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

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learning pattern mirrored the one found for the purely holistic model with the diametrically opposed development of connection weights for the two to-be-compared numbers. Importantly, for the connection between the hidden layer and the output layer the connection weights for the holistic part of the model developed similarly to the connection weights of the unit digits of the decomposed part of the model and diametrically opposed to the weights for the decade digits. Figure D: Development of connection weights as observed for the decomposed part of the hybrid model (after 1.000, 10.000 and 50.000 training cycles). Again, the first row depicts connection between input and hidden layer for the decade digits while the weights for the units are given in the second row. The third row reflects the connection weights between hidden and output layer separated for tens and units. 282
Figure E: Development of connection weights for the holistic part of the hybrid model (after 1.000, 10.000 and 50.000 training cycles). The first row reflects the connection weights between the input and the hidden layer for each of the two numbers whereas in the second row the connection weights between the input and the output layer are depicted. 283
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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
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 and 270: A Output Layer 11 99 11 #1 #2 99 In
Page 271 and 272: expected considering their magnitud
Page 273 and 274: the output of the right node would
Page 275 and 276: advantages for these numbers (see a
Page 277 and 278: Finally, a regression analysis only
Page 279 and 280: 236) = 417.38, p < .001] and incorp
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
Page 293 and 294: of overall distance necessarily req
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Page 297 and 298: activation of the output layer is a
Page 299 and 300: The nature of the representation of
Page 301 and 302: Thereby, possible interactions and
Page 303 and 304: decomposed model did not include an
Page 305 and 306: APPERNDIX B Development of connecti
Page 307: Figure C: 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 333 and 334: system and the ordinality of the me
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
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Karmiloff-Smith, Annette (1996). Be
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Pixner, S. (2009). How do children
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