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date: 10 December 2019

(p. xv) List of Figures

(p. xv) List of Figures

  1. 1.1 (a) A Circular Celtic Knot. (b) The Chased Chicken Sona pattern. 15

  2. 4.1 An illustration of the typically found modulation of the SiCE by the task-irrelevant intrapair numerical distance. 49

  3. 4.2 SiCE is larger for unit-decade compatible versus incompatible trials, but is unaffected by the decade size. 51

  4. 4.3 Congruency effect indicating automating processing of the irrelevant dimension of place-value. 52

  5. 4.4 SiCE in comparisons of a set of natural numbers to a given fraction (black lines) increases with the magnitudes of the natural numbers. 54

  6. 4.5 Mean RTs as a function of comparison type, congruency and (absolute) distance. 59

  7. B5.1 Grip aperture during the grasping of wooden blocks with small or large numbers printed on their visible face. 69

  8. B5.2 A finger-based method to multiply any pair of numbers between 5 and 10. 71

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

  10. B5.4 Finger-based method to subtract 1 from 5. 74

  11. 5.1 A fMRI experiment where a wooden block was placed in each hand of the participant. 81

  12. 6.1 Schematic depiction of the SNARC effect. 91

  13. 6.2 Schematic depiction of the number interval bisection task. 94

  14. 6.3 Depiction of the magnitude comparison distance effect in neglect patients. 96

  15. 6.4 Results of the Principal Component Analyses (PCA) of the association between numbers and space. 101

  16. 8.1 Illustration of the abstract-code model. 141

  17. 8.2 Illustration of triple-code theory. 142

  18. 8.3 Illustration of the encoding-complex model. 143

  19. 9.1 Each part of the Change situation. 173

  20. 9.2 Initial total amount of objects portioned into two parts. 174

  21. 10.1 Examples for symmetric and asymmetric dot patterns. 184

  22. 10.2 Examples for ratio bias task. 186 (p. xvi)

  23. 10.3 Illustration of rectangle task. 187

  24. 11.1 Monkeys show ratio dependence when ordering numerosities. 204

  25. 11.2 Infant numerical change detection tasks demonstrates ratio dependence. 205

  26. 12.1 Laboratory studies have shown the existence of quantity abilities in mammals, birds, basal vertebrates, and invertebrates. 216

  27. 12.2 Preferential looking time was recorded to assess whether dogs can spontaneously add numerical information. 220

  28. 12.3 Compared guppies and humans in small and large number discrimination. 227

  29. 12.4 Ordinal information seems to be already processed at birth in birds. 230

  30. 13.1 Example stimulus cards from the investigation of number concept formation in rhesus monkeys. 238

  31. 13.2. Stimuli used to show that squirrel monkeys could make relative numerousness judgments between polygons. 239

  32. 13.3 A macaque monkey is given an array of numerals and must select them in descending order using a cursor to contact the numerals in sequence. 243

  33. 13.4 Systematically controlled tests of relative numerousness judgment in rhesus monkeys. 244

  34. 15.1 Path model of development of SFON and mathematical skills from the age of 3.5 years to the age of 6 years. 281

  35. 15.2 Model illustrating significant direct effects between subitizing-based enumeration, verbal-counting skills, SFON, and mathematics achievement. 282

  36. 16.1 Aggregate data from 641 2–4-year-old children tested on the Give-N task. 297

  37. 16.2 Relations among expressive vocabulary, receptive vocabulary, and knower-level variables. 299

  38. 16.3 Heat map of data from 65 bilingual children tested on the Give-N task in each of their languages separately. 300

  39. 16.4 Blocks and Water task design. 302

  40. 16.5 Example of a Match-to-Sample trial where the correct response picture matches the sample picture on number. 303

  41. 17.1 Example of an animated sequence showing addition of dot arrays. 312

  42. 17.2 Example of an animated sequence for a non-canonical problem. 320

  43. 18.1 Two examples of the stimuli used in the non-symbolic number comparison task. 332

  44. 18.2 A schematic representation of the neural representation of number. 334

  45. 19.1 Age-related changes in number sense acuity with age. 346

  46. 19.2 Numerical distance effects in children, young and older adults, and in MCI patients. 347 (p. xvii)

  47. 19.3 Age-related changes in SNARC effects. 348

  48. 19.4 Age-related changes in counting and subitizing speed per item. 349

  49. 19.5 Age-related changes in arithmetic problem size effects. 351

  50. 19.6 Effects of arithmetic problem complexity with age. 352

  51. 19.7 Age-related changes in strategy repertoire. 354

  52. 19.8 Strategy selection in young and older adults’ strategy selection in a computational estimation task. 356

  53. 19.9 Strategy execution and strategy distribution in older adults compared to younger adults. 357

  54. 20.1 A 27-body part counting system. 373

  55. 21.1 A canonical Collecting gesture. 393

  56. 21.2 A canonical Path gesture. 394

  57. 21.3 Participants spontaneously produced Path and Collecting gestures. 395

  58. 22.1 Compressed 0–100 number scale and logarithmic estimation curve. 408–9

  59. 23.1 A test item assessing children’s place value understanding. 419

  60. 26.1 Representation of visual cardinality in rhesus monkeys. 458

  61. 26.2 Numerosity-selective neurons in the monkey. 460

  62. 26.3 Coding of sequentially presented numerosity. 462

  63. 26.4 Relation between monkey behaviour and numerosity-selective neurons. 464

  64. 26.5 Implementation of numerosity detectors. 466

  65. 26.6 Neuronal coding of continuous and discrete quantity. 469

  66. 26.7 Neuronal coding of proportions. 470

  67. 26.8 Semantic associations between signs and numerical categories by single neurons. 472

  68. 26.9 Temporal ordering task and single cell responses from the PFC. 476

  69. 26.10 Numerical rule coding. 478

  70. 27.1 Four-step-developmental model of numerical cognition. 488

  71. 27.2 Tentative developmental calculation model emphasizing the dynamic nature of numerical development. 489

  72. 27.3 Schematic representation depicting neural networks supporting the acquisition of number skills. 491

  73. 28.1 Schematic circuit diagram of basic neurocognitive processes involved in arithmetic. 503

  74. 28.2 Canonical brain areas involved in arithmetic problem solving. 506

  75. 28.3 Neuroanatomy of posterior parietal cortex regions involved in arithmetic. 507 (p. xviii)

  76. 28.4 Brain regions that showed significant differences in activation between arithmetic operations. 512

  77. 28.5 Neurodevelopmental changes in arithmetic. 515

  78. 28.6 Developmental changes in causal network interactions during arithmetic problem solving. 518

  79. 28.7 Comparison of ventral visual stream responses in 2nd and 3rd grade children. 520

  80. 28.8 Strategy differences in arithmetic problem solving. 522

  81. 29.1 Numerals 1–9 in the Thai script. 532

  82. 29.2 The three fields in the first phase of the neural network experiment. 535

  83. 29.3 Illustration of the second phase of the neural network experiment. 536

  84. 29.4 Brain regions commonly involved in number processing. 541

  85. 31.1 Illustration of the pathway for the processing of number from visual input to a number-selective code. 568

  86. 31.2 Activation of three different contrasts. 573

  87. 32.1 Reported brain sites whose stimulation through TMS has led to disruption in quantity. 592

  88. 32.2 Reported brain sites whose stimulation through TMS has led to disruption in spatial-numerical processes. 596

  89. 32.3 Scheme of the localization of calculation and language interference. 600

  90. 32.4 Examples of two patients showing the proximity of sites where Gerstmann symptoms were observed. 601

  91. 35.1 Causal model of possible interrelationships between biological, cognitive, and simple behavioural levels. 651

  92. 35.2 Structural abnormalities in young dyscalculic brains suggesting the critical role for the IPS. 653

  93. 35.3 Remediation using learning technology. 656

  94. 36.1 Deficits in IPS and AG leading to difficulties both in language and DD. 664

  95. 40.1 Summary of deficient brain function, gray matter, white matter, and brain metabolism in children with DD. 735

  96. 42.1 Fluency scores on the Number Sets Test comparing typically achieving children to children with mathematical learning disability. 772

  97. 42.2 Procedural competence scores for complex addition and the percentage of the simple addition problems that were correctly solved using retrieval. 774

  98. 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)

  99. 43.2 ERSP results illustrating the topography of electrode sites showing the distance effect in phase 1 and in phase 2. 793

  100. 43.3 Illustration of the experimental design. 794

  101. 43.4 Blind and sighted participants’ accuracy scores in the key press estimation task and in the auditory events estimation tasks. 797

  102. 43.5 Blind and sighted participants’ accuracy scores in the footstep estimation task and in the oral verbal estimation tasks. 799

  103. 45.1 Speed and accuracy improvements in complex multiplication and subtraction following training. 844

  104. 45.2 Schematic illustration of brain activation changes in complex calculation following training. 845

  105. 49.1 The working memory tasks. 923

  106. 55.1 The related MD&A diagrams for addition and multiplication. 1044

  107. 55.2 Mathematically-desirable and accessible algorithms and diagrams. 1045

  108. 55.3 Using the multiplication table for teaching and learning ratio and proportion. 1047

  109. 56.1 Scissors tool for creating doubles. 1066

  110. 56.2 Grouping boxes and number line. 1067

  111. 56.3 The Ghost Box grouping tool. 1068

  112. 56.4 Four pedagogical agents. 1068

  113. 58.1 Entry standardized score and number age gain. 1110

  114. 59.1 Potential components of an information-processing model for the relations between conceptual and procedural knowledge. 1128

  115. 60.1. The number and colour linear board games. 1144

  116. 60.2 Performance of preschoolers from low-income backgrounds on four numerical tasks. 1145

  117. 60.3 The circular number board games. 1146

(p. xx)