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

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al. 2006). Neglect not only impairs access to relatively smaller numbers in the one-digit or small two-digit range, it does so for the full range of two-digit numbers. The second hypothesis outlined in the introduction was that neglect patients may have particular difficulties with two-digit number triplets which cross a decade boundary compared to triplets from within one decade. Indeed, we did not only observe a general decade crossing main effect, but a reliable interaction with participant group: Neglect patients exhibited a larger decade crossing effect than their non-neglect counterparts; i.e. they had particular difficulties when the numbers of a triplet were from different decades. The effect of decade crossing on behavioural performance (Nuerk et al. 2002) and fMRI signal (Wood et al., 2008) has been interpreted as resulting from a deeper processing of number magnitude and the base-10 structure of the Arabic number system required when different decade digits are encountered in the task. Processing different decades seems to demand the integration of decomposed decade digit and unit digit magnitudes (cf. Nuerk & Willmes, 2005 for a discussion). Neglect patients may have problems mentally representing the magnitude of two-digit numbers in a decomposed fashion, as they may have problems representing the decade digit’s magnitude. This may lead to the more pronounced decade crossing effect observed for neglect patients in the current study. A perceptual account for this larger decade crossing effect is that neglect patients may have difficulties perceiving the decade digit which is located on the left within a two-digit number. This perceptual account is however not consistent with the diagnostic data of our patients: All neglect patients were able to read aloud the two-digit numbers of a triplet presented above each other without errors. Moreover, in the NET, the neglect patients were even able to read four-digit digital clock times without major problems (one single error in two of six patients). Moreover, this perceptual account is also disproved by more fine-grained analyses: (Decade digit driven) problem size was a highly significant predictor of neglect patients’ performance. So, neglect patients did not only perceive the visual presence of the 220
decade digits but they even processed their magnitude. Finally, we wish to point out that the visual presentation of the number triplets with the numbers above each other was identical in all experimental conditions. Only the mental spatial representation in the mental numerical interval differed between conditions. For all these reasons, it can be suggested that the differences between conditions are not due to perceptual impairments in neglect, but to impairments of the mental (spatial) representation of the presented two-digit numbers. Interestingly, the group difference in the decade crossing effect seemed to be modulated by the location of this decade crossing within the numerical interval: A detailed comparison of triplets with decade crossing occurring between the first and the central number (e.g. 28_31_34) and triplets with decade crossing between the central and the third number (e.g. 25_28_31) revealed that the deficit present for neglect patients was more pronounced in triplets with decade crossing between the first and the central number. In those triplets, the decade crossing is on the left side of the mental numerical interval to which access seems particularly impaired in neglect. Therefore, this detailed analysis of the decade crossing effect corroborates the assumption of an oriented MNL influencing performance in the NBT. In the remaining paragraphs, the results regarding hypotheses (iii) and (iv) will be discussed: With regard to the factors distance to the arithmetical middle and multiplicativity no group differences between neglect patients and non-neglect controls were observed. As the distance to the arithmetical middle effect is a magnitude-related effect, it may seem surprising at first that no group difference was observed because, for the other magnitude-related factors, neglect patients showed specific impairments as discussed above. For the distance to the middle effect, this was different. A possible reason for this lack of difference between the two groups may be that magnitude processing can be subserved by the IPS bilaterally (cf. Dehaene et al., 2003). General difficulties to process number magnitude (e.g. with regard to the distance effect) are usually only observed when both IPS areas are 221
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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
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 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: DISCUSSION The effects of neglect o
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
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
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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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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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