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

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the rules applied, participants needed to determine the mean of the interval and compare it with the magnitude of the central number 1 . However, in the case of non-bisectable triplets the number that correctly bisects the interval is not an integer. Therefore, per definition the central number of the triplet cannot be the correct mean of the interval. In this particular case, there is no need to compare the magnitudes of the central number and the correct mean of the interval. In other words, the procedure of comparing the magnitude of the central number with the correct mean of the interval becomes meaningless in the case of non-bisectable triplets. Changes in the relevance of mental rules are associated with the activation of a specific region in the ventrolateral prefrontal cortex. Ventrolateral prefrontal cortex Imaging and patient studies show that the ventrolateral prefrontal cortex is associated with controlled retrieval of mental rule meanings and generation of alternative solutions for a problem (Donohue,Wendelken, Crone, & Bunge, 2005, Miller & Tippet, 1996). Miller and 1 Note that the NBT can be solving using different strategies: participants may for instance subtract the first number from the central, add the result to the central number again and compare it with the third number. Alternatively, participants may calculate the mean of the two outer numbers and compare the result with the central one. In both cases, triplets with small bisection ranges may be solved by recruiting fact retrieval while trials large bisection ranges and/or decade crossing may be more demanding and may result in stronger activation of the fronto-parietal network. However, there are important similarities between the two strategies: in both cases, a distance is computed and compared with a standard. In the first case, the distance to the third number, in the other case the distance to the middle is calculated. Independently of the strategy used, however, in both cases an effect of decade crossing, problem size, range, bisection possibility and distance to mean are expected. Finally, it is possible that the underscore character used to separate the numbers in the NBT induced the use of subtraction strategies in the present study. Since subtraction strategies are associated with magnitude manipulation, the use of procedural rules may be even stronger when the underscore character is not utilized. In follow-up studies, this aspect should be investigated further. 200
Tippet (1996) have shown that patients with brain lesions extending to the right ventral prefrontal cortex were selectively impaired in tasks involving the generation of alternative solutions for a problem. Furthermore, Goel and Vartarian (2005) reported increased activation in this region when participants successfully generated alternative solutions for a task. In the present study, the right ventrolateral prefrontal cortex was the only region (see Figure 2B) showing more activation for non-bisectable triplets. These results suggest that the right ventrolateral prefrontal cortex is relevant for checking, whether the comparison between the mean of the interval and the central number should be completed or not. When the mean of the two outer numbers is not an integer number (e.g. for 21_25_30 the mean of 21 and 30 is 25.5), number magnitude comparison should be discontinued, because the central number cannot be the correct response of the bisection. Behavioural data are in line with neurofunctional evidence showing that activation in the ventrolateral prefrontal cortex is specific for cognitive set changes (Goel & Vartarian, 2005). While responses to nonbisectable triplets presented in isolation were significantly faster than to bisectable triplets, response latencies for non-bisectable triplets following another non-bisectable triplet did not differ from that for bisectable triplets. This increase in reaction time may reflect costs imposed by again activating the original cognitive set which specified a comparison between the mean of the interval and the central number. Since right ventrolateral prefrontal cortex is engaged in the rational use of cognitive resources and cognitive set changes, we suggest that this area may be considered part of the neural system subserving procedural and conceptual knowledge about numbers and their manipulations. CONCLUSIONS The results of the present study strongly suggest that different neurocognitive systems dedicated to (i) magnitude and symbolic base-10 processing, (ii) familiarity, monitoring and 201
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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
Page 175 and 176: (ii) Effect approach analyses of pr
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Page 179 and 180: Non-carry problems: For the percent
Page 181 and 182: from basic numerical concepts to ar
Page 183 and 184: transcoding and/or who showed relat
Page 185 and 186: A word on the validity of the curre
Page 187 and 188: 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
Page 193 and 194: 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
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: primarily necessary for solving the
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 and 246: DISCUSSION The effects of neglect o
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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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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
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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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