1. Introduction.- References.- 2. Pre-fraction Concepts of Preschoolers.- Method.- Problem Descriptions and Rationales.- Results.- Problem 1: Skipping Ropes.- Problem 2: Sharing the Crackers.- Problem 3: Sausages.- Problem 4: Swapcards.- Discussion.- References.- 3. Dimensions of Young Children's Conceptions of the Fraction One Half.- Theory.- Method.- Problem Descriptions.- Results.- Accuracy of Responses: One-half (Candysticks and Jellybabies).- Procedures Used.- Accuracy of Responses: Sharing (Snakes and Crackers).- Procedures Used.- Procedures Used by Most Accurate Children: One-half.- Procedures Used by Most Accurate Children: Sharing.- Case of John (4,11): A Strong "Halver".- Case of Julie (5,3): A Weak "Halver".- Case of Jamie (5,3): A Strong Sharer.- Case of Lisa (5,5): A Weak Sharer.- Discussion.- References.- 4. The Social Origins of Pre-fraction Knowledge in Three Year Olds.- Method.- Development of the Interview Plan.- Description of the Interview Tasks.- The Parent Questionnaire and Interview.- Results.- Performance on the Sharing Tasks.- Performance on the Recognition Tasks.- Data from the Parent Interview.- Relationships Between Sharing Competence and Home Factors.- General Discussion.- References.- 5. Higher Order Thinking in Young Children's Engagements with a Fraction Machine.- The Copy Cat Fraction Machine.- Method.- Results.- Copy Cat Performance.- Counting Performance.- Relationships Between Copy Cat Performance and Counting.- The Case of Jayme and One-third.- Discussion.- References.- 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.- Method.- The Counting Tasks.- The Sharing Tasks.- Classification Criteria.- Results.- Counting Tasks.- Sharing Tasks.- Observed Frequencies for Counting and Sharing.- Poor Counters but Good Sharers.- Good Counters but Weak Sharers.- General Discussion.- References.- 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.- Re-distribution by Dealing.- The Most Common Strategies.- Counting in the Re-distribution Task.- Apportioning by Ratios?.- Conclusions.- References.- 11. Cognitive Research on Early Fraction Learning Applied to Classrooms: Two Experiments.- I. An Experiment in Teacher-focused Curriculum Change.- Rationale.- Development Model.- Products and Purposes.- The Development Process.- Reflections and Recommendations.- II. Applying the Didactic Framework.- Method of the Constructivist Teaching Experiment.- Rationale for The Farm.- The Farm: A Didactic Framework.- The Lessons: Description and Commentary.- Discussion.- References.- Appendix: Fractions and Sharing Interview.- 12. The Interaction of Thought, Words, and Deeds in Children's Early Fraction Learning.- The Problem.- Why We Conducted the Teaching Experiment.- Why We Interviewed the Children After Session 2.- A Brief Comment on the Interview Method.- Detail of Rebecca's Solution.- What the Interviews Revealed.- What the Other Children Thought.- Concluding Comments.- References.- 13. On Clinical Methods for Studying Young Children's Mathematics.- Examples of Recent Adaptations.- Children Talking About Other Children's Mathematics.- Using Video Excerpts to Stimulate Recall of Cognitive Processes: Lachlan's Distribution Strategy.- Neutralizing Adult Interviewer Effects.- 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.

Early Fraction learning is centrally of interest to students
and researchersin mathematics education, tackling as it
does one of that discipline's most vexing problems: why are
fractions so difficult to learn and to teach?