Early Fraction Learning

1. Introduction.- References.- 2. Pre-fraction Concepts of Preschoolers.- Method.- Results.- Discussion.- References.- 3. Dimensions of Young Children's Conceptions of the Fraction One Half.- Theory.- Method.- Results.- Discussion.- References.- 4. The Social Origins of Pre-fraction Knowledge in Three Year Olds.- Method.- Results.- General Discussion.- References.- 5. Higher Order Thinking in Young Children's Engagements with a Fraction Machine.- The Copy Cat Fraction Machine.- Copy Cat Performance.- Counting Performance.- Relationships Between Copy Cat Performance and Counting.- The Case of Jayme and One-third.- 6. Fractions as Operators and as Cloning Machines.- The Rational Numbers as Operators.- Fraction Algorithms and Abstract Symbolism.- A More General Construction of Fields of Fractions via Operators.- References.- 7. Preschoolers' Knowledge of Counting and Sharing in Discrete Quantity Settings.- Counting.- The Theory of Counting Types.- Sharing in Discrete Quantity Settings.- The Link Between Counting and Sharing.- The Counting Tasks.- The Sharing Tasks.- Classification Criteria.- Counting Tasks.- Sharing Tasks.- Observed Frequencies for Counting and Sharing.- Poor Counters but Good Sharers.- Good Counters but Weak Sharers.- 8. Preschoolers' Spontaneous Partitioning of Discrete Items.- References.- 9. Cognitive Issues about Dealing.- Dealing and Counting.- Commentary on Tom's Responses.- Cognitive and Problem Solving Aspects of Tom's Responses.- General Discussion.- References.- 10. Sharing by Dealing as Problem Solving.- A Sharing Problem.- Children's Strategies.- Conclusions.- References.- 11. Cognitive Research on Early Fraction Learning Applied to Classrooms: Two Experiments.- I. An Experiment in Teacher-focused Curriculum Change.- II. Applyingthe Didactic Framework.- References.- Appendix: Fractions and Sharing Interview.- 12. The Interaction of Thought, Words, and Deeds in Children's Early Fraction Learning.- The Problem.- What the Interviews Revealed.- Concluding Comments.- References.- 13. On Clinical Methods for Studying Young Children's Mathematics.- Examples of Recent Adaptations.- Artifacts of the Clinical Interview.- References.- 14. A Fraction of Epistemology.- Relations and Reality.- Order and Symmetry Breaking.- Reflection, Disequilibrium, and the Transition to Order.- Information and Action.- The Idea of a Concept.- References.