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

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limited set of symbols (cf. Verguts & de Moor, 2005). The results of Studies 3 and 4 of the present thesis indicated that apart form inter-individual differences the development of decomposed processing of multi-digit numbers is also liable to inter-cultural differences. The structure of different number word systems may either hinder or corroborate the successful mastery of the place-value structure of the Arabic number system. Synced with the results of Study 7, in which the strictly decomposed model was found to account best for typical empirical effects in two-digit number comparison model, this may have important implications for the nature of the human representation of number magnitude. In particular, the fact that a holistic representation of two-digit number magnitude is not mandatory to simulate typical empirical effects in number comparison suggests that the decomposed representations of e.g., tens and unit do not seem to be integrated into one entity representing the overall magnitude of a given number, at least for symbolic notation. In turn, the current findings imply that at most the representations of single digit numbers up to 9 may be grounded on a holistic non-symbolic counterpart whereas representations of multi-digit numbers above 10 may involve no direct reference to a non-symbolic quantity representation. Please note that this interpretation fits nicely into the ongoing debate on notation dependent or notation independent / abstract representation of number magnitude (e.g., Cohen-Kadosh & Walsh, 2009; Piazza, Pinel, Le Bihan, & Dehaene, 2007; Cohen-Kadosh, Cohen-Kadosh, Kaas, Henik, & Goebel, 2007). PRÉCIS All in all, the results of the this thesis, obtained from a broad variety of numerical tasks administered to different populations (i.e., German- and Italian-speaking children, healthy adults, and neglect patients) and assessed using different experimental methodologies (i.e., reaction time, eye-tracking , and fMRI paradigms) lend further support to the notion that a mental representation of the place-value structure of the Arabic number system exists. 310
Furthermore, converging evidence is provided that the latter exhibits a task invariant influence on human numerical cognition and its development. Evidence on such a general validity of place-value influences comes from studies indicating (i) that the mental number line is not analogue holistic in nature but involves place-value information (cf. Studies 1, 5, and 6), (ii) that indicators of decomposed processing of two-digit numbers were observed from new basic numerical as well as basic arithmetical tasks [i.e., number line estimation (cf. Study 3) and addition (cf. Study 2), respectively], (iii) that decomposed processing seems to develop culture invariant but is nevertheless influenced by attributes of the respective language’s number word system (cf. Study 4), (iv) that models implying decomposed processing outperformed models of holistic processing in direct comparisons (cf. Studies 3 and 7), and (v) that specific neural correlates of processing place-value information can be identified. Hence, the results of this thesis can be summarized as suggesting decomposed processing of multi-digit numbers along place-value constraints to be a comprehensive, culture invariant, and developmentally important processing characteristic of human numerical cognition. Apart from this integrative view on the implications of the current thesis going beyond the scope of the individual studies, the results of each individual study are nevertheless important on its own. Therefore, the following paragraphs address these results in a less integrative manner, instead focussing on the scientific contribution of each study on a more individual level. EVALUATION AND DISCUSSION In the following section, the results of the previously presented studies will be discussed in respect to two major topics: First, it will be evaluated in what way behavioural, neuropsychological and computational evidence provided by the individual studies corroborated the understanding of how the place-value structure of the Arabic number system 311
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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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Page 291 and 292: Taken together, these results revea
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Page 299 and 300: The nature of the representation of
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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 339 and 340: methodological problems. In most of
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Page 345 and 346: Pollatsek, 1989) states that an obj
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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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