The article examines the implementation of blended learning strategy in teacher training programs as an innovation in online learning. The blended learning idea comes from blending elements which use online technology with more traditional face-to-face teaching in the same course. The article analyses teacher training programs offered by Chernivtsi Regional Institute of Postgraduate Pedagogical Education. Additional data were gathered through a questionnaire administered to teachers who attended training courses. The characteristics of blended learning strategy, its benefits and limitations for teacher training are supported by a review of literature. The article closes with the comparison of curriculum components (content delivery, learner activities, materials, and required competences) between traditional and blended learning teacher training programs. Having obvious benefits in teacher training programs, the implementation of blended learning strategy sets some additional requirements to a learner, as well as to course instructors and lectors.

The global pandemic COVID-19, which began in 2020, has brought about new challenges to Ukraine’s higher and postgraduate education institutions. Despite the previous experience of providing online education courses, one of the main challenges to these institutions now remains to perform effective online teaching, which requires a proper theoretical and instructional basis. The study aims to outline efficient online teaching principles and best practices of how online teaching can provide effective interaction in the virtual online classroom with the aid of online learning management systems, online educational technologies, and digital learning tools. The article suggests a framework of twelve principles of effective online teaching determined for higher and postgraduate education that is deployed at five levels grouped according to specific online teaching issues (Level 0. Lecturer readiness, Level 1. Course planning, Level 2. Students’ facilitation, Level 3. Course content dissemination, and Level 4. Course feedback). Each identified principle at every level is covered with examples of specific teaching methods and best practices of how it is used to enhance the effectiveness of online teaching in higher and postgraduate education. To perform most effectively in an online education context, every lecturer should use each level from the developed framework and each principle from the level. However, even a partial application of certain principles will increase the efficiency of the online teaching process. The suggested five-level framework of twelve principles will be helpful for lecturers, professors, and other academic staff of universities and institutes of higher and postgraduate education when providing online and distance education.

The problem of synthesis of the optimal control for a stochastic dynamic system of a random structure with Poisson perturbations and Markov switching is solved. To determine the corresponding functions for Bellman functional and optimal control the system of ordinary differential equation is investigated.

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