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

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early place-value understanding for the later development of numerical / arithmetical competencies (see also Study 2 for influences of the place-value structure of the Arabic number system on adult addition performance). Section 4: The neuro-cognitive underpinnings of place-value integration In Study 5, the neural correlates of place-value integration were investigated by functional MRI. In a previous study Nuerk, Geppert, van Herten, and Willmes (2002) found that in a verification version of the number bisection task triplets crossing a decade boundary were responded to significantly slower than triplets that stayed within the same decade (e.g., 25_28_31 vs. 23_26_29). The authors interpreted this as indicating the need for additional processing of place-value information to evaluate the distance between e.g., the second and the third number when crossing a decade boundary. The interest of the current study was to examine whether there is a specific neural correlate of place-value processing, implicating a cortex area with higher neural activity while processing triplets crossing into the next decade compared to triplets with no decade crossing. As previously observed by Wood, Nuerk, and Willmes (2006, see also Knops, Nuerk, Sparing, Foltys, & Willmes, 2006), processes of unitdecade integration were associated with more pronounced activation in the IPS, a cortex location affiliated with the processing of magnitude information in general. However, as overall range (i.e., the distance between the two outer numbers) was accounted for in the parametric analyses for triplets with and without a decade crossing this activation dissimilarity cannot be explained by differences in holistic magnitude measures. Instead, the data argue for specific processing demands associated with the required processing of placevalue information for triplets crossing into the next decade meaning that place-value information is not only retained in the representation of two-digit number magnitude but also explicitly processed when participants encounter two-digit numbers. Thereby, these results indicate a specific neural correlate of place-value processing / integration. 292
Study 6 examined the influence of the disruption of the mental number in neglect patients – as first reported by Zorzi, Priftis, and Umiltà (2002) – on the processing of placevalue information. Zorzi and co-workers (2002; see also Zorzi, Priftis, Meneghello, Marenzi, & Umiltà, 2006) observed that, when asked to indicate the midpoint of a numerical interval, patients’ perceived midpoints of the intervals were misplaced to the right (reporting 7 as being midway between 1 and 9) similarly to these patients impairments in line bisection. In the present study, interest was paid to the question whether neglect-related deficiencies also affect the processing of place-value information as conceptualized by decade crossings in a verification version of the number bisection task (e.g., 23_26_29 vs. 25_28_31; see also Study 5). Results showed that neglect patients’ performance for triplets crossing a decade boundary was significantly worse than performance of the control group. Furthermore, neglect patients were particularly impaired when the decade crossing occurred between the first and the second number of the triplet rather than when the decade crossing occurred between the second and the third number (e.g., 28_31_34 vs. 25_28_31). These results denote that not only the processing of number magnitude information but also the representation of the placevalue structure of the Arabic number system is impaired in neglect patients. Section 5: A computational model of place-value integration in two-digit number comparison Finally, Study 7 provided first computational evidence helping to differentiate between three different representational formats of two-digit numbers (i.e., holistic, strictly decomposed, or hybrid). Particular attention was paid to the dissociation of the strictly decomposed and the hybrid model as there is currently no empirical data that allows distinguishing between these two assumptions. To pursue these issues three computational models were programmed realizing an either holistic, decomposed or hybrid representation of two-digit number magnitude. In both the comparison of a given number to a fixed as well as a 293
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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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Page 269 and 270: A Output Layer 11 99 11 #1 #2 99 In
Page 271 and 272: expected considering their magnitud
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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
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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 and 314: Section 6 Summary 287
Page 315 and 316: may be determined by the presentati
Page 317: tens and units: German-speaking chi
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Page 323 and 324: GENERAL DISCUSSION In the last sect
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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
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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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