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| Авторизация |
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| David O.Tall — Advanced Mathematical Thinking |
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| Предметный указатель |
Oesterle, R. 141
Olson, D.R. 30
Operator, object-valued 83 87 93
Operator, point-wise 83 85 87 93 109
Operator, uniform 83 86 93
Organization of mathematical content 133—136
Organizing agent 187
Orton, A. 8 163 176 177
Ownership of the mathematical concepts 137
Oxford Dictionary 253
Papert, S. 8 9
PARAMETER 90
Parametric equation 87
Parametric function 87
Parkin, T.R. 232
Pascal 242
Paulos, J.A. 148
Peacock, G. 10
Pedantry 61
Peitgen, H. 29
Perlmutter, A. 214
Phillips, E.G. 171
Piaget J. 7—9 63 82 95 96 102 103 110 119 120 132 133 143 206 253
Piaget, notion of reflective abstraction 97—102
Piaget, theory of conservation of variable 145
Pivot (in a structural proof) 221
Poincare, H. 4 13—16 29 42 48 63 170 171 200
Point-wise dependence 85
Point-wise operator 83 85 87 93 109
Poison 151
Poisson, D. 189
Ponte, J. 147
Potential conflicting factors 155
Potentialist 156
Practical tangent 188 190
Pre-operational stage 8
Predicate calculus 106 109 114—116 121 123
Predicate calculus schema 114
Primary intuitions 14
Principle of continuity 10
Principle of selective construction of knowledge 258
Probability theory 149
Problem-solving 18 32 93 132 137 150 165 257
Problem-solving, tactics 151
Procept 254 258
Proceptual divide 255
Proceptual knowledge 255
Process 102
Process of representing 30
Process-concept duality 134 255
Processing load 83
Programming 197 231 241—248
Programming as a generic process 12 144
Programming in BASIC 241
Programming in ISETL 242 244—248
Programming in Mathematica 242
Programming the function concept 143
Programming to teach algebra 241
Programming, choice of language 242
PROLOG 242
proof 126 130 132
Proof and refutation 56
Proof and uncertainty 225
Proof as a contractual agreement 195
Proof as a mathematical activity 216
Proof as a necessary mathematical activity 215
Proof as a stylistic exercise 130
Proof as mechanistic deduction from axioms 234
Proof by computer checking 233
Proof by contradiction 5 130 163 217
Proof by induction 102 113 120 130
Proof debate, organisation 228—229
Proof in linear style 215 221—222
Proof in structural style 222—224
Proof of Jordan's Theorem 146
Proof to convince 20 226
Proof to show 226
Proof, acceptance of 58 162
Proof, analytic formalization 146
Proof, avoidance by using algebraic algorithms 186
Proof, careful reasoning 60
Proof, cognitive aspects 19
Proof, direct methods for equivalence of infinite sets 210
Proof, exposition by the structural method 219—221
Proof, formal 54 57 125 136 217 231
Proof, formal, its origins 55
Proof, formal, its validity 55
Proof, general 217
Proof, generic 19 216 229
Proof, indirect method for non-equivalence 210
Proof, interiorization of steps 104
Proof, mathematical 16 21 54—61
Proof, research on learning 215—230
Proof, schema 120
Proof, standards of rigour 56 129
Proof, structural 220 229
Proof, student understanding 216—219
Proof, the social process 59
Proof, using a generic example 217
Propositional calculus schema 114
Proving 20 41 220
Pseudo-empirical abstraction 97 99
Psychological barriers 152
Psychological characteristics of students 131
Psychology of advanced mathematical thinking 3—21
Psychology of Invention in the Mathematical Field 3
Psychology of Learning Mathematics 16
Psychology, behaviourist 7 142
Psychology, constructivist 7
Ptolemy 236
Purdue University 233
Quantification 115
Quantification, existential 116
Quantification, higher-level 116
Quantification, single-level 115 116
Quantification, three-level 116
Quantification, two-level 115 116
Quantification, universal 116
Quantifier 115 153 197 198
Quotient object 87
Rachlin, S. 144
Re-equilibrium 132
Reconstruction 114 136
Reconstruction of cognitive structure 9 159 164
Reconstruction of function schema 119
Reconstruction of knowledge 236 252
Reconstruction, through reflection on conflict 199
Recursion 286
Reding, A.H. 241
Reflection 25 61 252 257
Reflection on a function process 117
Reflection, meta-mathematical 131 137 138
Reflection, on infinity 125
Reflections 251—258
Reflective abstraction 95—124 134 166 253
Reflective abstraction as construction 99 101
Reflective abstraction in children's thinking 100
Reflective abstraction, its nature 99
Reification 82
Relational understanding 48 49
Representation 30 225
Representation, computer-implemented 39
Representation, concrete 31 38 39
Representation, mathematical 34
Representation, mental 31 32
Representation, switching 32
Representation, symbolic 31
Representation, visual 39
Representational difficulties 151—152
Representing 38 41
| Representing, relationships with abstracting 38
Research in teaching and learning 127—139
Reversal 103 105 106 143
Reversal, of a process 102 118
Review phase of problem-solving 18 19 20
Richard, F. 216
Riemann integral 227 237
Riemann, G.F.B. 4
Rigour 13—14 182—183 197
Riley, M.S. 118
Rival, I. 148
Robert, A. 125 127—139 155 164 165 180 195 257
Roberti, J.V. 210
Robinet, J. 129
Robins, B. 161
Robinson, A. 163 168 172 208
Robitaille, D.F. 241
Rolle's theorem 218
Royal Road to Geometry 236
Rucker, R. 207
Russell, B. 200 208 214
Ruthven, K. 29
Sandleson, R. 141
Sawyer, W.W. 16—17
Schema 102 106 109 143
Schema, organization of 106—110
Schoenfeld, A.H. 18 25 32 37 132 133 137 142—145 147 151 195
School Mathematics Project 171 175
Schwarz, B. 33 41
Schwarzenberger, R.L.E. 10 103 125 127—139 154 164 257
Schwingendorf, K. 242
Science students 73
Scientific debate 136 191—193 195 216 224—225
Scientific debate an example in analysis 224—227
Scientific debate, evaluating its role 229
Scientific debate, generating 225—226
Secondary intuition 14 203 205
See-saw 100
Selden, A. 27 32 146
Selden, J. 27 32 146
Selective nature of creativity 49
Semigroup 119
Sensori-motor stage 8
SEQUENCE 70
Sequence, 'final term' 164
Sequence, formal definition 155
Sequence, limit of 78—79
Sequence, student conceptions 134
Series 161 166
Set theory 98 199 203 205 207 208 212
Severi, F. 14
Sfard, A. 82
Sierpinska, A. 103 156 177 201 203
Simons, F.H. 241
Sinclaire, H. 140 144
Sixth form 131
Skemp, R.R. 3 9 16 48 49 131
Slope of straight lines 90
Small group work 138
Small, D. 235
Smith, J.P. 32 37 142
Smorgasbord 142
Social factors 128
Socio-cognitive conflict 132
Socratic mode, enhanced 187
Solution Sketcher 239
Soul of departed quantities 160
Southwell, B. 25
Specializing 138
Sperry, R.W. 13
Spontaneous conceptions 154—158 164 166
Stage theory 8
Staircase 163
Stationary (limit) 155
Stavy, R. 205 214
Steffet, L. 82
Stein, M. 148
Steiner, M. 216
Stolz, O. 168
Stone — Czech compactification 89
Strauss, S. 214
Structural proof 219—221 229
Structural proof, elevator 222 223
Student conceptions of derivative 175
Student conceptions of derivative, integral, tangent 174—175
Student conceptions of differential 180
Student conceptions of differentiation 180
Student conceptions of infinity 201—205
Student conceptions of integral 175
Student conceptions of integration 180
Student conceptions of sequences 134
Study of historical texts 138
Sub-object 87
subspace 88
Sullivan, K.A. 172
Sweller, J. 141
Sylow theorem 230
Symbol manipulator 231 235—236 258
Symbol manipulator as a tool 236
Symbol pushing 30 61
Symbolic representation 31
Symbolism 61
Symbolism, used for process and concept 253
Symbolism, used to solve problems formally 131
symbols 82—93
Symbols and structure of mathematical objects 91
Symbols, conventions 89
Symbols, elaborated 92
Symbols, nominal role 89 92
Symbols, tacit 92
Synthesis 37
Synthesis of knowledge 15
Synthesizing 34 35
Szwed, T. 178
Tacit notation 88 93
Tacit symbol 92
Tall, D.O. 3—21 33 37 41 68 73 78 103 121 122 126 128 134 136 139 145 147 148 154 155 157 162 165 166 175 178 187—190 193 197 198 201—203 207 208 216 217 219 231—248 251—258
Tall, R. 153
Tangent 73 168
Tangent as a close approximation to the curve 239
Tangent as limit of secants 76 188
Tangent linear functional 170
Tangent, concept image 75—78 174—175
Tangent, generic 76 78
Tangent, practical 188 190
Tangent, student drawings 77
Task-sequencing 140
Teacher, effect of 131
Teaching calculus/analysis in England 174
Teaching calculus/analysis in France 173
Teaching in analysis 186
Teaching in proof debates 228
Teaching of the limit concept 165
Teaching, consequences of complex conceptions 165
Teaching, consequences of concept images 79
Teaching, consequences of creativity 52
Teaching, consequences of practice of proof 60
Teaching, consequences of reflective abstraction 119
Teaching, inadequacy of traditional practices 120
Teaching, integration through scientific debate 191
Tenaud, I. 138
Tends to, meanings of the word 154
Tennessee Technological University 27
Terence Tao 39 253
Theoretical fantasy 140
Thinking Mathematically 20
Thom, R. 141
Thomas, H.L. 147
Thomas, M.O.J. 41 126 148 215—230 241 258
Thompson, P.W. 37 41 82 123
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