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

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place-value knowledge in numerical development, in particular, when evaluating specific effects related to the processing of place-value information at different stages of development and schooling. This is of specific interest as the results of Study 2 of the present thesis showed that influences of place-value processing are still observable in adult arithmetic (i.e., particular processing of the unit digits in two-digit addition when a carry is required). Thus, there is the possibility that difficulties in early place-value understanding may not only be a problem restricted to a certain time in numerical development. Instead, impaired place-value understanding at earlier stages of numerical development may determine later arithmetic performance even into adulthood. This assumption is further corroborated by the results of a recent eye-tracking study comparing the eye-fixation behaviour of children and adults in an addition verification paradigm (Moeller, Klein, & Nuerk, under revision). On the one hand, results indicated that there are considerable differences as regards the way children and adults select the correct probe. While children seemed to primarily rely on calculating the correct result, adults seemed to rely on both verifying the correct probe as well as rejecting the incorrect probe. On the other hand, the data also showed that the requirement of a carry specificall increased the processing times of the unit digits of the summands for both children and adults consistently. Synced with the findings of the current thesis this suggests that basic principles of place-value processing in mental arithmetic may be acquired early on during learning arithmetic may not change qualitatively. Thereby, the importance of successful mastery of place-value principles in early stages of numerical development (cf. discussion and Figure 1 above) is again emphasized. However, as already been reported by Miura and colleagues (1989; 1994) there are considerable differences in the accuracy of place-value processing between children but also between different countries and languages. Usually, a superiority of Asian over Western children is observed when it comes to explicitly producing tens and units of a given number. The authors attributed their finding to the fact that the number word systems of most East 326
Asian languages (e.g., Japanese, Korean) are entirely regular and transparent in their coding of units, tens, hundreds, etc. For instance, 86 is spoken as hachi-juu-roku (literally translated: eight-ten-six). Taking into account these differences it remains to be investigated whether the observed association between early place-value understanding and further numerical / arithmetical development represents the general presence and importance of the place-value structuring principle in the world wide dominant Arabic number system. On the other hand, the influence of place-value understanding may be moderated by language specificities such as the inversion property in number words in a way similar to the influence of place-value structuring on the development of the spatial representation of number magnitude (cf. Study 3; Moeller et al., 2009b). In summary, the results of the current thesis addressing influences of the place-value structure of the Arabic number system on numerical development are straight forward. Extending previous results suggesting place-value information to determine performance in a number line estimation task (cf. Moeller et al., 2009b) it was observed that this influence generalises to language with a number word system without inversion. Nevertheless, the data also indicated that the correspondence between the order of tens and units in a language’s number words and symbolical notation serves as a moderating factor of the influence of place-value structuring on the development of the spatial representation of number magnitude. However, apart from this cross-culturaal observation Study 4 of the current thesis also showed that that the influence of the place-value understanding is not limited to basic numerical tasks tapping only basic numerical representations. Instead, longitudinal evidence was reported that indicates a reliable influence of early place-value understanding on later numerical /arithmetical development, in particular, when also requiring processes of place-value integration as in solving carry addition problems. Thus, it can be concluded that on a crosscultural perspective the place-value ordering principle of the Arabic number system exhibits a 327
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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
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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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Page 305 and 306: APPERNDIX B Development of connecti
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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 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
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Page 345 and 346: Pollatsek, 1989) states that an obj
Page 347 and 348: computational models were programme
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Page 351: correspond to the order of tens and
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
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