The influence of the place-value structure of the Arabic number ...

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ZUSAMMENFASSUNG Das wissenschaftliche Interesse an der menschlichen Fähigkeit zur Repräsentation und Verarbeitung von Zahlen ist in den letzten Jahren stärker geworden. Die beteiligten Repräsentationen wie z.B. die der visuellen Zahlenform, des verbalen Faktenwissens und der analogen Größenrepräsentation, sowie deren neuronalen Korrelate werden in einem aktuellen Modell der Zahlenverarbeitung (z.B. Triple Code Model, Dehaene & Cohen, 1995; 1997) bereits sehr genau beschrieben. Jedoch fehlt bisher eine systematische Untersuchung zum Einfluss des dominierenden Organisationsprinzips unseres arabischen Zahlsystems, dem Platz x Wert System, auf die Zahlenverarbeitungfähigkeit. Die vorliegende Studie stellt dabei einen ersten Schritt in Richtung eines besseren Verständnisses von Platz x Wert Einflüssen, deren Auswirkungen auf die Endwicklung der Zahlenverarbeitungsfähigkeiten bei Kindern und ihrer neurokognitiven Grundlagen dar. Zusammenfassend lässt sich sagen, dass die einzelnen Studien der vorliegenden Arbeit übereinstimmend zeigen, dass Aufgaben übergreifende Einflüsse des Platz x Wert Systems bei Erwachsenen und Kindern nahelegen, dass (i) der mentale Zahlenstrahl Informationen zur Platz x Wert Kodierung von mehrstelligen Zahlen beinhaltet, (ii) dass sich die Verarbeitung von Platz x Wert Information kulturinvariant zu entwickeln scheint, aber doch Spracheinflüssen unterliegt, und (iii) dass sich auf neurofunktioneller Ebene spezifische neuronale Korrelate der Verarbeitung von Platz x Wert Information identifizieren lassen. Ergebnisse, die diese Schlussfolgerungen untermauern kommen dabei von einer Vielzahl numerischer Aufgaben, die von unterschiedlichen Stichproben bearbeitet wurden und deren Ergebnisse mittels verschiedener experimenteller Methoden erhoben wurden (u.a., Eye-Tracking und fMRT). Grundsätzlich lassen diese Ergebnisse den Schluss auf unfassende, Kultur übergreifende, aber von Sprache moderierte und entwicklungsrelevante Einflüsse der Platz x Wert Struktur des arabischen Zahlsystems auf die menschliche Fähigkeit zur Zahlenverarbeitung zu. II
TABLE OF CONTENTS Section 1: General Introduction GENERAL INTRODUCTION ............................................................................................... 3 The Arabic number system in a typology of notational systems ...................................... 4 Arabic place-value coding in current models of number processing ............................ 10 The Abstract Code Model ................................................................................................ 10 The Triple Code Model ................................................................................................... 11 An alternative account: decomposed processing of tens and units ................................ 14 Further evidence for decomposed processing of tens and units ..................................... 17 An independent representation of place-value structuring ............................................ 20 OVERVIEW OVER THE STUDIES INCORPORATED INTO THE PRESENT THESIS ................................................................................................................................... 27 Section 2: On task general influences of the place-value structure of the Arabic number system in human number representation .......................................................... 28 Study 1: Internal number magnitude representation is not holistic, either .................... 28 Study 2: On the cognitive instantiation of the carry effect – Evidence from eye-tracking ............................................................................................................................................. 28 Section 3: On the importance of place-value integration for arithmetic development 30 Study 3: Language effects on children’s non-verbal number line estimations ............... 30 Study 4: Early place-value understanding as a precursor for later arithmetic performance – a longitudinal study on numerical development. ........................................ 31 Section 4: The neuro-cognitive underpinnings of place-value integration ................... 32 III
Page 1 and 2: The influence of the place-value st
Page 3: Ich möchte mich an dieser Stelle b
Page 8 and 9: Study 5: All for one but not one fo
Page 10 and 11: Classification performance ........
Page 12 and 13: The Arabic place-value structure an
Page 14 and 15: Large problem size > small problem
Page 16 and 17: ABSTRACT ..........................
Page 18 and 19: Study 2: On the cognitive instantia
Page 20 and 21: LIST OF FIGURES Section 1: General
Page 22 and 23: Section 4: The neurocognitive under
Page 24 and 25: LIST OF TABLES Section 1: General I
Page 26: Section 5: A computational model of
Page 29 and 30: Number, the simplest and most unive
Page 31 and 32: limited set of symbols. To achieve
Page 33 and 34: interexponential structuring princi
Page 35 and 36: (i) separating power and base dimen
Page 37 and 38: Abstract Code Model Verbal number C
Page 39 and 40: of the Arabic number system within
Page 41 and 42: previous studies on two-digit numbe
Page 43 and 44: eview), questions concerning repres
Page 45 and 46: esponses are easier (i) for problem
Page 47 and 48: 2009c). Based on the framework of t
Page 49 and 50: scarce (but see Domahs et al., 2006
Page 51 and 52: y Studies 5 and 6 of the current th
Page 53 and 54: OVERVIEW OVER THE STUDIES INCORPORA
Page 55 and 56: a view was corroborated by a variet
Page 57 and 58:
would indicate influences of verbal
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2. Study 6: Impairments of the ment
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eplicate processes of unit-decade i
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determinants shall be reviewed. Mor
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Study 6: Hoeckner, S., Moeller, K.,
Page 67 and 68:
Section 2 Task invariant influences
Page 69 and 70:
ABSTRACT Over the last years, evide
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The authors used a magnitude compar
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Such congruency is important as mos
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Procedure: After ten randomly chose
Page 77 and 78:
compatibility effect for large unit
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2004). So, increasing the maximum u
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proportions in earlier studies. The
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CONCLUSIONS In summary, the conclus
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Study 2 On the cognitive instantiat
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INTRODUCTION One of the probably mo
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eye fixation behaviour for the eval
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(see Rayner, 1998 for a review). Tr
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Procedure: After being seated with
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adjustment of the degrees of freedo
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Eye fixation behaviour TRT in ms To
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or the decade digits was found (393
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egressions were directed to the sec
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the requirement of a carry is deter
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In line with this prediction, total
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Instead, it seems to involve differ
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Section 3 On the importance of plac
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ABSTRACT The mental number line of
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preschoolers aged 4). However, the
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still represented even when only th
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specificity. Yet, in adults it has
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METHOD The reported experiment was
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child separately to obtain estimati
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indicated that Italian children’s
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striking similarity between the ori
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Inversion influences on estimation
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% overall Estimation Error 25 20 15
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seem to be the origin of the perfor
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this system. In this case no langua
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two-linear model with one linear re
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and 40 is 10 times as large as the
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APPENDIX A A remark on how to evalu
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Splitting the overall interval in t
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much closer to the proposed breakpo
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the individual approach may be more
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Contrarily, data produced by a two-
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Study 4 Early place-value understan
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INTRODUCTION Dealing with (multi-di
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has been shown that other represent
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number system as formalized by its
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124 10024, see Deloche & Seron, 19
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Please note that in contrast to Miu
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Roid, Prifitera, & Weiss, 1993; see
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effect-based approach has never bee
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(ii) In the magnitude comparison ta
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problem of capitalizing on differen
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Error rate in % Error Categories (c
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positively correlated with overall
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(ii) Effect approach analyses of pr
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whether the carry effect for additi
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Non-carry problems: For the percent
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from basic numerical concepts to ar
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transcoding and/or who showed relat
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A word on the validity of the curre
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e adapted for the results of the pr
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Non-monotonous Model of the Relatio
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of the probes unit and/or decade di
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the place-value structure of the Ar
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Therefore, we believe that this stu
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APPENDIX A: Correlation tables of s
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Section 4 The neurocognitive underp
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ABSTRACT Mental arithmetic and calc
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manipulations cannot be distinguish
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numbers is assumed to be more preci
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correctly bisected by the second nu
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Table 1: Behavioural performance in
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were found bilaterally in the dorso
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Table 3: Brain areas activated by b
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portion of the superior frontal gyr
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Figure 3: fMRI signal for parametri
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The fronto-parietal network and the
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Finally, increased activation in th
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Inferior parietal cortex, the angul
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primarily necessary for solving the
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Tippet (1996) have shown that patie
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Study 6 Impairments of the mental n
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INTRODUCTION Patients suffering fro
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2005), triplets in which the interv
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Visual field assessment showed no s
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Table 1: Demographic and clinical d
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For 80 bisected triplets the factor
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Factorial Analyses (ANOVA and t-tes
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Errors: The ANOVA revealed a main e
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DISCUSSION The effects of neglect o
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decade digits but they even process
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strategic value of retrieving this
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APPENDIX A Test results of patients
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the middle the beneficial effect of
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229
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Study 7 Two-digit number processing
Page 259 and 260:
INTRODUCTION The internal represent
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all numbers the overlap for larger
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digit numbers were engaged. Additio
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single node in an ordered sequence
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modelling to evaluate the plausibil
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A Output Layer 11 99 11 #1 #2 99 In
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expected considering their magnitud
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the output of the right node would
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advantages for these numbers (see a
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Finally, a regression analysis only
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236) = 417.38, p < .001] and incorp
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Regression The stepwise regression
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compared numbers, unit distance, lo
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RT in ms 780 770 Compatible Number
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esponse latencies increased as dist
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Table 5: Predictors included in the
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Taken together, these results revea
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of overall distance necessarily req
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most important predictor in a multi
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activation of the output layer is a
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The nature of the representation of
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Thereby, possible interactions and
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decomposed model did not include an
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APPERNDIX B Development of connecti
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Figure C: Development of connection
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Figure E: Development of connection
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Interestingly, one might infer from
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Section 6 Summary 287
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may be determined by the presentati
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tens and units: German-speaking chi
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Study 6 examined the influence of t
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295
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GENERAL DISCUSSION In the last sect
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to-be-evaluated triplet crossed a d
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specific processing of place-value
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explanation of the decade consisten
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verbal recoding of tens and units d
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system and the ordinality of the me
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epresentation itself as well as its
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Furthermore, converging evidence is
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methodological problems. In most of
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jittering the positions of the sequ
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numbers within the triplets, proces
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Pollatsek, 1989) states that an obj
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computational models were programme
Page 349 and 350:
The developmental influence of plac
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correspond to the order of tens and
Page 353 and 354:
Asian languages (e.g., Japanese, Ko
Page 355 and 356:
indicating that decomposed processi
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REFERENCES Banks, W. P. (1977). Enc
Page 359 and 360:
Karmiloff-Smith, Annette (1996). Be
Page 361 and 362:
Pixner, S. (2009). How do children
Page 363 and 364:
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