The phenomenon of complex knowledge in teaching mathematics as a factor of mathematical literacy forming of school students

Introduction. Mathematics teaching based on the development of complex knowledge generalized constructs (for example, modern achievements in science) becomes an effective direction for the formation of school student’s mathematical literacy with a significant applied and mathematical-informational potential of personal development. The purpose of the study: to develop a technology for student’s mathematical literacy formation during the development of complex mathematical knowledge and in the context of universal educational actions actualization by means of mathematical and computer modeling. Materials and methods. The research materials are based on the historiogenesis and actualization of mastering processes of complex mathematical knowledge by students as an effective mechanism for personal development. A synergetic approach, digitalization tools and visual modeling methods are being implemented to adapt the mastering processes of complex knowledge to school mathematics with the effect of student’s mathematical literacy forming. The choice and justification of methods for personal experience founding create the effect of core actualization of universal educational actions, manifest themselves in the processes of students ' activities individualization. The results of the study. For the first time, a technology for student’s mathematical literacy formation based on the symbiosis of mathematical and computer modeling in mathematics development of complex knowledge has been developed. The founding clusters and research and adaptation technology of hierarchies of complex multi-level knowledge (including modern achievements in science) to school mathematics are constructed. The stages and means of visual modeling and personal experience founding with the effect of student’s mathematical literacy forming in a rich information and educational environment are clarified. Conclusion. Educational practices have shown the high efficiency of this method to school student’s mathematical literacy forming in the process of modern achievements mastering in science. Such didactic solutions and practices are characterized by the ability to fully meet the needs of each school student in self-education and self-actualization when complex knowledge constructs mastering and set the value imperative of personal development, including mathematical literacy.

The finding and researching algorithm for potentially oscillating enzymatic systems

Many processes in living organisms are subject to periodic oscillations at different hierarchical levels of their organization: from molecular-genetic to population and ecological. Oscillatory processes are responsible for cell cycles in both prokaryotes and eukaryotes, for circadian rhythms, for synchronous coupling of respiration with cardiac contractions, etc. Fluctuations in the numbers of organisms in natural populations can be caused by the populations’ own properties, their age structure, and ecological relationships with other species. Along with experimental approaches, mathematical and computer modeling is widely used to study oscillating biological systems. This paper presents classical mathematical models that describe oscillatory behavior in biological systems. Methods for the search for oscillatory molecular-genetic systems are presented by the example of their special case – oscillatory enzymatic systems. Factors influencing the cyclic dynamics in living systems, typical not only of the molecular-genetic level, but of higher levels of organization as well, are considered. Application of different ways to describe gene networks for modeling oscillatory molecular-genetic systems is considered, where the most important factor for the emergence of cyclic behavior is the presence of feedback. Techniques for finding potentially oscillatory enzymatic systems are presented. Using the method described in the article, we present and analyze, in a step-by-step manner, first the structural models (graphs) of gene networks and then the reconstruction of the mathematical models and computational experiments with them. Structural models are ideally suited for the tasks of an automatic search for potential oscillating contours (linked subgraphs), whose structure can correspond to the mathematical model of the molecular-genetic system that demonstrates oscillatory behavior in dynamics. At the same time, it is the numerical study of mathematical models for the selected contours that makes it possible to confirm the presence of stable limit cycles in them. As an example of application of the technology, a network of 300 metabolic reactions of the bacterium Escherichia coli was analyzed using mathematical and computer modeling tools. In particular, oscillatory behavior was shown for a loop whose reactions are part of the tryptophan biosynthesis pathway.

PROBLEMS OF BUILDING AND APPLICATION OF HYPERCOMPLEX NUMBERS

The article is dedicated to the problems of using multidimensional numbers for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices, including digital image processing. The emphasis is on the issues of building the methods for processing three-dimensional signals. It is proposed to use three-dimensional variables presented in the form of hypercomplex numbers to formulate the three-dimensional Fourier transformation forms, which allows to analyze and process three-dimensional signals.