Toward a Visually-Oriented School Mathematics Curriculum: Research, Theory, Practice, and Issues

TABLE OF CONTENTS0 A Reflection on Visual Studies in Mathematics Education: From Purposeful Tourism to a Traveling TheoryI PURPOSEFUL TOURISMII A TRAVELING THEORYIII OVERVIEW OF CHAPTER 1IV A NOTE ON PARTICIPANTS IN MY FOUR-YEAR STUDY 1 Visualization and Progressive Schematization 1.1 NATURE OF COGNITIVE ACTIVITY AND COGNITIVE ACTION1.2 BASIC ELEMENTS AND TENSIONS IN A VISUALLY-ORIENTED SCHOOL MATHEMATICS CURRICULUM1.3 GENERAL NOTION OF VISUALIZATION IN THIS BOOK1.4 THE MEANING OF REPRESENTATION IN THIS BOOK1.5 THREE FUNDAMENTAL PRINCIPLES OF VISUALIZATION1.6 CONTEXT-BASED VISUAL REPRESENTATIONS1.7 FORMS AND LEVELS OF VISUAL REPRESENTATIONS1.8 INSTRUCTIONAL IMPLICATIONS1.9 OVERVIEW OF CHAPTER 22 Visual Roots of Mathematical Cognitive Activity2.1 THREE CONTEXTS OF VISUALIZING ACTIVITY IN MATHEMATICS2.2 EXPLICIT AND IMPLICIT KNOWLEDGE IN MATHEMATICS2.3 MATHEMATICAL OBJECTS IN COGNITIVE ACTIVITY2.4 MATHEMATICAL CONCEPTS IN COGNITIVE ACTIVITY2.5 VISUAL THINKING IN MATHEMATICS2.6 REVISITING PRINCIPLES OF VISUALIZATION FROM A MATHEMATICAL POINT OF VIEW2.7 OVERVIEW OF CHAPTER 33 Visual Roots of Mathematical Symbols3.1 MATHEMATICAL SYMBOLS AND THEIR SIGNS AND MODES OF SIGNIFICATION3.2 A PROGRESSIVE FORMAL ACCOUNT OF SYMBOLS: FOCUS ON SCHOOL ALGEBRA3.3 SOME ISSUES IN INTER- AND INTRA-SEMIOTIC TRANSITIONS3.4 DEVELOPMENT OF VISUOALPHANUMERIC SYMBOLS IN PATTERN GENERALIZATION ACTIVITY IN A MIDDLE SCHOOL CLASSROOM3.5 OVERVIEW OF CHAPTER 44 Visuoalphanumeric Representations in Pattern Generalization Activity4.1 ASSUMPTIONS ABOUT PATTERNS4.2 ABDUCTIVE REASONING IN PATTERN GENERALIZATION ACTIVITY4.3 ENTRY LEVEL ABDUCTIONS 4.4 MATURE ABDUCTIONS: VISUAL TEMPLATES4.5 SITUATIONAL DISCORDS IN VISUAL TEMPLATES4.6 STRUCTURED VISUAL THINKING IN TRANSFORMATIONAL ACTIVITY4.7 OVERVIEW OF CHAPTER 55 Visual Thinking and Diagrammatic Reasoning (DR)5.1 PSYCHOPHYSICAL AND COGNITIVE PERSPECTIVES ON DIAGRAMS5.2 THREE BASIC ISSUES WITH DIAGRAMS5.3 DIAGRAM TYPES AND PROGRESSIVE DIAGRAMMATIZATION 5.4 EXTENDING DR BEYOND ESTABLISHING REGULARITIES AND CONSTRUCTING MATHEMATICAL KNOWLEDGE: SUPPORTING GROWTH IN MATHEMATICAL REASONING THROUGH DR 5.5 OVERVIEW OF CHAPTER 66 Cultural and Blind-Specific Issues and Implications to Visual Thinking in Mathematics6.1 CULTURAL PREFERENCES RELATIVE TO VISUAL ATTENTION AND THE CONSTRUCTION OF CONCEPTS6.2 SOCIOCULTURAL INFLUENCES IN SEEING AND IMAGING FROM A MATHEMATICAL PERSPECTIVE6.3 MATERIALIST VERSUS FORMALIST VIEWS ON MATHEMATICAL OBJECTS, CONCEPTS, AND PROCESSES6.4 CULTURAL VIEWS ON WHAT CONSTITUTES A MATHEMATICAL PROOF6.5 THE ROLE OF ARTIFACTS AND VISUAL LANGUAGE IN EARLY MATHEMATICAL DEVELOPMENT6.6 MULTISENSORY CONVERGENCE ON IMAGERY TASKS IN TOTALLY AND PARTIALLY BLIND SUBJECTS 6.7 BENEFITS OF VISUAL REPRESENTATIONS: THREE LESSONS FROM BLIND STUDIES AND IMPLICATIONS FOR SIGHTED LEARNERS 6.8 MODELS AND IMPLICATIONS OF MULTISENSORY LEARNING 6.9 OVERVIEW OF CHAPTER 77 Instructional Implications: Toward Visual Thinking in Mathematics7.1 SOME FINAL THOUGHTS ON VISUALIZATION FOR MATHEMATICS EDUCATION 7.2 SIGNIFYING INTERSECTING CIRCLES 7.3 PAST THE ICONIC AND THE INDEXICAL: THE SYMBOLIC MEDIATED NATURE OF VISUAL THINKING IN MATHEMATICS (A STRONGER VERSION) 7.4 VISUAL PRESENCE AND SEEING IN IN THE REFERENTIAL PHASE7.5 A CIRCLING ASCENT TO THE OPERATIVE 7.6 THE OPERATIVE CONTENT PHASE 7.7 THE PLEASURE OF VISUAL FORMS IN MATHEMATICAL KNOWING