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

More documents

Recommendations

Info

whose spatial representation of number magnitude is more precise also achieved higher scores in a mathematics achievement test. Additionally, a better spatial representation of number magnitude was also predictive of children’s learning of unfamiliar arithmetic problems. In summary, it can be concluded that for most of above introduced basic numerical representations their importance for numerical cognition in general has already been subject to several studies. However, so far a comprehensive evaluation of the importance of processes of unit-decade integration into the place-value structure of the Arabic number system is still missing. The current thesis aimed at closing this gap by evaluating influences and implications of place-value integration for questions addressing the (i) nature of two-digit number magnitude representation, (i) the development of place-value representations, (iii) the neurocognitive underpinnings of unit-decade integration/place-value understanding, as well as (iv) a computational model replicating basic numerical effects. The motivation behind each of these issues will be introduced briefly. Regarding the nature of two-digit number representation accumulating evidence suggests that decomposed processing of tens and units exists (see above). However, the generality of decomposed processing is still debatable. On the one hand, recent research (e.g., Ganor-Stern et al., 2009; Zhou et al., 2008) claimed that decomposed processing of tens and units in two-digit number comparison is limited to the case when the two to-be-compared numbers are presented simultaneously, i.e., externally when arguing with Zhang & Norman (1995; see also Zhang & Wang, 2005). On the other hand, so far, evidence for decomposed processing of two-digit number magnitude has only been reported for very basic numerical tasks such as number magnitude comparison (e.g., Nuerk et al., 2001; Moeller et al., 2009a; Ratinckx et al., 2006) and number bisection (Nuerk et al., 2002b) while evidence for decomposed processing of multi-digit numbers in more complex arithmetical tasks is still 22
scarce (but see Domahs et al., 2006; 2007). Therefore, the general aim of the current thesis was to provide further evidence for the generality of a decomposed representation of two-digit number magnitude. On the level of basic research on the representation of two-digit number magnitude this goal was pursued by two studies. Based on the origins of the notion of decomposed processing of two-digit numbers Study 1 investigated whether there are indeed representational differences between internal and external magnitude representations as claimed by Zhang and Wang (2005) with only the external representation being decomposed complying with the place-value structure of the Arabic number system while the internal representation of number magnitude were holistic instead. Going beyond the case of such basic numerical tasks Study 2 was intended to offer new evidence for the generalizability of decomposed processing of tens and units to the case of mental arithmetic by evaluating the case of two-digit addition. However, when interested in the general validity of the account of decomposed processing of multi-digit number magnitude an important aspect refers to the influence of place-value properties on the development of numerical cognition which will be addressed in Studies 3 and 4. Study 3 follows up on a recent study by Moeller et al. (2009b) in which the authors presented an alternative account for the development of the spatial representation of number magnitude up to 100 suggesting two separate representations of single- and two-digit numbers; thereby considering place-value properties of the Arabic number system. In Study 3 of the current thesis the validity of their account is evaluated in a trans-lingual study also allowing for an evaluation of influences determined by differences in the number word systems of German and Italian on a non-verbal numeric task, i.e., the number line estimation task. Thus, apart from the obvious investigation of place-value influences in numerical development Study 3 also aimed at extending the notion of decomposed processing of tens and units to the case of the number line task. 23
Page 1 and 2: The influence of the place-value st
Page 3: Ich möchte mich an dieser Stelle b
Page 6 and 7: ZUSAMMENFASSUNG Das wissenschaftlic
Page 8 and 9: Study 5: All for one but not one fo
Page 10 and 11: Classification performance ........
Page 12 and 13: The Arabic place-value structure an
Page 14 and 15: Large problem size > small problem
Page 16 and 17: ABSTRACT ..........................
Page 18 and 19: Study 2: On the cognitive instantia
Page 20 and 21: LIST OF FIGURES Section 1: General
Page 22 and 23: Section 4: The neurocognitive under
Page 24 and 25: LIST OF TABLES Section 1: General I
Page 26: Section 5: A computational model of
Page 29 and 30: Number, the simplest and most unive
Page 31 and 32: limited set of symbols. To achieve
Page 33 and 34: interexponential structuring princi
Page 35 and 36: (i) separating power and base dimen
Page 37 and 38: Abstract Code Model Verbal number C
Page 39 and 40: of the Arabic number system within
Page 41 and 42: previous studies on two-digit numbe
Page 43 and 44: eview), questions concerning repres
Page 45 and 46: esponses are easier (i) for problem
Page 47: 2009c). Based on the framework of t
Page 51 and 52: y Studies 5 and 6 of the current th
Page 53 and 54: OVERVIEW OVER THE STUDIES INCORPORA
Page 55 and 56: a view was corroborated by a variet
Page 57 and 58: would indicate influences of verbal
Page 59 and 60: 2. Study 6: Impairments of the ment
Page 61 and 62: eplicate processes of unit-decade i
Page 63 and 64: determinants shall be reviewed. Mor
Page 65 and 66: Study 6: Hoeckner, S., Moeller, K.,
Page 67 and 68: Section 2 Task invariant influences
Page 69 and 70: ABSTRACT Over the last years, evide
Page 71 and 72: The authors used a magnitude compar
Page 73 and 74: Such congruency is important as mos
Page 75 and 76: Procedure: After ten randomly chose
Page 77 and 78: compatibility effect for large unit
Page 79 and 80: 2004). So, increasing the maximum u
Page 81 and 82: proportions in earlier studies. The
Page 83 and 84: CONCLUSIONS In summary, the conclus
Page 86 and 87: Study 2 On the cognitive instantiat
Page 88 and 89: INTRODUCTION One of the probably mo
Page 90 and 91: eye fixation behaviour for the eval
Page 92 and 93: (see Rayner, 1998 for a review). Tr
Page 94 and 95: Procedure: After being seated with
Page 96 and 97: adjustment of the degrees of freedo
Page 98 and 99:
Eye fixation behaviour TRT in ms To
Page 100 and 101:
or the decade digits was found (393
Page 102 and 103:
egressions were directed to the sec
Page 104 and 105:
the requirement of a carry is deter
Page 106 and 107:
In line with this prediction, total
Page 108 and 109:
Instead, it seems to involve differ
Page 111 and 112:
Section 3 On the importance of plac
Page 113 and 114:
ABSTRACT The mental number line of
Page 115 and 116:
preschoolers aged 4). However, the
Page 117 and 118:
still represented even when only th
Page 119 and 120:
specificity. Yet, in adults it has
Page 121 and 122:
METHOD The reported experiment was
Page 123 and 124:
child separately to obtain estimati
Page 125 and 126:
indicated that Italian children’s
Page 127 and 128:
striking similarity between the ori
Page 129 and 130:
Inversion influences on estimation
Page 131 and 132:
% overall Estimation Error 25 20 15
Page 133 and 134:
seem to be the origin of the perfor
Page 135 and 136:
this system. In this case no langua
Page 137 and 138:
two-linear model with one linear re
Page 139 and 140:
and 40 is 10 times as large as the
Page 141 and 142:
APPENDIX A A remark on how to evalu
Page 143 and 144:
Splitting the overall interval in t
Page 145 and 146:
much closer to the proposed breakpo
Page 147 and 148:
the individual approach may be more
Page 149 and 150:
Contrarily, data produced by a two-
Page 151 and 152:
Study 4 Early place-value understan
Page 153 and 154:
INTRODUCTION Dealing with (multi-di
Page 155 and 156:
has been shown that other represent
Page 157 and 158:
number system as formalized by its
Page 159 and 160:
124 10024, see Deloche & Seron, 19
Page 161 and 162:
Please note that in contrast to Miu
Page 163 and 164:
Roid, Prifitera, & Weiss, 1993; see
Page 165 and 166:
effect-based approach has never bee
Page 167 and 168:
(ii) In the magnitude comparison ta
Page 169 and 170:
problem of capitalizing on differen
Page 171 and 172:
Error rate in % Error Categories (c
Page 173 and 174:
positively correlated with overall
Page 175 and 176:
(ii) Effect approach analyses of pr
Page 177 and 178:
whether the carry effect for additi
Page 179 and 180:
Non-carry problems: For the percent
Page 181 and 182:
from basic numerical concepts to ar
Page 183 and 184:
transcoding and/or who showed relat
Page 185 and 186:
A word on the validity of the curre
Page 187 and 188:
e adapted for the results of the pr
Page 189 and 190:
Non-monotonous Model of the Relatio
Page 191 and 192:
of the probes unit and/or decade di
Page 193 and 194:
the place-value structure of the Ar
Page 195 and 196:
Therefore, we believe that this stu
Page 197 and 198:
APPENDIX A: Correlation tables of s
Page 199 and 200:
Section 4 The neurocognitive underp
Page 201 and 202:
ABSTRACT Mental arithmetic and calc
Page 203 and 204:
manipulations cannot be distinguish
Page 205 and 206:
numbers is assumed to be more preci
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
Page 247 and 248:
decade digits but they even process
Page 249 and 250:
strategic value of retrieving this
Page 251 and 252:
APPENDIX A Test results of patients
Page 253 and 254:
the middle the beneficial effect of
Page 255 and 256:
229
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
Page 279 and 280:
236) = 417.38, p < .001] and incorp
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
Page 293 and 294:
of overall distance necessarily req
Page 295 and 296:
most important predictor in a multi
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
Page 303 and 304:
decomposed model did not include an
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 and 318:
tens and units: German-speaking chi
Page 319 and 320:
Study 6 examined the influence of t
Page 321 and 322:
295
Page 323 and 324:
GENERAL DISCUSSION In the last sect
Page 325 and 326:
to-be-evaluated triplet crossed a d
Page 327 and 328:
specific processing of place-value
Page 329 and 330:
explanation of the decade consisten
Page 331 and 332:
verbal recoding of tens and units d
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?