Inversion Property in German
Number Words.
Univ.Doz.Dr.SilviaPixner
UMIT,Institutof Psychology,HallinTyrol,Austria
WhatistheinversionpropertyinGerman
numberwords?
25 à nottwentyfivebutfive andtwenty inGerman
234à twohundredfour andthirty
3456à threethousandfourhundred..
65 321à five andsixty thousand…
• NotonlyinGermanbutalsoinDutch,Danish,
Maltese,SlovenianandCzechlanguage
Arabicvs.RomanNotation
• ArabicNotation
à place-valuesysteme.g.32 vs.28
• RomanNotation
à symbol– valuesysteme.g.X vs.XIII
Numberwordtransparency
• 11(eleven,elf)and12(twelve,zwölf)but13
(thirteen,dreizehn /3 10/),14…
• InversionpropertybyteensinGermanbutalsoinEnglish
language
• inItalyonlybynumbersfrom11-16(undici /1 10/,
tredici..)butnotfrom17(diciasette /10 7/,
diciotto…)
• inSlovaklanguage:ownsuffixsedemnast (7)
• InHungary/Japanese:tizenegy /10 1/,tizenharom
/10 3/
Numberwordtransparency
differencebetweenteensandtens
• English:“fourteen”and“forty”
• German:“vierzehn /410/”and“vierzig /4 zig/”
• Slovak:“trinast /3 nast/”and“tridsat /3 dsat/”
• Japanese:“ju san /103/”and“san ju /310/”
Otherdifficulties:
• French
e.g.97quatre - vingt - dix - sept (4x20 107)
Swedishnumberwords?
Longitudinalstudyinthreecountries
German
sample
with
Inversion
Italian
sample
without
Inversion
Czech sample
with and without
Inversion
Curricula'sinthe3countries
• InItaly,AustriaandalsoCzechRepublic
• Firstgrade
• Numbersbetween0-20(30)
• additionandsubtraction
• Secondgrade
• Numbersbelong100
• additionandsubtraction
• multiplicationanddivision
• Thirdgrade
• Numbersbelong1000
• writtencalculationinallarithmeticoperations
Transcoding
• transformationfromonenumericalcode(verbal
numberword)toothernumericalcode(Arabic
notation)
• necessaryforcalculation
• Lochy andcolleagues(2003)describedatfirsttime
thedifficultywiththeinversionerrorsinchildren
Study1
• Participants:130Germanspeakingchildreninthe
firstgrade
• Material:
• 64Arabicnumbers(4singledigitnumbers,20
two-digitnumbersand40three-digitnumbers)
• verbalandvisuo-spatialworkingmemory
• centralexecutive
Lexicalerrors
(TaxonomyadaptedofDeloche andSeron,1982)
Lexicalerrorsinvolvethesubstitutionofalexicalelementby
anotherlexicalelement.
90à 91
25à 24
90à 19
Syntacticalerrors
Syntacticalerrorsareconsideredaserrorswhereelementsof
thenumberarecorrectlyproducedbutoverallnumber
magnitudeiswrong.
120à 10020
200à 2100
95à 59
205à 502
Combinationerrors
245à 200540
Methodologicalbenefit
• Nuerk andColleague(2005)shows,thatJapanese
childrenmade6xfewertranscodingerrorsas
Germanspeakingchildren
• itisthebenefitfromthetransparentnumberword
system?
• ormaybetheJapaneseteachermightfocusearlierand
moreintensivelyontheacquisitionoftheverbalnumber
wordsystem?
• theCzechlanguageallowsanalyzedtheinfluenceof
theinversionpropertyinthesamesample
Study2
• Participants:118Czechchildreninthefirstgrade
• Material:
• 64Arabicnumbers(4singledigitnumbers,20
two-digitnumbersand40three-digitnumbers)
• BlockAwithInversionandblockBwithout
Inversion->thesameoveragedifficulty
Results
• averageerrorrateintheinvertedform(blockA)
à 49,20%
• averageerrorrateinthenon- invertedform(block
B)
à 37,23%
Results
Whydoesgoodplace-valueunderstandingare
importantforcomplexarithmetic?
124+251=375
132+259=391
Distanceeffect
• Distanceeffect(Moyer&Landauer,1967)reflect
theintegrityofthementalnumberline.Sodigitare
comparedfaster,whentheoveralldistance
betweenislarger.
• 5and9arecomparedfasteras6and7
• Isnotrestrictedtoprocessingsingledigitnumbers
butratheralsobecomesevidentintwodigit
numberprocessing
• reflectthemoreholisticallyprocessingoftwodigit
numbers
Compatibilityeffect
Thecompatibilityeffectreflectthekindofprocessing
oftwodigitnumbers(Nuerk etal.,2001)
Overall Distance 15
42
ΛΛ
57
47
vs.
ΛV
62
Study3
• Participants:94Austrianchildrenintheelementary
school
• Material:
• 1.grade:transcoding,twodigitnumber
comparison(120items)
• 3.grade.additionperformance(48problem/24
withcarryand24withoutcarry)
Results
• inthefirststepwewereinterestedwhether
additionperformanceingeneralwasdeterminedby
thegeneraltranscodingperformanceandthe
performanceinthetwodigitnumbercomparison
task
• intheregressionanalysisweincludedadditionally
theIQ,verbalWM,visuo-spatialWMandCE
• onlythemagnitudecomparisontaskturnedonasa
reliablepredictorfortheadditioncompetencies2
yearslater
Results
• inthesecondstepmoredifferentialanalysisofthe
relevanceofspecificnumericaleffects(distance
effect,compatibilityeffectandpureinversion
errors)wouldbecomputed
• Distanceeffectasreflectingthenumbermagnitude
understanding
• Compatibilityeffectasreflectingtheintegratingof
decadeandunitdigit
• Pureinversionerrorsasreflectingofplacevalue
understanding
Results
• Thefinalregressionmodelincludedthreereliable
variables:thepureinversionerrors,compatibility
effectanddistanceeffectasreliableprecursorsof
thelateradditionperformanceinchildren
• Theresultsindicatedthatchildrenwhocommitted
morepureinversionerrorsinthefirstgradealso
exhibitedhigheroverallerrorsinadditiontask.
• Childrenwithrelativelylargecompatibilityeffectin
thefirstgradecommittedmoreerrorsinthe
addition
• Childrenwithlargedistanceeffectimpliedbetter
performanceintheaddition
Compatibilityeffect
positivevs.negativecompatibilityeffect
Overall Distance 15
42
ΛΛ
57
47
vs.
ΛV
62
Study4
• Participants:130Austrianchildreninthefirstgrade
• Material:
• 120twodigitnumberpairs/RTexperiment
• 40withindecadetrials(24an27)
• 80trialwith4differentdigits(42and57)
• Decadedistancewasmanipulated(4-8large)
• Unitdistanceconstant,inallconditionslarge
• Compatibilitycontrolledinthedesign(2x2)
Results
• 2(distance)x2(compatibility)ANOVA
• strongeffectofdecadedistanceà inaccuracyand
inlatencieschildrenarebetterinthelargedistance
condition
• reliablecompatibilityeffectà thechildrenare
fasteranderrorlessbycompatibletrialcompared
withincompatibleones
Mentalnumberline
• Relativelysmallernumbersareassociatedwiththe
leftside,whereasthelargernumbersareonthe
rightside
0 12345678910 ...
Mentalnumberline
Itissuggestedthatthecodingofnumbermagnitude inchildren
graduallychangefromlogarithmictolinear
100
0
7
26
Österreich
100
90
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
120
Twolinearrepresentationarebetterasa
logarithmic?
Languageeffectsonthenumberline
• Inthecontextofplacevalueintegration,Germanspeakingchildren´sestimationsshouldbe
particularlyerroneousfornumbersforwhichmixing
updecadesandunitsresultsinalargeestimation
error(e.g.82and28)comparedwithnumbers
leadingtoonlysmallestimationerror(e.g.54and
45)
Study5
• Participants:130Austrianand107Italianchildren
inthefirstgrade
• Material:
• thenumberlineestimationtaskwiththerange
from0to100wasused
• with18differentnumberswerepresentedin
visualform(Arabicnumbers)
Results
Takentogether:
• Theresultsindicatethatestimationaccuracyof
Germanspeakingchildrenisparticularlypoorfor
theitemsforwhichinversionplaysamajorrolein
theaccuraterepresentationofnumbermagnitude
• Theresultsshowstheimportanceoftranslingual
studiesalsoinnon-verbaltasks,suchasthenumber
linetasktobetterunderstandingofthenumerical
cognitionandtherelationtothelanguage
Takentogether:
• TheinversionpropertyintheGermannumber
wordsisparticularlydifficultfortheAustrian
(Germanspeaking)childrennotonlyintheverbal
tasksasthetranscodingbutalsoinnon-verbaltasks
asthetwodigitnumbercomparisonorthenumber
lineestimationtask
Actuallyresearch
• Influenceofthelanguageandthespatialabilities
ontheearlynumberprocessingcompetenciesby
smallchildrenfrom3to6years
• Differentcompetencieswouldbeinvestigatedsuch
ascounting,cardinality,wholepartunderstanding,
patternoffingersanddicesandspecialfocusinthis
studywetookofthezero– ontheothersidethe
vocabulary,spatialprepositions(ason,inbetween)
andquantifier(asmoreas,lessas)andspatial
competencieswouldbetestedinthesample
Actuallyresearch
• Inanotherworkwetrytodescribethefirststeps,
whatmadethewordorstoryproblemsdifficultfor
thechildren.
• Thefirstpilotstudyshowalotofspectacular
results.Infocuswealsotookthequantifiersmore
as,lessas,halfofanddoubleof…