Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations, Softcover reprint of the original 1st ed. 2017

In this book, Raymond Duval shows how his theory of registers of semiotic representation can be used as a tool to analyze the cognitive processes through which students develop mathematical thinking. To Duval, the analysis of mathematical knowledge is in its essence the analysis of the cognitive synergy between different kinds of semiotic representation registers, because the mathematical way of thinking and working is based on transformations of semiotic representations into others. Based on this assumption, he proposes the use of semiotics to identify and develop the specific cognitive processes required to the acquisition of mathematical knowledge. In this volume he presents a method to do so, addressing the following questions:

? Why use a semio-cognitive analysis of the mathematical activity to teach mathematics

? How to distinguish the different types of registers

? How to organize learning tasks and activities which take into account the registers of representation

? How to make an analysis of the students? production in terms of registers

?Understanding the Mathematical Way of Thinking ? The Registers of Semiotic Representations is an essential work for mathematics educators and mathematics teachers who look for an introduction to Raymond Duval?s cognitive theory of semiotic registers of representation, making it possible for them to see and teach mathematics with fresh eyes.?

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

3.2 Peirce: the classification of representation types

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

2.1 How do we see a figure?

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

1. How to isolate and recognize mathematically relevant units of meaning in the content of a representation?

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 

3.2 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

Raymond Duval is a philosopher and psychologist, devoted to researching mathematics education since the 1970s. He has worked at the Research Institute on Mathematical Education (IREM) in Strasbourg, France, from 1970 to 1995, where he developed an important research in cognitive psychology. Today, he is honorarius professor of the University du Litoral Côte d'Opale, France. His book Sémiosis et pensée humaine (1995) is a milestone in the theory of registers of semiotic representations, and his research papers over the years have greatly influenced research in mathematics education. 

Presents a new way to teach and learn mathematics

Provides a method for teachers to use the registers of semiotic representations theory in their everyday work at the classroom

Explains how to develop the cognitive processes required to the acquisition of mathematical knowledge

Includes supplementary material: sn.pub/extras