This article focuses on the problem of implementing the integrated learning of Mathematics and English by future Mathematics teachers based on the Content and Language Integrated Learning (CLIL) model. Our review of the literature confirms that this method is considered to be an effective approach to the formation and development of students' subject knowledge and foreign language skills. This technology is significant for the professional training of future Mathematics teachers since they should be competent in both Mathematics and foreign language to effectively teach today's schoolchildren. A student questionnaire was conducted and its analysis shows that students believe that CLIL will benefit them in the process of their professional training. Since the CLIL approach is multidimensional process, its implementation requires thorough preparation. This paper demonstrates and describes some specific steps in the preparatory stage for implementing the CLIL approach for the professional training of future Mathematics teachers: determining mathematical course, choosing its specific themes (units), planning for expected results, designing the organizational features of the educational process using the CLIL method and designing the appropriate teaching and methodological support. The possible variant for designing the preparatory stage for the integrated teaching and learning of "Number Theory" and English is suggested. It is demonstrated that the Mathematical and foreign language component of the expected educational outcomes need to be distinguished. The educational process should involve a special system of multilevel bilingual assignments. It also should have some special preconditions and joint activities of Mathematics and English teachers.

The article analyzes the results of determination of students'-future math teachers' beliefs about the state of their preparation for future professional activities. The measurement and qualitative analysis of students' beliefs concerning their future profession is an important pedagogical problem. The measurement analysis of Ukrainian students' value beliefs (future Mathematics teachers) with the help of the methods proposed in the TEDS-M project are compared with the results of analogical investigations in Russia. The beliefs about the nature of mathematics; beliefs about learning mathematics and students' mathematics abilities; students' beliefs about their preparedness level to professional activity were compared. Also "problem zones" for Ukrainian students' value beliefs were reviled.

This article considers the issue of on-line training of Master students, majoring in Mathematics (Speciality code in Ukrainian educational system "014.04 Secondary Education. Mathematics") and preparing them for teaching at University. The research papers and resources, focusing on the vocational bent of on-line training of undergraduate students majoring in Mathematics have been analyzed in this paper. The present paper describes designing the structure and components for the on-line course "Methods for Teaching Mathematics to Students at Technical Universities", which is in open access on the educational platform "Higher School Mathematics Teacher". This article presents the analysis of the survey results, in which 107 students in their first and second year of studying took part, which allowed defining the content of the course. The model is presented as a polyhedron, vertices, faces and edges of which show structural elements of the model and relations between them. The paper also introduces the analysis of the internship syllabi at Ukrainian universities and aims at defining the criteria for assessing learning performance of the on-line course users. It also considers the impact of the course on the efficiency of training 112 Master students in their second year of study, when they do their assistant internship at technical universities. The qualitative and quantitative analysis of the results of the experiment proved the efficiency of introducing the course into the process of practical training of Master students. The interns who joined the course, showed a better understanding of how to motivate students when teaching Mathematics to them. They also showed better results in applying active learning tools and techniques for teaching Mathematics, as well as in managing the audience through information technologies and communication strategies.

No abstract

The article considers the certain features of the development of learners' logical reasoning in classes with different subject-specialization at secondary school. A short author's review of literature suggests that three main directions have been formed in the theory of teaching mathematics in the last 50-60 years in Ukraine. The author investigates them in brief. The analysis shows that the current approach to forming pupils' logical thinking, significantly varies depending on the subject-specialization and the level of mathematics training. More detailed investigation of particular methods and techniques of teaching the elements of mathematical logic in classes with advanced study of mathematics are given in the article.