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

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one even before having seen the second number. Thereby, when the second number is presented this expectation is either confirmed or disconfirmed. In the first case, no further processing of the two numbers magnitude is required and only in the latter case additional evaluations of the magnitude relations between tens and units may be initiated. Thereby, the results for simultaneous and sequential presentation conditions may stem from two completely different underlying processes and thus direct comparisons should be treated with caution. Moreover, the issue of confirming / disconfirming expectations about the second number may be even aggravated by the realization of the sequential presentation condition. From a methodological point of view it seems problematic that to-be-compared numbers were presented sequentially at exactly the same position and with an inter-number-interval of only 200 ms without using a backward mask. Even when about one-third within-decade trials such as 43_49 were included to prevent participants from focusing on the tens position only (see Experiment 2 of Ganor-Stern et al., 2008), at this speed the task is more like a visual discrimination task on the tens position than a number magnitude comparison task requiring the processing of both tens and units. Imagine the case of a within-decade trial, with an internumber-interval of just 200 ms and both numbers being presented at exactly the same position such an item is identified immediately by a change of the visual input at the unit position only. On the other hand, between-decade trials can be identified 100 % correct by a change of visual stimulus properties at the tens position. Thereby, the sequential condition as realized by Ganor-Stern et al. (2008) includes visual cues coding which digit is relevant for which trial that are 100 % valid. Based on this methodological flaw synced with above considerations on possible representational differences between sequential and simultaneous presentation conditions the claim of a holistic magnitude representation engaged in the sequential processing condition seems much too strong. A future study preventing visual cues to be 100 % valid by e.g., employing a backward mask presented after each individual number or 314
jittering the positions of the sequentially presented numbers may indeed provide evidence for a decomposed processing of tens and units even when the to-be-compared numbers are presented sequentially. Finally, Wood, Mahr, and Nuerk (2005) showed that when the to-becompared numbers are presented one after the other (with a 100 ms delay) the individual digits of the number presented first are compared by their magnitude. The authors also found that these intra-number comparisons worked against the overall compatibility effect. Again, this questions the conclusions drawn by Ganor-Stern et al. (2008). Taking into account above arguments as well as the results of Study 1 it seems reasonable to assume that the internal representation of number magnitude may not be holistic as claimed by Zhang and Wang (2005) and Ganor-Stern et al. (2008; see also Zhou, Chen, Chen, & Dong, 2008) but is decomposed as well, thereby retaining the place-value structure of the Arabic number system. The latter proposition that the place-value structure of the Arabic number system is an integral constituent of the human representation of number magnitude is corroborated by the variety of tasks for which an influence of place-value knowledge on performance was observed in the current thesis. Apart from the very basic number comparison task further evidence for the importance of processes of place-value integration come from the number bisection task, mental addition as well as a number line task which shall be discussed in the following. 315
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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
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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
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
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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
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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
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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: methodological problems. In most of
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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