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

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contrasted. In the current study it was observed that the model fit of the two-linear model was reliably better than that of the logarithmic model for Italian-speaking children (see also Moeller et al., 2009a for a more detailed description of identical results for the Germanspeaking sample). (b) Alternatively, one may also be interested in the results of separate analyses for one- and two-digit numbers as described above on an individual bases. (c) It is also possible to compute the optimal break-point of the two-linear model for each individual participant and then evaluate the distribution of these break-points. Only if the optimal break-points of the vast majority of participants fall close to the hypothetisized breakpoint the two-linear model should be considered. Again, this was given in the present data set: the majority of Italian-speaking participants had an optimal break-point around 10 (see also Moeller et al., 2009a for similar results concerning the German-speaking participants). However, as already pointed out in the introduction even in a situation where the data undoubtedly follow a two-linear function, a logarithmic model would provide a very good model fit and vice versa. Therefore, we evaluated this issue by comparing the distributions of individual breakpoints of the empirical data and 100 simulated participants produced by a 100% logarithmic model. Figure A, Panel I depicts the distribution of breakpoints as observed for the simulated data whereas the individual breakpoints empirically found for both Germanas well as Italian-speaking children are to be found in Figure A, Panels II and III, respectively. As can be observed from these figures the pattern differs considerably: For the simulated data the vast majority of breakpoints was found to be located between 13 and 17 with the median at 14.37 and the mean being very close at 15.39. Contrarily the majority of breakpoints for the empirical data was found to be located between 8 and 12 (median at 9.8 and 11.8 for Italianand German-speaking children, respectively; Means: 21.2 and 25.3, respectively) and thus 118
much closer to the proposed breakpoint at 10 reflecting two different representations for single- and two-digit numbers. On the other hand, we are not aware of any theoretical assumption suggesting differing representations of numbers below and above 14. Another interesting finding is that of the 100 individual breakpoints of the simulated data only 4 were to be found below 10 whereas the number of breakpoints below 10 was higher for the empirical data (German-speaking: 34; Italian-speaking: 42) of which the majority was located on 8 and 9 (German-speaking: 21; Italian-speaking: 26). N 80 Distribution of individual breakpoints I Figure A: Distribution of breakpoints for the data simulated by a logarithmic 60 model (Panel I), Germanspeaking (Panel II) and Italian- 40 speaking (Panel III) children 20 0 37 Breakpoint (rounded) N 60 Distribution of individual breakpoints II 40 20 0 37 Breakpoint (rounded) 119
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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
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eye fixation behaviour for the eval
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(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 140: and 40 is 10 times as large as the
Page 141 and 142: APPENDIX A A remark on how to evalu
Page 143: Splitting the overall interval in t
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
Page 191 and 192: of the probes unit and/or decade di
Page 193 and 194: 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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