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

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Implications for the theoretical understanding of numerical cognition As outlined above we observed particular language differences in estimation accuracy in the number-line task. In the following paragraph the implications of this finding concerning the theoretical conceptualization of number processing as proposed by the most influential model, i.e., the so-called Triple Code model (Dehaene & Cohen, 1995; 1997; Dehaene, Piazza, Pinel, & Cohen, 2003) shall be addressed. The Triple Code model postulates three distinct representation systems for numerical information that are directly interlinked: a visual system representing the visual identities of the digital input, a verbal system involving number words and arithmetic facts and a quantity system coding numerical magnitude information. Numerical input can directly access all three of the systems dependent on the format in which it is presented. The input information is then transcoded into the appropriate representation for solving the task. This means, depending on task demands one or more of the representation systems are involved. From the beginning, the representation of number magnitude in the quantity system was assumed to be organized as an internal MNL with numbers being arranged in ascending order from left to right upon this MNL (cf. Dehaene et al., 1993; Dehaene & Cohen, 1995). Consequently, number magnitude seems to be associated with physical space (see Bueti & Walsh, 2009; de Hevia, Vallar, & Girelli, 2008; Umiltà, Priftis, & Zorzi, 2009 for recent reviews on different aspects of spatial-numerical associations; Wood, Willmes, Nuerk, & Fischer, 2008; for a review and a meta-analysis). In this context, the number-line task intends to measure the nature and stability of (linear) spatial-numerical associations. How can language properties exert an influence on the spatial representation of number magnitude? According to the model, at least three processing steps would be required for number-line estimation: (i) access to the numbers to-be-estimated through the visual Arabic system; (ii) transmission to the spatial representation system where magnitude is processed analogously upon a mental number-line; and (iii) direct retrieval of the output from 108
this system. In this case no language-specific effects would be predicted. In order to explain the influence of language properties on the spatial representation system an interaction with the verbal system needs to be assumed. Such an interaction has been proposed by Nuerk et al. (2005) who reported language effects in a task as basic as magnitude comparison. Their results indicated that the verbal representation of numbers interacts with the magnitude representation, even for the case that numbers are presented in Arabic digital notation. Despite the fact that magnitude comparisons should be possible to perform without involving the verbal representations of the respective numbers, Nuerk et al. (2005) suggested that the Triple Code model does not per se exclude the possibility of a co-activation of the verbal representation even when it is not of primary relevance for the task at hand. Considering the current study, the data clearly corroborate the assumption that the verbal number representation influences the number line task, which is usually assumed to exclusively assess the spatial representation of number magnitude. A possible explanation of this influence in the present study may stem from the development of the neural circuitry underlying numerical cognition. Recent functional MRI studies indicated that the cortex areas associated with the three representational codes of numerical information are much less distinct in children than in adults (Kaufmann et al., 2006; Kaufmann et al., 2008; Rivera, Reiss, Eckert, & Menon, 2005). This means, for the same numerical task children activate a more wide-spread neural network possibly including activation of several representations in parallel, even when they are not primarily needed for solving the task. Thus, in number-line estimation the to-be-estimated numbers seem to co-activate their verbal word frame. In German-speaking children inverted two-digit number words on the verbal and non-inverted digital Arabic representation may both be elicited and thus compete for use. When the influence of the verbal representation is larger for one item it would necessarily determine a misplacement of the item on the hypothetical number-line. As such an influence would be 109
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The influence of the place-value st
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Ich möchte mich an dieser Stelle b
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ZUSAMMENFASSUNG Das wissenschaftlic
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Study 5: All for one but not one fo
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Classification performance ........
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The Arabic place-value structure an
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Large problem size > small problem
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ABSTRACT ..........................
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Study 2: On the cognitive instantia
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LIST OF FIGURES Section 1: General
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Section 4: The neurocognitive under
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LIST OF TABLES Section 1: General I
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Section 5: A computational model of
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Number, the simplest and most unive
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limited set of symbols. To achieve
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interexponential structuring princi
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(i) separating power and base dimen
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Abstract Code Model Verbal number C
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of the Arabic number system within
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previous studies on two-digit numbe
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eview), questions concerning repres
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esponses are easier (i) for problem
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2009c). Based on the framework of t
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scarce (but see Domahs et al., 2006
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y Studies 5 and 6 of the current th
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OVERVIEW OVER THE STUDIES INCORPORA
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a view was corroborated by a variet
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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.,
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Section 2 Task invariant influences
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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
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compatibility effect for large unit
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2004). So, increasing the maximum u
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proportions in earlier studies. The
Page 83 and 84: CONCLUSIONS In summary, the conclus
Page 86 and 87: Study 2 On the cognitive instantiat
Page 88 and 89: INTRODUCTION One of the probably mo
Page 90 and 91: eye fixation behaviour for the eval
Page 92 and 93: (see Rayner, 1998 for a review). Tr
Page 94 and 95: Procedure: After being seated with
Page 96 and 97: adjustment of the degrees of freedo
Page 98 and 99: Eye fixation behaviour TRT in ms To
Page 100 and 101: or the decade digits was found (393
Page 102 and 103: egressions were directed to the sec
Page 104 and 105: the requirement of a carry is deter
Page 106 and 107: In line with this prediction, total
Page 108 and 109: Instead, it seems to involve differ
Page 111 and 112: Section 3 On the importance of plac
Page 113 and 114: ABSTRACT The mental number line of
Page 115 and 116: preschoolers aged 4). However, the
Page 117 and 118: still represented even when only th
Page 119 and 120: specificity. Yet, in adults it has
Page 121 and 122: METHOD The reported experiment was
Page 123 and 124: child separately to obtain estimati
Page 125 and 126: indicated that Italian children’s
Page 127 and 128: striking similarity between the ori
Page 129 and 130: Inversion influences on estimation
Page 131 and 132: % overall Estimation Error 25 20 15
Page 133: seem to be the origin of the perfor
Page 137 and 138: two-linear model with one linear re
Page 139 and 140: and 40 is 10 times as large as the
Page 141 and 142: APPENDIX A A remark on how to evalu
Page 143 and 144: Splitting the overall interval in t
Page 145 and 146: much closer to the proposed breakpo
Page 147 and 148: the individual approach may be more
Page 149 and 150: Contrarily, data produced by a two-
Page 151 and 152: Study 4 Early place-value understan
Page 153 and 154: INTRODUCTION Dealing with (multi-di
Page 155 and 156: has been shown that other represent
Page 157 and 158: number system as formalized by its
Page 159 and 160: 124 10024, see Deloche & Seron, 19
Page 161 and 162: Please note that in contrast to Miu
Page 163 and 164: Roid, Prifitera, & Weiss, 1993; see
Page 165 and 166: effect-based approach has never bee
Page 167 and 168: (ii) In the magnitude comparison ta
Page 169 and 170: problem of capitalizing on differen
Page 171 and 172: Error rate in % Error Categories (c
Page 173 and 174: positively correlated with overall
Page 175 and 176: (ii) Effect approach analyses of pr
Page 177 and 178: whether the carry effect for additi
Page 179 and 180: Non-carry problems: For the percent
Page 181 and 182: from basic numerical concepts to ar
Page 183 and 184: 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
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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
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The developmental influence of plac
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correspond to the order of tens and
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Asian languages (e.g., Japanese, Ko
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indicating that decomposed processi
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REFERENCES Banks, W. P. (1977). Enc
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Karmiloff-Smith, Annette (1996). Be
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Pixner, S. (2009). How do children
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