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

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However, unlike in previous studies (e.g., Holloway & Ansari, 2009) the use of two-digit stimuli allowed for the evaluation of place-value processing. In Study 3 of the current thesis it was observed that the worse children’s place-value understanding was in first grade the worse they performaned in an addition task administered in third grade. Notably, a higher number of inversion transcoding errors in first grade was associated with a larger number of errors in carry addition problems and a larger carry effect in third grade, both indicating impaired processes of place-value integration. Thus, these data not only suggested a reliable influence of place-value understanding on numerical development but also indicated rather direct developmental trajectories for the representation of the place-value structure of the Arabic number system. Specificities in processing place-value information Further indication for the validity of decomposed processing to be a rather general characteristic of multi-digit number processing comes from modelling results. On the one hand a two-linear model suggesting two separate representations of single- and two-digit numbers fitted children’s estimation pattern in the number line task reliably better than did a holistic logarithmic model (see Moeller, Pixner, Kaufmann, & Nuerk, 2009b; Study 3). On the other hand, directly contrasting all three proposed models of two-digit number magnitude representation (i.e., holistic, strictly decomposed, and hybrid) using computational modelling revealed that the compatibility effect could not be simulated by the holistic model but by the strictly decomposed and the hybrid model. Taking into account model proximity it was found that all empirical effects observed for symbolic two-digit number comparison could be simulated by a strictly decomposed model. Against this background the question arises whether there is a specific neural correlate associated with the processing of place-value properties. Assessed by the number bisection task, increased activation in posterior parts of the IPS was found for the evaluation of triplets crossing a decade boundary that required 300
specific processing of place-value information (cf. Study 5). This again is in line with the interpretation suggesting the mental number line to not only represent analogue magnitude information but to also involve information about place-value properties. Taken together, the results of the current thesis provide further evidence for decomposed processing of multi-digit numbers (reflecting a crucial role of the place-value structure of the Arabic number system) to be a general processing principle rather than a specificity to particular presentational formats, tasks, and/or participant populations. In turn, this strongly suggests that place-value information is an inherent part of the mental representation of number magnitude (possibly along a mental number line) as it seems to be retained in numerical cognition in general. Future perspectives on the generalizability of place-value processing On a more abstract level, the assumption of place-value structuring as a general processing characteristic raises important questions regarding the overall validity of previous findings in number processing research. In particular, when acknowledging the results of the computational modelling study further research is needed to investigate the full range of consequences of the fact that typical empirical effects of two-digit number comparison were accounted for best by a strictly decomposed model, not incorporating any holistic magnitude representation at all. Based on these findings, is it justified to operationalize bisection range in a number bisection task in terms of the overall distance between the two numbers defining the to-be-bisected interval (e.g., Nuerk et al., 2002; Study 5; Göbel, Calabria, Farnè, & Rossetti, 2006; Loftus, Nicholls, Mattingly, & Bradshaw, 2008), or the distance between the correct middle number and a given probe by giving the overall distance between the probe and the correct middle number (e.g., Nuerk et al., 2002; Study 5; Loftus et al., 2008)? The current data suggest that it might be more appropriate to distinguish between the decade and the unit 301
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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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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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Page 299 and 300: The nature of the representation of
Page 301 and 302: Thereby, possible interactions and
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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: to-be-evaluated triplet crossed a d
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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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