Theory of Didactical Situations in Mathematics: Didactique des mathématiques, 1970?1990

Theory of Didactical Situations in Mathematics: Didactique des mathématiques, 1970?1990

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English

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This is a collection of texts which give the best presentation of the principles and key concepts of the theory of didactical situations that Guy Brousseau developed in the period from 1970 to 1990. These texts have been edited and organized so that they provide a comprehensive presentation of the theory. Concepts such as didactical contract, didactical variable and epistemological obstacle are presented in detail. At the heart of the book, one will find two works which demonstrate in detail the articulation between theoretical work and experimental research which is the source of the richness of the theory. In order to facilitate the reading of certain points, or elucidate them for the reader, footnotes have been added to the original text, as well as preludes and interludes to place in context the chosen texts and clarify the construction of the book.

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Published 01 January 1997
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EAN13 0306472112
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TABLE OF CONTENTS
Editors’ Preface Biography of Guy Brousseau
Prelude to the Introduction Introduction. Setting the scene with an example: The race to 20 1. Introduction of the race to 20 1.1.The game 1.2. Descriptionofthe phasesofthe game 1.3.Remarks 2. First phaseofthe lesson: Instruction 3.Action—situation, pattern, dialectic 3.1.First partofthe game (one against the other) 3.2. Dialecticofaction 4. Formulation situation, pattern, dialectic 4.1.Second part of the game (group against group) 4.2. Dialectic of formulation 5.1. Thirdpart of the game (establishment of theorems) 5.Validationsituation, pattern, dialectic 5.2. The attitude of proof, proof and mathematical proof 5.3.Didactical situationofvalidation 5.4. Dialecticofvalidation
Chapter1prelude
Chapter1.Foundations and methodsofdidactique 1.Objects of studyofdidactique 1.1.Mathematical knowledge and didactical transposition 1.2.The work of themathematician 1.3.The student’s work 1.4. The teacher’s work 1.5.Afew preliminary naïve and fundamental questions 2. Phenomena ofdidactique 2.1. The Topaze effect and the control of uncertainty 2.2. The Jourdain effect or fundamental misunderstanding v
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Metacognitive shift The improper useofanalogy The agingofteaching situations
3. Elements foramodelling 3.1. Didactical and adidactical situations 3.2.The didactical contract 3.3.An exampleofthe devolutionofan adidactical situation 3.4. The epistemologyofteachers 3.5. Illustration: the Diénès effect 3.6. Heuristics anddidactique
4. Coherence and incoherenceofthe modelling envisaged: The paradoxes ofthe didactical contract 4.1. The paradoxofthe devolutionofsituations 4.2. Paradoxesofthe adaptationofsituations 4.2.1.Maladjustment to correctness 4.2.2. Maladjustment toalater adaptation 4.3. Paradoxesoflearning by adaptation 4.3.1. Negation of knowledge 4.3.2. Destructionofits cause 4.4. The paradox of the actor
5. Ways and means of modelling didactical situations 5.1. Fundamental situation corresponding to an itemofknowledge 5.1.1.With respect to the target knowledge 5.1.2. With respect to teaching activity 5.2. The notion of “game” 5.3. Game and reality 5.3.1. Similarity 5.3.2. Dissimilarity 5.4. Systemic approachofteaching situations 6. Adidactical situations 6.1. Fundamental subsystems 6.1.1.Classical patterns 6.1.2. First decomposition proposed 6.1.3. Necessity of the “adidacticalmilieusubsystem 6.1.4. Statusofmathematical concepts 6.2. Necessity of distinguishing various types of adidactical situations 6.2.1. Interactions 6.2.2. The forms of knowledge 6.2.3. The evolution of these forms of knowledge: learning 6.2.4. The subsystems of themilieu
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6.3.three typesFirst study of ofadidactical situations 6.3.1.Action pattern 6.3.2.Communicationpattern 6.3.3.Explicit validation pattern
Chapter2prelude
Chapter 2. Epistemological obstacles, problems and didactical engineering
1. Epistemological obstacles and problems in mathematics 1.1. The notionofproblem 1.1.1.Classical conceptionofthe notionofproblem 1.1.2. Critiqueofthese conceptions 1.1.3.Importance of the notion of obstacle in teaching by means of problems 1.2.The notionofobstacle 1.2.1.Epistemologicalobstacles 1.2.2.Manifestation of obstacles indidactiqueof mathematics 1.2.3.Originofvarious didactical obstacles 1.2.4. Consequences for the organization of problemsituations 1.3. Problems in the construction of the concept of decimals 1.3.1.History ofdecimals 1.3.2.History of the teaching of decimals 1.3.3.Obstaclestodidactiqueofa constructionof decimals 1.3.4. Epistemological obstacles—didactical plan 1.4. Comments after a debate
2.Epistemological obstacles anddidactiqueofmathematics 2.1. Why isdidactiqueof mathematics interested in epistemological obstacles? 2.2.Do epistemological obstacles existinmathematics? 2.3.Search for an epistemological obstacle: historical approach 2.3.1.The case of numbers 2.3.2.Methods and questions 2.3.3.Fractions in ancient Egypt 2.3.3.1.Identification of pieces of knowledge 2.3.3.2.What are the advantages of using unit fractions? 2.3.3.3.Does the systemofunit fractions constitute an obstacle? 2.4. Search for an obstacle from school situations:Acurrent unexpected obstacle, the natural numbers. 2.5. Obstacles and didactical engineering 2.5.1. Local problems: lessons. How can an identified obstacle be dealt with?
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2.5.2. “Strategic” problems: the curriculum. Which obstacles can be avoided and which accepted? 2.5.3. Didactical handling of obstacles 2.6. Obstacles and fundamental didactics 2.6.1. Problems internal to the class 2.6.2. Problems externaltothe class
Chapter 3 prelude
Chapter 3. Problems with teaching decimal numbers 1. Introduction 2. The teaching of decimals in the 1960s in France 2.1. Description of a curriculum 2.1.1. Introductory lesson 2.1.2. Metric system. Problems 2.1.3. Operations with decimal numbers 2.1.4. Decimal fractions 2.1.5. Justifications and proofs 2.2. Analysis of characteristic choices of this curriculum and of their consequences 2.2.1. Dominant conception of the school decimal in 1960 2.2.2. Consequences for the multiplication of decimals 2.2.3.The two representations of decimals 2.2.4.Theorderof decimal numbers 2.2.5. Approximation 2.3. Influence of pedagogical ideas on this conception 2.3.1.Evaluation of the results 2.3.2. Classical methods 2.3.3. Optimization 2.3.4.Other methods 2.4. Learning of “mechanisms” and “meaning” 2.4.1. Separationofthis learning and what causesit 2.4.2. Algorithms 3. The teaching of decimals in the 1970s 3.1. Descriptionofa curriculum 3.1.1. Introductory lesson 3.1.2. Other bases. Decomposition 3.1.3. Operations 3.1.4. Order 3.1.5. Operators. Problems 3.1.6. Approximation 3.2. Analysis of this curriculum 3.2.1. Areas
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3.2.2. The decimal point 3.2.3.Order 3.2.4. Identification and evaporation 3.2.5. Product 3.2.6. Conclusion 3.3.Studyof atypical curriculumofthe ‘70s 3.3.1. The choices 3.3.2. Properties of the operations 3.3.3. Product 3.3.4.Operators 3.3.5. Fractions 3.3.6. Conclusion 3.4. Pedagogical ideas of the reform 3.4.1.The reform targets content 3.4.2.Teaching structures 3.4.3. Diénès’s psychodynamic process 3.4.4. The psychodynamic process and educational practice 3.4.5.Influence of the psychodynamic process on the teaching of decimals, critiques and comments 3.4.6. Conceptions and situations
Chapters3and4interlude
Chapter4.Didactical problems with decimals 1. General design of a process for teaching decimals 1.1. Conclusions from the mathematical study 1.1.1.Axioms and implicit didactical choices 1.1.2. Transformations of mathematical discourse 1.1.3. Metamathematics and heuristics 1.1.4. Extensions and restrictions 1.1.5. Mathematical motivations 1.2.Conclusion of the epistemological study 1.2.1. Different conceptions of decimals 1.2.2. Dialectical relationships between D andQ 1.2.3. Typesofrealized objects 1.2.4. Different meaningsofthe productoftwo rationals 1.2.5. Need for the experimental epistemological study 1.2.6. Cultural obstacles 1.3. Conclusions of the didactical study 1.3.1.Principles 1.3.2. The objectivesofteaching decimals 1.3.3. Consequences: types of situations 1.3.4.New objectives 1.3.5. Options
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1.4.Outline of the process 1.4.1.Noticetothe reader + 1.4.2. Phase II: From measurement to the projections of D 1.4.3. Phase I: From rational measures to decimal measures 2. Analysis of the process and its implementation 2.1. The pantograph 2.1.1.Introduction to pantographs: the realization of Phase 2.6 2.1.2. Examples of different didactical situations based on this schemaof a situation 2.1.3. Place of this situation in the process 2.1.4.Composition of mappings (two sessions) 2.1.5. Mathematical theory/practice relationships 2.1.6. Different “levels of knowledge” relativetothe compositions of the linear mappings 2.1.7.About research ondidactique 2.1.8.remainder of the process (2 sessions)Summary of the 2.1.9.process of repriseLimits of the 2.2. The puzzle 2.2.1.The problemsituation 2.2.2. Summary of the rest of the process 2.2.3.of mathematical proofAffective and social foundations 2.3.Decimal approach to rational numbers(five sessions) 2.3.1. Location of a rational number within a naturalnumber interval 2.3.2. Rationalnumber intervals 2.3.3. Remainder of the process 2.4. Experimentation with the process 2.4.1. Methodological observations 2.4.2. The experimental situation 2.4.3. School results 2.4.4.Reproducibility—obsolescence 2.4.5. Brief commentary 3. Analysis of a situation: The thickness of a sheet of a paper 3.1.Description of the (Session 1, Phase 1.1)didactical situation 3.1.1. Preparation of the materials and the setting 3.1.2. First phase: search foracode (about 2025 minutes) 3.1.3.to 15 minutes)Second phase: communication game (10 3.1.4. Third phase: result of the games and the codes (20 to 25 minutes) [confrontation] 3.1.5. Results 3.2.Comparison of thicknesses and equivalent pairs (Activity1,Session 2) 3.2.1. Preparation of materials and scene 3.2.2. First phase (25 30 minutes)
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3.2.3.Secondphase: Completion oftable; search for missing values (2025 minutes) 3.2.4. Third phase: Communication game (15 minutes) 3.2.5.Results 3.2.6.Summary of the rest of the sequence (Session 3) 3.2.7. Results 3.3. Analysis of the situation—the game 3.3.1. The problemsituation 3.3.2. The didactical situation 3.3.3. The maintenance of conditions of opening and their relationship with the meaning of the knowledge 3.3.4.The didactical contract 3.4.Analysis of didactical variables. Choice of game 3.4.1.The type of situation 3.4.2.The choice of thicknesses: implicit model 3.4.3.From implicit model to explanation 4.Questions aboutdidactiqueof decimals 4.1. The objects of didactical discourse 4.2. Some didactical concepts 4.2.1.The components of meaning 4.2.2.The didactical properties of a problemsituation 4.2.3.Situations, knowledge, behaviour 4.3.Return to certain characteristics of the process 4.3.1.Inadequacies of the process 4.3.2.Return to decimalmeasurement 4.3.3.Remarks about the number of elements that allow the generation of a set 4.3.4.Partitioning and proportioning 4.4.Questions about methodology of research ondidactique (on decimals) 4.4.1.Models of errors 4.4.2. Levels of complexity 4.4.3.Dependencies and implications
Chapters3 and 4 postlude—Didactiqueand teaching problems
Chapter 5 prelude
Chapter 5. The didactical contract: the teacher, the student and themilieu 1. Contextualization and decontextualization of knowledge 2.Devolution of the problem and “dedidactification” 2.1.The problem of meaning of intentional knowledge 2.2.Teaching and learning 2.3. The concept of devolution
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3.Engineering devolution: subtraction 3.1. The search for the unknowntermof a sum 3.2.First stage: devolution of the riddle 3.3.Second stage: anticipation of the solution 3.4. Third stage: the statement and the proof 3.5.Fourth stage: devolution and institutionalization of an adidactical learning situation 3.6.Fifth stage: anticipation of the proof 4.Institutionalization 4.1. Knowing 4.2. Meaning 4.3. Epistemology 4.4. The student’s place 4.5. Memory, time 5. Conclusions
Chapter 6 prelude
Chapter 6.Didactique: What use is it to a teacher? 1. Objects ofdidactique 2.Usefulnessofdidactique 2.1.Techniques for the teacher 2.2.Knowledge about teaching 2.3.Conclusions 3.Difficulties with disseminatingdidactique 3.1. How one research finding reached the teaching profession 3.2. What lesson can we draw from thisadventure? 4.Didactiqueand innovation
Appendix. The center for observations: The école Jules Michelet at Talence Bibliography References Index of names Index of subjects
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