The Development of a Mathematical Model of an Algorithm for Constructing an Individual Educational Trajectory for the Development of Methodological Competence among IT Discipline Teachers

Abstract

This article explores contemporary models of the development of methodological competence, focusing on the needs of IT discipline teachers. The challenges in implementing these features within modern educational conditions are identified, underscoring the necessity for creating innovative teaching methods adapted to the requirements of IT teachers. By analyzing current trends in the educational environment, the authors highlight key stages in the continuous education system for teachers, including the mastering of basic education, adapting young teachers, and fostering their professional development. The article reveals the actual possibilities of developing the methodological competence of teachers as an ongoing endeavor to elevate their professional pedagogical culture. In the article, the authors propose a conceptual model within the domain of education, serving as the basis for constructing an efficient mathematical model which is specifically designed to create individualized learning trajectories for IT discipline teachers with the focus on managing the process of methodological competence development during the synthesis of training courses. The authors propose an innovative approach to teacher retraining, centered around individualized needs and abilities, with the aim of enhancing the quality of education in the field of information technology.

1. Introduction

In the modern educational context, there is a constant complexity of tasks presented to educational institutions, driven by increased demands from society and the state. This dynamic process requires adaptability from the education system to ever-changing socio-cultural and technological challenges. Consequently, there is a growing need to fulfill the social order, which includes the formation of a competitive graduate capable of continuous personal and professional growth.

One of the key factors in solving this problem is the professional competence of the teacher, since the quality of the staff largely depends on them. Teachers’ willingness to successfully address modern educational challenges, adapt to new pedagogical approaches and technologies, and their ability to effectively interact with a diverse student audience, are becoming important aspects of a teacher’s professional activity.

In the context of higher education in Kazakhstan, this takes on special significance in light of the increasing demand for graduates with higher levels of education. This demand is a consequence of the development of technology-intensive sectors of the economy that require the implementation of digital technologies. According to data from the Ministry of Digital Development, Innovation, and Aerospace Industry of Kazakhstan, the annual additional demand for IT professionals in the country is estimated at approximately 30,000 people [1]. Of course, these professionals must be competitive and competent.

Taking into account the needs of modern society, the emphasis must be placed on the quality of training in universities, particularly focusing on the development of methodological competence among teachers. To achieve this, it is essential to develop skills in applying innovative educational methods and technologies, as they play a significant role in shaping the quality of education and the success of student learning.

The analysis of the content and methods of competence-based training for specialists in various fields highlights the importance of an activity-based approach. Ryabykhina [2] emphasizes that responsibility for educational activities and the search for effective solutions to pedagogical problems are key values for teachers. Mâţă [3] asserts that competence is always manifested in activities related to solving professional tasks. Liu Y, Zhao L, and Su Y-S [4] stress that the role of teachers is to organize student learning activities, emphasizing that teaching is primarily about organization and facilitating learning.

From this, we can conclude that a lack of methodological competence among IT discipline teachers may lead to a number of problems in teaching IT specialties as follows:

• Limited effectiveness of educational practices: insufficient training of teachers from a methodological point of view may lead to suboptimal use of modern methodological tools, thereby reducing the effectiveness of educational practices and limiting the students’ learning of material;
• Resistance to outdated pedagogical approaches: a lack of methodological readiness leads to the use of outdated pedagogical strategies that do not align with the dynamics and requirements of the contemporary field of information technology;
• Inadequate adaptation to IT realities: instructors’ inability to adapt teaching methods to the specificities of IT specialties can hinder the transmission of relevant knowledge and the development of appropriate skills;
• Lack of psychological support: a low methodological level in the areas of psychology and pedagogy can impede the provision of necessary psychological support to students, especially given the high levels of stress and uncertainty inherent in the IT field;
• Insufficient objectivity in assessment: teachers may encounter difficulties in designing objective assessment methods, which, in turn, increases the likelihood of subjectivity and unfairness in evaluation procedures;
• Challenges in implementing innovations: a low methodological level may restrict the adoption and use of innovative educational technologies—an essential aspect in IT education for preparing students to meet dynamic market demands;
• Lack of selection and use of inclusive technologies: insufficient training of teachers may lead to insufficient selection and use of inclusive technologies, making it difficult to create a favorable educational environment for different categories of students;
• Impact of methods and means on students’ motivation: students may face a problem related to the impact of methods, means, and organizational forms of learning on their motivation in the process of learning IT technologies.

To overcome the identified challenges, it is essential to establish a unified digital space that fosters the continuous development of methodological competence among teachers of IT disciplines. This space should also serve as a proactive platform for professional growth, aimed at cultivating competencies necessary for navigating the complexities of modern challenges and constant changes. As such, an information system represents an adaptive and versatile tool, enabling participation in the learning process for an unlimited number of individuals, regardless of their geographical location. With its help, it will be possible to build an individual trajectory for the development of a teacher’s methodological competence.

The objective of this article is to develop a mathematical model for constructing an individualized learning trajectory focused on enhancing the methodological competence of IT discipline teachers, which will serve as a fundamental module within the existing information system [5].

By tailoring the model to each teacher’s unique competency profile, considering their personal challenges and growth prospects, we can mitigate the risks of uniformity and ensure flexibility and personalization in the teaching and development process.

2. Literature Review

An analysis of theoretical aspects in the scientific literature on higher education and the professional–pedagogical activities of university teachers reveals that there is currently no unified interpretation of the terms “methodological competence” and “teacher’s methodological competency”. Overall, researchers agree that competence is an objective condition encompassing a set of powers, rights, and responsibilities, as well as a generalized approach to actions that contribute to effective problem-solving. In turn, competency is defined as an integrative characteristic of an individual, reflecting a formed system of attitudes and internal qualities that enable a person to realize their competencies [6].

Researcher Liliana Mâţă [3] asserts that the formation of competence occurs through generalization, involving multiple abilities within a single context. She claims that every individual is capable of independently developing their competence by leveraging specific internal resources, such as intellectual abilities, practical skills, individual qualities, and motivation. According to the researcher, these internal resources encompass knowledge, skills, abilities, competencies, and value orientations.

Kansanen and Pertti [7], define competence as a quality influenced by external factors, developed over a lifetime, and manifested through activities and interactions with others. It is based on knowledge and individual characteristics that evolve within the educational process and become its outcome.

Given the general understanding of competencies in the educational field and drawing from scientific research on the specifics of a teacher’s methodological competence, as well as considering the nuances of professional–pedagogical training and their teaching activities, we can conclude that methodological competence is a structural component of professional competence. It represents an integrative characteristic of a teacher’s personality, encompassing knowledge and skills related to the development, selection, and application of appropriate teaching technologies and methods for solving educational tasks related to instruction, upbringing, and student development. Additionally, it involves awareness of these approaches as valuable orientations and proficiency in reflection and continuous improvement of one’s own methodological practices.

According to Karl Schweizer, Merle Steinwascher, Helfried Moosbrugger, and Siegbert Reiss [8], the foundation for the formation, emergence, and manifestation of a university teacher’s methodological competence lies in a combination of developed didactic, organizational, scientific, socio-psychological, and instrumental competencies.

Considering the research findings of these authors and reviewing an extensive list of works by both domestic and international scholars on the given topic, particularly the works of Görlich A., Ebert T., Bauer D., Grasl M., Hofer M., Lammerding-Köppel M., and Fabri G. [9], which focus on the development of methodological competence among medical university teachers, we can highlight the model proposed by the author of the article. This model identifies the following six competency areas: educational activities in medicine, student-centered teaching approaches, socio-communicative skills of teachers, role modeling and professionalism, reflection and advancement of personal pedagogical practices, and systemic teaching and training within professional contexts.

Based on the highlighted areas of competence identified by researchers, it is evident that these aspects contribute equally to the development of methodological competence. For each competency area, specific components can be defined, aligned with educational goals, and illustrated with practical examples to facilitate their application in practice.

The aforementioned studies serve as a foundation for identifying the key components of methodological competence among IT discipline teachers, where the main components were identified as follows [10]:

• The cognitive component encompasses the foundational knowledge and understanding of essential concepts, methods, and principles related to the teaching and learning process.
• The didactic component is one of the key elements of professional competence. It covers knowledge, skills, and abilities associated with designing, organizing, and conducting educational processes.
• The design component plays a crucial role in the preparation and organization of the educational process. This component includes the knowledge, skills, and abilities necessary to develop courses, training materials and methodological strategies.
• The information component is vital in the educational context. This component involves knowledge of current educational theories, scientific research, teaching methods and educational materials.
• The communicative component plays a significant role in an effective educational process. This component covers skills and abilities in the field of communication, interaction, and creation of an enabling educational environment.
• The reflective component represents the ability for self-analysis and self-reflection, as well as the capacity to make informed decisions based on this analysis. This component enables teachers to continually refine their teaching methods and adapt them to changing conditions and student needs.
• The monitoring component is essential for ensuring quality education. It involves an instructor’s ability to systematically track and evaluate the teaching process, student learning outcomes, and the effectiveness of their teaching methods.
• The personal–motivational component plays an important role in the formation of high-quality education and the impact on the success of students’ education. It is linked to a teacher’s personality traits, motivation, and psychological characteristics, which enable them to effectively fulfill their educational responsibilities.
• Further, the soft skills were combined into one component (communicative, reflective, personally motivational). Research on professional education, pedagogical competence, instructional activities of teachers, teaching and upbringing methodologies (Hoffman, J. V., Svrcek, N., Lammert, C., Daly-Lesch, A. [11]) has demonstrated that experts highly value these components as the most important and useful in the context of actual pedagogical practice.

The choice of these specific components of the methodological competence of an IT teacher is justified by the fact that many studies in the field of education and pedagogy (Hikida, M., Chamberlain, K., Tily, S. [12]) identify these components as key to successful learning [5]. They are based on long-term observations and analysis of the effectiveness of teaching methods. Observations have shown that many educational systems and curricula include these components in their standards and recommendations for teachers of IT disciplines, since the components are interconnected and mutually reinforcing, providing an integrated and effective methodological approach to teaching.

Numerous pedagogical theories and concepts, such as constructivism and the sociocultural approach, emphasize the importance of these components for successful training and the development of IT discipline teachers [13].

Consequently, the selection of these components is driven by their recognized significant contribution to education quality and the effectiveness of teaching IT disciplines. Moreover, it contributes to the formation of a comprehensive methodological competence of the teacher, enabling them to successfully carry out the educational process.

Given the ambiguity, integrativity, and complexity of such education as the methodological competence of a teacher, it can be assumed that an exhaustive and universal structuring of it is hardly possible. However, it is evident that across various frameworks for pedagogical competence, certain consistent components emerge, namely the personal, cognitive (knowledge-based), and activity-related components.

This is confirmed by the work of scientists studying the issue of developing the competence of teachers in various applied branches of science and education. For example, Rohr-Mentele S. Forster-Heinzer S [14] were able to empirically divide methodological competence into two dimensions as follows: understanding-based and action-based. Action-based competencies are mainly acquired in the course of work and reflect the training programs of the customer (employer).

Research in the field of assessment of methodological qualifications, the development of methodological skills, and professional assessment of pedagogical competence, etc. [15,16], contributed to the justification of the choice of specific levels of proficiency in the skills of methodological competence of teachers of IT disciplines and confirmation of their relevance for a specific educational system or organization.

Research and articles in the field of pedagogy and professional education, discussing the levels of teacher competence and criteria for their determination (Antera S. [17], Yermolenko, A., Kulishov, V. [18], Aleksieienko-Lemovska, L. [19]); textbooks, instructional manuals, and standards that describe the requirements for teachers’ methodological competence (Di Donato-Barnes, N., Fives, H., & Krause, E. S. [20]); and the study of methods and practices from other countries in assessing and developing teachers’ methodological competence (Di Donato-Barnes, N., Fives, H., & Krause, E. S. [21]) have contributed to defining criteria that will be used to assess the level of proficiency in methodological competence.

Knowledge, skills, adaptability and innovation, assessment skills, and others were accepted as criteria. Further, an assessment scale was developed that would reflect the levels of proficiency in methodological competence skills. The scale was chosen numerically and contained descriptions of levels (advanced, experienced, expert, etc.).

The analysis of the presented articles allows us to identify several key aspects related to the methodological competence of teachers of IT disciplines.

Firstly, the components of methodological competence are highlighted, such as professional education, pedagogical competence, educational and methodological activities, as well as teaching and upbringing methods. These components are recognized by experts as the most important for successful teaching activities.

In their work, Kuk and Hatala [22] discuss issues related to measuring and assessing methodological competence among university teachers. The article emphasizes that existing assessment methods tend to focus more on measuring subject-specific knowledge rather than determining the level of methodological competence. Additionally, many of these methods rely on surveys, which may lead to subjective and unreliable results due to potential biases in teachers’ responses.

To improve the situation, it is essential to develop more objective and reliable methods for assessing methodological competence. These methods could include not only surveys but also other approaches, such as observing teaching activities or analyzing student outcomes. Such approaches can provide a more accurate picture of teachers’ methodological competence and the effectiveness of their instructional practices within the university.

Secondly, the importance of choosing these components of methodological competence based on long-term observations and analysis of the effectiveness of teaching methods is emphasized. This confirms their importance and relevance in modern educational systems.

The third aspect concerns the approach to assessing the methodological competence of teachers. Numerous studies are highlighted here, where competence levels, criteria for their determination, and the development of assessment scales are discussed. This indicates the need for a systematic approach to assessment and the development of methodological competence, as well as the desire for objectivity and reliability of the assessment process.

Finally, an important conclusion is the recognition of the complex and integrative nature of the teacher’s methodological competence. This confirms the need to consider not only knowledge and skills, but also personal qualities, abilities to adapt and innovate, as well as evaluative skills. Such an integrated approach makes it possible to assess and develop the methodological competence of teachers, which contributes to improving the quality of education in general more fully.

Thus, to determine the level of proficiency in the components of methodological competence among IT discipline instructors, we propose using five levels as follows: expert, advanced, developing, situational, and zero. The assignment of a specific level to a particular percentage range will depend on the context and specific assessment criteria. Here, is the suggested schema for defining the levels and their percentage distribution:

Expert level—at this level, the teacher has a high level of methodological competence. They demonstrate a deep understanding of methodological principles and possess a wide range of skills in all components of methodological competence. The teacher is able to innovatively and creatively apply methods and approaches to learning, effectively use a variety of methodological techniques and technologies, and adapt them to different learning contexts.

Advanced level—the teacher at this level demonstrates high methodological competence. They have in-depth knowledge and a wide range of skills in most components of methodological competence. The teacher successfully applies a variety of teaching methods, adapts educational materials and resources to the needs of students, effectively evaluates students, and analyzes learning outcomes.

Developing level—at this level, the teacher has basic knowledge and skills in most components of methodological competence. They can apply basic teaching methods and techniques, select, and create learning materials, adapt them to the needs of students, and conduct a basic assessment of students.

Situational level—at this level, the teacher has limited knowledge and skills in some components of methodological competence. They can apply basic teaching methods and techniques, but with limited flexibility and innovation. The teacher needs to further develop their methodological skills and replenish their knowledge in the field of teaching methods.

Level zero—at this level, the teacher has extremely limited or missing knowledge and skills in many components of methodological competence. They find it difficult to apply teaching methods and techniques, select and adapt educational materials, and evaluate students. The teacher requires significant training and support to develop their methodological competence.

It is important to note that the percentage determination for each level is relative and can be contextual. This scheme represents the general concept of assessing the levels of methodological competence and can be adapted in accordance with specific criteria and expectations of an educational organization or system.

In the scientific research conducted by [23,24], a structural model for constructing a system to develop methodological competence and methods for assessing its effectiveness are examined.

One of the important studies devoted to the assessment of the methodological competence of university teachers is presented in [25]. The author identifies several key skills and competencies that teachers need to successfully apply active teaching methods and create a stimulating learning environment. They also emphasize the importance of developing educational materials and using innovative technologies to make the educational process more exciting and accessible to students.

Another noteworthy study is [26], which is devoted to the theoretical foundations of the formation of methodological competence of teachers of higher educational institutions. The author offers a comprehensive approach to the assessment and development of methodological readiness of teachers, emphasizing the importance of supporting and assisting the university administration in this process.

In a number of works [27,28], the scientific and methodological foundations of designing information and educational portals are considered.

In the scientific research conducted by X. Yang, W. Wang, W. Zhang, J. Xu [8], and others [29], various aspects of subject identification models within educational spaces are explored, including the development of knowledge identification systems for learners, methods for analyzing and modeling knowledge, and the identification of learning styles and recommendation systems.

However, there is no review of the research concerning the theory and practice of using a single digital space in the format of an information system to improve the methodological competence of teachers of IT disciplines with the possibility of fine-tuning to an individual profile of competencies and subsequent tutor support, which led to the conclusion that the aspect of its use has not been sufficiently studied, particularly from this point of view. The problem of developing these systems is poorly covered. This justifies the urgency in the need to develop a mathematical model for building an individual learning trajectory for the development of methodological competence of teachers, the result of which will be an improvement in the quality of learning outcomes.

3. Research Methodology

To determine the range of problems, the following steps were taken: (1) theoretical analysis of scientific psychological–pedagogical and specialized literature related to the research topic (works by domestic and foreign scholars); (2) analysis of legislative and normative documents regarding the organization of professional development for teachers (laws and regulatory acts of the Republic of Kazakhstan). For this purpose, qualitative methods such as comparative, aspect-oriented, and content analysis were employed. Overall, more than 200 scientific papers were analyzed for the purposes of this study, with more than 40 being used directly.

To achieve the objectives of the study, project vector management methodology was used to build a learning trajectory for teachers of IT disciplines. To calculate the optimal learning trajectory of students, the method for calculating the achievement of target points was used. The Monte Carlo method was implemented to model probabilistic models in the methodology of design vector control.

For the research, the following methods were employed: analysis of pedagogical, psychological, and methodological literature; pedagogical modeling; observation of IT discipline teachers’ and students’ activities during the educational process; surveys and interviews with teachers; testing of teachers; pedagogical experiments; and statistical methods for data processing.

The implementation of the model developed by [23] involves the following stages enhancing the methodological competence of IT discipline teachers during professional development: initial preparatory (pre-course), theoretical and practical (courses), and implementation (post-course). Therefore, in the process of experimental work, the methods were divided into three categories as follows:

(a)

Methods of installation, input diagnostics, which are used at the beginning of teacher training in advanced training courses to differentiate course participants by the level of formation of methodological competence;

(b)

Methods of current, in-depth diagnostics to track intermediate results and the effectiveness of the course preparation process, and to identify problems and difficulties faced by students;

(c)

The methods used in the final diagnosis after completion of advanced training courses to assess the success of students in completing course training programs and to determine the levels of formation of methodological competence of IT discipline teachers. The evaluation of diagnostic tools was carried out taking into account the quality criteria (according to K. Ingenkamp) as follows: validity, reliability, and objectivity.

4. Main Part

Against the backdrop of rapid changes in the field of education, driven by the development of technologies and new socio-cultural trends, the formation of the methodological competence of the teacher is of particular relevance. Change is affecting all countries worldwide, transforming their economies and labor markets, albeit at different speeds. Uncertainties, opportunities, and risks that need to be managed lead to change, and in order to make progress in building a prosperous society, there is a need to build a new model of methodological competence focused on the skills of the future needed to successfully meet the challenges of today’s educational landscape.

In their scientific paper, the authors of [10] determined the new structure of the model of methodological competence of IT teachers. The possession of the components specified in the model is considered as a necessary condition for improving the level of pedagogical activity aimed at achieving new educational results stated in the state educational standards of education and involves the continuous development of methodological competence.

The authors of [10] defined the structure of the model of methodological competence of IT teachers. Possession of the components specified in the model is considered as a necessary condition for increasing the level of pedagogical activity aimed at achieving new educational results stated in the educational standards of education and implies continuous development of methodological competence. Initially, six components of teachers’ methodological competence were identified, such as «Knowledge», «Design», «Monitoring», «Personal–motivational», «Reflexive», and «Communication». Further, content analysis of theoretical material, analysis of professional activity of a teacher, the structure of teacher’s methodological competence proposed in modern English-language studies and research papers [30,31] revealed the need to revise, clarify the names, and integrate the components of the model to more accurately reflect their impact on the educational process (

Table 1. Description of components of the structural model of methodological competence of teachers of IT disciplines.

Components			Functions
After		Before
Knowledge		Instructional delivery	The “Instructional delivery” component combines aspects of didactic competence and focuses on the effective transfer of learning material and the development of students’ academic skills, namely: - The design of teaching materials—the ability to develop teaching aids, lectures and practical tasks to ensure structured and logical learning for students; - The application of modern educational technologies—the ability to apply advanced teaching methods, including interactive technologies, online resources and virtual environments to maximize student engagement; - The assessment of comprehension—the ability to develop mechanisms for assessing the effectiveness of learning and adapting methods depending on the reaction of students.
Design		Curriculum design	The “Curriculum design” component integrates the elements of the educational program design component related to the development of courses, lectures and teaching methodologies and includes: - The analysis of the needs of the industry—continuous study of the current and future requirements of the IT industry in order to adapt the content of training programs; - The integration of new technologies—the skill of introducing relevant technologies into educational courses, ensuring their compliance with the dynamics of industry development; - The development of educational modules—the ability to create a modular structure of curricula for flexibility and the possibility of updating according to requirements.
Monitoring		Assessment Skill	The “Assessment skill” component covers aspects of monitoring methodological competence, including systematic control, assessment of the educational process and students’ success: - The creation of evaluation criteria—the ability to develop objective and clear criteria for assessing student achievements within the framework of educational tasks; - The use of a variety of assessment methods—apply various forms of assessment, including testing, practical tasks, projects and collective work; - Feedback—the ability to provide effective feedback to help students understand their strengths and areas for improvement.
Personal–motivational		Soft skills	The “Soft skills” component is a combination of communicative, reflexive, and personal–motivational aspects. This component of methodological competence includes the professional and pedagogical orientation of the individual (values, needs, motives, goals), awareness of the value of continuous development of methodological competence; communicative skills of the teacher; independence, intuition, which allows you to effectively solve pedagogical problems on the basis of insufficient information; manifestation of individuality in methodological activities; empathy, as well as: - Communication—developing effective communication skills to establish an open dialogue with students and colleagues; - Leadership—the ability to inspire and motivate students, stimulate their professional development; - Teamwork—the ability to effectively collaborate with other teachers and IT specialists.
Reflexive
Communication

).

Thus, by choosing a more capacious name, the most frequently used for the designation of terms, the component «Knowledge» was renamed into «Instructional delivery», «Design» into «Curriculum design» (as it exclusively concerns the educational process), and «Monitoring» into «Assessment Skill». In this study, we proceed from the position that methodological competence is based on the correlation of interrelated motivation, value orientations, theoretical knowledge, practical skills, professional qualities of personality, and reflection. As mentioned above, «Personal–motivational», «Reflexive», and «Communication» were united under the generalizing term «Soft skills».

Many scientists [32,33] include the following components in “soft skills”: emotional intelligence, persuasion skills, finding an approach to people, communicative and managerial talents, the ability to resolve conflict situations, emotional intelligence, readiness for well-thought-out risks, loads, stresses and conflicts, creativity, flexibility, and the inclination to search for alternative solutions. Having analyzed the terms “soft skills”, “personal–motivational”, “reflexive”, and “communication”, we came to the conclusion that they are in many ways similar in meaning, and these components can be considered as one of the “soft skills” of teachers necessary for effective professional activity in the constantly changing conditions of the professional environment.

The proposed configuration of the model of methodological competence of teachers of IT disciplines is optimal, since it covers all aspects of the teacher’s pedagogical activity. This structure of methodological competence (

Table 2. Levels of methodological competence formation.

Level	Characteristics
Level zero (extremely low)	Low values of indicators reflecting various aspects and components of methodical activity. The teacher uses ready-made methodological support to organize the learning process without thinking about the expediency and effectiveness of its application, about the need and possibility of even its slight modification. The teacher is characterized by complete formalism in maintaining methodological support (they care only about the formal purity of educational and methodological documentation). There is a complete lack of connection between methodological activity and research; the latest or relatively new achievements of science are not reflected in the content of training. Didactic modeling of the training course does not correspond to the logic of teaching an isomorphic academic discipline; innovative didactic methods and technologies (corresponding to the competence approach) are practically not applied. Advanced methodological and pedagogical experience is not analyzed at all; there is no aspiration to improve methodological skills.
Situational level (low)	There is an improvement of information and methodological support of the educational process, replenishment of its content, but neither the latest achievements of science (in the content of training) nor innovative teaching methods and technologies are taken into account; there is practically no development of pedagogical tasks corresponding to the competence paradigm of training; the development of tasks of the traditional type dominates.
Developing level (average)	Active improvement of methodological support of the educational process (constant modernization of its content), primarily by increasing the number and quality of pedagogical tasks. There is an application of traditional and innovative didactic methods and technologies. The constantly replenished fund of assessment means contains a sufficient number of tasks of traditional type and control-competence assessment tasks. The tasks vary in difficulty level, fully corresponding to the academic discipline and assessed competencies of students. The design of the learning process is characterized by rationality; the teacher is aware of the logic of teaching the academic discipline, and fully takes it into account when designing the course. Information technologies are used mainly in the formation of the educational process support. The quality of electronic educational resources is at an average level. The teacher indicates (in the methodological support) references to the information resources available to students (the arsenal of references is wide), the use of which will help students to better master the course.
Advanced level (high, systematic)	The teacher’s research activity becomes the leading factor of permanent (rather than episodic) modification of teaching content. The quality of electronic educational resources is at a high or very high level. The teacher actively uses the potential of the scientific and educational environment to improve their methodological activity through all possible ways of analyzing and adapting the advanced experience of pedagogical and methodological activity accumulated by the society, but does not contribute to the replenishment of this experience accumulated by the society (i.e., does not broadcast their experience).
Expert level (highest, optimal)	It is characterized by the teacher’s activity in broadcasting their own positive experience of methodical and pedagogical activity, its transfer to other scientific–pedagogical workers. The teacher actively assists other scientific–pedagogical workers in improving methodological and psychological–pedagogical competence, participates in the monitoring of methodological activities of scientific–pedagogical teams, in the work of juries and expert commissions, scientific and methodological associations, etc. Actively conducts research in the field of pedagogical sciences (with the publication of results), enriching the theory and methodology of teaching, promotes the results in order to implement them in practice in the scientific and pedagogical community, etc. Methodological activity of a teacher acquires a supra-disciplinary character.

) involves the process of changing it in moving from one level to another [34].

Based on the analysis of scientific papers, we believe that methodological competence, like other personal and professional qualities [35,36], can be formed at one of five levels (Table 2). Since the scheme described in the Literature review section represents the general concept of assessing the levels of methodological competence, we adapted it to the updated configuration of the model of methodological competence of teachers of IT disciplines. As can be seen, each subsequent higher level acquires some qualitative characteristics that are (especially) absent in the previous one.

Based on the updated components of methodological competence proposed above and their proficiency levels, it is possible to visualize the reflection of this process as follows (Figure 1). The central element is the student–teacher relationship of IT disciplines. There are various levels of competencies around the student, which are presented in the form of a diagram.

At each level, the teacher has certain competencies corresponding to their professional training and experience. The key components at each level are knowledge, skills, adaptability and innovation, and assessment skills. The structure of the model makes it possible to systematize and structure the process of developing the methodological competence of the teacher, providing a clearer idea of how a certain level of competence can be achieved and what competencies need to be developed at each stage of professional growth.

Such a structure is a valuable tool for assessing and developing the methodological competence of IT discipline teachers, contributing to improving the quality of education in this area.

The development of methodological competence among IT discipline teachers is integrated within the lifelong learning concept. The modern structure of the continuing education system for teachers in the global context includes the following stages: mastering fundamental education in Kazakhstan [6,37], the period of adaptation, and formation of professional activity of young teachers under the guidance of more experienced colleagues, professional development, and self-education for practicing teachers.

However, the analysis of the subject area revealed that professional development or retraining of teachers has a number of key factors as follows:

• The availability of different levels of education (on the basis of college, bachelor’s, master’s or doctoral studies);
• Different categories of students, with different levels of need for the development of methodological competence;
• Availability of residual knowledge on certain topics of study;
• Short-term nature and possible interruption during the training period.

Therefore, an important condition for the development of the teacher’s methodological competence in the conditions of retraining or professional development lies in the design of individual educational trajectories of the teacher within the framework of the subject area studied by them.

An individual educational trajectory is defined as a purposefully designed differentiated educational program that enables teachers to choose its development and implementation while providing pedagogical support for their self-realization, considering professional and educational needs, abilities, and personal interests.

To manage competence development, it is necessary to calculate the optimal learning trajectory. For determining the optimal learning trajectory of students, a method of calculating the achievement of target points is proposed.

The method for calculating the optimal learning trajectory (referred to as «movement») of IT discipline teachers (referred to as «learners») to achieve the target points of educational programs involves using the Monte Carlo method [38]. We chose this group of numerical methods because they rely on generating a large number of realizations of a stochastic (random) process. This process is designed to match the probabilistic characteristics of the relevant quantities in the problem being solved. In other words, it models processes with uncertainty, e.g., in our case, constructing teachers’ learning trajectories, accounting for individual variations [39]. Since each IT discipline teacher has their own set of personal competencies, we need to tailor a unique learning trajectory for each teacher based on the specific methodological competencies they lack from a pedagogical perspective. In this article, we apply the method of calculating the optimal trajectory of learners within the project-vector management methodology, specifically utilizing the Monte Carlo method to calculate probabilistic learning trajectories for IT discipline teachers based on their required methodological competencies.

Project-vector management methodology has been tested in several systems implemented for managing educational processes and universities as a whole [38,40].

At the same time, the distribution of probabilities when choosing the learning trajectory of students will be calculated through the priority of the components of the student’s training for the formation of certain competencies and the formation of the student’s knowledge and skills.

In the method of determining the learning goals that correspond to the maximum acquisition of the necessary competencies in the educational program of the course, the endpoints of the students’ movement are calculated.

$$\forall {\Pi }_k,C_j:A_k^{\left(j\right)}\left(\overline{T_k^{\mathrm{d}ir}}\right)=⌊x_{k1}^{\left(j\right)}\left(\overline{T_k^{dir}}\right),x_{k2}^{\left(j\right)}\left(\overline{T_k^{\mathrm{d}ir}}\right),\dots ,x_{kp}^{\left(j\right)}\left(\overline{T_k^{dir}}\right)⌋,$$

(1)

where $x_{k1}^{\left(j\right)}\left(\overline{T_k^{dir}}\right),\dots ,x_{kp}^{\left(j\right)}\left(\overline{T_k^{dir}}\right)$ learner’s end coordinates $C_j^k$ of educational program of the Pk course at the planned time of completion of the student’s training $\overline{T_k^{dir}}$.

In the method of calculating the optimal trajectory of movement, additional information is the relationship of students along the course of movement in the educational environment. This interconnection (or, more precisely, interactions) determines how much energy (time, credits, finances) must be additionally spent so that a certain student shifts by one unit of distance in the educational environment, taking into account the acquisition of specific knowledge and skills for the formation of competencies necessary for the student at the end of training and, accordingly, how much and what resources are needed for this.

In other words, the movement of stakeholders in the educational environment should be linked to the movement of all learners in such a way that existing interactions contribute to the achievement of goals (movement towards endpoints) rather than hinder it. To carry this out, it is necessary to note the interconnection of the students themselves (hereinafter referred to as subjects) within the framework of different educational programs of the course. The movement of one subject overcomes the resistance zones caused by the movement of other subjects (students, course trainers, etc.). Therefore, it is exceedingly difficult to find the optimal trajectory of movement in the total amount of interacting subjects.

To solve this problem, we will set the structure of interactions between subjects in the educational space. Let

$$F⌊Q_j\left(A_k^{\left(j\right)}\right(t\left)\right)/Q_i\left(A_k^{\left(i\right)}\right(t\left)\right)⌋,$$

(2)

where the impact of Q_i subject with $A_k^{\left(i\right)}\left(t\right)$ coordinates to subject Q_j with $A_k^{\left(j\right)}\left(t\right)$ coordinates. This influence leads either to resistance to the movement of the subject of the educational process, or to the promotion of this movement.

The determination of the influence of educational environment subjects is crucial for establishing priorities in the displacement of subjects within the educational environment (EE). The significance of these subjects reflects their impact on other components of the EE. After all, the location (coordinates) of influential subjects within the EE will determine how quickly other subjects will shift to acquire necessary educational competencies.

$${\theta }_{jkp}=\frac{\sum _{i=1}^k{\phi }_{jip}}{K},$$

(3)

where ${\theta }_{jkp}$—a coefficient that determines the average magnitude of the subjects’ exposure EE Q_j of P_k educational program(EP) in the N_p direction;

K—number of subjects affected by EE subjects Q_j P_k EP.

It is also important to consider the impact on each of the EE subjects. This impact is equal to:

$${\rho }_{ikp}=\frac{\sum _{j=1}^k{\phi }_{jip}}{K},$$

(4)

where ${\rho }_{ikp}$—coefficient that determines the average value of the impact about the EE Q_i P_k EP with other entities of EE in the N_p direction.

The determination of the EE subjects’ goals in present within the fixed asset (the final coordinates of the movement). The final coordinates correspond to the goals of implementing the training of an individual trajectory of movement for obtaining certain competencies by students. They can be obtained by using the vector method of goal achievement and are presented in the form of

$${\mathrm{P}}_k:\forall C_{jk}:x_{k1}^{\left(j\right)}\left(t_{max}\right),\dots ,x_{kp}^{\left(j\right)}\left(t_{max}\right),$$

(5)

where $C_{jk}$—EE subject of P_k EP;

$x_{k1}^{\left(j\right)}\left(t_{max}\right),\dots ,x_{kp}^{\left(j\right)}\left(t_{max}\right)-\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s} \mathrm{o}\mathrm{f}$ EE subject $C_j^k$ of P_k EP at the time of completion of the training t_max; t_max—moment of completion.

The conditions for achieving the goals of students are determined in the EE (restrictions). The final coordinates of the movement should not be less than the directive (initially) set and should be reached before the planned completion date of training. In addition, the cost of training (time, finances) should not exceed the planned one as follows:

$$\begin{gathered} t_{max}\le t_{fin}; \\ \forall i=\overline{1,p}:x_{ki}^{\left(j\right)}\left(\overline{T_k^{\mathrm{д}\mathrm{и}\mathrm{p}}}\right)\le x_{ki}^{\left(j\right)}\left(t_{max}\right); \\ {E}_{fact}^k\le E_{plan}^k. \end{gathered}$$

(6)

Establishing the initial conditions for calculating the trajectories of students, the initial conditions are as follows:

• The time of the start of training (start of movement) is t₀.
• The starting point of the movement of students in the EE. When accepted, we obtain the following:

$${\mathrm{P}}_k:\forall Q_j:x_{k1}^{\left(j\right)}\left(t_0\right),\dots ,x_{kp}^{\left(j\right)}\left(t_0\right),$$

where $x_{k1}^{\left(j\right)}\left(t_0\right),\dots ,x_{kp}^{\left(j\right)}\left(t_0\right)$ the initial coordinates of the students in EE Q_j of educational program P_k.

The end point of the subjects’ movement in the EE as far as possible from the initial one.

When accepted, we obtain the following:

$${\mathrm{P}}_k:\forall C_{jk}:x_{k1}^{\left(j\right)}\left(t_{fin}\right)+x^{\prime },\dots ,x_{kp}^{\left(j\right)}\left(t_{fin}\right)+x^{\prime },$$

(7)

3.

The number of motion modeling options in EE—$N_v^{\left(max\right)}$.

To determine a rational trajectory for students following their individual learning paths, we will conduct modeling of the movement options (Figure 2). These options will be defined by prioritization and interaction among students to achieve the necessary competencies based on learning outcomes. The best alternatives will be proposed to students within the framework of the selected educational program project, allowing them to choose the optimal learning trajectory.

The initial simulation option is set to 0: N_v = 0, where N_v is the number of the simulation option.

• The transition to the next version of the simulation.

The next number of the simulation option is set N_v = N_v + 1.

If $N_v>N_v^{max}$, transition to paragraph 6.

The movement step is set—N_d = 0, where N_d—the number of the movement step.

The initial coordinates of the EE students (subjects) and the initial moment of time are set t₀ as follows:

$${\mathrm{P}}_k:\forall Q_j:x_{k1}^{\left(j\right)}\left(t_0\right),\dots ,x_{kp}^{\left(j\right)}\left(t_0\right)$$

(8)

where $x_{k1}^{\left(j\right)}\left(t_0\right),\dots ,x_{kp}^{\left(j\right)}\left(t_0\right)$ is the initial coordinates of the subjects of the Q_j educational program of the P_k course.

The initial costs necessary to achieve certain competencies (time, finances) within the framework of the educational program of the course are fixed:

$$\forall {\mathrm{P}}_k:E_{fact}^k=e_0^k$$

(9)

where $E_{fact}^k$ is the actual cost of P_k training;

$e_0^k$ is the initial cost of a P_k project (incurred at the beginning of P_k training).

2.

Moving on to the next step of the movement, we obtain the following:

Nd = Nd + 1.

(10)

The calculation of the next moment in time is as follows:

$${\mathrm{t}}_{N_d}=\left(N_d-1\right)\ast ∆t+t_0$$

(11)

If the coordinates of all subjects exceed the target, or the actual costs are higher than planned, go to paragraph 1.

3.

Calculation of forces when moving in directions at a time ${\mathrm{t}}_{N_d}$

The direction of movement of subjects is estimated by the specific efforts to shift the subject in each direction, the priority of this direction and the magnitude of the impact on the subject in this direction, determined through δ_jki. To reduce the cost of moving subjects, there may be a subject whose displacement will reduce the cost of learning subjects. Moreover, it will reduce more significantly than the cost of movement of the subject of training, i.e., for the N_i direction:

$$\begin{gathered} {\mathrm{P}}_k:Q^{\mathrm{*}}=\left\{Q_s\right\},s=\overline{1,K^{\mathrm{*}}},K^{\mathrm{*}}>0\wedge {\delta }_{jki}^S<{\delta }_{jki}: \\ {\delta }_{jki}^S=K_i^{jk}+{\rho }_{jki}^S\left(x_{k1}^{\left(j\right)}\left(t_{N_d}\right)=x_{k1}^{\left(j\right)}\left(t_{N_{d-1}}\right)+∆x,\dots ,x_{kp}^{\left(j\right)}\left(t_{N_d}\right)=x_{kp}^{\left(j\right)}\left(t_{N_{d-1}}\right)+∆x\right); \\ {\delta }_{jki}=K_i^{jk}+{\rho }_{jki}\left(x_{k1}^{\left(j\right)}\left(t_{N_d}\right)=x_{k1}^{\left(j\right)}\left(t_{N_{d-1}}\right)+∆x,\dots ,x_{kp}^{\left(j\right)}\left(t_{N_d}\right)=x_{kp}^{\left(j\right)}\left(t_{N_{d-1}}\right)+∆x\right); \\ {\delta }_{ski}=K_i^{sk}+{\rho }_{ski}\left(x_{k1}^{\left(s\right)}\left(t_{N_d}\right)=x_{k1}^{\left(s\right)}\left(t_{N_{d-1}}\right)+∆x,\dots ,x_{kp}^{\left(s\right)}\left(t_{N_d}\right)=x_{kp}^{\left(s\right)}\left(t_{N_{d-1}}\right)+∆x\right), \end{gathered}$$

(12)

where K*—the number of subjects whose displacement leads to a reduction in the cost of shifting subjects in the EE;

Q_s—the number of subjects whose displacement leads to a reduction in the cost of shifting subjects in the EE;

${\mathrm{\delta }}_{jki}$—the generalized coefficient of resistance to the movement of the subject C_j OP P_k in the N_i direction;

${\delta }_{jki}^S$—the generalized coefficient of resistance to the movement of the subject C_j OP P_k in the N_i direction;

${\delta }_{ski}$—the generalized coefficient of resistance to the movement of subjects is Q_s,

and the cost of shifting subjects is less than the cost compensation due to the reduced impact on this subject:

$${\delta }_{jki}^S\ast {\left(∆x\right)}^3+{\delta }_{ski}\ast {\left(∆x\right)}^3<{\delta }_{jki}\ast {\left(∆x\right)}^3⟹{\delta }_{jki}^S+{\delta }_{ski}<{\delta }_{jki}$$

(13)

The need for prioritizing the displacement of subjects within the educational environment arises. This involves subsequent recalculations of the displacement possibilities for these subjects. Achieving this condition is feasible when increasing the coordinates for certain subjects beyond the coordinates of the reference subject results in a change of the interaction coefficient sign from “negative” to “positive” within the educational environment.

4.

Selection of shifting subjects. If K* = 0, then:

If there are objects shifted by ∆x in this step of the movement, then proceed to step 2. Otherwise, the selection for displacement among the educational environment subjects occurs randomly based on a probability distribution according to the following formula:

$$p_{jk}=\frac{{\sigma }_{jk}}{\sum _l{\sigma }_{jk}}$$

(14)

where $p_{jk}$—the probability of selection to a bias in the direction N_i of the C_j subject of P_k EP;

Otherwise, the objects within the educational environment that undergo displacement are selected from the set $Q_U^{\mathrm{*}}=\left\{Q_b^U\right\},b=\overline{1,U},{where Q}_U^{\mathrm{*}}\subseteq Q^{\mathrm{*}}$. The displacement of these objects by an amount ∆x reduces the resistance within the educational environment relative to other educational subjects (according to Formula (1)). If the set Q_U^ is empty, proceed to Step 2. The selection of the object is then carried out randomly based on a probability distribution using the following formula:

$$p_{jki}=\frac{{\theta }_{jki}}{\sum _{b=1}^U{\theta }_{bki}}$$

(15)

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} p_{jki}$—the probability of selection to a bias in the direction of N_i of the subject Q_j EP P_k;

5.

Calculation of the displacement. If:

$$E_{plan}^k-E_{fact}^k\ge {\delta }_{jkp}\ast {\left(∆x\right)}^3,$$

Then, the following is accepted:

$$\begin{gathered} x_{kp}^{\left(j\right)}\left(t_{N_{d-1}}\right)+∆x; \\ {E}_{fact}^k=E_{fact}^k+{\delta }_{jkp}\ast {\left(∆x\right)}^3. \end{gathered}$$

Otherwise:

$$x_{kp}^{\left(j\right)}\left(t_{N_d}\right)=x_{kp}^{\left(j\right)}\left(t_{N_{d-1}}\right).$$

(16)

Return to paragraph 3.

6.

Estimation of the obtained target coordinates of movement in PVP.

The evaluation of the received variants of the learning trajectory (movement) is carried out in order to achieve the learning goals and obtain the necessary competencies. If the values do not satisfy the students, then the initial data are corrected, and everything is repeated from point 1. If they satisfy–completion.

To ensure the implementation of training, an information system has been developed for training, taking into account the individual learning trajectories of students.

For a course posted in an information system, learning outcomes must be recorded at the time of registration of this content on the platform and automatically recorded in the digital profile of the teacher upon completion of training.

Movement along the track is reflected in the form of progress in the levels of formation and manifestation of each component of methodological competence, the formation of which is aimed at the track.

The overall progress on the track consists of the progress on the track component.

The progress in the levels of formation and manifestation of the component of methodological competence is calculated as follows:

• The total weight of the competence component is equal to the sum of the weights of all its elements.
• The weight of an element of the competence component is equal to the product of its indices in the matrix of target results of the competence component.
• The current progress is equal to the sum of the weights of the fixed elements of the competence component.

Let us consider the criteria for the formation of weighting coefficients of methodological competence, as shown below:

$$K={\left\{K_i\right\}}_I,$$

where $K_i$—$i$-th criteria for the formation of weighting coefficients of methodological competence; $I$—components of methodological competence.

$$K_{i}iI$$

Then, the main statement of the task of multi-criteria formation of weighting coefficients of methodological competence takes the following form:

$$K\left(Comp\right)\to max$$

Let us define the main criteria of methodological competence as follows:

• «Instructional delivery» component;
• «Curriculum design» component;
• «Assessment skill» component;
• «Soft skills» component.

The next step is to create a square matrix and, using the method of expert assessments, calculate the weight coefficients for each of the criteria as follows:

	K1,1	…	Ki,j
K1,1	1	v1,m	v1,j
…	vn,1	1	vn,j
Ki,j	vi,1	vi,m	1

The next step is to create a square matrix and, using the method of expert assessments, calculate the weight coefficients for each of the criteria:

Taking into account the opinion of experts, we will determine the weight of each methodological competence according to the following formula:

$$\sum _{\mathrm{q}=1}^{\mathrm{Q}}\left({\mathrm{r}}_{\mathrm{q}}\times {\mathrm{K}}_{\mathrm{q}}\left(\mathrm{C}\mathrm{o}\mathrm{m}p\right)\right)\to \mathrm{m}\mathrm{a}\mathrm{x},$$

(17)

where the weighting factor of the optimization criterion is as follows:

$$r_{q}q$$

Let us take an example.

Table 3. General scheme of the matrix of target results of the components of methodological competence N.

Component N Total Weight: 100 Levels of Formation	Levels of Manifestation
	Low	Acceptable	Average	High	Optimal
Zero	Element 1 1	Element 1 2	Element 1 3	Element 1 4	Element 1 5
	Weight = 0.5	Weight = 1	Weight = 1.5	Weight = 2	Weight = 2.5
Situational	Element 2 1	Element 2 2	Element 2 3	Element 2 4	Element 2 5
	Weight = 1	Weight = 3	Weight = 3.5	Weight = 4	Weight = 4.5
Developing	Element 3 1	Element 3 2	Element 3 3	Element 3 4	Element 3 5
	Weight = 1.5	Weight = 3.5	Weight = 5	Weight = 5.5	Weight = 6
Advanced	Element 4 1	Element 4 2	Element 4 3	Element 4 4	Element 4 5
	Weight = 2	Weight = 4	Weight = 5.5	Weight = 7	Weight = 7.5
Expert	Element 5 1	Element 5 2	Element 5 3	Element 5 4	Element 5 5
	Weight = 2.5	Weight = 4.5	Weight = 6	Weight = 7.5	Weight = 8.5

shows the general scheme of the matrix of the target results of one component of methodological competence N. The elements of the matrix are the actions that the user must master as part of the formation of the competence component when moving along the track. The fixed elements of the competence component are highlighted in gray, namely those topics that the user has already mastered.

Applying the above-described procedure for calculating progress by levels of formation and manifestation of the methodological competence component, we reach the overall progress of the development of this component at 36%.

In this example, attention should be paid to the “missing” element in the process of forming the component. Such a situation is possible when moving along the track, since this element of the competence component can be formed, for example, only in the process of completing a project that the user abandoned when building an individual educational trajectory. In this case, even after the complete completion of the movement along the track, the component of methodological competence will not be formed by 100% progress, since the experience of manifesting all the actions provided for by the model for the formation of this component has not been accumulated. Nevertheless, this element of the methodological competence component can eventually be covered by other educational courses.

Thus, movement along the track can be non-linear and depends on the user’s choice of how to build an individual educational trajectory.

An information system was developed to implement the training (Figure 3).

The system is designed to enable efficient interoperability between its components, which include the backend, interface, and database. The backend of the system is developed using Java Spring Boot, which provides a comprehensive platform for server-side development. This allows the system to process numerous user requests at the same time, which is an important feature in the context of an information system intended for educational purposes.

The system’s interface is designed using React 17 JS, which offers a robust and user-friendly interface for system users. It provides an intuitive interface, which is a crucial requirement for an educational system that needs to be easy to use. The interface supports various functionalities of the system, which include training, forecasting, and post-course support. PostgreSQL 16.3 is used to develop the database, which provides a scalable and reliable data management system for the proposed information system. The database is used to store the data from the system, which include user details, course information, and all other relevant information. The database is designed to ensure the integrity and confidentiality of the system’s data.

The backend and frontend communicate with each other through an application programming interface (API) to ensure that all the necessary data are processed and transferred efficiently between the two components. The interface passes user requests to the backend, which then processes the requests and provides the necessary responses. The database also communicates with the server component to provide the necessary data, and the server component updates the database with any new information generated by the system.

The initial stage of the process of forming a digital profile of a teacher in the forecasting system and continuous support for the development of methodological competence, as the first step in building their educational trajectory involves collecting data on the following aspects:

• Personal data—full name, date of birth, education, work experience and other personal data of the teacher;
• Pedagogical experience—periods and place of teaching in educational institutions;
• Specialization and subject areas—indication of the names of the disciplines taught by the user, indicating the periods;
• Scientific activity—data on scientific publications, research, participation in scientific conferences and projects;
• Results of the questionnaire on the input testing of the system;
• Additional data—information about internships, advanced training (taking place in the system or other educational institutions), etc.

For data collection, various sources are used, including educational institution databases, teacher surveys, and current information system, which considers the teacher’s activity in the educational process. After filling out the initial questionnaire, the teacher’s answers are stored in the database, and the level of their competencies is recorded and displayed on the competency map in their profile (Figure 4).

Personal data can only be changed by the user of the account. The data of the competence card are stored in the database and are changed when the teacher fills out the questionnaire in the digital profile (Figure 5).

After collecting data on teachers, the next step is to analyze and assess the current competence, which makes it possible to assess the current level of competence of each teacher in various aspects and based on the data. From this, it is possible to identify areas in which teachers need additional support and development. This may include learning new teaching methods or deepening knowledge in a subject area.

As part of the study, questionnaires were compiled to assess the level of methodological competence of teachers, for the development of which the following methods were used [41,42]:

• Likert-based questionnaires are one of the most common methods. Respondents answer the questions by choosing one of the proposed answer options (for example, from “strongly disagree” to “strongly agree”). These questionnaires can cover different aspects of competence, such as knowledge, skills, and abilities.
• Case study methodology—respondents are asked to consider specific scenarios or cases related to pedagogical practice and give their assessment for each case.

The methodology of situational tasks was used as an example for creating questions in a questionnaire aimed at assessing the level of methodological competence in IT discipline teachers [43,44]. This technique allows evaluating a teacher’s abilities in communication and interaction with students in various situations, which is a crucial aspect of methodological competence.

This process can be described as follows:

• Definition of competencies and identification of the necessary competencies for teachers of IT disciplines.
• Creating scenarios and the development of scenarios reflecting typical situations that a teacher may encounter in the process of learning and interacting with students. Using the example of a communicative component, these may be situations when students ask a lot of questions or when there is a conflict in the audience.
• Formulation of questions, specifically, drawing up questions that would allow you to assess the teacher’s reaction to various situations. The questions are structured in such a way that they reflect several aspects of the communicative component, such as the ability to listen, adapt to the audience, emotional intelligence, etc.
• Definition of evaluation criteria for evaluating the answers that were defined for each question. These criteria include the level of activity and flexibility in communication, emotional responsiveness, the ability to resolve conflicts and other important aspects.

In the period from March to May 2024, a group of IT discipline teachers from Kazakhstani universities underwent testing to assess their level of methodological competence. Let us denote this group as the pre-course evaluation group (EG₁). The post-course evaluation group (EG₂) consisted of teachers who were part of the EG₁ but had completed additional courses to enhance their methodological competence, allowing for the measurement of changes in their level of competence.

After conducting the survey, the responses were evaluated according to established criteria, and the results were analyzed. The outcomes following the initial entrance testing were determined as the baseline level for EG₁ (see

Table 4. Initial levels of methodological competence formation among teachers of EG₁ (in %).

The Levels of Formation of the Teacher’s Methodological Competence	Instructional Delivery		Curriculum Design		Assessment Skill		Soft Skills
	Ind.	%	Ind.	%	Ind.	%	Ind.	%
Zero	26	42.6	36	59	41	67.2	29	47.5
Situational	18	29.5	7	11.4	12	19.6	19	31.1
Developing	11	18	10	16.3	5	8.1	7	11.4
Advanced	4	6.5	2	3.2	2	3.2	4	6.5
Expert	2	3.2	6	9.8	1	1.6	2	3.2

).

The data obtained from the survey indicate that within the framework of teacher professional development courses, it is necessary to focus on the development of skills for analyzing program and methodological materials, evaluating the quality of educational resources, identifying methodological issues and determining solutions.

In the digital profile of the “Courses” module, a list of training courses registered in the system, which are necessary for the teacher to pass, is formed. In the process of training the student, the algorithm for building an educational trajectory can be based on any change in the current level of development in the digital profile. The system will generate recommendations, and the user can use them when choosing a further path.

Administrators can create courses, assign a trainer to a course, create a schedule for the start and end of the course, and fill in the certificate of successful completion of the course in their accounts. Throughout the course, they can view the current results of the instructors, monitor the organization and delivery of the course, and make adjustments. The course trainer, in turn, can contribute to the course materials, and add course sections, presentations, and practical and test tasks (Figure 6).

In the “Post-Course Support” module, the user of the system gains access to the course materials, including recorded video lessons, seminars, trainings, etc. (Figure 7).

This module is necessary for managing courses, and creating and adding video lectures, tests, and open-ended assignments to the course.

The principles on the basis of which all teacher training programs are developed as follows:

-

Modularity—each training program of a topic or an independent section within a topic should contain interrelated modules (parts) for different levels of retraining;

-

Individuality and variability—for different categories of students there may be different aspects of studying topics (options for training programs);

-

Uniqueness—the content of the programs of training modules should not overlap both by levels within the same topic (section of the topic) and between topics (sections of topics);

-

Minimization of costs for retraining of students—the training course should be minimal in duration, provided that a given set of knowledge and skills is achieved, and based on the standards for advanced training of teaching staff in Kazakhstan.

Based on the principles listed above, it is necessary to decompose the subject area of study and then build a model in the form of a discrete space Q—a space for the formation of methodological competence for the provision and organization of the learning space. Methodological competence is a set of knowledge and skills (practical skills; X), each element of which x_i characterizes a certain state of competence in space Q as follows:

X = {x₁, x₂, …, x_n}.

(18)

The transition from one state of competence x_i to another x_j in space Q can be carried out with the help of one or more training modular training programs p_t ∈ P, which complement the set of knowledge and skills of state x_i to the set of knowledge and skills of state x_j.

The structure of modular training programs is developed on the basis of the following step-by-step decomposition of the subject area of study:

• The allocation of retraining cycles;
• The allocation of independent topics and sections of training within each cycle;
• The separation of aspects of the study of the topic (section) depending on the target audience;
• The distribution of target competence states depending on the levels of immersion—zero, situational, basic, advanced, or expert.

In the space Q, we define an incident relation between sets X and P using a weighted oriented hypergraph H (X, P), where X is the set of vertices modeling different competence states, P is the set of arcs modeling the retraining process (program modules).

Let us consider the features of the model that reflect the specifics of this subject area and make it possible to develop an effective method for managing the process of building a personal learning trajectory for the formation of competence in the course of the synthesis of training courses from program modules.

In order to implement the principle of minimizing the cost of retraining personnel, the weights of arcs L (p_i) are introduced, corresponding to the number of academic hours of the corresponding training module.

The decomposition of the domain is reflected in the hypergraph by dividing the original set X into a set of non-overlapping subsets as follows:

X = {X1, X2, …, Xn},

(19)

where n is the number of different topics (sections of topics) to be studied; X_i is a subset of competence states related to a given topic (topic section).

The presence during the topic (section of the topic) of training modules with different aspects and levels of training allows you to establish an order relation on a subset of vertices X_i. Each vertex x_j ∈ X_i is marked with the ordinal number s(x_j) = 1, 2, …, k.

Such vertex marking establishes an order relation in each subset of X_i: the zero level of training in a given topic (section of the topic) corresponds to number 1, the situational level corresponds to 2, and so on.

Thus, it is possible to establish the incidence between the vertex of the target competence and the ordered subset of vertices X_i of the initial competence, while linking the vertex of the target competence with an arc only to the vertex x_j ∈ X_i that corresponds to the minimum permissible initial competence for a given topic (section of the topic).

Figure 8 shows a fragment of the graph in which the arc (x_i, x_j) also indicates the presence of an incident relationship with other vertices of the subset X_i, the ordinal number of which is higher (in the figure, the arcs (x_i, x_k) and (x_i, x_m) are shown as dashed lines).

In the hypergraph of the domain, the incidence relation can be as follows (Figure 9):

-

Binary, if there is such a training module (arc p_i = (x_i, x_j)) in the course, which complements the set of knowledge and skills from the state of initial competence (vertex of graph x_j) to the target state of competence of this module (vertex of graph x_j);

-

N-ary, if there is such a training module (oriented hyperrebro p_i = {x_i, x_k, …, x_m, x_j}) in a course that complements the set of knowledge and skills from the necessary set of states of initial competence (vertices of graph x_i ∈ X) to the target state of competence of this module (vertex of graph x_j).

Let us highlight the characteristic features of the model as follows:

• There is such a vertex x₀ ∈ X, which has a half-step of entry p + (x₀) = 0; this vertex corresponds to zero methodological competence (within this course).
• There are such vertices x_i ∈ X, in which the half-degree of the outcome is p − (x₀) = 0; these vertices correspond to the final states of the target methodological competence (within this course).
• The set of vertices of the hypergraph X represents ordered subsets X₁, X₂, …, X_n with a given order ratio at the vertices of the subset (ordered components of methodological competence in terms of aspects and levels of training).
• The binary arc establishes the relationship as follows: initial competence (not lower)—training module—target competence.
• A directed hyperedge has one important property; there is always one, and only one vertex of drain (the vertex of target competence) and several vertices of the source (a set of vertices of minimum initial competence).
• The transition to the top of x_j ∈ X (the state of target competence) along the hyperline p_i = {x_i, x_k, …, x_m, x_j} is possible only if all vertices x_i ∈ {p_i\x_j} are reached (the necessary set of states of minimum initial competencies for the implementation of this training module).
• Each arc of the hypergraph has a weight (the number of academic hours of the module).

The method for synthesizing an educational course involves the following scenario: Suppose we have a list of requirements for methodological competence (a set of knowledge and skills to be achieved during the learning process). Additionally, we have a list of initial competency states for the learners, established through an entrance test. The goal is to find a sequence of course modules that

• Augments the initial set of knowledge and skills to match the desired competency level;
• Minimizes the training costs associated with achieving this competency.

Using the developed domain model in the form of a weighted directed hypergraph H(X,P), this problem in terms of graph theory was formulated.

In a hypergraph H(X,P), two sets are defined as follows: the set of initial competence states X_initial ∈ X and multiple states of target competence X_target ∈ X, such as

X_initial, X_target ≠ ∅, X_initial ∩ X_target = ∅.

For the lack of knowledge and skills, the state of x₀ is given—in this case, X_initial = x₀. Then, in the connected hypergraph, H(X,P) is given. It is always possible to find such a counterway μ_t,

∀x_k ∈ X_target ∃μ_t (x_k, …, x_i): x_i ∈ X_initial.

To find such a set of edges P_course ⊂ P,

P_course =∪ μ_t и ∀p_i ∈ P_course (ΣLp_i → min).

Thus, the task of synthesizing the P_course curriculum with minimal costs (minimum duration in academic hours) is reduced to finding a combination of counter way that are minimal in terms of total weight, connecting the vertices of target competence with one of the peaks of initial competence. It is important to note that achieving all specified initial competency vertices is not mandatory.

The structure of a typical educational module for the course consists of a passport with the program’s content and an educational–methodical complex. Its composition and volume correspond to the types of tasks and their complexity specified in the program’s passport. It includes presentations, assignments for practical sessions, independent exercises, and a list of literature and internet references.

The next stage of the training involved assessing the effectiveness of the implemented organizational and pedagogical conditions for developing a teacher’s methodological competence during professional development.

During the analysis, the following tasks needed to be addressed:

(1)

Correction of the process of developing a teacher’s methodological competence during professional development, taking into account previously obtained results;

(2)

Final assessment of the level of development of the teacher’s methodological competence.

The purpose of the final assessment was to determine the effectiveness of the work carried out in enhancing a teacher’s methodological competence. To address this task, test assignments and surveys were used. These tests and surveys were developed based on the same principles that were used during the initial analysis when conducting similar diagnostic methods.

Table 5. The results of the experiment on the formation of components of methodological competence among teachers of EG₁ and EG₂ (in %).

The Levels of Formation of the Teacher’s Methodological Competence	Instructional Delivery				Curriculum Design				Assessment Skill				Soft Skills
	EG1		EG2		EG1		EG2		EG1		EG2		EG1		EG2
	Ind.	%	Ind.	%	Ind.	%	Ind.	%	Ind.	%	Ind.	%	Ind.	%	Ind.	%
Zero	26	42.6	1	1.6	36	59	6	9.8	41	67.2	25	40.9	29	47.5	18	29.5
Situational	18	29.5	9	14.7	7	11.4	6	9.8	12	19.6	6	9.8	19	31.1	18	29.5
Developing	11	18	22	36	10	16.3	23	37.7	5	8.1	10	16.3	7	11.4	13	21.3
Advanced	4	6.5	21	34.4	2	3.2	11	18	2	3.2	16	26.2	4	6.5	5	8.1
Expert	2	3.2	8	13.1	6	9.8	15	24.5	1	1.6	4	6.5	2	3.2	7	11.4
The average value %	19.96		28.75		39.87		25.41		49.81		27.69		34.11		23.91

presents the results.

During the assessment, it was found that EG₂ demonstrated statistically significant improvement in the level of teachers’ methodological competence components compared to EG₁.

Notably, significant enhancements were observed across advanced, high, average and low levels. However, the differences between groups did not reach statistical significance at an acceptable level.

Based on this, continuing the professional development program to further enhance teachers’ competence in EG₂ is recommended. Also, conducting additional research to identify specific factors contributing to competence improvement and adapting the program based on the discovered results would maximize effectiveness in future educational and professional contexts.

5. Discussion

The conducted research has shown that didactic competencies are significantly superior to design, monitoring, and personal competencies. The results were based on the procedure of assessing the potential for changing the level of digital competencies and building a trajectory of formation of digital competencies of higher education teachers based on the project-vector methodology. This allowed higher education administrations to determine the administrative vector of teacher’s movement to the desired indicator of the competence level due to the proposed method of calculating the optimal learning trajectory. This allowed us to determine the level change potential for each of the digital competency groups of the faculty member within the range of 1.869 to 1.770, and the potential for a comprehensive assessment of the digital competency level of higher-education faculty members. That is, the potential for changing the level of digital competencies of the teacher has a value greater than 1; thus, the potential for changing the competencies has positive dynamics.

The creation of a universal complex method for calculating the assessment of the level of digital competencies of teachers of HEE and determining the training trajectory allows for flexible changes and corrections of the competency system. This allows us to use the research method to study the dynamics of change in the level of competencies of respondents. In contrast to the studies of other scholars, where different approaches of static assessment of digital competencies level are considered, in this study, in addition to assessment, the trajectory of competency-level change is constructed. This allows us to assess the potential of each teacher in the process of achieving the appropriate level of competence. The obtained result is made possible by integrating project-vector management methodology into the process of assessing the level of digital competencies.

However, a number of factors have been identified that influence the effectiveness of teachers’ professional development, such as different levels of education, the nature of learners, and the short-term nature of training. Successful development of methodological competence requires individualized planning of the educational process, adapted to the needs and capabilities of teachers.

The developed method of assessing the level of digital competence and building an individual learning trajectory of higher education teachers allows scientifically based determination of the level of digital competence, which contributes to improving the effectiveness of higher education institutions.

The limitations of the study are related to the fundamental complexity of determining the competencies of teachers. The survey used in the study contains open-ended questions and therefore requires interpretation of the answers.

The proposed mathematical model of individual educational trajectories for teachers allows taking into account their level of education, interests, and professional demands. The methodology of individual trajectory training is based on the four components of methodological competence and the matrix of target results. This approach makes it possible to systematize and track progress in the development of methodological competence, ensuring effective and personalized training of teachers in a rapidly changing pedagogical environment.

The study also describes the information-analytical system developed by the authors, in which the scientific methods of the authors’ team are implemented and tested.

The main drawback of the study is the small sample size of the validation of the procedure for assessing the potential for changing the level of digital competencies and building a trajectory of formation of digital competencies of higher education teachers on the basis of project-vector methodology. To eliminate it, it is necessary to conduct procedures for assessing the potential for changing the level of digital competencies of teachers in other universities of the Republic of Kazakhstan and beyond.

In this study, the method of building an individual learning trajectory on the basis of project-vector management methodology was verified. The testing of 62 teachers at Astana IT University, Karaganda Buketov University, and Toraygyrov Universitet was taken as a basis. The analysis provided for measuring the achievement of their level of competence was divided into four categories: didactic, design, monitoring, and personal. The tests in these areas were conducted from 20 September to 7 October 2023 and from 1 May to 15 May 2024. The obtained results show that didactic competencies of teachers of WEE of the Republic of Kazakhstan significantly exceeded designing, monitoring, and personal competencies. Based on this study, the vector of administrative influence was aimed at the development of designing, monitoring and personal competencies of teachers was determined. The potential of a comprehensive assessment was based on the level of digital competencies of higher education teachers. This method does not need to form a separate list of weighting factors to take into account one or another category of competence assessments to a greater or lesser extent in the overall assessment. The competency levels are in equilibrium. In addition, if the list of competencies to each category is changed, the form of calculation of a comprehensive competency level assessment will not change.

This study is a pilot study and shows the main advantages of using the method of competency level assessment and building an individualized learning trajectory for teachers. In the future, it is planned to test and apply this approach for use in other institutions of higher education in the Republic of Kazakhstan.

6. Conclusions

The main process in the formation of an individual learning trajectory is the development of a course program that forms an individual educational learning trajectory, the definition of invariant and variable modules for the development of methodological competence, their content, the form of conducting classes, the didactic complex, based on the needs of the university teacher, their capabilities, goals and objectives to be achieved by the teacher, and the indication of the necessary time for mastering the invariant and variable part of the route.

The developed mathematical model was implemented and used in the educational information system as a built-in module, which has the ability to take into account not only the goals and constraints predetermined by the standards, but also to optimally meet the individual needs of teachers.

According to the analysis conducted, it was found that professional development courses for teachers significantly improved their levels of methodological competence. This demonstrates the high effectiveness of these courses and their importance for the professional growth of teachers. Enhancing teachers’ methodological competence is a key factor in improving educational quality. Regular professional development courses, support, motivation for IT discipline teachers, and the implementation of modern educational technologies contribute to their professional growth and, as a result, improve the educational results of students.