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

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strictly decomposed model revealed a reliable main effect of compatibility [F(1, 39) = 6.03, p < .05] meaning that estimated RTs for compatible comparisons were significantly shorter than estimated RTs for incompatible comparisons (496 ms vs. 506 ms, respectively). Moreover, both the main effect of unit distance and the interaction of compatibility and unit distance were not reliable [both F(1, 39) < 1]. Yet, when evaluating the compatibility effect separately for comparisons with either a large or a small unit distance it was found that the compatibility effect was present only for comparisons with a large unit distances [496 ms vs. 508 ms respectively; F(1, 9) = 13.16, p < .01] but missed significance for the comparisons with small unit distances [498 ms vs. 504 ms respectively; F(1, 29) = 2.02, p = .17, see Figure 3, Panel C]. Comparably to the analyses on the data produced by the holistic model the influence of the covariate problem size was reliable in all analyses [two-way ANOVA: F(1, 39) = 15.27, p < .001; large unit distance: F(1, 9) = 7.71, p < .05; small unit distance: F(1, 29) = 10.79, p < .01]. These results not only indicated that the strictly decomposed model accounted for a larger part of the variance of the empirical RTs in general, but also showed that the compatibility effect was simulated successfully. Regression The final model of the stepwise regression analysis on the modelled RT data of the strictly decomposed incorporated the predictors difference between the logarithms of the tobe-compared numbers, unit distance and logarithmic unit distance [adj. R 2 = .80, R = .90, F(3, 40) = 59.49, p < .001, see Table 5]. A closer look at the beta weights revealed that reaction times increased as the overall distance between the to-be-compared numbers decreased and problem size increased. More particularly, response latencies also increased as unit distance decreased. As unit distance is positive for compatible pairs but negative for incompatible number pairs the inclusion of this predictor reflects a reliable influence of unit-decade compatibility on response latencies estimated by the strictly decomposed model. 262
Table 5: Predictors included in the final regression model for the decomposed data Predictor B b t sign. Change in R 2 Raw correlation Partial correlation Constant 491.63 - 78.30 < .001 - - - Overall Distance -2.04 - .69 10.16 < .001 .48 - .69 - .85 Problem size .74 .50 7.42 < .001 .25 .50 .76 Unit distance - 1.14 - .29 4.33 < .001 .09 - .30 - .57 At last, the regression analysis including only the two predictors identified to significantly predict item RT in the study by Moeller et al. (2009b) resulted in a reliable model [adj. R 2 = .51, R = .73, F(2, 41) = 23.23, p < .001]. However, descriptive adequacy of the model was considerably reduced as compared to the final model of the stepwise regression analysis. Nevertheless, both variables predicted item RTs reliably. Inspection of the beta weights revealed that item RTs increased as logarithmic overall distance [constant = 575.48; B = -63.81; b = - .66, t = 6.20, p < .001] as well as unit distance [B = -1.12; b = - .29, t = 2.68, p < .01] decreased. Again, the reliable negative influence of unit distance indicated item RTs to be driven by unit-decade compatibility. In summary, these analyses consistently indicated a good fit between the data produced by the strictly decomposed model and the empirical results, not only regarding descriptive adequacy, but also concerning the reliable simulation of the relevant effects (i.e., distance and compatibility effect). Hybrid model ANCOVA Overall, predicting empirical RTs by the RT estimates produced by the hybrid model resulted in a R 2 of .43 [r = .65; F(2, 70) = 52.24, p < .01]. However, unlike previously observed for the comparisons to a variable standard (see above) the strictly decomposed and the hybrid model did not perform similarly for comparisons to an internal standard. No 263
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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
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 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
Page 287: esponse latencies increased as dist
Page 291 and 292: Taken together, these results revea
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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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Page 333 and 334: system and the ordinality of the me
Page 335 and 336: epresentation itself as well as its
Page 337 and 338: 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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