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

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is reflected in the human number magnitude representation. In a second paragraph, the influence of place-value understanding on the development of numerical competencies will be elaborated on. On the representation of the place-value structure of the Arabic number system Addressing the question whether and if so, in what way the place-value structure of the Arabic number system is retained in our representation of two-digit numbers, the results of Studies 1, 2, 5, 6, and 7 are meaningful regarding (i) the influence of presentational format, (ii) place-value processing in more complex numerical tasks such as the number bisection task and mental addition, and (iii) the nature of two-digit number magnitude representation (i.e., holistic vs. decomposed vs. hybrid). These facets will be discussed in turn. Presentational format and place-value processing Recent criticism suggested that decomposed processing of tens and units may be determined by either presentational format or mode of presentation. In 2005, Zhang and Wang claimed decomposed processing of tens and units to be limited to the case when both to-becompared numbers (i.e., probe and standard) are presented and thus represented externally – an argument also put forward by Ganor-Stern, Pinhas, and Tzelgov (2008). On the other hand, these authors presented evidence suggesting holistic processing of two-digit numbers in which the place-value structure of the Arabic number system is not retained when (i) a given number has to be compared to a fixed internally memorized standard (cf. Zhang & Wang, 2005) and when (ii) the to-be-compared numbers are presented sequentially instead of simultaneously (cf. Ganor-Stern et al., 2008). In this context, the results of Study 1 clearly indicate that decomposed processing did not depend on an external and simultaneous presentation of both probe and standard. Rather, we were able to show that the lack of unit based effects (such as the compatibility effects) in studies using a fixed internal standard may be attributed to 312
methodological problems. In most of these studies (e.g., Zhang & Wang, 2005; Dehaene, Dupoux, & Mehler, 1990) a fixed standard ending on five such as 55 or 65 was used. Thereby, unit distance was limited to rather small values with a maximum of 4 (e.g., 55_79). However, already Nuerk et al. (2001) observed that the compatibility effect was more pronounced for large unit distances (i.e., 4 - 8) and weaker or even not significant for small unit distances (i.e., 1 - 4). Thus, it is not very surprising that unit-based effects such as the compatibility effect have not been observed in studies using a fixed standard ending on 5 (see also Nuerk & Willmes, 2005 as well as Study 1 for a more detailed discussion of this point). However, by the time Study 1 was conducted the paper by Ganor-Stern and colleagues (2008) was not yet published so the issue of simultaneous vs. sequential presentation was not tackled in this study but should nevertheless be discussed. Sequential vs. parallel presentation of to-be-compared numbers Ganor-Stern et al., (2008) observed a regular compatibility effect only when the to-becompared numbers were presented simultaneously but not when they were presented sequentially. From this finding the authors concluded that the internal representation of towdigit number magnitude representation is holistic in nature rather than decomposed. However, there are a number of theoretical and methodological problems concerning the validity of the sequential presentation condition as used by Ganor-Stern et al. (2008). From a theoretical point of view it may not be entirely appropriate to directly compare simultaneous and sequential processing conditions without referring to the problem of processing stages involved in either of the two conditions. Already back in 1977 Banks argued in his influencing paper on semantic processing in comparative judgements that differences between simultaneous and sequential presentation conditions may be determined by the direction of what he called relative coding of the to-be-compared numbers. In the sequential condition participants may have build up expectations about the first number being the large or smaller 313
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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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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
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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
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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: Furthermore, converging evidence is
Page 341 and 342: jittering the positions of the sequ
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Page 345 and 346: Pollatsek, 1989) states that an obj
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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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