Contributors |
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xi | |
Foreword |
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xiii | |
Preface |
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xxi | |
Part I Number and Magnitude in Non-Human Animals |
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1 At the Root of Math: Numerical Abilities in Fish |
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3 | (32) |
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Maria Elena Miletto Petrazzini |
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3 | (1) |
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4 | (1) |
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Problems and Methods of Study |
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4 | (9) |
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6 | (7) |
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Number vs. Continuous Quantities: Is Number More Cognitively Demanding? |
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13 | (3) |
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One or Multiple Systems of Numerical Representation? |
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16 | (3) |
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Ontogeny of Numerical Abilities |
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19 | (2) |
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Similarities in Numerical Abilities among Fish and between Fish and Other Vertebrates |
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21 | (4) |
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Conclusions and Future Directions |
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25 | (2) |
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27 | (1) |
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27 | (8) |
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2 Foundations of Number and Space Representations in Non-Human Species |
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35 | (32) |
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35 | (2) |
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Core Knowledge of Objects |
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37 | (2) |
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39 | (12) |
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Core Knowledge of Geometry |
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51 | (8) |
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59 | (1) |
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59 | (1) |
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60 | (7) |
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3 Numerical Concepts: Grey Parrot Capacities |
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67 | (24) |
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67 | (3) |
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Alex's Non-numerical Capacities |
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70 | (1) |
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Alex's Early Numerical Abilities |
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70 | (4) |
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Training and Testing Methods |
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71 | (1) |
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Labeling of Basic Quantities and Simple Heterogeneous Sets |
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71 | (1) |
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Complex Heterogeneous Sets |
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72 | (2) |
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Alex's More Advanced Numerical Abilities |
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74 | (11) |
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74 | (2) |
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76 | (1) |
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Addition of Small Quantities |
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77 | (3) |
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Ordinality and Equivalence |
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80 | (3) |
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83 | (1) |
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83 | (2) |
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85 | (1) |
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85 | (1) |
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85 | (6) |
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4 Numerical Cognition and Quantitative Abilities in Nonhuman Primates |
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91 | (32) |
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91 | (1) |
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The Question of Animal Counting |
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92 | (5) |
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Relative Quantity Judgments |
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97 | (5) |
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102 | (2) |
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104 | (1) |
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Mechanisms for Representing Quantity and Number |
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105 | (6) |
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111 | (1) |
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112 | (1) |
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112 | (11) |
Part II Number and Magnitude in Infants and Young Children |
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5 Evolutionary and Developmental Continuities in Numerical Cognition |
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123 | (22) |
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123 | (1) |
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Cross-Species Comparisons |
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124 | (4) |
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Numerical Representations in Infancy |
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128 | (4) |
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132 | (1) |
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Causal Relation Between ANS Acuity and Mathematics |
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133 | (4) |
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137 | (2) |
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139 | (1) |
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139 | (6) |
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6 On the Relation between Numerical and Non-Numerical Magnitudes: Evidence for a General Magnitude System |
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145 | (30) |
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145 | (3) |
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Analog Format for Number and Other Magnitudes |
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148 | (2) |
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A General Magnitude System: Three Types of Supporting Evidence |
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150 | (12) |
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Cross-Magnitude Interactions |
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151 | (5) |
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Correlations in Precision of Numerical and Non-Numerical Representations |
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156 | (1) |
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157 | (3) |
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Potential Neural Mechanisms |
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160 | (2) |
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Do Mathematical Concepts Have a Basis in Nonsymbolic Magnitudes? |
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162 | (3) |
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165 | (3) |
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168 | (1) |
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168 | (7) |
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7 Foundations of the Formal Number Concept: How Preverbal Mechanisms Contribute to the Development of Cardinal Knowledge |
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175 | (26) |
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175 | (2) |
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Two Core Mechanisms for Representing Number |
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177 | (8) |
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System 1: Analog Magnitude System (ANS) |
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177 | (4) |
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System 2: Object Tracking System (OTS) |
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181 | (4) |
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185 | (1) |
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Does the ANS and/or OTS Measure Up? |
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186 | (4) |
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186 | (1) |
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187 | (1) |
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188 | (1) |
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189 | (1) |
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Foundations of the Formal Number Concept |
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190 | (2) |
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192 | (1) |
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193 | (8) |
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8 Intuitive Nonsymbolic Arithmetic |
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201 | (24) |
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201 | (3) |
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204 | (2) |
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Adding and Subtracting Nonsymbolic Magnitudes |
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206 | (2) |
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Multiplication and Division of Nonsymbolic Quantity |
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208 | (8) |
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Ratio and Proportion Tasks |
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208 | (4) |
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Serial Multiplication and Division in Infants and Children |
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212 | (4) |
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216 | (1) |
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217 | (8) |
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9 Analog Origins of Numerical Concepts |
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225 | (28) |
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225 | (1) |
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226 | (5) |
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227 | (1) |
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228 | (2) |
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230 | (1) |
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Neural Basis of Analog Number |
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231 | (3) |
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Analog Origins of Number Symbols |
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234 | (3) |
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Neural Substrate of Human Number Symbols |
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237 | (4) |
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The Analog System and Formal Mathematics Achievement |
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241 | (4) |
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245 | (1) |
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246 | (1) |
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246 | (7) |
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10 The Small—Large Divide: A Case of Incompatible Numerical Representations in Infancy |
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253 | (24) |
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253 | (1) |
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Distinct Systems of Representation: Evidence of Continuity across Development and Phylogeny |
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254 | (3) |
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Evidence for Two Systems in Infancy |
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257 | (2) |
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Small versus Small Discriminations |
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258 | (1) |
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258 | (1) |
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Tracking Small Sets with the ANS: Exceptions to the Rule |
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259 | (5) |
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Signal Clarity of the ANS |
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261 | (1) |
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262 | (2) |
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Overcoming the Small—Large Divide |
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264 | (3) |
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Children's Acquisition of Numerical Language |
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264 | (2) |
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Increasing Precision in the ANS across Development |
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266 | (1) |
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Open Questions and Future Directions |
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267 | (2) |
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What Parameters Help Infants Succeed at Small-Large Comparisons? |
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268 | (1) |
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Is the Small-Large Incompatibility Receptive to Feedback or Training? |
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268 | (1) |
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269 | (1) |
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269 | (8) |
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11 The Influence of Multisensory Cues on Representation of Quantity in Children |
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277 | (28) |
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277 | (1) |
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Representation of Number without Language |
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278 | (3) |
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Development of Intersensory Integration |
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281 | (3) |
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Beneficial Effects of Multisensory Stimulation on Cognition and Learning |
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284 | (2) |
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Does Multisensory Information Improve Numerical Abilities? |
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286 | (6) |
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How Does Multisensory Information Improve Numerical Competence? |
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289 | (3) |
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Unanswered Questions and Future Directions |
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292 | (2) |
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294 | (1) |
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295 | (10) |
Part III Number Judgments: Theoretical Perspectives and Evolutionary Foundations |
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12 The Precision and Internal Confidence of Our Approximate Number Thoughts |
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305 | (30) |
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305 | (1) |
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Behavioral and Neural Signatures of the ANS |
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306 | (5) |
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A Psychophysical Model for ANS Representations |
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311 | (16) |
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How to Think of a Weber Fraction (w) in the Approximate Number System (ANS) |
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315 | (5) |
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Weber Fraction Conceptualized as a Scaling Factor |
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320 | (4) |
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How a Weber Fraction (w) Indexes Individual Differences in ANS Precision |
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324 | (3) |
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The Relation between the Weber Fraction and Internal Confidence |
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327 | (2) |
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329 | (1) |
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330 | (5) |
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13 The Evolution of Number Systems |
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335 | (20) |
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335 | (1) |
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The Evolutionary Functions of Number Sense in Nonhuman Species |
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336 | (6) |
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336 | (4) |
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340 | (2) |
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Quantitative Processing in the Evolution of Human Decision Making |
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342 | (6) |
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The Evolutionary Emergence of Symbolic Representation |
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342 | (2) |
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The Evolution of Numerical Heuristics and Decision Making |
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344 | (1) |
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From Continuous Magnitude Perception to Discrete Numerical Representation |
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345 | (1) |
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From Approximation to Numerical Thinking |
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346 | (2) |
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348 | (1) |
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349 | (6) |
Index |
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355 | |