On the teaching of linear algebra:
Gespeichert in:
| Weitere Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
c2000
|
| Schriftenreihe: | Mathematics education library
23 |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents only Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XXII, 288 S. Ill. |
| ISBN: | 0792365399 |
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| adam_text | TABLE OF CONTENTS
FOREWORD TO THE ENGLISH EDITION xiii
PREFACE xv
INTRODUCTION xix
PART I. EPISTEMOLOGICAL ANALYSIS OF THE GENESIS
OF THE THEORY OF VECTOR SPACES 1
J.-L. Dorier
1. INTRODUCTION 3
2. ANALYTICAL AND GEOMETRICAL ORIGINS. 6
2.1. Systems of Numerical Linear Equations: First Local Theory of Linearity 6
2.1.1. Euler and the Dependence of Equations 6
2.1.2. Cramer and the Birth of the Theory of Determinants 8
2.1.3. The Concept of Rank 9
2.1.4. Georg Ferdinand Frobenius 10
2.1.5. Further Development 11
2.2. Geometry and Linear Algebra: a Two-Century-Long Process of Complex
Reciprocal Exchanges 11
2.2.1. Analytic Geometry 13
2.2.2. Leibniz s Criticism 13
2.2.3. Geometrical Representation of Imaginary Quantities 14
2.2.4. Mobius s Barycentric Calculus 15
2.2.5. Bellavitis s Calulus of Equipollences 16
2.3. Hamilton s Quaternions 16
2.4. Hermann Grassmann s Ausdehnungslehre 18
2.4.1. The Context 18
2.4.2. The Philosophical Background 20
2.4.3. The Mathematical Content 22
2.4.4. The 1862 Version 24
2.5. First Phase of Unification of Finite-Dimensional Linear Problems 27
2.5.1. Analytical Origins 27
2.5.2. Geometrical Context 27
2.5.3. New Perspectives 28
vi
3. TOWARDS A FORMAL AXIOMATIC THEORY 30
3.1. First Axiomatic Presentations of Linear Algebra 31
3.1.1. Giuseppe Peano 31
3.1.2. Salvatore Pincherle 33
3.1.3. Cesare Burali-Forti and Roberto Marcolongo 35
3.1.4. Hermann Weyl 37
3.1.5. Modern Algebra 38
3.2. Infinite-dimensional linear problems 42
3.2.1. Origins 43
3.2.2. Fredholm Integral Equation 45
3.2.3. David Hilbert 47
3.2.4. A Topological Approach 49
3.2.5. Frigyes (Frederic) Riesz, Ernst Fischer and Erhard Schmidt 49
3.2.6. Stefan Banach, Hans Hahn and Norbert Wiener 51
3.2.7. Maurice Frechet 53
3.2.8. John von Neumann and Quantum Mechanics 54
3.2.9. Banach s Decisive Contribution 55
3.2.10. Conclusion 55
3.3. Epilogue 56
4. CONCLUSION 59
5. NOTES 60
REFERENCES - PART 1 73
PART II. TEACHING AND LEARNING ISSUES 83
CHAPTER 1. THE OBSTACLE OF FORMALISM
IN LINEAR ALGEBRA 85
J.-L. Dorier, A. Robert, J. Robinet and M. Rogalski
1. INTRODUCTION 85
2. FIRST DIAGNOSIS, A 1987 STUDY 88
2.1. The Questionnaire 88
2.2. Rapid Analysis of the Contents of the Questionnaire 89
2.3. Relativization of the Results Obtained 89
2.4. Analysis of the Questionnaire 90
2.4.1. Question 1 90
2.4.2. Question 2 91
2.4.3. Question 3 91
2.4.4. Question 4 92
2.4.5. Question 5 93
2.4.6. Question 6 93
2.4.7. Question 7 94
vii
2.4.8. Conclusion 94
3. A SECOND SURVEY MADE IN 1990 : IS FORMALISM A DIDACTICAL
OBSTACLE RELATED TO THE LACK OF PREREQUISITES ? 95
3.1. Introduction 95
3.2. The Methodology and the Hypotheses 95
3.3. The Results 98
3.3.1. Global Analysis of the Contents of the Teaching 98
3.3.2. The Main Statistical Results 100
3.3.3. Correlations With the Pretest 100
4. THE LAST SURVEYS FROM 1991 TO 1994: THE OBSTACLE STILL
HAS NOT BEEN OVERCOME. 103
4.1. The Present State in an Experimental Teaching Method from Responses on
Four Tests 103
4.2. Ranking 103
4.3. Vector Subspaces 105
4.3.1. Questions on the Equality or Inclusion of Subspaces 105
4.3.2. Questions on the Representation of Subspaces 107
4.4. Difficulties Stemming from Weakness in Logic and Set Theory: Inclusion
and Implication, the Role of Parameters, the Notion of an Algebraic
Equation of a Geometric Object 111
4.4.1. The February 1992 Exam 112
4.4.2. The 1994 Workshop 116
4.4.3. Hypotheses and Suggestions 121
5. ANNEX - MATHEMATICAL SOLUTION OF THE HYPERPLANES
PROBLEM (1992, QUESTION (a) ) 123
6. NOTES 124
CHAPTER 2. LEVEL OF CONCEPTUALIZATION AND SECONDARY
SCHOOL MATH EDUCATION 125
A. Robert
1. WHAT DO WE CALL LEVEL OF CONCEPTUALIZATION ? 125
1.1. Seeking Magic Squares of the Third Order 125
1.2. A different Example: Complex Numbers and their Use in Geometry 127
1.3. Outcome 12 8
2. LEVELS OF CONCEPTUALIZATION
IN SECONDARY EDUCATION 129
3. CONCEPTUALIZATION 131
CHAPTER 3. THE TEACHING EXPERIMENTED IN LILLE 133
M. Rogalski
1. THE UNDERLYING PRINCIPLES OF THE COURSE DESIGN
IN LILLE 133
1.1. Take Into Account the Formalizing, Unifying, Generalizing, and
Simplifying Nature of the Concepts of Linear Algebra 134
1.2. The Prerequisite Hypothesis 134
viii
1.2.1. Comprehension of the True Stakes in Mathematics and a Certain
Practice of Elementary Logic and Language of Set Theory are of
Primary Importance 135
1.2.2. Accepting the Algebraic and Axiomatic Process 135
1.2.3. A Specific Practice of Geometry within the Sphere of Cartesian
Geometry 136
1.3. Hypothesis (3) on Long-Term Course Design, Use of Meta
and Changes in Point of View 136
2. THE DETAILED PLAN OF THE COURSE DESIGN 137
2.1. First Step 137
2.2. Second Step 138
2.3. Third Step 139
2.4. Fourth Step 139
2.5. Further Points 139
3. THE POSITIVE EFFECTS OF THE LILLE EXPERIMENT 142
4. ANNEXES 143
4.1. Annex 1: An Example of a Workshop on the Epistemology of Axiomatic
Linear Algebra 143
4.2. Annex 2: Examples of the Meaning of Methods and Methodology in
Linear Algebra 144
4.3. Annexe 3: Chronology of the Course Design (Example in 1991) 146
4.4. Annex 4 : A Test About the Notion of Rank, (1990/91) 148
4.5. Annex 5: Capacity for Modeling by an Equation T(u) = v 148
5. NOTES 149
CHAPTER 4. THE META LEVER 151
J.-L. Dorier, A. Robert, J. Robinet and M. Rogalski
1. DEFINITIONS AND EDUCATIONAL POSITIONS 151
2. INTRODUCTION TO THE STRUCTURE OF VECTOR SPACE. 155
2.1. Introduction 155
2.2. Presentation of the Teaching Sequence 156
2.3. Results 160
2.4. General Issues 161
2.5. Institutionalization 162
2.6. Long Term Evaluation 162
3. AN EXAMPLE OF THE USE OF META FOR TEACHING THE
GREGORY FORMULA 163
3.1. Presentation of the Mathematical Content 163
3.2. Use of the Gregory Formula in a Classic Setting 164
3.3. Reformulating the Problem 165
3.4. Results 16 8
4. UNDERSTANDING LINEARITY AND NON-LINEARITY 169
4.1. Presentation of the exercise 170
4.2. Rapid analysis of the exercise 171
ix
4.3. Situating the Exercise 171
4.4. The Type of Students 172
4.5. Analysis of the Results 172
5. CONCLUSION: AN IMPORTANT METHODOLOGICAL PROBLEM,
PERSPECTIVES. 173
5.1. Evaluation of Meta-Type Teaching 173
5.2. Perspectives on Teacher Intervention and Meta-Lever Analyses 174
5.2.1. The Assumed Function of Teacher Comments Related to Teaching
Phases: Various Scenarios and a Historical, Epistemological, and
Pedagogical Study to Pursue 174
5.2.2. Form and Range of Meta-Type Comments: a More Pedagogical
Problem 175
6. NOTES 776
CHAPTER 5. THREE PRINCIPLES OF LEARNING AND TEACHING
MATHEMATICS 177
G. Hard
1. BACKGROUND 177
1.1. Proof 178
1.2. Time Allocated to Linear Algebra 178
1.3. Technology 179
1.4. Core Syllabus 179
2. THE CONCRETENESS PRINCIPLE 180
2.1. Vector Space of Functions 181
2.2. Spatial Symbol Manipulation: The Case of Matrix Arithmetic 181
2.3. Evidence and Instructional Treatment 182
2.4. Some new Observation 184
3. THE NECESSITY PRINCIPLE 185
4. GENERALIZIB1LITY PRINCIPLE 187
5. SUMMARY 188
6. NOTES 189
CHAPTER 6. MODES OF DESCRIPTION AND THE PROBLEM OF
REPRESENTATION IN LINEAR ALGEBRA 191
J. HUM
1. INTRODUCTION 191
2. MODES OF DESCRIPTION IN LINEAR ALGEBRA 192
2.1. The Geometric Level: 1-, 2-, and 3-Dimensional Spaces 193
2.1.1. Coordinate-Free Geometry 193
2.1.2. Coordinate Geometry 194
2.1.3. Vectors As Points 195
X
2.2. The Algebraic Mode: En 195
2.3. The Abstract mode: Vector Space V defined by a set of axioms 195
2.4. Some Students Difficulties with the Geometric Mode 196
2.4.1. Arrows Versus Points 196
2.4.2. Standard Versus Non-Standard Basis 197
2.5. The General Use of a Geometric Language 197
3. THE PROBLEM OF REPRESENTATIONS 199
3.1. Moving Between Modes 200
3.1.1. Geometric - Algebraic Modes 200
3.1.2. Algebraic-Abstract Modes 201
3.2. The Problem of Representation in Terms of Basis 201
3.3. A Persistent Student Difficulty with Representation 202
4. APPLICABILITY OF THE GENERAL THEORY 205
5. CONCLUDING COMMENTS. 205
6. NOTES 206
CHAPTER 7. ON SOME ASPECTS OF STUDENTS THINKING IN
LINEAR ALGEBRA 209
A. Sierpinska
1. INTRODUCTION 209
2. PRACTICAL, AS OPPOSED TO THEORETICAL, THINKING - A
SOURCE OF DIFFICULTY TN LINEAR ALGEBRA STUDENTS 211
2.1. Students Practical Thinking in a Linear Algebra with Cabri Course...212
2.1.1. Transparency of Language 212
2.1.2. Lack of Sensitivity to the Systemic Character
of Scientific Knowledge 217
2.1.3. Thinking of Mathematical Concepts in Terms of their Prototypical
Examples Rather than Definitions 222
2.1.4. Reasoning Based on the Logic of Action, and Generalization from
Visual Perception 229
2.2. General Reflection on the Distinction Between Theoretical and Practical
Thinking 232
3. SYNTHETIC-GEOMETRIC, ANALYTIC-ARITHMETIC AND ANALYTIC-
STRUCTURAL MODES OF THINKING IN LINEAR ALGEBRA 232
3.1. A Characterization 233
3.2. The Tension Between the Synthetic, the Analytic-Arithmetic and the
Structural Modes of Thinking in Students of Linear Algebra 236
3.3. Intermediate Forms of Argumentation Used by Students 238
3.4. General Remarks on the Students Ways of Thinking in Linear Algebra.244
4. CONCLUSIONS. 244
5. NOTES 246
xi
CHAPTER 8. PRESENTATION OF OTHER RESEARCH WORKS 247
M. Artigue, G. Chartier andJ.-L. Dorier
1. COORDINA TION OF SEM1OTIC REPRESENTATION REGISTERS ....247
2. ARTICULATION PROBLEMS BETWEEN CARTESIAN AND
PARAMETRIC VIEWPOINTS IN LINEAR ALGEBRA 252
3. STUDIES WITH FIRST-YEAR UNIVERSITY STUDENTS IN VANNES.256
3.1. The 1991/92 study 256
3.2. The 1992/93 study 257
4. STRUCTURING OF KNOWLEDGE 259
5. USING GEOMETRY TO TEACH AND LEARN LINEAR ALGEBRA 262
6. NOTES 264
REFERENCES - PART II 265
CONCLUSION 273
1. Diagnoses - Epistemological Issues 273
2. Cognitive Flexibility 274
3. Long-Term And Evaluation 275
4. Perspectives 276
NOTES ON CONTRIBUTORS 277
INDEX 281
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| indexdate | 2024-07-09T22:44:39Z |
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| isbn | 0792365399 |
| language | English French |
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| spelling | Enseignement de l'algèbre linéaire en question On the teaching of linear algebra ed. by Jean-Luc Dorier Dordrecht [u.a.] Kluwer c2000 XXII, 288 S. Ill. txt rdacontent n rdamedia nc rdacarrier Mathematics education library 23 Includes bibliographical references and index Algebras, Linear Study and teaching Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Mathematikunterricht (DE-588)4037949-8 s DE-604 Dorier, Jean-Luc edt Mathematics education library 23 (DE-604)BV000020735 23 http://www.loc.gov/catdir/enhancements/fy0813/00061288-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0813/00061288-t.html Table of contents only HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020553613&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | On the teaching of linear algebra Mathematics education library Algebras, Linear Study and teaching Mathematikunterricht (DE-588)4037949-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4037949-8 (DE-588)4035811-2 |
| title | On the teaching of linear algebra |
| title_alt | Enseignement de l'algèbre linéaire en question |
| title_auth | On the teaching of linear algebra |
| title_exact_search | On the teaching of linear algebra |
| title_full | On the teaching of linear algebra ed. by Jean-Luc Dorier |
| title_fullStr | On the teaching of linear algebra ed. by Jean-Luc Dorier |
| title_full_unstemmed | On the teaching of linear algebra ed. by Jean-Luc Dorier |
| title_short | On the teaching of linear algebra |
| title_sort | on the teaching of linear algebra |
| topic | Algebras, Linear Study and teaching Mathematikunterricht (DE-588)4037949-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algebras, Linear Study and teaching Mathematikunterricht Lineare Algebra |
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