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

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APPENDIX B Factorial analyses (ANOVA and t-tests) differentiating between all three participant groups: Non-bisected triplets (RT and error data): Response latencies: An analysis of variance (ANOVA) incorporating the factors distance to the middle, size relative to the middle and participant group was conducted on z-transformed RTs. The ANOVA revealed that the interaction between size relative to the arithmetical middle and patient group was marginally significant [F(2, 15) = 3.01, p = .08]. Subsequently, Bonferroni-Holm corrected t-tests were conducted to directly test the hypothesis that neglect patients should benefit less from a central number smaller than the arithmetical middle of the interval: Neglect patients did benefit reliably less (-426 ms) from a second number smaller than the triplet’s middle than did either patient controls [-527 ms, t(10) = 1.82, p = .05, onesided] or healthy controls [-747 ms, t(10) = 2.15, p < .05, one-sided a ]. The three-way interaction of distance to the middle, size relative to the middle and participant group [F(2, 15) < 1] was not reliable indicating that the disadvantage for neglect patients was not modulated by distance to the arithmetical middle. Additionally, the ANOVA showed strong main effects for of distance to the middle [F(1, 15) = 48.29, p < .001] and size relative to the middle [F(1, 15) = 38.71, p < .001]. So, triplets in which the central number was numerically far from the arithmetical middle were rejected faster than triplets with a central number numerically close to the middle (4424 ms vs. 5043 ms). Moreover, triplets with a second number smaller than the arithmetical middle were responded to faster than triplets with a second number larger than the middle (4451 ms vs. 5017 ms). Furthermore, the interaction of distance to the middle and size relative to the middle was reliable [F(1, 15) = 13.94, p < .01]. This means that for far numerical distances to a Note: Similar results were obtained when directly contrasting the effects of decade crossing in an one-way ANOVA with subsequent post-hoc Dunnett t-tests: neglect vs. patient control p = .05, neglect vs. healthy control p < .05. 226
the middle the beneficial effect of a central number smaller than the arithmetical middle was more pronounced than for close distances (930 ms vs. 203 ms). Finally, the three participant groups did not differ reliably in terms of RT [F(2, 15) < 1]. Errors: In the error analysis a main effect of group was observed [F(2,15) = 6.83, p < .01]. Bonferroni-Holm corrected t-tests showed that neglect patients committed reliably more errors than control patients [+5.0 %, t(10) = 3.41, p < .01, one-sided] as well as healthy controls [+5.0 %, t(10) = 3.12, p < .01, one-sided]. Additionally, the interaction of distance to the middle and size relative to the middle was significant [F(1, 15) = 5.23, p < 05.]. This indicates that for triplets with a central number numerically close to the arithmetical middle rejection was more error prone when than when the central number was smaller than the middle compared to when it was larger than the middle (6.4 % vs. 3.9 % errors, respectively). However, this pattern was reversed for triplets with a central number far from the middle (smaller: 1.9 % vs. larger: 6.1 %). There were no further significant main effects of the manipulated item characteristics [all F < 1.16, all p > .30 or interactions of group with any of them [all F < 1.11, all p > 13]. Bisected triplets (RT and error data): Decade crossing: In line with the hypothesis, the interaction of decade crossing and participant group was reliable for errors [F(2, 15) = 6.53, p < .01], but not for response latencies [F(2, 15) < 1]. Bonferroni-Holm corrected t-tests were conducted to directly test the hypothesis of neglect patients’ performance being impaired most for triplets crossing into the next decade. Therefore, the absolute effects of decade crossing were contrasted for neglect patients vs. patient controls [t(10) = 0.90, p = .20, one-sided] as well as for neglect patients vs. healthy controls [t(10) = 2.82, p < .05, one-sided b ]. This indicates that the increase of errors b Note: Again, similar results were obtained when directly contrasting the effects of decade crossing in an oneway ANOVA with subsequent post-hoc Dunnett t-tests: neglect vs. patient control p = .37, neglect vs. healthy control p < .01. 227
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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
Page 201 and 202: ABSTRACT Mental arithmetic and calc
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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
Page 225 and 226: primarily necessary for solving the
Page 227 and 228: 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
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Page 249 and 250: strategic value of retrieving this
Page 251: APPENDIX A Test results of patients
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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
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Page 281 and 282: Regression The stepwise regression
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
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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
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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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337
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