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

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Further evidence for place-value influences from more complex tasks and mental arithmetic Place-value information in the number bisection task In the number bisection task triplets involving a decade crossing (e.g., 23_27_31) were more difficult to evaluate than triplets staying within the same decade (e.g., 21_25_29; Nuerk et al., 2002; see also the following discussion on Study 5 below). This is particularly remarkable as this specific item property is not primarily important for solving the task. Instead, the number bisection task was originally assumed to rely on a default solution strategy of magnitude manipulations upon the holistic mental number line, only (e.g., Dehaene & Cohen, 1997; Cohen & Dehaene, 2000). Building on this considerations of performance in the number bisection task to be driven primarily by analogue number magnitude representations (seeing each triplet as a cut-out part of the mental number line), thus, may indicate that place-value information is nevertheless represented by the mental number line. This interpretation is supported by the results of the patient study (Study 5). Not only did the neglect patients exhibit performance impairments associated with them neglecting the left part of the mental number line (cf. Zorzi, Priftis, & Umiltà, 2003) and thus the numerically left part of each triplet also for the whole range of two-digit numbers. More specifically, they had particular problems for decade crossing triplets. The decrease of performance accuracy due to a decade crossing was more pronounced for the neglect patients than for the participants of the control group. On a representational level this may suggest that the processes of integration place-value information are particularly impaired in hemispatial neglect. Furthermore, it was observed that the decade crossing effect for neglect patients interacted with the position of the decade crossing within the triplet. Neglect patients’ performance was reliably worse in triplets in whom the decade crossing occurred between the first and the second number (e.g., 28_32_36) compared to the second and third number (e.g., 24_28_32). So, in addition to their impairment determined by neglecting the relatively smaller 316
numbers within the triplets, processes of place-value, i.e., unit-decade, integration also deteriorated the further left they occurred in the neglected area upon the mental number line. This finding is of particular interest as the similar gradients of impaired magnitude as well as impaired place-value processing again suggested that place-value characteristics of numbers may be inherent in the human number magnitude representation. Moreover, this evidence from neuropsychological patients was further corroborated by a functional MRI study presenting a similar number bisection task to healthy adult participants to evaluate the neural correlates of processes of place-value integration such as the decade crossing effect. As previously observed for the case of number magnitude comparison and in particular the compatibility effect reflecting demands on place-value processing (cf. Wood, Nuerk, & Willmes, 2006) increased activation of the intraparietal sulcus (IPS), bilaterally was associated with specific processing of decade crossings. Interestingly, the IPS is also generally agreed to be essentially involved in the representation of number magnitude in general (see Dehaene, Piazza, Pinel, & Cohen, 2003; Nieder & Dehaene, 2009, for reviews). And again, these data suggest that the place-value structuring principle of the Arabic number system may be implemented into the human representation of numerical magnitude. Nevertheless, triplets crossing into the next decade were also associated with increased neural activation in prefrontal cortex areas. Previously, increased activation of prefrontal sites was observed in mental addition in general and for carry addition problems / borrowing subtraction problems, in particular (e.g., Kong et al, 2005; Imbo, Vandierendonck, & De Rammelaere, 2007). Both, the carry as well as the borrowing procedure in addition and subtraction, respectively, require processes of place-value integration (e.g., updating the sum of the decade digits of the summand by the carry). Generally, this increased prefrontal activation is interpreted to reflect increased demands of cognitive control / working memory / or attentional processes in carry addition / borrowing subtraction problems required for keeping intermediate results in mind, monitoring the columnwise solution process and finally 317
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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
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
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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
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: jittering the positions of the sequ
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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