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

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unit and decade distance must be processed separately to result in the antidromic influence just mentioned (see also Nuerk & Willmes, 2005 for a more detailed discussion of this point. Yet, to account for the compatibility effect a model is necessary that allows us to differentiate between processing the tens digits and processing the unit digits complying with the place-value structure of the Arabic number system: a strictly decomposed or at least a hybrid model of two-digit number representation. In the strictly decomposed model the magnitudes of tens and units are represented separately (possibly on concomitant number lines, cf. Verguts & De Moor, 2005; see Nuerk and Willmes, 2005 for a review) and the overall magnitude of a two-digit number is achieved by integrating the single digits’ of tens and units magnitudes into their respective position within the place-value structure of the Arabic number system (henceforth referred to as place-value integration, see also Poltrock & Schwartz, 1984 for an early inclusion of place-value knowledge into the decision process in number comparison). Thereby, the place-value structure is a necessary prerequisite for correctly representing multi-digit numbers. In the hybrid model (cf. Nuerk et al., 2001; Nuerk & Willmes, 2005) above described decomposed representation is accompanied by a holistic representation of the number’s overall magnitude, meaning each number to be represented as an integrated entity of all its constituting digits without retaining the place-value structure. So, brought up by the observation of the compatibility effect (Nuerk et al., 2001) decomposed processing of tens and units in two-digit number comparison and on a broader level the possibility of an (additional) representation of number magnitude in terms of separate coding of each power level (cf. McCloskey, 1992) had to be considered as an alternative to the traditional assumption of an analogue holistic representation of all numbers disregarding the place-value structure of the Arabic number system (cf. Dehaene et al., 1990). Nevertheless, although there is cumulating evidence for the notion of decomposed processing of multi-digit numbers from more and more numerical tasks (see below for a brief 16
eview), questions concerning representational universality / generality versus specificity are still under debate. Therefore, it was the aim of the current thesis to provide converging evidence for the existence and the importance of the place-value structure of the Arabic number system being retained in the human representation of number magnitude. This issue was pursued by evaluating effects possibly driven by separate representations of the individual power levels of multi-digit numbers in a variety of numerical tasks, using different methodologies and assessing diverse populations. However, before turning to a more explicit description of the present thesis recent evidence suggesting the place-value structure of the Arabic number system to be represented in the human understanding of number semantics will be reviewed briefly. Further evidence for decomposed processing of tens and units In recent years evidence for the existence of decomposed processing of number magnitude has accumulated. Extending the original results recent studies also using a number comparison paradigm revealed that the influence of the place-value structure of the Arabic number system on numerical cognition is generally language invariant (Moeller, Fischer, Nuerk, & Willmes, 2009a; Verguts & De Moor, 2005), although slightly moderated by characteristics of a language’s number word system (Nuerk et al., 2005; Pixner, 2009), starts early in numerical development (Nuerk, Kaufmann, Zoppoth, & Willmes, 2004b; Pixner, Moeller, Zuber, & Nuerk, 2009), and is relatively independent from literacy (Wood, Nuerk, Freitas, Freitas, & Willmes, 2006a for data of Brazilian semi illiterates). Furthermore, Moeller et al., 2009a showed that separate processing of tens and units is performed in parallel rather than sequentially as suggested by Poltrock and Schwartz (1984). Moreover, Wood, Nuerk, and Willmes (2006b) were able to pinpoint the neural correlates of place-value integration (i.e., the compatibility effect) to the depths of the intraparietal sulcus (IPS, see also Goebel, 17
Page 1 and 2: The influence of the place-value st
Page 3: 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: previous studies on two-digit numbe
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 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
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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
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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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