THEORY OF DIDACTICAL SITUATIONS IN MATHEMATICS DIDACTIQUE DES MATHEMATIQUES,

Transcription

1 THEORY OF DIDACTICAL SITUATIONS IN MATHEMATICS DIDACTIQUE DES MATHEMATIQUES, by GUY BROUSSEAU Edited and translated by NICOLAS BALACHEFF, MARTIN COOPER, ROSAMUND SUTHERLAND AND VIRGINIA WARFIELD KLUWER ACADEMIC PUBLISHERS DORDRECHT/ BOSTON / LONDON

2 Editors' Preface Biography of Guy Brousseau xiii xv Prelude to the Introduction 1 Introduction. Setting the scene with an example: The race to Introduction of the race to The game Description of the phases of the game Remarks 5 2. First phase of the lesson: Instruction 6 3. Action situation, pattern, dialectic First part of the game (one against the other) Dialectic of action 9 4. Formulation situation, pattern, dialectic Second part of the game (group against group) Dialectic of formulation Validation situation, pattern, dialectic Third part of the game (establishment of theorems) The attitude of proof, proof and mathematical proof Didactical situation of validation Dialectic of validation. 17 Chapter 1 prelude 19 Chapter 1. Foundations and methods of didactique Objects of study of didactique Mathematical knowledge and didactical transposition The work of the mathematician The student's work The teacher's work A few preliminary naive and fundamental questions Phenomena of didactique The Topaze effect and the control of uncertainty The Jourdain effect or fundamental misunderstanding 25

3 vi TABLE OF CONTENTS 2.3. Metacognitive shift 2.4. The improper use of analogy 2.5. The aging of teaching situations 3. Elements for a modelling 3.1. Didactical and adidactical situations 3.2. The didactical contract 3.3. An example of the devolution of an adidactical situation 3.4. The epistemology of teachers 3.5. Illustration: the Dienes effect 3.6. Heuristics and didactique 4. Coherence and incoherence of the modelling envisaged: The paradoxes of the didactical contract 4.1. The paradox of the devolution of situations 4.2. Paradoxes of the adaptation of situations Maladjustment to correctness Maladjustment to a later adaptation 4.3. Paradoxes of learning by adaptation Negation of knowledge Destruction of its cause 4.4. The paradox of the actor 5. Ways and means of modelling didactical situations 5.1. Fundamental situation corresponding to an item of knowledge With respect to the target knowledge With respect to teaching activity, 5.2. The notion of "game" 5.3. Game and reality Similarity Dissimilarity 5.4. Systemic approach of teaching situations 6. Adidactical situations 6.1. Fundamental sub-systems Classical patterns First decomposition proposed Necessity of the "adidactical milieu" sub-system Status of mathematical concepts 6.2. Necessity of distinguishing various types of adidactical situations Interactions The forms of knowledge The evolution of these forms of knowledge: learning The sub-systems of the milieu

4 vii 6.3. First study of three types of adidactical situations Action pattern Communication pattern Explicit validation pattern 69 Chapter 2 prelude 77 Chapter 2. Epistemological obstacles, problems and didactical engineering Epistemological obstacles and problems in mathematics The notion of problem Classical conception of the notion of problem Critique of these conceptions Importance of the notion of obstacle in teaching by means of problems The notion of obstacle Epistemological obstacles Manifestation of obstacles in didactique of mathematics Origin of various didactical obstacles Consequences for the organization of problem-situations Problems in the construction of the concept of decimals History of decimals History of the teaching of decimals Obstacles to didactique of a construction of decimals Epistemological obstacles didactical plan Comments after a debate Epistemological obstacles and didactique of mathematics Why is didactique of mathematics interested in epistemological obstacles? Do epistemological obstacles exist in mathematics? Search for an epistemological obstacle: historical approach The case of numbers Methods and questions Fractions in ancient Egypt Identification of pieces of knowledge What are the advantages of using unit fractions? Does the system of unit fractions constitute an obstacle? Search for an obstacle from school situations: A current unexpected obstacle, the natural numbers Obstacles and didactical engineering Local problems: lessons. How can an identified obstacle be dealt with? 110

5 viii TABLE OF CONTENTS "Strategic" problems: the curriculum. Which obstacles can be avoided and which accepted? Ill Didactical handling of obstacles Obstacles and fundamental didactics Problems internal to the class Problems external to the class 114 Chapter 3 prelude 117 Chapter 3. Problems with teaching decimal numbers Introduction The teaching of decimals in the 1960s in France Description of a curriculum Introductory lesson Metric system. Problems Operations with decimal numbers Decimal fractions Justifications and proofs Analysis of characteristic choices of this curriculum and of their consequences Dominant conception of the school decimal in Consequences for the multiplication of decimals The two representations of decimals The order of decimal numbers Approximation Influence of pedagogical ideas on this conception Evaluation of the results Classical methods Optimization Other methods Learning of "mechanisms" and "meaning" Separation of this learning and what causes it Algorithms The teaching of decimals in the 1970s Description of a curriculum Introductory lesson Other bases. Decomposition Operations Order Operators. Problems Approximation Analysis of this curriculum Areas 134

6 ix The decimal point Order Identification and evaporation Product Conclusion Study of a typical curriculum of the '70s The choices Properties of the operations Product Operators Fractions Conclusion Pedagogical ideas of the reform' The reform targets content Teaching structures Dienes's psychodynamic process The psychodynamic process and educational practice Influence of the psychodynamic process on the teaching of decimals, critiques and comments Conceptions and situations 144 Chapters 3 and 4 interlude 147 Chapter 4. Didactical problems with decimals General design of a process for teaching decimals Conclusions from the mathematical study Axioms and implicit didactical choices Transformations of mathematical discourse Metamathematics and heuristics Extensions and restrictions Mathematical motivations Conclusion of the epistemological study Different conceptions of decimals Dialectical relationships between D and Q Types of realized objects Different meanings of the product of two rationals Need for the experimental epistemological study Cultural obstacles Conclusions of the didactical study Principles The objectives of teaching decimals Consequences: types of situations New objectives Options 163

7 1.4. Outline of the process Notice to the reader Phase II: From measurement to the projections of D Phase I: From rational measures to decimal measures Analysis of the process and its implementation The pantograph Introduction to pantographs: the realization of Phase Examples of different didactical situations based on this schema of a situation Place of this situation in the process Composition of mappings (two sessions) Mathematical theory/practice relationships Different "levels of knowledge" relative to the compositions of the linear mappings About research on didactique Summary of the remainder of the process (2 sessions) Limits of the process of reprise The puzzle The problem-situation Summary of the rest of the process Affective and social foundations of mathematical proof Decimal approach to rational numbers (five sessions) Location of a rational number within a natural-number interval Rational-number intervals Remainder of the process Experimentation with the process 182 " Methodological observations The experimental situation School results Reproducibility obsolescence Brief commentary Analysis of a situation: The thickness of a sheet of a paper Description of the didactical situation (Sessionl,Phasel.l) Preparation of the materials and the setting First phase: search for a code (about minutes) Second phase: communication game (10 to 15 minutes) Third phase: result of the games and the codes (20 to 25 minutes) [confrontation] Results Comparison of thicknesses and equivalent pairs (Activity 1, Session 2) Preparation of materials and scene First phase (25-30 minutes) 200

8 xi Second phase: Completion of table; search for missing values (20-25 minutes) Third phase: Communication game (15 minutes) Results Summary of the rest of the sequence (Session 3) Results Analysis of the situation the game The problem-situation The didactical situation The maintenance of conditions of opening and their relationship with the meaning of the knowledge The didactical contract Analysis of didactical variables. Choice of game The type of situation The choice of thicknesses: implicit model From implicit model to explanation Questions about didactique of decimals The objects of didactical discourse Some didactical concepts The components of meaning The didactical properties of a problem-situation Situations, knowledge, behaviour Return to certain characteristics of the process Inadequacies of the process Return to decimal-measurement Remarks about the number of elements that allow the generation of a set ' Partitioning and proportioning Questions about methodology of research on didactique (on decimals), Models of errors Levels of complexity Dependencies and implications 221 Chapters 3 and 4 postlude Didactique and teaching problems 223 Chapter 5 prelude 225 Chapter 5. The didactical contract: the teacher, the student and the milieu Contextualization and decontextualization of knowledge Devolution of the problem and "dedidactification" The problem of meaning of intentional knowledge Teaching and learning The concept of devolution 229

9 xii TABLE OF CONTENTS 3. Engineering devolution: subtraction The search for the unknown term of a sum First stage: devolution of the riddle Second stage: anticipation of the solution Third stage: the statement and the proof Fourth stage: devolution and institutionalization of an adidactical learning situation Fifth stage: anticipation of the proof Institutionalization Knowing Meaning Epistemology The student's place Memory, time Conclusions 246 Chapter 6 prelude 251 Chapter 6. Didactique: What use is it to a teacher? Objects of didactique Usefulness of didactique Techniques for the teacher Knowledge about teaching Conclusions Difficulties with disseminating didactique " How one research finding reached the teaching profession What lesson can we draw from this adventure? Didactique and innovation 267 Appendix. The center for observations: The ecole Jules Michelet at Talence 275 Bibliography 283 References 287 Index of names 295 Index of subjects 299