Mathematical Training Research Articles

Solving the problem of insuffi cient continuity of school and university education in the fi eld of mathematics (experience of Ural Federal University)

The relevance of the article is determined by the insufficient continuity of school and university mathematical training of modern students studying engineering. The purpose of the article is to develop adaptive educational materials for studying the basics of mathematical statistics by students of the Institute of Construction and Architecture of Ural Federal University. Practical skills formed in the process of applying the manual are used by students in the processing of experimental data in the course of preparing final qualification works and master’s theses. As part of the development of the manual, trends in changes in various systems of engineering and construction education were studied, an analysis of curricula and manuals for the preparation of students of architectural and construction training was carried out. The manual discusses frequently used statistical methods, the study of which ends with a step-by-step algorithm for solving a specific numerical problem, implemented by using the MS Excel spreadsheet.

Проведення аналогій у процесі формування математичних понять у студентів технічних спеціальностей

The article demonstrates the idea of applying the method of analogies in the process of forming mathematical concepts among students of technical specialties. Different approaches to the interpretation of the process of making analogies in the educational process are analyzed. Based on the analysis of scientific studies devoted to the specified issue, the method of conducting analogies is defined as a method of scientific knowledge that allows you to form conclusions about the researched object by comparing it with an already known object. Among other things, the main consequence of mental actions when conducting an analogy is an inference – the result of conducting an analogy, which allows you to draw a conclusion about the similarity or difference of the features (elements) of the studied objects or subjects, as well as their properties. Therefore, the result of analogizing (making an analogy) is students' acquisition of new knowledge about an unknown object (subject).
 The method of analogies has two important functions: the transition from a known information object to an unknown one, and the transition from a “simple” information object to its “complex” counterpart. An important feature of the analogy method is its potential for application in the formation of mathematical concepts. Thus, taking into account the dialectical unity of the functional properties of the method of analogies (explanatory and searching functions), the explanatory function of the method of analogies, in particular when forming concepts, is realized by building models for carrying out analogies, which will become the basis for forming an idea of a specific concept (object), searching the function of the method of analogies during the formation of concepts is based on the fact that students carry out an analysis and synthesis of the received information, build hypotheses regarding a specific information structure (concept).
 The purpose of the article is to reveal the essence of the method of analogies, to demonstrate its implementation in the process of forming mathematical concepts among students of technical specialties.
 The conducted study of the method of analogies allows us to characterize the basic factors of its implementation in the educational process.
 Given the multifacetedness of the chosen research topic, the prospects for further research include theoretical and practical developments in the implementation of the concept of the method of conducting analogies in the mathematical training of future technical specialists.

Цифровые инструменты визуализации в подготовке студентов экономических вузов

The study is devoted to the issue of improving the quality of training of students of economic universities using visualization tools for educational material. The importance for future economists of the ability to formalize economic problems is emphasized. An important role is given to graphical models as a means of visualizing quantitative data. t is noted that within the framework of mathematical training, students of economic universities study the basics of differential and integral calculus, which, in turn, allows them to explore, analyze and interpret a wide range of graphical models. The advantages of digital technologies for the visualization of educational material and the study of dynamic economic and mathematical models in the process of mathematical training of students of an economic university are considered. The capabilities of the MS Excel spreadsheet processor and the Python programming language are demonstrated, which make it possible to formalize specific economic problems and perform an intra-model solution. An online survey of students of economic universities in the city of Omsk was conducted on the problematic aspects of using digital tools for visualizing economic and mathematical models. The findings indicate the need for further development of integrative links between the disciplines of the subject areas “Economics”, “Mathematics”, “Computer Science and Information Technology” in the context of the digitalization of modern society.

Проблемний метод у системі фундаменталізацї математичної підготовки майбутніх технічних фахівців

The article presents the idea of applying the problem method in the system of fundamentalization ofmathematical training of future technical specialists. The stages of application of the problem method in theprocess of mathematical training of future technical specialists are analyzed, the main advantages anddisadvantages of using the proposed method in the educational process are given, the main stages of theapplication of the problem method in the educational process during the mathematical training of future48technical specialists are given. The work demonstrates an example of the application of the problem methodduring the study of higher mathematics topics by future technical specialists.The purpose of the article is to reveal the essence of the problematic method in the process offundamentalizing the mathematical training of future technical specialists,The conducted study of the problematic method and its implementation in the system offundamentalization of mathematical training of future technical specialists provides grounds for building ascheme for its implementation in the educational process. It consists of the following stages: receiving a taskcontaining a problem; finding ways to solve the problem, realizing the limitations of available resources andinformation for solving the task, performing the task taking into account the new knowledge obtained.The advantages of implementing the problem-based method in the educational process include: focusingstudents' attention on the problem, activation of students' mental activity, independent search for ways to solvethe problem, better memorization of ways to solve the problem compared to the traditional explanation of thematerial.Taking into account the multifacetedness of the chosen research topic, future explorations include: theneed for further theoretical and practical research, as well as the development of the concept of implementinga problem-based method in the process of fundamentalizing mathematical training.

Interdisciplinary connections of Mathematics and Literature in the preparation for External Independent Assessment of Humanities students

This article examines the issue of finding new methodological approaches to the development of problems that can contribute to the preparation of ‘Humanities’ students to pass the state final certification in the format of External independent assessment (EIA). The study analyzes the existing development of methods of qualitative mathematical training of students of humanities classes who study Mathematics according to the basic level program. In particular, the authors of the article were interested in the use of interdisciplinary connections between Mathematics and humanitarian disciplines. The authors of the article identified cross-curricular problems as a means of forming students’ motivation to study Mathematics, as well as a toolkit for measuring their level of readiness to solve typical problems of EIA. The researchers set a goal to develop cross-curricular problems following the EIA program based on the plot of Bulgakov’s novel ‘The Master and Margarita’. An experiment was conducted to confirm the effectiveness of the use of the system of interdisciplinary problems in the preparation for the EIA of humanitarian students. The obtained results confirmed that the implementation of problems in which the intersubject connections of Mathematics and Literature are realized has a more significant effect on both increasing the motivation of students to study Mathematics and their mathematical preparation.

Assessment of pre-arithmetic relations in children and adolescents with visual impairment

Mathematics training is a challenge for students with visual impairments (VI). However, there are few protocols to assess math skills for them, especially the content related to counting and measurement. The objective of this article was to develop and test the applicability of a protocol to assess math skills in counting and measuring for children and adolescents with VI. The work was organized into two studies. The first dealt with the development of a protocol for assessing pre-arithmetic skills and knowledge and the second study assessed its applicability. In the first study, the protocol was developed from the few papers found on a literature review. In the second study, the protocol was applied to 12 participants with VI aged 5 and 16 years. The results showed that 8 of the 12 participants had performance ranging from 83% to 100% of correct responses in the protocol application, both for counting and measuring skills. The protocol applicability and difference in participants’ performance are discussed. The study concludes that the protocol can be used as a tool to assess math knowledge for children at the end of Early Childhood Education or at the beginning of Elementary School, with or without VI.

Computer simulation as one of the ways to increase the efficiency of training in higher mathematics in a technical university

Introduction. The article considers one of the ways to improve the effectiveness of teaching higher mathematics taught at a technical university using the Maple 18 computer program as a computer simulation. The use of procedures, some functions of the program and materials increases visibility, however, in our opinion, there are other methods for improving the effectiveness of teaching, which are discussed in pedagogical practice, but nevertheless are not widely used in the classroom. The research methods of this work are the analysis of the learning outcomes of students of technical universities and teachers taught this subject using computer modeling, who are interested in improving the effectiveness of learning and teaching using the Maple 18 computer program. The basis of this study is the process of mastering knowledge in higher mathematics of students of engineers of a technical university using computer simulation. The use of computer models, including the Maple 18 computer program, showed the best results in higher mathematics classes and, with the use of modeling, had a positive effect on the quality of teaching knowledge of engineering students and the results of assessing their learning were observed for several years. The most important stages of computer modeling and research questions with engineers in order to check the mastery of the subject in the classroom of higher mathematics are considered. To increase the effectiveness of training and the use of computer simulation, including the Maple 18 program, it is advisable to switch to an active learning method, for which it is advisable to use computer simulation and the Maple 18 computer program.

Fundamentalisation of professional training of future teachers of mathematics and computer science in the conditions of digitalisation

Relevance. Teachers of mathematics and computer science play an important role in shaping the fundamental knowledge of students. This knowledge manifests itself in the form of in-depth theoretical knowledge, readiness to apply it in practice, critical attitude to information, creativity in their actions. In the conditions of digital transformation, fundamental knowledge enables an individual to take an active part in the life of the modern society and generate new decisions in unfamiliar situations. The aim of the article is to identify due strategies for professional training of mathematics and computer science teachers in the conditions of digitalisation of education, those who are ready to implement the educational process at school with a focus on transfer of fundamental knowledge. Methodology and methods. The methodological basis of the research involves: systemic approach in defining the new role of the teacher as a subject of digital educational ecosystems; environmental approach in characterising the potential of digital educational environment; activity-based approach in training future teachers of mathematics and computer science towards enhancing their ability to form a system of fundamental knowledge in students. The results of the study are presented by describing digitalisation as a modern trend towards modernisation of education, defining the essence of fundamentalisation of education, establishing the interrelation between these processes. The priorities of digital educational space helping to improve the work of mathematics and computer science teachers aimed at the formation of students’ system of fundamental knowledge, to make it more efficient, rational, dynamic and creative, include: continuous online support of educational process, availability of high-quality verified and variable educational content, personalisation of educational trajectory of each student, operational feedback and monitoring of learners’ academic achievements, flexibility of educational programmes, due potential to create sustainable motivation, support of students’ involvement in active learning through the use of new educational tools (virtual simulators and laboratories), interactivity, multimediality and hypertextuality of the environment, automation of teacher’s routine activities, etc. Conclusion. Fundamentalisation of education is viewed as a process of formation of solid, profound basic knowledge defining a person’s scientific worldview and allowing for integration of this knowledge through interdisciplinary links, enabling a person to continuously engage in self-education. The presented results can be used for the development of educational programmes of pedagogical baccalaureate, programmes of supplementary professional education providing for due training of teachers of mathematics and computer science, development of their readiness to form a system of fundamental knowledge in students relying of the digital learning environment resource.

Satisfaction of Postgraduate Students With Hybrid Flipped Learning to Develop Their Mathematical Competencies

The objective of the research is to assess the effectiveness of flipped learning in the development of mathematical competencies of postgraduate students to determine the level of satisfaction achieved during academic training in hybrid modality. The methodological approach was a quantitative a descriptive scope. The questionnaire was the instrument used with a cross-sectional design. The sample was 32 postgraduate students. The results are that hybrid flipped learning promotes the usefulness of the contents; integrates through discussions, practical cases or problems what has been learned for work performance. The mathematical competencies with the highest level of development are the use of mathematical terminology; use digital tools for the presentation of the strategy. The relevance of the research is to contribute about the effectiveness and use of flipped learning with the support of digital tools to promote quality mathematical training.

Мониторинг предметных образовательных результатов в интересах устойчивого развития

The article examines the problem of improving the quality of subject educational results at an economic university in the interests of sustainable development. The authors noted that the modern unstable world requires a professional in any field to be able to solve new problems in changing circumstances. This makes it necessary to develop a strategy for preparing a competent, mobile and creative worker in the field of education. In this regard, the emphasis in the organization of the educational process naturally shifts from the procedural component to the productive component. At the same time, the significance of the student’s subjective position increases: the integration of his motivational attitudes, social and market demand as a future professional. Based on the analysis of the most important international documents, the principles of education for sustainable development are identified, which are grouped according to their functional orientation. The first group determines the development of the individual (continuity of education, professional culture of interaction, the effectiveness of education, transformative intelligence). The second group provides conditions for personal development (fundamentalization, openness of education, advanced learning, active learning, informatization, monitoring of learning outcomes, etc.). The requirements for the results of subject mathematical training at the University of Economics are determined in the context of the principles of education for sustainable development. Based on the results of the study, a pedagogical experiment was conducted, during which the monitoring of educational results within the disciplines of the subject area “Mathematics” was carried out. The influence of experimental learning (use of case-tasks, project-type tasks) in the course of mathematical training on the level of formation of the components of mathematical competence in the context of ESD principles was studied. The results of pedagogical measurements showed a positive dynamics of the components of mathematical competence, in particular, the motivational and value qualities of future bachelors and masters.

Результаты мониторинга интеграции учебных дисциплин в условиях цифрового образования

The authors study the problem of implementing integrative links between academic disciplines at an economic university using digital tools. The importance of digital technologies as a means of integration for the study of economic and mathematical models in the process of mathematical training of students of an economic university is determined. A practical problem is considered, demonstrating the constructive possibilities of economic and mathematical modeling and digital tools in the implementation of integrative links between academic disciplines. When solving the problem, a double integral is used as a mathematical model, which makes it possible to formalize and study the socio-economic process. The capabilities of the Python programming language, which provides the implementation of an intra-model solution, are demonstrated. In the course of the study, the authors monitored the dynamics of the level of integrative connections of academic disciplines within the framework of the continuity of educational programs for undergraduate and graduate programs. An online survey was conducted among students of an enlarged group of areas of training in Economics and Management on the problematic aspects of using digital tools to study economic and mathematical models and obtain integrated educational results. The findings indicate the need for further development of integrative links between the disciplines of the subject areas “Economics”, “Mathematics”, “Computer Science and Information Technology” in the context of digital education, which, in turn, provides the necessary level of quality of modern educational results.

Application and Integration of Stochastic Mathematical Decision Tree Modeling Algorithm in Higher Vocational Mathematics Training Intelligent Platform

Application and Integration of Stochastic Mathematical Decision Tree Modeling Algorithm in Higher Vocational Mathematics Training Intelligent Platform

ИЗКУСТВЕН ИНТЕЛЕКТ И ЛОГИЧЕСКО ПРОГРАМИРАНЕ В STEM ОБУЧЕНИЕТО

In this article, the authors present one possibility for introducing artificial intelligence training, specifically Prolog logic programming, into the secondary school. The choice of Prolog logic programming as an appropriate artificial intelligence topic for secondary school study is motivated. The training is planned to take place within the school’s STEM centers. The introduction is carried out through the training of science, mathematics and informatics teachers for the subsequent training of students in the school’s STEM centers. The authors describe the structure and content of the curriculum created. The first results of the implementation of this program in teacher training are presented. One of the goals is to create a network of STEM centers offering training in artificial intelligence. The preferred form is project-based learning and multidisciplinary lessons. For the better organization of this training, a project called „Digital Bulgaria in Prolog“ has been proposed. The content of the project is demonstrated with a selected concrete example. Within this project, during the training courses, training resources are created that can be used by the participants of the network. Our experience shows that such organized training stimulates the teamwork of teachers from different specialties.

Learning platforms in the training of future mathematics teachers: analysis of foreign experience

Learning platforms in the training of future mathematics teachers: analysis of foreign experience

Mathematics Teaching in Youth and Adult Education in Manaus

Background: Youth and Adult Education (YAE) is a teaching modality aimed at young people, adults and older adults who need to complete their studies so that their specificities are respected within this teaching and learning process. Objectives: Analyse aspects related to mathematics teaching in the 3rd phase of Youth and Adult Education (YAE) through the continuing education of teachers who work in the segment, proposing new methodologies based on the results of the ADE (Student Performance Assessment). Design: The research carried out was of a qualitative exploratory nature, using psychometric theories as aids in the composition of the research, these being the classical test theory (CTT) and the item response theory (IRT). Setting and participants: From the universe of schools that compose the Semed - Manaus, a group of schools from the East Zone of Manaus was chosen; this choice was motivated by the fact that the area has the highest number of teachers (nine) who work in the 3rd phase of the YAE of Manaus. Data collection and analysis: The CTT/IRT was applied to the ADE, and the most difficult-level items were selected. For data collection, interviews, observation and application of forms on the Google Forms platform were used. Results: The research showed that teachers have many doubts regarding the curricular competencies that involve specific mathematics topics, mainly algebra. In the investigated group, none of the teachers had training in mathematics. Also, the textbook is not contextualised for the segment and does not include these new National Base skills. Schools also do not have digital resources to teach mathematics more playfully. Conclusions: Research shows that more mathematics training is needed for teachers in the segment. The Semed qualification sector needs to create a format aimed at YAE. Research is needed to assess the segment after the 3rd phase to verify the impact of mathematics teaching by teachers with no qualification in the area.

ВЫСШАЯ МАТЕМАТИКА В ТЕХНИЧЕСКИХ ВУЗАХ С УЧЕТОМ НАПРАВЛЕНИЯ ПОДГОТОВКИ

The article discusses the problems arising in the implementation of mathematical training of bachelors and masters in technical areas that meet the requirements of the time in terms of digitalization of education, including the use of computer technology

Improving the Teaching of Geometric Construction Problems in the Process of Training Future Mathematics Teachers

The article examines the problem of improving the teaching of geometric construction problems using information and communication technology (ICT) tools in the process of training future mathematics teachers at pedagogical universities. During the clarifying experiment, methodological problems in teaching geometric construction problems were identified, and as one of the ways to solve them, the need for methodological training of future mathematics teachers to use applied information and computer resources in teaching geometric construction problems was based. Analysis of information and computer technologies and programs intended for use in teaching geometry courses was performed, and the effectiveness of using the “Live Geometry” program in creating geometric construction problems was shown. Learners are more interested in geometry and construction problems, and can save all work in electronic format for future use. While studying the topic, the tools in the program will help to draw figures and use these figures in solving problems, formulate the properties of geometric figures, and use them to model geometric bodies. In the given article, the authors analyzed the state programs and concepts, the works of domestic and foreign scientists, and the results of an online survey and interviews with students. 54 students of the mathematics education program were involved in a pedagogical experiment, and the results were shown in detail. As the results of the experiment show, after using the “Living Geometry” program in the lesson, students were recognized from a certain personal subjective point of view, which regulates their cognitive and practical activities, and an increase in theoretical knowledge and practical skills was determined. It has been proven by mathematical and statistical methods that the use of the “Living Geometry” program in the teaching of structural problems allows effective and efficient organization of education and increases the methodological training of future mathematics teachers.

“Let Someone Who is Not a Mathematician Not Read Me” (Natural-Scientific Views in the Works of Leonardo da Vinci)

Introduction. It is noted the originality of Leonardo da Vinci’s natural-scientific views in the context of the Renaissance. The objectives of the article are outlined: the systematization of Leonardo’s natural-scientific views, the identification of methods used by him to cognize the reality, the analysis of the set of social conditions that influenced the formation of the thinker’s views. It is also indicated the relevance of the researched argument; the object and subject of the study are determined. Materials and Methods. It is indicated that the main source used in the article are the notes of Leonardo, reflecting his ideas about nature. The methods used in the study are related to the specifics of this source: a concrete historical approach, a comparative historical method, logical and textual methods. Results. The circumstances that gave rise to an interest in natural-scientific problems are indicated – painting and its interpretation as a reflection of the real world, on the one hand; good mathematical training received by Leonardo in his youth, on the other. It is proved, using primary sources, that Leonardo da Vinci considered experience to be the main method of cognition, and the ultimate goal of cognitive activity was the practical application of knowledge. The peculiarity of the humanistic orientation of Leonardo’s thought is revealed. On the basis of primary sources, the thinker’s ideas about nature, human, and the purpose of science are revealed. It is shown how Christian ideas and the humanistic Renaissance view were combined in a peculiar way in the views of the thinker. Attention is drawn to the inconsistency of Leonardo da Vinci’s relationship with Christianity. Discussion and Conclusion. It is shown the significance of Leonardo’s philosophical views, his role in the history of philosophy as a predecessor of the empiricist philosophers of Modern Time. It is noted that the comprehensive activity of the thinker, in fact, embodied the ideal of a person, great creative opportunities that feed, firstly, the desire to fully know the world and, secondly, the awareness of the usefulness of their activities for others. It is noted that the cultivation of these motivations has a huge educational potential in our time.

Benefits of using matrix methods in teach-ing mechanics in higher education institu-tions

Problem. Basic concepts, concepts, principles of mechanics are mainly taught from simple to complex in institutions of higher technical education, which to some extent resembles a course on the history of mechanics. In today's conditions, the education system is required to take into account current trends in scientific and engineering fields. The use of inherently geometric methods of vector algebra and vector analysis as a basis for studying mechanics is losing its relevance. To a certain extent, this approach interferes with the use of multidimensional spaces (configurational and phase), tensor algebra, and computer technologies in modeling complex mechanical systems. Goal. The goal is to expand the field of application of the matrix formalism to infinite-dimensional spaces, justify its advantages compared to other approaches used in teaching the basics of mechanical disciplines, and demonstrate its capabilities in solving complex practical problems, both in terms of modeling and in terms of using computer technologies. Methodology. The methodological basis for choosing matrix methods is the use of infinite-dimensional and abstract, in physical terms, spaces when studying the movement of mechanical systems. The fundamental impossibility of accurately describing the state of a mechanical system requires the use of statistical modeling methods and the extension of the so-called law of conservation of matter in the form of a balance equation to these models. Results. It is shown that the matrix method, which is widely used in quantum mechanics and very limited in classical mechanics, allows you to use the advantages of the transition to phase space when obtaining canonical equations and when solving problems of kinematics and dynamics of a solid body. Originality. The paper proposes certain improvements of the matrix apparatus, related to the use of geometric parameters of the mechanical system as indexes. That is, infinite-dimensional matrices with real coefficients are used. Practical value. The proposed method allows you to effectively solve the problems of the mechanics of an absolutely rigid body without using any theorems of the general course, remaining within the limits of standard mathematical training.

The role of technical university in the system of schoolchildren additional education in technical directions

Introduction. Additional education for high school students is considered as an integral part of the sustainable education system. In addition, during the school years there is a period of professional self-determination, so students have an increased interest in classes, which allows them to decide on their future specialty and prepare for admission to university.Purpose setting. To explore the role of a technical university in additional education for high school students in mathematics and computer science and show the priority areas of work of the university with schoolchildren.Methodology and methods of the study. The scientific research is based on legislative and regulatory documents, publications of domestic specialists and personal experience of the authors on extended school studies at the Novosibirsk State Technical University.Results. Additional education is considered from the point of view of the implementation of career guidance for students by a technical university. It is shown that universities play an important role in the scientific and methodological support and staff maintenance of additional education of schoolchildren in the technical direction of training. Based on the results of the analysis of a survey of students and their parents is shown that the participation of students in additional classes increases their self-esteem, which is objectively manifested in the growth of motivation in the study of mathematics and computer science at school, in achievements at conferences of various levels, in the choice of technical specialties. All students attending classes as part of additional education entered technical universities, moreover, none of the guys changed their decision to become a technical specialist.Conclusions. The specificity of additional education allows a technical university toeffectively participate in the system of additional mathematical training of schoolchildren, identify talented students, increase their interest in technical sciences and engage them as applicants to technical universities.