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

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von Cramon, 2004) and inhibition of a cognitive set (Konishi, Jimura, Asari, & Miyashita, 2003). Such mechanism may monitor the occurrence of information dispensable or redundant for solving the NBT, but which can be useful to solve it more efficiently. We will elaborate more on this point in the next section and relate it to the processing of multiplicativity and distance to the correct mean. Monitoring numerical information from different sources and inhibition Activation of the superior frontal gyrus and the angular gyrus have been associated with the monitoring of information from different sources (Brass and von Cramon, 2004), inhibition of a cognitive set (Konishi et al., 2003) and verification of rules (Skosnik et al., 2002), respectively. In the current study, the angular gyrus was activated when numerical distance to the mean was large, when no decade crossing occurred, and when triplets were part of a multiplication table (Figures 1B, 2A and 3, respectively). The superior frontal gyrus bilaterally and the posterior cingulate gyrus, were also activated in these contrasts as well as the supramarginal cortex for the contrasts small distance to the mean > large distance to the mean and no decade crossing > decade crossing. Despite a great disparity between the numerical properties represented in these contrasts, there is a crucial similarity common to all of them. They all convey a procedural rule, which, when correctly applied, were useful for solving a triplet. When the middle number was numerically close to one of the outer numbers – being consequently far from the correct mean- (e.g. 32_33_41), computations might be interrupted, as it is very unlikely that the central number is the correct mean of the interval. Similarly, when no decade crossing occurred, the decade digits might be ignored, since they were all in the same decade (e.g. 92_95_98). Furthermore, when a triplet was part of a multiplication table, the central number is per definition the mean of the interval. Always when these rules can be applied, further computations involving magnitude comparison were unnecessary. However, information about decade crossing and multiplicativity is not 198
primarily necessary for solving the NBT. Therefore, participants need to verify the applicability of these rules for each trial. For that, it is necessary to monitor in parallel the different numerical properties of the triplet as well as to inhibit the magnitude processing. In the present study, participants were able to monitor for multiplicativity, presence of decadecrossing distance from the central number to the mean and number magnitude. Furthermore, part of the activation observed in the superior frontal gyrus may be due to inhibition processes (Konishi et al., 2003) suppressing magnitude manipulations as these are not required in triplets for which the correct response can be inferred by using a procedural rule. For instance, in triplets without decade crossing participants may have inhibited the magnitude of tens because the comparison of unit distances is sufficient for responding correctly. Furthermore, in multiplicative triplets the central number is per definition the correct mean of the interval. Therefore, no magnitude processing would be needed at all. Additionally, in triplets with a large distance to the mean, the central number is numerically too close to one of the outer numbers to be the correct mean of the interval. Again, magnitude manipulations may be discontinued in these cases 1 . In the particular case of the stronger activation in the angular gyrus found for multiplicatively related triplets, a second account for the current data is possible. Apart from the above-mentioned monitoring and inhibitory mechanisms, activation of the left angular gyrus has also been associated with arithmetic fact knowledge (Delazer et al. 2003; 2005). Such an interpretation is also compatible with the data pattern observed for multiplicativity. Therefore, the increased activation in the (left) angular gyrus may also indicate table-related multiplication facts are accessed while performing the NBT (Cohen et al., 2000). In summary, when numbers keep well-learned relations with each other and these relations are more frequently activated and manipulated, activation in the left inferior parietal cortex, especially in the angular gyrus increased. Moreover, activation of the superior frontal gyrus increased when a triplet could be solved by applying procedural rules. When none of 199
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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
Page 173 and 174: positively correlated with overall
Page 175 and 176: (ii) Effect approach analyses of pr
Page 177 and 178: whether the carry effect for additi
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: Inferior parietal cortex, the angul
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 and 246: DISCUSSION The effects of neglect o
Page 247 and 248: decade digits but they even process
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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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
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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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