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

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corroborated by a variety of observations indexing decomposed processing of tens and units in more complex numerical tasks such as the number bisection task and even locating a specific area within the intraparietal sulcus associated with processes of place-value integration (cf. Study 5). These imaging results were accompanied by patient data (cf. Study 6) again suggesting separate processing of tens and units as well as a representation of placevalue information to be impaired by hemi-spatial neglect in a way previously assumed to be affect the spatial representation of number magnitude (i.e., the mental number line) only. Thereof, we inferred the mental number line representation of number magnitude to involve at least some kind of place-value information. In this vein, the results of Study 2 imply that place-value information is not only an important aspect / cue in basic numerical tasks but also specifically processed in basic arithmetic such as addition, in particular, when processes of place-value integration are required to calculate the correct result as in carry addition problems. Finally, these conclusions drawn on the current empirical data are backed by the computational results provided by a neural network model of two-digit number comparison. Here, it was found that a strictly decomposed model accounted best for the empirical data (cf. Study 7), thereby, indicating no need for an additional holistic representation as proposed by the hybrid model of two-digit number representation (cf. Nuerk & Willmes, 2005). Taken all these evidence together, an essential role of the place-value structuring principle of the Arabic number system for the adult human representation of number magnitude seems undisputed. Nevertheless, recent findings from studies investigating the development of numerical competencies reported evidence suggesting an important role of place-value understanding even for early numerical development. The findings from the current thesis addressing the influence of the place-value structure of the Arabic number system on the development of numerical competencies as well as the developmental trajectories of place-value understanding will be discussed in the following section. 322
The developmental influence of place-value knowledge In discussing the influence of place-value understanding on the development of numerical competencies in children two different lines of evidence pursued in the current thesis shall first be evaluated and then integrated. On the one hand, the place-value knowledge structure of the Arabic number system may influence the way basic numerical representations are shaped (cf. Zuber et al., 2009). This aspect of early place-value knowledge was addressed by Study 3. On the other hand, early place-value understanding may be an important predictor of later arithmetical skills as investigated in Study 4. Such a predictive association has already been observed for other basic numerical representations such as the representation of number magnitude (cf. Holloway & Ansari, 2009). These two aspects of the developmental influence of the place-value structure of the Arabic number system will be discussed in turn. The influence of early place-value processing on the development of the mental number line Only recently, Zuber and colleagues (2009) found that even a task as basic as transcoding a number from spoken to written format is influenced by aspects relating to placevalue. The authors found that in German-speaking children about half of all transcoding errors committed were related to the inversion property inherent in German number words [e.g., writing down 72 instead of 27 when dictated “siebenundzwanzig” (literally: seven and twenty)] and thereby possibly determined by confusion in place-value integration of the single digits, i.e., difficulties in assigning each digit its corresponding power level as these are inverted in German number words as compared to symbolic notation. Thus, this indicated an influence of the place-value structure of the Arabic number system on other numerical representations such as the verbal representation of number words and related task such as transcoding. In this context, Moeller et al. (2009b) observed that place-value influences may not be limited to the development of transcoding skills but also to building up the spatial 323
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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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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 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
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Page 345 and 346: Pollatsek, 1989) states that an obj
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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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