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

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linear model as well. In line with Moeller et al. (2009a) we argue that this corroborates the notion that performance improvements with age and experience may be driven in by progress in the children’s understanding of the base-10 place-value structure of the Arabic number system. In this context, the current results indicate that the process of mastering the base-10 property of the place-value structure may be language invariant as we observed no differences between German- and Italian-speaking children. On the other hand, despite the fact that the underlying processes may be the same in both languages language differences in the number word system nevertheless influenced the execution of these processes. Estimation performance for two-digit numbers was particularly worse in German-speaking children when confusing tens and units in integrating the individual digits into the overall number (due to their inverted order in German number words) results in large differences between the actual and the misunderstood number (i.e., seven and twenty 72 instead of 27). This indicated that although German-speaking children refer to the same underlying process of integrating tens and units into one coherent representation of a two-digit number, the fact that the input to this process is diverse (verbal vs. digital notation) takes its toll on estimation accuracy. To conclude, we want to emphasize the importance of translingual studies for investigating the development of numerical cognition. The present study indicated that there are language-independent as well as language-dependent influences. However, in both cases general as well as language specific processing within a language could only be identified with the reference to at least one other language. Without such a direct contrast between two languages with e.g., different number word systems it is almost impossible to dissociate language specificities from more general developments of numerical competencies. In this vein, future translinguistic studies provide the possibility to better understand which aspects of numerical development follow rather language invariant trajectories as well as which language-specific properties influence these trajectories and how this influence looks like. 114
APPENDIX A A remark on how to evaluate the logarithmic vs. two-linear issue There are different ways of evaluating whether the logarithmic or the two-linear model provide a better fit of the empirical data. In the following, these different approaches shall be summarized. Generally, (i) approaches averaging data across participants can be distinguished from (ii) approaches contrasting individual performance measures. Averaging across participants (fixed effects analysis): (a) After averaging the estimates for each to-be-located number across all participants a comparison may be of interest which of the two models (logarithmic vs. two-linear) provides a better fit of the empirical data (e.g., in terms of adjusted R 2 ). This can be done for instance by a stepwise multiple regression analysis in which both a linear as well as a logarithmic predictor is entered. By the logic of the stepwise regression analysis the predictor which accounts for the largest part of the variance is included in the model first. Any further predictor will only be incorporated into the model when it adds significantly to the variance explained by the final regression model. Thereby, it would be possible to identify which of the two predictors (i.e., linear or logarithmic) is incorporated into the regression model first and to evaluate whether the inclusion of the other predictor would add reliably to the explanatory power of the model or not. When running this stepwise regression analysis on mean estimates of all Italian-speaking children it could be observed that only the two-linear predictor was considered in the final regression model while the logarithmic predictor was not incorporated [R = .99, adj. R 2 = .97, F(1, 16) = 553.59, p < .001, b two-lin = .99, p < .001, b log = - .35, p = .18]. This observation was substantiated by the results of two forward regression analyses. When entering the logarithmic and the two-linear predictor successively the logarithmic predictor on its own was found to be a reliable predictor of estimation performance [b = .96, p < .001]. However, when entering the two-linear predictor the logarithmic predictor was no 115
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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
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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
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 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 40 is 10 times as large as the
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
Page 185 and 186: A word on the validity of the curre
Page 187 and 188: e adapted for the results of the pr
Page 189 and 190: 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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