(p. xv) List of Figures
(p. xv) List of Figures

1.1 (a) A Circular Celtic Knot. (b) The Chased Chicken Sona pattern. 15

4.3 Congruency effect indicating automating processing of the irrelevant dimension of placevalue. 52

4.5 Mean RTs as a function of comparison type, congruency and (absolute) distance. 59

B5.2 A fingerbased method to multiply any pair of numbers between 5 and 10. 71

B5.3 A fingerbased method to prove that 9 × n = 10(n – 1) + (10 – n). 72

5.1 A fMRI experiment where a wooden block was placed in each hand of the participant. 81

6.2 Schematic depiction of the number interval bisection task. 94

6.3 Depiction of the magnitude comparison distance effect in neglect patients. 96

6.4 Results of the Principal Component Analyses (PCA) of the association between numbers and space. 101

9.2 Initial total amount of objects portioned into two parts. 174

10.1 Examples for symmetric and asymmetric dot patterns. 184

10.2 Examples for ratio bias task. 186 (p. xvi)

10.3 Illustration of rectangle task. 187

11.1 Monkeys show ratio dependence when ordering numerosities. 204

11.2 Infant numerical change detection tasks demonstrates ratio dependence. 205

12.3 Compared guppies and humans in small and large number discrimination. 227

12.4 Ordinal information seems to be already processed at birth in birds. 230

13.1 Example stimulus cards from the investigation of number concept formation in rhesus monkeys. 238

13.4 Systematically controlled tests of relative numerousness judgment in rhesus monkeys. 244

16.1 Aggregate data from 641 2–4yearold children tested on the GiveN task. 297

16.2 Relations among expressive vocabulary, receptive vocabulary, and knowerlevel variables. 299

16.4 Blocks and Water task design. 302

17.1 Example of an animated sequence showing addition of dot arrays. 312

17.2 Example of an animated sequence for a noncanonical problem. 320

18.1 Two examples of the stimuli used in the nonsymbolic number comparison task. 332

18.2 A schematic representation of the neural representation of number. 334

19.1 Agerelated changes in number sense acuity with age. 346

19.2 Numerical distance effects in children, young and older adults, and in MCI patients. 347 (p. xvii)

19.4 Agerelated changes in counting and subitizing speed per item. 349

19.5 Agerelated changes in arithmetic problem size effects. 351

19.8 Strategy selection in young and older adults’ strategy selection in a computational estimation task. 356

19.9 Strategy execution and strategy distribution in older adults compared to younger adults. 357

20.1 A 27body part counting system. 373

21.1 A canonical Collecting gesture. 393

21.2 A canonical Path gesture. 394

21.3 Participants spontaneously produced Path and Collecting gestures. 395

22.1 Compressed 0–100 number scale and logarithmic estimation curve. 408–9

23.1 A test item assessing children’s place value understanding. 419

26.1 Representation of visual cardinality in rhesus monkeys. 458

26.4 Relation between monkey behaviour and numerosityselective neurons. 464

26.6 Neuronal coding of continuous and discrete quantity. 469

26.7 Neuronal coding of proportions. 470

26.8 Semantic associations between signs and numerical categories by single neurons. 472

26.9 Temporal ordering task and single cell responses from the PFC. 476

26.10 Numerical rule coding. 478

27.1 Fourstepdevelopmental model of numerical cognition. 488

27.2 Tentative developmental calculation model emphasizing the dynamic nature of numerical development. 489

27.3 Schematic representation depicting neural networks supporting the acquisition of number skills. 491

28.1 Schematic circuit diagram of basic neurocognitive processes involved in arithmetic. 503

28.2 Canonical brain areas involved in arithmetic problem solving. 506

28.3 Neuroanatomy of posterior parietal cortex regions involved in arithmetic. 507 (p. xviii)

28.4 Brain regions that showed significant differences in activation between arithmetic operations. 512

28.6 Developmental changes in causal network interactions during arithmetic problem solving. 518

28.7 Comparison of ventral visual stream responses in 2nd and 3rd grade children. 520

28.8 Strategy differences in arithmetic problem solving. 522

29.1 Numerals 1–9 in the Thai script. 532

29.2 The three fields in the first phase of the neural network experiment. 535

29.3 Illustration of the second phase of the neural network experiment. 536

29.4 Brain regions commonly involved in number processing. 541

32.1 Reported brain sites whose stimulation through TMS has led to disruption in quantity. 592

32.3 Scheme of the localization of calculation and language interference. 600

32.4 Examples of two patients showing the proximity of sites where Gerstmann symptoms were observed. 601

35.2 Structural abnormalities in young dyscalculic brains suggesting the critical role for the IPS. 653

36.1 Deficits in IPS and AG leading to difficulties both in language and DD. 664

43.1 Illustration of the effect of adaptation on numerosity perception as a function of the number of dots in the probe. 789 (p. xix)

45.1 Speed and accuracy improvements in complex multiplication and subtraction following training. 844

45.2 Schematic illustration of brain activation changes in complex calculation following training. 845

49.1 The working memory tasks. 923

55.1 The related MD&A diagrams for addition and multiplication. 1044

55.2 Mathematicallydesirable and accessible algorithms and diagrams. 1045

55.3 Using the multiplication table for teaching and learning ratio and proportion. 1047

56.1 Scissors tool for creating doubles. 1066

56.2 Grouping boxes and number line. 1067

56.3 The Ghost Box grouping tool. 1068

56.4 Four pedagogical agents. 1068

60.1. The number and colour linear board games. 1144

60.2 Performance of preschoolers from lowincome backgrounds on four numerical tasks. 1145

60.3 The circular number board games. 1146