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Foreword; Preface; Acknowledgments; Contents; List of Figures ; Chapter 1: Representation and Knowledge: The Semiotic Revolution; 1.1 The Fundamental Epistemological Distinction and the First Knowledge Analysis Scheme; 1.1.1 The Cognitive Issue of Access to the Objects Themselves and Role of Representations; 1.1.2 Sign and Representation: The Cognitive Divide; 1.2 The Semiotic Revolution: Towards a New Knowledge Analysis Scheme; 1.3 The Three Models of Sign Analysis That Have Founded Semiotics: Contributions and Limits; 1.3.1 Saussure: Structural Analysis of Semiotic Systems
1.3.2 Peirce: The Classification of Kinds of Representations1.3.3 Frege: Semiotic Substitution and Production of New Knowledge in Mathematics; 1.4 Conclusion: The Semiotic Representations; Annex; Chapter 2: Mathematical Activity and the Transformations of Semiotic Representations; 2.1 Two Epistemological Situations, One Irreducible to the Other, in the Access to Objects of Knowledge; 2.1.1 The Juxtaposition Test with a Material Object: The Photo Montage of Kosuth; 2.1.2 The Juxtaposition Test with the Natural Numbers; 2.1.3 How to Recognize the Same Object in Different Representations?
2.1.4 A Fundamental Cognitive Operation in Mathematics: One-to-One Mapping2.2 The Transformation of Semiotic Representations at the Heart of the Mathematical Way of Working; 2.2.1 Description of an Elementary Mathematical Activity: The Development of Polygonal Unit Marks Configuration; 2.2.2 Representational Transformations Specific to each Kind of Semiotic Representation: The Case of Representation of Numbers; 2.2.2.1 Operations with Small Natural Numbers; 2.2.2.2 Operations with Relative Integers; 2.2.2.3 The Operations with Rational Numbers
2.3 Conclusion: The Cognitive Analysis of the Mathematical Activity and the Functioning of the Mathematical ThinkingChapter 3: Registers of Semiotic Representations and Analysis of the Cognitive Functioning of Mathematical Thinking; 3.1 Semiotic Registers and Cognitive Functioning of Thought; 3.1.1 Two Heterogeneous Kinds of Semiotic Systems: The Codes and Registers; 3.1.2 The Three Types of Discursive Operations and the Cognitive Functions of Natural Languages; 3.1.3 The Relationship Between Thought and Language: Discursive Operations and Linguistic Expression
3.1.4 Conclusion: What Characterizes a Register of Semiotic Representation3.2 Do the Various Forms of Representation Used in Mathematics Depend on Registers?; 3.2.1 How do We see a Figure?; 3.2.2 The Two Types of Figural Operations Proper to the Geometrical Figures; 3.2.3 Concealment of the Register of Figures in the Teaching of Geometry and Didactic Analyses; 3.2.4 Geometric Visualization and Problems from Reality: Direct Passage or Need for Intermediary Representations?; 3.3 Conclusions; Chapter 4: The Registers: Method of Analysis and Identification of Cognitive Variables
1.3.2 Peirce: The Classification of Kinds of Representations1.3.3 Frege: Semiotic Substitution and Production of New Knowledge in Mathematics; 1.4 Conclusion: The Semiotic Representations; Annex; Chapter 2: Mathematical Activity and the Transformations of Semiotic Representations; 2.1 Two Epistemological Situations, One Irreducible to the Other, in the Access to Objects of Knowledge; 2.1.1 The Juxtaposition Test with a Material Object: The Photo Montage of Kosuth; 2.1.2 The Juxtaposition Test with the Natural Numbers; 2.1.3 How to Recognize the Same Object in Different Representations?
2.1.4 A Fundamental Cognitive Operation in Mathematics: One-to-One Mapping2.2 The Transformation of Semiotic Representations at the Heart of the Mathematical Way of Working; 2.2.1 Description of an Elementary Mathematical Activity: The Development of Polygonal Unit Marks Configuration; 2.2.2 Representational Transformations Specific to each Kind of Semiotic Representation: The Case of Representation of Numbers; 2.2.2.1 Operations with Small Natural Numbers; 2.2.2.2 Operations with Relative Integers; 2.2.2.3 The Operations with Rational Numbers
2.3 Conclusion: The Cognitive Analysis of the Mathematical Activity and the Functioning of the Mathematical ThinkingChapter 3: Registers of Semiotic Representations and Analysis of the Cognitive Functioning of Mathematical Thinking; 3.1 Semiotic Registers and Cognitive Functioning of Thought; 3.1.1 Two Heterogeneous Kinds of Semiotic Systems: The Codes and Registers; 3.1.2 The Three Types of Discursive Operations and the Cognitive Functions of Natural Languages; 3.1.3 The Relationship Between Thought and Language: Discursive Operations and Linguistic Expression
3.1.4 Conclusion: What Characterizes a Register of Semiotic Representation3.2 Do the Various Forms of Representation Used in Mathematics Depend on Registers?; 3.2.1 How do We see a Figure?; 3.2.2 The Two Types of Figural Operations Proper to the Geometrical Figures; 3.2.3 Concealment of the Register of Figures in the Teaching of Geometry and Didactic Analyses; 3.2.4 Geometric Visualization and Problems from Reality: Direct Passage or Need for Intermediary Representations?; 3.3 Conclusions; Chapter 4: The Registers: Method of Analysis and Identification of Cognitive Variables