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| David O.Tall — Advanced Mathematical Thinking |
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| Предметный указатель |
Defining 20 41
Definition 132 254 See
Definition in technical contexts 69
Definition, cognitive situation 67
Definition, formal 125
Definition, some common assumptions 65
Delens, P. 170
Deligne, P. 220
Derivative 85 107 167 176
Derivative as a first order approximation 195
Derivative as a limit of slopes of secants 188
Derivative as affine approximation 173
Derivative as gradient of locally straight curve 136 175
Derivative as limit of gradient of secants 165
Derivative of the second order 170
Derivative, concept images 175
Derivative, dy/dx as an indivisible symbol 171
Derivative, partial 169
Describing 20
Development of concepts 102—103
Development, biological 100
Development, intellectual 100
Didactic contract 132 137
Didactic engineering 186 195 197
Didactic engineering in teaching differential equations 193—194
Didactic engineering, through scientific debate 191
Dienes, Z.P. 15
Dieudonne, J.A. 48 54
Differences between elementary and advanced maths 20 26 127 133
Differentiable manifold 136
Differential 171
Differential and local approximation 181
Differential and related notions 181
Differential as a component of the tangent vector 239
Differential calculus 160
Differential calculus, based on derivative as a limit 171
Differential equation 119 129 135 238
Differential equation to predict the weather 232
Differential equation with no symbolic solution 239
Differential equation, algebraic solution 173
Differential equation, existence of solution 239 258
Differential equation, higher order 239
Differential equation, qualitative theory of 135 193
Differential equation, simultaneous 239
Differential equation, solved symbolically or visually 236
Differential formula 170
Differential in education 170
Differential in physics as infinitesimal increase 169
Differential in terms of linear tangent map 168 173
Differential in terms of tangent linear approximation 180
Differential of Leibniz 169
Differential, algorithmic calculation 181
Differential, analyst's view 6
Differential, difficulties with symbols and meaning 178
Differential, its survival in analysis 169
Differential, student explanations 181
Differential, student lack of understanding 181 184—186
Differential, visualized pictorially through local straightuess 239
Differentiation 174
Differentiation as an algorithm for formulas 85
Differentiation operator 85
Differentiation, algorithms 174
Difficulties 186
Difficulties in the beginnings of analysis 196
Difficulties with actual infinity 200
Difficulties with cardinal infinity 201
Difficulties with continuity 178
Difficulties with differentiability 178
Difficulties with graphical representations 178
Difficulties with infinity 161
Difficulties with limits of sequences 178
Difficulties with symbols 178
Difficulties with unencapsulated limit concept 165
Difficulties, due to formalization 196
Difficulties, representational 151—152
Discontinuity between elementary and advanced math 125
discovering 40 41
Discovery 231
Discrete mathematics 123
Disequilibrium 132
Doerfler, W. 37
Domino stones in mathematical induction 38
Donaldson, M. 176
Douady, R. 43 133—135 165 225
Double limit 86
Dreyfus, T. 11 23 25 25—41 63 103 116 123 128 131 139 142 145 147 148 217
Drinking coffee 257
Duality theory 109
Dubinsky, E. 12 13 37 41 63 82 95—124 126 131 139 143 144 148 162 166 197 231—248 253 254 258
Duffin, J. 41
Duval, r. 201 203
Dynamical systems 232
Education Reform Act (U.K.) 174
Edwards, E.M. 5
Ehrlich, G. 141
Einstein, A. 31
Eisen, Y. 205
Eisenberg, T. 32 41 116 125 140—152 217 258
Elaborated notation 88 91 93 See elaborated"
Elaborated symbol 91 92
Electronic notebooks 236
Elementary mathematical thinking, differences from advanced mathematical thinking 20 26 127 133
Elevator in structural proof 222 223
Ellerton, N.F. 8
Elterman, F. 106
Empirical abstraction 97 99 121
Encapsulation 13 21 63 82 101 103 105 106 108 112 115 116 136 143 144 253 258
Encapsulation of a function 108
Encapsulation of addition 100
Encapsulation of getting small as an infinitesimal 162
Encapsulation of implication process 113
Encapsulation of the function process 143
Encapsulation, failure to encapsulate 105
Encyclopedic Methodique 169
Endomorphism 119
Enhanced Socratic mode 187
Enseignement Mathematique 170
Entification 92 93
Entry phase of problem-solving 18 20
Environments for learning 133
Epistemological analysis 118
Epistemological obstacles 134 258
Epistemological obstacles and didactical transmission 163—164
Epistemological obstacles in historical development 159—162
Epistemological obstacles in history and education 158
Epistemological obstacles in modem mathematics 162—163
Epistemological obstacles in the limit concept 162 165
Epistemological obstacles, important characteristics 158
Epistemological obstacles, using a computer 166
Epistemology 225
Epistomological problems 141
Ergodic theory 149
Error, arbitrary 176
Error, executive 176
Error, structural 176
Ervynck, G. 23 42—53
Euclid 222 236
Euclid's Elements 159
Euclidean ring 97
Eudoxus of Cnidos 159
Euler, definition of tangent 174
Euler, L. 56 160 161 232
Exhaustion 159—162 168
Expansion of cognitive structure 9
Explaining 220
Explanation 135
Fallibility of mathematics 56
Familiarisation 135
Fehr, H. 141 149
Fermat, definition of tangent 174
| Fermat, P. 59
Ferrara, R.A. 25
Field 84
Fields medal 220
Final term of a sequence 164
Finite and infinite sets learning unit 206 209
First order propositions 111
Fischbein, E. 14 201—203 205 214 216 219
Fitness of creativity 49
Fluid levels 100
Fluxion 169
Focus of attention 83 93
Fodor, J.A. 67 68 80
Formal operations 8
Formal proof 136
Formalism 55 56 61 146
Formalist school 54
Formalization 129
Formalization, gradual initiation 198
FORTRAN 242
Foundations of mathematics 54
Four colour theorem 234
Fourier, J. 35
Frankel, A.A. 208
Frehet, M. 168
Freudenthal, H. 145 149 151
Full cycle of advanced mathematical thinking 42 132 136 252 259
Function 68 70 73 98 106 109 125 167
Function and its graph 147
Function as a black input-output box 141
Function as a conceptual entity 82 85 86 92
Function as a formula 32 74 104 116
Function as a graph 118
Function as a process 12 31 82 108
Function as a rule 74
Function as a set of ordered pairs 118 141
Function as a set of ordered pairs, its pedagogical weakness 141
Function as a table of values 141
Function as an algebraic formula 141
Function as an algebraic term 74
Function as an algorithm 104
Function as an arithmetical manipulation 74
Function as an arrow diagram 141
Function as an equation 74
Function as an object 87 108 118
Function as data 104
Function as process and object 119 143
Function as uniform operator 92
Function as unifying factor in school mathematics 140
Function schema 108 114
Function through interiorization of actions 117
Function, composite as a concept 84
Function, composite as a process 84 92
Function, composite of two functions 104
Function, composite through substitution 117
Function, concept image 74
Function, considered graphically 148
Function, continuous 107 160
Function, continuous linear 108
Function, continuous, properties of 163
Function, determined by numerical data 90
Function, different representations 32
Function, differentiable 188 238
Function, geometric conception 146
Function, historical background 140
Function, inverse as graph 147
Function, its complexity 140
Function, limited to one representation 33
Function, linear 90 107
Function, logical versus algebraic 146
Function, multiple embodiment 141
Function, non-differentiable 188 238
Function, one-to-one 117
Function, onto 117
Function, parametric 87
Function, proposition valued 112—114 116 120
Function, schema 114 117
Function, space 82
Function, visual versus analytic 146
Functional 84
Functional analysis 167 168 170
Functions and learning difficulties 140—152
functor 82
Fundamental theorem of calculus 105 107
Gagne, R.M. 142 144
Gagnean hierarchy 144
Gal-Ezer, J. 29
Galileo, G. 199 200
Galois theory 149
Galois, E. 48
Garcia, R. 97 120
Gauss, C.F. 200
Gautschi, W. 233
Gazit, A. 205
Gazzigna, M.S. 13
General coordinations 97 99 101
General triangle 51
Generalization 11—12 36 37 97 103 105 106 143
Generalization as concept 11
Generalization as process 11
Generalization in mathematical creativity 48
Generalization of a schema 101
Generalization of function schema 113
Generalization, constructive 97
Generalization, disjunctive 12
Generalization, expansive 12 48
Generalization, extensional 102
Generalization, generic 12
Generalization, reconstructive 12 48
Generalizing 34 35 138
Generic abstraction 13
Generic example 122 217
Generic extension principle 10
Generic limit 10 162 164
Generic organizer 187—190
Generic organizer for differential equations 188
Generic organizer for gradient (derived function) 187
Generic organizer for integration 188
Generic proof 19 216 229
Generic tangent 76 78
Genetic decomposition 96 102 106 110 123 143 144 166
Genetic decomposition of dual vector space 108
Genetic decomposition of function 110 116—120
Genetic decomposition of predicate calculus 114—116
Genetic decomposition of three schema 109—119
Geometry 138
Gleick, J. 232
Goedel's incompleteness theorem 98 162
Goedel, K. 5 98 162
Goldbach's conjecture 233
Goldin, G. 90
Gong, C. 106
Graphic Calculus 187 238
Graphs, interpretation of 147
Gray, E.M. 255
Greek mathematics 159
Greeno, G.J. 82 83 85 89 118
Grenier, D. 216
Grenoble 19 136 191 192 195 216 224 225
Grogono, P. 242
Group 87 106
Group as formal structure 129
Hadamard, J. 3 14—15 29 31 42 146
Hadar-Moscovitz, N. 147 See
Hahn, H. 200 208
Haifa University, Israel 216
Haken, W. 16 233
Halmos, P.R. 137
Hamilton, W. 11
Hammersley, J.M. 141
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