Advances in the Anthropological Theory of the Didactic

Acknowledgements
A Word of Introduction
Contents
Dialogue Between Theories in Didactics:
On the Genesis and Progress of the ATD
1 Theory and Theorization
2 ``Anthropological´´
3 A Recent Finding
4 Epistemological Break
5 Humpty Dumpty
6 Cognition
7 Instantial Relativity
8 The Didactic Revisited
9 Possibly Didactic Situations
10 By Way of Conclusion
References
ATD on Relationships Between Research and Teaching. The Case of a Didactic Problem Concerning Real Numbers
1 Introduction
2 Basic Assumptions and the Object of Study in ATD
2.1 ATD as a Theory of Human Activity
2.2 Research Ends and Teaching Ends Are Postulates
2.3 Relative Autonomy of Didactics
2.4 Personal Praxeologies and Institutional Praxeologies
2.5 The Praxeological Analysis as an Entrance to Didactic Analysis
2.6 Transpositive Phenomena Are at the Heart of Didactic Problems
2.7 The Minimal Unit of Analysis of Didactic Processes
2.8 Relativity of Epistemological Roles and Purpose of the REM
2.9 Levels of Conditions Which Affect the Life of Praxeologies
2.10 Object of Study and Admissible Research Results
3 Teaching Ends and Didactic Paradigms Considered by the ATD
3.1 Characterisation of a Didactic Paradigm
3.2 On the Paradigm of Questioning the World (PQW)
3.3 On the Paradigm of Mathematical Modelling (PMM)
4 The Case of a Didactic Problem Concerning Real Numbers
4.1 A Teaching Problem Concerning Real Numbers, and the Institutional Answer
4.2 A REM on Real Numbers for Teachers Education
4.3 Economy of Real Numbers in the Step from Secondary to Tertiary Education
4.4 Ecology of Real Numbers in the Step from Secondary to Tertiary Education
5 Towards a Third Step of the Dialogue Between Theories
References
From the Networking of Theories to the Discussion of the Educational Implications of Research
1 Introduction
2 Theories as Components of Research Praxeologies
3 The Scale of Networking Strategies
4 Some Lessons from the Networking Enterprise
5 Research, Theories and Educational Action
6 Recovering Research Praxeologies and Relationship to Educational Action from Publications
References
Theory of Didactical Situations in Mathematics: An Epistemological Revolution
1 Scientific Foundations of Theory of Didactical Situations in Mathematics
1.1 Brousseau´s Scientific Project
1.2 The Ambition to Understand Teaching and Learning Mathematics
2 Epistemological Principles
2.1 Theory of Mathematical Situations (TMS)
2.2 Situational and Institutional Knowledge
3 Interactions Between Institutional Knowledge and Situational Knowledge: An Example in the Context of Didactic Engineering
3.1 Didactic Engineering
3.2 From Institutional to Situational Knowledge in Didactic Engineering
3.3 Action Situation
3.4 During an Action Situation, No Formulation Is Necessary, Situational Knowledge Is Mostly Implicit. Formulation Situation
3.5 Validation Situation
3.6 From Situational Knowledge to Institutional Knowledge: Roles of the Different Situations in the Institutionalization Proce...
4 Didactical Consequences of Theory of Mathematics Situations on Teaching
4.1 Engineering and ``Task Design´´
4.2 Mathematic Problem, Situations and Observation
4.3 From Situational to Institutional Knowledge: Student´s Struggle
References
The Onto-semiotic Approach in Mathematics Education. Analysing Objects and Meanings in Mathematical Practice
1 Introduction
1.1 Didactics as Science and as Technology
2 Onto-semiotic Approach: A Modular and Inclusive Theoretical Framework for Mathematics Education
3 Objects and Meanings Analysis of Mathematical Practices
3.1 Workshop Aims and Method
3.2 Meaning in the Onto-semiotic Approach
3.3 Pragmatist and Referential Meanings
3.4 Workshop Development
3.5 Some Conclusions
4 Final Reflections
Annex. Solutions to a Proportionality Problem
References
APOS Theory and the Role of the Genetic Decomposition
1 Introduction
2 APOS Theory and Its Relation to Teaching
2.1 Mental Structures and Mechanisms
3 The Genetic Decomposition: Its Role in Research and Teaching
3.1 How Is a Genetic Decomposition Designed?
3.2 How Is a Genetic Decomposition Used?
4 An Example: The Role of the Genetic Decomposition in the Design of Activities and in Teaching
4.1 The Genetic Decomposition
4.1.1 Previous Constructions
4.1.2 Construction of the Matrix-Vector Product and the Matrix Transformation
4.1.3 Construction of the Matrix Product and the Composition of Matrix Transformations
4.2 Teaching and Research Methodology
4.3 Use of Activities Based on the Genetic Decomposition in Students´ Construction of Knowledge
4.4 The Role of the Teacher
5 Final Reflections
References
Mathematics Teacher Education and the Professionalisation of Teaching:
Challenges and Advances in Teacher Education Within the ATD
1 By Way of Warning
2 A Typical Example
3 Starting from Scratch
4 The Teacher as Servant
5 A Semiprofession
6 Towards a Profession?
7 A Few Reminders
8 Some Questions
9 The Paradigm of Visiting Works and Its Limitations
10 The Paradigm of Questioning the World
11 A Foreseeable Future
References
Analyzing Mathematics Teachers´ Collective Work in Terms of the Inquiry
1 Introduction
2 Teacher´s Work Inside and Outside Mathematics Classroom
2.1 Paradidactic Praxeologies
2.2 Paradidactic Infrastructure
3 Analysis of Teachers´ Collective Work in Terms of the Inquiry
3.1 Teachers´ Work as an Inquiry
3.2 The Case of Lesson Study
3.3 The Case of Collaborative Design of Lessons
4 Discussion and Conclusion
References
On the Problem Between Devices and Infrastructures in Teacher Education Within the Paradigm of Questioning the World
1 Introduction
2 Devices and Infrastructures: an ATD Perspective
3 Devices and Infrastructures for Teacher Education in the Paradigm of Questioning the World of the Teaching Profession
4 Lesson Study as a Teacher Education Device Within the Paradigm of Questioning the World: An ATD Perspective
5 Lesson Study and the Problem of the Infrastructure
6 Conclusions
References
Introduction of Ordinal Number at the Beginning of the French Curriculum: A Study of Professional Teaching Problem
1 Introduction
2 A Praxeological Analysis of the Mathematical Coordinate System´s Knowledge
3 A Didactic Engineering for Research
4 Towards Training: Teacher´s Mathematical and Didactical Knowledge Needs
5 Conclusion
References
Study and Research for Teacher Education: Some Advances on Teacher Education in the Paradigm of Questioning the World
1 Introduction: Study and Research Paths for Teacher Education
2 General Structure of the Study and Research Paths for Teacher Education
3 The Herbartian Schema as a Didactic Model of Reference for the Analysis of the SRP-TE
4 Analysis of Two Case Studies with SRP-TE
4.1 Case Study 1: The SRP-TE About Modelling the Evolution of Facebook Social Network
4.2 Case study 2: the SRP-TE about modelling the cake box
5 Conclusions and Discussion
References
Transpositive Phenomena of Didactics in Teacher Training
1 Introduction
2 A Didactic Organisation Produced with the ATD: An Example of Infrastructure
3 Archididactic Transposition Process
4 Conditions and Constraints of Didactic Transposition Phenomena of Didactics in Teacher Training
References
Prospective Teachers´ Narrative Analysis Using the Didactic-Mathematical Knowledge and Competences Model (DMKC)
1 Introduction
2 Didactic-Mathematical Knowledge and Competences Model
2.1 The Notion of Competence
2.2 The Notion of Knowledge of the Mathematics Teacher
2.3 Competence in the Analysis and Didactic Intervention and Its Relation to the Model of Analysis of Instructional Processes ...
3 Methodology
4 Results
5 Considerations
References
Exploring the Paradidactic Ecosystem: Conditions and Constraints on the Teaching Profession
1 Introduction: An Epistemological Obstacle in the Research on Teachers´ Work
2 Didactic Systems Involved in Complex Institutional Ecosystems
2.1 A Triptych of Didactic Systems
2.2 The Scale of Levels of Didactic Co-determinacy
2.3 The Ecology and Economy of Didactic Systems
3 Towards Paradidactic Research
3.1 A Nested Triptych of Paradidactic Systems
3.2 Economic Usage of the Scale of Levels of Didactic Co-determinacy
3.3 The Scale of Levels of Paradidactic Determinacy
4 Relation of the Paradidactic Theory to the Fundamental Theory in ATD
5 Final Remarks: What Could Be the raison d´être of Paradidactic Research?
References
Teacher Learning in Collaborative Settings: Analysis of an Open Lesson
1 Teacher Knowledge Within the Anthropological Theory of the Didactic
2 Teacher Learning: Devices and Infrastructures
3 The Teacher Education Device ``Open Lessons´´ and the Paradidactic Infrastructure in Japan
4 Workshop´s Rationale and Guiding Questions
5 Discussion and Perspectives
References
The Curriculum Problem and the Paradigm of Questioning the World:
Toward a Scientific Understanding of a Possibly Upcoming Civilizational Revolution
1 Prefatory Remarks
2 Revisiting Some Basics of the ATD
2.1 Persons, Institutions, and Positions
2.2 Relations to Objects
2.3 Who´s Speaking?
2.4 The Case of Peer Review
2.5 The Case of Algebra
2.6 The Praxeological Analysis of Action
2.7 Praxeological Obsolescence
2.8 Praxeologies and Relations
2.9 Conditions and Constraints
2.10 ``Unknown Unknowns´´
2.11 Within Reach or out of Reach?
2.12 The Possibly Didactic
3 Humanity, Civilizations, and Societies
3.1 The Assertedly (or Allegedly) Didactic
3.2 The Infrastructure/Superstructure Dialectic
3.3 Homo sapiens and the Didactic
3.4 Humankind: We Versus They, and Way Beyond
3.5 Humankind: An Answer to Every Question?
3.6 Societies and Civilizations
4 Questioning and Studying: The Human Art of Learning
4.1 The Notion of School
4.2 The Level of Pedagogies
4.3 The Pedagogies of Inquiry
4.4 Study Paradigms
4.5 The Disciplines and the Professor
4.6 Steps Towards Questioning the World
4.7 Questioning the World? Really?
4.8 The Question of Questions
4.9 Conducting an Inquiry
4.10 An Endless Analysis
References
The Analysis of Dominant Praxeological Models with a Reference Praxeological Model: A Case Study on Quadratic Equations
1 Introduction
2 A Reference Epistemological Model of Elementary Algebra
3 A Comparative Study of the Teaching of Quadratic Equations in Two Curricula
4 Students´ Personal Relation with Solving Quadratic Equations in both Institutions
5 Textbooks Analysis: Which Are the Dominant Praxeological Models in France and Brazil for Solving Quadratic Equations?
6 Conclusion
References
Study and Research Paths, Ecology and In-service Teachers
1 Introduction
2 The Context and In-service Teachers
3 Reflections Upon Course Design
References
Analysing the Dialectic of Questions and Answers in Study and Research Paths
1 Introduction: A Dialectical Generalisation of the Notion of ``Study´´
2 Didactic Systems and Their Study-and-Research Functioning
2.1 The Name of ``Study and Research Path´´ and Its Two Referents
2.2 A Generic Model of Didactic Systems
2.3 The Herbartian Schema as a Model of Inquiry Processes
2.4 The Three Genetic Facets of SRP
3 Investigation into the Chronogenesis of SRP
3.1 Questions-and-Answers Map
3.2 Two Dimensions of Answers
3.3 Praxeological Functions of Answers
4 Final Remarks: How to Analyse the Question on Inquiry?
References
Experimentation of a Study and Research Path: Didactic-Mathematical Indicators of Dialectics
1 Study and Research Path´s Generative Question and Its Hypotheses
2 SRP´s Implementation Contexts
3 Didactic-Mathematical Indicators of the Dialectics
4 Final Thoughts
References
Research in Didactics of Mathematics at the University Level:
Institutional Transitions in University Mathematics Education
1 Introduction
2 The ATD Perspective on Transition Issues
3 Investigating the Secondary/University Transition with ATD: Pioneering Theses
4 The Evolution of ATD Research Praxeologies on Transition Issues
4.1 The Evolution of the Praxis Block
4.2 The Evolution of the Theoretical Block
5 Final Reflections
References
Examining Individual and Collective Level Mathematical Progress
1 Theoretical and Methodological Background
1.1 Classroom Mathematical Practices
1.2 Disciplinary Practices
1.3 Mathematical Conceptions
1.4 Participation in Mathematical Activity
2 Background and Setting
3 Sample Analysis
4 Conclusion
References
Mathematical Analysis at University
1 What Is Analysis and Why Is It Important?
2 The Transition from Calculus into Analysis
3 Task Design in Analysis
4 Analysis and the Teaching Profession
5 Problems for Future Research
References
Using Tools from ATD to Analyse the Use of Mathematics in Engineering Tasks: Some Cases Involving Integrals
1 Introduction
2 Theoretical Framework
3 Methodology
4 Some Results
5 Final Comments
References
A Workshop on the Epistemology and Didactics of Mathematical Structuralism
1 Epistemology of Mathematical Structuralism
2 Modeling Mathematical Structuralism Within ATD
3 The Thread on (Z/32Z)* and Abelian Group Theory
References
About Two Epistemological Related Aspects in Mathematical Practices of Empirical Sciences
1 Introduction
2 ATD-Research on the Use of Mathematics in Other Sciences
3 Epistemological Considerations Regarding the Relationship of Mathematics and Empirical Sciences
4 About Mathematical Practices in Electrical Engineering
5 About Mathematical Practices in Psychology
6 Outlook
References
Describing Mathematical Activity: Dynamic and Static Aspects
1 Describing Inquiry: A Challenge for Research
2 Fostering This Transition: Study and Research Paths
3 Modelling and Describing Knowledge Within the ATD: Praxeological Analysis
4 Describing Study Processes: Herbartian Schema
5 Describing Study Processes: Question-Answer Maps
6 A Possible Raison d´Être for Elemental Differential Calculus
7 The Workshop: Describing Mathematical Activity with ATD Tools
8 Results
References