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

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SUMMARY Before evaluating the implications of the preceding results in the General Discussion a short summary of each of the seven studies shall be given in turn to recapitulate the most important results. Please note that the summary will be focused on illustrating the most important results of the individual studies while the interpretation of the results will be kept at a minimum here and will be given in the Synopsis and Evaluation / Discussion sections of the General Discussion. The current thesis aimed at investigating the influence of the place-value structure of the Arabic number system on numerical cognition in three different respects: (i) In a first set of studies questions regarding the cognitive representation / instantiation of the place value structure were addressed (cf. Section 2). (ii) Attention was paid to the importance of children’s early place-value understanding and its developmental trajectories on the development of numerical / arithmetical competencies such as addition (cf. Section 3). (iii) The neuropsychological underpinnings of place-value processing were examined in a functional MRI study as well as in a patient study evaluating its impairments caused by hemispatial neglect (cf. Section 4). Apart from these empirical data this thesis provided a first evaluation of the representational structure of two-digit numbers (i.e., holistic, strictly decomposed, or hybrid) using a computational modelling approach (cf. Section 5). The summary will follow this structuring as the results of the respective sections will be illustrated in turn Section 2: On task general influences of the place-value structure of the Arabic number system in human number representation Study 1 of this thesis addressed the nature of the internal representation of two-digit number magnitude. Recent research (e.g., Zhang & Wang, 2005; Ganor-Stern, Pinhas, & Tzelgov, 2008) proposed that the decomposed representation of two-digit number magnitude 288
may be determined by the presentational format employed in the respective studies. These authors argue that evidence suggesting separate processing of decade and unit digit magnitudes comes from studies in which both to-be-compared numbers were presented externally (e.g., 37_53). Additionally, they presented empirical evidence indicating holistic rather than decomposed processing of tens and units when only one number is presented and had to be compared to an internally memorized standard (Zhang & Wang, 2005). In the first study, when participants had to compare a given probe to an internal standard, empirical evidence for decomposed processing of two-digit number magnitude was observed as indicated by a standard unit-decade compatibility effect and its characteristic interaction with unit distance (i.e., a more pronounced compatibility effect for large unit distances). These results are hard to reconcile with the notion of a holistic internal representation of two-digit number magnitude in which the place-value structure of the Arabic number system is not retained. In Study 2 the influence of place-value information on the solving of basic arithmetic addition problems was investigated. It was hypothesized that place-value information is especially important in addition problems in which the execution of a carry operation is needed. In such trials the decade digit of the unit digit sum has to be carried to the result’s decade digit to calculate the correct result, thereby, requiring processes of place-value, i.e., unit-decade integration. More particularly, it was expected that the basic processes of (i) calculating the unit sum to determine whether a carry is needed or not, and (ii) the updating of the decade digit of the result should be dissociable by evaluating participants’ eye fixation behaviour. Based on the eye-mind and immediacy assumption (cf. Rayner & Pollatsek, 1989), a more pronounced increase in the number of fixations on the unit digits of the summands should be found if the difficulty of carry addition problems arises from computing the twodigit sum of these unit digits. Otherwise, if updating the decade digit of the result causes the difficulty of carry addition problems, there should be a steeper increase in the number of 289
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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
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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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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 299 and 300: The nature of the representation of
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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: Section 6 Summary 287
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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Page 333 and 334: system and the ordinality of the me
Page 335 and 336: epresentation itself as well as its
Page 337 and 338: Furthermore, converging evidence is
Page 339 and 340: methodological problems. In most of
Page 341 and 342: jittering the positions of the sequ
Page 343 and 344: numbers within the triplets, proces
Page 345 and 346: Pollatsek, 1989) states that an obj
Page 347 and 348: computational models were programme
Page 349 and 350: The developmental influence of plac
Page 351 and 352: correspond to the order of tens and
Page 353 and 354: Asian languages (e.g., Japanese, Ko
Page 355 and 356: indicating that decomposed processi
Page 357 and 358: 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: 337
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