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

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

Springer International Publishing AG






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Introduction Chapter I - Representation and knowledge: the semiotic revolution 1. The fundamental epistemological distinction and the first analytical model of knowledge 1.1 Cognitive question of access modes to the objects themselves: the role of representations 1.2 Sign and representation: the cognitive divide 2. The semiotic revolution: towards a new model of analysis of knowledge 3. The three models of sign analysis that are the basis of semiotics: contributions and limits 3.1 Saussure: structural analysis of semiotic systems 3.2 Peirce: the classification of representation types 3.3 Frege: the semiotic process as the producer of new knowledge Conclusion: the semiotic representations Annex Chapter II - Mathematical activity and the transformations of semiotic representations 1. 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 Kosuth 1.2 The juxtaposition test with the natural numbers 1.3 How to recognize the same object in different representations? 1.4 A fundamental cognitive operation in mathematics: put in correspondence 2. The transformation of semiotic representations in the center stage of the mathematical work 2.1 Description of an elementary mathematical activity: the development of polygonal configuration from the unit marks 2.2 The specific transformations of each type of semiotic representation: the case of representation of numbers Conclusion: 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 thinking 1. Semiotic registers and functioning of thought 1.1 Two types of heterogeneous semiotic systems: the codes and registers 1.2 The three types of discursive operations and cognitive functions of natural languages 1.3 The relationship between thought and language: discursive operations and linguistic expression 1.4 Conclusion: what characterizes a register of semiotic representation 2. 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 figures 2.3 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. Conclusions Chapter IV - The registers: method of analysis and identification of cognitive variables 1. 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 produced 1.2 Analysis method to isolate the mathematically relevant units of meaning in the content of representations 1.3 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 registers 2.1 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 reasoning 2.3 The understanding of the problem statements and the need for transitional auxiliary representations 2.4 The problem of cognitive connection between natural language and other registers 3. 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 representations 3.2 The computer monitors: another phenomenological mode of production of representations 4. Method of analysis of the activities given in class and student productions: the problem of didactically relevant variables 4.1 The organization of sequences of activities always has two sides 4.2 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 reactions 4.4 Interactions and cognitive impact of three types of verbalization on understanding 5. Conclusions Annex: Analysis of an example of introduction of the linear function concept in a textbook for students aged 13-14 years old Index of terms and expressions
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