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INDEX

A abbreviations 223, 224, 225, 226, 227

numerical 227 schematic 225

abstraction 323, 324 acquisition 191 action-theory 312, 313 additive reasoning

incorrect 143, 153, 163, 213, 223 algebra 321 algorithm 267, 270 analysis 339 application

diagrams 138 ofschemas 147, 156, 166, 177, 185,

196,205 approach

algorithmic 282 empiristic 22 mechanistic 16 realistic 19 rule-oriented 283 structuralistic 15, 18

arguments external 40 internal 40

arithmetic blind 296

attitude algorithmic 267 mathematical 263, 268, 302, 324

B Babylonians 7 Bartjens 8 basis of orientation 315, 324 Bouman and VanZelm 8,9, 10

c

direct 267,271,274 cancelling 313 clinical interview 36 concept 191 concept acquisition 136, 142, 152, 162,

173, 182, 203, 211 extentional 197 intentional 197

conflict cognitive 142, 174, 219

considerations external 215 internal 215

construction course 246

constructions 157, 167, 178, 186, 214, 218, 241, 248, 261, 263

own 302 contextual frameworks 7 Cristianus van Varenbraken 7

D Dejong 258 De Thiende 7 describer 330 diagrams 234 Diels and Nauta 9, 10 distance 39 distributing 51, 52 distribution

French 179, 199, 233, 237, 238, 241, 242, 245

global 233 dividing 328, 330 division

distribution- 340 French 73, 78, 81, 168, 170 ratio- 340

calculation E clever 270, 274, 279, 280, 282, 286 Erlwanger 5, 11

351

352 INDEX

error IN-distractor 136, 170, 175, 178, 188,

191,204,206,246,262,265, 268,277,278,281,282,286, 288,298,299,300,304,316

evaluation

F

external 250 fonnative 249 internal 218 surnmative 250, 253

fonnalization 216 fonning

hypotheses 241 fractions

decimal 323, 327, 341 named 327

framework conceptual 312

Freudenthal 11, 18, 30, 324, 339

G general educational theory 313 generalization 323, 324

I images

mental 136, 140, 147, 156, 166, 177,186,197,206

infonnation processing 312 instability 218, 250 interaction 31, 248 interactive 20 intertwined 21 intertwinement 233,323

J justification 40

K knowledge

declarative 312, 313 procedural 312, 313

Krutetskii 139

L Land of Together 60, 117, 159, 170, 180,

188,246 learning process 19, 20

individual 135 long-tenn 135, 218

Leinhardt 312, 313 Leontev 313 level

classroom 30, 31 concrete 46,48,223,224,234,315 constructions 136, 140, 148 course 30, 31, 246, 247 curriculum 34 first 236 fonnal47 global 30, 32 intennediate 47 lesson 246, 247 local 30, 32 ofschematization 224 productions 136, 140, 148 second 215, 225, 227, 235, 236 symbolic 48, 50, 58, 59, 69, 99,

136, 140, 148, 157, 167, 178, 186, 198,207,241

theoretical 30, 33 third 235, 237, 243

level-theory 246, 248, 250, 312 lines of instruction 21 lines of learning intertwined 21

M magnitude

mediating 54, 327, 335 natural 329, 336 values of 333

material generative 30, 247 monographic 50, 149, 158, 200,

234, 235, 246 mathematization

horizontal 16, 17, 19, 21, 22, 30, 31, 142, 162, 193, 211, 234, 236, 246, 247, 260, 267

progressive 53, 236, 248, 261 vertical 16, 17, 18, 21, 22, 31, 32,

51, 152, 197, 212, 218, 239, 244, 246, 247, 314, 316, 319, 322

measuring 82, 328, 332

INDEX 353

model mathematical 326 mediating 231, 237 situation 237 visual 267, 289, 291, 294, 295, 326

model formation 236 progressive 236, 245, 320, 321

models 234 cognitive process 239 situation 242 visual 316

monograph 32, 33, 59, 99, 148, 168, 180, 188, 200, 207, 209, 241, 243, 245, 248, 263, 316, 320

monotonous 159 notebooks 109 of order 59, 149, 158, 169, 187,

199,208 monographic treatment 8

N IN-distractor 70, 79, 81, 142, 152, 162, 173,

174, 182, 188, 192, 198,203,211, 214,219,222,223,224,238,240, 244,245,248,275,276,283,285, 322

levels of resistance 219 networks

semantic 312 New Math 10, 15, 16 number-line 139, 158, 169, 179, 187,

200, 208, 235, 237

o observation

participatory 39 operating

blind 273 fractions 54, 60, 61

operator 50, 330, 333, 339 division 22 ratio- 339 two-stage 339

orientation algorithmic 283, 285, 304

originator 330

p Papyrus Rhind 7, 8, 32, 326

part of 48, 49, 329 participation 39 percentage 323, 327, 341 phenomenology

didactical 314 problems

formal 136, 140, 147, 156, 166, 177,186,197,206

process abbreviation 224 schematization 226

processes cognitive 324

processing information 313, 315

production 157,167,178,186,214,218, 241,248,263,302

free 31, 157, 167,207,244,261 rules of 312

proportional reasoning 51, 52, 53, 75, 76 incorrect additive 137

prototype 30 pseudonym 59, 60, 74, 85, 150, 158,

170, 180, 187,201,209,241, 242, 244, 316

Q quantities

mediating 54, 317

R ratio table 58, 60, 78, 97, 98, 99, 289,

304 rational analysis 32 reflection

didactical 313 relation

division 22 representations 239 reproduction 18 research

comparative 36 developmental 12, 24, 33, 34, 234,

241,248,312 re-teaching

telescoped 233

S schemas 234

354 INDEX

schematization 212, 278 progressive 51, 52, 136, 139, 143,

153, 163, 175, 176, 183, 193, 204, 223, 225, 232, 236, 238, 244, 247, 248, 249,317

seating arrangement 51, 53, 59, 80, 225 Simon Stevin 7 Smith 312,313 solutions

direct 280 stability 218, 223 standardization 323 standpoint

relative 323, 324 state 330, 333 structuralism

pragmatic 17 system

formal 235

T table-symbol 51, 52, 75, 76, 243, 247 teaching experiment 35 Terwel39 test

average 264 CITO 253,254,264,265,301,310 final 265

theory cultural-historical 313 domain-specific 312

Tract of Ghent 7, 8 Treffers 16, 19, 314

U unification 323, 324 unknown 336 use

V

of models 138, 147, 156, 166, 177, 185, 196, 205

Van Hiele 11, 17,46 Van Oers 313, 314 variable

concept 227 effect 35 process 35

Versluys method 8

W Wiskobas 24 Wiskobas project 10, 11, 12, 24, 29

Mathematical Education Library

Managing Editor: A.I. Bishop, Cambridge, U.K.

1. H. Freudenthal: Didactical Phenomenology o/Mathematical Structures. 1983 ISBN 90-277-1535-1; Pb 90-277-2261-7

2. B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives on Mathe­matics Education. Papers submitted by Members of the Bacomet Group. 1986. ISBN 90-277-1929-1; Pb 90-277-2118-1

3. A. Treffers: Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction - The Wiskobas Project. 1987 ISBN 90-277-2165-3

4. S. Mellin-Olsen: The Politics o/Mathematics Education. 1987 ISBN 90-277-2350-8

5. E. Fischbein: Intuition in Science and Mathematics. An Educational Ap-proach. 1987 ISBN 90-277-2506-3

6. A.J. Bishop: Mathematical Enculturation. A Cultural Perspective on Mathematics Education. 1988

ISBN 90-277-2646-9; Pb (1991) 0-7923-1270-8

7. E. von Glasersfeld (ed.): Radical Constructivism in Mathematics Education. 1991 ISBN 0-7923-1257-0

8. L. Streefland: Fractions in Realistic Mathematics Education. A Paradigm of Developmental Research. 1991 ISBN 0-7923-1282-1

9. H. Freudenthal: Revisiting Mathematics Education. China Lectures. 1991 ISBN 0-7923-1299-6

KLUWER ACADEMIC PUBLISHERS - DORDRECHT I BOSTON I LONDON