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| David O.Tall — Advanced Mathematical Thinking |
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| Предметный указатель |
Abel, N.H. 35
abstracting 36—38 41
Abstracting, relationships with representing 38—39
Abstraction 11—12 36 37 97 98 132 139 144 148—151 217
Abstraction as concept 11
Abstraction as process 11
Abstraction of properties 129
Abstraction, empirical 97 99 121
Abstraction, generic 12 13
Abstraction, processes in 34
Abstraction, pseudo-empirical 97 99
Abstraction, reflective 13 21 95—124 134 253
Abstraction, using a generic organizer 187
Accommodation 9 103
Acquisition of knowledge 132 133
Acquisition of specific concepts 134
Adler, C. 140
Advanced mathematical thinking as a process 26
Advanced mathematical thinking processes 25—41
Advanced mathematical thinking, differences from elementary mathematical thinking 20 26 127 133
Advanced mathematical thinking, full cycle of 42 132 136 252 259
Advanced mathematical thinking, Psychology of 3—22
Advanced mathematical thinking, taught as a finished theory 215
Affine approximation 173
Algebra 220
Algebra, learning difficulties 144
Algebraic permanence, principle of 10
Algorithm 5 43 61 104 137 163 193
Algorithm as a replacement for proof 186
Algorithm to solve a problem 125 131
Algorithm, algebraic differentiation 180
Algorithm, premature algebraic use 186
Algorithm, procedures in analysis 186
Algorithmisation 197
Alibert, D. 19 41 126 136 180 191 215—230 258
Analysis 167—198
Analysis of knowledge 15
Analysis of several variables 167
Analysis, arithmetization of 168
Analysis, complex 167
Analysis, constructive 5
Analysis, epistemological 118
Analysis, functional 167 168 170
Analysis, mathematical 125 153
Analysis, non-standard 6 172 187 196 197 202
Analysis, real 131
Analysis, Weierstrassian 162
Analytic thinking 147
Anton, H. 92
Anxiety 148—152
APL 242
Appel, K. 16 233
Appollonins of Pergs 174
Approximation in reasoning 182—183
Arbitrarily small 162
Arcavi, A. 32 37 142 145
Archimedes' axiom 256
Aristotle 200
Arithmetization of analysis 168
Arithmetization of mathematics 146
Artigue, M. 41 125 135 167—198 258
Assessment 130
Assimilation 9
Assimilation, generalizing 102
Athens, Georgia 144
Atiyah, M.F. 231
Attack phase of problem-solving 18—20
Attainment, variation in 131
Ausubel, D.P. 8
Authier, H. 180
Axiom of Choice 163
Axiomatic method 54
Ayers, T. 82 83 103 104 117 118 242
Bachelard, G. 134 154 158
Balacheff, N. 215 225
Ballistics 168
Barrier to advanced mathematical thinking 129
Barrier(limit) 155
BASIC 241 242
Bautier, E. 131
Beberman, M. 140
Begle, E. 140
Behaviourist psychology 7
Beke, E. 170 171
Ben-Chaim, D. 148
Ben-Gurion University 149
Berkeley, G. 169
Beth, E.W. 82 95 97 99
Bieberbach's conjecture 233
Biggs J. 8
Biological development 100
Birkhoff, G.D. 151
Bishop, A. 225
Bishop, E. 5 172
Blackett, N. 148
Blancmange function 188
Boas, R. 152
Bolzano, B. 200 208
Borasi, R. 201
Boschet, F. 131 132
Bourbaki 16 54 98 140 141 149
Boyer, C.B. 160 168
Bransford, J.D. 25
Breuer, S. 29
Brousseau, G. 133 134 159 224
Brown, A. 25
Brown, A.L. 141
Bruckheimer, M. 33 41
Buck, R.C. 141
Built-in knowledge generator 255
Bulletin of the American Mathematical Society 172
Cajori, F. 91 161
Calculus 8 16 27 105 107 142 147 148 153 160 161 163 165 220
Calculus, based on limits 169
Calculus, history of 168—198
Calculus, infinitesimal 168
Calculus, introduction into secondary education 170
Calculus, its metaphysical difficulties 161
Calculus, rigorously based on infinitesimals 172
Calculus, supplemented by programming in BASIC 241
Calculus, using graphic and symbol manipulating software 237
Calgary, Canada 144
Cambridge Conference 140
Campbell, R. 30
Campione, J.C. 25
Cantor — Bernstein theorem 230
Cantor, G. 4—6 200 201 208 214
Cantorian set theory 199 205 207 208 212
Cantorian set theory, constructing intuitive background 203
Cardinality of sets 105
Carter, H.C. 141
Case, R. 141
Category theory 98
Cauchy sequence 168
Cauchy, A.L. 10 35 56 129 160—162 168 169
Celestial mechanics 168
Cesaro's lemma 129
Chaos theory 232
Char, B.W. 235
checking 40 41
Cheshire, F.D. 241
Chunking complex ideas 88 252
Cipra, B. 148
Clark, C. 150
Clement, J. 122 205 214
Clements, M.A. 39 146 253
Cobb, P. 82
Codidactic situation 226
Cognitive characteristics of students 131
Cognitive conflict 134 206 236
| Cognitive development 3 7—8
Cognitive mechanisms in learning 132
Cognitive obstacles 9—11 21 158—159 164 165 199 256
Cognitive re-construction 9 114 136 159 164
Cognitive root 136
Cognitive theory 63
Cohen, L. 141
Collis, K. 8
Commission Internationale, pour l'Enseignement de Mathematiques 170
Commission Internationale, pour l'Enseignement des Mathematiques 171
Commutativity of addition 100
Comparison of infinite quantities 203
Completeness axiom 6 196
Complex analysis 167
Complex number 5
complexity 151
Complexity of analysis 163
Complexity of function concept 140
Complexity, encountered by students 131 139
Comprehension of complex concepts 83
Comprehension of object-valued operators 86—88
Comprehension of point-wise operators 84
Compression of ideas 35
Computer 126
Computer algebra system 235
Computer and the need for finite algorithms 163
Computer as an experimental tool 29 166 189
Computer as environment for exploration of ideas 238—240
Computer for conceptual development 237
Computer for implementing processes 123
Computer for linking representations 33
Computer for programming 197 241—248
Computer for providing concrete representations 104 187
Computer for visualizing differential equations 193 239
Computer for visualizing graphic representations 193 232
Computer generated experiments 232
Computer in advanced mathematical thinking 231—248
Computer in mathematics education 234—235
Computer in mathematics research 231—234
Computer to construct solution of a differential equation 239
Computer to perform algorithms 236
Computer, aversion displayed by teaching staff 241
Computer, didactic advantages in analysis 197
Computer, used in mathematical proof 233
Concept acquisition 6
Concept definition 6—7 21 70—73 103 122 125 130 145 196—198
Concept definition in teaching and learning 65—80
Concept definition of a limit 156
Concept definition, operational deficiency 197
Concept definition, theory and practice 69
Concept formation 69
Concept formation, long term processes 71
Concept frame 68
Concept image 6—7 14 17 21 68—73 76 78 83 103 122 123 125 127 134 145 166 196—198
Concept image in geometry 134
Concept image of a function 74
Concept image of a function as a graph 146
Concept image of a limit 155 156
Concept image of a limit of a sequence 78 164
Concept image of a limit of a series 166
Concept image of a tangent 75—78 174—175
Concept image of continuity 156—158
Concept image of derivative 175 188
Concept image of infinity 156 199
Concept image of rigorous proof 197
Concept image, changing 70
Concept image, construction compatible with formal mathematics 187
Concept image, evoked 68 73 83 144
Concept image, evoked of a limit 155
Concept image, theory and practice 69
Concept image, three illustrations 73
Concept image, weakness of geometric image of differential 184
Conceptual entities 21 82 82—93 134 143 150 255
Conceptual entities and symbolism 88
Conceptual entities as aids to focus 88
Conceptual entities, construction of 83
Conceptual entities, three roles 83
Conceptual obstacles 133 153 251
Conceptualisation 197
Concrete operations 8
Concrete representations 38
Condensing power of creativity 50
Conflict 129 155
Conflict between actual infinity and finite experiences 201 205
Conflict between concept image and definition 125 158
Conflict between different student conceptions 175
Conflict between different theoretical paradigms 203
Conflict between differential and derivative 169—171
Conflict between infinity in limits and set theory 125 203
Conflict between limit as a process and its definition 156
Conflict between mathematics and cognition 65
Conflict between previous experience and formal theory 199 205
Conflict between secondary intuitions and primitive convictions 203
Conflict between spontaneous conceptions and definitions 158 196
Conflict between two conceptions of a differential 185
Conflict in comprehending cardinal infinity 206
Conflict in learning continuity 134
Conflict in learning limits 134 164
Conflict with infinity 199 204
Conflict, cognitive 134 206 236
Conflict, concerning limits and infinity 156
Conflict, lack of awareness of 180
Confusion in first year university 129
Conjecture 132 136 191 224—225 227 229 252 257 258
Constructivism 224
Constructivist psychology 7
Continuity 156—158 167
Continuity, conceptual difficulties 178
Continuous function, definition of Cauchy 160
Convergence of sequences and series 129
Convergence of series 159
Convergence, via epsilon-delta methods 129
Convincing 20 130
Coordination 103 104 106 114 143
Coordination of actions 97 99 101
Coordination of function schema 113
Coordination of processes 101 107 113 115 119
Coordination of quantifications 116
Coordination of schemas 104
Cornu, B. 9 17 41 103 122 125 134 153—166 177 255 258
Coset 87
Counter-example 226
Counter-example, generated by computer 232
Cours d'analyse (Cauchy) 160
Cramer, G., definition of tangent 174
Creative activities, absent in students 132
Creativity 21 42—53 257
Creativity, a tentative definition 46
Creativity, characteristics of 49
Creativity, fallibility 52
Creativity, ingredients 47
Creativity, motive power 47
Creativity, results of 50
Creativity, stages of development 42
Curriculum design 17 165
Cybernetic environment 236
D'Alembert, definition of tangent 174
D'Alembert, J.L. 160—162 169
D'Hallum, C. 189
Dalen, D. See "Van Dalen D."
Dauben, J. 207
Davis, G. 82
Davis, P.J. 44 56 57 59 146 148
Davis, R.B. 27 68 73 78 94 164
de Branges, L. 233
Debating forum 56
Decapsulation 119
Dedekind cut 168
Dedekind, J.W.R. 200
Deep-end principle 15
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