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

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eliable main effect of compatibility was present in the RT data produced by the hybrid model [F(1, 39) = 1.24, p = .27, see Figure 3, Panel D]. This indicated that modelled RTs for compatible comparisons were not reliably faster than RTs for incompatible comparisons (501 ms vs. 505 ms, respectively). Also, the interaction of compatibility and unit distance was not statistically reliable [F(1, 39) < 1]. In line with this, separate analyses for comparisons with large and small unit distances did not reveal the expected dissociation: neither for comparisons with a large unit distance [F(1, 9) = 1.38, p = .27] nor for comparisons with a small unit distance [F(1, 29) < 1] a reliable compatibility effect was found. This means that the hybrid model could not account for the compatibility effect as observed for comparisons to a fixed standard. Problem size as incorporated by a covariate did not exhibit a reliable influence in any ANCOVA [F(1, 39) = 1.85, p = .18; large unit distance: F(1, 9) = 3.44, p = .10; small unit distance: F(1, 29) < 1]. Regression The only reliable predictor incorporated into the final regression model for the RT data modelled by the hybrid model was overall distance between the to-be-compared numbers [adj. R 2 = .46, R = .69, F(1, 38) = 37.56, p < .001]. Looking at the beta weight showed that reaction times increased as the distance between the two numbers decreased [constant = 533.22; B = - 2.03; b = - .69, t = 6.13, p < .001]. Rerunning the regression analysis with incorporation of the two reliable predictors as identified by Moeller et al. (2009b) did result in a statistically reliable model without considerable loss of predictive power [adj. R 2 = .42, R = .67, F(2, 41) = 16.39, p < .001]. However, inspection of the beta weights showed that only the influence of the predictor logarithmic overall distance was significant [constant = 575.50; B = -62.72; b = - .66, t = 5.62, p < .001] while unit distance did not predict item RT reliably [B = - .43; b = - .11, t = 0.95, p = .35]. 264
Taken together, these results revealed that the compatibility effect for comparisons to a fixed standard could not be simulated by the hybrid model. RT in ms A RT in ms B 520 Compatible comparisons Incompatible comparisons 520 Compatible comparisons Incompatible comparisons 510 510 500 500 490 490 480 480 Small Unit Distance Large Unit Distance Small Unit Distance Large Unit Distance Empirical RT Holistic Model RT in ms C RT in ms D 520 Compatible comparisons Incompatible comparisons 520 Compatible comparisons Incompatible comparisons 510 510 500 500 490 490 480 480 Small Unit Distance Large Unit Distance Small Unit Distance Large Unit Distance Decomposed Model Hybrid Model Figure 3: Response latencies for compatible and incompatible comparisons to a fixed standard separated for small and large unit distances. Again, Panel A depicts the empirically observed RT pattern (Moeller et al., 2009b), whereas Panels B, C, and D indicate the RT pattern for the data produced by either the holistic, the strictly decomposed, or the hybrid model, respectively. Error bars reflect 1 SEM. Contrasting the individual models’ correlations with empirical data Finally, individual correlation coefficients reflecting the interrelation between modelled and empirical data were transformed into Fisher’s Z-values to directly contrast 265
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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
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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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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 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
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Page 275 and 276: advantages for these numbers (see a
Page 277 and 278: Finally, a regression analysis only
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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
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Page 289: Table 5: Predictors included in the
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Page 299 and 300: The nature of the representation of
Page 301 and 302: Thereby, possible interactions and
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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
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Page 333 and 334: system and the ordinality of the me
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Page 337 and 338: Furthermore, converging evidence is
Page 339 and 340: 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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