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

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analyses Ebersbach and colleagues (2009) observed that the breakpoint of the two-linear representation was correlated to children’s counting abilities. The authors interpreted this finding to suggest two separate representations of numbers within and outside the children’s counting range with an identifiable change point reflected by the breakpoint of a two-linear fitting function. Picking up on the idea of an initially two-linear representation of number magnitude, Moeller and coworkers (2009a) argued that performance changes with age and experience can also be interpreted as an improvement in integrating the single digits’ magnitudes of tens and units in compliance with the place-value structure of the Arabic number system. That is, to correctly solve the number line task, one has to be aware that the distance between 0 and 40 is ten times as large as the distance between 0 and 4, as this is an attribute of the base-10 structure of the Arabic number system. However, when children have not yet acquired a comprehensive understanding of this relationship, they may consequently overestimate one-digit number intervals (i.e., misplacing them towards the right). Consequently, the remaining two-digit numbers (in the case of the 0-to-100 interval is assessed) have to be located in a compressed manner upon a relatively short segment of the hypothetical number line. This would suggest two linear representations in a number line task from 0-to-100 with a theoretically fixed breakpoint at 10: one for single digit numbers and one for two-digit numbers. In segmented regression analyses Moeller et al. (2009a) observed that a two-linear model with an assumed breakpoint at 10 (reflecting two separate representations for single- and two-digit numbers) fits the empirical data even better than a logarithmic model. Based on this intriguing finding, Moeller et al. (2009a) concluded that to estimate the magnitude of a given number in the number line task correctly, integration of the single digits’ magnitude of tens and units complying with the place-value structure of the Arabic number system is inevitable. In this view the magnitude of a two-digit number is not represented holistically but built up by integrating tens and units into a coherent representation: This decomposed account states that the magnitude of the constituting digits is 90
still represented even when only the overall magnitude of the number is task relevant; this has been demonstrated successfully in other tasks such as number magnitude comparison (see also Nuerk, Kaufmann, Zoppoth, & Willmes, 2004; Pixner, Moeller, Zuber, & Nuerk, 2009, for children data; Nuerk, Weger, & Willmes, 2001; 2004; Moeller, Nuerk, & Willmes, 2009b, for adult data; Nuerk & Willmes 2005, for a review) and has now also been applied to the number line task (cf. Moeller et al., 2009a). Taken together, Moeller et al. (2009a) specified the basic two-linear model as proposed by Ebersbach et al. (2008) by the assumption of a fixed breakpoint at 10 driven by theoretical consideration regarding the place-value structure of the Arabic number system as retained by decomposed processing of tens and units. Against this common background of the Ebersbach et al. (2008) and Moeller et al. (2009a) model one aim of the current study was to differentiate the notion of decomposed processing from the assumption of a holistic representation of overall magnitude as reflected by the logarithmic model. However, the implementation of the place-value system differs in important aspects between the number word systems of various languages regarding for instance the order in which tens and units are referenced in number words (see Comrie, 2005, 2006 for a more detailed discussion). Thus, one might ask whether these specific differences may affect the acquisition of the mental representation of number magnitude. For example, more complex number word systems might be expected to impede this acquisition process whereas transparent number word systems, in particular regarding place-value structures, might ease the acquisition. To further pursue this question, we will briefly review to what extent language specificities have been found to influence numerical understanding. Subsequently, it will be outlined in which way the notion of language-specific effects may affect results in a number line task. 91
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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
Page 65 and 66: 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
Page 71 and 72: The authors used a magnitude compar
Page 73 and 74: Such congruency is important as mos
Page 75 and 76: Procedure: After ten randomly chose
Page 77 and 78: compatibility effect for large unit
Page 79 and 80: 2004). So, increasing the maximum u
Page 81 and 82: 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: preschoolers aged 4). However, the
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 and 134: seem to be the origin of the perfor
Page 135 and 136: this system. In this case no langua
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
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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
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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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