Toward a Visually-Oriented School Mathematics Curriculum

TABLE OF CONTENTS 0 A Reflection on Visual Studies in Mathematics Education: From Purposeful Tourism to a Traveling Theory I PURPOSEFUL TOURISM II A TRAVELING THEORY III OVERVIEW OF CHAPTER 1 IV A NOTE ON PARTICIPANTS IN MY FOUR-YEAR STUDY 1 Visualization and Progressive Schematization 1.1 NATURE OF COGNITIVE ACTIVITY AND COGNITIVE ACTION 1.2 BASIC ELEMENTS AND TENSIONS IN A VISUALLY-ORIENTED SCHOOL MATHEMATICS CURRICULUM 1.3 GENERAL NOTION OF VISUALIZATION IN THIS BOOK 1.4 THE MEANING OF REPRESENTATION IN THIS BOOK 1.5 THREE FUNDAMENTAL PRINCIPLES OF VISUALIZATION 1.6 CONTEXT-BASED VISUAL REPRESENTATIONS 1.7 FORMS AND LEVELS OF VISUAL REPRESENTATIONS 1.8 INSTRUCTIONAL IMPLICATIONS 1.9 OVERVIEW OF CHAPTER 2 2 Visual Roots of Mathematical Cognitive Activity 2.1 THREE CONTEXTS OF VISUALIZING ACTIVITY IN MATHEMATICS 2.2 EXPLICIT AND IMPLICIT KNOWLEDGE IN MATHEMATICS 2.3 MATHEMATICAL OBJECTS IN COGNITIVE ACTIVITY 2.4 MATHEMATICAL CONCEPTS IN COGNITIVE ACTIVITY 2.5 VISUAL THINKING IN MATHEMATICS 2.6 REVISITING PRINCIPLES OF VISUALIZATION FROM A MATHEMATICAL POINT OF VIEW 2.7 OVERVIEW OF CHAPTER 3 3 Visual Roots of Mathematical Symbols 3.1 MATHEMATICAL SYMBOLS AND THEIR SIGNS AND MODES OF SIGNIFICATION 3.2 A PROGRESSIVE FORMAL ACCOUNT OF SYMBOLS: FOCUS ON SCHOOL ALGEBRA 3.3 SOME ISSUES IN INTER- AND INTRA-SEMIOTIC TRANSITIONS 3.4 DEVELOPMENT OF VISUOALPHANUMERIC SYMBOLS IN PATTERN GENERALIZATION ACTIVITY IN A MIDDLE SCHOOL CLASSROOM 3.5 OVERVIEW OF CHAPTER 4 4 Visuoalphanumeric Representations in Pattern Generalization Activity 4.1 ASSUMPTIONS ABOUT PATTERNS 4.2 ABDUCTIVE REASONING IN PATTERN GENERALIZATION ACTIVITY 4.3 ENTRY LEVEL ABDUCTIONS 4.4 MATURE ABDUCTIONS: VISUAL TEMPLATES 4.5 SITUATIONAL DISCORDS IN VISUAL TEMPLATES 4.6 STRUCTURED VISUAL THINKING IN TRANSFORMATIONAL ACTIVITY 4.7 OVERVIEW OF CHAPTER 5 5 Visual Thinking and Diagrammatic Reasoning (DR) 5.1 PSYCHOPHYSICAL AND COGNITIVE PERSPECTIVES ON DIAGRAMS 5.2 THREE BASIC ISSUES WITH DIAGRAMS 5.3 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 6 6 Cultural and Blind-Specific Issues and Implications to Visual Thinking in Mathematics 6.1 CULTURAL PREFERENCES RELATIVE TO VISUAL ATTENTION AND THE CONSTRUCTION OF CONCEPTS 6.2 SOCIOCULTURAL INFLUENCES IN SEEING AND IMAGING FROM A MATHEMATICAL PERSPECTIVE 6.3 MATERIALIST VERSUS FORMALIST VIEWS ON MATHEMATICAL OBJECTS, CONCEPTS, AND PROCESSES 6.4 CULTURAL VIEWS ON WHAT CONSTITUTES A MATHEMATICAL PROOF 6.5 THE ROLE OF ARTIFACTS AND VISUAL LANGUAGE IN EARLY MATHEMATICAL DEVELOPMENT 6.6 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 7 7 Instructional Implications: Toward Visual Thinking in Mathematics 7.1 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 PHASE 7.5 A CIRCLING ASCENT TO THE OPERATIVE 7.6 THE OPERATIVE CONTENT PHASE 7.7 THE PLEASURE OF VISUAL FORMS IN MATHEMATICAL KNOWING