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

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Thereby, the present data allowed for particularly pinpointing measures reflecting understanding of the place-value structure of Arabic numbers for both tasks to be most relevant for the prediction of overall arithmetic performance in later numerical development. (iii) Finally, examining the influence of specific numerical effects in first grade (e.g., compatibility effect, pure inversion errors) on corresponding effects in third grade arithmetic (e.g., carry effect, error rate in carry problems) revealed several meaningful interrelations. The size of the carry effect in third grade addition was predicted reliably by the first grade measure of CE functioning and the number of pure inversion transcoding errors: A higher number of pure inversion errors in first grade transcoding reliably predicted a more pronounced carry effect in third grade addition. Thus, flawed understanding of the place-value positions of tens and units in first grade is related to difficulties in later arithmetic performance in additions requiring a carry operation in which correct integration of tens and units is essential. We suggest that both variables tap the same underlying concept of placevalue understanding in a basic numerical task in first grade (i.e., pure inversion errors in transcoding) as well as in a complex numerical task in third grade (i.e., addition involving carry over). Thereby, the present data indicate that if the place-value concept is not mastered in early schooling difficulties in the successful application of this concept will evolve in later numerical development. To our knowledge, these are the first longitudinal data which directly evidence the importance of early place-value understanding. Additionally, the CE component of WM was also positively correlated with the size of the carry effect: the better the CE in first grade the larger the carry effect in third grade indicating an influence of WM on complex addition which has previously also been observed in adults (e.g., Imbo et al., 2007a; 2007c). The regression model predicting the number of errors committed in carry addition problems incorporated three reliable predictors: a) the number of pure inversion errors in transcoding, b) visuo-spatial WM span and c) the size of the compatibility effect in number comparison. So, children who had problems to identify tens and units successfully in 156
transcoding and/or who showed relatively larger interference effects due to a worse integration of tens and units in two-digit number comparison also committed particularly more errors on addition problems requiring a carry operation. Again, poor performance in a task specifically involving place-value updating and integration is predicted best by basic numerical competencies reflecting exactly these requirements. Moreover, a higher visuospatial WM span in first grade was associated with an increased number of errors for carry problems. This positive correlation is not in line with standard hypotheses as one would assume a larger WM span to be associated with better performance. We will elaborate on this finding below. Finally, for the number of errors for non-carry addition problems the number of combination errors in transcoding and the size of the (decade) distance effect turned out to be the only reliable predictors: In line with the results of the regression analysis for overall addition performance a larger distance effect in first grade was associated with fewer errors for non-carry addition problems indicating a major influence of the representation of number magnitude on easier problems. Furthermore, children with a higher number of combination errors also committed more errors in non-carry addition problems. Importantly, the fact that combination errors and not pure inversion errors were the best predictor of performance for non-carry problems may indicate that for these simpler problems demands on place-value understanding are not as specific as for carry problems. Obviously, place-value understanding is required for all two-digit addition problems. However, incorrect ordering of tens and units, as reflected by pure inversion errors, is particularly detrimental for carry problems: To correctly solve a carry problem the units must be summed first. Subsequently, the tens and the carry from the units must be summed to obtain the correct result. Contrarily, the order of computations is irrelevant for non-carry problems as there is no carry from the units to the tens. So, whether the tens or the units are summed first yields identical results as long as the position of the single digits within the place-value system is applied correctly. For this reason, 157
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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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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
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: from basic numerical concepts to ar
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
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
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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
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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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