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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Article
Research Synergy of Multi-Stage Mathematical and Informational Tasks on Fractal Geometry
Author(s)
Valery S. Sekovanov
Eugeny I. Smirnov
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DOI:10.17265/2161-623X/2017.08.003
Affiliation(s)
Kostroma State University (KSU), Kostroma, Russia
Yaroslavl State Pedagogical University, Yaroslavl, Russia
ABSTRACT
The paper emphasized the importance
of students’ creativity development based on the principle of variability and synergetic
approach. The basic characteristics of the variability principle are also presented.
The contribution of Russian teachers to studying the concepts of “mental flexibility”
and the “principle of variability” is noted. This paper presents the ways of students’
mental flexibility development when elaborating the algorithms of constructing the
Julia sets with numerous applications in various areas of the real world in different
environments. The Julia sets of quadratic polynomials are considered first and the
algorithms of constructing the Julia sets are proposed. During studying
Chebyshev polynomials and elaborating their Julia sets, the possibility of developing
students’ mental flexibility is substantiated on the specific problems. Attention
to the filling Julia sets is given. The algorithms
of constructing the Julia sets of rational functions are considered and the examples
are given. A description of the algorithms
is given with the aid of computer program. The parameter values, at which the Julia sets become smooth for quadratic polynomials,
are specified. It is established that the segments of a real straight line are the
Julia sets of Chebyshev polynomials. The examples of the Julia sets with a fractal
structure are given. The relationship of
the Julia sets with random maps is indicated. Construction of the Julia sets provides an opportunity to develop students’ mental
flexibility—one of the most important creative qualities of personality.
KEYWORDS
creativity, Julia set, chaotic displaying, mental flexibility, multi-stage mathematical and informational task, attractor
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