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.

On the 26th, 27th, and 28th of February of 1988, a conference was held on the epistemological foundations of mathematical experience as part of the activities of NSF Grant No. MDR-8550463, Child Generated Multiplying and Dividing Algorithms: A Teaching Experiment. I had just completed work on the book Construction of Arithmetical Meanings and Strategies with Paul Cobb and Ernst von Glasersfeld and felt that substantial progress had been made in understanding the early numerical experiences of the six children who were the subjects of study in that book. While the book was in preparation, I was also engaged in the teaching experiment on mUltiplying and dividing algorithms. My focus in this teaching experiment was on investigating the mathematical experiences of the involved children and on developing a language through which those experiences might be expressed. However, prior to immersing myself in the conceptual analysis of the mathematical experiences of the children, I felt that it was crucial to critically evaluate the progress that we felt we had made in our earlier work. It was toward achieving this goal that I organized the conference. When trying to understand the mathematical experiences of a child, one can do no better than to interact with the child in a mathematical context guided by the intention to specify the child's current knowledge and the progress the child might make.