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

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meaning that larger problem size was associated with longer response latencies. This suggests that neglect patients were not only able to perceive the decade digits of the involved two-digit numbers but also processed their magnitude. Furthermore, a strong main effect for decade-crossing was found for both speed and accuracy [RT: F(1, 16) = 34.59, p < .001, accuracy: F(1, 16) = 15.95, p < .001]. Triplets that did not cross a decade boundary were responded to faster (4620 ms) and more accurately (5.4 % errors) than triplets which crossed into the next decade (5195 ms and 16.3 % errors). Exclusively for response latencies this effect was even further differentiated: while nonneglect controls benefited (-168 ms) from a decade crossing occurring between the first and the central number of a triplet (e.g. 28_31_34 vs. 25_28_31), this had a deteriorating effect on neglect patients [+908 ms, t(16) = 2.79, p < .05]. This suggests that the cognitively more demanding processing of a decade crossing is especially impaired when it occurs in the neglected part of the numerical interval. Moreover, the ANOVA revealed a reliable difference between neglect patients and nonneglect controls for RT [F(1, 16) = 5.85, p < .05] as well as errors committed [F(1, 16) = 8.06, p < .05]. So, neglect patients exhibited longer response latencies (6330 ms) and made more errors (17.4 %) compared to non-neglect controls (3485 ms and 4.2 % errors). Multiplicativity: A main effect for multiplicativity was observed neither for RT nor for errors [RT: F(1, 16) < 1, errors: F(1, 16) = 2.02, p = .17]. However, neglect patients and nonneglect controls differed significantly in respect to errors committed [F (1, 16) = 11.23, p < .01] but not for response latencies [F (1, 16) = 1.09, p = 31]. Neglect patients’ performance was more error prone (24.2 % errors) than that of non-neglect controls (7.3 % errors). Finally, multiplicativity and participant group did not interact reliably for either RT or errors [RT: F(1, 16) < 1, errors: F(1, 16) < 1]. 218
DISCUSSION The effects of neglect on the bisection of numerical intervals in the present study can be summarized as follows: First, the ANOVA revealed that neglect patients benefited reliably less when the central number was smaller than the arithmetical middle (e.g. 21_22_35) compared to when it was larger (e.g. 21_34_35). Second, it was observed that neglect patients were especially impaired for triplets crossing a decade boundary (e.g. 25_28_31) compared to triplets within the same decade (e.g. 22_25_28). Finally, neglect patients were not generally impaired in all numerical representations: Neither the effect of distance to the arithmetical middle nor that of multiplicativity differed between neglect patients and non-neglect controls. According to Nuerk et al. (2002; see also Moeller, 2006), participants’ response latencies are shorter for triplets with the central number being numerically smaller than the arithmetical middle compared to triplets with a central number larger than arithmetical middle. This general beneficial effect was replicated in the present study. However, neglect patients benefited reliably less than non-neglect controls when the central number was smaller than the arithmetical middle. From this it can be inferred that neglect patients neglect the left side of a given numerical interval. It can be hypothesized that the effect of neglect on the mental numerical interval is stronger for numbers farther towards the left side of that interval than for numbers which are on the left side of the interval but relatively close to the middle. Although there was no reliable three-way interaction between group, size relative to the middle, and distance to the middle, fine-grained analyses revealed that this hypothesis tended to be true. While for numbers close to the arithmetical middle, neglect patients and controls differed only marginally; for numbers far from the middle, there was a significant group effect: For those far distances, neglect patients benefited significantly less when the central number was smaller than the arithmetical middle than when it was larger. These findings extend previous results from Zorzi et al. (2002; and later Priftis et al., 2006; Rossetti et al., 2004 and Zorzi et 219
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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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Page 195 and 196: Therefore, we believe that this stu
Page 197 and 198: APPENDIX A: Correlation tables of s
Page 199 and 200: Section 4 The neurocognitive underp
Page 201 and 202: ABSTRACT Mental arithmetic and calc
Page 203 and 204: manipulations cannot be distinguish
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Page 207 and 208: correctly bisected by the second nu
Page 209 and 210: Table 1: Behavioural performance in
Page 211 and 212: were found bilaterally in the dorso
Page 213 and 214: Table 3: Brain areas activated by b
Page 215 and 216: portion of the superior frontal gyr
Page 217 and 218: Figure 3: fMRI signal for parametri
Page 219 and 220: The fronto-parietal network and the
Page 221 and 222: Finally, increased activation in th
Page 223 and 224: Inferior parietal cortex, the angul
Page 225 and 226: primarily necessary for solving the
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
Page 233 and 234: 2005), triplets in which the interv
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: Errors: The ANOVA revealed a main e
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Page 249 and 250: strategic value of retrieving this
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 and 270: A Output Layer 11 99 11 #1 #2 99 In
Page 271 and 272: expected considering their magnitud
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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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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
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Thereby, possible interactions and
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decomposed model did not include an
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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
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Section 6 Summary 287
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may be determined by the presentati
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tens and units: German-speaking chi
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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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