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

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ventral “what” and the “dorsal” where stream of processing visual stimuli: In this way the one dimensional representation of single-digit numbers is augmented by a second dimension coding the base-10 property of the Arabic number system. However, Grossberg and Repin (2003) did not specify whether this two-dimensional coding of multi-digit numbers also manifests itself in separate representations of e.g., tens and units or rather converges on one holistic representation, upon which the comparison process is based. Nevertheless, inspection of Figures 22B as well as 23A (Grossberg & Repin, 2003, p. 1130) provides at least descriptive evidence suggesting compatibility like effects for both error and RT data produced by the model. From the distribution of relative errors committed by the model (see Figure 22B, Grossberg & Repin, 2003, p. 1130) one can derive that for number smaller as well as larger than the standard (i.e., 55) incompatible comparisons (e.g., 38_55 or 55_72) were associated with more errors than compatible comparisons with even smaller overall distance (e.g., 41_55 or 55_69, see also Dehaene et al., 1990 for a discussion about these discontinuities, i.e., reversed distance effects at decade boundaries). A similar effect is evident for the original simulated RT data (mainly for probes smaller than the standard) with the effect being most pronounced at the transition from the second to the third decade (see Figure 23A, Grossberg & Repin, 2003, p. 1130). Taken together, these observations implicate that the model by Grossberg & Repin (2003) may be based on the assumption of decomposed processing of tens and units but in the paper there is no explicit statement on this issue. Most importantly for the current study, Grossberg and Repin (2003) did not aim at differentiating between the strictly decomposed and the hybrid processing approach. The present study Nevertheless, in the present study we were not interested to further add to the ongoing debate about possible coding and scaling characteristics of the mental number line and their implementation in computational models. Instead, we aimed at employing computational 240
modelling to evaluate the plausibility and validity of above introduced models of two-digit number processing (i.e., holistic, decomposed, and hybrid processing) for the case of twodigit number comparison. So far, there is ample evidence for decomposed processing of twodigit numbers (e.g., Nuerk et al., 2001; Moeller et al., 2009a). However, at the moment there is no empirical variable allowing a reliable differentiation between the strictly decomposed and the hybrid model. Therefore, particular interest was paid to the question whether the assumption of an additional holistic representation of overall magnitude as proposed by the hybrid model (cf. Nuerk & Willmes, 2005) is actually necessary to account for the empirical data. This question is specifically interesting as the hybrid model is the most complex one and following the principle of parsimony/theoretical economy it should only be chosen when the other models and in particular the strictly decomposed model (cf. Verguts & De Moor, 2005) do not account for the data at hand. In summary, the current study set off to distinguish between three proposed models of number magnitude representation (i.e., holistic, decomposed, and hybrid model) in a computational modelling approach. In a first step, the validity of the computational models is evaluated by their ability to account for the distance effect, a standard effect observed in numerical cognition research. It is expected that all three models should be able to account for the distance effect. In a second step, it is aimed to evaluate the influence of decomposed processing of tens and units as indicated by the compatibility effect. Here, it is hypothesized that the holistic model should fail to account for the unit-decade compatibility effect while both the strictly decomposed and the hybrid model should be capable of accounting for the compatibility effect. Third, it shall be appraised whether incorporation of a predictor reflecting a measure of overall distance [i.e., (logarithmic) overall distance or the distance between the logarithms of the to-be-compared numbers] necessarily requires a holistic magnitude representation as previously argued by Nuerk and Willmes (2005, see also Nuerk et al., 2001; Knops, 2006). If so, such a predictor should only be included in the final 241
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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
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
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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 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: single node in an ordered sequence
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 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 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
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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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