Introduction. The era of post-industrial society of knowledge differs from the previous socio-economic period in many important aspects: total digitalisation, automation and robotisation, and furthermore, cross-disciplinary integration of various scientific fields, among which mathematics is of particular relevance. Over the last decades, the language, tools and methods of the digital era have emerged as the basic universal research tools in physics, biology, chemistry, and engineering, organisation of production and in many other theoretical and applied spheres of activity. Against the background of progressively advanced mathematisation, which covers larger functional intellectual space, it is extremely important to provide the fundamental, advancing nature of mathematical training of students, including in the first place the students of the pedagogical directions. The graduates of pedagogical higher education institutions will have not only to convey certain academic educational information, but to coordinate educational trajectories according to a training maxim in their lifetime that is hardly possible without understanding of basic mathematics.The aim of the present publication is to discuss methodological, culturological and didactic aspects of mathematical training of students of the pedagogical directions.Methodology and research methods. In the course of the work, the authors used the provisions and requirements of the new Federal State Educational Standards (FSES). The psychological and pedagogical framework of the research is based on the concepts of developing learning and upbringing, vocational and pedagogical orientation of training of future teachers and the idea of lifelong education. The analytical review and synthesis of the content of philosophical, mathematical, pedagogical, methodological literature and normative documents was conducted. The state of mathematical components of pedagogical education and its ways of fundamentalisation were considered according to the theory of training, comparative, culturological and system-based types of analysis.Results and scientific novelty. The authors note the imbalance between the fundamental mathematical preparation, which is relevant for a modern person, and the competency-based approach, which currently dominates in education and constraints to students’ holistic understanding of mathematical area of knowledge, even within the boundaries of a particular discipline. The situation is exacerbated by an avalanche distribution of information technologies. One of the negative consequences of this situation is replacement of meaningful learning of mathematics with “coaching” of school students and university students for formal test performance. To address these challenges, culturological educational models, which form common cultural ideas of modern mathematics according to the principle of a cultural conformity, have to be the predominant in education, but not competency-based attitudes focused on narrow specialisation. Mathematical modelling, discrete mathematics and computation processes are identified as the most significant components of a new stage of universal mathematical culture evolution. As a result of a digital era, general-education concepts have already become the key concepts of various sections of mathematics, therefore, it is necessary to start a general introduction to the study of these concepts at the stages of school education.It is highlighted that fundamentalisation of mathematical preparation plays the leading role in the development of logical and professional innovative thinking of students through the understanding of multifunctional structures and schemes (as means and knowledge methods) similar to mathematical structures and schemes. It is appropriately useful to include special courses into a variable part of the master’s programmes (according to education directions), implying students’ research activities: “Mathematical Modelling in Professional Education”, “Bases of Modern Mathematical Culture”, “Discrete Mathematics”, “Mathematical Bases of the System Analysis”, etc.The authors are convinced that the introduction of the concept of mathematical preparation, based on the culturological and system-based approaches, will help to resolve an apparent contradiction between, on the one hand, the necessary integration and fundamentalisation of knowledge acquired by students, and, on the other hand, the disciplinary dissociations of educational material, shortcomings of competency-based training and formal use of information technologies.Practical significance. The materials of the publication can be useful for future and practicing teachers of mathematics, computer science and related academic disciplines. Moreover, the present publication can be used by education professionals interested in improving the education quality.

Abstract. The spatial representation problems of the educational sphere, of the information educational milieu, media milieu are analyzed; the problems of the media education, of Smarteducation, media competence and media culture of education subjects are examined. The descriptions of the educational cyberspace and interaction characteristics with the educational subjects in the context of modal logic are offered.Educational space is a wide and multifaceted concept. It is an ordered representation of the real system, rather metasystem, education existing in a variety of forms and organizational structures. This is general educationsecondary, higher, tertiary and career development system, steadily expanding in modern information society. In the era of global informatization the distance learning becomes more available and in-demand. It is also different forms of self-education, provided by society's educational and socio-cultural resources and technologies of its informatization.The general education structure has been thoroughly studied and substantiated with reference to the current sets of objective factors: the education conditions, the requirements for it -scientific, social, ideological, cultural, economic, technological, etc., as well as based on trends in education and its intestinal necessity. Therefore, it is relatively invariant accurate to changing the conditions and requirements themselves by the corresponding evolution. Nevertheless, as the education is social, then the subjective factor, subjectivism occurs in its organization and development. There may be revolutionary changes, affecting not only the structure, but also education system architecture not always obtaining the identical degree of positivity.The spatial representation of the education sector is informative and, therefore, it is a part of the informational space in social and informational milieu. First, it includes the education subjects (carriers, distributors and recipients of educational information) forming the basic "dimension" of informational and educational space. The second "dimension" is built up by teaching techniques, information and educational resources, electronic and educational resources (IER and EER), the media milieu resources. The third "dimension" is built up by electronic technologies and communications. Each of these dimensions has its own representation in a variety of constant and variable "dimensions" with own relevance.No less important "dimension" of the space is the science of education, pedagogy, and didactics. In the era of IT development when computer science concepts become general scientific notions, then they become a part of the pedagogics, subject methods. The education methodology becomes informational, pedagogic systems are actualized in pedagogical techniques. Consequently, the personal information culture including its information security culture, media competence and media culture turns into the subjects of pedagogics.Another structure of the information educational space is a representation...

Cloud technologies are one of the manifestations of smart technologies, a product of Smart society, which many people consider to be the state of modern society. The development of Smart education implies the expansion of the area of cloud technologies use in education. Mobile learning provides the opportunity to introduce the interactive forms of learning in the educational processes to provide easier and faster access to knowledge, on the one hand, and the formation of the competence of proficiency in modern ITtechnologies for solving the communication tasks, on the other hand. The article discusses the structure and content of cloud space, its communication with Smart environment and Smart education. Cloud technology can be used both by teachers and students in the classroom and in extracurricular learning activities, in work with the information-educational environment, with media resources. The authors discuss the content and problems of educational work with cloud technologies in teachers and students' cooperation; the content of the required competencies of the modern teacher.

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