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

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easily solve. As no backup strategies are necessary in such simple experiments in normally developing children, a larger distance effect was observed to be associated with worse access to the mental number line and – consequently – worse performance in complex arithmetic task. In contrast, consider now the studies by Rousselle and Noël (2007) as well as the present study. As laid out above, Rousselle and Noël (2007) argued that some dyscalculic children may have used back-up strategies even for simple single-digit number comparison. These back-up strategies increased error variance with respect to the distance effect and thus, the distance effect decreased (see Figure 2). Comparably, in the current study the comparison task involved two-digit numbers, of which not all have been taught at school by the time children were assessed in first grade. Nevertheless, most children could do the task. However, for such a complex task even normally developing children may have resorted to back-up strategies similar to that employed by dyscalculic children in more simple tasks. Notably, Nuerk et al. (2004a) report similar findings for children attending grades two to five. In particular, children with a higher error rate (in the number comparison task) were found to exhibit a smaller (decade) distance effect than children with a comparably lower error rate (see Fig. 4). Following the above argument, the children relying on backup-strategies may have boosted error variance and thereby, the distance effect diminished or even disappeared. Put differently, those children able to produce an analogue distance effect in first grade could represent twodigit numbers in an analogue way without major problems (or back-up strategies) and such early magnitude representation of two-digit numbers may corroborate future arithmetic development. To summarize, we suggest that the seemingly contradictory data can be integrated if one postulates a curvilinear relation between distance effect and arithmetic performance which relies on both, explained magnitude variance and disturbing error variance with the latter depending on interactions of task difficulty and individual capability (and especially the need for backup-strategies). 162
Non-monotonous Model of the Relation between the Distance Effect and Arithmetic Performance Arithemtic Performance Low numerical capability High task complexity Size of the Distance Effect High numerical capability Low task complexity Capability/complexity range Rousselle & Noel (2007) Current Study Capability/complexity range Holloway & Ansari (in press) Kaufmann & Nuerk (2008) Figure 2: A hypothetical curvilinear model of a non-monotone relationship between the distance effect and arithmetic performance (see text for details). It is suggested that the size of the distance effect might be determined by two factors: First, increasingly precise and automatic access to the magnitude representation (i.e., higher capability) decreases the explained variance of the distance effect. Second, the distance effect also depends on the error variance which is increased by backup strategies (e.g., counting) necessary for tasks of very high complexity. However, too much of such non-systematic error variance may diminish the distance effect or even make it disappear. The model in Figure 2 accounts for the seemingly antidromic correlations of the distance effect and arithmetic capabilities by assuming that the numerical capability as well as task complexity differ between the cited studies. In Rousselle and Noël’s (2007) study, capability is very low (dyscalculic children) while in our study relative task complexity is very high (two-digit number comparison for first graders). In this capability/complexity range, a higher distance effect is particularly associated with less error variance (e.g., less backup strategies) so that a larger distance effect indicates better numerical competencies and better arithmetic performance. Other studies (e.g., Holloway & Ansari, 2009) employed a less complex task (single-digit comparison) and the participants should have had higher individual capabilities (as older children were assessed). For such individually simple tasks, no backup strategies (which boost error variance) are to be expected in normally developing children. Therefore, a smaller distance effect might then indicate more efficient access to the magnitude representation leading to a negative relation between the distance effect and arithmetic performance in this capability/complexity range. Please note that this model remains to be tested empirically, however, at the moment it offers a possible explanation for seemingly contradictory results. Working memory and later arithmetic performance There was one other somewhat unexpected finding in our data. CE and visuo-spatial WM capacity were positively correlated with the carry effect or the error rate for carry 163
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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
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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
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
Page 185 and 186: A word on the validity of the curre
Page 187: e adapted for the results of the pr
Page 191 and 192: of the probes unit and/or decade di
Page 193 and 194: the place-value structure of the Ar
Page 195 and 196: Therefore, we believe that this stu
Page 197 and 198: APPENDIX A: Correlation tables of s
Page 199 and 200: Section 4 The neurocognitive underp
Page 201 and 202: ABSTRACT Mental arithmetic and calc
Page 203 and 204: manipulations cannot be distinguish
Page 205 and 206: numbers is assumed to be more preci
Page 207 and 208: correctly bisected by the second nu
Page 209 and 210: Table 1: Behavioural performance in
Page 211 and 212: were found bilaterally in the dorso
Page 213 and 214: Table 3: Brain areas activated by b
Page 215 and 216: portion of the superior frontal gyr
Page 217 and 218: Figure 3: fMRI signal for parametri
Page 219 and 220: The fronto-parietal network and the
Page 221 and 222: Finally, increased activation in th
Page 223 and 224: Inferior parietal cortex, the angul
Page 225 and 226: primarily necessary for solving the
Page 227 and 228: Tippet (1996) have shown that patie
Page 229 and 230: Study 6 Impairments of the mental n
Page 231 and 232: INTRODUCTION Patients suffering fro
Page 233 and 234: 2005), triplets in which the interv
Page 235 and 236: Visual field assessment showed no s
Page 237 and 238: 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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