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

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prominent role of the unit digits of the summands in mental addition (Klein et al., under revision). Rather, the data also served as a generalization of the findings by Moeller and coworkers (2009a) as they provide further eye tracking evidence validating the assumption of parallel but decomposed processing of two-digit numbers not only in a task as basic as number magnitude comparison but also in mental addition. So far, the evidence provided by the current thesis regarding the role of the place-value structure of the Arabic number system within the human representation of multi-digit number magnitude was empirical in nature. However, in Study 7 the question in which way two-digit number magnitude is represented (i.e., either holistically thereby not retaining the place-value structuring or decomposed or hybrid and thus referencing place-value principles) was pursued by a computational approach. Computational evidence on the nature of two-digit number magnitude representation On a more theoretical level the current thesis also added significantly to our understanding of how two-digit numbers are represented. As already elaborated on above, the empirical findings of the present thesis consistently provide evidence for the notion of twodigit number magnitude being represented and processed in a decomposed manner separating tens and units; and thus explicitly complying with the place-value structure of the Arabic number system. However, currently there are two theoretical frameworks capable of accounting for such decomposed processing: The notion of a strictly decomposed representation of multi-digit numbers separated into their constituting digits (cf. Verguts & De Moor, 2005) as well as the hybrid model of multi-digit number representation assuming an additional holistic representation of the overall magnitude of a number despite the separate representations of units, tens, hundreds, etc. (cf. Nuerk & Willmes, 2005; Nuerk et al., 2001). As there is currently no empirical paradigm allowing for differentiating between these two theoretical models, we addressed this question by a computational approach. Three 320
computational models were programmed realizing (i) holistic (ii) strictly decomposed, and (iii) hybrid representations of number magnitude. When evaluating the descriptive adequacy of the data produced by each of these models it was found that the data by the strictly decomposed model fitted the empirical data best. In particular, Nuerk and colleagues (2001; see also Nuerk & Willmes, 2005) initially assumed a representation of overall number magnitude to accompany the separate representations of tens and units as in most regression analysis a measure coding the overall distance between the to-be-compared numbers was observed to be the most important predictor of task performance. However, assuming such an additional holistic representation, because a measure of overall distance between probe and standard was included as a reliable predictor in a regression analysis, was proved unnecessary by the current data. A measure of overall distance (i.e., difference between the logarithms) was found to be the most important predictor of item RT even for the data produced by the strictly decomposed model. Thereby, indicating that overall a strictly decomposed model of two-digit number representation accounted best for the empirical data. Please note that the superiority of the strictly decomposed model was evident even apart from the fact that the hybrid model is the least parsimonious model and should thus only be considered when controlling for the higher number of free parameters. Nevertheless, the computational data clearly argue for the human representation of two-digit number magnitude to comply with the power levels of the Arabic number system, thereby, retaining its place-value structure. In summary, the results of the present thesis were meaningful in addressing the question whether and if so in what way the place-value structure of the Arabic number system is reflected in the human representation of two-digit numbers. Empirical evidence suggested that decomposed processing of tens and units is not driven by an external presentational format. Instead, the present data indicate that the internal representation of number magnitude is decomposed as well (cf. Study 1). Furthermore, this important initial finding was 321
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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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For 80 bisected triplets the factor
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Factorial Analyses (ANOVA and t-tes
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Errors: The ANOVA revealed a main e
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DISCUSSION The effects of neglect o
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decade digits but they even process
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strategic value of retrieving this
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APPENDIX A Test results of patients
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the middle the beneficial effect of
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229
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Study 7 Two-digit number processing
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INTRODUCTION The internal represent
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all numbers the overlap for larger
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digit numbers were engaged. Additio
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single node in an ordered sequence
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modelling to evaluate the plausibil
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A Output Layer 11 99 11 #1 #2 99 In
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expected considering their magnitud
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the output of the right node would
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advantages for these numbers (see a
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Finally, a regression analysis only
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236) = 417.38, p < .001] and incorp
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Regression The stepwise regression
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compared numbers, unit distance, lo
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RT in ms 780 770 Compatible Number
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esponse latencies increased as dist
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Table 5: Predictors included in the
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Taken together, these results revea
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of overall distance necessarily req
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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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Page 313 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
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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 335 and 336: epresentation itself as well as its
Page 337 and 338: Furthermore, converging evidence is
Page 339 and 340: methodological problems. In most of
Page 341 and 342: jittering the positions of the sequ
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Page 353 and 354: Asian languages (e.g., Japanese, Ko
Page 355 and 356: indicating that decomposed processi
Page 357 and 358: REFERENCES Banks, W. P. (1977). Enc
Page 359 and 360: Karmiloff-Smith, Annette (1996). Be
Page 361 and 362: Pixner, S. (2009). How do children
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