Epistemological Foundations of Mathematical Experience

1 Philosophical and Psychological Aspects of Constructivism.- Foundationalism and Constructivism.- Psychological Constructivism.- Foundations Revisited.- Overview of Chapters.- 2 The Import of Fodor's Anti-Constructivist Argument.- Fodor's Argument.- Reconstructing the Argument.- What's Wrong with Contemporary Models of Representation?.- Why Should We Care?.- Interactivism: Outline of a Solution.- Conclusions.- 3 The Learning Paradox: A Plausible Counterexample.- A Guiding Analogy.- An Example of the Learning Paradox.- A Weak Form of the Innatist Hypothesis.- Learning the Initial Number Sequence.- A More General Reformulation of Learning.- Final Comments.- 4 Abstraction, Re-Presentation, and Reflection: An Interpretation of Experience and Piaget's Approach.- Reflection.- Abstraction.- Re-Presentation.- The Power of Symbols.- Piaget's Theory of Abstraction.- Form and Content.- Scheme Theory.- Four kinds of Abstraction.- The Question of Awareness.- Conclusion.- Philosophical Postscript.- 5 A Pre-Logical Model of Rationality.- Rationality as Logic.- Some Thoughts About Thought.- The Nature of Rationality.- The Necessity of Rationality.- The Rationality of Rationality.- The Rationality of Necessity.- The Nurturance of Rationality.- Conclusion.- 6 Recursion and the Mathematical Experience.- What is Recursion?.- What Contributes to Children's Mathematical Experience?.- Mathematical Knowledge Building as a Recursive Activity.- Summary.- 7 The Role Mathematical Transformations and Practice in Mathematical Development.- Transfer Problem Illustration.- Plan of Chapter.- Acquiring Mental Maps as a Metaphor for Interactive Knowing.- Infant Number Skills.- Addition and Subtraction.- Preschoolers' Understanding of One-to-One Correspondence.- Children's Acquisition of Some Algebraic Manipulations.- Summary and Conclusions.- 8 The Concept of Exponential Functions: A Student's Perspective.- The Traditional Account of Exponential Expressions and Exponential Functions.- Method.- Results.- Conclusions.- Discussion: Implications of Findings for Mathematics Education.- 9 Constructive Aspects of Reflective Abstraction in Advanced Mathematics.- Mathematical Knowledge and its Acquisition.- The Constructive Aspect of the Reflective Abstraction.- My Research Program.- Specific Mathematics Topics.- 10 Reflective Abstraction in Humanities Education: Thematic Images and Personal Schemas.- Piaget, Education, and Hermeneutics.- Reflective Abstraction of "Being Romantic".- Equilibration and the Cognitive Compensations.- An Integrated Guiding Image.- 11 Enhancing School Mathematical Experience Through Constructive Computing Activity.- Aspects of Significant School Mathematics Experiences.- Experiential Aspects of Constructive Computing Activities.- Summary.- 12 To Experience is to Conceptualize: A Discussion of Epistemology and Mathematical Experience.- Construction of Mathematical Thought.- Curriculum and Pedagogy.- Methodology.- Mathematics Education as Paideia.- Postscript.- References.- Author Index.