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

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of the summands for carry compared to non-carry addition problems, the latter should lead to particular increase of fixations on the decade digit of the result. Section 3: On the importance of place-value integration for arithmetic development 1. Study 3: Language effects on children’s non-verbal number line estimations. In a recent study Zuber et al. (2009) showed the importance of place-value understanding for transcoding numbers from one notation into another as well as the specific difficulties arising in a language (such as German) in which the order of tens and units is reversed in number words as compared to their symbolic instantiation when written in Arabic digits. Moreover, Moeller et al. (2009b) showed that in German speaking children place-value understanding is related to the development of the spatial representation of number magnitude: mastery of the place-value structure of the Arabic number system seems to be associated with the precision of children’s estimates when asked to mark the position of a given number on a hypothetical number line. Children’s estimation performance was accounted for best by a twolinear model suggesting two separate representations of single- and two-digit numbers indicating different processing of numbers which require or do not require place-value integration to correctly understand their overall magnitude, thereby, again highlighting the impact of place-value properties. However, from these studies it remains questionable whether this relation is specific for the German language (maybe driven by its inversion property) or rather culture invariant. This question was pursued in a cross cultural study replicating the results of Moeller et al. (2009b) in Italian speaking children who use a number word system without inversion and contrasting the results of German and Italian-speaking children in regard to the influence of place-value understanding on the results in a number line task.. This is particularly interesting as it 30
would indicate influences of verbal number word representations on performance in a non-verbal numerical task and thus raising the question of the influence of verbal recoding on performance in non-verbal task. 2. Study 4: Early place-value understanding as a precursor for later arithmetic performance – a longitudinal study on numerical development. In a cross-sectional approach Holloway and Ansari (2009) showed that an elaborate basic numerical representation of number magnitude as indicated by the performance in a number comparison task is associated with children’s performance in a standardized mathematics achievement test. The better children performed in the comparison task the higher did they score in the achievement test (see Booth & Siegler, 2008 for a similar interrelation of the spatial representation of number magnitude and mathematics achievement). However, the possible importance of early place-value understanding for later numerical/mathematical development has not been evaluated yet. Based on the notion of multiple numerical representation to determine numerical cognition (see above) it is of interest on a theoretical level whether measures of early place-value understanding such as the transcoding performance (cf. Zuber et al., 2009) and more specifically the number of transcoding errors related to inversion in German-speaking children as well as the compatibility effect (cf. Pixner et al., 2009) is predictive of later arithmetic performance in general and for effects of place-value integration in later arithmetic (i.e., the carry effect in addition), in particular. The current study addressed this question by conducting a longitudinal investigation of whether and how transcoding and/or number comparison performance (in particular inversion errors and/or the compatibility effect) observed in first grade assessment are reliable predictors of addition performance in third grade and more particularly the carry effect (being driven specific and prominent processing of the 31
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The influence of the place-value st
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Ich möchte mich an dieser Stelle b
Page 6 and 7: ZUSAMMENFASSUNG Das wissenschaftlic
Page 8 and 9: Study 5: All for one but not one fo
Page 10 and 11: Classification performance ........
Page 12 and 13: The Arabic place-value structure an
Page 14 and 15: Large problem size > small problem
Page 16 and 17: ABSTRACT ..........................
Page 18 and 19: Study 2: On the cognitive instantia
Page 20 and 21: LIST OF FIGURES Section 1: General
Page 22 and 23: Section 4: The neurocognitive under
Page 24 and 25: LIST OF TABLES Section 1: General I
Page 26: Section 5: A computational model of
Page 29 and 30: Number, the simplest and most unive
Page 31 and 32: limited set of symbols. To achieve
Page 33 and 34: interexponential structuring princi
Page 35 and 36: (i) separating power and base dimen
Page 37 and 38: Abstract Code Model Verbal number C
Page 39 and 40: of the Arabic number system within
Page 41 and 42: previous studies on two-digit numbe
Page 43 and 44: eview), questions concerning repres
Page 45 and 46: esponses are easier (i) for problem
Page 47 and 48: 2009c). Based on the framework of t
Page 49 and 50: scarce (but see Domahs et al., 2006
Page 51 and 52: y Studies 5 and 6 of the current th
Page 53 and 54: OVERVIEW OVER THE STUDIES INCORPORA
Page 55: a view was corroborated by a variet
Page 59 and 60: 2. Study 6: Impairments of the ment
Page 61 and 62: eplicate processes of unit-decade i
Page 63 and 64: determinants shall be reviewed. Mor
Page 65 and 66: Study 6: Hoeckner, S., Moeller, K.,
Page 67 and 68: Section 2 Task invariant influences
Page 69 and 70: ABSTRACT Over the last years, evide
Page 71 and 72: The authors used a magnitude compar
Page 73 and 74: Such congruency is important as mos
Page 75 and 76: Procedure: After ten randomly chose
Page 77 and 78: compatibility effect for large unit
Page 79 and 80: 2004). So, increasing the maximum u
Page 81 and 82: proportions in earlier studies. The
Page 83 and 84: CONCLUSIONS In summary, the conclus
Page 86 and 87: Study 2 On the cognitive instantiat
Page 88 and 89: INTRODUCTION One of the probably mo
Page 90 and 91: eye fixation behaviour for the eval
Page 92 and 93: (see Rayner, 1998 for a review). Tr
Page 94 and 95: Procedure: After being seated with
Page 96 and 97: adjustment of the degrees of freedo
Page 98 and 99: Eye fixation behaviour TRT in ms To
Page 100 and 101: or the decade digits was found (393
Page 102 and 103: egressions were directed to the sec
Page 104 and 105: 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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Thereby, possible interactions and
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decomposed model did not include an
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APPERNDIX B Development of connecti
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Figure C: Development of connection
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Figure E: Development of connection
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Interestingly, one might infer from
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Section 6 Summary 287
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may be determined by the presentati
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tens and units: German-speaking chi
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Study 6 examined the influence of t
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295
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GENERAL DISCUSSION In the last sect
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to-be-evaluated triplet crossed a d
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specific processing of place-value
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explanation of the decade consisten
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verbal recoding of tens and units d
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system and the ordinality of the me
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epresentation itself as well as its
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Furthermore, converging evidence is
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methodological problems. In most of
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jittering the positions of the sequ
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numbers within the triplets, proces
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Pollatsek, 1989) states that an obj
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computational models were programme
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The developmental influence of plac
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correspond to the order of tens and
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Asian languages (e.g., Japanese, Ko
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indicating that decomposed processi
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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:
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