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

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Again, a multiple stepwise regression analysis was conducted to evaluate the differential influences of the variables depicted in Table 2, Panel B on overall addition performance in third grade. The final model [R = .44, adjusted R 2 = .16, F(4, 90) = 6.99, p < .001, f 2 = .23] included the three predictors a) number of pure inversion transcoding errors, b) compatibility effect, and c) (decade) distance effect (see Table 3). Inspection of the beta weights indicated that children who committed more pure inversion transcoding errors in first grade also exhibited a higher overall error rate in the third grade addition task. Similarly, children with a relatively large compatibility effect in the first grade magnitude comparison task committed more errors in the addition task two years later. This indicated that in both cases flawed mastery of the place-value concept in first grade was associated with an increased overall error rate in the addition task in third grade. Finally, in contrast to the findings of Holloway and Ansari (2009) a larger (decade) distance effect in first grade implied better overall performance in the third grade arithmetic task. Table 3: Regression model predicting the overall number of errors in the addition task on the basis of specific numerical effects Predictor B 95% CI b Change in R 2 t sig. raw partial Constant .45 .34 – .56 8.19 < .001 Inversion errors (1 st grade) .30 - .13 – .48 .33 .12 3.47 < .001 .35 .34 Compatibility effect (1 st grade) .09 - .003 – .18 .18 .04 1.93 < .05 ( * ) .24 .20 Distance effect (1 st grade) - .18 - .38 – .01 - .18 .03 1.87 < .05 ( * ) - .17 - .19 CI = Confidence interval; ( * ) = tested one-sided (iii) Effect approach analyses of predictors and criterion variables: Influence of specific basic numerical competencies on related effects in mental arithmetic Finally, taking a step beyond Holloway and Ansari (2009) the effect approach was generalized to both, predictor and criterion variables. That is, we examined in how far mastery of specific basic numerical competencies, in particular place-value understanding, serves as a predictor for specific effects in later arithmetic performance. In particular, it was investigated 150
whether the carry effect for addition (as requiring correct place-value integration) in grade three can be traced back to indices of basic numerical competency in first grade, such as the distance effect and, more particularly, to indices of place-value understanding such as the compatibility effect or the rate of inversion errors. Additionally, we were interested whether the percentage of errors in carry problems is accounted for best by the same basic numerical competencies as is the error rate in non-carry problems. To evaluate these most differential aspects of longitudinal numerical development three stepwise regression analyses were conducted. All regression analyses incorporated the same predictors but varied with regard to the dependent variable (i.e., carry effect for addition errors vs. error rate in carry problems vs. error rate in non-carry problems). The included predictors were: (i) the (decade) distance as well as (ii) the compatibility effect for errors in the magnitude comparison task, the five disjoint subgroups of transcoding errors as introduced by Zuber et al. (2009): (iii) lexical errors, (iv) inversion errors (v) additive composition errors, (vi) multiplicative composition errors and (vii) combination errors. In addition to these predictors reflecting basic numerical competencies number-unrelated measures of IQ, verbal as well as visuo-spatial WM and CE were also incorporated as predictors. Thereby, it was aimed to distinguish possible nonnumerical influences of intelligence or WM capacity from effects attributable to children’s basic numerical competencies such as transcoding, number magnitude as well as place-value understanding. Please note that for this last step only the results of the regression analyses will be reported in the main text body for reasons of lucidity and brevity. To provide the reader with a full overview over all results nevertheless, all correlation analyses are given in Appendix A. Carry effect: The stepwise multiple regression analysis produced a significant model [R = .29, adjusted R 2 = .06; F(3, 91) = 4.08, p < .05, f 2 = .09] incorporating the first grade CE score and the percentage of pure inversion transcoding errors as the only reliable predictors (see Table 4, Panel A). The positive beta weight for both predictors indicated that a relatively 151
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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
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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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Page 127 and 128: striking similarity between the ori
Page 129 and 130: Inversion influences on estimation
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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 and 162: Please note that in contrast to Miu
Page 163 and 164: Roid, Prifitera, & Weiss, 1993; see
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: (ii) Effect approach analyses of pr
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
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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
Page 213 and 214: Table 3: Brain areas activated by b
Page 215 and 216: portion of the superior frontal gyr
Page 217 and 218: Figure 3: fMRI signal for parametri
Page 219 and 220: The fronto-parietal network and the
Page 221 and 222: Finally, increased activation in th
Page 223 and 224: Inferior parietal cortex, the angul
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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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