Children's Fractional Knowledge

Chapter I. A New Hypothesis Concerning Children's Fractional KnowledgeThe Interference HypothesisThe Separation HypothesisNext Steps Chapter II. Perspectives on Children's Fraction KnowledgeOn Opening the TrapFractions as SchemesMathematics of Living Rather Than Being Chapter III. Operations that Produce Numerical Counting SchemesComplexes of Discrete UnitsRecognition Templates of Perceptual Counting SchemesRecognition Templates of Figurative Counting SchemesNumerical Patterns and the Initial Number SequenceThe Tacitly Nested Number SequenceThe Explicitly Nested Number SequenceAn Awareness of Numerosity: A Quantitative PropertyThe Generalized Number SequenceAn Overview of the Principal Operations of the Numerical Counting SchemesFinal Comments Chapter IV: Articulation of the Reorganization HypothesisPerceptual and Figurative LengthPiaget's Gross, Intensive, and Extensive QuantityComposite Structures as Templates for FragmentingPartitioning and IteratingFinal Comments Chapter V: The Partitive and the Part-Whole SchemesThe Equi-Partitioning SchemeSegmenting to Produce a Connected NumberMaking a Connected Number SequenceAn Attempt to Use Multiplying Schemes in the Construction of Composite Unit FractionsLaura's Simultaneous Partitioning SchemeJason's Partitive and Laura's Part-Whole Fraction SchemesEstablishing Fractional Meaning for Multiple Parts of a StickContinued Absence of Fractional NumbersAn Attempt to Use Units-Coordinating to Produce Improper FractionsDiscussion of the Case Study Chapter VI. The Unit Composition and the Commensurate SchemesThe Unit Fraction Composition SchemeProducing Composite Unit FractionsProducing Fractions Commensurate with One-HalfProducingFractions Commensurate With One-ThirdProducing Fractions Commensurate With Two-ThirdsAn Attempt to Engage Laura in the Construction of the Unit FractionComposition SchemeDiscussion of the Case Study Chapter VII. The Partitive, the Iterative, and the Unit Composition SchemesJoe's Attempts to Construct Composite Unit FractionsAttempts to Construct a Unit Fraction of a Connected NumberPartitioning and Disembedding OperationsJoe's Construction of a Partitive Fraction SchemeJoe's Production of an Improper FractionPatricia's Recursive Partitioning OperationsThe Splitting Operation: Corroboration in Joe and Contraindication in PatriciaA Lack of Distributive ReasoningEmergence of the Splitting Operation in PatriciaEmergence of Joe's Unit Fraction Composition SchemeJoe's Reversible Partitive Fraction SchemeJoe's Construction of the Iterative Fraction SchemeA Constraint in the Children's Unit Fraction Composition SchemeFractional Connected Number SequencesEstablishing Commensurate FractionsDiscussion of the Case Study Chapter VIII. Equi-Partitioning Operations for Connected Numbers: Their Use and InteriorizationMelissa's Initial Fraction SchemesA Reorganization in Melissa's Units-Coordinating SchemeMelissa's Construction of a Fractional Connected Number SequenceTesting the Hypothesis that Melissa Could Construct a CommensurateFractional SchemeMelissa's Use of the Operations that Produce Three Levels of Units in Re-presentationA Child-Generated Fraction Adding SchemeAn Attempt to Bring Forth a Unit Fraction Adding SchemeDiscussion of the Case StudyChapter IX. The Construction of Fraction Schemes Using the Generalized Number SequenceThe Case of Nathan During his 3rd GradeMultiplication of Fractions and Nested FractionsEqual