Steffe / Olive Children's Fractional Knowledge

1. Auflage 2009
ISBN: 978-1-4419-0591-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 364 Seiten

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.

Autoren/Hrsg.

Steffe, Leslie P.

Olive, John

Inhaltsverzeichnis

1;Preface;6
2;Acknowledgments;8
3;Foreword;9
4;Contents;13
5;List of Figures;19
6;A New Hypothesis Concerning Children’s Fractional Knowledge;22
6.1;The Interference Hypothesis;23
6.2;The Separation Hypothesis;26
6.3;A Sense of Simultaneity and Sequentiality;27
6.3.1;Establishing Two as Dual;28
6.3.2;Establishing Two as Unity;29
6.4;Recursion and Splitting;30
6.4.1;Distribution and Simultaneity;31
6.4.2;Splitting as a Recursive Operation;32
6.5;Next Steps;33
7;Perspectives on Children’s Fraction Knowledge;34
7.1;On Opening the Trap;35
7.1.1;Invention or Construction?;36
7.1.2;First-Order and Second-Order Mathematical Knowledge;37
7.1.3;Mathematics of Children;37
7.1.4;Mathematics for Children;38
7.2;Fractions as Schemes;39
7.2.1;The Parts of a Scheme;41
7.2.2;Learning as Accommodation;42
7.2.3;The Sucking Scheme;42
7.2.4;The Structure of a Scheme;43
7.2.5;Seriation and Anticipatory Schemes;45
7.3;Mathematics of Living Rather Than Being;46
8;Operations That Produce Numerical Counting Schemes;47
8.1;Complexes of Discrete Units;47
8.2;Recognition Templates of Perceptual Counting Schemes;49
8.2.1;Collections of Perceptual Items;49
8.2.2;Perceptual Lots;50
8.3;Recognition Templates of Figurative Counting Schemes;52
8.4;Numerical Patterns and the Initial Number Sequence;55
8.5;The Tacitly Nested Number Sequence;58
8.6;The Explicitly Nested Number Sequence;61
8.7;An Awareness of Numerosity: A Quantitative Property;62
8.8;The Generalized Number Sequence;63
8.9;An Overview of the Principal Operations of the Numerical Counting Schemes ;65
8.9.1;The Initial Number Sequence;65
8.9.2;The Tacitly Nested and the Explicitly Nested Number Sequences;65
8.10;Final Comments;67
9;Articulation of the Reorganization Hypothesis;68
9.1;Perceptual and Figurative Length;69
9.2;Piaget’s Gross, Intensive, and Extensive Quantity;70
9.3;Gross Quantitative Comparisons;71
9.4;Intensive Quantitative Comparisons;71
9.5;An Awareness of Figurative Plurality in Comparisons;72
9.6;Extensive Quantitative Comparisons;74
9.7;Composite Structures as Templates for Fragmenting;76
9.8;Experiential Basis for Fragmenting;77
9.9;Using Specific Attentional Patterns in Fragmenting;78
9.10;Number Sequences and Subdividing a Line;83
9.11;Partitioning and Iterating;86
9.12;Levels of Fragmenting;87
9.13;Final Comments;89
9.14;Operational Subdivision and Partitioning;91
9.15;Partitioning and Splitting;92
10;The Partitive and the Part-Whole Schemes;94
10.1;The Equipartitioning Scheme;94
10.1.1;Breaking a Stick into Two Equal Parts;94
10.1.2;Composite Units as Templates for Partitioning;95
10.2;Segmenting to Produce a Connected Number;97
10.2.1;Equisegmenting vs. Equipartitioning;97
10.2.2;The Dual Emergence of Quantitative Operations;99
10.3;Making a Connected Number Sequence;99
10.4;An Attempt to Use Multiplying Schemes in the Construction of Composite Unit Fractions;102
10.4.1;Provoking the Children’s use of Units-Coordinating Schemes;102
10.4.2;An Attempt to Engender the Construction of Composite Unit Fractions;105
10.4.3;Conflating Units When Finding Fractional Parts of a 24-Stick;106
10.4.4;Operating on Three Levels of Units;108
10.4.5;Necessary Errors;109
10.5;Laura’s Simultaneous Partitioning Scheme;111
10.5.1;An Attempt to Bring Forth Laura’s Use of Iteration to Find Fractional Parts;114
10.6;Jason’s Partitive and Laura’s Part-Whole Fraction Schemes ;117
10.6.1;Lack of the Splitting Operation;117
10.6.2;Jason’s Partitive Unit Fraction Scheme;119
10.6.3;Laura’s Independent Use of Parts;121
10.6.4;Laura’s Part-Whole Fraction Scheme;126
10.7;Establishing Fractional Meaning for Multiple Parts of a Stick;129
10.7.1;A Recurring Internal Constraint in the Construction of Fraction Operations;131
10.8;Continued Absence of Fractional Numbers;132
10.8.1;An Attempt to Use Units-Coordinating to Produce Improper Fractions;133
10.8.2;A Test of the Iterative Fraction Scheme;135
10.9;Discussion of the Case Study;137
10.9.1;The Construction of Connected Numbers and the Connected Number Sequence;137
10.9.2;On the Construction of the Part- Whole and Partitive Fraction Schemes;138
10.9.3;The Splitting Operation;140
11;The Unit Composition and the Commensurate Schemes;142
11.1;The Unit Fraction Composition Scheme;143
11.1.1;Jason’s Unit Fraction Composition Scheme;144
11.1.2;Corroboration of Jason’s Unit Fraction Composition Scheme;145
11.1.3;Laura’s Apparent Recursive Partitioning;147
11.2;Producing Composite Unit Fractions;148
11.2.1;Laura’s Reliance on Social Interaction When Explaining Commensurate Fractions;152
11.2.2;Further Investigation into the Children’s Explanations and Productions;155
11.3;Producing Fractions Commensurate with One-Half;157
11.4;Producing Fractions Commensurate with One-Third;161
11.5;Producing Fractions Commensurate with Two-Thirds;166
11.6;An Attempt to Engage Laura in the Construction of the Unit Fraction Composition Scheme;167
11.6.1;The Emergence of Recursive Partitioning for Laura;170
11.6.2;Laura’s Apparent Construction of a Unit Fraction Composition Scheme;172
11.6.3;Progress in Partitioning the Results of a Prior Partition;176
11.7;Discussion of the Case Study;180
11.7.1;The Unit Fraction Composition Scheme and the Splitting Operation;181
11.7.2;Independent Mathematical Activity and the Splitting Operation;182
11.7.3;Independent Mathematical Activity and the Commensurate Fraction Scheme;182
11.7.4;An Analysis of Laura’s Construction of the Unit Fraction Composition Scheme;183
11.7.5;Laura’s Apparent Construction of Recursive Partitioning and the Unit Fraction Composition Scheme;188
12;The Partitive, the Iterative, and the Unit Composition Schemes;189
12.1;Joe’s Attempts to Construct Composite Unit Fractions;190
12.2;Attempts to Construct a Unit Fraction of a Connected Number;192
12.3;Partitioning and Disembedding Operations;194
12.4;Joe’s Construction of a Partitive Fraction Scheme;198
12.5;Joe’s Production of an Improper Fraction;203
12.6;Patricia’s Recursive Partitioning Operations;206
12.6.1;The Splitting Operation: Corroboration in Joe and Contraindication in Patricia;206
12.7;A Lack of Distributive Reasoning;209
12.8;Emergence of the Splitting Operation in Patricia;211
12.9;Emergence of Joe’s Unit Fraction Composition Scheme;213
12.10;Joe’s Reversible Partitive Fraction Scheme;215
12.10.1;Fractions Beyond the Fractional Whole: Joe’s Dilemma and Patricia’s Construction;217
12.11;Joe’s Construction of the Iterative Fraction Scheme;222
12.12;A Constraint in the Children’s Unit Fraction Composition Scheme;226
12.13;Fractional Connected Number Sequences;229
12.14;Establishing Commensurate Fractions;232
12.15;Discussion of the Case Study ;235
12.15.1;Composite Unit Fractions: Joe;235
12.15.2;Joe’s Partitive Fraction Scheme;236
12.15.3;Emergence of the Splitting Operation and the Iterative Fraction Scheme: Joe;237
12.15.4;Emergence of Recursive Partitioning and Splitting Operations: Patricia;238
12.15.5;The Construction of the Iterative Fraction Scheme;239
12.15.6;Stages in the Construction of Fraction Schemes;240
13;Equipartitioning Operations for Connected Numbers: Their Use and Interiorization;242
13.1;Melissa’s Initial Fraction Schemes;242
13.1.1;Contraindication of Recursive Partitioning in Melissa;244
13.1.2;Reversibility of Joe’s Unit Fraction Composition Scheme;245
13.2;A Reorganization in Melissa’s Units-Coordinating Scheme;248
13.3;Melissa’s Construction of a Fractional Connected Number Sequence;253
13.4;Testing the Hypothesis that Melissa Could Construct a Commensurate Fraction Scheme;258
13.5;Melissa’s Use of the Operations that Produce Three Levels of Units in Re- presentation;264
13.5.1;Repeatedly Making Fractions of Fractional Parts of a Rectangular Bar;264
13.5.2;Melissa Enacting a Prior Partitioning by Making a Drawing;268
13.5.3;A Test of Accommodation in Melissa’s Partitioning Operations;271
13.5.4;A Further Accommodation in Melissa’s Recursive Partitioning Operations;273
13.6;A Child-Generated Fraction Adding Scheme;277
13.7;An Attempt to Bring Forth a Unit Fraction Adding Scheme;280
13.8;Discussion of the Case Study;283
13.9;The Iterative Fraction Scheme;285
13.9.1;Melissa’s Interiorization of Operations that Produce Three Levels of Units;286
13.9.2;On the Possible Construction of a Scheme of Recursive Partitioning Operations;288
13.9.3;The Children’s Meaning of Fraction Multiplication;290
13.9.4;A Child-Generated vs. a Procedural Scheme for Adding Fractions;292
14;The Construction of Fraction Schemes Using the Generalized Number Sequence;293
14.1;The Case of Nathan During His Third Grade;293
14.1.1;Nathan’s Generalized Number Sequence;294
14.1.2;Developing a Language of Fractions;295
14.1.3;Reasoning Numerically to Name Commensurate Fractions;300
14.1.4;Corroboration of the Splitting Operation for Connected Numbers;302
14.1.5;Renaming Fractions: An Accommodation of the IFS: CN;304
14.1.6;Construction of a Common Partitioning Scheme;305
14.1.7;Constructing Strategies for Adding Unit Fractions with Unlike Denominators;307
14.2;Multiplication of Fractions and Nested Fractions;311
14.3;Equal Fractions;314
14.3.1;Generating a Plurality of Fractions;315
14.3.2;Working on a Symbolic Level;317
14.4;Construction of a Fraction Composition Scheme;319
14.4.1;Constraining How Arthur Shared Four-Ninths of a Pizza Among Five People;320
14.4.2;Testing the Hypothesis Using TIMA: Bars;323
14.5;Discussion of the Case Study;326
14.5.1;The Reversible Partitive Fraction Scheme;326
14.5.2;The Common Partitioning Scheme and Finding the Sum of Two Fractions;327
14.5.3;The Fractional Composition Scheme;329
15;The Partitioning and Fraction Schemes;331
15.1;The Partitioning Schemes ;331
15.1.1;The Equipartitioning Scheme;331
15.1.2;The Simultaneous Partitioning Scheme;332
15.1.3;The Splitting Scheme;333
15.1.4;The Equipartitioning Scheme for Connected Numbers;335
15.1.5;The Splitting Scheme for Connected Numbers;336
15.1.6;The Distributive Partitioning Scheme;337
15.2;The Fraction Schemes;338
15.2.1;The Part-Whole Fraction Scheme;338
15.2.2;The Partitive Fraction Scheme;339
15.2.3;The Unit Fraction Composition Scheme;344
15.2.4;The Fraction Composition Scheme;346
15.2.5;The Iterative Fraction Scheme;349
15.2.6;The Unit Commensurate Fraction Scheme;351
15.2.7;The Equal Fraction Scheme;352
15.3;School Mathematics vs. “School Mathematics”;353
16;Continuing Research on Students’ Fraction Schemes;357
16.1;Research on Part-Whole Conceptions of Fractions;358
16.2;Transcending Part-Whole Conceptions;360
16.3;The Splitting Operation;361
16.4;Students’ Development Toward Algebraic Reasoning;364
16.5;References;369
17;Index;374