Students' mathematical training for professional self-realization in a modern, inventive society

In the framework of their professional self-realization on the contemporary job market, the paper explores the issues with professionals' training for mathematics at universities. The structure of students' mathematical proficiency is taken into consideration, and its key elements are substantively presented (the ability to ask and answer questions in and with mathematics, the ability to deal with mathematical language and tools). The initial (ascertaining) step of the pedagogical experiment allowed for the establishment of the level of students' mathematical proficiency and formation in accordance with the predetermined structure. It has been demonstrated that students, both of mathematical and non-mathematical majors, lack the mathematical knowledge and abilities necessary to solve complicated issues, particularly those involving modeling in the context of future professional engagement. The following fundamental strategies for developing students' mathematical competence are supported: research-focused instruction, an interdisciplinary approach with a practical focus, and the use of digital learning environments and related resources. The implementation strategies for these techniques are described in the authors' optional course "Methods of formation of students' mathematical competence for the future successful career" for math specialist students. The research will continue with the introduction of the aforementioned discipline to the educational system, which will include the subsequent assessment of future teachers' readiness to employ the suggested methods in their professional work.In the framework of their professional self-realization on the contemporary job market, the paper explores the issues with professionals' training for mathematics at universities. The structure of students' mathematical proficiency is taken into consideration, and its key elements are substantively presented (the ability to ask and answer questions in and with mathematics, the ability to deal with mathematical language and tools). The initial (ascertaining) step of the pedagogical experiment allowed for the establishment of the level of students' mathematical proficiency and formation in accordance with the predetermined structure. It has been demonstrated that students, both of mathematical and non-mathematical majors, lack the mathematical knowledge and abilities necessary to solve complicated issues, particularly those involving modeling in the context of future professional engagement. The following fundamental strategies for developing students' mathematical competence are supported: research-focused instruction, an interdisciplinary approach with a practical focus, and the use of digital learning environments and related resources. The implementation strategies for these techniques are described in the authors' optional course "Methods of formation of students' mathematical competence for the future successful career" for math specialist students. The research will continue with the introduction of the aforementioned discipline to the educational system, which will include the subsequent assessment of future teachers' readiness to employ the suggested methods in their professional work

Problem оf Function Minimization іn Theory оf Management

В останні десятиліття теорія оптимального керування інтенсивно розвивається, що пояснюється не лише наявністю складних і цікавих суто математичних проблем, а й широким спектром прикладних задач у різних галузях науки і людської діяльності: фізиці, економіці, біології, екології, медицині, енергетиці та ін. Нові наукові (теоретичні) й реальні (прикладні) задачі відрізняються своєю складністю, що зумовлює не лише розширення сфери застосування математичного моделювання, а й удосконалення самих моделей у напрямі більшої їх точності та повноцінності. Особливо гостро в сучасних умовах стрімкого розвитку науки, техніки, інформаційних технологій постає проблема керованості системи (процесу). Як відомо, кожна задача оптимального керування містить такі складові: 1) математичну модель об’єкта керування; 2) мету керування (т. зв. критерій якості); 3) певні обмеження на стан (траєкторію) системи, тривалість процесу керування та ін., при яких має бути забезпечена мета керування. Пропонована стаття присвячена одній із оптимізаційних задач математичної теорії керування, у якій еволюційний процес описується лінійними диференціальними рівняннями, а функція керування задається невласним інтегралом.In recent decades, optimal control theory is intensively developed, which is explained not only by the presence of complex and interesting pure mathematical problem, but also a wide range of applied problems in various fields of science and human activity: physics, Economics, biology, ecology, medicine, energy etc. New scientific (theoretical) and real (application) tasks differ in their complexity, resulting in not only expanding the scope of mathematical modeling and improve the models themselves in the direction of greater their accuracy and usefulness. Particularly acute in modern conditions of rapid development of science, technology, information technology arises the problem of controllability of the system (process). As you know, each optimal control problem contains the following components: 1) a mathematical model of control object; 2) goal management (T. N. The quality criterion); 3) constraints on the state (trajectory) of the system, the process time control, etc., which must be provided for the purpose of control. The article is devoted to one of the optimization problems of mathematical control theory in which the evolutionary process is described by linear differential equations and the control function is specified by an improper integral

The Problems Role in the Formation of the Mathematical Competence of Schoolchildren

Закон України «Про освіту» визначає ключові компетентності, необхідні кожній сучасній людині для успішної життєдіяльності (стаття 12). Серед них – на чільному місці математична компетентність. Державний стандарт шкільної математичної освіти основною метою і завданням визначає формування в учнів математичної компетентності на рівні, достатньому для забезпечення життєдіяльності в сучасному світі, успішного оволодіння знаннями з інших освітніх галузей у процесі шкільного навчання, забезпечення інтелектуального розвитку учнів, розвитку їх уваги, пам’яті, логіки, культури мислення та інтуїції [1]. Зокрема, як зазначається у пояснювальній записці навчальної програми з математики для учнів 10–11 класів загальноосвітніх навчальних закладів (рівень стандарту), щоб бути успішним в сучасному суспільному житті, треба володіти певними прийомами математичної діяльності та навичками їх застосування до розв’язування практичних задач. А без доброї шкільної математичної підготовки сьогодні неможливо продовжити навчання на наступних етапах в багатьох галузях, отримати якісну професійну освіту, стати фахівцем, здатним до математичного моделювання в різних сферах, щоб бути затребуваним на ринку праці [2]. У пропонованій статті розглянуто зміст математичної компетентності учня сучасної школи, висловлено і, на основі існуючих досліджень (зокрема й власних) та власного досвіду, аргументовано точку зору про провідну роль математичних задач у її формуванні. Розглянуто типи задач, які якнайкраще надаються для досягнення зазначеної мети. До них, зокрема належать: задачі на доведення; геометричні задачі на побудову; так звані «цікаві» задачі або задачі з нестандартним змістом; компетентнісно-орієнтовані задачі або задачі з практичним змістом, найчастіше, з нематематичної галузі. Наведено деякі методичні рекомендації для учителів та приклади задач.The Law of Ukraine "On Education" defines the key competencies that are necessary for every modern person to succeed (Article 12). Among them - at the forefront the mathematical competence. The state standard of school mathematical education determines the formation of students' mathematical competence at a level sufficient for life in the modern world, the successful acquisition of knowledge from other educational branches in the process of school education, ensuring the intellectual development of students, the development of their attention, memory, logic, culture of thinking and intuition [1]. In particular, as stated in the explanatory memorandum of the curriculum for Maths for students of grades 10-11 of general education institutions (standard level), in order to be successful in modern social life, one must possess certain techniques of mathematical activity and skills of their application while solving practical problems. And without good school mathematical training today it is impossible to continue education in the following stages in many industries, receive high-quality professional education, become a specialist capable of mathematical modeling in various fields in order to be in demand on the labor market [2]. The article deals with the content of the mathematical competence of the student of a modern school, expressed and, on the basis of existing researches (including own ones) and own experience, the point of view on the leading role of mathematical problems in its formation is argued. The types of tasks that are best suited to achieve this goal are considered. These include, in particular, the problems for proof; geometric problems for construction; so-called "interesting" problems or problems with nonstandard content; competency-oriented problems or problems with practical content, most often, from non-mathematical field. Some methodological recommendations for teachers and examples of problems are given

Experience in Implementing IBME at the Borys Grinchenko Kyiv University

In this chapter, we learn that an educational community was formed at the university, including colleagues from several universities, to address low motivation of students in choosing mathematics programmes and to share understand�ings of IBME. This Community of Inquiry (CoI) addressed issues related to conceptual versus procedural learning approaches drawing on a range of literature. PLATINUM members led the CoI in suggesting inquiry-based approaches to teaching and learning and one member led a course in Mathematical Analysis to enable the community to observe and address processes and issues. An open questioning approach was taken by this teacher with encouragement for students to address the questions and to ask their own questions. Extracts from the teaching and examples of students’ responses suggested that teaching had motivated students and engaged their interest, thus also motivating the CoI to engage further with IBME approaches

Formation of High School Students’ Resistance to Destructive Information Influences

Today, information attacks occupy a prominent place in hybrid wars and their influence and significance are not inferior to armed aggression. Furthermore, information warfare begins long before the invading tanks and launching missiles. It continues during the so-called “hot” war, taking on more and more “sophisticated” forms, using a wide range of tools, and improving the distribution network. One of the primary targets of information attacks is civil society, which, being misinformed, disoriented, and processed by enemy propaganda, becomes an enemy ally in the “hot” phase of the war. Therefore, strengthening society’s resilience to information threats is an urgent issue for global security. This article analyzes the experience of European countries, in particular, Eastern Europe and the Baltic States, which shows that citizens’ immunity to disinformation and hostile propaganda can be strengthened through: their media and information education; development of critical thinking; and civic education. As a result of theoretical analysis of scientific sources and the results of a questionnaire survey of Borys Grinchenko Kyiv University students, the course “Counter-manipulation Strategies in Information Security” was substantiated and developed to build university students’ resistance to destructive information influences

Inquiry in University Mathematics Teaching and Learning

The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students` own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of “modelling” in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

Inquiry in University Mathematics Teaching and Learning. The Platinum Project

The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students‘ own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of „modelling“ in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

Онтологическое моделирование интеллектуальных систем обучения с элементами геймификации

An ontological model of an intelligent learning system with elements of gamification has been built, which captures and structures knowledge common to the corresponding subject area. This allows you to reuse it as the basis of a single knowledge model, which ensures logical consistency between individual ontologies when combined to create a training course with a wider list of topics and tasks. The application of the ontological approach is a very effective way to design intelligent learning systems. The constructed separate ontological models (for topics, training courses, etc.) contribute to the design of a unified information learning environment within which intelligent learning systems can operate using gamification elements. Ontological modeling of intelligent learning systems based on multidimensional models is proposed. The proposed approach allows the development of an infological model of any learning system (information or intelligence), which fully reflects the pragmatics of the studied subject area.Побудовано онтологічну модель інтелектуальної навчальної системи з елементами гейміфікації, яка фіксує та структурує знання, спільні для відповідної предметної області. Це дозволяє повторно використовувати його як основу єдиної моделі знань, що забезпечує логічну узгодженість між окремими онтологіями при поєднанні для створення навчального курсу з ширшим переліком тем і завдань. Застосування онтологічного підходу є дуже ефективним способом проектування інтелектуальних систем навчання. Побудовані окремі онтологічні моделі (для тем, навчальних курсів тощо) сприяють розробці єдиного інформаційного навчального середовища, в якому можуть працювати інтелектуальні навчальні системи з використанням елементів гейміфікації. Запропоновано онтологічне моделювання інтелектуальних систем навчання на основі багатовимірних моделей. Запропонований підхід дозволяє розробити інфологічну модель будь-якої системи навчання (інформаційної чи інтелектуальної), яка повністю відображає прагматику досліджуваної предметної області.Построена онтологическая модель интеллектуальной системы обучения с элементами геймификации, которая фиксирует и структурирует знания, общие для соответствующей предметной области. Это позволяет повторно использовать его как основу единой модели знаний, что обеспечивает логическую согласованность между отдельными онтологиями при объединении для создания учебного курса с более широким перечнем тем и задач. Применение онтологического подхода — очень эффективный способ проектирования интеллектуальных систем обучения. Построенные отдельные онтологические модели (по темам, учебным курсам и т.п.) способствуют проектированию единой информационной среды обучения, в которой могут работать интеллектуальные системы обучения с использованием элементов геймификации. Предложено онтологическое моделирование интеллектуальных обучающих систем на основе многомерных моделей. Предложенный подход позволяет разработать инфологическую модель любой системы обучения (информационной или интеллектуальной), которая полностью отражает прагматику изучаемой предметной области

Mathematical preparation of students for their professional self-realization in modern innovative society

The article analyzes the problems of mathematical university preparation of specialists in the context of their professional self-realization in the modern labour market. The structure of mathematical competence of students is considered, its most important components are substantively described (the ability to ask and answer questions in and with mathematics, the ability to deal with mathematical language and tools). As a result of the first (ascertaining) stage of pedagogical experiment allowed to establish the level of students’ mathematical competence formation according to the specified structure was established. It is proved that students of both mathematical and non-mathematical majors have poor mathematical knowledge and skills for solving problems that require their complex application, in particular, modeling problems in the context of future professional activity. The basic approaches to formation of mathematical competence of students are substantiated: research-oriented teaching; interdisciplinary approach and practical orientation; use of digital learning environments and related tools. The methods of implementation of these approaches are disclosed in the authors’ elective discipline for the students of mathematical specialty “Methods of formation of students’ mathematical competence for the future successful career”. The continuation of the research will be the introduction of the discipline above to the educational process which contains the next monitoring of future teachers’ readiness to use indicated approaches in their professional activity