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

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systems differ between languages and even within the same language different regional influences can be observed. In this context it is interesting that the left-to-right order of tens and units in two-digit numbers is not retained in their corresponding number words in all languages (e.g., 27 twenty seven), rather the units are spoken before the tens in some languages’ number words (e.g., 27 achtundzwanzig in German [seven and twenty]). This so-called inversion property is found among others in German, Maltese, Arabic, Norwegian, etc. (see Comrie, 2005 for an overview). On the other hand, there are also intra-language variations. For instance, in the Czech language the non-inverted number words are officially used and taught at school. Nevertheless, in some parts of Czech inverted number words are primarily used in everyday life (see Pixner, 2009; see also Seron, Deloche, & Noël, 1992; Seron & Fayol, 1994 for differences in French number words used in Belgium and France). Taken together, these examples illustrate quite nicely that although there are correspondences between the numerical notation system and the number words used to vocalize symbolic numbers, most number word systems involve a lot irregularities that are not inherent in the numerical notation system they actually refer to (i.e., the Arabic number system). In this vein, the generality of the Arabic number system is striking. In the following, a typology of numerical notation systems shall be introduced (cf. Chrisomalis, 2004). Subsequently, the most important characteristics of the currently dominating Arabic number system shall be outlined. The Arabic number system in a typology of notational systems Apart from very simple tally-like notational approaches relying on one-to-one correspondence between the to-be-counted objects and an equal number of markings or body parts all numerical notation systems have to be structured in a specific way as this is a prerequisite of their actual function: to represent any given number (in the best case) by a 4
limited set of symbols. To achieve this goal numerical notation systems are dimensionally organized into bases and powers, for instance, the Arabic number system is a base-10 system. In all notational systems the powers of the base are of special designation (as can be mathematically indicated by an exponent 10 0 = 1; 10 1 = 10, 10 2 = 100 etc. for the Arabic number system). However, please note that there are also numerical notation systems with more than one base for which a comprehensive mathematical description is more complex (e.g., the Roman notation system with the main base 10 and its power levels, i.e., 10 0 = I; 10 1 = X, 10 2 = C etc. and the sub-base 5 and its power levels 5 1 x 10 0 = V; 5 1 x 10 1 = L, 5 1 x 10 2 = D). When expressing e.g., the number of items in a set by their corresponding number there are different possibilities to code the quantity to-be-represented at each individual power level but also in the way the single power levels are integrated into a coherent overall representation of a given number are possible. In his recent typology for numerical notation, Chrisomalis (2004) used exactly such intra- as well as interexponential (referring to the exponent coding the power dimension) characteristics of notation systems to develop a clearly structured typology (see also Guitel, 1975 as well as Zhang & Norman, 1995 for possible typologies relying on different structural aspects of numerical notation systems). Both of these dimensions differentiate between a number of principles determining the way numerical representation is organized. According to Chrisomalis (2004) the intraexponetial dimension can be subdivided into cumulative, ciphered, and multiplicative organization subtypes. In systems with a cumulative intraexponential structure any power of the base is indicated by repeating number symbols until the sum of these symbols equals the to-be-represented quantity at the particular power level (e.g., CCC represents 300 in the Roman number system, literally meaning 100 + 100 + 100). Contrarily, in ciphered numerical notation systems each power level is represented by only one number symbol. Thus, coding a given number requires as many number symbols as 5
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: Number, the simplest and most unive
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 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
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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
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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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