search for books and compare prices
Tables of Contents for Theory of Didactical Situations in Mathematics
Chapter/Section Title
Page #
Page Count
Editors' Preface
xiii
2
Biography of Guy Brousseau
xv
Prelude to the Introduction
1
2
Introduction. Setting the scene with an example: The race to 20
3
16
1. Introduction of the race to 20
3
3
1.1. The game
3
1
1.2. Description of the phases of the game
3
2
1.3. Remarks
5
1
2. First phase of the lesson: Instruction
6
2
3. Action-situation, pattern, dialectic
8
2
3.1. First part of the game (one against the other)
8
1
3.2. Dialectic of action
9
1
4. Formulation-situation, pattern, dialectic
10
3
4.1. Second part of the game (group against group)
10
2
4.2. Dialectic of formulation
12
1
5. Validation-situation, pattern, dialectic
13
6
5.1. Third part of the game (establishment of theorems)
13
2
5.2. The attitude of proof, proof and mathematical proof
15
1
5.3. Didactical situation of validation
15
2
5.4. Dialectic of validation
17
2
Chapter 1 prelude
19
2
Chapter 1. Foundations and methods of didactique
21
56
1. Objects of study of didactique
21
4
1.1. Mathematical knowledge and didactical transposition
21
1
1.2. The work of the mathematician
21
1
1.3. The student's work
22
1
1.4. The teacher's work
23
1
1.5. A few preliminary naive and fundamental questions
23
2
2. Phenomena of didactique
25
4
2.1. The Topaze effect and the control of uncertainty
25
1
2.2. The Jourdain effect or fundamental misunderstanding
25
1
2.3. Metacognitive shift
26
1
2.4. The improper use of analogy
27
1
2.5. The aging of teaching situations
27
2
3. Elements for a modelling
29
11
3.1. Didactical and adidactical situations
29
2
3.2. The didactical contract
31
2
3.3. An example of the devolution of an adidactical situation
33
2
3.4. The epistemology of teachers
35
1
3.5. Illustration: the Dienes effect
35
2
3.6. Heuristics and didactique
37
3
4. Coherence and incoherence of the modelling envisaged: The paradoxes of the didactical contract
40
7
4.1. The paradox of the devolution of situations
41
1
4.2. Paradoxes of the adaptation of situations
42
2
4.2.1. Maladjustment to correctness
42
1
4.2.2. Maladjustment to a later adaptation
43
1
4.3. Paradoxes of learning by adaptation
44
1
4.3.1. Negation of knowledge
44
1
4.3.2. Destruction of its cause
45
1
4.4. The paradox of the actor
45
2
5. Ways and means of modelling didactical situations
47
7
5.1. Fundamental situation corresponding to an item of knowledge
47
1
5.1.1. With respect to the target knowledge
47
1
5.1.2. With respect to teaching activity
48
1
5.2. The notion of "game"
48
3
5.3. Game and reality
51
2
5.3.1. Similarity
51
1
5.3.2. Dissimilarity
52
1
5.4. Systemic approach of teaching situations
53
1
6. Adidactical situations
54
23
6.1. Fundamental sub-systems
54
6
6.1.1. Classical patterns
54
2
6.1.2. First decomposition proposed
56
1
6.1.3. Necessity of the "adidactical milieu" sub-system
57
1
6.1.4. Status of mathematical concepts
58
2
6.2. Necessity of distinguishing various types of adidactical situations
60
5
6.2.1. Interactions
61
1
6.2.2. The forms of knowledge
62
1
6.2.3. The evolution of these forms of knowledge: learning
63
2
6.2.4. The sub-systems of the milieu
65
1
6.3. First study of three types of adidactical situations
65
12
6.3.1. Action pattern
65
2
6.3.2. Communication pattern
67
2
6.3.3. Explicit validation pattern
69
8
Chapter 2 prelude
77
2
Chapter 2. Epistemological obstacles, problems and didactical engineering
79
38
1. Epistemological obstacles and problems in mathematics
79
19
1.1. The notion of problem
79
4
1.1.1. Classical conception of the notion of problem
79
2
1.1.2. Critique of these conceptions
81
1
1.1.3. Importance of the notion of obstacle in teaching by means of problems
82
1
1.2. The notion of obstacles
83
7
1.2.1. Epistemological obstacles
83
1
1.2.2. Manifestation of obstacles in didactique of mathematics
84
2
1.2.3. Origin of various didactical obstacles
86
1
1.2.4. Consequences for the organization of problem-situations
87
3
1.3. Problems in the construction of the concept of decimals
90
3
1.3.1. History of decimals
90
1
1.3.2. History of the teaching of decimals
90
1
1.3.3. Obstacles to didactique of a construction of decimals
91
1
1.3.4. Epistemological obstacles-didactical plan
92
1
1.4. Comments after a debate
93
5
2. Epistemological obstacles and didactique of mathematics
98
19
2.1. Why is didactique of mathematics interested in epistemological obstacles?
98
1
2.2. Do epistemological obstacles exist in mathematics?
99
1
2.3. Search for an epistemological obstacle: historical approach
100
7
2.3.1. The case of numbers
100
1
2.3.2. Methods and questions
101
1
2.3.3. Fractions in ancient Egypt
101
6
2.3.3.1. Identification of pieces of knowledge
102
2
2.3.3.2. What are the advantages of using unit fractions?
104
3
2.3.3.3. Does the system of unit fractions constitute an obstacle?
107
1
2.4. Search for an obstacle from school situations: A current unexpected obstacle, the natural numbers.
107
3
2.5. Obstacles and didactical engineering
110
2
2.5.1. Local problems: lessons. How can an identified obstacle be dealt with?
110
1
2.5.2. "Strategic" problems: the curriculum. Which obstacles can be avoided and which accepted?
111
1
2.5.3. Didactical handling of obstacles
111
1
2.6. Obstacles and fundamental didactics
112
5
2.6.1. Problems internal to the class
113
1
2.6.2. Problems external to the class
114
3
Chapter 3 prelude
117
2
Chapter 3. Problems with teaching decimal numbers
119
30
1. Introduction
119
2
2. The teaching of decimals in the 1960s in France
121
10
2.1. Description of a curriculum
121
2
2.1.1. Introductory lesson
121
1
2.1.2. Metric system. Problems
122
1
2.1.3. Operations with decimal numbers
122
1
2.1.4. Decimal fractions
123
1
2.1.5. Justifications and proofs
123
1
2.2. Analysis of characteristic choices of this curriculum and of their consequences
123
3
2.2.1. Dominant conception of the school decimal in 1960
123
1
2.2.2. Consequences for the multiplication of decimals
123
2
2.2.3. The two representations of decimals
125
1
2.2.4. The order of decimal numbers
125
1
2.2.5. Approximation
126
1
2.3. Influence of pedagogical ideas on this conception
126
2
2.3.1. Evaluation of the results
126
1
2.3.2. Classical methods
127
1
2.3.3. Optimization
127
1
2.3.4. Other methods
128
1
2.4. Learning of "mechanisms" and "meaning"
128
3
2.4.1. Separation of this learning and what causes it
128
1
2.4.2. Algorithms
129
2
3. The teaching of decimals in the 1970s
131
16
3.1. Description of a curriculum
131
3
3.1.1. Introductory lesson
131
1
3.1.2. Other bases. Decomposition
132
1
3.1.3. Operations
132
1
3.1.4. Order
132
1
3.1.5. Operators. Problems
133
1
3.1.6. Approximation
134
1
3.2. Analysis of this curriculum
134
2
3.2.1. Areas
134
1
3.2.2. The decimal point
134
1
3.2.3. Order
134
1
3.2.4. Identification and evaporation
134
1
3.2.5. Product
135
1
3.2.6. Conclusion
135
1
3.3. Study of a typical curriculum of the `70s
136
2
3.3.1. The choices
136
1
3.3.2. Properties of the operations
136
1
3.3.3. Product
136
1
3.3.4. Operators
136
1
3.3.5. Fractions
136
1
3.3.6. Conclusion
137
1
3.4. Pedagogical ideas of the reform
138
9
3.4.1. The reform targets content
138
1
3.4.2. Teaching structures
138
1
3.4.3. Dienes's psychodynamic process
139
3
3.4.4. The psychodynamic process and educational practice
142
1
3.4.5. Influence of psychodynamic process on the teaching of decimals, critiques and comments
143
1
3.4.6. Conceptions and situations
144
3
Chapters 3 and 4 interlude
147
2
Chapter 4. Didactical problems with decimals
149
76
1. General design of a process for teaching decimals
149
18
1.1. Conclusions from the mathematical study
149
3
1.1.1. Axioms and implicit didactical choices
149
1
1.1.2. Transformations of mathematical discourse
149
1
1.1.3. Metamathematics and heuristics
150
1
1.1.4. Extensions and restrictions
150
1
1.1.5. Mathematical motivations
151
1
1.2. Conclusion of the epistemological study
152
8
1.2.1. Different conceptions of decimals
152
1
1.2.2. Dialectical relationships between D and Q
153
1
1.2.3. Types of realized objects
154
1
1.2.4. Different meanings of the product of two rationals
154
6
1.2.5. Need for the experimental epistemological study
160
1
1.2.6. Cultural obstacles
160
1
1.3. Conclusions of the didactical study
160
4
1.3.1. Principles
160
1
1.3.2. The objectives of teaching decimals
161
1
1.3.3. Consequences: types of situations
161
1
1.3.4. New objectives
162
1
1.3.5. Options
163
1
1.4. Outline of the process
164
3
1.4.1. Notice to the reader
164
1
1.4.2. Phase II: From measurement to the projections of D(+)
164
2
1.4.3. Phase I: From rational measures to decimal measures
166
1
2. Analysis of the process and its implementation
167
28
2.1. The pantograph
167
10
2.1.1. Introduction to pantographs: the realization of Phase 2.6
167
1
2.1.2. Examples of different didactical situations based on this schema of a situation
168
1
2.1.3. Place of this situation in the process
169
1
2.1.4. Composition of mappings (two sessions)
169
1
2.1.5. Mathematical theory practice relationships
169
3
2.1.6. Different "levels of knowledge" relative to the compositions of the linear mappings
172
3
2.1.7. About research on didactique
175
1
2.1.8. Summary of the remainder of the process (2 sessions)
176
1
2.1.9. Limits of the process of reprise
176
1
2.2. The puzzle
177
3
2.2.1. The problem-situation
177
2
2.2.2. Summary of the rest of the process
179
1
2.2.3. Affective and social foundations of mathematical proof
179
1
2.3. Decimal approach to rational numbers (five sessions)
180
2
2.3.1. Location of a rational number within a natural-number interval
181
1
2.3.2. Rational-number intervals
181
1
2.3.3. Remainder of the process
182
1
2.4. Experimentation with the process
182
13
2.4.1. Methodological observations
182
2
2.4.2. The experimental situation
184
1
2.4.3. School results
185
7
2.4.4. Reproducibility--obsolescence
192
2
2.4.5. Brief commentary
194
1
3. Analysis of a situation: The thickness of a sheet of a paper
195
17
3.1. Description of the didactical situation (Session 1, Phase 1.1)
195
5
3.1.1. Preparation of the materials and the setting
195
1
3.1.2. First phase: search for a code (about 20-25 minutes)
195
2
3.1.3. Second phase: communication game (10 to 15 minutes)
197
1
3.1.4. Third phase: result of the games and the codes (20 to 25 minutes) [confrontation]
198
2
3.1.5. Results
200
1
3.2. Comparison of thicknesses and equivalent pairs (Activity 1, Session 2)
200
4
3.2.1. Preparation of materials and scene
200
1
3.2.2. First phase (25-30 minutes)
200
2
3.2.3. Second phase: Completion of table; search for missing values (20-25 minutes)
202
1
3.2.4. Third phase: Communication game (15 minutes)
202
1
3.2.5. Results
203
1
3.2.6. Summary of the rest of the sequence (Session 3)
204
1
3.2.7. Results
204
1
3.3. Analysis of the situation--the game
204
4
3.3.1. The problem-situation
204
1
3.3.2. The didactical situation
205
1
3.3.3. The maintenance of conditions of opening and their relationship with the meaning of the knowledge
206
1
3.3.4. The didactical contract
207
1
3.4. Analysis of didactical variables. Choice of game
208
4
3.4.1. The type of situation
208
1
3.4.2. The choice of thicknesses: implicit model
209
1
3.4.3. From implicit model to explanation
210
2
4. Questions about didactique of decimals
212
11
4.1. The objects of didactical discourse
212
1
4.2. Some didactical concepts
213
2
4.2.1. The components of meaning
213
1
4.2.2. The didactical properties of a problem-situation
214
1
4.2.3. Situations, knowledge, behaviour
215
1
4.3. Return to certain characteristics of the process
215
6
4.3.1. Inadequacies of the process
215
2
4.3.2. Return to decimal-measurement
217
1
4.3.3. Remarks about the number of elements that allow the generation of a set
218
1
4.3.4. Partitioning and proportioning
218
3
4.4. Questions about methodology of research on didactique (on decimals)
221
2
4.4.1. Models of errors
221
1
4.4.2. Levels of complexity
221
1
4.4.3. Dependencies and implications
221
2
Chapter 3 and 4 postlude--Didactique and teaching problems
223
2
Chapter 5 prelude
225
2
Chapter 5. The didactical contract: the teacher, the student and the milieu
227
24
1. Contextualization and decontextualization of knowledge
227
1
2. Devolution of the problem and "dedidactification"
227
3
2.1. The problem of meaning of intentional knowledge
227
1
2.2. Teaching and learning
228
1
2.3. The concept of devolution
229
1
3. Engineering devolution: subtraction
230
5
3.1. The search for the unknown term of a sum
230
1
3.2. First stage: devolution of the riddle
231
1
3.3. Second stage: anticipation of the solution
232
1
3.4. Third stage: the statement and the proof
232
1
3.5. Fourth stage: devolution and institutionalization of an adidactical learning situation
233
1
3.6. Fifth stage: anticipation of the proof
234
1
4. Institutionalization
235
11
4.1. Knowing
235
2
4.2. Meaning
237
2
4.3. Epistemology
239
4
4.4. The student's place
243
2
4.5. Memory, time
245
1
5. Conclusions
246
5
Chapter 6 prelude
251
2
Chapter 6. Didactique: What use is it to a teacher?
253
22
1. Objects of didactique
253
2
2. Usefulness of didactique
255
8
2.1. Techniques for the teacher
256
3
2.2. Knowledge about teaching
259
4
2.3. Conclusions
263
1
3. Difficulties with disseminating didactique
263
4
3.1. How one research finding reached the teaching profession
263
2
3.2. What lesson can we draw from this adventure?
265
2
4. Didactique and innovation
267
8
Appendix. The center for observations: The ecole Jules Michelet at Talence
275
8
Bibliography
283
4
References
287
8
Index of names
295
4
Index of subjects
299