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

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it is conclusive that the number of inversion errors (indexing wrong order in transcoding) specifically predicted performance for carry problems while the less pure category of combination errors became a predictor for non-carry problems. Such an interpretation corroborates our argument that understanding the place-value structure of Arabic numbers should be of particular relevance in situations requiring place-value updating and/or integration. Taken together, the results of the present longitudinal evaluation clearly indicated early place-value understanding to be a highly predictive precursor competency of later arithmetic performance in general and in particular for the capability to integrate tens and units into the Arabic place-value system as required in basic arithmetic operations such as addition problems requiring a carry. Yet, at the same time intelligence was not considered to be a reliable predictor of task performance in any of our analyses. At a first look this might seem awkward. However, this observation is in line with recent data that also showed no reliable interrelation between intelligence and basic numerical tasks such as number magnitude comparison (De Smedt et al., 2009b; Halberda, Mazzocco, & Feigenson, 2008) as well as number reading (De Smedt et al., 2009b) and transcoding (Zuber et al., 2009). Most importantly, this finding seems invariant to the fact which intelligence scale was used. While Zuber and colleagues (2009) also used the CFT-1 and De Smedt and co-workers (2009b) employed a comparable matrices test (i.e., Raven’s standard progressive matrices, Raven, Court, & Raven, 1992), Halberda et al. (2008) measured intelligence by the WASI (Wechsler abbreviated scale of intelligence, Wechsler, 1999). Synced with the fact that there were no peculiarities during the testing sessions at any of the involved schools, the current data do not corroborate the notion of a direct link between intelligence and basic numerical concepts such as place-value understanding, thereby replicating previous results (De Smedt et al., 2009b, Halberda et al., 2008). 158
A word on the validity of the current study Although statistically reliable influences of place-value understanding in first grade on arithmetic performance in third grade were observed in the current study, the results tell little about the external validity of these influences. To evaluate the validity of the current results we ran an additional regression analysis involving third grade mathematics school grades as the dependent variable. Unfortunately, it was not possible to get the mathematics grades from all children due to reasons of data protection. Nevertheless, we were able to collect the mathematics grades of a subsample of 54 children. A stepwise regression analysis was conducted predicting the mathematics grade by the same predictors as incorporated in all other analyses with one exception. Due to the reduced sample we only differentiated between inversion related and inversion unrelated transcoding errors. The group of inversion related errors comprised all errors (including combination errors) that involved an error in digit order in at least one position, while all other errors were pooled in the category inversion unrelated errors. In line with our argument that place-value understanding may be a crucial building block for later arithmetic competencies it was observed that inversion related transcoding errors assessed in first grades were the only reliable predictor of mathematics grade two years later (R = .50, adjusted R 2 = .22; F(2, 51) = 8.59, p < .001, f 2 = .33; b = .40, t = 3.15, p < .01) whereas inversion unrelated errors did not account for any additional variance (b = .15, t = 1.13, p = .27). Based on this result we are confident that our tasks provide a reliable (see above) and valid estimate of children’s numerical competencies. The fact that the compatibility effect was not found to be a reliable predictor of mathematics grade (b = - .05, t = 0.38, p = .71) might be caused by the lower reliability of the magnitude comparison task as compared to the transcoding task. The reliability of the magnitude comparison task might be lower because it involved numbers up to 100 and was thus rather difficult in general. In this context, it has to be noted that by the end of grade one (i.e., the time of assessing our participants) children in Austria are supposed to just master the 159
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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
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 and 182: from basic numerical concepts to ar
Page 183: transcoding and/or who showed relat
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
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
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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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