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

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epresentation but rather progress in children’s understanding of the place-value structure that underlies performance improvements with age and experience. Moeller and colleagues (2009a) presented evidence indicating a better fit of the twolinear compared to the logarithmic model in various indices for German-speaking children only. However, it is known that the intransparent German number word system, in particular its inversion property, has a reliable influence on children’s numerical development (cf. Zuber et al., 2009). Therefore, it was still to be shown that the results reported by Moeller and coworkers (2009a) were not determined by this peculiarity of the German number word system. In the present study, we evaluated the spatial representation of number magnitude in 107 Italian-speaking first graders. As there is no inversion of tens and units in the Italian number word system, replicating the better two-linear as compared to a logarithmic fit would corroborate the notion of children’s mastery of the place-value structure of the Arabic number system to determine children’s spatial representation of number magnitude irrespective of differences in the number word systems. The current data support the latter assumption. The results of the Italian-speaking children mirrored those of the German-speaking ones in each and every aspect: (i) comparing logarithmic to two-linear fittings on an individual basis showed that the two-linear model provided a reliably better fit; (ii) the vast majority of children exhibited a breakpoint around ten, and (iii) the slope of the linear segment representing one-digit numbers was negatively correlated to the slope of the segment for two-digit numbers. Taken together, the results for the Italian-speaking children were identical to those previously observed for Germanspeaking children (cf. Moeller et al., 2009a). In our view, this data indicate that the mastery of the base-10 place-value structure of the Arabic number system to be a corner stone in the development of the MNL in different cultures. Even more, the present results imply that the general way in which children learn to represent numbers complying with the base-10 property of the place-value structure of the Arabic number system (i.e., the distance between 0 112
and 40 is 10 times as large as the distance between 0 and 4) is not influenced by language differences. Finally, Booth and Siegler (2008) also showed that performance in the number line task is associated with actual mathematics achievement and in particular the ability to solve unlearned problems. To substantiate these findings we evaluated whether the size of the individual mean estimation error in first grade predicted the mathematics mark two years later by the end of grade three for a subsample of 50 of the German-speaking children. And indeed, in line with the results of Booth and Siegler (2008) we observed that third grade mathematics mark became better, the smaller the number line estimation error was in first grade (r = .40; n = 50). Synced with above findings suggesting two separate representations of single- and twodigit numbers one could as well assume that as an alternative account to the logarithmic-tolinear shift, place-value understanding serves as a building block for later arithmetic competencies. SUMMARY AND CONCLUSIONS This study set off to gain new insights in the development of the spatial representation of number magnitude and the role of language properties on this representation system by evaluating performance of Italian- and German-speaking first grade children in a number-line estimation task. The latter languages were chosen as they differ in the organization of their number word systems. While the order of tens and units in German number words is reversed as compared to the digital notation (i.e., 27 seven and twenty), no such inversion is present in Italian. Recently, the inversion property of German number words was shown to pose specific problems to children when acquiring the Arabic notation of multi-digit numbers (Zuber et al., 2009). The present data are informative in two respects reflecting language independent as well as language dependent development: First, it was observed that comparable to previous results for German-speaking children estimation performance of Italian-speaking children was accounted for best by a two- 113
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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
Page 88 and 89: 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: two-linear model with one linear re
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 and 188: e adapted for the results of the pr
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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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