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

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place-value structure. For instance, consider the problem 132 + 257. Here, unit, decade and hundred digits need to be manipulated and arranged correctly to obtain the result 389, i.e., overwriting of zeros: 300 and 89 389. Moreover, when addition problems become more difficult, e.g., by involving a carry operation, correct integration of digits into the place-value system gets more demanding as well. Considering the problem 132 + 259, this means that not only the correct result (i.e., 391) has to be integrated into the place-value structure. Additionally, one needs to keep track of the carry required at the unit position (i.e., 9 + 2 = 11; 11 ≥ 10 carry 1). Thus, it is reasonable to assume that apart from the use of procedures, strategy selection and strategy efficiency (Imbo & Vandierendonck, 2008b) multi-digit addition involves a profound knowledge of basic numerical skills, in particular, the understanding of the place-value structure of the Arabic number system. Interestingly, Gervasconi and Sullivan (2007) reported that in first and second grade about 10% and 27% of children, respectively, exhibited problems in understanding the place-value structure of the Arabic number system. Hence, it is sensible to suppose that a better understanding of basic numerical concepts and especially of the place-value structure of the Arabic number system during early school years may have an influence on calculation performance later on 1 . Objectives and hypotheses To test this prediction, the current study investigated whether third graders’ addition performance as a measure of calculation ability was affected by their basic numerical knowledge, i.e., their place-value understanding as indexed by transcoding and magnitude comparison in first grade. It is generally agreed that calculation competencies are determined by both individual intelligence (e.g., Aiken, 1971; Hale, Fiorello, Bertin, & Sherman, 2003; 1 Please note that we do not claim that the ability to name a number correctly to be a necessary prerequisite for calculation. However, we propose that the capability of naming a number correctly early on, involving mastery of the place-value structure of the Arabic number system, shall be a valid predictor of later arithmetic competencies such as addition performance. 136
Roid, Prifitera, & Weiss, 1993; see Sattler, 2001 for a review) as well as working memory capabilities (e.g., Geary, Hoard, Byrd-Craven, & DeSoto, 2004; McLean & Hitch, 1999; Swanson & Sachse-Lee, 2001; see Zago, Petit, & Turbelin, 2008 for adult imaging data). Moreover, influences of working memory capabilities have also been associated with children’s development of numerical representations in recent models of typical as well as atypical development of numerical cognition (von Aster & Shalev, 2007; Rubinsten & Henik, 2009, respectively). Against this background it is important to account for rather unspecific effects of intelligence and working memory in our longitudinal analyses because we were only interested in the specific effects of early place-value understanding on later addition performance. So, by the inclusion of these variables (i.e., intelligence and working memory) we aimed at isolating these specific effects as the beta weights observed in the multiple regression analyses are then free from influences of intelligence and/or working memory and can be considered of reflecting the particular influence of children’s early place-value understanding. To particularly investigate influences of the Arabic place-value structure, this study focused on German speaking children. As outlined above, the German language inherits an inversion of tens and units in number words compared to digital notation. In the context of the current study this specificity may allow for a more comprehensive investigation of children’s competencies in place-value understanding. As understanding the principle of reversed sequences of unit and decade-digits poses a particular complicacy (Zuber et al., 2009), a higher error variance for German-speaking participants can be expected compared to participants with other first languages (e.g., Brysbaert, Fias, & Noël, 1998). Since some precursors of mathematical achievement might remain undetected due to low-variance performances, the more not transparent German number-word system may thus allow for a more solid identification of basic numerical precursor competencies as it increases performance variance. 137
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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
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-
Page 151 and 152: Study 4 Early place-value understan
Page 153 and 154: INTRODUCTION Dealing with (multi-di
Page 155 and 156: has been shown that other represent
Page 157 and 158: number system as formalized by its
Page 159 and 160: 124 10024, see Deloche & Seron, 19
Page 161: Please note that in contrast to Miu
Page 165 and 166: effect-based approach has never bee
Page 167 and 168: (ii) In the magnitude comparison ta
Page 169 and 170: problem of capitalizing on differen
Page 171 and 172: Error rate in % Error Categories (c
Page 173 and 174: positively correlated with overall
Page 175 and 176: (ii) Effect approach analyses of pr
Page 177 and 178: whether the carry effect for additi
Page 179 and 180: Non-carry problems: For the percent
Page 181 and 182: from basic numerical concepts to ar
Page 183 and 184: transcoding and/or who showed relat
Page 185 and 186: A word on the validity of the curre
Page 187 and 188: e adapted for the results of the pr
Page 189 and 190: Non-monotonous Model of the Relatio
Page 191 and 192: of the probes unit and/or decade di
Page 193 and 194: the place-value structure of the Ar
Page 195 and 196: Therefore, we believe that this stu
Page 197 and 198: APPENDIX A: Correlation tables of s
Page 199 and 200: Section 4 The neurocognitive underp
Page 201 and 202: ABSTRACT Mental arithmetic and calc
Page 203 and 204: manipulations cannot be distinguish
Page 205 and 206: numbers is assumed to be more preci
Page 207 and 208: correctly bisected by the second nu
Page 209 and 210: Table 1: Behavioural performance in
Page 211 and 212: 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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