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

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or the decade digits was found (393 ms vs. 507 ms, respectively; t(18) = 1.46, p = .16). Finally, the two-way interaction of carry and digit identity was significant [F(1, 18) = 4.85, p < .05] indicating that the requirement of a carry led to a reliably stronger increase of total reading times on the unit digits than on the decade digits of the problem (+ 111 ms vs. + 65 ms). Interestingly, this two-way interaction was further specified by the reliable three-way interaction of carry, problem element and digit identity [F(2, 36) = 12.40, p < .001]. Breaking down this interaction in its constituting two-way interactions revealed a significant interaction of carry and digit identity for the first summand [F(1, 18) = 13.58, p < .01] and the result [F(1, 18) = 24.35, p < .001] while it was only marginally significant for the second summand [F(1, 18) = 4.46, p = .06]. However, a closer inspection of the marginal means showed that for the first summand as well as for the second summand a required carry operation resulted in a more pronounced increase of total reading time on the unit digits compared to the decade digits (first summand: + 119 ms vs. ± 0 ms; second summand: + 205 ms vs. + 101 ms, see Figure 2, Panel B). Contrarily, for the result the carry effect on total reading time was stronger on the decade digit than on the unit digit (+ 80 ms vs. + 2 ms fixations, respectively, see Figure 2, Panel B). Origin of regressions: The ANOVA evaluating the number of regressions from the second summand back to the first summand showed that carry addition problems were associated with a reliably higher number of regressions [0.87 vs. 0.61 regressions; F(1, 18) = 27.58, p < .001]. Moreover, the significant interaction of carry and digit identity [F(1, 18) = 4.26, p = .05] indicated that, in particular, the number of regressions from the unit digit of the second summand back to the first summand (as compared to regressions from the decade digit of the second summand) was increased in carry addition problems (+ 0.37 vs. + 0.15 regressions). The main effect of digit identity was not significant [F(1, 18) = 2.37, p = .14]. 74
Comparably, also the number of regressions from the result back to the summands was moderated by the requirement of a carry operation [F(1, 18) = 8.68, p < .01]: in carry addition problems participants looked back to the summands significantly more often than in non-carry problems (0.54 vs. 0.46 regressions, respectively). Additionally, the main effect of digit identity indicated that more regressions to the summands started from the decade digit as compared to the unit digit of the result. This was further specified by the reliable two-way interaction of carry and digit identity [F(1, 18) = 17.24, p < .001]. The requirement of a carry specifically increased the number of regressions from the decade digit of the result back to the summands (+ 0.15 vs. ± 0 regressions). Target of regressions: Examining the regressions from the second summand back to the first summand, the ANOVA showed a reliable effect of carry [F(1, 18) = 11.72, p < .01] with more regressions to the first summand for carry addition problems as compared to noncarry problems (0.58 vs. 0.44 regressions, respectively). Additionally, the main effect of digit identity was significant as well [F(1, 18) = 42.20, p < .001] indicating that the majority of regressions was directed to the unit digit when looking back to the first summand (0.83 vs. 0.19 regressions). Furthermore, the reliable interaction of these two factors [F(1, 18) = 6.55, p < .05] implied that the increase of regressions towards the first summand due to the requirement of a carry was more pronounced on the unit digit than on the decade digit of the first summand (+ 0.13 vs. + 0.05 regressions, respectively). In line with these results, the ANOVA evaluating the regressions from the result back to the two summands also revealed a significant carry effect [F(1, 18) = 27.22, p < .001]: more regressions from the results to the summands were observed for carry than for non-carry problems. Furthermore, also the two main effects of problem element [F(1, 18) = 18.42, p < .001] and digit identity [F(1, 18) = 26.21, p < .001] were reliable. This indicated that more 75
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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
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 and 56: a view was corroborated by a variet
Page 57 and 58: would indicate influences of verbal
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 102 and 103: egressions were directed to the sec
Page 104 and 105: the requirement of a carry is deter
Page 106 and 107: In line with this prediction, total
Page 108 and 109: Instead, it seems to involve differ
Page 111 and 112: Section 3 On the importance of plac
Page 113 and 114: ABSTRACT The mental number line of
Page 115 and 116: preschoolers aged 4). However, the
Page 117 and 118: still represented even when only th
Page 119 and 120: specificity. Yet, in adults it has
Page 121 and 122: METHOD The reported experiment was
Page 123 and 124: child separately to obtain estimati
Page 125 and 126: indicated that Italian children’s
Page 127 and 128: striking similarity between the ori
Page 129 and 130: Inversion influences on estimation
Page 131 and 132: % overall Estimation Error 25 20 15
Page 133 and 134: seem to be the origin of the perfor
Page 135 and 136: this system. In this case no langua
Page 137 and 138: two-linear model with one linear re
Page 139 and 140: and 40 is 10 times as large as the
Page 141 and 142: APPENDIX A A remark on how to evalu
Page 143 and 144: Splitting the overall interval in t
Page 145 and 146: much closer to the proposed breakpo
Page 147 and 148: the individual approach may be more
Page 149 and 150: 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
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Karmiloff-Smith, Annette (1996). Be
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Pixner, S. (2009). How do children
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