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

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Two-digit number processing Basically, Arabic numbers are structured in a two-dimensional place-value system. First, the base dimension is reflected by the ten digits from 0 to 9, whereas, second, the power dimension is coded by the positions of the digits with a base ten. The position within a digit string defines the value of each digit on a power of ten dimension (cf. Zhang & Norman, 1995; Chrisomalis, 2004). However, the way in which these two dimensions of numbers are represented psychologically on the mental number line is subject to controversial debates. At the moment three models accounting for the organization of this number line can be discerned. 1. The holistic model According to the holistic model (e.g., Dehaene, Dupoux, & Mehler, 1990) numbers are transformed into an integrated entity mapped onto a holistic magnitude representation before any magnitude comparison or numerical manipulation is performed. In the context of number magnitude comparison this property implies that performance should be exclusively determined by the overall distance between the numbers to-be-compared, thereby, overriding the base-10 structure of two-digit numbers. However, to account for the problem size effect additional assumptions are required. In this context, either linear coding of numbers along the mental number line with scalar variability or logarithmic coding of numbers with fixed variability have been suggested (for a recent discussion see Brannon, Wusthoff, Gallistel, & Gibbon, 2001; Dehaene, 2001). The latter model implies that magnitudes of larger numbers may be compared more slowly because magnitude is assumed to be represented along a logarithmically compressed mental number line. Consequently, the distance between the logarithmic magnitudes of two relatively larger numbers is smaller than the distance between the logarithmic magnitudes of two relatively smaller numbers. As variability is assumed to be constant for the representation of 234
all numbers the overlap for larger numbers is suggested to be gets relatively larger. Due to this larger overlap more interference with the magnitude representations of neighbouring numbers occurs slowing down magnitude comparison of larger numbers compared to magnitude comparison of two smaller numbers with the same absolute distance. Alternatively, when linear coding with scalar variability is assumed, numbers are mapped with equal separations onto the mental number line but the variability of their entries increases with their size. Again, the overlap between two larger numbers is relatively larger and the comparison is slowed down correspondingly. 2. The decomposed model The decomposition model (e.g., Verguts & De Moor, 2005) predicts that the magnitudes in two-digit numbers are represented separately for each of the constituting digits. Thus, the magnitude of a two-digit number is not mapped onto one mental number line as an entity. Instead, each digit constituting a two-digit number is mapped onto its own number line. Considering the base-10 structure of the Arabic number system, these mental number lines must then be labelled by referring to the place-value of the digit represented. In number comparison, separate comparisons of corresponding digits at equivalent positions (e.g., units, tens, etc.) within the to-be-compared numbers are assumed. The representation of the individual numbers (i.e., 0 - 9) on the separate number lines is supposed to be organized in an analogue way comparable to the one postulated by the holistic model of magnitude representation. This assumption allows accounting for the problem size effect based on the increasing magnitude of the individual digits on their respective number lines (see above for the distinction between linear coding with scalar variability and logarithmic coding with fixed variability). 235
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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
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
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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
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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: INTRODUCTION The internal represent
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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 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 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
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Page 309 and 310: 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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