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

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Ansari, 2009; Halberda, Mazzocco, & Feigenson, 2009 for the case of number magnitude representation). The current data support such a view since children’s early place-value understanding serves as a reliable predictor of their later numerical / arithmetical performance (see Study 4) and also accounts for language differences concerning the transparency of number word systems. Acknowledging its reliable influence on numerical development a representation of the place-value structure of the Arabic number system should be incorporated in any model capturing the development of numerical / arithmetical competencies in children. In the following paragraph different models of numerical development shall be evaluated regarding their comprehension of place-value representations. Place-value representation in a current model of numerical development As already described in the general introduction the dominant structuring principle of the Arabic number system apart from its ciphered nature (see Chrisomalis, 2004 and introduction for details) is the principle of place-value organization. Additionally, the results obtained in the individual studies of the current thesis argue for an important role of placevalue information in the representation of multi-digit numbers as well as for basic arithmetic involving such numbers. Nevertheless, to the best of our knowledge there is currently no (neuro)cognitive model of numerical development explicitly incorporating place-value understanding. For instance, von Aster and Shalev (2007) proposed a four-staged developmental model of numerical cognition in which they associated specific numerical/calculation competencies to cognitive representations underlying these capabilities on the one hand, and to brain areas subserving these abilities. Place-value understanding is not explicitly conceptualized within this model as well. Yet, it seems plausible to assume that place-value understanding comes into play at stages three and/or four (i.e., Arabic number 306
system and the ordinality of the mental number line, respectively) of the von Aster and Shalev (2007) model as it is a general ordering principle of the Arabic number systems and was also shown to influence the development of a spatial mental number line representation (cf. Moeller et al., 2009b; Study 3). This interpretation is corroborated by the authors’ description of typical impairments of children with dyscalculia for each of the four developmental stages. For stage three, transcoding errors related to the inversion property of e.g., the German number word system are mentioned, clearly tapping place-value processing. Nevertheless, the authors did not specify the issue of place-value representations within the context of their model of number acquisition. Moreover, even when considering other theoretical conceptualizations of numerical development and developmental dyscalculia, in particular, the influence of e.g., insufficient place-value understanding is widely neglected. In line with Karmiloff-Smith’s developmental theory (1996; 1998), Rubinsten and Henik (2009) suggested different underlying deficits and their interactions to determine developmental dyscalculia and/or mathematical learning disorders. Amongst others, the authors proposed deficits in magnitude/quantity processing, working memory, attentional resources, and executive functioning to be relevant for mathematics impairments, but again did not specifically address processing of place-value information. Taken together, there is so far no comprehensive theory of numerical development, either typical or atypical, explicitly considering place-value understanding to be a relevant and necessary step in emerging numerical cognition. Taking into account the finding obtained in the current thesis, in particular the longitudinal influence of early place-value understanding on later arithmetic performance it would be desirable to further conceptualize the developmental impact of place-value knowledge by explicitly incorporating it into a model of numerical development such as the model by von Aster and Shalev (2007). Within this framework, implementing a representation of the Place-value structure of the Arabic number 307
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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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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
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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 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
Page 343 and 344: numbers within the triplets, proces
Page 345 and 346: Pollatsek, 1989) states that an obj
Page 347 and 348: computational models were programme
Page 349 and 350: The developmental influence of plac
Page 351 and 352: correspond to the order of tens and
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
Page 363 and 364: 337
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