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

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found that only the influence of logarithmic overall distance [constant = 1060.03; B = - 126.03; b = - .72, t = 25.29, p < .001] and problem size [B = -2.25; b = - .58, t = 20.53, p < .001] was significant. Item RTs increased as logarithmic overall distance and problem size decreased. Contrarily, the influence of unit distance [B = 0.26; b = .02, t = 0.72, p = .47] as well as absolute unit distance [B = 1.24; b = .05, t = 1.67, p = .10] was not reliable. Again, these results indicated item RTs as produced by the holistic model not to be determined by unit-decade compatibility, but rather by measures of overall distance and problem size. Decomposed model ANOVA Corroborating our expectations predicting the empirical RTs by the modelled RT data showed a reliable R 2 of .50 [r = .71; F(1, 238) = 241.13.99, p < .001] indicating a shared variance of 50%. More particularly, analyzing the RT data produced by the strictly decomposed model revealed a reliable main effect of compatibility [F(1, 232) = 23.68, p < .001] with response latencies following a compatible number pair being significantly shorter than latencies following a incompatible pair (735 ms vs. 753 ms, respectively). Moreover, this main effect was moderated by the interaction of compatibility and unit distance in the expected direction [F(1, 232) = 6.28, p < .05, see Figure 2, Panel C]: the compatibility effect was reliably more pronounced for number pairs with a large unit distance as compared to pairs with a small unit distance (28 ms vs. 9 ms, respectively). Thereby, these results clearly corroborate the notion of decomposed processing of tens and units. Finally, the main effect of decade distance indicated that number pairs with a large decade distance were responded to faster than number pairs with a small decade distance [699 ms vs. 789 ms, respectively; F(1, 232) = 573.17, p < .001]. No further main effect or interaction was statistically reliable [all F < 1.35, all p > .25]. 254
Regression The stepwise regression analysis on the modelled RT data of the strictly decomposed model resulted in a final model including the predictors difference between the logarithms of the to-be-compared numbers and unit distance [adj. R 2 = .91, R = .95, F(3, 237) = 1199.20, p < .001, see Table 2]. Closer inspection of the beta weights indicated that reaction times increased as the difference between the logarithms of the two numbers decreased and, most importantly, response latencies also increased as the unit distance decreased. As unit distance is negative for incompatible number pairs but positive for compatible pairs this clearly indexed an influence of unit-decade compatibility on item RTs estimated by the strictly decomposed model. Table 2: Predictors included in the final regression model for the decomposed data Predictor B b t sign. Change in R 2 Raw correlation Partial correlation Constant 834.62 - 354.23 < .001 - - - Difference of logarithms -303.40 - .94 48.20 < .001 .89 - .94 - .95 Unit distance - 1.84 - .13 6.90 < .001 .02 - .17 - .41 Again, a multiple regression incorporating the four significant predictors of item RTs as observed by Nuerk et al. (2001) accounted for a considerable amount of variance [adj. R 2 = .85, R = .92, F(5, 235) = 325.56, p < .001] without a substantial loss of predictive power. Moreover, inspecting the beta weights revealed that apart from logarithmic overall distance [constant = 938.76; B = -168.35; b = - .87, t = 34.18, p < .001] and problem size [B = 0.86; b = .20, t = 7.93, p < .001], unit distance [B = -2.02; b = - .15, t = 5.78, p < .001] reliably predicted item RT in the expected direction. Item RT increased as logarithmic overall distance decreased, problem size increased and, most importantly, as unit distance increased; thereby, indicating a significant influence of unit-decade compatibility on Item RT. On the other hand, absolute unit distance was not a reliable predictor of item RTs [B = 1.27; b = .04, t = 1.73, p = 255
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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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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
Page 229 and 230: Study 6 Impairments of the mental n
Page 231 and 232: INTRODUCTION Patients suffering fro
Page 233 and 234: 2005), triplets in which the interv
Page 235 and 236: Visual field assessment showed no s
Page 237 and 238: Table 1: Demographic and clinical d
Page 239 and 240: For 80 bisected triplets the factor
Page 241 and 242: Factorial Analyses (ANOVA and t-tes
Page 243 and 244: Errors: The ANOVA revealed a main e
Page 245 and 246: DISCUSSION The effects of neglect o
Page 247 and 248: decade digits but they even process
Page 249 and 250: strategic value of retrieving this
Page 251 and 252: APPENDIX A Test results of patients
Page 253 and 254: the middle the beneficial effect of
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Page 257 and 258: Study 7 Two-digit number processing
Page 259 and 260: INTRODUCTION The internal represent
Page 261 and 262: all numbers the overlap for larger
Page 263 and 264: digit numbers were engaged. Additio
Page 265 and 266: single node in an ordered sequence
Page 267 and 268: modelling to evaluate the plausibil
Page 269 and 270: A Output Layer 11 99 11 #1 #2 99 In
Page 271 and 272: expected considering their magnitud
Page 273 and 274: the output of the right node would
Page 275 and 276: advantages for these numbers (see a
Page 277 and 278: Finally, a regression analysis only
Page 279: 236) = 417.38, p < .001] and incorp
Page 283 and 284: compared numbers, unit distance, lo
Page 285 and 286: RT in ms 780 770 Compatible Number
Page 287 and 288: esponse latencies increased as dist
Page 289 and 290: Table 5: Predictors included in the
Page 291 and 292: Taken together, these results revea
Page 293 and 294: of overall distance necessarily req
Page 295 and 296: most important predictor in a multi
Page 297 and 298: activation of the output layer is a
Page 299 and 300: The nature of the representation of
Page 301 and 302: Thereby, possible interactions and
Page 303 and 304: decomposed model did not include an
Page 305 and 306: APPERNDIX B Development of connecti
Page 307 and 308: Figure C: Development of connection
Page 309 and 310: Figure E: Development of connection
Page 311 and 312: Interestingly, one might infer from
Page 313 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
Page 317 and 318: tens and units: German-speaking chi
Page 319 and 320: Study 6 examined the influence of t
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Page 323 and 324: GENERAL DISCUSSION In the last sect
Page 325 and 326: to-be-evaluated triplet crossed a d
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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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