The article describes the problem of preparing a modern teacher of mathematical disciplines for a higher technical educational institution in the system of higher pedagogical and classical education. The modern scientific research focusing on training a highly-qualified mathematics teacher has been analyzed. The results of the questionnaire for mathematics teachers at higher technical education institutions and students studying in 014 'Secondary Education (Mathematics)' specialties are presented. The current state of preparedness of master's degree holders to teach mathematicians at higher technical educational institutions is evaluated. One of the ways to solve the problem is to develop the 'For Higher School Mathematics Teacher' educational platform. The concept of developing this online environment and filling its components with content has been highlighted. The rationale for creating a community of practitioners consisting of teachers of higher technical and pedagogical educational institutions, the cooperation of which should contribute to solving the problem, is provided. The possibilities of using the platform for further training of both young and experienced teachers are considered.

In the era of the fourth industrial revolution – Industry 4.0 – developing key competences (digital, multilingual and mathematical competences in particular) is of paramount importance. The purpose of this work is to investigate the content of key competences of a secondary school student and to develop a method of teaching for the integrated development of multilingual and mathematical competences in the process of teaching Programming Basics with the help of distant technologies. The objectives of the research include generalizing and systematizing theoretical data on the structure and the content of key competences and the potential of informatics lessons for the development of separate components of multilingual and mathematical competences; generalizing and systematizing theoretical data on the ways of arranging distant support for informatics learning, Programming Basics in particular; to investigate the content and the methods of teaching Programming Basics in 7th-11th grades; to develop the e-learning Moodle course using Python for Programming Basics on the principles of integrated approach to developing separate components of multilingual and mathematical competence with determining some methodical special features while using it. The object of the study is to teach informatics to junior high school and high school students. The subject of the study is the means and the methods of realizing distant support in the process of teaching Programming Basics using Python on the principles of an integrated approach to developing multilingual and mathematical competences.

The article deals with the problem of introducing cloud calculations into 10th–11th graders’ training to solve optimization problems in the context of the STEM-education concept. After analyzing existing programmes of optional courses on optimization problems, the programme of the optional course Optimization Problems has been developed and substantiated implying solution of problems by the cloud environment CoCalc. It is a routine calculating operation and not a mathematical model that is accentuated in the programme. It allows considering more problems which are close to reality without adapting the material while training 10th–11th graders. Besides, the mathematical apparatus of the course which is partially known to students as the knowledge acquired from such mathematics sections as the theory of probability, mathematical statistics, mathematical analysis and linear algebra is enough to master the suggested course. The developed course deals with a whole class of problems of conventional optimization which vary greatly. They can be associated with designing devices and technological processes, distributing limited resources and planning business functioning as well as with everyday problems of people. Devices, processes and situations to which a model of optimization problem is applied are called optimization problems. Optimization methods enable optimal solutions for mathematical models. The developed course is noted for building mathematical models and defining a method to be applied to finding an efficient solution.

The article addresses the issue of implementing the usability principles of educational internet resources. The paper debates the latest researches on the question concerning the search for the factors that influence the results of online education. The analysis, which we carried out, allowed us to focus on such known six criteria of usability design as Information Quality, System Navigation, System Learnability, Visual Design, Instructional Assessment, and System Interactivity and suggest the existence of the seventh criterion named Responsiveness. The research considers the principles of usability implementation following the example of the open platform of online education "Higher School Mathematics Teacher". The answers given by 203 respondents during the survey allowed defining the direction of implementing the usability criteria on the platform. We were eager to know the opinion of teachers and students, who became the first users of the platform. The article discusses the criteria implementation while developing online courses on the platform. There was ground to conclude that when designing on-line platform courses, all seven usability subcategories are important.

This paper is aimed at creating a model of a Personal e-Learning Environment (PLE) of a University Mathematics teacher. A special emphasis is put on the analysis of the approaches to understanding the notion Personal e-Learning Environment (PLE) and studies into creating a PLE model. The Deductive Content Analysis of the research papers and other academic sources in this article allowed defining the structure of the PLE model of University Mathematics teachers. The authors of the present paper see the bottom line of the modelling process as giving content to the PLE according to the types of teaching activities, which Mathematics teachers do to perform the Instructive, Cognitive, Designing, Planning, Social, Managerial roles. With the help of The Inductive Content Analysis Method the researchers in this paper analysed the data on ranking web-services. The data were taken from the web-resources and helped to select web 2.0 tools, which can be used to give content to the PLE. The conducted research made it possible to conclude that the idea to develop a PLE model of a teacher is based on such factors as the ranking of web-tools, IK competency of a teacher and his/her willingness to develop professionally. The authors of the present article assessed the answers of the academic staff at Ukrainian Universities concerning their level of awareness about using web-tools in teaching. The analysis of the results proved the insufficient level of such awareness and justified the relevance of designing an on-line course in order to support a Mathematics teacher in creating and using Personal e-Learning Environment.