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

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have claimed that basic numerical knowledge serves as an important precursor for later addition performance and arithmetic achievement in general (e.g., Butterworth, 2005; Dehaene, 1997), systematic investigation of this relationship has only began recently. In a cross-sectional approach, Holloway and Ansari (2009) found children’s early ability to compare symbolically coded numerical magnitudes to be a predictor of their mathematical skills. The authors assessed children’s ability to compare symbolic (i.e., digital notation) and non-symbolic (i.e., dot patterns) magnitudes. They found that only the distance effect observed in symbolic magnitude comparison reliably correlated with their performance in a standardized mathematical test. These findings provide direct empirical evidence that complex calculation performance relies upon basic numerical skills, such as magnitude understanding (see also Kaufmann, Handl, & Thoeny, 2003). However, apart from this important observation, the study is innovative for another aspect: Unlike the majority of previous studies (e.g., Booth & Siegler, 2008) Holloway and Ansari (2009) did not index one (numerical) representation by one numerical task (henceforth: task approach; e.g., indexing the quality of magnitude representation by overall error rate in a number comparison task). Rather, Holloway and Ansari (2009) used a specific numerical effect, i.e., the numerical distance effect, as a more stringent index for the underlying representation, i.e., number magnitude representation (henceforth: effect approach). As the distinction between task approach and effect approach is crucial for the current study and beyond, we will elaborate on that distinction a bit more. Consider the following as an example for the task approach: Dehaene and Cohen (1997) concluded that the magnitude representation of their patient MAR was impaired as he was moderately to severely impaired (i.e., exhibiting an abnormally high error rate) in a number of quantitative numerical tasks including magnitude comparison and number bisection amongst others. Thus, a specific task (in this case number magnitude comparison) is used to index a specific representation (in this case number magnitude representation). At first sight, this may seem convincing. However, it 128
has been shown that other representations influence task performance in such tasks as well. For instance, Nuerk, Weger, and Willmes (2005) have shown non-magnitude-related language effects in the magnitude comparison task whereas Nuerk, Geppert, van Herten, and Willmes (2002, see also Korvorst & Damian, 2008; Wood et al., 2008) observed reliable nonmagnitude effects of parity and multiplication fact knowledge in the number bisection task. On the other hand, following the effect approach an underlying cognitive representation (e.g., magnitude representation) is reflected by the size of a specific effect associated with this representation (e.g., the numerical distance effect). As the distance effect is one of the most robust effects in numerical cognition (observed even in non-human primates, e.g., Nieder, Diester, & Tudusciuc, 2006) its disappearance, reversion, or inflation can be used to indicate impairments of the underlying number magnitude representation. In this vein, Delazer and colleagues (Delazer, Karner, Zamarian, Donnemiller, & Benke, 2006) observed an increased distance effect to indicate impaired number magnitude processing capabilities (despite the fact that overall error rate in a magnitude comparison task did not seem conspicuous). Thereby, the effect approach allows for a more fine-grained evaluation of performance even in situations where overall performance measures (e.g., overall error rates) may be too unsubtle and thus undifferentiated. Therefore, focusing on a specific numerical effect within a task instead of overall performance in this task may provide additional information when investigating mastery of a particular underlying representation (see also Hoeckner et al., 2008; Korvorst & Damian, 2008; Nuerk et al., 2002; Wood et al., 2008). Such an effect approach has already been successfully applied to the domain of numerical development by Holloway and Ansari (2009). They observed that children who exhibited a comparatively larger symbolic distance effect (i.e., more errors when comparing two numbers close to each other than two numbers further apart, e.g., 3_4 vs. 1_6) scored lower on a standardized mathematics achievement test (see also De Smedt, Verschaffel, & Ghesquière, 2009a; Landerl & Kölle, 2009 for similar results). 129
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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
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 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: INTRODUCTION Dealing with (multi-di
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
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
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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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