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

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system would not be too problematic. As von Aster and Shalev (2007) propose children get confronted with the issue of place-value coding first as they experience number words above 10 (or 20 when acknowledging the special character of teen numbers). Therefore, the postulated representation of place-value information should be triggered in step 2 of the von Aster and Shalev (2007) model. However, as the authors suggest for the role of working memory (dotted arrow in Figure 1) it may be the case that a place-value representation develops not in an all-or-nothing manner but gets more and more established and relevant over the four developmental steps: For representations of concrete quantity as proposed in step 1 no place-value representations should be acquired. However, with the experience of number words within and/or above the teen range in step 2 place-value issues become more and more important to build up a successful connection between the respective number word and its quantitative meaning. In step 3, the role of place-value information is even more prominent as within the to-be-learned Arabic number system it reflects one of the core organization principles (together with ciphering, see Chrisomalis, 2004). Thus, to correctly reflect a given number in symbolic Arabic notation place-value coding has to be understood (see Zuber, Pixner, Moeller, & Nuerk, 2009). Yet, the development of place-value representations may still be in progress in step 4 when the spatial mental number line representation is established. The results of Study 3 clearly show that even non-verbal representations of spatial number magnitude representations are influence by place-value properties (see also Moeller et al., 2009b). Additionally, the longitudinal influences of early place-value understanding on arithmetic performance (cf. Study 4) suggest place-value influences to actually generalize to children’s arithmetic thinking. To sum up, a representation of the place-value structure of the Arabic number system within the model of von Aster and Shalev (2007) may be conceptualized similarly to the influence of working memory (see Figure 1, dotted arrow) as an arrow starting in step 2 and ascending until step 4 (see Figure 1, broken arrow and shaded area below) representing both the development of the place-value 308
epresentation itself as well as its increasing importance for the numerical capabilities associated with the consecutive developmental steps. Working Memory Capacity Cognitive Representation Step 1 Step 2 Step 3 Step 4 Core system of magnitude (cardinality) Verbal number system /one/ /two/ … Arabic number system …, 45, 46, 47, … Mental number line (ordinality) Concrete quantity Number words Digits Spatial image Brain Area Bi-parietal Left prefrontal Bi-occipital Bi-parietal Ability Subitizing, approximation, comparison Verbal counting, counting strategies, fact retrieval Written calculations, odd/even Approximate calculation, Arithmetic thinking Place-value understanding Infancy Preschool School Time Figure 1: Model of numerical development by von Aster and Shalev (2007, p. 870) adapted and extended by a representation of the place-value structure of the Arabic number system (broken arrow and shaded area below) On a closer inspection of the quantity representations proposed for Steps 1, 2, and 3 it has to be noted that only from step 2 on symbolic coding (verbal and digital) is assumed while step 1 quantity representations rely on non-symbolic coding of concrete quantities. This distinction seems to be motivated by observations suggesting that even preverbal infants are able to discriminate between numerical quantities (e.g., Feigenson, Carey, & Hauser, 2002; Feigenson, Dehaene, & Spelke, 2004; Xu, Spelke, & Goddard, 2005; for a review see Cordes, & Brannon, 2008). The place-value structure of the Arabic number system first comes into play with the development of symbolic quantity representations in step 2 when tens and units have to be discerned when confronted with the respective number words. Decomposed processing of tens and units can be assumed to start at this developmental stage as well and to get more and more automated with increasing age and experience (cf. Pixner, Moeller, Zuber, & Nuerk, 2009) as it allows for representing any given number in a very efficient way by a 309
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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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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
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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
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 and 326: to-be-evaluated triplet crossed a d
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Page 333: system and the ordinality of the me
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
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