Teaching Fractions through Situations: A Fundamental Experiment (eBook)

Preface 6
Contents 8
Acronyms 14
Chapter 1: Why These Adventures? 15
A Few Words by the Anglophone Author 17
First an Introduction to All Three Authors 17
Next the Background of the Teaching Project Itself: How and Why It Came to Exist 18
Introductory Remarks by Guy Brousseau 21
Chapter 2: The Adventure as Experienced by the Students 23
Module 1: Introducing Rational Numbers as Measurements 24
Lesson 1: Measurement of the Thicknesses of Sheets of Paper by Commensuration 24
The Set Up 24
The Search for a Code 24
The Communication Game 25
Result of the Games and Comparison of the Coding Systems 27
Different Types of Inconsistencies 27
Lesson 2: Comparison of Thicknesses and Equivalent Pairs (Summary of Lesson) 29
Lesson 3: Equivalence Classes – Rational Numbers (Summary of Lessons) 30
Module 2: Operations on Rational Numbers as Thicknesses 32
Lesson 1: The Sum of Thicknesses (Summary of Lesson) 32
Remarks on This Step: The Choice of Values 32
Lesson 2: Practicing the Sum of Thickness. What Should We Know Now? 33
Lesson 3: The Difference of Two Thicknesses (As Measure) 35
Lesson 4: The Thickness of a Piece of Cardboard Composed of Many Identical Sheets: Product of a Rational Number and a Whole Number 36
Lesson 5: Calculation of the Thickness of One Sheet: Division of a Rational Number by a Whole Number 38
Lesson 6: Assessment 39
Module 3: Measuring Other Quantities: Weight, Volume and Length 40
Lesson 1: Making Measurements 40
The Situation 41
Conclusions 42
Lesson 2: Construction of Fractional Lengths: A New Method Appears 42
Communication Game and Building Lengths Corresponding to a Pair 43
Assignment 43
Development 43
Lesson 3: Comparison of Methods, and Demonstration of Equivalence 44
Summary of the Lesson 44
Lesson 4: Fractions of Collections 45
Module 4: Groundwork for Introducing Decimal Numbers 47
Lesson 4: Whole Number Intervals Around a Fraction 47
First Phase: Introduction to the Game 47
Second Phase: Playing Two Against Two 49
Third Phase: Collective Synthesis 49
Module 5: Construction of the Decimal Numbers 50
Lesson 1: Bracketing a Rational Number with Rational Numbers: Chopping up an Interval 50
First Phase 50
Second Phase: The Search for a Smaller Interval 50
Third Phase: Search for Smaller and Smaller Intervals 51
Lesson 2: Bracketing a Rational Number Between Rational Numbers, Shrinking the Intervals, and Observing Decimal Filters 53
First Phase: Return to the Game from the Previous Session 53
Lesson 3: Representation on the Rational Number Line 55
Lesson 4: From Writing Decimal Rational Numbers as Fractions to Writing Them as Decimal Numerals 59
Starting a Table 59
Writing Fractions in Table A 60
Passage to Decimal Notation 60
Module 6: Operations with Decimal Numbers (Summary) 62
Module 7: Brackets and Approximations (Summary) 63
Module 8: Similarity 65
Lesson 1: Enlargement of a Puzzle 65
Strategies and Behaviors Observed 65
Lesson 2: The Image of a Whole Number 67
Lesson 3: The Image of a Fraction 69
Collective Synthesis of Methods 73
Exercises for Practice 73
Lesson 4: The Image of a Decimal Number 74
Phase 2: Comparison of Methods 75
Phase 3: Making the Pieces 75
Lesson 5: Division of a Decimal Number by 10, 100, 1,000, … (Summary) 76
Module 9: Linear Mappings 77
Lesson 1: Another Representation of the Optimist (Lesson Summarized) 77
Lesson 2: (Summary of Lesson) 79
Lesson 3: Lots of Representations of the Optimist (Summary of Lesson) 79
Lesson 4: Good Representations, Not So Good Representations 82
Lesson 5: Change of Model 84
Conclusion 85
Calculations with Other Images 85
Images and Reproductions 86
Lesson 6: Reciprocal Mappings 87
Presentation of the Problem 87
Module 10: Multiplication by a Rational Number 88
Lesson 1: Multiplication by a Rational Number 88
Lesson 2: Multiplying by a Decimal (Summary of Lessons) 92
Lesson 3: Methods of Solving Linear Problems (Summary of Lessons) 92
Lesson 4: The Search for Linear Situations (Summary of Lessons) 93
Module 11: The Study of Linear Situations in “Everyday Life” 95
Lesson 1: Fraction of a Magnitude 95
Exercises in Formulating Fractions in Terms of Linear Mappings 97
Summary of the Remaining Paragraphs and Sections of Module 11 101
Module 12: More on the Problem Statement Contest 104
Lesson 1 104
Phase 1: Research 104
Phase 4: Production of New Problems and Use of the Criteria 104
Module 13: New Division Problems in the Rationals 107
Lesson 2: (Extract) Division as Reciprocal Mapping of Multiplication (The Term Is Not Taught to the Students) 108
Division (by a Number), a Linear Mapping 108
Division by a Fraction: Calculation of the Image 108
Extracts from the Original Text 110
Module 14: Composition of Linear Mappings 115
Lesson 1: The Pantograph 115
Introduction of the Pantographs 116
Lesson 2: Composition of Mappings: First Session 117
Materials 117
Presentation of the Situation 117
Presentation of a Game: First Try 118
Game: Second Try 119
Lesson 3: Composition of Linear Mappings: Designation of Composed Mappings 121
Lesson 4: Different Ways of Writing the Same Mapping 124
Conclusion 126
Lesson 5: Rational Linear Mappings 128
Presentation of the Problem 128
The Teacher Writes 129
A search for all the rational linear mappings the pantograph can produce 129
Making a Table 130
Application exercises, done individually in mathematics notebooks 131
Module 15: Decomposition of Rational Mappings. Identification of Rational Numbers and Rational Linear Mappings 132
Lesson 1: Decomposition of Rational Mappings 132
Decomposition of a Rational Mapping into Natural Number Mappings 132
Decomposition of the Reciprocal 132
Lesson 2: The Meaning of “Division by a Fraction” (Summary of Lessons) 135
Lesson 3: Division of Decimals 138
A Mapping For the Calculation of Decimal Numbers 138
Conclusions and Installation of the Algorithm 139
Chapter 3: The Adventure as Experienced by the Teachers 140
Background of the Project 140
The Relationship with the Theory of Situations 144
The Perspective of the Teacher 147
Observable Aspects of Connaissances 153
Manifestation of Savoirs 153
What Then Are the Causes of Learning and the Reasons for Knowing? 157
How Does the Teacher Use Assessment of and Within the Curriculum? 158
The Assessment of Students and Groups of Students 159
The Types of Situations That Appear in the Lessons 160
The Types of Didactical Situation and How They Are Conducted 161
Situations of Institutionalization 161
Situations of Devolution of an A-didactical Situation 162
Situations of Evaluation 162
The Types of A-didactical Situation and How They Are Conducted 163
Situations of Validation (or Proof) 164
Situations of Formulation 164
Situations of Action 165
Presentation of the Rules of the Game 166
Evaluation in A-didactical Situations 167
Obsolescence 168
Isolated Evaluation of Savoirs and Constant Evaluation the Necessity of the Uncertain and the Implicit
The Play of the Real and the Fictional 169
The Inexpressible, the Said and the Unsaid 170
Further Aspects of the Teachers’ Adventures 170
The Mathematical Organization of the Curriculum 172
Mathematical Commentary on Chap. 2 173
The Temporary Replacement of Fractions by Commensurations 174
Chapter 4: The Adventure as Experienced by the Researchers 177
Warfield Introduction Concerning the History of and Voice in Chapter 4 177
Brousseau Introduction to Chapter 4 178
Prelude (1960–1970) 182
The Sources 182
The Adventure of “Modern Mathematics” 183
The Subject of the Studies Proposed by Lichnerowicz 184
The Background of the Future Research 185
Experimentation: How and in What Form? 187
Our Experiments 189
From Experiments to Theories: And a Science? 190
The Framework 191
Observation 193
Reflections on This Ambitious Project 193
The Foundations (1970–1975) 194
The IREM [Instituts de Recherches pour l’Enseignement des Mathématiques] 12 the Bordeaux IREM
The COREM (1973) 198
Further Developments over Time 198
The Diplôme d’Études Avancées de Didactique des Mathématiques (DEA) 1975 198
The Doctorate of Didactique of Mathematics, Part of Mathematical Sciences 199
Documentation 200
Research Organizations in Didactique of Mathematics 200
The Current State of Didactique of Mathematics 201
Further Commentary on Professor Lichnerowicz’s Challenge 201
Institutional Difficulties 201
Difficulties in Experimentation 202
Possibilities for Experiments 202
A New Conception of What It Means to Teach 203
Conception of Teaching 203
Construction of Alternatives 204
The Contributions of Piaget 204
The Notion of Situation 205
First Questions 206
A Child and a Concept 207
Models of a Genesis 207
The Standard Presentation of Mathematical Concepts 208
Connaissances and Savoirs 209
The Place for Connaissances: The Types of Mathematical Situation s and Theories 210
Chapter 5: Expansions and Clarifications 212
Connaissances and Savoirs 212
Didactical Situations 213
Institutionalization 214
Didactical Contract 215
Connaissances and Epistemological Obstacles 216
Metadidactical Slippage 217
Various Examples of Uncontrolled Chains of Metadidactical Slippages 219
The Slippages Studied in This Curriculum 219
Evaluations 220
Bibliography 222
Index 224