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

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Table 4: Regression model predicting the carry effect in third grade (Panel A), the number of errors for carry problems (Panel B) as well as the number of errors in non-carry problems (Panel C) on the basis of specific numerical effects A Predictor B 95% CI b Change in R 2 t sig. raw partial Constant - .43 - .81 – - .09 2.27 < .05 CE (1 st grade) .15 .03 – .27 .26 .04 2.47 < .05 .20 .25 Inversion errors (1 st grade) .25 .003 – .49 .21 .04 2.00 < .05 .14 .21 Distance effect (1 st grade) - - .06 0.56 .58 .05 .06 Compatibility effect (1 st grade) - - .02 0.14 .89 - .01 .02 B Predictor B 95% CI b Change in R 2 t sig. raw partial Constant - . 06 - .49 – .38 .26 .80 Inversion errors (1 st grade) .42 .21 – .62 .39 .14 4.03 < .001 .38 .39 Visuo-spatial WM (1 st grade) .10 .01 – .20 .20 .04 2.11 < .05 .14 .22 Compatibility effect (1 st grade) .10 - .004 – .20 .18 .03 1.91 < .05 ( * ) .23 .20 Distance effect (1 st grade) - - - .15 - 1.60 .12 - .13 - .17 C Predictor B 95% CI b Change in R 2 t sig. raw partial Constant .43 .30 – .55 6.76 < .001 Combination errors (1 st grade) .22 .08 – .35 .31 .10 3.20 < .01 .31 .32 Distance effect (1 st grade) - .23 - .49 – .02 - .18 .13 1.82 < .05 ( * ) - .17 - .19 Inversion errors (1 st grade) - - .10 - 0.94 .35 .21 .10 Compatibility effect (1 st grade) - - .11 - 1.07 .29 .23 .11 CI = Confidence interval; ( * ) = tested one-sided DISCUSSION The aim of the current study was to investigate the influence of early basic numerical competencies such as transcoding or number comparison on later arithmetic capabilities in typically developing children. Therein, we set out to resolve several limitations in the hitherto investigation of this developmental trajectory: (i) most of the existing evidence supporting this notion comes from studies employing a simpler task approach in which one particular representation is indexed by an overall performance measure (e.g., error rates) of one particular task. (ii) The stimulus sets used to assess basic numerical competencies in previous studies mostly comprised one-digit numbers only instead of multi-digit numbers (e.g., Holloway & Ansari, 2009). (iii) Consequently, when only single-digit stimuli were used examination of the influence of place-value understanding as an important precursor of later arithmetic ability was not possible in past research. (iv) Finally, a developmental trajectory 154
from basic numerical concepts to arithmetic has never been observed in a longitudinal approach to date. In addressing these issues our proceeding was three-staged: (i) We were interested whether overall performance in number comparison and/or transcoding as assessed in first grade were reliable predictors of overall addition performance in third grade. (ii) More specifically, attention was paid to whether mastery of specific basic numerical concepts in first grade (e.g., place-value understanding) as indexed by related numerical effects (e.g., compatibility effect) predicts overall addition performance two years later (see Holloway & Ansari, 2009, for a similar approach). (iii) Most specifically, the quality of such specific basic numerical effects (e.g., compatibility effect) serving as predictors for specific effects in later addition (e.g., carry effect) that can be attributed to the same underlying concept of placevalue understanding, was of interest. Our results were straightforward on each of these questions: (i) We clearly identified overall transcoding and number comparison performance in first grade to be precursor competencies of general arithmetic performance in third grade. (ii) When following the approach suggested by Holloway and Ansari (2009) we were not able to replicate their findings concerning the (decade) distance effect but took a successful next step in identifying precursors of arithmetic capability. On the one hand, we observed a more pronounced (decade) distance effect in first grade to be associated with a lower error rate in third grade arithmetic, whereas Holloway and Ansari (2009) found that a relatively smaller distance effect was associated with better mathematics performance (see below for detailed interpretation of these differences). On the other hand, we extended the results of Holloway and Ansari (2009) in that reliable influences of both the compatibility effect as well as of the number of pure inversion errors in first grade on overall arithmetic performance were present in our data: a larger compatibility effect and more pure inversion errors in transcoding implied a higher error rate in third grade arithmetic performance. 155
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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
Page 129 and 130: 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: Non-carry problems: For the percent
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
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
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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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