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Children's Fractional Knowledge elegantly tracks the construction of knowledge, both by children learning new methods of reasoning and by the researchers studying their methods. The book challenges the widely held belief that children's whole number knowledge is a distraction from their learning of fractions by positing that their fractional learning involves reorganizing-not simply using or building upon-their whole number knowledge. This hypothesis is explained in detail using examples of actual grade-schoolers approaching problems in fractions including the schemes they construct to relate parts to a whole, to produce a fraction as a multiple of a unit part, to transform a fraction into a commensurate fraction, or to combine two fractions multiplicatively or additively.

These case studies provide a singular journey into children's mathematics experience, which often varies greatly from that of adults. Moreover, the authors' descriptive terms reflect children's quantitative operations, as opposed to adult mathematical phrases rooted in concepts that do not reflect-and which in the classroom may even suppress-youngsters' learning experiences.

Highlights of the coverage:

• Toward a formulation of a mathematics of living instead of being
• Operations that produce numerical counting schemes
• Case studies: children's part-whole, partitive, iterative, and other fraction schemes
• Using the generalized number sequence to produce fraction schemes
• Redefining school mathematics

This fresh perspective is of immediate importance to researchers in mathematics education. With the up-close lens onto mathematical development found in Children's Fractional Knowledge, readers can work toward creating more effective methods for improving young learners' quantitative reasoning skills.

Children's Fractional Knowledge

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