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

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4 blocks of 40 trials each, which the participant initiated by pressing one of the response keys. Trial order was randomized for each participant. Before each trial a fixation cross was presented in the centre of the screen for 500ms. Then the three numbers were presented for 15,000 ms followed by a ISI of 5,000 ms. The experiment took approximately 15 to 25 minutes, depending on how long participants rested between blocks. RESULTS For statistical analyses RT and error data were used. Only RTs followed by a correct response were considered for further analysis. A trimming procedure eliminated trials with RTs shorter than 200 ms and longer than 15,000 ms and – in a second step – RTs outside +/- 3 SD of each participant’s individual mean. In total 92.33 % of all trials fulfilled these criteria and were used for further analyses. As stroke patients show a greater variability in their RTs than healthy participants, a z-transformation on individual item RT, using the mean and standard deviation per participant for standardization, was carried out. Error rates were arcsine transformed prior to the analyses. There was no speed accuracy trade-off present in our data as indicated by the absence of a reliable negative correlation of error rate and RT in any item condition (all r > - .40, all p > .10, N = 18). The main interest of the current study was to contrast patients suffering from neglect and those without neglect in terms of their performance in the NBT. Therefore, the two control groups (patient controls and healthy controls) were pooled into one group for the main analyses of this study. For the interested reader Appendix B reports all results separated for the single control groups. As neglect patients do not only neglect the left side of space but also the left parts of numerical intervals (e.g. Zorzi et al., 2002), processing numbers on the left side of a numerical interval (i.e. the central number being smaller than the arithmetical middle of the interval) should be relatively more impaired. 214
Factorial Analyses (ANOVA and t-tests) Non-bisected triplets (RT and error data): Response latencies: An analysis of variance (ANOVA) incorporating the factors distance to the middle, size relative to the middle and group was conducted on z- transformed RTs. Most importantly, and in accordance with the hypothesis, a reliable interaction between size relative to the arithmetical middle and patient group was obtained [see Figure 1A, F(1, 16) = 6.40, p < .05]. This indicates that neglect patients did benefit reliably less from a central number smaller than the triplet’s arithmetical middle than did control participants (426 ms vs. 637 ms). However, this disadvantage for neglect patients did not seem to be modulated by distance to the arithmetical middle. This was indicated by the non-significant three-way interaction of distance to the middle, interval side and group [F(1, 16) < 1]. However, from the notion of a spatially oriented MNL (e.g. Dehaene et al., 1993), it can be inferred that neglect patients should exhibit a more pronounced disadvantage in triplets such as 21_22_35 compared to e.g. 21_27_35. On the one hand, both central numbers are smaller than the arithmetical middle of the interval (e.g. 28) in these examples. But on the other hand, they differ in their distance to the middle: while 27 is numerically close to the arithmetical middle of the interval (e.g. 28), 22 is far smaller than the middle. Assuming an ascending left-to-right orientation of the MNL, 22 would be located further to the left than 27 and should thus be even more neglected. To directly test this hypothesis the absolute effects of the factor size relative to the arithmetical middle were contrasted for neglect patients vs. non-neglect controls separately for triplets with a central number either numerically close to the arithmetical middle or far from it using Bonferroni-Holm corrected t-tests (Holm, 1979). The beneficial effect of the central number being smaller than the arithmetical middle only tended to be less pronounced for neglect patients [t(16) = 1.95, p = .07, one-sided], whereas neglect patients benefited reliably less from a second number far smaller than the arithmetical middle [t(16) = 2.19, p < .05, one-sided]. This indeed indicates a greater relative disadvantage for 215
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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
Page 189 and 190: Non-monotonous Model of the Relatio
Page 191 and 192: 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
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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: For 80 bisected triplets the factor
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
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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
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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
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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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