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

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Large problem size > small problem size ..................................................................... 185 Small Problem size > large problem size...................................................................... 187 Small distance to the mean > large distance to the mean ............................................. 187 Large distance to the mean > small distance to the mean ............................................ 188 Possible > impossible.................................................................................................... 189 Impossible > possible .................................................................................................... 189 Multiplicative > non-multiplicative .............................................................................. 189 Non-multiplicative > multiplicative .............................................................................. 190 DISCUSSION ....................................................................................................................... 192 The fronto-parietal network and the intraparietal cortex ........................................... 193 Magnitude processing ................................................................................................... 193 Symbolic base-10 processing ........................................................................................ 195 Other areas associated with the fronto-parietal network ............................................. 196 Inferior parietal cortex, the angular gyrus and fact knowledge .................................. 197 Familiarity with arithmetic problems ........................................................................... 197 Monitoring numerical information from different sources and inhibition .................... 198 Ventrolateral prefrontal cortex ....................................................................................... 200 CONCLUSIONS................................................................................................................... 201 Study 6: Impairments of the mental number line for two-digit numbers in neglect ABSTRACT .......................................................................................................................... 204 INTRODUCTION ................................................................................................................ 205 MATERIALS AND METHODS......................................................................................... 208 X
Participants ....................................................................................................................... 208 Stimuli and design ............................................................................................................ 210 Procedure .......................................................................................................................... 213 RESULTS .............................................................................................................................. 214 Factorial Analyses (ANOVA and t-tests) ....................................................................... 215 Non-bisected triplets (RT and error data) ..................................................................... 215 Bisected triplets (RT and error data) ............................................................................ 217 DISCUSSION ....................................................................................................................... 219 CONCLUSIONS................................................................................................................... 223 APPENDIX A ....................................................................................................................... 225 Test results of patients suffering from neglect in all sub tests of the NET .................. 225 Test results of patients suffering from neglect in all sub tests of the EC 301 R ......... 225 APPENDIX B ........................................................................................................................ 226 Factorial analyses (ANOVA and t-tests) differentiating between all three participant groups ................................................................................................................................ 226 Non-bisected triplets (RT and error data) ..................................................................... 226 Bisected triplets (RT and error data) ............................................................................ 227 Section 5: A computational model of place-value integration in two-digit number comparison Study 7: Two-digit number processing – holistic, decomposed or hybrid? A computational modelling approach XI
Page 1 and 2: The influence of the place-value st
Page 3: Ich möchte mich an dieser Stelle b
Page 6 and 7: ZUSAMMENFASSUNG Das wissenschaftlic
Page 8 and 9: Study 5: All for one but not one fo
Page 10 and 11: Classification performance ........
Page 12 and 13: The Arabic place-value structure an
Page 16 and 17: ABSTRACT ..........................
Page 18 and 19: Study 2: On the cognitive instantia
Page 20 and 21: LIST OF FIGURES Section 1: General
Page 22 and 23: Section 4: The neurocognitive under
Page 24 and 25: LIST OF TABLES Section 1: General I
Page 26: Section 5: A computational model of
Page 29 and 30: Number, the simplest and most unive
Page 31 and 32: limited set of symbols. To achieve
Page 33 and 34: interexponential structuring princi
Page 35 and 36: (i) separating power and base dimen
Page 37 and 38: Abstract Code Model Verbal number C
Page 39 and 40: of the Arabic number system within
Page 41 and 42: previous studies on two-digit numbe
Page 43 and 44: eview), questions concerning repres
Page 45 and 46: esponses are easier (i) for problem
Page 47 and 48: 2009c). Based on the framework of t
Page 49 and 50: scarce (but see Domahs et al., 2006
Page 51 and 52: y Studies 5 and 6 of the current th
Page 53 and 54: OVERVIEW OVER THE STUDIES INCORPORA
Page 55 and 56: a view was corroborated by a variet
Page 57 and 58: would indicate influences of verbal
Page 59 and 60: 2. Study 6: Impairments of the ment
Page 61 and 62: eplicate processes of unit-decade i
Page 63 and 64: determinants shall be reviewed. Mor
Page 65 and 66:
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
Page 259 and 260:
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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most important predictor in a multi
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activation of the output layer is a
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The nature of the representation of
Page 301 and 302:
Thereby, possible interactions and
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decomposed model did not include an
Page 305 and 306:
APPERNDIX B Development of connecti
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Figure C: Development of connection
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Figure E: Development of connection
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Interestingly, one might infer from
Page 313 and 314:
Section 6 Summary 287
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may be determined by the presentati
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tens and units: German-speaking chi
Page 319 and 320:
Study 6 examined the influence of t
Page 321 and 322:
295
Page 323 and 324:
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
Page 333 and 334:
system and the ordinality of the me
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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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