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

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The detailed description of the three neural network models will first focus on depicting the characteristics common to all three models before covering the specific features of each model in turn. Features common to all three models The number zero was not implemented in any of the three models as the mental representation of zero seems to be special as compared to the other single digits: For instance, Brysbaert (1995) observed that reading zero took participants reliably longer than reading any other single digit number. Moreover, Nuerk, Iversen and Willmes (2004b) provided additional evidence based on nonmetric multidimensional scaling analyses that the number zero may not be represented on the same number line as the other numbers. However, these studies cannot clarify whether the specificity of zero originates from the fact that zero occurs less frequently in daily life or from difficulties with the conception of a number symbol representing nothing (e.g., Frobisher, 1999; Levenson, Tsamir, & Tirosh, 2007). Taken together, it is still debateable whether the mental number line starts at zero or at one. Finally, because all computational models of number magnitude representation start with a representation of one (e.g., Verguts et al., 2005; Grossberg & Repin, 2003; Zorzi & Butterworth, 1999) – zero was also left out in our modelling study. Basically there would be two different possibilities of how to implement zero, either by a separate node preceding the node associated with one or by no specific node at all. Because the latter alternative would result in no specific neural activation when semantically processing the number zero, this conceptualization seems problematic because the missing representation of zero would also result in missing representations of single-digit numbers and decade numbers: At least in the fully decomposed model a representation of zero is inevitable to account for numbers such as 04, 08, etc as well as 20, 30, etc, respectively. Please note that in contrast to the prolonged processing times observed for the number zero processing multiples of 10 is faster than 244
expected considering their magnitude. To guarantee a valid representation of all numbers in all three models the number range implemented in the current models goes from 11 to 99 with exclusion of all multipliers of ten. The empirical data about the compatibility effect were also obtained without the employment of decade numbers. Nevertheless, modelling the role of zero in multi-digit numbers poses a challenge for the future. The representation of number magnitude as assumed in the present study was identical to Verguts et al. (2005) in their models of small number representation. Each node represents one number, with numbers being aligned in ascending order from left to right. For instance, when presenting the number 5, the fifth node from the left gets maximally activated, with all preceding and adjacent nodes being co-activated symmetrically to some degree, depending on their distance to the maximally activated node. Nevertheless, activation always reaches its maximum of for the number actually presented. All three models were programmed as feedforward networks with the decomposed as well as the hybrid model comprising an input layer, one hidden layer and an output layer. The holistic model is set up without a hidden layer (see below for a discussion). For all three models the hyperbolic tangent [a commonly used transfer function in neural network modelling, e.g., Harrington, 1993; Hérvas-Martínez et al., 2009; Ngaopitakkul & Kunakorn, 2009] was chosen as the activation function of the hidden layer and the output layer. The propagation function is given in equation (1) with τ representing a model constant determining how fast the maximum activation of a node is reached and net j (t) reflects the average input over time for a respective node j: a ( t) = net ( t) = τ net ( t) + (1 − τ ) net ( t −1) (1) j j j j 245
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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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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 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 and 266: single node in an ordered sequence
Page 267 and 268: modelling to evaluate the plausibil
Page 269: A Output Layer 11 99 11 #1 #2 99 In
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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
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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Page 319 and 320: 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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