Understanding the Mathematical Way of Thinking - The Registers of Semiotic Representations

IntroductionChapter I - Representation and knowledge: the semiotic revolution1. The fundamental epistemological distinction and the first analytical model of knowledge1.1 Cognitive question of access modes to the objects themselves: the role of representations1.2 Sign and representation: the cognitive divide2. The semiotic revolution: towards a new model of analysis of knowledge3. The three models of sign analysis that are the basis of semiotics: contributions and limits3.1 Saussure: structural analysis of semiotic systems3.2 Peirce: the classification of representation types3.3 Frege: the semiotic process as the producer of new knowledgeConclusion: the semiotic representationsAnnexChapter II - Mathematical activity and the transformations of semiotic representations1. Two epistemological situations, one irreducible to the other, in the access to objects of knowledge 1.1 The juxtaposition test with a material object: the photo montage of Kosuth1.2 The juxtaposition test with the natural numbers1.3 How to recognize the same object in different representations?1.4 A fundamental cognitive operation in mathematics: put in correspondence2. The transformation of semiotic representations in the center stage of the mathematical work2.1 Description of an elementary mathematical activity: the development of polygonal configuration from the unit marks2.2 The specific transformations of each type of semiotic representation: the case of representation of numbersConclusion: The cognitive analysis of the mathematical activity and the functioning of the mathematical thinking Chapter III - Registers of semiotic representations and analysis of the cognitive functioning of mathematical thinking1. Semiotic registers and functioning of thought1.1 Two types of heterogeneous semiotic systems: the codes and registers1.2 The three types of discursive operations and cognitive functions of natural languages1.3 The relationship between thought and language: discursive operations and linguistic expression1.4 Conclusion: what characterizes a register of semiotic representation2. Do other forms of representation used in mathematics depend on registers?2.1 How do we see a figure?2.2 The two types of figural operations proper to the geometrical figures2.3 The reasons for concealment of the register of figures in the teaching of geometry and didactic analyses  2.4 Geometric visualization and reality problems: direct passage or need for intermediate representations?  3. ConclusionsChapter IV - The registers: method of analysis and identification of cognitive variables1. How to isolate and recognize mathematically relevant units of meaning in the content of a representation? 1.1 Production of graphical representations and the visualization mistakes produced1.2 Analysis method to isolate the mathematically relevant units of meaning in the content of representations1.3 The development of the recognition of mathematically relevant units of meaning: what kind of task?  2. The analysis of mathematical activity based on the pairs of mobilized registers2.1 The congruence and non-congruence phenomena in the conversion of the representations 2.2 The particular place of natural language in the cognitive functioning subjacent to the mathematical reasoning2.3 The understanding of the problem statements and the need for transitional auxiliary representations2.4 The problem of cognitive connection between natural language and other registers3. Functional variations of phenomenological production methods and semiotic representation registers  3.1 Leaving behind the confusion between functional and structural analysis of the production of representations3.2 The computer monitors: another phenomenological mode of production of representations4. Method of analysis of the activities given in class and student productions: the problem of didactically relevant variables4.1 The organization of sequences of activities always has two sides4.2 The field of work cognitively required for a geometry class at primary school 4.3 The observations of the students work and the analysis of their productions and reactions4.4 Interactions and cognitive impact of three types of verbalization on understanding5. ConclusionsAnnex: Analysis of an example of introduction of the linear function concept in a textbook for students aged 13-14 years oldIndex of terms and expressions... Mehr