I Babenko M. V. On the polynomial semiring over a Bezout semiring
Babenko Marina − Senior Lecturer of the Department of applied mathematics and computer science, Vyatka State University, e-mail: firstname.lastname@example.org
The article examines a polynomial semiring over a Bezout Rickart semiring. Namely, let all left annihilator ideals of the semiring S be ideals. Then the semiring of polynomials R = S[x] is a semiring without nilpotent elements and every finitely generated left monic ideal from R is principal iff S is a left Rickart left Bezout semiring and any non-zero divisor of the semiring S is convertible to S. This result is analogous to the statement for rings, if the condition each finitely generated left monic ideal of R is principal replaced by R is left Bezout ring. The left monic ideal of a polynomial semiring is a left ideal that contains each monomial of its polynomial. The principal left monic ideals over a left Rickart left Bezout semiring are described.
Keywords: polynomial semiring, Rickart semiring, Bezout semiring, monic ideal.
- Tuganbaev A. A. Kol’ca Bezu, mnogochleny i distributivnost’ (Bezout rings, polynomials and distributivity) Mathematical notes, 2001, 70:2, pp. 270288.
- Dale L. Monic and monic free ideals in polynomial semirings, Proc. Amer. Math. Soc., 1976, No 56, pp. 45-50.
- Dale L. The structure of monic ideals in a noncommutative polynomial semirings, Acta Math. Acad. Sci. Hungar, 1982, 39:1-3, pp. 163-168.
- Golan J. S. Semirings and their applications, Kluwer Acad. Publ., Dordrecht, 1999.
- Chermnykh V. V. Functional representations of semirings, J. Math. Sci., New York, 2012, 187:2, pp. 187-267.
- Maslyaev D. A., Chermnykh V. V. Polukol’ca kosyh mnogochlenov Lorana (Semirings of skew Laurent polynomials), Siberian electronic mat. reports., 2020, Vol. 17, pp. 512-533.
For citation: Babenko M. V. On the polynomial semiring over a Bezout semiring, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 5-15.
II Efimov D. B. A method for computing the hafnian
Efimov Dmitry − Ph. D., research associate, Institute of Physics and mathematics of the Komi national research center of the Ural branch of the Russian Academy of Sciences, e-mail: email@example.com
The hafnian was initially introduced by E.R. Caianiello, by analogy with the Pfaffian, as a convenient mathematical apparatus for working with certain quantum-mechanical quantities. From a combinatorial point of view, the hafnian of a symmetric matrix is equal to the sum of weights of perfect matchings of a graph with the given incidence matrix. In contrast to the Pfaffian, the hafnian has a smaller set of good properties, and determining its value is an example of a complex computational problem. We consider a new method for calculating hafnian of a matrix in terms of permanents of its submatrices. We also give a comparison with other methods in terms of computational complexity. The property underlying the method could also be used outside the context of the computation speed, for example, to estimate the hafnian of a nonnegative matrix based on known estimates of the permanent.
Keywords: hafnian, permanent, computational complexity
- Caianiello E. R. On quantum field theory – I: Explicit solution of Dyson’s equation in electrodynamics without use of Feynman graphs, IL Nuovo Cimento, 1953, V. 10(12), pp. 1634-1652.
- Caianiello E. R. Theory of coupled quantized fields, Supplemento Nuovo Cimento, 1959, V. 14(1), pp. 177191.
- Caianiello E. R. Regularization and Renormalization, IL Nuovo Cimento, 1959, V. 13(3), pp. 177-191.
- Mink H. Permanenty (Permanents), M.: Mir, 1982, 216 p.
- Valiant L. G. The complexity of computing the permanent, Theoretical Computer Science, 1979, V. 8(2), pp. 187-201.
- Bjorklund A., Gupt B., Quesada N. A faster hafnian formula for complex matrices and its benchmarking on a supercomputer, ACM Journal of Experimental Algorithmics, 2019, V. 24(1), 17 p.
- Aaronson S., Arkhipov A. The computational complexity of linear optics, Proceedings of the Annual ACM Symposium on Theory of Computing, 2011, pp. 333-342.
- Kruse R., Hamilton C. S., Sansoni L., Barkhofen S., Silberhorn C., Jex I. Detailed study of Gaussian boson sampling, Physical Review A, 2019, V. 100(3), 032326.
For citation: Efimov D. B. A method for computing the hafnian, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 16-25.
III Gabova M. N., Muzhikova A. V. Ñontext approach in the teaching of mathematics future engineers
Gabova Maria − Senior Lecturer of the Department of the Department of higher mathematics, Ukhta state technical University, e-mail: firstname.lastname@example.org
Muzhikova Alexandra − Ph. D., associate Professor of the Department of the Department of higher mathematics, Ukhta state technical University, e-mail: email@example.com
There is a problem of reducing the mathematical education of school graduates, and as a result, the lack of motivation and cognitive activity of first-year students when studying mathematics in higher school. Mathematics, devoid of professional direction, is not of interest to most students of a technical higher school. The efectiveness of the teaching process can be achieved by using a context approach. Context teaching is teaching in which the subject and social content of students’ professional activity is modeled in the language of science and with the help of the entire system of forms, methods and means of teaching. Considering context teaching as an integral system that meets the corresponding principles, the article presents the developed methodological and organizational support for educational activities.
The main idea in developing the content is a gradual transition from abstract mathematical concepts to their applied meaning in related sciences, and then to their application in professional fields. The principles of context teaching are best implemented when using active and interactive forms of teaching and their corresponding methods. The most efective methods in terms of achieving the goals of teaching, development and education were shown by such methods as problem-based lecture format, swapping of topics in pairs and partners rotation, paragraph-by-paragraph study of theoretical material in small groups, task swapping in practical classes, etc.
The use of a context approach allows students to develop social interaction, motivation and cognitive activity, mathematical literacy, the ability to apply mathematics in their educational and professional activities and contribute to the formation of a modern engineer capable of creative activity and self-realization.
Keywords: mathematics for engineer, context approach, active and interactive methods of teaching.
- Kostenko I. P. Evolyuciya kachestva matematicheskogo obrazovaniya (1931-2009 gg.) (Evolution of the quality of mathematics education (1931-2009)), Izvestija VGPU, 2013, No 2 (261), pp. 81-87.
- Muzhikova A. V. Matematicheskaya obrazovannost’ studentov: problemy i perspektivy (Mathematical education of students: problems and prospects), XIX mezhdunarodnaja nauchno-prakticheskaja konferencija Kommunikacii. Obshhestvo. Duhovnost’ 2019, Ukhta: UGTU,2019, Vol. 3, pp. 141-144.
- Muzhikova A. V., Gabova M. N. Razvitie gramotnoj matematicheskoj rechi studentov v tekhnicheskom vuze (Development of Competent Mathematical Speech of Students in a Technical University), Vysshee obrazovanie v Rossii, 2019, Vol. 28, no. 12, pp. 66-75,.
- Rozanova S. A. Matematicheskaja kul’tura studentov tehnicheskih universitetov (Mathematical culture of students of technical universities), Moscow: FIZMATLIT Publ., 2003, 176 p.
- Bogomolova E. P. Diagnoz: matematicheskaya malogramotnost’ (Diagnosis: mathematical illiteracy), Matematika v shkole, 2010, No 4, pp. 3-9.
- Senashenko V. S., Vostrikova N. A. O preemstvennosti srednego i vysshego matematicheskogo obrazovaniya (On the continuity of secondary and higher mathematics education), Mezhdunarodnaja konferencija Obrazovanie, nauka i jekonomika v vuzah. Integracija v
mezhdunarodnoe obrazovatel’noe prostranstvo¿, Plock (Poland), 2006, pp. 103-106.
- Zajniev R. M. Preemstvennost’ matematicheskoj podgotovki v inzhenerno-tehnicheskom obrazovanii (Continuity of mathematical training in engineering and technical education), Kazan: Kazan State University Publ., 2009, 366 p.
- Egorova I. P. Proektirovanie i realizacija sistemy professional’nonapravlennogo obuchenija matematike studentov tehnicheskih vuzov (Design and implementation of a system of professional-oriented learning mathematics to students of technical universities:Cand. Sci.
Thesis), Tolyatti, 2002, 24 p.
- Verbickij A. A. Aktivnoe obuchenie v vysshej shkole: kontekstnyj podhod (Active learning in higher education: context approach), Moscow: Vysshaja shkola Publ., 1991, 207 p.
- Grebenkina A. S. Osobennosti kontekstnogo obucheniya vysshej matematike studentov tekhnicheskih special’nostej (Features of context learning of higher mathematics to students of technical specialties), II mezhdunarodnaja nauchno-prakticheskaja konferencija Psihologija
i pedagogika XXI veka: teorija, praktika i perspektivy, Cheboksary: CNS Interaktiv pljus Publ., 2015, pp. 2430.
- Kolbina E. V. Metodika formirovanija matematicheskoj kompetentnosti studentov tehnicheskih vuzov v problemno-prikladnom kontekste obuchenija. Kand. Diss. (Methods of forming mathematical competence of students of technical universities in the problem-applied context of learning: Cand. Diss.), Barnaul, 2016, 221 p.
- Janushhik O. V., Sherstnjova A. I., Pahomova E. G. Kontekstnye zadachi kak sredstvo formirovaniya klyuchevyh kompetencij studentov tekhnicheskih special’nostej (Context tasks as a means of forming key competencies of students of technical specialties), Sovremennye problemy nauki i obrazovanija, 2013, No. 6, p. 376.
- Pidkasistij P. I. Pedagogika (Pedagogics: textbook for students of pedagogical universities and pedagogical colleges), Moscow: Pedagogicheskoe obshhestvo Rossii Publ., 1998, 640 p.
- Nizhnikov A. I., Rastopchina O. M. Obuchenie vysshej matematike: kontekstnyj podhod (Learning higher mathematics: the context approach), Vestnik Moskovskogo gosudarstvennogo oblastnogo universiteta, 2018, No 3, ðð. 184-193, doi:10.18384/2310-7219-2018-3-184-
- Sorokopud Ju. V. Pedagogika vysshej shkoly (Pedagogy of Higher School), Rostov-on-Don: Feniks Publ., 2011, 541 p.
- Mkrtchjan M. A. Metodiki kollektivnyh uchebnyh zanyatij (Methods of Collective Training), Spravochnik zamestitelja direktora shkoly, 2011, No 1, pp. 5564.
- Prudnikova O. M., Gabova M. N., Kaneva E. A. K voprosu formirovaniya u studentov kriticheski-refeksivnogo stilya myshleniya (To the Question of Formation of Students Critical-Refiexive Style of Thinking), Nauchno-tehnicheskaja konferencija, Ukhta: UGTU, 2011, Vol. 3, pp. 226-229.
- Muzhikova A. V. Interaktivnoe obuchenie matematike v VUZe (Interactive Teaching of Mathematics in Higher School), Vestnik Syktyvkarskogo universiteta. Serija 1: Matematika. Mehanika. Informatika, 2015, Vol. 1 (20), pp. 74-90.
- Muzhikova A. V. Issledovanie efektivnosti kollektivnyh uchebnyh zanyatij po vysshej matematike (Study the Interactive Teaching Efiectiveness in Higher Mathematics), Vestnik Tomskogo gosudarstvennogo pedagogicheskogo universiteta, 2018, No 7 (197), pp. 174-181.
- Lobos E., Macura J. Mathematical competencies of engineerin students, In ICEE-2010, International Conference on Engineering Education, July 18-22, 2010, Gliwice, Poland, Silestian University of Technology.
- Zeidmane A., Rubina T. Student Related factor for dropping out in the first year of studies at LLU engineering programmes, Engineering for Rural Development, 2017, N 16, pp. 612-618,.
- Steyn T., Plessis I. D. Competence in mathematics-more than mathematical skills?, International Journal of Mathematical Education in Science and Technology, 2007, Vol. 38, Issue 7, pp. 881-890,doi:10.1080/00207390701579472.
- Ravn O., Bo Henriksen L. Engineering mathematics in context – learning university mathematics through problem based learning, International Journal of Engineering Education, 2017, Vol. 33, Issue 3, pp. 956-962.
- Firouzian S., Kashe H., Yusof Y. M., Ismail Z., Rahman R. A. Mathematical competencies as perceived by engineering students, lecturers, and practicing engineers, International Journal of Engineering Education, 2016, Vol. 33, Issue 6, pp. 2434-2445.
For citation: Gabova M. N., Muzhikova A. V. Ñontext approach in the teaching of mathematics future engineers, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 26-50.
IV Odyniec W. P. The Fate of two mathematicians: Perelman and Perelman Jr.
Odyniec Vladimir − Doctor of Physical and Mathematical Sciences, Professor, Syktyvkar state University named after Pitirim Sorokin, e-mail: W.P.Odyniec@mail.ru
In the article the work of Jacob I. Perelman (1882-1942) in the area of mathematics and its application to the theory of elasticity is described for the first time at the ever of the Grand Patriotic War. Also described the life and work of his son Michael J. Perelman (1919-1942).
Keywords: J. I. Perelman, M. J. Perelman, Galerkin method, continuity modulus, the least power and pseudo-power of a topological space.
- Mishkevich G. I. Doktor zanimatelnyh nauk (Doctor of Entertaining Sciences), M.: Znanie, 1986, 192 p.
- Matematika v SSSR za tridcat’ let (The URSS Mathematics for thirty years: 1917-1947), Pod. red. A. G. Kurosha, A. I. Markushevicha, P. K. Rashevskogo, M.-L.: OGIZ, Izd-vo tehn-teor. lit-ry, 1948, 1045 p.
- Matematika v SSSR za sorok let (The URSS Mathematics for forty years: 19171957), Biobibliografiya, Vol. 2, M.: Fizmatlit, 1959, 819 p.
- Leibenson L. S. Variacionnye metody resheniya sadach teorii uprugosti (Variational methods of solving the problems of the theory of elasticity), M.-L.: 1943, 287 p.
- Kniga pamyati. Leningrad. 1941-1945. Primorskii raion T. 12 (The Book of Memory. Leningrad. 1941-1945. The Primorsky District. Vol. 12), SPb: Notabene, 1997, 557 p.
- Perelman J. I. Metod Galerkina v variacionnom ischislenii i v teorii uprugosti (Galerkin method in calculus of variations and in the theory of elasticity), Prikladnaya matematika I mehanika, T. V, vyp. 3, 1941, 345-358 p.
- Galerkin B. G., Perelman J. I. Napryazheniya i peremeshtcheniya v krugovom zylindricheskom truboprovode (Tensions and Displacements in Cylindrical Pipe Line), Izvestiya nauchno- issledovatelskogo instituta gidrotehniki, T. 27, 1940, pp. 160-192.
- Odyniec W. P. O leningradskih matematikah, pogibshih v 1941-1944 godah (On some Leningrad based Mathematicians perished in 1941 -1944), Syktyvkar: Izd- vo SGU im. Pitirima Sorokina, 2020, 122 p.
- Blokada 1941-1944. Kniga pamyati, Leningrad, T. 23 (Blockade 1941-1944. The book of Memory, Leningrad, Vol. 23), SPb.:Stella, 2005, 717 p.
- Odyniec W. P. K 125-letiu reformatora matematicheskogo obrazovaniya O.A. Volberga (1895-1942) (On the 125 anniversary of reformer of the birth of mathematical education O.A .Volberg (1895-1942), Matematika v shkole (Mathematics in School), 4, 2020, p. 54-59.
- Perelman M. J. O module nepreryvnosti analiticheskih funkcii (On the Continuity Modulus of Analytical Functions), Uchenye zapiski LGU. Seriya mat. nauk, vyp. 12, 1941, 62-82 p.
- Trudy Pervogo Vsesouznogo s’ezda matematikov (Proceedings of the First All-Union congress of mathematicians), M.-L.: ONTI NKTP SSSR, 1936, 376 p.
- Fomin D. V. Sankt- Peterburgskie matematicheskie olimpiady (Saint Petersburg mathematics olympiades), SPb.: Politehnika, 1994, 309 p.
- Matematicheskii enciklopedicheskii slovar’ (The Mathematical Encyclopaedia), M.: Soviet Encyclopaedia, 1988, 848 p.
- Perelman M. J. Ob odnom svoistve posledovatelnosti polinomov (On one property of sequences of polynomials), Uchenye zapiski LGU. Seriya mat. nauk, vyp. 12, 1941, 83-91 p.
For citation: Odyniec W. P. The Fate of two mathematicians: Perelman and Perelman Jr., Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 51-65.
V Pevnyi A. B., Yurkina M. N. Sieve of Eratosphenes complexity and distribution of primes
Pevny Alexander − Doctor of Physics and Mathematics, Professor, Department of Applied Mathematics and Information Technologies in Education, Pitirim Sorokin Syktyvkar State University, e-mail: firstname.lastname@example.org
Yurkina Marina − Senior Lecturer, Department of Applied Mathematics and Information Technologies in Education, Syktyvkar State University named after Pitirim Sorokin, e-mail: email@example.com
Primes are widely used not only in pure mathematics, but also in related disciplines. And although they have been known for a long time, many problems concerning prime numbers are still open and the questions of their study do not lose their relevance. One of the well-known algorithms for finding all primes not exceeding a given N is the sieve of Eratosthenes. To estimate the number of operations required to execute this algorithm, the authors used one result of P. L. Chebyshev. In 1849 P. L. Chebyshev proved a two-sided estimate for the number of primes not exceeding a given N. Based on these estimates, the article establishes that the number of operations in the Eratosthenes algorithm is estimated as O (N ln ln N).
Keywords: sieve of Eratosthenes, primes, Chebyshev.
- Leandro M., Antonio J. J., Antonio S. F. Multiplication and Squaring with Shifting Primes on OpenRISC Processors with Hardware Multiplier, Journal of Universal Computer Science, 2013, vol. 19, no. 16, pp. 2368-2384.
- Krishan K., Deepti S. D. Eratosthenes sieve based key-frame extraction technique for event summarization in videos, Multimedia Tools and Applications, 2018, 77, pp. 7383-7404.
- Durfian R. D., Masque M. Optimal strong primes, Information Processing Letters, 2015, 93 (1), pp. 47-52.
- Samir B. B., Zardari M. A. Generation of prime numbers from advanced sequence and decomposition methods, International Journal of Pure and Applied Mathematics, Vol. 85 No. 5, 2013, pp. 833-847.
- Mohammad G., Ali K. A novel secret image sharing scheme using large primes, Multimedia Tools and Applications, 2018, 77, pp. 11903-11923.
- Barzu M., Tiplea F. L., Drfiagan C. C. Compact sequences of coprimes and their applications to the security of CRT-based threshold schemes, Information Sciences, 2013, 240, pp. 161-172.
- Popov V. A., Kaneva E. A. Dlinnaya arifmetika v issledovaniyah statistiki pervyh cifr stepenej dvojki, chisel Fibonachchi i prostyh chisel (Long arithmetic in studies of statistics of the first digits of powers of two, Fibonacci numbers and primes), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 2(31), pp. 91-107.
- Kudrina E. V., Kuzmina V. R. Algoritmy nakhozhdeniya prostykh chisel: ot shkoly do vuza (Algorithms for finding prime numbers: from school to university), E-learning in lifelong education: Collection of scientific papers of the III International scientific-practical conference,
Ulyanovsk: UlSTU, 2016, pp. 1106-1113.
- Chebyshev P. L. Izbrannye matematicheskie trudy (Selected mathematical works), M., L.: Ogiz. GOS. Izd-vo tehn.-theoretical lit., 1946, 200 p.
- Buchstab A. A. Teoriya chisel (Number theory), M.: Uchpedgiz, 1960, 376 p.
For citation: Pevnyi A. B., Yurkina M. N. Sieve of Eratosphenes complexity and distribution of primes, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 66-72.
VI Popov N. I., Yakovleva E. V. Use of the schematization method in teaching students and pupils in math
The publication objective is to highlight and generalize the features of using the schematization method in teaching mathematics as a means of developing thinking and mathematical abilities of learners. The research is based on analyzing scientific and methodical works of Russian and foreign scientists on both activity theory, pedagogy, and author’s researches on applying the schematization method in teaching mathematics. The article proposes a schematic model to teach pupils and students solve mathematical problems. The methodological approaches developed in the research can be used in teaching mathematics at diferent levels of education. We believe that the method described in this paper can be successfully applied in studying natural sciences.
Keywords: a method of schematization; teaching mathematics; stages of mathematical problems solving; schematic model.
- Robert I. V. Didaktika epokhi tsifrovykh informatsionnykh tekhnologiy (Didactics of the digital information technology era), Professional’noye obrazovaniye. Stolitsa, 2019, no 3, pp. 16-26.
- Krutetskiy V. A. Psikhologiya matematicheskikh sposobnostey shkol’- nikov (Psychology of mathematical abilities of schoolchildren), M.: Institut prakticheskoy psikhologii, 1998, 416 p.
- Dalinger V. A. Teoreticheskiye osnovy kognitivno-vizual’nogo podkhoda k obucheniyu matematike: monografiya (Theoretical foundations of the cognitive-visual approach to teaching mathematics: monograph), Omsk: Izd-vo OmGPU, 2006, 144 p.
- Popov N. I. Metodika obucheniya trigonometrii na osnove kognitivnovizual’nogo podhoda (Methods of teaching trigonometry on the basis of cognitive-visual approach), Sibirskiy pedagogicheskiy zhurnal, 2008, no 11, pp. 34-42.
- Christochevskaya A. S., Christochevsky S. A. Kognitivizatsiya – sleduyushchiy etap informatizatsii obrazovaniya (Ñognitivization – the next stage of informatization of education), Informatika i obrazovaniye, 2018, no 9, pp. 5-11.
- Tchoshanov M. A. Digital age didactics: from teaching to engineering of learning (Part 1), Informatika i obrazovaniye, 2018, no 9, pp. 53-62.
- Khenner E. K. Vychislitel’noye myshleniye (Ñomputational thinking), Obrazovanie i nauka, 2016, no 2, pp. 18-33.
- Van Kesteren M. T. R., Rijpkema M., Ruiter D. J., Fernandez G. Consolidation Differentially Modulates Schema Effects on Memory for Items and Associations, PLOS ONE, 2013, Vol. 8, Issue 2.
- Dakhin A. N. Kognitivnaya garmoniya matematiki (Cognitive harmony of mathematics), Narodnoye obrazovaniye, no 6-7, 2017, pp. 81-88.
- Anderson R. K., Boaler J., Dieckmann J. Achieving Elusive Teacher Change through Challenging Myths about Learning: A Blended Approach, Education Sciences, 2018, Vol. 8, Issue 3: 98.
- Popov N. I. Teoretikometodologicheskiye osnovy obucheniya resheniyu tekstovykh algebraicheskikh zadach (Theoretical and methodological foundations of teaching to solve text-based algebraic problems), Obrazovaniye i nauka. Izvestiya Ural’skogo otdeleniya Rossiyskoy
akademii obrazovaniya, 2009; no 3(60), pp. 88-96.
- Popov N. I. Ob efiektivnosti ispol’zovaniya modeli obuchayushchey tekhnologii po trigonometrii pri obuchenii studentov-matematikov
(Education of students-mathematicians: the effectiveness of implementation of the educational technology when teaching trigonometry),
Obrazovaniye i nauka, 2013, no 9, pp. 138-153.
- Bacabac M. A. A., Lomibao L. S. 4S Learning Cycle on Students’ Mathematics Comprehension, American Journal of Educational Research, 2020, Vol. 8, Issue 3, pp. 182-186.
- Burte H., Gardony A. L., Hutton A., Taylor H. A. Think3d!: Improving mathematics learning through embodied spatial training, Cognitive Research: Principles and Implications, 2017, Vol. 2, Issue 1.
- Hoogland K., Pepin B., Koning J., Bakker A., Gravemeijer K. Word problems versus image-rich problems: an analysis of effects of task characteristics on students’ performance on contextual
mathematics problems, Research in Mathematics Education, 2018, Vol. 20, Issue 1, pp. 3752.
- Bernikova I. K. Skhemy kak sredstva organizatsii myshleniya v protsesse obucheniya matematike (Schemes as means of organizing thinking in the process of teaching mathematics), Vestnik OmGU, 2015, no 1(75), pp. 2327.
- Rahmawati D., Purwantoa, Subanji, Hidayanto E., Anwar R. B. Process of Mathematical Representation Translation from Verbal into Graphic, International Electronic Journal of Mathematics Education, 2017, Vol. 12, Issue 3, pp. 367-381.
- Zlotnikov I. V. Psikhologicheskoye i psikho-zicheskoye obespecheniye protsessa obucheniya studentov: metodicheskiye rekomendatsii (Psychological and psychophysical support of the student learning process: guidelines), Riga: Izdatel’stvo RPI, 1988, 36 p.
- Poya D. Kak reshat’ zadachu / Pod red. YU. M. Gayduka (How to solve a problem / Ed. Yu. M. Gaiduk), M., 1959, 208 p.
- Kolyagin U. M. Zadachi v obuchenii matematike (Problems in teaching mathematics), M: Prosveschenie, 1977, Ch. 1,113 p.
- Mordkovich A. G. Besedy s uchitelyami matematiki: ucheb.-metod. Posobiye (Conversations with teachers of mathematics: textbookmethod. allowance), M.: Oniks, 2007, 334 p.
- Sarantsev G. I. Uprazhneniya v obuchenii matematike (Exercises in teaching mathematics), M., 2005, 254 p.
- Neshkov K. I., Semushin A. D. Funkcii zadach v obuchenii (Task functions in training), Matematika v shkole, 1971. no 3, pp. 4-7.
- Popov N. I., Yakovleva E. V. Aktual’nyye problemy obucheniya matematike inostrannykh studentov v vuze (Topical issues of teaching mathematics to international students at a university), Vestnik Moskovskogo gosudarstvennogo oblastnogo universiteta, Series: Pedagogika, 2019, no 3, pp. 144-153.
- Marasanov A.N. Sistema zadach po trigonometrii v obuchenii matematike uchaschihsya srednih obscheobrazovatelnih uchrejdenii (System of problems in trigonometry in teaching mathematics to students of secondary educational institutions): diss. . . . kand. ped. nauk., Saransk, 2012, 180 p.
For citation: Popov N.I., Yakovleva E.V. Use of the schematization method in teaching students and pupils in math, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 74-87.