Children's Fractional Knowledge

Abstract

Chapter I. A New Hypothesis Concerning Children's Fractional Knowledge The Interference Hypothesis The Separation Hypothesis Next Steps Chapter II. Perspectives on Children's Fraction Knowledge On Opening the Trap Fractions as Schemes Mathematics of Living Rather Than Being Chapter III. Operations that Produce Numerical Counting Schemes Complexes of Discrete Units Recognition Templates of Perceptual Counting Schemes Recognition Templates of Figurative Counting Schemes Numerical Patterns and the Initial Number Sequence The Tacitly Nested Number Sequence The Explicitly Nested Number Sequence An Awareness of Numerosity: A Quantitative Property The Generalized Number Sequence An Overview of the Principal Operations of the Numerical Counting Schemes Final Comments Chapter IV: Articulation of the Reorganization Hypothesis Perceptual and Figurative Length Piaget's Gross, Intensive, and Extensive Quantity Composite Structures as Templates for Fragmenting Partitioning and Iterating Final Comments Chapter V: The Partitive and the Part-Whole Schemes The Equi-Partitioning Scheme Segmenting to Produce a Connected Number Making a Connected Number Sequence An Attempt to Use Multiplying Schemes in the Construction of Composite Unit Fractions Laura's Simultaneous Partitioning Scheme Jason's Partitive and Laura's Part-Whole Fraction Schemes Establishing Fractional Meaning for Multiple Parts of a Stick Continued Absence of Fractional Numbers An Attempt to Use Units-Coordinating to Produce Improper Fractions Discussion of the Case Study Chapter VI. The Unit Composition and the Commensurate Schemes The Unit Fraction Composition Scheme Producing Composite Unit Fractions Producing Fractions Commensurate with One-Half ProducingFractions Commensurate With One-Third Producing Fractions Commensurate With Two-Thirds An Attempt to Engage Laura in the Construction of the Unit Fraction Composition Scheme Discussion of the Case Study Chapter VII. The Partitive, the Iterative, and the Unit Composition Schemes Joe's Attempts to Construct Composite Unit Fractions Attempts to Construct a Unit Fraction of a Connected Number Partitioning and Disembedding Operations Joe's Construction of a Partitive Fraction Scheme Joe's Production of an Improper Fraction Patricia's Recursive Partitioning Operations The Splitting Operation: Corroboration in Joe and Contraindication in Patricia A Lack of Distributive Reasoning Emergence of the Splitting Operation in Patricia Emergence of Joe's Unit Fraction Composition Scheme Joe's Reversible Partitive Fraction Scheme Joe's Construction of the Iterative Fraction Scheme A Constraint in the Children's Unit Fraction Composition Scheme Fractional Connected Number Sequences Establishing Commensurate Fractions Discussion of the Case Study Chapter VIII. Equi-Partitioning Operations for Connected Numbers: Their Use and Interiorization Melissa's Initial Fraction Schemes A Reorganization in Melissa's Units-Coordinating Scheme Melissa's Construction of a Fractional Connected Number Sequence Testing the Hypothesis that Melissa Could Construct a Commensurate Fractional Scheme Melissa's Use of the Operations that Produce Three Levels of Units in Re-presentation A Child-Generated Fraction Adding Scheme An Attempt to Bring Forth a Unit Fraction Adding Scheme Discussion of the Case Study Chapter IX. The Construction of Fraction Schemes Using the Generalized Number Sequence The Case of Nathan During his 3rd Grade Multiplication of Fractions and Nested Fractions Equal