Children's Fractional Knowledge (eBook)

Preface 6
Acknowledgments 8
Foreword 9
Contents 13
List of Figures 19
A New Hypothesis Concerning Children’s Fractional Knowledge 22
The Interference Hypothesis 23
The Separation Hypothesis 26
A Sense of Simultaneity and Sequentiality 27
Establishing Two as Dual 28
Establishing Two as Unity 29
Recursion and Splitting 30
Distribution and Simultaneity 31
Splitting as a Recursive Operation 32
Next Steps 33
Perspectives on Children’s Fraction Knowledge 34
On Opening the Trap 35
Invention or Construction? 36
First-Order and Second-Order Mathematical Knowledge 37
Mathematics of Children 37
Mathematics for Children 38
Fractions as Schemes 39
The Parts of a Scheme 41
Learning as Accommodation 42
The Sucking Scheme 42
The Structure of a Scheme 43
Seriation and Anticipatory Schemes 45
Mathematics of Living Rather Than Being 46
Operations That Produce Numerical Counting Schemes 47
Complexes of Discrete Units 47
Recognition Templates of Perceptual Counting Schemes 49
Collections of Perceptual Items 49
Perceptual Lots 50
Recognition Templates of Figurative Counting Schemes 52
Numerical Patterns and the Initial Number Sequence 55
The Tacitly Nested Number Sequence 58
The Explicitly Nested Number Sequence 61
An Awareness of Numerosity: A Quantitative Property 62
The Generalized Number Sequence 63
An Overview of the Principal Operations of the Numerical Counting Schemes 65
The Initial Number Sequence 65
The Tacitly Nested and the Explicitly Nested Number Sequences 65
Final Comments 67
Articulation of the Reorganization Hypothesis 68
Perceptual and Figurative Length 69
Piaget’s Gross, Intensive, and Extensive Quantity 70
Gross Quantitative Comparisons 71
Intensive Quantitative Comparisons 71
An Awareness of Figurative Plurality in Comparisons 72
Extensive Quantitative Comparisons 74
Composite Structures as Templates for Fragmenting 76
Experiential Basis for Fragmenting 77
Using Specific Attentional Patterns in Fragmenting 78
Number Sequences and Subdividing a Line 83
Partitioning and Iterating 86
Levels of Fragmenting 87
Final Comments 89
Operational Subdivision and Partitioning 91
Partitioning and Splitting 92
The Partitive and the Part-Whole Schemes 94
The Equipartitioning Scheme 94
Breaking a Stick into Two Equal Parts 94
Composite Units as Templates for Partitioning 95
Segmenting to Produce a Connected Number 97
Equisegmenting vs. Equipartitioning 97
The Dual Emergence of Quantitative Operations 99
Making a Connected Number Sequence 99
An Attempt to Use Multiplying Schemes in the Construction of Composite Unit Fractions 102
Provoking the Children’s use of Units-Coordinating Schemes 102
An Attempt to Engender the Construction of Composite Unit Fractions 105
Conflating Units When Finding Fractional Parts of a 24-Stick 106
Operating on Three Levels of Units 108
Necessary Errors 109
Laura’s Simultaneous Partitioning Scheme 111
An Attempt to Bring Forth Laura’s Use of Iteration to Find Fractional Parts 114
Jason’s Partitive and Laura’s Part-Whole Fraction Schemes 117
Lack of the Splitting Operation 117
Jason’s Partitive Unit Fraction Scheme 119
Laura’s Independent Use of Parts 121
Laura’s Part-Whole Fraction Scheme 126
Establishing Fractional Meaning for Multiple Parts of a Stick 129
A Recurring Internal Constraint in the Construction of Fraction Operations 131
Continued Absence of Fractional Numbers 132
An Attempt to Use Units-Coordinating to Produce Improper Fractions 133
A Test of the Iterative Fraction Scheme 135
Discussion of the Case Study 137
The Construction of Connected Numbers and the Connected Number Sequence 137
On the Construction of the Part- Whole and Partitive Fraction Schemes 138
The Splitting Operation 140
The Unit Composition and the Commensurate Schemes 142
The Unit Fraction Composition Scheme 143
Jason’s Unit Fraction Composition Scheme 144
Corroboration of Jason’s Unit Fraction Composition Scheme 145
Laura’s Apparent Recursive Partitioning 147
Producing Composite Unit Fractions 148
Laura’s Reliance on Social Interaction When Explaining Commensurate Fractions 152
Further Investigation into the Children’s Explanations and Productions 155
Producing Fractions Commensurate with One-Half 157
Producing Fractions Commensurate with One-Third 161
Producing Fractions Commensurate with Two-Thirds 166
An Attempt to Engage Laura in the Construction of the Unit Fraction Composition Scheme 167
The Emergence of Recursive Partitioning for Laura 170
Laura’s Apparent Construction of a Unit Fraction Composition Scheme 172
Progress in Partitioning the Results of a Prior Partition 176
Discussion of the Case Study 180
The Unit Fraction Composition Scheme and the Splitting Operation 181
Independent Mathematical Activity and the Splitting Operation 182
Independent Mathematical Activity and the Commensurate Fraction Scheme 182
An Analysis of Laura’s Construction of the Unit Fraction Composition Scheme 183
Laura’s Apparent Construction of Recursive Partitioning and the Unit Fraction Composition Scheme 188
The Partitive, the Iterative, and the Unit Composition Schemes 189
Joe’s Attempts to Construct Composite Unit Fractions 190
Attempts to Construct a Unit Fraction of a Connected Number 192
Partitioning and Disembedding Operations 194
Joe’s Construction of a Partitive Fraction Scheme 198
Joe’s Production of an Improper Fraction 203
Patricia’s Recursive Partitioning Operations 206
The Splitting Operation: Corroboration in Joe and Contraindication in Patricia 206
A Lack of Distributive Reasoning 209
Emergence of the Splitting Operation in Patricia 211
Emergence of Joe’s Unit Fraction Composition Scheme 213
Joe’s Reversible Partitive Fraction Scheme 215
Fractions Beyond the Fractional Whole: Joe’s Dilemma and Patricia’s Construction 217
Joe’s Construction of the Iterative Fraction Scheme 222
A Constraint in the Children’s Unit Fraction Composition Scheme 226
Fractional Connected Number Sequences 229
Establishing Commensurate Fractions 232
Discussion of the Case Study 235
Composite Unit Fractions: Joe 235
Joe’s Partitive Fraction Scheme 236
Emergence of the Splitting Operation and the Iterative Fraction Scheme: Joe 237
Emergence of Recursive Partitioning and Splitting Operations: Patricia 238
The Construction of the Iterative Fraction Scheme 239
Stages in the Construction of Fraction Schemes 240
Equipartitioning Operations for Connected Numbers: Their Use and Interiorization 242
Melissa’s Initial Fraction Schemes 242
Contraindication of Recursive Partitioning in Melissa 244
Reversibility of Joe’s Unit Fraction Composition Scheme 245
A Reorganization in Melissa’s Units-Coordinating Scheme 248
Melissa’s Construction of a Fractional Connected Number Sequence 253
Testing the Hypothesis that Melissa Could Construct a Commensurate Fraction Scheme 258
Melissa’s Use of the Operations that Produce Three Levels of Units in Re- presentation 264
Repeatedly Making Fractions of Fractional Parts of a Rectangular Bar 264
Melissa Enacting a Prior Partitioning by Making a Drawing 268
A Test of Accommodation in Melissa’s Partitioning Operations 271
A Further Accommodation in Melissa’s Recursive Partitioning Operations 273
A Child-Generated Fraction Adding Scheme 277
An Attempt to Bring Forth a Unit Fraction Adding Scheme 280
Discussion of the Case Study 283
The Iterative Fraction Scheme 285
Melissa’s Interiorization of Operations that Produce Three Levels of Units 286
On the Possible Construction of a Scheme of Recursive Partitioning Operations 288
The Children’s Meaning of Fraction Multiplication 290
A Child-Generated vs. a Procedural Scheme for Adding Fractions 292
The Construction of Fraction Schemes Using the Generalized Number Sequence 293
The Case of Nathan During His Third Grade 293
Nathan’s Generalized Number Sequence 294
Developing a Language of Fractions 295
Reasoning Numerically to Name Commensurate Fractions 300
Corroboration of the Splitting Operation for Connected Numbers 302
Renaming Fractions: An Accommodation of the IFS: CN 304
Construction of a Common Partitioning Scheme 305
Constructing Strategies for Adding Unit Fractions with Unlike Denominators 307
Multiplication of Fractions and Nested Fractions 311
Equal Fractions 314
Generating a Plurality of Fractions 315
Working on a Symbolic Level 317
Construction of a Fraction Composition Scheme 319
Constraining How Arthur Shared Four-Ninths of a Pizza Among Five People 320
Testing the Hypothesis Using TIMA: Bars 323
Discussion of the Case Study 326
The Reversible Partitive Fraction Scheme 326
The Common Partitioning Scheme and Finding the Sum of Two Fractions 327
The Fractional Composition Scheme 329
The Partitioning and Fraction Schemes 331
The Partitioning Schemes 331
The Equipartitioning Scheme 331
The Simultaneous Partitioning Scheme 332
The Splitting Scheme 333
The Equipartitioning Scheme for Connected Numbers 335
The Splitting Scheme for Connected Numbers 336
The Distributive Partitioning Scheme 337
The Fraction Schemes 338
The Part-Whole Fraction Scheme 338
The Partitive Fraction Scheme 339
The Unit Fraction Composition Scheme 344
The Fraction Composition Scheme 346
The Iterative Fraction Scheme 349
The Unit Commensurate Fraction Scheme 351
The Equal Fraction Scheme 352
School Mathematics vs. “School Mathematics” 353
Continuing Research on Students’ Fraction Schemes 357
Research on Part-Whole Conceptions of Fractions 358
Transcending Part-Whole Conceptions 360
The Splitting Operation 361
Students’ Development Toward Algebraic Reasoning 364
References 369
Index 374