrorsforanumericaldistancebetween magnitudeassymbolicnumerosities.
tactile and symbolic numerosity of 3 compared to a distance of Whilestartingpreferencesdidnotmodulatethecross-modal
1, t(22)=−4.58, p<0.001, as well as for a distance of 2 com- distanceeffect,itwasmodulatedbythesetoffingersstimulated.
pared to a distance of 1, t(22)=−5.10, p<0.001. There was Interestingly, the effect was stronger for counting habit incom-
no significant difference between a distance of 1 and a distance patiblefingersetsthanforcountinghabitcompatiblefingersets.
of 0, t(22)=0.76, p=0.46. No main effects were observed for This appears counter-intuitive as one would have expected the
thefactorsSetof Fingers,F(1,21)=0.02,p=0.89,η2 = 0.001, exactoppositepatterniffingerrepresentationswereconnectedto
p
and Hand,F(1,21)=1.86,p=0.19,η2 = 0.08. There were no ananalognumericalmagnituderepresentation,thatis,astronger
p
significant effects for the interactions Semantic Distance×Set effect for counting habit compatible finger sets. Furthermore,it
of Fingers, F(1.52, 32.14)=0.39, p=0.63, η2 = 0.02, Seman- has to be noticed that the vast majority of our subjects showed
p
ticDistance×Hand,F(2.27,47.66)=1.94,p=0.15,η2 = 0.09, a prototypicalWestern finger counting habit (Lindemann et al.,
Semantic Distance×Starting Hand, F(3, 63)=0.44,pp=0.73, 2011)andstartedcountingwiththemedialfingersfromthumb
η2 = 0.02,theSetof Fingers×Hand,F(1,21)=0.16,p=0.69, topinkie.Consequently,thedissociationbetweencountinghabit
p compatibleandincompatiblefingersetsgoesinthepresentstudy
η2 = 0.008, Set of Fingers×Starting Hand, F(1, 21)=0.22,
p along with the dissociation between medial (i.e., starting from
p=0.65, η2 = 0.01, Hand×Starting Hand, F(1, 21)=0.11,
p thumb)andlateralfingers(i.e.,startingfrompinkie),whichseems
p=0.74,η2p =0.005,SemanticDistance×SetofFingers×Hand, tobeaproblematicconfoundfortheinterpretationofourfind-
F(1.87, 39.32)=0.28, p=0.75, η2 = 0.01, Semantic Dis- ings.Consequently,itremainsunclearif thedifferencesbetween
p
tance×SetofFingers×StartingHand,F(3,63)=0.42,p=0.74, thestimulatedfingersetsandthemodulationof thenumerosity
η2 = 0.02, Semantic Distance×Hand×Starting Hand, F(3, distance effect were driven by differences in the finger counting
p
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Krauseetal. Afeelingfornumbers
preferences or whether they merely reflected differences in the finger is stimulated, while when seeing a 4, one would need to
hand physiology between the medial and lateral finger sets and wait until all four finger have been stimulated). Tactile stimula-
resulting differences in touch acuity and cortical representation tion always started with the stimulation of a single finger.After
(cf.Elbertetal.,1995).Tobemoreprecise,amoredevelopedcor- each 100ms the next finger in the sequence was added to the
ticalrepresentationofthemedialfingerscouldaccountforafaster stimulation.Whenallfingerswereaddedthestimulationcontin-
and more precise detection of a tactile stimulation of these fin- ued on all fingers until a total stimulation time of 600ms was
gers,comparedtothelateralfingerswithalessdevelopedcortical reached.
representation. To specifically investigate the influence of finger
counting habits in our setting,independent of such physiologi- RESULTS
cal differences,we conducted a second experiment in which the Fingercountingpreferences
samesetof fingerswassequentiallystimulated.Importantly,the Theanalysisof fingercountinghabitsyieldedthat57.1%of the
type of sequence and direction of the tactile stimulations, not participantspreferredtostartcountingwiththeirleftand42.9%
thesetof fingers,definedthecompatibilitywithfingercounting withtheirrighthand.Crucially,allparticipantsreportedtostart
habits. countingwiththethumb.
EXPERIMENT2 Numerositycomparisons
Thesecondexperimenttestsapotentialinfluenceoffingercount- Erroneous responses (8.46%) as well as responses that deviated
inghabitsforthedetectionandrepresentationoftactilenumerosi- morethanthreeSDfromindividualmeanresponsetimes(only
ties.Sinceitcannotbeexcludedthattheeffectofthesetoffingers fast responses: 0.22%) were excluded from the response time
in Experiment 1 might be driven by physiological differences, analysisofthenumerositycomparisons.Sinceallinvestigatedpar-
Experiment 2 aimed to introduce finger counting compatible ticipantsstartedcountingwiththethumb,tactilestimulationsin
and incompatible tactile numerosities while keeping the set of forwarddirectioncouldbeconsideredasfingercountingcompat-
stimulated fingers constant. This has been achieved by sequen- ibleandbackwardstimulationsasfingercountingincompatible.
tialstimulationsintwodifferentdirections;eitherstartingfrom Errorsandmedianresponsetimeswereeachenteredintoasep-
the thumb or starting from the ring finger. If finger counting araterepeatedmeasuresANOVAwiththewithin-subjectfactors
habits influence the analog representation of numerical magni- Semantic Distance (0, 1, 2, 3), Direction of Stimulation (finger
tude,participantsthatstartcountingwiththethumbareexpected countingcompatible,fingercountingincompatible),Hand(left,
toshowadifferentcross-modalnumerositydistanceeffectifthe right),andthebetween-subjectfactorStartingHand(left,right).
sequenceofstimulationwasnotcompatibletotheirdirectionof ReporteddegreesoffreedomfortheFstatisticswereHuynh–Feldt
counting. correctedwhennecessary.
AsinExperiment1,theresponsetimeanalysisrevealedasig-
METHOD nificantmaineffectof SemanticDistance,F(2.01,52.21)=8.09,
Participants p<0.001, η2 = 0.24, confirming our main hypothesis of
p
Twenty-eightstudents(eightmale,oneleft-handed)between18 an interaction of tactile and symbolic numerosities. That is,
and25yearsofage(mean=20.07,SD=2.37)participatedinthe responses were faster for a numerical distance between a tac-
study in return of C5 or credit points. None of them partici- tile and symbolic stimulus of 3 compared to a distance of 1,
pated in Experiment 1.All of them reported to have normal or t(27)=−4.73, p<0.001, as well as for a distance of 2 com-
corrected-to-normalvision. pared to a distance of 1, t(27)=−4.36, p<0.001. There was
no significant difference between a distance of 1 and a dis-
Setupandmaterial
tanceof0(samenumerosityinbothmodalities),t(27)=−0.29,
ThesetupandmaterialwereidenticaltothatofExperiment1.The
p=0.77 (see also Figure 2). There was only a trend for an
experimentwascontrolledusingthesoftwareExpyriment(Krause
effect of Direction of Stimulation, F(1, 26)=4.10, p=0.053,
andLindemann,2012).Participantswereaskedtoindicatestart- η2 = 0.14, with descriptively slightly shorter reaction times
ingpreferenceandspecificfingercountinghabits(cf.Lindemann p
for finger counting compatible stimulation sequence (732ms)
etal.,2011).
compared to incompatible stimulations (739ms). That is, in
contrast to Experiment 1, we did not observe a reliable advan-
Procedureanddesign
tageof finger countingcompatiblestimulations. Nomaineffect
The procedure and design were similar to Experiment 1, with
of the factor Hand was observed, F(1, 26)=0.15, p=0.70,
twoexceptions.First,tactilestimuliconsistedofastimulationof
one to four fingers (1=[Thumb] to 4=[Thumb,Index Finger, η2p = 0.01. Importantly, there was no interaction between the
factors Semantic Distance and Direction of Stimulation, F(3,
Middle Finger,Ring Finger]). Crucially,all fingers were sequen-
78)=0.12,p=0.95,η2 = 0.01, showing that, unlike in Exper-
tiallystimulatedintwodifferentdirections:aforwarddirection, p
iment 1, the numerical distance effect was not modulated by
startingfromthethumb,andabackwarddirection,startingfrom
fingercountingcompatibility.Nosignificanteffectswereobserved
the last finger ending with the thumb. Second, the onset of a
for the interactions Semantic Distance×Hand, F(3, 78)=1.23,
visual stimulus was equivalent to the offset of the tactile stim-
p=0.31, η2 = 0.05, Semantic Distance×Starting Hand, F(3,
ulation. This was done to ensure that response times were not p
confoundedwithdifferencesinsequencelength(e.g.,whensee- 78)=0.75, p=0.52, η2p = 0.03, Direction of Stimulation×
ing the digit 1, a response could already be given after one Hand, F(1, 26)=0.12, p=0.73, η2 = 0.01, Direction of
p
www.frontiersin.org January2013|Volume4|Article7|5
Krauseetal. Afeelingfornumbers
0.07, Semantic Distance×Direction of Stimulation×Starting
900 Hand, F(3, 78)=0.62, p=0.61, η2 = 0.02, Semantic Dis-
p
tance×Hand×Starting Hand, F(3, 78)=0.07, p=0.98, η2 =
p
0.003,andDirectionofStimulation×Hand×StartingHand,F(1,
800 26)=0.39,p=0.54,η2 =0.02.
p
s)
m
( DISCUSSION
RT700 Experiment2confirmedthefindingofthecross-modalnumeros-
n itydistanceeffectfromExperiment1.Again,theeffectwaspresent
a
Me inbothresponsetimesanderrorrates.
However,thecross-modalnumerositydistanceeffectwasnot
600
modulated by any finger counting preferences (Starting Hand
or Direction of Stimulation), as would have been expected, if
counting habits influence the analog representation of numer-
500
0 1 2 3 ical magnitude. We interpret this as evidence that a common
Semantic Distance
metric shared by the representation of tactile and symbolic
numerosityinformationreflectsamagnituderepresentationthat
FIGURE2|Thecross-modalsemanticdistanceeffectbetweentactile
is independent of finger representations and analog numeros-
andsymbolicnumerositiesforsequentialtactilestimulation.Mean
ity representations acquired while learning to count with the
responsetimesaresignificantlyshorterforalargesemanticdistanceof2
and3thanforasmallsemanticdistanceof1.Errorbarsrepresent95% fingers.
within-subjectconfidenceintervals(cf.LoftusandMasson,1994). In contrast to Experiment 1, in which finger counting com-
patibilityledtofasterresponses,butwasconfoundedwithhand
physiology,neitherthestimulationdirectionnorthestartingpref-
Stimulation×Starting Hand, F(1, 26)=0.16, p=0.69, η2 =
p erencesignificantlyinfluencedtheperceptionofthetactilestim-
0.01, Hand×Starting Hand, F(1, 26)=0.12, p=0.73, η2p = ulus. While there was a trend for a main effect of Direction of
0.01,SemanticDistance×DirectionofStimulation×Hand,F(3, StimulationnomaineffectforStartingHandcouldbeobserved.
78)=0.83, p=0.48, η2p = 0.03, Semantic Distance×Direction Thus, while counting habits do not influence a shared magni-
of Stimulation×Starting Hand,F(3,78)=1.07,p=0.37,η2 = tuderepresentation,theymighthaveamarginalinfluenceonthe
p
0.04,SemanticDistance×Hand×StartingHand,F(3,78)=0.85, perceptionofatactilestimulus.
p=0.47,η2 = 0.03,DirectionofStimulation×Hand×Starting
p
Hand, F(1, 26)=3.35, p=0.08, η2 = 0.11, and Semantic Dis- GENERALDISCUSSION
p
tance×Direction of Stimulation×Hand×Starting Hand, F(3, The current study demonstrates an interference between fin-
78)=2.30,p=0.08,η2 =0.08. gers and numbers on the level of analog numerical magnitude
p
TheerroranalysisrevealedasignificantmaineffectofSeman- representations.Intwoexperimentsweinvestigatedtherelation
tic Distance,F(2.28,59.17)=30.45,p<0.001,η2 = 0.54, with betweentactileandsymbolicnumerosities,andtheinfluenceof
p
fewer errors for a distance of 3 compared to a distance of 1, fingercountinghabitsthereon.Ourdataprovidefirstevidencefor
t(27)=−4.07,p<0.001,aswellasadistanceof2comparedtoa theexistenceofacross-modalsemanticdistanceeffectinpartici-
distance of 1, t(27)=−4.27, p<0.001. The difference between pantscomparingtactilepresentednumerositieswithsymbolically
a distance of 1 and a distance of 0 was significant as well, presented numerosities. More specifically, responses were faster
t(27)=−2.93,p<0.01.Nosignificantmaineffectswereobserved andlesserror-pronewhenjudgingtwodistantnumerosities(e.g.,
forthefactorsDirectionofStimulation,F(1,26)=1.58,p=0.22, 1 and 4) than when judging two close-by numerosities (e.g., 1
η2 = 0.06, and Hand, F(1, 26)=0.44, p=0.51, η2 = 0.02. and2).
p p
The 4-way interaction Semantic Distance×Direction of Stimu- Importantly, all numerosities used in the current study were
lation×Hand×Starting Hand was significant, F(3, 78)=3.28, withintherangeofsubitizingandarethusassumedtobeperceived
p<0.05,η2 = 0.11.Sinceourhypothesesareindependentfrom directlyandaccuratelywithoutrelyingonaserialcountingprocess
p
thisobserved4-wayinteractionbetweenallfactors,wedidnotfur- (Riggs et al.,2006; Plaisier and Smeets,2011).We can therefore
theranalyzeandinterpretthiscomplexeffect.Therewerenosig- assumethatourresults(atleastinExperiment1,wherethestimu-
nificanteffectsfortheinteractionsSemanticDistance×Direction lationwasnon-sequential)arenotmediatedbyverballycounting
ofStimulation,F(2.27,59.07)=1.39,p=0.28,η2 =0.05,Seman- thestimulatedfingers.Rather,sincethenumericaldistanceeffect
p
ticDistance×Hand,F(2.26,68.19)=1.07,p=0.36,η2 = 0.04, hasbeenconsistentlyinterpretedtoreflectarepresentationalover-
Semantic Distance×Starting Hand, F(3, 78)=0.13,pp=0.94, lapbetweenneighboringitemsonananalogcontinuum(Moyer
η2 =0.01,theDirectionofStimulation×Hand,F(1,26)=1.81, andLandauer,1967;Restle,1970),ourresultssuggestthattactile
p presentednumerositieswereautomaticallymappedontothesame
p=0.19,η2 = 0.07, Direction of Stimulation×Starting Hand,
p analogrepresentationassymbolicnumerosities.Thisinterpreta-
F(1,26)=0.74,p=0.40,η2 =0.03,Hand×StartingHand,F(1,
p tion receives further support by the fact that participants made
26)=0.34, p=0.56, η2p = 0.01, Semantic Distance×Direction a same-different judgment (and not a magnitude judgment),as
of Stimulation×Hand, F(2.43, 63.12)=2.09, p=0.12, η2 = it has been shown that the numerical distance effect resulting
p
FrontiersinPsychology|CognitiveScience January2013|Volume4|Article7|6
Krauseetal. Afeelingfornumbers
fromasame-differentjudgmentcruciallydependsonoverlapping effect between the sets of fingers found in Experiment 1, our
analogrepresentations(VanOpstalandVerguts,2011).Thus,the data does also not provide any evidence for this. We observed
factthatwefindanumericaldistanceeffectallowsustoexclude a stronger numerical distance effect for the fingers which are
the possibility that the comparison of the tactile and symbolic not used to represent the numerosities during counting. How-
numerositieswasdonebymerelycomparingverbalcodes(since ever,iffingerrepresentationswereindeedconnectedtoananalog
this would not have led to a distance effect). It is also unlikely numerical magnitude representation, one would have expected
thatthemagnituderepresentationof bothnumerositieswasnot the opposite, namely, a stronger numerical distance effect for
activated directly, but through a preceding verbal code, since it thosefingerscompatibletothisrepresentation.Consideringfur-
has been shown that already preschoolers use surface features thermore that no influence of finger counting habits on the
of numerical stimuli instead of a magnitude representation to numerical judgments could be found when the same set of
solve a same-different judgment,when available (Defever et al., fingers was stimulated in different sequential orders (Exper-
2012). This means that if a verbal code preceded a magnitude iment 2), it seems very likely that physiological differences
representationinoursetting,judgmentscouldhavealreadybeen between the medial and lateral sets of fingers were responsi-
solved on this more direct verbal level, without the need for a ble for the observed differences in the judgment latencies of
more abstract representation of the numerical magnitude. Cru- Experiment1.
cially,again,asame-differentjudgmentonthebasisofsuchverbal In contrast to our study, some previous studies reported an
codes would not have led to a numerical distance effect. Taken influence of finger counting habits on the processing of sym-
together,the current finding suggests that tactile and numerical bolic numbers (e.g., Di Luca et al., 2006; Di Luca and Pesenti,
numerositiesshareacommonanalogrepresentationofnumerical 2008). The question arises therefore why finger counting habits
magnitude. didnotaffectthecross-modalnumerositycomparisonasinves-
Thefindingof across-modalnumerositydistanceeffectisin tigated in the present paradigm. First, it is important to note
line with the notion of a generalized magnitude system (Walsh, that most of the existing literature demonstrated associations
2003),whichhypothesizesthatthebrainprocessesgeneralmag- betweenfingerpatternsandnumbersbymeansofafasterdetec-
nitude information according to a shared metric, independent tion or stronger number activations for canonical finger pat-
from the domain this magnitude information comes from. In terns.Theseeffectsmightbemediatedbyaperceptualfamiliarity
our study, magnitude information from two different modali- of canonical finger patterns. While we observed a similar pat-
ties (tactile, visual) and with two different notations (symbolic, ternoffacilitationinExperiment2wherestimulationsequences
non-symbolic) had to be processed and compared. The obser- compatible with the participants’ finger counting pattern were
vation that the judgment latencies and accuracies depended on detectedslightlyfasterandprocessedmorefluently,thiseffectwas,
the cross-modal numerosity distance suggests that both types however, not statistically significant. Second, the current study
of numerosity information were mapped onto the same ana- is one of the first to investigate the influence of finger count-
log magnitude representation which were then utilized for the ing habits on an analog representation of numerical magnitude
actual cognitive comparison. It has to be furthermore men- in the subitizing range. Following the literature on subitizing,
tioned that the current study is focusing on the processing of this should have resulted in a very automatic and fast activa-
small numerosities. It is therefore unclear whether visual and tionofthenumberconcept(Kaufmanetal.,1949).Theabsence
tactile numerosities share also cognitive codes for larger num- of any influence of finger counting habits under these circum-
bers. Taking into account the possibility that common rep- stances suggests that differently preferred patterns of fingers are
resentations are shaped while using the fingers to count, it not differently coupled to an analog representation of numer-
is an important open question for further research whether ical magnitude. Typical finger counting patterns might instead
thesecross-domainassociationsarealsopresentfornumerosities constitute an additional independent numerical representation
larger10. (see also Moeller et al., 2012 for a similar proposal) and rep-
The conclusion that processing of sensory and symbolic resent verbally and perceptually mediated associations between
numerosityinformationleadstoanactivationofcommonanalog posturesandnumbermeaningthatareacquiredwhilelearningto
codessupportstheideaof embodiednumerosities.Theembod- count.
ied cognition view claims that abstract cognitive concepts are Whilethepresenceof cross-modalnumericaldistanceeffects
“grounded”in sensorimotor experiences (Barsalou, 2008). That supports the view of an embodied representation of numeri-
is,thecontentof abstractconcepts,likenumbers,isassumedto cal magnitude, we argue that the fact that this phenomenon
become meaningful by being coupled to bodily representations is independent of acquired finger counting preferences shows
(Lindemannetal.,2009).Here,thecross-modalsemanticdistance that finger counting postures serve as the function of motor
effect reveals a direct relationship between tactile and abstract symbols and reflect probably the individuals’ cognitive strat-
numerosities and the presence of a magnitude metric shared by egy to offload numerical information (Lindemann and Krause,
bothmodalities.Representationsofsensoryexperiencesaboutsize 2012).
andnumerositymightthiswayprovideagroundingforthemean- Takentogether,thecurrentstudyprovidesevidenceforashared
ingof symbolicnumbersandmightthereforeplayacrucialrole metric for tactile and symbolic numerosities, as an instance of
inthedevelopmentofnumberconcepts. anembodiedrepresentationof numbers.Crucially,theunderly-
While we cannot entirely exclude that finger counting habits ing analog representation of numerical magnitude information
are responsible for the differences in the numerical distance appearedtobeindependentfromfingerrepresentations.
www.frontiersin.org January2013|Volume4|Article7|7
Krauseetal. Afeelingfornumbers
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