Development of Creative Qualities of Students when Performing a Multi-Stage Mathematical and Information Tasks "Newton's Method for Complex Polynomials"

Abstract

The proposed article describes the methodology for performing a multi-stage mathematical-informational task "Newton's Method for Complex Polynomials", aimed at developing the creative qualities of students. It notes creative activities that the student performs in the course of solving multifaceted problems. A scheme-plan for the implementation of a multi-stage mathematical and information task has been built. Examples of Julia sets of rational functions obtained by applying Newton's method to polynomials are given. The aesthetics of Julia sets are indicated, with the help of which students are invited to create artistic compositions. The integration of mathematics and programming is mentioned. The creative qualities have been identified, which are formed in students in the process of performing multi-stage mathematical and information task. When performing the tasks, students establish connections between iterations of rational functions of a complex variable and quadratic polynomials, discover for themselves Julia sets having complex fractal structure. They analytically calculate the fixed points of rational functions, determining their character. At the next stages, the study is visualized: with the help of a computer program, the orbits of the points of the studied functions and their Julia set are built. The introduction of color classification for different points allows you to get a more informative image of the fractal. The resulting image at the final stage is used to create some artistic compositions. The proposed method of function study makes it possible to develop flexibility of thinking, intuition, overcome stereotypes of thinking, develop aesthetic ideas, which positively affects the development of students' creative qualities.


 


Author Biographies

Valeriy Sergeevich Sekovanov, Kostroma State University

Head of the Chair of Applied Mathematics and Information Technology, Institute of Physics and Mathematics and Natural Sciences, Dr. Sci. (Ped.), Professor

Vladimir Anatolyevich Ivkov, Kostroma State University

Associate Professor of the Chair of Applied Mathematics and Information Technology, Institute of Physics and Mathematics and Natural Sciences, Cand. Sci. (Econ.), Associate Professor

Alexey Alexandrovich Piguzov, Kostroma State University

Associate Professor of the Chair of Applied Mathematics and Information Technology, Institute of Physics and Mathematics and Natural Sciences, Cand. Sci. (Ped.), Associate Professor

Larisa Borisovna Rybina, Kostroma State Agricultural Academy

Associate Professor of the Higher Mathematics Department, Cand. Sci. (Philos.), Associate Professor

Irina Vadimovna Shaposhnikova, Surgut State University

Associate Professor of the Chair of Applied Mathematics, Cand. Sci. (Eng.)

References

1. Douady A. Julia Sets and the Mandelbrot Set. In: Peitgen H. O., Richter P.H. (eds.) The Beauty of Fractals. Berlin, Heidelberg: Springer; 1986. p. 161-174. https://doi.org/10.1007/978-3-642-61717-1_13
2. Milnor J. Dynamics in One Complex Variable. Third Edition. Princeton: Princeton University Press; 2006. Available at: http://www.jstor.org/stable/j.ctt7rnxn (accessed 11.02.2023).
3. Sekovanov V.S. On Julia set of some rational functions. Vestnik of Kostroma State University. 2012;18(2):23-28. (In Russ., abstract in Eng.) EDN: PYNQJR
4. Peitgen H.O., Richter P.H. Julia Sets and their Computergraphical Generation. In: In: Peitgen H. O., Richter P.H. (eds.) The Beauty of Fractals. Berlin, Heidelberg: Springer; 1986. p. 27-52. https://doi.org/10.1007/978-3-642-61717-1_2
5. Ivkov V.A., Sekovanov V.S., Smirnov E.I. Attractors of nonlinear mappings in the framework learning of multi-stage mathematical and information tasks as a means of students' creativity developing. Mathematical Forum. (Results of Science. South of Russia). 2018;12:150-164. (In Russ., abstract in Eng.) EDN: PAKBEA
6. Sekovanov V.S., Ivkov V.A., Piguzov A.A., Fateev A.S. Execution of mathematics and information multistep task "Building a Fractal Set with L-systems and Information Technologies" as a means of creativity of students. CEUR Workshop Proceedings. 2016;1761:204-211. Available at: http://ceur-ws.org/Vol-1761/paper26.pdf (accessed 11.02.2023). (In Russ., abstract in Eng.)
7. Sekovanov V.S., Smirnov E.I., Ivkov V.A., Selezneva E.M., Shlyahtina S.M. Visual Modeling and Fractal Methods in Science. In: 2014 International Conference on Mathematics and Computers in Sciences and in Industry. Bulgaria: Varna; 2014. p. 94-98. https://doi.org/10.1109/MCSI.2014.28
8. Sekovanov V.S., Smirnova A.O. development of students' ideation flexibility when studying structure of fixed points of polynomials of a complex variable. Vestnik of Kostroma State University. Series: Pedagogy. Psychology. Sociokinetics. 2016;33(3):189-192. (In Russ., abstract in Eng.) EDN: WTOYOL
9. Smirnov E.I., Sekovanov V.S., Mironkin D.P. Multi-stage mathematic-information tasks as a means to develop pupils creativity in profile mathematical classes. Yaroslavl Pedagogical Bulletin. 2014;2(1):124-129. (In Russ., abstract in Eng.) EDN: RZLXHB
10. Sekovanov V., Ivkov V., Piguzov A., Seleznyova Y. Designing Anticipation Activity of Students When Studying Holomorphic Dynamics Relying on Information Technologies. In: Sukhomlin V., Zubareva E. (eds.) Modern Information Technology and IT Education. SITITO 2018. Communications in Computer and Information Science. vol. 1201. Cham: Springer; 2020. p. 59-68. https://doi.org/10.1007/978-3-030-46895-8_4
11. Sekovanov V.S. On some discrete nonlinear dynamical systems. Fundamental and Applied Mathematics. 2016:21(3):185-199. (In Russ., abstract in Eng.) EDN: YPVWJV
12. Sekovanov V.S., Smirnov E.I., Ivkov V.A. Motivacii v izuchenii nelinejnyh otobrazhenij fraktal'nosti i haosa metodom nagljadnogo modelirovanija [Motivation in the study of nonlinear mappings of fractality and chaos by the method of visual modeling]. Eurasian Scientific Association. 2015;2(2):302-305. (In Russ.) EDN: TQLTYZ
13. Sekovanov V.S., Salov A.L., Samokhov E.A. O vychislenii konstanty Fejgenbauma [On the calculation of the Feigenbaum constant]. Modern Information Technologies and IT-Education. 2010;6(1):364-371. (In Russ.) EDN: UIZHWR
14. McCartney M., Glass D.H. Computing Feigenbaum's δ constant using the Ricker map. International Journal of Mathematical Education in Science and Technology. 2014;45(8);1265-1273. https://doi.org/10.1080/0020739X.2014.920534
15. Briggs K.M. A precise calculation of the Feigenbaum constants. Mathematics of Computation. 1991;57:435-439. https://doi.org/10.1090/S0025-5718-1991-1079009-6
16. Hertling P., Spandl Ch. Computing a Solution of Feigenbaum s Functional Equation in Polynomial Time. Logical Methods in Computer Science. 2014;10(4):1-9. https://doi.org/10.2168/LMCS-10(4:7)2014
17. Sekovanov V.S., Rybina L.B., Strunkina K.Y. The study of frames of Mandelbrot sets of polynomials of the second degree as a means of developing the originality of students' thinking. Vestnik of Kostroma State University. Series: Pedagogy. Psychology. Sociokinetics. 2019;25(4):193-199. (In Russ., abstract in Eng.) https://doi.org/10.34216/2073-1426-2019-25-4-193-199
18. Raba N.O. Realization of Algorithms of Quaternion Julia and Mandelbrot Sets Visualization. Differential Equations and Control Processes. 2007;(3):25-59. EDN: MVKEJD
19. Rosa A. Methods and Applications to Display Quaternion Julia Sets. Differential Equations and Control Processes. 2005;(4):1-22. EDN: VHRQLH
20. Hubbard J., Schleicher D., Sutherland S. How to find all roots of complex polynomials by Newton's method. Inventiones mathematicae. 2001;146(1):1-33. https://doi.org/10.1007/s002220100149
21. Schleicher D., Stoll R. Newton's method in practice: Finding all roots of polynomials of degree one million efficiently. Theoretical Computer Science. 2017;681:146-166. https://doi.org/10.1016/j.tcs.2017.03.025
22. Sekovanov V.S. On Julia sets of functions having fixed parabolic points. Fundamental and Applied Mathematics. 2021;23(4):163-176. (In Russ., abstract in Eng.) EDN: NZZZRV
23. Sekovanov V.S. Smooth Julia sets. Fundamental and Applied Mathematics. 2021;21(4):133-150. Available at: https://www.mathnet.ru/links/44c77fdf5b9365ae378626d1d37a5661/fpm1751.pdf (accessed 11.02.2023). (In Russ., abstract in Eng.)
24. Sekovanov V.S., Ivkov V.A., Piguzov A.A., Rybina L.B. Performing a Multi-Stage Mathematical and Informational Task "Dynamics of Iteration of Piecewise Linear Functions" as a Means of Developing Students Creativity. Modern Information Technologies and IT-Education. 2020;16(3):711-720. (In Russ., abstract in Eng.) https://doi.org/10.25559/SITITO.16.202003.711-72
25. Sekovanov V.S., Ivkov V.A., Piguzov A.A. Rybina L.B. Development of Thinking Flexibility of Students when Calculating the Feigenbaum Constant Using Information and Communication Technologies. Modern Information Technologies and IT-Education. 2021;17(2):415-422. (In Russ., abstract in Eng.) https://doi.org/10.25559/SITITO.17.202102.415-422
Published
2023-03-30
How to Cite
SEKOVANOV, Valeriy Sergeevich et al. Development of Creative Qualities of Students when Performing a Multi-Stage Mathematical and Information Tasks "Newton's Method for Complex Polynomials". Modern Information Technologies and IT-Education, [S.l.], v. 19, n. 1, p. 141-151, mar. 2023. ISSN 2411-1473. Available at: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/945>. Date accessed: 09 sep. 2025. doi: https://doi.org/10.25559/SITITO.019.202301.141-151.
Section
Educational resources and best practices of IT Education