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

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epresentation of number magnitude. In a number line estimation task Moeller and co-workers (2009b) found that children’s estimates of the spatial position of a given number upon a hypothetical number line were accounted for best by a two-linear function with a break point at 10. From this data, Moeller et al. (2009b) concluded that children’s early mental number line representation seems to differentiate between single- and two-digit numbers. In Study 3 of the present thesis we were able to show that this influence of the structuring principle of the Arabic number system exhibits its influence on the development of the spatial representation of number magnitude occurs not only in German-speaking children (cf. Moeller et al., 2009b) but also in Italian-speaking children, thereby suggesting it to be a rather general principle. Nevertheless, we also found evidence for a specific influence of the structure of the languages’ number word systems. This was possible because unlike in German number words the order of tens and units in Italian number words corresponds to the left-to-right order of e.g., tens and units in symbolic notation. Base on this difference in the number word systems of German and Italian we hypothesized the estimates of German-speaking children to be less accurate as they were more prone to confuse tens and units because of their antidromic order in verbal and symbolic notation. The results confirmed our hypothesis: The estimates of Italian-speaking children were reliably more precise as those of their German-speaking counterparts. Even more interestingly, the finding that this advantage was specifically driven by the mislocalizing of items for which the confusion of tens and units leads to a large deviation between the correct position of the number and the actually marked position (e.g., 82 as compared to 65). Taken together, these results indicate that there is a rather general influence of the place-value structuring principle of the Arabic number system on the development of the spatial representation of number magnitude which is valid independent of cultural influences as long as the Arabic number system is used. However, the data also imply that this influence is moderated by the language specific attributes such as the number word system. When the order of tens and units in a language’s number word system does not 324
correspond to the order of tens and units in symbolic Arabic notation this is detrimental for children developing a precise and accurate spatial representation of number magnitude. On a broader level, this is interesting as an accurate spatial representation of number magnitude seems to be a reliable predictor of actual mathematics achievement as well as children’s ability to solve unknown arithmetic problems (Booth & Siegler, 2008). The influence of early place-value understanding on further numerical development At this point and based on the observed influence of the place-value structure of the Arabic number system on the development of a spatial representation of number magnitude (cf. Moeller et al., 2009b; Study 3 of the present thesis) the question arises whether early place-value knowledge itself may serve as a building block for further numerical arithmetical development. This issue was pursued in Study 4 of the current thesis in a longitudinal approach. As already described above, results indicated that early place value knowledge in first class (as operationalized by inversion errors committed in transcoding as well as by the compatibility effect in a magnitude comparison task) reliably predicted performance in an addition task two years later in third grade. Thereby, Study 4 provided first direct and longitudinal evidence for early place-value knowledge to determine the future development of children’s numerical / arithmetical competencies. Apart from this very important finding Study 4 may be considered innovative for another reason. The predictive value of early place value understanding on later arithmetic performance could be specified in an effect based approach. Therein, it is not overall performance in third grade addition that is predicted by overall performance in magnitude comparison and transcoding assessed in first grade. Instead, specific measures reflecting the accuracy of processes of place-value integration in addition (i.e., the number of errors in carry addition problems and the carry effect) were reliably predicted by inversion transcoding errors and the compatibility effect, again being a measure of processes of place-value integration. Thus, Study 4 suggested a persistent influence of 325
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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
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
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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
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 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
Page 349: The developmental influence of plac
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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