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Table of Contents
Intro; Preface; Contents; Chapter 1: From Biological Brain to Mathematical Mind: The Long-Term Evolution of Mathematical Thinking; Introduction; Differing Conceptions of Mathematics and a Multi-Contextual Overview; The Biological Brain; How the Brain Makes Sense of Spatial Information and Number; How the Eyes Read Text and Symbolic Expressions; How the Brain Interprets Spoken and Aural Symbolic Expressions; Flexible Use of Symbolism Dually Representing Process or Concept; Making Sense of Mathematical Expressions Dually Representing Operation or Object; Equations
Building a New Framework for Long Term of MathematicsExtending the Framework; How the Brain Makes Sense of More Sophisticated Mathematics; How the Eye Follows a Moving Object, Giving Meaning to Constants and Variables; How Dynamic Movement Can Represent Infinitesimals as Process and Concept; How the Eye Reads Through a Written Proof to Make It Meaningful; The Role of the Limbic System in Enhancing and Inhibiting Mathematical Thinking; Strategies for Enhancing Long-Term Mathematical Thinking; Moving to the Future in Different Communities of Practice; Reflections; References
Chapter 2: Compression and Decompression in Mathematics1References; Chapter 3: How Technology Has Changed What It Means to Think Mathematically; Early Mathematics; The Growth of Mathematics; The Nineteenth-Century Mathematical Revolution; Mathematics in the Digital Age; Experimental Mathematics; Mathematics Education; The Symbol Barrier; References; Chapter 4: Machine Versus Structure of Language via Statistical Universals; Introduction; Statistical Universals Underlying Human Language; Layers of Universals; Mathematical Reasoning on Statistical Universals
Number as RankNumber as Amount; Number as Reification; Revising the Maps; And So ...?; References; Chapter 6: The Body of/in Proof: An Embodied Analysis of Mathematical Reasoning; Mathematical Proof and Logical Deduction; How Is Proof Conceptualized?; Evidence from Emerging Mathematicians; Metaphors for Proof; Evidence from Gesture; Conceptual Roots of Logical Deduction; Cognitive Continuity in Conditionals; Discussion and Conclusions; References; Chapter 7: Math Puzzles as Learning Devices; Introduction; Puzzles, Problems, and Games; The Aha, Gotcha, and Eureka Effects; Objectives of ERM; Psychological and Pedagogical Aspects
Building a New Framework for Long Term of MathematicsExtending the Framework; How the Brain Makes Sense of More Sophisticated Mathematics; How the Eye Follows a Moving Object, Giving Meaning to Constants and Variables; How Dynamic Movement Can Represent Infinitesimals as Process and Concept; How the Eye Reads Through a Written Proof to Make It Meaningful; The Role of the Limbic System in Enhancing and Inhibiting Mathematical Thinking; Strategies for Enhancing Long-Term Mathematical Thinking; Moving to the Future in Different Communities of Practice; Reflections; References
Chapter 2: Compression and Decompression in Mathematics1References; Chapter 3: How Technology Has Changed What It Means to Think Mathematically; Early Mathematics; The Growth of Mathematics; The Nineteenth-Century Mathematical Revolution; Mathematics in the Digital Age; Experimental Mathematics; Mathematics Education; The Symbol Barrier; References; Chapter 4: Machine Versus Structure of Language via Statistical Universals; Introduction; Statistical Universals Underlying Human Language; Layers of Universals; Mathematical Reasoning on Statistical Universals
Number as RankNumber as Amount; Number as Reification; Revising the Maps; And So ...?; References; Chapter 6: The Body of/in Proof: An Embodied Analysis of Mathematical Reasoning; Mathematical Proof and Logical Deduction; How Is Proof Conceptualized?; Evidence from Emerging Mathematicians; Metaphors for Proof; Evidence from Gesture; Conceptual Roots of Logical Deduction; Cognitive Continuity in Conditionals; Discussion and Conclusions; References; Chapter 7: Math Puzzles as Learning Devices; Introduction; Puzzles, Problems, and Games; The Aha, Gotcha, and Eureka Effects; Objectives of ERM; Psychological and Pedagogical Aspects