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A R T I C L E I N F O A B S T R A C T Article history: Received: 30 August, 2020 Accepted: 09 January, 2021 Online: 22 January, 2021 In this paper, one of the "problem zones" in mathematics concerned with the surface area of three-dimensional bodies is presented by computer and mathematical modeling with synergetic effects using intelligent system support. The problems of personal selforganization and technological stages of student’s research activity in the process of modern achievements in science adaptation in teaching mathematics by means of computer and mathematical modeling based on intelligent system support are considered. The technology of intelligent system support to choose and investigate the complex generalized constructs in student’s research activity (on the example of multifaceted surfaces of Schwartz cylinder (or "boot")) using computer and mathematical support are considered. So, the technological stages and personal quality parameters of student’s research activity and content of adaptation the most important generalized constructions of surface area to the actual positions of mathematical knowledge and fractal parameters are considered with synergetic effects and extend Vygotsky’ zones of immediate development as a finding. Intelligent management of student’s cognitive activity on the base of expert systems during the complex knowledge development creates the zones of immediate development and selfdevelopment of students as a new methodology of cognitive student’s activity. The original computer design and mathematical modeling of nonlinear dynamics of synergetic effects manifestation are developed step by step and dynamic invariants are revealed in the course of surface area mastering as a realization of student’s educational trajectory. The model of hybrid intelligent system for supporting of student’s research activities is constructed. Details of concretization are described on the example of Schwartz cylinder so it is fractal complex surface with problems zones of student’s understanding on complex concept “surface area”. Moreover, all constructs of mathematical and computer modeling with Schwartz cylinder are new and based on means of the QT Creator environment and GeoGebra. Namely, the means of the QT Creator environment are used for computer design of non-linear dynamics of functional and fractal parameters and technological constructs and students reveal the regularities and irregularities of lateral surface approximations of Schwartz cylinder. The construction of Schwartz cylinder’s fractal surface as an indirect synergetic effect of student’s research activities is new mathematical and computer design result. Conclusion: new methods of intelligent management of student’s research activity and thinking development using neural networks support was carried out on the basis of lateral surface triangulations of the Schwartz’ cylinder with regular and irregular limiting grinding by layered polyhedral complexes with computer and mathematical modeling tools and as one of the educational trajectories of student’s research activity.

Technological Stages of Schwartz Cylinder’s Computer and Mathematics Design using Intelligent System Support