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

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distance. And indeed, such a distinction may be useful. Consider the latter case of distance to the middle number when the following interval should be bisected e.g., 29_52_75. In such an instance rejecting an incorrect middle number should be harder when the probe differs at the unit digit only as when it differs at the tens position as well; even when overall difference were the same (e.g., 29_56_75 vs. 29_48_75, respectively; please note that absolute overall distance to the middle is 4 for both probes). Thus, it may be more appropriate to operatinalize the distance between the correct middle number and the probe in terms of its distance at the tens as well as the unit position rather than computing an overall distance measure (i.e., 29_56_75 decade distance: 0, unit distance 4; 29_48_75 decade distance: 1, unit distance: 6). Similarly, in addition verification (cf. Study 2) it might be the case that for the problem 18 + 24 = 32 the probe 35 may be harder to reject as incorrect when compared to the probe 29 since it shares the correct tens digit even though overall split between both probes and the correct result is identical (i.e., 3 in this example). Again, the results of the current thesis imply that it should be more suitable to account for the split in terms of decade and unit distance as compared to an overall distance measure not accounting for place-value related differences of the probes. In light of the longitudinal results of Study 4 the latter distinction becomes even more relevant as it would provide an additional way of analyzing place-value influences not only for the correctly solved problems (e.g., 18 + 24 = 32, requiring a “yes” response) but also for those solved incorrectly (e.g., 18 + 24 = 35, requiring a “no” response; actually representing 50% of all items). So far, incorrect probes have been chosen to differ at either the tens or the unit position but did not discern between above introduced decadeconsistent and decade-inconsistent probes. However, Domahs, Nuerk, and Delazer (2006; Domahs et al., 2007) were already able to show that such kind of decade consistency had a reliable effect on performance in simple multiplication: participants produced more decadeconsistent than inconsistent faulty responses and it took them longer to reject a decadeconsistent as compared to a decade-inconsistent probe (see introduction for a more elaborate 302
explanation of the decade consistency effect in multiplication). Considering this recent evidence on place-value influences in mental arithmetic, closer investigation differentiating effects of decade and unit distance seems a promising approach to get further insights into the generalizability of the influence of the place-value structure of the Arabic number system on numerical cognition. Anyway, apart from the results suggesting place-value related influences to reflect a comprehensive processing principle rather than specific effects associated with a particular task, the current thesis also provided evidence suggesting that place-value understanding is nevertheless subject to individual and cultural differences. Aspects concerning interindividual as well as inter-cultural variation in processing place-value information will be addressed in the following paragraph. Individual and cultural differences in processing place-value information Even when accepting the assumption that representations of the Arabic number system’s place-value structure are vitally involved in numerical cognition, the question of possible inter-individual and inter-cultural differences remains. As regards the latter, Miura and colleagues (Miura & Okamoto, 1989; Miura et al., 1994) showed that the transparency of the number word system of a given language concerning the way in which units, tens, etc. are verbally recoded is an important determinant of children’s mastery of the place-value structure. Against this background Study 3 of the current thesis was conducted to evaluate the influence of such language differences of number word systems in a totally non-verbal task, the number line estimation task. Despite the fact that performance of both German- and Italian speaking children was accounted for best by a two-linear model with a fixed breakpoint at 10 (see above for a discussion), German- and Italian-speaking children differed reliably in the accuracy of their estimates upon the 303
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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
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Therefore, we believe that this stu
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APPENDIX A: Correlation tables of s
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Section 4 The neurocognitive underp
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ABSTRACT Mental arithmetic and calc
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manipulations cannot be distinguish
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numbers is assumed to be more preci
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correctly bisected by the second nu
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Table 1: Behavioural performance in
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were found bilaterally in the dorso
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Table 3: Brain areas activated by b
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portion of the superior frontal gyr
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Figure 3: fMRI signal for parametri
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The fronto-parietal network and the
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Finally, increased activation in th
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Inferior parietal cortex, the angul
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primarily necessary for solving the
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Tippet (1996) have shown that patie
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Study 6 Impairments of the mental n
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INTRODUCTION Patients suffering fro
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2005), triplets in which the interv
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Visual field assessment showed no s
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Table 1: Demographic and clinical d
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For 80 bisected triplets the factor
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Factorial Analyses (ANOVA and t-tes
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Errors: The ANOVA revealed a main e
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DISCUSSION The effects of neglect o
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decade digits but they even process
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strategic value of retrieving this
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APPENDIX A Test results of patients
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the middle the beneficial effect of
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229
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Study 7 Two-digit number processing
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INTRODUCTION The internal represent
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all numbers the overlap for larger
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digit numbers were engaged. Additio
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single node in an ordered sequence
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modelling to evaluate the plausibil
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A Output Layer 11 99 11 #1 #2 99 In
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expected considering their magnitud
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the output of the right node would
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advantages for these numbers (see a
Page 277 and 278: Finally, a regression analysis only
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Page 283 and 284: compared numbers, unit distance, lo
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Page 289 and 290: Table 5: Predictors included in the
Page 291 and 292: Taken together, these results revea
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Page 299 and 300: The nature of the representation of
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Page 305 and 306: APPERNDIX B Development of connecti
Page 307 and 308: Figure C: Development of connection
Page 309 and 310: Figure E: Development of connection
Page 311 and 312: Interestingly, one might infer from
Page 313 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
Page 317 and 318: tens and units: German-speaking chi
Page 319 and 320: Study 6 examined the influence of t
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Page 323 and 324: GENERAL DISCUSSION In the last sect
Page 325 and 326: to-be-evaluated triplet crossed a d
Page 327: specific processing of place-value
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Page 333 and 334: system and the ordinality of the me
Page 335 and 336: epresentation itself as well as its
Page 337 and 338: Furthermore, converging evidence is
Page 339 and 340: methodological problems. In most of
Page 341 and 342: jittering the positions of the sequ
Page 343 and 344: numbers within the triplets, proces
Page 345 and 346: Pollatsek, 1989) states that an obj
Page 347 and 348: computational models were programme
Page 349 and 350: The developmental influence of plac
Page 351 and 352: correspond to the order of tens and
Page 353 and 354: Asian languages (e.g., Japanese, Ko
Page 355 and 356: indicating that decomposed processi
Page 357 and 358: REFERENCES Banks, W. P. (1977). Enc
Page 359 and 360: Karmiloff-Smith, Annette (1996). Be
Page 361 and 362: Pixner, S. (2009). How do children
Page 363 and 364: 337
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