www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com 7892 tp.new.indd 2/24/11 9:11 AM SERIES ON MATHEMATICS EDUCATION Series Editors: Mogens Niss (Roskilde University, Denmark) Lee Peng Yee (Nanyang Technological University, Singapore) Jeremy Kilpatrick (University of Georgia, USA) Published Vol How Chinese Learn Mathematics Perspectives from Insiders Edited by: L Fan, N.-Y Wong, J Cai and S Li Vol Mathematics Education The Singapore Journey Edited by: K Y Wong, P Y Lee, B Kaur, P Y Foong and S F Ng Vol Russain Mathematics Education History and World Significance Edited by: A Karp and B R Vogeli Vol Russian Mathematics Education Programs and Practices Edited by A Karp and B R Vogeli www.EngineeringBooksPDF.com ZhangJi - Russian Math Education - Programs.pmd 4/5/2011, 9:51 AM Series on Mathematics Education Vol RUSSIAN M ATHEM ATICS EDUCATION Programs and Practices Edited by Alexander Karp Bruce R Vogeli Columbia University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI www.EngineeringBooksPDF.com 7892 tp.new.indd 2/24/11 9:11 AM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Series on Mathematics Education — Vol RUSSIAN MATHEMATICS EDUCATION Programs and Practices Copyright © 2011 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN-13 978-981-4322-70-6 ISBN-10 981-4322-70-9 Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore www.EngineeringBooksPDF.com ZhangJi - Russian Math Education - Programs.pmd 4/5/2011, 9:51 AM March 9, 2011 15:5 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm Contents Introduction Chapter vii On the Mathematics Lesson Alexander Karp and Leonid Zvavich Chapter The History and the Present State of Elementary Mathematical Education in Russia 37 Olga Ivashova Chapter On the Teaching of Geometry in Russia 81 Alexander Karp and Alexey Werner Chapter On Algebra Education in Russian Schools 129 Liudmila Kuznetsova, Elena Sedova, Svetlana Suvorova and Saule Troitskaya Chapter Elements of Analysis in Russian Schools 191 Mikhael Jackubson Chapter Combinatorics, Probability, and Statistics in the Russian School Curriculum 231 Evgeny Bunimovich Chapter Schools with an Advanced Course in Mathematics and Schools with an Advanced Course in the Humanities 265 Alexander Karp v www.EngineeringBooksPDF.com March 9, 2011 15:5 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm vi Russian Mathematics Education: Programs and Practices Chapter Assessment in Mathematics in Russian Schools 319 Alexander Karp and Leonid Zvavich Chapter Extracurricular Work in Mathematics 375 Albina Marushina and Maksim Pratusevich Chapter 10 On Mathematics Education Research in Russia 411 Alexander Karp and Roza Leikin Notes on Contributors 487 Name Index 493 Subject Index 501 www.EngineeringBooksPDF.com March 9, 2011 15:5 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm Introduction This volume is a continuation of the previously published work Russian Mathematics Education: History and World Significance As its title indicates, its primary focus is on Russian programs and practices in school mathematics education Thus, it deals mainly with the contemporary situation, although this does not rule out a historical perspective, without which it is often impossible to understand what is happening today Practices that are widespread and established at the time of the book’s publication may change in the near future More profound characteristics, positions, and traditions, however, not change quickly: the aim of this volume is to help readers to become acquainted with them and to understand them These traditions, however, may be understood in different ways More precisely, it may be said that a genuine understanding of what has happened and what is happening in Russian mathematics education requires a recognition of the fact that Russian mathematics education includes different traditions and different perspectives on these traditions The editors of these two volumes have strived to represent this variety of perspectives Thus, invited contributors include well-known figures in Russian education, the authors of widely used and sometimes competing textbooks, as well as mathematics educators who are currently working outside of Russia The chapters in this volume are devoted to different aspects of mathematics education in Russia and to different processes taking place in it First chapter by Alexander Karp and Leonid Zvavich discusses mathematics lessons and the traditional approaches to structuring mathematics lessons in Russia This chapter also contains basic information about the Russian system of mathematics education that may be useful to the readers vii www.EngineeringBooksPDF.com March 9, 2011 15:5 viii 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm Russian Mathematics Education: Programs and Practices Mathematical subjects and courses taught in Russian schools are addressed in special chapters in this volume Olga Ivashova analyzes the elementary school mathematics program in chapter two The next chapter, by Alexander Karp and Alexey Werner, is devoted to the course in geometry — that is distributed over five years in Russia (USSR), in contrast to many other countries Liudmila Kuznetsova, Elena Sedova, Svetlana Suvorova, Saule Troitskaya discuss the teaching of algebra; Mikhael Jackubson describes instruction in elementary calculus (which is a required course for all students in the higher grades); and Evgeny Bunimovich addresses the teaching of topics that are new to Russian schools — combinatorics, probability, and statistics Subsequent chapters are devoted to the structures and systems in Russian mathematics education, important for students of different ages and for the teaching of different mathematical subjects Alexander Karp traces the history, practices, and distinctive features of so-called schools with an advanced course of study in mathematics and schools specializing in the humanities, which are of relatively recent provenance The next chapter, written by Alexander Karp and Leonid Zvavich, is devoted to mathematics assessment in Russian schools and the chapter that follows it, written by Albina Marushina and Maksim Pratusevich, addresses extracurricular work in mathematics Finally, the last chapter, written by Alexander Karp and Roza Leikin, differs somewhat from the preceding ones Its subject is not the school itself, but academic studies devoted to mathematics education The authors characterize the directions, goals, and styles of academic studies in mathematics education in Russia over the last twenty years (mainly by analyzing dissertational studies) As in the first volume, several chapters were originally written in Russian and subsequently translated into English The editors wish to thank Ilya Bernstein and Sergey Levchin for help in preparing the manuscript for publication The editors also express their gratitude to Heather Gould and Gabriella Oldham for their assistance in proofreading manuscripts www.EngineeringBooksPDF.com March 9, 2011 15:0 9in x 6in Russian Mathematics Education: Programs and Practices b1073-ch01 On the Mathematics Lesson Alexander Karp Teachers College, Columbia University New York, USA Leonid Zvavich School #1567, Moscow, Russia Introduction The basic form of mathematics instruction in Russia is the classroom lesson Of course, other forms exist, such as individual instruction, which is used with poorly performing students after classes or with seriously ill children at home Naturally, teaching also takes place outside of class — through extracurricular activities, homework, and so on — so it cannot be equated with that which goes on in class Nonetheless, it would be no mistake to repeat that the basic form of mathematics instruction is the classroom lesson Not for nothing were those who in Soviet times were most concerned with teaching students, and not with what was conceived of as communist character-building work, contemptuously labeled “lesson providers.” Every day in the upper grades, six or seven required classes are followed by optional activities outside of the standard schedule: elective www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-listofcont Notes on Contributors 491 at the Russian Academy of Education Troitskaya graduated from the mechanics-mathematics department of Moscow State University and completed this department’s graduate school as well, receiving a Ph.D in the physical-mathematical sciences For a number of years, Troitskaya was an associate professor and taught various courses in higher mathematics at the mathematics department of the Moscow State Institute of Steel and Alloys (technological university) At present, she is conducting research along two fronts: one pedagogical and the other strictly mathematical Troitskaya is the author of over 30 publications, many of which have appeared in leading Russian mathematics journals Bruce R Vogeli is the Clifford Brewster Upton Professor and Director of the Program in Mathematics at Teachers College, Columbia University Before joining the Columbia University faculty in 1965, Professor Vogeli served as a Visiting Professor at the Lenin Pedagogical Institute in Moscow He also lectured at the Herzen Pedagogical Institute in Leningrad and worked as a “curriculum specialist” in the Mathematics Section of the Academy of Pedagogical Science in Moscow Professor Vogeli is the author of Soviet Schools for the Mathematically Talented and Special Mathematics Schools — An International Panorama as well as more than 200 books for students and teachers Alexey Werner belongs to A D Aleksandrov’s geometrical school He is a professor at the Geometry Department of Herzen Pedagogical University in St Petersburg Werner received a candidate’s degree (Ph.D.) in the physical-mathematical sciences from Leningrad University and a doctoral degree (Dr Hab.) in the physical-mathematical sciences from Herzen University in 1969 He has written over 150 works on modern geometry (on the theory of convex and saddle surfaces) and on the problems of geometry education; in addition, he has written (together with coauthors, including A D Aleksandrov, A P Karp, V I Ryzhik, and others) several series of school textbooks in mathematics for schools of general education, as well as for schools with an advanced course of study in mathematics and for schools with a humanities profile Werner has sponsored 30 Ph.D dissertations; two of his students have become doctors (Dr Hab.) of sciences www.EngineeringBooksPDF.com March 9, 2011 15:6 492 9in x 6in Russian Mathematics Education: Programs and Practices b1073-listofcont Russian Mathematics Education: Programs and Practices Leonid Zvavich — teacher and head of the mathematics department at school no 1567 in Moscow Zvavich graduated from Moscow Pedagogical University He has been teaching classes with an advanced course of study in mathematics for over 40 years, He has participated in the creation of a program for mathematical schools For many years, Zvavich wrote problems for State written examinations in mathematics for grades and 11; he was also a member of the Ministry of Education’s council of experts Zvavich was elected vice president of the Association of Mathematics Teachers of the USSR, and subsequently of Russia He was a member of the editorial board of the journal Matematika v shkole and is a many-time winner of the Soros Prize and the Moscow Government Prize Zvavich holds the title of Honored Teacher of Russia and is the author of over 100 publications, including over 30 books, many of which have received recommendations from the Ministry of Education and are widely used in mathematical schools www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Name Index Abel, N.H., 154 Abramov, A.M., 92, 99, 103, 104, 125, 188, 219, 227–229, 368, 370, 471 Abramov, A.V., 455 Afanasiev, V.V., 457, 458, 471 Ahmed, A., 316 Alekseev, V.B., 283, 314 Alexandrov, A.D., 82, 86, 88, 89, 104, 105, 108–112, 116, 117, 120, 122–126, 128, 303, 314, 491 Alexandrova, E.I., 63, 64, 66–68, 71, 73, 75, 77, 444, 471 Alimov, Sh.A., 200–204, 206, 208–213, 218, 219, 227 Altynov, P.I., 333, 371 Ammosova, N.V., 458, 471 Andronov, I.K., 42, 77, 78, 96 Arginskaya, I.I., 63, 67, 74, 77, 78 Ariutkina, S.V., 423, 485 Arnold, I.V., 415 Arsentieva (Kochetova), I.V., 467, 471 Arutiunyan, E.B., 353, 371 Arzhenikov, K.P., 46 Ashkinuze, V.G., 316, 318 Atanasyan, L.S., 21, 35, 104, 107, 108, 115, 126, 218, 227, 297, 314 Avdeeva, N.N., 238, 259 Avdeeva, T.K., 426, 427, 471 Averchenko, A.T., 323, 371 Averyanov, D.I., 318, 374 Balk, M.B., 376, 407 Bantova, M.A., 55, 56, 59, 60, 78, 79 Barbin, E., 108, 126 Barsukov, A.N., 207, 227 Bashmakov, M.I., 62–67, 74, 78, 200, 221, 222, 227, 265, 298, 301, 308, 310, 311, 314 Bayes, T., 233 Bekker, B.M., 314 Beklemishev, D.V., 448, 472 Bellustin, V.K., 46 Beloshistaya, A.V., 443, 472 Beltiukova, G.V., 59, 60, 67, 78, 79 Benenson, E.P., 77 Berdiugina, O.N., 468, 472 Bereday, G., 267, 314 Berlov, S., 384, 406, 407 Berman, A., 408 Bernoulli, J., 233, 237 Bernstein, I., viii Bertrand, J.L.F., 129, 130 Bevz, G.P., 105 Bevz, V.G., 105 Bézout, E., 135 Blonsky, P., 267, 315 Bloom, B., 321, 371, 418, 472 Bobynin, V.V., 38, 40 493 www.EngineeringBooksPDF.com March 9, 2011 15:6 494 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Russian Mathematics Education: Programs and Practices Bolotov, V., 368, 371 Boltyansky, V.G., 96, 126, 322, 371, 376, 398, 408, 410, 451 Borodkina, V.V., 254, 257, 264 Bourbaki, N., 426 Breitigam, E.K., 446, 472 Brezhnev, L.I., 274, 278, 325 Brianchon, C.J., 290 Brickman, W., 267, 314 Buffon, G.-L.L., 233 Bulychev, V.A., 240, 246, 247, 252, 259, 260 Bunimovich, E.A., v, viii, 188, 228, 231, 246, 247, 250, 252, 259, 260, 372, 487 Busev, V.M., 428, 472 Busse, F.I., 196 Butuzov, V.F., 104, 308–310, 314, 315 Bychkova, L.O., 239, 260 Cauchy, A.-L., 213, 222 Ceva, G., 297 Chebyshev, P.L., 8, 233 Chekanov, Yu V., 292, 315 Chekin, A.L., 66, 68, 69, 76, 78 Chentsov, N.N., 409 Cherkasov, R.S., 127 Chinkina, M.V., 8, 36, 357, 374 Chubarikov, V.N., 281, 315 Chudovsky, A.N., 278, 315, 326, 332, 364, 371 Clinkenbeart, P.R., 281, 316 Dadayan, A.A., 355, 371 Dalinger, V.A., 434, 466, 472, 484 Davidovich, B.M., 282, 292, 315 Davydov, V.V., 58–60, 63, 64, 66–68, 70, 71, 78, 411, 440, 444, 451 de Moivre, A., 135 Delone (Delaunay), B.N., 376 Demidova, T.E., 64, 66, 67, 71, 76, 78 Depman, I.Ya., 393, 408 Descartes, R., 8, 124, 394 Dobrova, O.N., 206, 227 Dograshvili, A.Ya., 236, 260 Donoghue, E.F., 275, 315 Dorofeev, G.V., 73, 75, 78, 113, 121, 126, 140, 145–149, 151, 153, 154, 156, 159–165, 175–181, 185, 187, 188, 200, 203, 204, 206, 228, 241, 245, 248, 251, 252, 260, 261, 321, 330, 366, 371 Dorofeev, S.N., 458, 473 Doroshevich, V.M., 350, 371 Drobysheva, I.V., 460, 473 Dubrovsky, V., 270, 274, 315 Dudnitsyn, Yu.P., 188, 228, 229 Dyadchenko, G.G., 245, 261 Dynkin, E.B., 293, 315, 376, 408 Dzhurinsky, A.N., 267, 315 Efimov, V.F., 441, 473 Egorchenko, I.V., 422, 473 Egorov, F.I., 46, 47, 78 Elkonin, D., 440, 444, 451 Emenov, V.L., 52, 80 Engels, F., 267, 317 Episheva, O.B., 437, 473 Erganzhieva, L.N., 113, 127 Ermak, E.A., 446, 473 Ern, F.A., 48, 49, 78 Escher, M.C., 114 Euclid, 81, 86, 107, 108, 118, 122, 124 Euler, L., 8, 40, 42, 115, 195, 297 Evstafieva, L., 310, 312, 316, 390, 408 Evtushevsky, V.A., 41, 43, 44, 78 Ewald, G.F., 43 Ezersky, S.N., 286 Falke, L.Ya., 379, 380, 408 Fedorova, N.E., 227, 229, 246, 264, 315 Fedorovich, L.V., 12, 35 Fedoseev, V.N., 245, 261 Fehr, H.F., 81, 126 Fermat, P., 8, 224 Fetisov, A.I, 98 Fikhtengolts, G.M., 213, 228, 294, 315 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Name Index Filichev, S.V., 330, 371 Firsov, V.V., 238, 240, 261, 346, 371 Fomenko, A., 114 Fomin, D., 284, 315, 377, 390, 391, 397, 401, 408 Freudenthal, H., 440, 451 Friedman, L.M., 422, 473 Frobenius, F.G., 454 Frolov, P.S., 232, 233 Fursenko, A.A., 270, 315 Fursenko, Andrey, 192 Fuss, N., 42, 195 Gaidar, Ye.T, 272, 315 Gaisinskaya, I.M., 236, 261 Galanin, D.D., 42, 43, 78 Galitsky, M.L., 296, 298, 303, 315 Galois, E., 283 Galperin, P.J., 438 Gamidov, S.S., 461, 462, 473 Ganeev, Kh.Zh., 419, 420, 474 Gauss, C.F., Gelfand, I.M., 395 Gelfman, E.G., 433, 474 Genkin, S., 315, 408 Gerver, M.L., 292, 293, 316 Glazkov, Yu.A., 89, 126, 371 Glebova, L., 325, 371 Gnedenko, B.V., 235, 261, 273, 284 Gol’khovoy, V.M., 314 Goldenberg, A.I., 41, 44, 45, 78 Goldman, A.M., 296, 315 Golikov, A.I., 443, 474 Golovin, A.N., 302, 317 Golubev, V.I., 392, 409 Goncharov, V.L., 129, 188 González, G., 85, 126 Gorbachev, M.S., 240, 266, 278, 304, 429 Gorbov, S.F., 78 Gordin, R.K., 292, 316 Gould, H., viii Gravemeijer, K.P.E., 307, 316 Grigorenko, E.L., 281, 316 Grube, A., 41, 43, 44 Grudenov, Ya.I., 423, 474 495 Grushevsky, S.P., 433, 474 Gugnin, G., 267, 268, 316 Guriev, P.S., 41, 42, 78 Guriev, S.E., 42 Gurvits, Yu.O., 330, 371 Gusak, A.A., 449, 474 Gusev, V.A., 388, 408, 429, 451, 474 Gutenmacher, V.L., 409 Guter, R.S., 318 Gutsanovich, S.A., 429, 430, 474 Gzhesyak, Ya., 442, 475 Herbst, P., 85, 126 Hilbert, D., 8, 95, 118 Horner, W.G., 135 Huygens, C., 233 Igoshin, V.I., 464, 465, 475 Ilyin, V., 286 Imranov, B.G.O., 436, 475 Ionin, Yu., 295, 314, 316 Isaeva, R.I., 371 Istomina (Kastrovskaya), N.B., 60, 62–69, 71, 73, 75, 78, 443, 444, 475 Itenberg, I., 315, 408 Itina, L.S., 77 Ivanel’, A.V., 371 Ivanov, O.A., 395, 408, 459, 475 Ivanov, S., 407 Ivanovskaya, E.I., 78 Ivanova, T.A., 417, 418, 475 Ivashev-Musatov, O.S., 302, 318 Ivashova, O.A., v, viii, 37, 62–67, 69, 73–76, 78, 487 Ivlev, B.M., 228, 229 Jackubson, M.Ya., v, viii, 191, 488 Kabekhova, L.M., 236, 261 Kadomtsev, S.B., 104, 314 Kagan, V.F., 49 Kalinin, S.I., 463 Kalinina, M.I., 65, 78 Kalmykova, Z.I., 24, 35 Kapustina, T.V., 465, 476 www.EngineeringBooksPDF.com March 9, 2011 15:6 496 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Russian Mathematics Education: Programs and Practices Karp, A.P., v–viii, 1, 5, 20, 28, 35, 36, 44, 81, 96, 113, 122, 125, 126, 227, 238, 261, 262, 265, 267, 275, 281, 284–287, 290, 291, 293, 294, 296, 303, 307–312, 314–316, 318, 319, 323–325, 332, 333, 342, 357, 360, 364, 366, 372, 383, 390, 405, 406, 408, 411, 424, 427, 476, 488, 491 Karsavin, L., 286, 318 Kavun, I.N., 52, 54, 78, 79 Khalamaizer, A.V., 359, 372 Khanish, Ya., 441, 476 Khazankin, R.G., 359, 360, 372 Khinchin, A.Ya., 235, 261 Khodot, T.G., 116, 128 Khrushchev, N.S., 270–272, 274, 278, 280, 366 Kikoin, I.K., 270 Kilpatrick, J., 411, 470, 476 Kirik of Novgorod, 38 Kirillov, A.A., 295, 316 Kirshner, L., 267, 268, 316 Kiselev, A.P., 95–102, 106–108, 115, 119, 122, 123, 126, 197, 427 Klakla, M., 430, 476 Klein, F., 124, 290 Klopsky, V.M., 103, 126 Koichu, B., 408 Kokhas’, K., 407 Kokhuzheva, R.B., 467, 476 Kolmogorov, A.N., 8, 88, 89, 95, 98–106, 119, 124, 127, 139, 174–177, 184, 188, 198–200, 213–215, 219, 220, 225, 228, 229, 232, 235–237, 262, 270, 273–275, 278, 281–285, 290, 317, 376, 490 Kolyagin, Yu.M., 37–41, 59, 60, 79, 208, 219, 227, 229, 315, 422, 425, 426, 476 Kondakov, A.M., 117 Kondratieva, G.V., 427, 428, 476 Konkol, H., 439, 477 Konstantinov, N.N., 291–293, 316 Kormishina, S.N., 78 Kornilov, V.S., 449, 477 Kostitsyn, V.N., 464, 477 Kostrikina, N.P., 388, 408 Koval’dzhi, A., 281, 317 Kovalevskaya, S.V., Kozlov, V.V., 117 Kozlova, S.A., 78 Kozlova, V.A., 442, 477 Kozlovska, A., 436, 477 Kraevich, K.D., 231, 262 Krasnianskaya, K., 260 Krupich, V.I., 422, 423, 477 Krutetskii, V.A., 411, 440 Krylov, I., 11 Kuchugurova, N.D., 453, 477 Kulagina, I.I., 365, 374 Kurganov, N.G., 40 Kuryndina, K.N., 239, 262 Kushnerenko, A.G., 292, 293, 316 Kuz’minov, Ya., 368, 372 Kuznetsova, E.P., 371 Kuznetsova, G.M., 283, 317, 371 Kuznetsova, L.V., v, viii, 129, 166, 171, 175, 176, 179, 181, 187, 188, 200, 228, 347, 365, 371, 372, 374, 488 Kuznetsova, T.I., 436, 478 Lagrange, J.-L., 129, 130, 216, 217, 220, 221 Lamszus, W., 11 Lankov, A.V., 42–44, 51, 52, 79 Larichev, P.A., 266, 317 Latyshev, V.A., 41, 44 Lavrentiev, M.A., 271, 273 Lebedeva, S.V., 467, 478 Legendre, A., 86 Leibniz, G.W., 8, 194, 217, 222 Leikin, R., vi, viii, 408, 411, 489 Leman, A.A., 408 Lenin, V.I., 267, 271, 317, 426 Leontiev, A.N., 438, 440, 451 Lermantov, V.V., 49 Levchin, S., viii Lester, F., 373 Levitas, G.G., 371, 438, 439, 478 Ligachev, Ye, K., 278, 317 Lipatnikova, I.G., 421, 478 Liubicheva, V.F., 453, 478 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Name Index Lobachevsky, N.I., 8, 308 Lomonosov, M.V., 32, 40, 41, 275 Lukankin, G.L., 315 Lukicheva, E.Yu., 358, 372 Lungardt, R.M., 206, 227 Lyapin, M.P., 297, 317 Lyusternik, L.A., 376 Magnitsky, L.F, 39, 40, 79 Magomeddibirova, Z.A., 435, 478 Maizelis, A.R., 30, 36, 266, 316, 356, 357, 372 Makarov, A.A., 247, 260, 264 Makarychev, Yu.N., 139, 145, 149, 151, 159, 163–165, 174, 188, 189, 200, 229, 246, 262 Malikov, T.S., 421, 478 Malova, I.E., 455, 478 Manvelov, S.G., 17, 18, 36, 432, 433, 479 Markushevich, A.I., 228, 235, 270 Marushina, A.A., vi, viii, 375, 489 Marx, K., 267, 317 Mazanik, A.A., 371 Medvedev, D.A., 280 Melnikov, I., 426 Menchinskaya, N.A., 55, 79, 411 Menelaus, 297 Metel’sky, N.V., 479 Mikulina, G.G., 78 Mindyuk, N.G., 188, 189, 229, 246, 262 Mirakova, T.N., 73, 75, 78 Monakhov, V.M., 316 Monchinsky, A., 324 Mordkovich, A.G., 200, 223, 224, 229, 246, 263 Moro, M.I., 55–60, 63, 66, 68, 70, 72, 74, 79 Moshkov, A., 51, 79 Moshkovich, M.M., 303, 315 Moskalenko, K., 340, 372 Münchhausen, K., 309 Muravin, G.K., 224, 225, 229, 371 Muravina, O.V., 224, 225, 229 Mushtavinskaya, I.V., 358, 372 497 Naziev, A.Kh., 456, 479 Nefedova, M.G., 62–67, 69, 74, 78 Nekrasov, P.A., 233, 275 Nekrasov, V.B., 90, 126, 366, 372 Neshkov, K.I., 188, 189, 229 Newton, I., 129, 130, 135, 194, 217, 221, 222, 233, 386 Nikandrov, N.D., 117 Nikitin, N.N., 88, 98, 99, 127 Nikolskaya, I., 387, 408 Nikolsky, S.M., 246, 263 Novikov, S.P., 275, 317 Ochilova, Kh., 238, 263 Okhtemenko, O.V., 181, 188 Oldham, G., viii Orlov, A.I., 408 Orlov, V.V., 413, 427, 445, 479, 483 Orwell, G., 223 Ovchinsky, B.V., 318 Pardala, A., 420, 421, 479 Pascal, B., 290 Pchelko, A.S., 47, 53–55, 79 Perelman, Ya.I., 376 Perevoschikova, E.N., 456, 457, 479 Perminov, E.A., 448, 479 Pestalozzi, I.G., 43 Peter the Great, 40 Peterson, L.G., 60, 66, 67, 71, 75, 76, 80 Petrova, A.I., 428, 480 Petrova, E.S., 459, 460, 480 Piaget, J., 426, 440 Pichugin, A.G., 50, 80 Pichurin, L.F., 394, 408 Pigarev, B.P., 318, 372, 374 Plotsky, A., 447 Podkhodova, N.S., 78, 320, 373, 444, 445, 480, 483 Pogorelov, A.V., 87, 104–108, 124, 127 Poincaré, H., Polevschikova, A.M., 78 Polya, G., 91, 127, 451 www.EngineeringBooksPDF.com March 9, 2011 15:6 498 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Russian Mathematics Education: Programs and Practices Polyakova, T.S., 38, 39, 80, 425, 480 Popova, N.S., 52, 54, 55, 78–80 Potapov, M.K., 263 Potapov, V.G., 236, 263 Potoskuev, E.V., 303, 317, 334, 360, 372, 374 Pozdnyakov, S.N., 439, 480 Poznyak, E.G., 104, 107, 126, 227, 315 Pratusevich, M.Ya., vi, viii, 302, 317, 375, 489 Printsev, N.A., 330, 372 Prokofiev, A.A., 431, 480 Prudnikov, V.E., 425, 481 Pushkar’, P.E., 292, 315 Pushkin, A.S., 252 Putin, V.V., 280 Pyryt, M.C., 281, 315 Pyshkalo, A.M., 55 Rabbot, Zh.M., 409 Ragulina, M.I., 440, 481 Read, W., 267, 314 Reshetnikov, N.N., 263 Reznik, N.I., 420, 481 Rozanova, S.A., 450, 481 Rozental’, A.L., 408 Rudnitskaya, V.N., 63–67, 70, 73, 75, 80 Rybkin, N.A., 96, 97, 119, 126 Ryzhik, V.I., 2, 36, 88, 104, 105, 116, 117, 125, 126, 128, 268, 303, 314, 317, 336, 373, 423, 424, 481, 491 Sadovnikov, N.V., 454, 481 Safuanov, I.S., 462, 481 Salekhova, L.L., 450, 451, 481 Samigulina, Z.N., 236, 263 Sanina, E.I., 434, 482 Sarantsev, G.I., 304, 317, 423, 482 Saul, M., 377, 408 Savvina, O.A., 195, 229, 428, 482 Scheglov, N.T., 231, 263 Schwarz, K.H.A., 115 Schweitzer, A., 113 Sedova, E.A., v, viii, 129, 175–179, 181, 185, 187, 188, 200, 228, 371, 490 Sekovanov, V.S., 448, 482 Seliutin, V.D., 239, 260, 263, 264 Semenov, E.E., 394, 408 Semenov, P.V., 246, 247, 260, 263, 392, 409 Semenovich, A.F., 127 Shabunin, M.I., 227, 229, 315 Shagilova, E.V., 423, 466, 482 Sharygin, I.F., 113–115, 121, 126–128, 140, 145–148, 175, 188, 200, 241, 245, 248, 261, 368, 373, 392, 409, 451 Shatalov, V.F., 355, 373 Sheinina, O., 386, 409 Shestakov, S.A., 314, 365, 366, 373 Shevkin, A.V., 263 Shibasov, L.P., 409 Shibasov, Z.F., 409 Shiryaeva, E.B., 321, 373 Shkerina, L.V., 462, 482 Shkliarskii, D.O., 376, 377, 397, 398, 409 Shlyapochnik, L.Ya., 365, 374 Shneider, R.K., 10–12, 20, 36 Shokhor-Trotsky, S.I., 46–48, 80 Shvartsburd, S.I., 228, 229, 268, 269, 288, 290, 291, 293, 301–303, 315, 317, 318 Sidorov, Yu.V., 227, 229, 315, 447, 482 Silaev, E.V., 461, 483 Simpson, T., 290, 297 Sivashinsky, I.Kh., 301, 318 Skanavi, M.I., 93, 128, 354, 373 Skatkin, M., 10–12, 20, 36 Skopets, Z.A., 126 Smirnov, E.I., 452, 453, 483 Smirnov, V.A., 115, 128 Smirnova, I.M., 115, 128, 200, 223, 224, 229, 430, 431, 483 Smykovskaya, T.K., 437, 483 Sobolevsky, A.I., 38 Solovieva, G., 386, 409 Somova, L.A., 278, 315, 326, 364, 371 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Name Index Sossinsky, A., 274, 275, 285, 286, 318 Sotnikova, O.A., 463, 483 Spinoza, B., 87, 128 Stalin, I.V., 2, 198, 267, 271, 286, 325, 366 Stefanova, N.L., 320, 373, 419, 451, 453, 483 Stepanov, V.D., 378, 379, 409 Stepanova, S.V., 79 Stolbov, K.M., 302, 317 Stolyar, A.A., 371, 422, 484 Suvorova, S.B., v, viii, 129, 140, 151, 153, 154, 156, 160, 162, 164, 165, 175, 188, 189, 200, 203, 204, 206, 228, 229, 251, 264, 347, 372, 374, 490 Talyzina, N.F., 440 Tazhiev, M., 446, 484 Telyakovsky, S.A., 189 Temerbekova, A.A., 320, 322, 373 Testov, V.A., 413, 423, 484, 485 Thales, 84 Tikhonov, A.N., 104 Timofeeva, I.L., 465, 484 Tkacheva, M.V., 227, 229, 246, 264, 315 Tkachuk, V.V., 10, 36 Tkhamofokova, S.T., 466, 484 Tokar, I., 281, 318 Tolstoy, L., 31, 44 Tonkikh, A.P., 78 Toom, A.L., 409 Totsky, E., 445, 484 Troitskaya, S.D., v, viii, 129, 175–179, 185, 188, 491 Tropin, I.T., 273, 317 Trushanina, T.N., 318, 374 Tsukar’, A.Ya., 420, 484 Turkina, V.M., 78, 435, 484 Tyurin, Yu.N., 246, 247, 250, 260, 264 Uspenskii, V.A., 408 Uvarov, S., 196 Vaneev, A., 286, 318 Vasarely, V., 114 499 Vasiliev, N.B., 395, 396, 409 Vavilov, V.V., 273, 317 Veliev, B.V., 236, 264 Venttsel, E.S., 236, 264 Verebeychik, I.Ya., 287, 318, 376 Verzilova, N.I., 380, 409 Viète, F., 8, 133, 342 Vilenkin, N.Ya., 60, 139–141, 145, 174, 190, 236, 264, 269, 301, 302, 318, 393, 394, 408, 409, 451 Vinogradov, I.M., 103, 104 Vladimirova, N.G., 105 Vlasov, A.K., 50, 80 Vogeli, B.R., 125, 227, 274, 275, 314–316, 318, 372, 408, 476, 491 Volkova, S.I., 79 Volkovksy, D.L., 43 Volovich, M.B., 371, 438, 439, 484 Voron’ko, T.A., 424, 485 Vygotsky, L., 24, 36, 86, 119, 128, 419, 435, 438, 440, 451 Vysotsky, I.R., 247, 254, 257, 260, 264, 373 Weierstrass, K., 213, 222 Wenninger, M., 8, 36, 357, 373 Werner, A.L., v, viii, 81, 82, 88, 104, 105, 107, 110, 116, 117, 125, 126, 128, 303, 308–310, 312, 314, 316, 491 Williams, H., 316 Wilson, L.D., 329, 373 Yaglom, A.M., 410 Yaglom, I.M., 96, 126, 228, 235, 297, 318, 376, 398, 408, 409 Yagodovsky, M.I., 126 Yakimanskaya, I.S., 321, 373, 411, 419, 485 Yaroslav the Wise, 38 Yaschenko, I.V., 247, 260, 264, 380, 410 Yaskevich, V., 432, 485 Yeltsin, B.N., 272 Yudacheva, T.V., 63–67, 70, 73, 75, 80 Yudina, I.I., 90, 104, 126, 314 www.EngineeringBooksPDF.com March 9, 2011 15:6 500 9in x 6in Russian Mathematics Education: Programs and Practices b1073-nameindex Russian Mathematics Education: Programs and Practices Zaguzov, N.I., 415, 485 Zaikin, M.I., 423, 432, 485 Zaks, A.Ya., 375, 410 Zalgaller, V.A., 276, 283, 318 Zankov, L.V., 58–60, 451 Zaretsky, M., 13, 36 Zenchenko, S.V., 52, 80 Zharov, V.K., 428, 485 Zhgenti, M., 404 Zhokhov, A.L., 418, 485 Zhokhov, V.I., 278, 315, 326, 371 Zhurbenko, I.G., 235, 264 Zilberstein, H., 324 Zinchenko, P.I., 420 Ziv, B.G., 331, 352, 373, 374, 389, 410 Zlotsky, G.V., 454, 486 Zvavich, L.I., v–viii, 1, 8, 36, 296, 299, 303, 315, 317–319, 334, 347, 351, 357, 360, 365, 372–374, 492 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-subjectindex Subject Index Axiomatic approach, 102, 114, 115, 129, 244 Academy of Pedagogical Sciences, 55 Academy of Sciences, 40, 117, 269 Advanced level, 83, 131, 132, 135, 137, 166, 170, 173, 174, 181, 302 Algebra, 5, 9, 55, 66, 73, 77, 85, 87, 100, 118, 129–132, 135–141, 144, 145, 149, 164–166, 183, 186, 191, 195, 199–201, 208, 231, 232, 241, 283, 288–290, 295, 302, 306, 307, 310, 312, 333, 342, 345, 364, 365, 369, 389, 393–395, 420, 435, 448, 456, 463, 464, 467 Algebraic expression, 25, 27, 133–135, 137, 166, 171, 175, 176, 186 All Russian Congress of Mathematics Teachers, 196 All-Russia Olympiad, 384 Antiderivative, 192, 194, 195, 217, 218, 220, 222, 225, 365 Arithmetic, 13, 38–48, 50, 53–55, 114, 129, 134–136, 141, 143–145, 147, 149, 151, 154, 160, 161, 164, 165, 179, 184, 256, 327, 348, 386, 388, 393, 415 Attestation, 175, 186, 414, 415, 417 Axiom, 88, 102, 105, 112, 114, 164, 289, 290, 402 Basic education, 241, 447 Basic level, 83, 131, 132, 134, 137, 166, 173, 181, 186, 224, 235, 368 Basic school, 82, 85, 99, 103, 116, 121, 130–133, 136, 138, 139, 154, 160, 164–166, 173, 177, 179–181, 183, 191, 200, 201, 259, 364, 401, 421, 436 Calculus, 5, 130, 192–200, 208, 213–216, 219, 221, 222, 225–227, 283, 288, 289, 292, 294, 295, 297, 299, 302, 307, 309, 342, 365, 395, 402, 428, 446, 463 Candidate of Science, 413 Central Committee of the CPSU, 270 Combinations, 43, 74, 247–249 Combinatorics, 66, 74, 231, 236, 237, 239, 241, 243–249, 254, 259, 288, 308, 309 Complex numbers, 132, 135, 154, 238, 288, 308, 311, 336 Complex programs, 50 Congruence, 83, 100, 106–108, 112, 123 501 www.EngineeringBooksPDF.com March 9, 2011 15:6 502 9in x 6in Russian Mathematics Education: Programs and Practices b1073-subjectindex Russian Mathematics Education: Programs and Practices Constructions, 83, 85, 97, 117, 290, 387, 396, 419 Continuity, 67, 179, 194, 215, 217, 220, 224, 225, 245, 294, 296, 297, 402, 434, 435, 443, 447, 448, 455, 456, 468 Continuity in Education, 435, 443 Control, 2, 278, 319, 433, 443, 444, 453, 454, 457, 458, 469 Coordinates, 83, 85, 86, 98, 114, 117, 125, 134, 136, 155, 157, 160, 170, 202–205, 210, 217, 222, 289, 430 Correspondence Mathematics Olympiad, 395 Correspondence school, 395, 396 Curriculum, 35, 40, 47, 50–53, 55, 56, 62, 68, 70, 72, 75, 76, 125, 175, 178, 179, 192, 196–200, 206, 214, 216, 220, 231, 232, 234–238, 240–247, 253, 254, 259, 269, 277, 282, 283, 287, 290, 296, 297, 299, 300, 302, 307, 310, 311, 322, 326, 334, 342, 388, 389, 394–396, 405, 418, 432, 470 Deduction, 88, 164, 465 Derivative, 138, 191, 192, 194, 197, 199, 200, 213–226, 289, 295, 296, 301, 360, 396 Differentiation in Education, 429 Divisibility, 27, 135, 137, 150, 178–181, 183, 243, 379, 390, 463 Doctor of science, 413 Domain of a function, 204, 206 Electives, 237, 243, 281, 283, 383, 385, 387–389, 391, 392, 458 Elementary School, 4, 5, 11, 43, 44, 47, 50, 53, 55, 57, 58, 60, 62, 65, 68, 73, 76, 77, 82, 130, 141, 325, 440–443, 458, 459, 461, 462 Equation, 27, 57, 73, 74, 130–138, 141–145, 147–163, 165, 166, 168–172, 176, 178, 182, 184–186, 191, 194, 201, 203, 204, 208, 210–213, 215,216, 219, 221, 223, 241, 288, 289, 291, 293, 296, 299, 301, 308, 309, 312, 330–332, 334, 342, 345, 347, 348, 365, 366, 383, 386, 388–390, 394, 396 Equivalence, 135, 136, 144, 345 Exponential functions, 192, 193, 208–211, 218, 220, 225, 297 Factorization, 133, 165, 171, 182 Federal Educational Standards, 37, 61 Festival of Young Mathematicians, 404 Final test, 362, 363 Finite mathematics, 231, 234, 240, 247, 259, 358 Foundations of geometry, 95, 96, 105, 120 Function, 84, 124, 129, 135–138, 160, 169, 177, 185, 191–195, 197, 199–227, 241, 288, 289, 294–297, 301–304, 308–310, 313, 322, 332, 336, 342, 345, 357, 360, 365, 388, 396, 401, 418–421, 433, 436, 438, 453, 455 Geometric transformation, 66, 83, 100, 124, 289, 296, 387, 402 Geometry, 5, 9, 10, 14, 21, 29, 30, 38–41, 50, 53, 55, 66, 77, 81–83, 85–89, 91–100, 102–105, 107–122, 124, 125, 130, 159, 194–197, 211, 218, 243, 283, 288–290, 292, 294, 295, 297, 303, 306–309, 312, 331–335, 342, 345, 348, 352, 357, 360, 361, 363, 369, 388, 389, 393, 394, 402, 414, 420, 421, 424, 428, 430, 431, 435, 443–446, 448, 456, 461, 468 Gymnasium (gymnasia), 96, 196–198, 271, 324 Herzen State Pedagogical University, 37, 81, 116, 191, 487–489, 491 Higher Attestation Commission, 414, 415, 417 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-subjectindex Subject Index Higher mathematics, 191, 192, 195–199, 210, 220 Homework, 1, 12–15, 17, 18, 41, 109, 207, 341, 349, 352, 355, 380, 381, 395 Humanitarization, 304, 306, 417, 418, 441, 448, 449, 456, 469 Imperial Petersburg Academy of Sciences, 40 Independent University of Moscow (IUM), 104, 113 Inequalities, 57, 73, 132–138, 151, 160, 161, 184–186, 191, 201, 206–208, 210, 212, 221, 223, 225, 241, 288, 296, 298, 299, 301, 308, 309, 312, 334, 388, 402 Informal geometry, 121 Informatics, 61, 62, 76, 259 Institute for the Continuing Education of Teachers, 286 Institute on Educational Content and Methods, 129, 241, 488, 490, 491 Integral, 86, 129, 135, 150, 152, 166, 168, 171, 191, 192, 194, 196–200, 213, 214, 217, 218, 220–222, 225, 251, 289, 294, 301, 308, 311, 320, 463 Integral sums, 218, 225 Intellectual-cultural habits, 48 Interval method, 135, 136, 185, 206, 207, 215, 225 Invariant, 390, 391, 396 Irrational equations, 134, 136, 137, 176, 178, 184, 208 Irrational inequalities, 136, 137, 185, 186, 208 Kolmogorov boarding school (#18), 270, 273–275, 278, 281, 282, 285, 290 Kolmogorov reform, 99, 103, 198, 213, 214 Komsomol, 277, 328, 398 Kvant, 9, 270, 393, 404 503 Lagrange’s theorem, 216, 217 Learning activity, 62, 322, 419, 437, 438, 444, 462, 468 Lenin State Pedagogical University, 488, 490, 491 Lesson planning, 17 Limit, 45, 48, 93, 98, 120, 191, 193, 194, 196, 198–200, 209, 213–215, 218–227, 242, 249, 266, 269, 287, 289, 294, 296, 301–303, 305, 311, 353, 365, 402, 405, 466 Linear equation, 133, 134, 136, 152, 154, 155, 158, 168, 170, 171, 203, 332, 347 Linear function, 191, 194, 201–204, 206, 288, 396 Linear inequality, 134, 136, 160, 206, 207, 288 Logarithm, 134–138, 175, 210, 211, 303, 323 Logarithmic functions, 192, 193, 199, 208, 210, 211, 218, 220, 288, 297, 301, 310 Long-term assignments, 356, 357 Magnitudes, 59, 66–68, 70–72, 75, 134, 137, 204, 311 Matematika v shkole, 98, 231, 246, 413, 487 Math battle, 284, 376, 402 Mathematical abstraction, 422 Mathematical circle, 239 Mathematical dictation, 33, 353 Mathematical logic, 283, 289, 387 Mathematical statistics, 234, 236, 237, 239, 247, 309, 457 Mathematical tables, Mathematical tournament, 380 Mathematical wall newspaper, 378 Mathematics classroom, 2, 6, 8, 10, 32, 327, 340 Mathematics schools, 274, 275, 278–282, 284–287, 305, 306, 313, 400 Method of learning operations, 42 www.EngineeringBooksPDF.com March 9, 2011 15:6 504 9in x 6in Russian Mathematics Education: Programs and Practices b1073-subjectindex Russian Mathematics Education: Programs and Practices Methodology, 4, 10, 17, 28, 40–44, 46, 48, 77, 92, 103, 111, 140, 203, 234, 236, 239, 241, 245–248, 250, 251, 345, 359, 375, 387, 414, 415, 418, 426, 427, 430, 433–436, 438–441, 445, 446, 448, 453, 458, 460, 461, 466–468 Ministry of education, 5, 19, 62, 76, 92, 94, 113, 131, 232, 241, 283, 304, 320, 326, 346, 363, 365 Modeling, 61, 67, 75, 88, 132, 161, 216, 237, 246, 248, 253, 310, 312, 422, 429, 439, 443, 454, 457, 464 Monitoring, 31, 246, 319, 320, 322, 433, 435 Monographical method, 41–43 Moscow Center of Continuing Education, 280 Moscow State University, 104, 113, 400 Multiple-choice test, 329, 333, 334 Municipal Olympiad, 384 National Congresses of Teachers of Mathematics, 50 October revolution, 50, 52, 197 Olympiad, 96, 179, 243, 277, 283–285, 287, 358, 376, 380, 384–386, 388, 395–402, 404, 406 Oral mathematics journal, 379 Oral questioning, 337, 338, 352 Oral test, 359–362 Pedology, 329 Perestroika, 240, 266, 278, 304, 333, 429 Performance requirements, 61, 242 Pigeonhole principle, 388, 390 Plane geometry, 21, 30, 41, 81–83, 96–99, 108, 115, 116, 122, 334, 335, 389, 446 Polynomials, 25, 133, 135, 136, 138, 150, 165, 166, 171, 178, 181–184, 195, 197, 288, 295, 303, 463 Polytechnic education, 55, 270 Polytechnization, 97, 198, 267 Popular Lectures in Mathematics, 9, 376, 393 Portfolio, 325, 358 Power functions, 192, 193, 200, 206, 208, 209, 218, 225 Practical applications, 17, 41, 59, 97, 110, 111, 197, 231, 232, 234, 235, 237, 240, 244 Probability, 66, 130, 186, 231–247, 250–254, 259, 288, 308, 309, 311, 457 Problem solving, 15, 20, 21, 24, 161, 289, 291, 292, 313, 343, 364, 380, 387, 401, 402, 421–423, 441, 451, 457, 462, 466, 468 Productive labor, 198, 267, 269, 283 Profile, 114, 130, 173, 265, 276, 307, 312, 313, 430, 431 Profile differentiation, 430, 431 Projects, 8, 277, 282, 325, 356–358, 401, 429, 436, 447 Proof, 10, 16, 21–24, 27, 85–87, 89, 91, 93, 96, 99–102, 106–108, 111, 117, 118, 121–124, 160, 163, 164, 177–180, 182, 183, 202, 207, 209, 210, 215, 216, 220, 221, 224, 226, 232, 300, 309, 331, 342, 362, 390, 438, 446, 456, 465 Psychology of mathematics education, 419 Pythagorean theorem, 10, 84, 87, 109, 334, 387 Quadratic equation, 133, 136, 152–154, 162, 168, 170, 204, 296, 299, 332, 342 Quadratic function, 160 Quiz, 331, 350–352, 354 Range of a function, 212, 295 Real number, 208–213, 289, 294, 301, 302, 402, 469 Real schools, 95, 96, 196, 197, 233, 234, 470 www.EngineeringBooksPDF.com March 9, 2011 15:6 9in x 6in Russian Mathematics Education: Programs and Practices b1073-subjectindex Subject Index Real-world processes, 132, 138, 191, 193, 312 Remainders, 388, 390 Russian Academy of Education, 117, 240, 241, 412, 425, 487, 488, 490, 491 School # 30, 276–278, 283–286, 294, 376 School # 38, 268, 277, 278 School # 45, 277 School # 57, 282, 285, 292, 315 School # 121, 275–278, 287 School # 139, 276, 277 School # 239, 276, 277, 280, 302, 375, 400, 401, 489 School # 470, 276, 277 School # 566, 404 School-level Olympiad, 385 Schools specializing in mathematics, 275 Schools with a humanities orientation, 304 Schools with an advanced course in mathematics, 32, 193, 265, 268, 273, 298, 383, 396, 397, 405 Schools with an advanced course of study in humanities, 94 Schools with an advanced course of study in mathematics, 6, 94, 113, 265, 266, 269, 270, 280, 290, 294, 300, 301, 303, 304, 376, 400 Set theory, 75, 100 Short answer question, 166 Soviet Studies in the Psychology, 411 Spatial imagination, 61, 73, 77, 85, 88, 116, 421, 464 Stagnation, 82, 274, 313 Standards, 37, 61, 65, 82, 83, 85, 86, 88, 120, 125, 131, 164, 193, 194, 238, 241, 246, 259, 324, 346, 417, 431, 432 Statistical thinking, 238–240 505 Stochastic, 77, 235, 238–247, 250, 253, 447 Structured system of problems, 24 Summer Camp, 285, 403 Supplemental Pages for the Textbook, 393 System of equations, 155 Teaching aids, 354, 438, 439 Test, 7, 37, 105, 107, 165, 168, 170, 202–204, 239, 254, 256, 258, 277, 291, 321, 334, 345–350, 354, 359–361, 369, 441, 443 The Little Kvant Library, 9, 393 Thematic test, 361 Topic plan, 19 Tournament of Towns, 406 Trigonometric functions, 84, 137, 138, 192, 193, 197, 199, 201, 208, 211, 212, 216, 220, 222, 288, 301 Trigonometry, 41, 98, 114, 303, 306, 310, 360, 388 Types of Lessons, 17–19, 432 Unified labor school, 197 Uniform State Exam (USE), 8, 94, 181, 186, 224, 298, 307, 322, 328, 331, 366 Vector, 83, 86, 98, 99, 103, 111, 117, 125, 222, 289, 290, 299, 300, 363, 430 Visual geometry, 53, 82, 83, 113, 121, 243 Visual-empirical conception, 114 Written Problem-Solving Contest, 284, 378, 382 Young Pioneer Organization, 403 Zone of proximal development, 435 www.EngineeringBooksPDF.com ... www.EngineeringBooksPDF.com March 9, 2011 15:5 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm vi Russian Mathematics Education: Programs and Practices Chapter Assessment in Mathematics. .. viii 9in x 6in Russian Mathematics Education: Programs and Practices b1073-fm Russian Mathematics Education: Programs and Practices Mathematical subjects and courses taught in Russian schools... www.EngineeringBooksPDF.com March 9, 2011 15:0 9in x 6in Russian Mathematics Education: Programs and Practices b1073-ch01 Russian Mathematics Education: Programs and Practices State Exam (USE), and so on