Rivera Toward a Visually-Oriented School Mathematics Curriculum

Research, Theory, Practice, and Issues
1. Auflage 2011
ISBN: 978-94-007-0014-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 316 Seiten

What does it mean to have a visual representation of a mathematical object, concept, or process? What visualization strategies support growth in mathematical thinking, reasoning, generalization, and knowledge? Is mathematical seeing culture-free? How can information drawn from studies in blind subjects help us understand the significance of a multimodal approach to learning mathematics? Toward a Visually-Oriented School Mathematics Curriculum explores a unified theory of visualization in school mathematical learning via the notion of progressive modeling. Based on the author’s longitudinal research investigations in elementary and middle school classrooms, the book provides a compelling empirical account of ways in which instruction can effectively orchestrate the transition from personally-constructed visuals, both externally-drawn and internally-derived, into more structured visual representations within the context of a socioculturally grounded mathematical activity. Both for teachers and researchers, a discussion of this topic is relevant in the history of the present. The ubiquity of technological tools and virtual spaces for learning and doing mathematics has aroused interest among concerned stakeholders about the role of mathematics in these contexts. The book begins with a prolegomenon on the author’s reflections on past and present visual studies in mathematics education. In the remaining seven chapters, visualization is pursued in terms of its role in bringing about progressions in mathematical symbolization, abduction, pattern generalization, and diagrammatization. Toward a Visually-Oriented School Mathematics Curriculum views issues surrounding visualization through the eyes of a classroom teacher-researcher; it draws on findings within and outside of mathematics education that help practitioners and scholars gain a better understanding of what it means to pleasurably experience the symmetric visual/symbolic reversal phenomenon – thatis, seeing the visual in the symbolic and the symbolic in the visual."

Autoren/Hrsg.

Rivera, Ferdinand

Inhaltsverzeichnis

1;Contents;7
2;1 Introduction;10
2.1;A Reflection on Visual Studies in Mathematics Education: From Purposeful Tourism to a Traveling Theory;10
2.2;1 Purposeful Tourism;14
2.2.1;1.1 Psychological Dimension;16
2.2.2;1.2 Curriculum/Instruction Dimension;17
2.2.3;1.3 Technological and Semiotic Dimensions;18
2.3;2 A Traveling Theory;19
2.3.1;2.1 Exploring Implications of Purposeful Tourism;19
2.3.2;2.2 Exploring Implications of Instrumental Genesis;20
2.3.3;2.3 Exploring Implications of O'Halloran's Grammar of Mathematical Visual Images;25
2.4;3 Overview of Chapter 2;27
2.5;4 A Note on Participants in My 4-Year Study;27
3;2 Visualization and Progressive Schematization: Framing the Issues;29
3.1;1 Nature of Cognitive Activity and Cognitive Action;30
3.2;2 Basic Elements and Tensions in a Visually Oriented School Mathematics Curriculum;39
3.3;3 General Notion of Visualization in this Book;43
3.4;4 The Meaning of Representation in this Book;48
3.5;5 Three Fundamental Principles of Visualization;51
3.6;6 Context-Based Visual Representations;59
3.7;7 Forms and Levels of Visual Representations;61
3.8;8 Instructional Implications;64
3.9;9 Overview of Chapter 3;65
4;3 Visual Roots of Mathematical Cognitive Activity;67
4.1;1 Three Contexts of Visualizing Activity in Mathematics;69
4.2;2 Explicit and Implicit Knowledge in Mathematics;78
4.3;3 Mathematical Objects in Cognitive Activity;86
4.4;4 Mathematical Concepts in Cognitive Activity;94
4.5;5 Visual Thinking in Mathematics;97
4.6;6 Revisiting Principles of Visualization from a Mathematical Point of View;102
4.7;7 Overview of Chapter 4;105
5;4 Visual Roots of Mathematical Symbols;106
5.1;1 Mathematical Symbols and Their Signs and Modes of Signification;115
5.2;2 A Progressive Formal Account of Symbols: Focus on School Algebra;121
5.3;3 Some Issues in Inter- and Intra-semiotic Transitions;127
5.4;4 Development of Visuoalphanumeric Symbols in Pattern Generalization Activity in a Middle School Classroom;138
5.5;5 Overview of Chapter 5;150
6;5 Visuoalphanumeric Representations in Pattern Generalization Activity;151
6.1;1 Assumptions About Patterns;152
6.2;2 Abductive Reasoning in Pattern Generalization Activity;160
6.3;3 Entry-Level Abductions;167
6.4;4 Mature Abductions: Visual Templates;177
6.5;5 Situational Discords in Visual Templates;191
6.6;6 Structured Visual Thinking in Transformational Activity;199
6.7;7 Overview of Chapter 6;203
7;6 Visual Thinking and Diagrammatic Reasoning;204
7.1;1 Psychophysical and Cognitive Perspectives on Diagrams;213
7.2;2 Three Basic Issues with Diagrams;220
7.3;3 Diagram Types and Progressive Diagrammatization;233
7.4;4 Extending DR Beyond Establishing Regularities and Constructing Mathematical Knowledge: Supporting Growth in Mathematical Reasoning Through DR;239
7.5;5 Overview of Chapter 7;244
8;7 Cultural and Blind-Specific Issues and Implications to Visual Thinking in Mathematics;246
8.1;1 Cultural Preferences Relative to Visual Attention and the Construction of Concepts;249
8.2;2 Sociocultural Influences in Seeing and Imaging from a Mathematical Perspective;252
8.3;3 Materialist Versus Formalist Views on Mathematical Objects, Concepts, and Processes;253
8.4;4 Cultural Views on What Constitutes a Mathematical Proof;258
8.5;5 The Role of Artifacts and Visual Language in Early Mathematical Development;261
8.6;6 Multisensory Convergence on Imagery Tasks in Totally and Partially Blind Subjects;263
8.7;7 Benefits of Visual Representations: Three Lessons from Blind Studies and Implications for Sighted Learners;266
8.8;8 Models and Implications of Multisensory Learning;269
8.9;9 Overview of Chapter 8;273
9;8 Instructional Implications: Toward Visual Thinking in Mathematics;274
9.1;1 Some Final Thoughts on Visualization for Mathematics Education;277
9.2;2 Signifying Intersecting Circles;280
9.3;3 Past the Iconic and the Indexical: The Symbolic Mediated Nature of Visual Thinking in Mathematics (a Stronger Version);283
9.4;4 Visual Presence and Seeing in the Referential Phase;283
9.5;5 A Circling Ascent to the Operative;285
9.6;6 The Operative Content Phase;287
9.7;7 The Pleasure of Visual Forms in Mathematical Knowing;288
10;References;292
11;Author Index;309
12;Subject Index;314