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

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Comparisons to a variable standard In the study originally reporting the compatibility effect Nuerk et al. (2001) observed a compatibility effect with response latencies following incompatible number pairs being reliably longer than latencies for compatible pairs (759 ms vs. 728 ms). Additionally, the compatibility effect was reliably moderated by unit distance as it was more pronounced for large unit distances as compared to small unit distances (52 ms vs. 12 ms; see Figure 2, Panel A). Holistic model ANOVA On the one hand predicting empirical RTs on the basis of the holistic model data yielded a R 2 of .42, which was reliable [r = .65; F(1, 238) = 171.71, p < .001]. However, closer inspection revealed that important empirical effects cannot be accounted for by the holistic model. In the item ANOVA for the data produced by the holistic model there was only a reliable main effect of decade distance [F(1, 232) = 211.04, p < .001]. This indicated that according to the model data number pairs with large decade distance were responded to significantly faster than number pairs with a small decade distance (710 ms vs. 778 ms). Yet, neither the compatibility effect nor its interaction with unit distance were statistically reliable [main effect: F(1, 232) < 1; interaction: F(1, 232) < 1, see Figure 2, Panel B]. Moreover, no other main effect or interaction was found to be significant [all F < 1.59, all p > .21]. Taken together, this indicated that a reliable correlation of modelled and empirical RT data of its own is not a sufficient index of model validity. Regression The final model of the stepwise regression analysis on the model data produced by the holistic model accounted for a considerable amount of variance [adj. R 2 = .84, R = .92, F(4, 252
236) = 417.38, p < .001] and incorporated the three predictors: problem size, difference between the logarithmic magnitudes of the two to-be-compared numbers, and logarithm of problem size (see Table 1). Evaluation of the beta weights revealed that response latencies increased as (i) distance between the logarithms of the numbers decreased, (ii) absolute problem size as well as (iii) logarithmic problem size decreased. Thus, comparable to the results of the ANOVA no indication of an influence of unit-decade compatibility was observed in the RT data modelled by the holistic model. Please note the unexpected direction of the influence of the predictors absolute and logarithmic problem size in the final model. However, the reversed problem size effect does not necessarily disprove the holistic model as it is probably driven by the interrelation of problem size and the difference between the logarithms of the two numbers. As explained in greater detail in Appendix C this interrelation is not linear as assumed in the regression analysis, but curvilinear instead meaning that up to a problem size of about 65, problem size is positively correlated with distance, which in turn drives the reversed problem size effect. Table 1: Predictors included in the final regression model for the holistic data Predictor B b t sign. Change in R 2 Raw correlation Partial correlation Constant 1226.38 - 10.16 < .001 - - - Inclusion of absolute overall distance .50 Problem size -1.53 - .39 2.08 < .05 .33 - .55 - .13 Diff Log distance - 216.83 - .74 23.91 < .001 .01 - .55 - .84 Exclusion of absolute distance - .001 Log problem size - 185.89 - .37 1.99 < .05 .003 - .61 - .13 Finally, the multiple regression including the four reliable predictors of item RT identified by Nuerk et al. (2001) produced a model predicting modelled RTs reliably [adj. R 2 = .81, R = .90, F(5, 235) = 254.29, p < .001] and without a substantial loss of descriptive adequacy. However, when looking at the beta weights of the individual predictors it was 253
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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
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seem to be the origin of the perfor
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this system. In this case no langua
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two-linear model with one linear re
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and 40 is 10 times as large as the
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APPENDIX A A remark on how to evalu
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Splitting the overall interval in t
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much closer to the proposed breakpo
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the individual approach may be more
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Contrarily, data produced by a two-
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Study 4 Early place-value understan
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INTRODUCTION Dealing with (multi-di
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has been shown that other represent
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number system as formalized by its
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124 10024, see Deloche & Seron, 19
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Please note that in contrast to Miu
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Roid, Prifitera, & Weiss, 1993; see
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effect-based approach has never bee
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(ii) In the magnitude comparison ta
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problem of capitalizing on differen
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Error rate in % Error Categories (c
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positively correlated with overall
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(ii) Effect approach analyses of pr
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whether the carry effect for additi
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Non-carry problems: For the percent
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from basic numerical concepts to ar
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transcoding and/or who showed relat
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A word on the validity of the curre
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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
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
Page 233 and 234: 2005), triplets in which the interv
Page 235 and 236: Visual field assessment showed no s
Page 237 and 238: Table 1: Demographic and clinical d
Page 239 and 240: For 80 bisected triplets the factor
Page 241 and 242: Factorial Analyses (ANOVA and t-tes
Page 243 and 244: Errors: The ANOVA revealed a main e
Page 245 and 246: DISCUSSION The effects of neglect o
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Page 249 and 250: strategic value of retrieving this
Page 251 and 252: APPENDIX A Test results of patients
Page 253 and 254: the middle the beneficial effect of
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Page 257 and 258: Study 7 Two-digit number processing
Page 259 and 260: INTRODUCTION The internal represent
Page 261 and 262: all numbers the overlap for larger
Page 263 and 264: digit numbers were engaged. Additio
Page 265 and 266: single node in an ordered sequence
Page 267 and 268: modelling to evaluate the plausibil
Page 269 and 270: A Output Layer 11 99 11 #1 #2 99 In
Page 271 and 272: expected considering their magnitud
Page 273 and 274: the output of the right node would
Page 275 and 276: advantages for these numbers (see a
Page 277: Finally, a regression analysis only
Page 281 and 282: Regression The stepwise regression
Page 283 and 284: compared numbers, unit distance, lo
Page 285 and 286: RT in ms 780 770 Compatible Number
Page 287 and 288: esponse latencies increased as dist
Page 289 and 290: Table 5: Predictors included in the
Page 291 and 292: Taken together, these results revea
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Page 297 and 298: activation of the output layer is a
Page 299 and 300: The nature of the representation of
Page 301 and 302: Thereby, possible interactions and
Page 303 and 304: decomposed model did not include an
Page 305 and 306: APPERNDIX B Development of connecti
Page 307 and 308: Figure C: Development of connection
Page 309 and 310: Figure E: Development of connection
Page 311 and 312: Interestingly, one might infer from
Page 313 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
Page 317 and 318: tens and units: German-speaking chi
Page 319 and 320: Study 6 examined the influence of t
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Page 323 and 324: GENERAL DISCUSSION In the last sect
Page 325 and 326: to-be-evaluated triplet crossed a d
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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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