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

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executing the carry by updating the decade digit of the result by the decade digit of the unit sum. Increased activation at similar sites associated with decade crossing triplets may thus implicate processes of unit decade integration to require executive / attentional as well as working memory resources supplementing the more demanding processing of magnitude information in these triplets. More particularly, this indicated that the structural organizing place-value principle of the Arabic number system requires specific processing. Thus, processing place-value information is not an inherent aspect of number magnitude representation but may involve different magnitude-based as well as more general cognitive processes. Moreover, which of these processes are actually performed or not may also depend on the task to be performed. While in number magnitude comparison tens and units seem to be encoded and processed in parallel (cf. Moeller et al., 2009a) performance in a more complex task such as the number bisection task was suggested to involve quite different basic numerical processes including bottom-up processing of specific stimulus characteristics as well as top-down mediated processes of e.g., plausibility evaluations (cf. Moeller, Fischer, Nuerk, & Willmes, 2009c). Eye-tracking evidence on place-value processing in mental addition As described currently, Moeller and colleagues (2009a; 2009c) were able to evaluate processing characteristics for numerical tasks. The authors chose the eye-tracking methodology to differentiate between different processing strategies (Moeller et al., 2009a) or the temporal specificities of different numerical processes (Moeller et al., 2009c). The evaluation of participants’ eye fixation behaviour while engaged in a given task seems to be a valid tool to distinguish between different basic numerical processes because of two general assumptions about eye fixations: (i) According to the eye mind assumption (e.g., Rayner & Pollatsek, 1989) the fixation location of the eyes serves as a reliable indicator of what part of a stimulus is processed at the moment. And (ii) the immediacy assumption (e.g., Rayner & 318
Pollatsek, 1989) states that an object or stimulus is processed as long as it is fixated. Thereby, the number of fixations as well as their durations is a valid predictor of how important information from a specific part of a stimulus (e.g., a particular digit) is in the context of performing the task at hand. Based on above considerations, the eye-tracking methodology should also offer the possibility to differentiate between different basic processes underlying place-value integration. A question pursued in Study 2 of the current thesis. Here, participants’ eye fixation behaviour in an addition verification task was evaluated to investigate possible origins of difficulty in carry addition problems. We were able to identify two distinct processes associated with the increased difficulty of carry addition problems each reflecting processes of place-value integration. On the one hand, performing carry addition problems seemed to be associated with specific unit based calculations indicating the requirement of a carry when reaching or exceeding a certain level (i.e., a unit sum of ≥ 10). This suggested that particular processing of the unit digits of the summand was necessary whenever the sum of these digits becomes a two-digit number and thus the decade digit of the unit sum has to be carried to the decade position to update the decade digit of the result. On the other hand, the results indexed a process of carry execution to add to overall difficulty as well. In carry addition problems the decade digit of the result required specific processing, possibly being associated by the updating of the sum of the decade digits of the summands by the carry from the sum of the unit digits. Thus, being able to dissociate these different processes and to associate them with the processing of specific digits (i.e., units of the summands vs. tens of the result) again argues for a decomposed representation of number magnitude. When assuming the magnitude of a two-digit number to be represented as an integrated entity not retaining any place-value information there should performance in two-digit addition should not be determined by processes that can certainly be attributed to the processing of either tens or units. Thereby, the results of Study 2 not only corroborated earlier findings suggesting a 319
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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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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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Page 305 and 306: APPERNDIX B Development of connecti
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Page 309 and 310: Figure E: Development of connection
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Page 313 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
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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
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Page 337 and 338: Furthermore, converging evidence is
Page 339 and 340: methodological problems. In most of
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Page 343: numbers within the triplets, proces
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Page 351 and 352: correspond to the order of tens and
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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
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