INTEGRATIVE APPROACH TO TEACHING COMPUTER-BASED MATHEMATICAL SYSTEM

RESEARCH AND DEVELOPMENT TADQIQOT VA TARAQQIYOT ISSN: 3030 – 3281 Volume II, Issue 11 (2025) © 2025, the Author(s). Published by IMFAKTOR. This is an open access article under the CC BY license http://creativecommons.org/licenses/by/4.0/ 76 INTEGRATIVE APPROACH TO TEACHING COMPUTER-BASED MATHEMATICAL SYSTEM Kayum KARIMOV¹. ¹ Ph.D., Acting Professor, Karshi State University, Uzbekistan. https://doi.org/10.5281/zenodo.17726143 ANNOTATION The article provides detailed methodological recommendations for carrying out mathematical calculations by leveraging the capabilities of modern computer algebra systems and network-based technologies. It emphasizes not only the technical advantages of these tools – such as increased computational accuracy, efficiency, and visualization – but also their pedagogical value in contemporary mathematics education. Scientifically grounded arguments are presented to substantiate that the integration of information technologies into the teaching of mathematics significantly enhances students’ ability to acquire, comprehend, and apply mathematical knowledge. The article highlights that interdisciplinary approaches foster deeper conceptual understanding, develop digital literacy, and promote problemsolving skills that are essential in the digital era. Additionally, the findings suggest that such integration encourages independent learning, increases student engagement, and supports the development of higher-order thinking competencies. Key words: algebraic equations, graphics, integral, interactive method, package, network technologies. KOMPYUTERLI MATEMATIKA TIZIMLARINI O‘QITISHDA INTEGRATIV YONDASHUV ANNOTATSIYA Maqolada zamonaviy kompyuter algebra tizimlari va tarmoqqa asoslangan texnologiyalarning imkoniyatlaridan foydalangan holda matematik hisob-kitoblarni amalga oshirish bo‘yicha batafsil metodik tavsiyalar beriladi. Unda ushbu vositalarning texnik afzalliklari – hisoblash aniqligining oshishi, samaradorlik va vizualizatsiya imkoniyatlari – bilan bir qatorda, zamonaviy matematika ta’limidagi pedagogik ahamiyati ham ta’kidlanadi. Matematikani o‘qitish jarayoniga axborot texnologiyalarini integratsiya qilish talabalarning matematik bilimlarni egallashi, anglash va amaliy qo‘llashi samaradorligini sezilarli darajada oshirishi ilmiy asoslangan dalillar bilan isbotlab beriladi. Maqolada fanlararo yondashuvlar talabalarda tushunchalarning chuqurroq shakllanishi, raqamli savodxonlikning rivojlanishi va zamonaviy raqamli davr uchun zarur bo‘lgan muammolarni hal qilish ko‘nikmalarining mustahkamlanishiga xizmat qilishi ta’kidlanadi. Bundan tashqari, olingan natijalar bunday integratsiya mustaqil ta’limni qo‘llab-quvvatlashi, talabalarning o‘quv jarayoniga jalb etilishini oshirishi va yuqori darajadagi fikrlash kompetensiyalarining rivojlanishiga ko‘maklashishini ko‘rsatadi. Kalit soʻzlar: algebraik tenglamalar, grafika, integral, interaktiv usul, paket, tarmoq texnologiyalari. ИНТЕГРАТИВНЫЙ ПОДХОД К ОБУЧЕНИЮ КОМПЬЮТЕРНОЙ МАТЕМАТИЧЕСКОЙ СИСТЕМЕ АННОТАЦИЯ В статье представлены подробные методические рекомендации по выполнению математических вычислений с использованием возможностей современных систем компьютерной алгебры и сетевых технологий. Подчёркивается не только техническое превосходство этих инструментов – повышение точности вычислений, эффективности и возможности визуализации, – но и их педагогическая ценность в современном математическом образовании. Приводятся научно обоснованные аргументы, подтверждающие, что интеграция информационных технологий в процесс преподавания математики существенно повышает способность студентов усваивать, понимать и применять математические знания. В статье отмечается, что междисциплинарные подходы способствуют более глубокому формированию понятий, развитию цифровой грамотности и укреплению навыков решения проблем, которые являются ключевыми в цифровую эпоху. Кроме того, полученные результаты показывают, что такая интеграция стимулирует самостоятельное обучение, повышает вовлечённость студентов и способствует развитию навыков мышления высокого уровня. Ключевые слова: алгебраические уравнения, графика, интеграл, интерактивный метод, пакет, сетевые технологии. TADQIQOT VA TARAQQIYOT | II-JILD | 11-SON | 2025 \ 77 Today, the global education system demands the use of advanced innovative technologies. In our country, particular attention is being given to the development of the education sector. The experiences of the education systems of Singapore, Russia, France, Japan, Korea, the United States and other developed nations are being studied and adapted to the educational process. Issues related to shaping the personality of future teachers, developing their professional skills and competencies in the educational process of our republic have been examined in the works of N. Muslimov, M. Ochilov, Z. F. Sharapova, U. Yuldashev, O. Musurmonova, M. T. Kadirova, M. M. Vakhobov, Kh. Sh. Kodirov, O. Tolipov, N. N. Azizkhodjaeva, D. I. Yunusova, T. A. Orlova, T. J. Oknazarov, D. Z. Khidirova, and others. The works of A. A. Abdukodirov, A. G. Khayitov, M. Aripov, B. B. Muminov, M. Mamarajabov, F. Zakirova, U. Sh. Begimkulov, I. Isakov, S. Kulmamatov, D. Toshtemirov and other scholars focus on the use of information and communication technologies in education and on improving the quality of teaching. However, their research has not sufficiently explored students’ mastery of knowledge regarding the capabilities of modern computer mathematical programs for performing calculations through interdisciplinary integration and the use of local area networks. During an event held on September 30, 2020, the President of the Republic of Uzbekistan, Sh. M. Mirziyoyev, emphasized the following: “We have set ourselves the main goal of creating the foundation for a new era of awakening – the Third Renaissance – in Uzbekistan through large-scale democratic reforms, including educational transformation. In this regard, it is essential that each of us, and our entire society, deeply understand the meaning and essence of the Third Renaissance. We consider preschool and school education, higher and secondary specialized education, as well as scientific and cultural institutions, to be four integral pillars of the future Renaissance. We regard our kindergarten teachers, schoolteachers, professors, and scientific and creative intellectuals as the four pillars of this new era.” In this context, it has become a pressing requirement of modern times for future teachers to master the use of modern computer programs in their professional activities. For university students, it is particularly important to understand the capabilities of contemporary computer mathematics systems (application software packages) and to acquire practical skills in working with them. Teaching mathematical knowledge through interdisciplinary integration with information technologies yields highly effective educational outcomes. In today’s rapidly evolving information environment, the primary task of education is to equip students with the ability to work independently and to use the ever-growing flow of information wisely. To achieve this, students must be provided with opportunities and conditions for continuous, self-directed learning. It is therefore essential to teach general professional subjects using information and communication technologies and to focus on improving students’ knowledge and skills to a higher level. Creating the necessary conditions and developing new methodological requirements for fostering students’ creative activity through the use of modern computer mathematics systems and network technologies has become a demand of the time. Among computer mathematics systems, Mathematica and Maple are designed for professional mathematicians and are distinguished by their broad functionality and ease of use. Maple includes a core system for symbolic computation in mathematics, incorporating hundreds of symbolic manipulation algorithms and predefined functions. It also features a library of functions, commands, and operators, along with numerous downloadable packages [1]. The introduction of modern information and communication technologies into the educational process, alongside traditional teaching methods, has led to the emergence of instruction based on computer networks. An analysis of local and international sources (D. Corneli, C. Danoff, D. McGregor, E. D. Patarakin, V. A. Polyakova, B. Smith, etc.) [2; 3; 4; 5] shows that the concept of the peer model is regarded as a promising direction in higher professional education. In online learning, this model is implemented through information and communication technologies, whereby students contribute equally to solving common problems and interact within a learning community. In general, instruction supported by computer networks pursues the following objectives: a) meeting the educational needs of learners, b) ensuring a new level of core education while maintaining its quality. The introduction of network-based teaching in educational institutions is beneficial in various aspects. As practice shows, network learning encompasses two fundamental approaches: personal and individual. Personal learning refers to the student’s attitude toward his or her own education – that is, the recognition of the need to study a particular subject and the pursuit of information that meets personal interests. RESEARCH AND DEVELOPMENT | VOLUME II | ISSUE 11 | 2025 \ 78 Individual learning, on the other hand, is a model of organizing the educational process in which the instructor interacts with a single student, taking into account the learner’s personal characteristics and creating the necessary psychological and pedagogical conditions for development. This model presupposes the presence of a coach or facilitator who constructs the student’s learning trajectory [6]. All the necessary conditions exist for the implementation of these approaches in the higher education system. For students specializing in Mathematics and Informatics, Applied Mathematics, and Methodology of Informatics, it is particularly important to enhance their knowledge of the computational capabilities of computer mathematics systems through the use of computer networks during instruction. Whereas the content of education was previously understood as a set of ready-made knowledge and algorithms that a graduate needed to master for professional practice, the new educational paradigm views its content as a process of acquiring knowledge and applying it directly in practice. Refusing to provide students with pre-packaged information means fostering their ability to independently search for, create, discover, and acquire knowledge [7]. Issues related to the use of computer mathematics systems in the teaching of mathematics have been examined in the works of B. I. Glisburg, B. A. Dalinger, B. P. Dyakonov, Yu. G. Ignatyev, T. B. Kapustina, M. P. Lapchik, B. R. Mayer, M. I. Ragulina, E. K. Henner, and others. However, the analysis of cases involving the graphical and computational capabilities of the Maple computer mathematics system, as well as the study of effective aspects of organizing lessons using network technologies, has not been sufficiently addressed. Considering the relevance of these issues, this research aims to examine and analyze the use of computer mathematics systems, develop a methodology for the effective application of network technologies, and prepare appropriate methodological recommendations. The aim of the research is to analyze the capabilities of modern computer mathematics systems, identify their effective aspects, and study the state of their application through network technologies. Research tasks a) to determine the possibilities and potential of using computer mathematics systems in performing mathematical calculations, b) to study the current state of the use of network technologies in the educational process. In the study of practical software packages such as Maple, Mathcad, and MATLAB, interactions naturally arise with geometry, algebra, mathematical analysis, and differential equations. When solving problems in these areas – for example, algebraic exercises – application software packages can be effectively used to obtain solutions. Integration becomes especially evident in constructing graphs of mathematical functions using Mathcad or Maple during instruction on application software packages. For instance, plotting function graphs ensures interdisciplinary integration when deriving the properties of functions. An analysis of the capabilities of computer mathematics systems shows that their effective application in the educational process can be characterized as follows: 1. Students acquire skills in using the advanced capabilities of programming and symbolic computation languages; 2. When using software packages, it becomes possible to analyze all potential solutions to a given practical problem and select the most efficient method of solving it; 3. Subject material is mastered by students in a systematic and logical manner; 4. Application software packages serve as software libraries essential for future scientific research; 5. The ability to modify or expand software packages as needed directs the student’s cognitive activity toward specific educational goals; 6. Students’ confidence in their knowledge and ability to solve practical problems increases, thereby enhancing motivation for new creative research. Thus, a specific software package can be used to solve any given problem [8]. Training students to use technologies aimed at independent learning and the continuous intensification of their academic activity is essential. Organizing the educational process with the help of computer technologies and information and communication tools has a positive impact on the effectiveness of learning. At present, there is a strong need to prepare competent, intellectually mature specialists capable of organizing the educational process through the use of network technologies and information and communication tools. The use of network technologies in higher education institutions is especially effective for improving students’ knowledge in working with application software packages. The Maple application software package is a system for both analytical and numerical solutions of mathematical problems encountered in mathematics and applied sciences. Commands and structures used in Maple allow users to solve algebraic and differential equations, perform symbolic and numeric operations, and construct graphs. Maple has the capacity to address a wide variety of problems related to mathematical analysis and linear algebra. TADQIQOT VA TARAQQIYOT | II-JILD | 11-SON | 2025 \ 79 The plots package in the Maple computer mathematics system offers extensive opportunities for working with graphical representations. With this package, it is possible to plot several functions simultaneously in different colors (e.g., black, red, blue) within the same coordinate system [9, 10]. Graphs can also be used to identify equation roots and to determine approximate values of those roots. In teaching mathematical calculations using computer mathematics systems, it is advisable to develop students’ skills in using Maple’s capabilities for constructing two-dimensional graphical representations. For example, the following assignment can be given to a student over a local network: Task 1. It is necessary to find the face S of the figure bounded by the parabola y =x2+1 and the straight-line y=x+3. Performs the task as follows: Determines the roots of the equation x2+1=x+3. This is defined in Maple 11 as: > So, the roots of the equation are x1=-1 and x2=2. Graphs the functions y =x2+1 and y=x+3 using Maple 11. The construction of the graph is carried out with the following command (Fig. 1): Figure 1. Graphs of the functions y =x2+1 and y=x+3 in Maple 11. It can be seen from this graph that the first face is bounded from above by the straight-line y=x+3, and the second by the parabolic arc y =x2+1, and [-1; 2] can be found as the difference of the faces of two trapezoids S1 and S2: s1=∫(𝑥 + 3)𝑑𝑥 2 −1 S2=∫(𝑥2 2 −1 + 1)𝑑𝑥 because S=S1-S2 So, the result is S=4.5. NetSupport School is one of the modern programs that provides a wide opportunity to work under the control of the teacher's computer using the network in computer classes. RESEARCH AND DEVELOPMENT | VOLUME II | ISSUE 11 | 2025 \ 80 The network environment offers a wide range of capabilities that enhance the organization and management of the educational process. It allows instructors to start and shut down students’ computers remotely, as well as to divide them into small working groups for collaborative tasks. Teachers can send files and assignments directly to any student’s computer and display information from a student’s screen on their own monitor in order to review completed work. In addition, the system provides tools for enabling or restricting students’ access to Internet resources and for monitoring or disabling Internet activity when necessary, ensuring a controlled and productive learning environment. Another similar feature is provided through the NetSupport School software. By installing this program on the teacher’s computer, all of the above operations can be performed efficiently through built-in commands. When conducting classes via a local area network, the teacher uses the program’s built-in journal function. To use it, the teacher activates the “Journal” button on the toolbar and selects the “Start” command. The “Start Journal” window will then appear on the screen. This window displays the instructor’s name, the lesson topic (e.g., “Calculations in Maple”), the lesson objective (e.g., “Learning mathematical calculations in Maple”), as well as the group being taught (for example, 023-43). To explain the content of the lesson, the teacher uses the program’s whiteboard feature. The whiteboard is launched through the horizontal menu. Learning materials related to the lesson are created using the “Whiteboard” menu item. To save the prepared materials as a file, the “File → Save ‘Whiteboard’” command is used. The teacher may select the necessary tools in the “Tools” section of the “Whiteboard” menu to prepare instructional materials. When drawing, the teacher selects line thickness using the “Line Width” command and creates the required shapes or curves. When typing text, the font style and size are adjusted using the “Font” command. The color of lines and text is selected through the “Color” command, enabling the creation of clear and visually effective learning materials. The “Show Whiteboard” command allows the teacher to display the whiteboard content on students’ computer screens. In this way, all students see the teacher’s materials on their own monitors and follow the explanation in real time. During the reinforcement stage of the lesson, the teacher can send practice tasks related to the topic to students’ computers via the network. Before doing so, the teacher groups students’ computers into small teams and assigns a specific task to each group. NetSupport School is a convenient tool for enabling communication and collaboration between teachers and students over the network. Using the “File Transfer” command (button on the panel), the teacher sends an assignment to the students. Students study the task and complete it using the Maple computer mathematics system. The teacher checks the completed task by viewing the students’ results through the network. Using the teacher’s desktop interface, various interactions can be carried out during the lesson via the local area network. If the teacher has multiple monitors, the computer screen can be broadcast to any chosen monitor. The teacher may also send one student’s screen to another student or to the entire class. Studying the capabilities of computer mathematics systems through a local computer network provides several important outcomes. First, students gain an in-depth understanding of the functionalities of modern computer mathematics systems and develop creative and analytical thinking skills while solving mathematical problems. This contributes to their ability to choose effective solution strategies and to justify their reasoning scientifically. Second, the use of network technologies strengthens students’ independence and increases their activity in the learning process. When learners receive tasks through the network, complete them using specialized software, and present results digitally, they become more responsible for their own learning and more motivated to master complex concepts. Third, students’ knowledge, skills, and competence in using the NetSupport School network program increase significantly. They learn how to engage in collaborative tasks, exchange information efficiently, and make productive use of digital communication tools during lessons. This prepares them for modern educational and professional environments where network-based collaboration is essential. Fourth, methodological knowledge and pedagogical skills also improve. Students learn how different teaching tools – such as computer mathematics systems, virtual whiteboards, and digital monitoring tools – can be used effectively in the classroom. As a result, their understanding of modern teaching methodologies expands, contributing to improved learning outcomes and professional readiness. Overall, the results of the research confirm that integrating computer mathematics systems and network technologies into the educational process plays a vital role in forming the competencies required of future specialists. In higher education institutions, it is crucial to ensure that graduates possess not only strong theoretical foundations but also the practical digital skills demanded by the current era. TADQIQOT VA TARAQQIYOT | II-JILD | 11-SON | 2025 \ 81 Such specialists must be able to think independently, solve complex problems, confidently navigate the labor market, and demonstrate a high level of professional competence in their field. The findings of this study show that the effective use of Maple, Mathcad, MATLAB, and NetSupport School in teaching mathematics and related disciplines significantly enhances the quality of education. Therefore, it is recommended that higher education institutions further develop digital infrastructure, expand the use of computer mathematics systems, and provide systematic training for both teachers and students. This will ensure the formation of a technologically literate generation capable of meeting contemporary scientific, technological, and economic challenges. RESEARCH AND DEVELOPMENT | VOLUME II | ISSUE 11 | 2025 \ 82 References 1. Dilmurodov, N., Karimov, Q. M., & Eshkarayeva, N. G. (2010). Working in the Maple environment: Study guide. QARSHI: Nasaf. 2. Corneli, D., & Danoff, Ch. Paragogics: Synergy of independent and organized learning. Retrieved February 15, 2013, from http://www.connected-learning.ru/home/ravnogogika/lst-paper 3. Patarakin, E. D. (2006). Setevye soobshchestva i obuchenie. Moscow: PER SE. 4. Polyakova, V. A. (2008). Model' formirovaniya gotovnosti uchitelya k dialogovomu vzaimodeystviyu v setevykh pedagogicheskikh soobshchestvakh. Sovremennye problemy nauki i obrazovaniya, (6), 100–106. 5. Smith, W. L., & MacGregor, J. T. (1992). What Is Collaborative Learning? National Center on Postsecondary Teaching, Learning, and Assessment, Pennsylvania State University. Retrieved February 17, 2013, from http://www.evergreen.edu/washcenter/natlc/pdf/collab.pdf 6. Nagaeva, I. A. (2013). Setevoe obuchenie: stanovlenie i perspektivy razvitiya. Nauchnoe obespechenie sistemy povysheniya kvalifikatsii kadrov: Nauchno-teoreticheskiy zhurnal, 3–4(16–17), 31–37. 7. Ergashev, J. B. (2016). An integrative approach to teaching mathematics. Tashkent: Science and Technology. 8. Olimov, M. (2021). Package of practical mathematical programs: Textbook. Namangan. 9. Karimov, Q. M. (2023). Development of students' knowledge and skills using a local computer network. Pedagogy: Scientific-Theoretical and Methodical Journal, (06/2023), 203–205. 10. Karimov, Q. M. (2023). Acceleration of mathematics education by means of computer mathematics systems. In Modern Problems of Analysis: Proceedings of the Republican Scientific Conference (pp. 358–359). Qarshi, June 2–3, 2023. 11. NetSupport School, Version 12. User Manual, p. 23. TADQIQOT VA TARAQQIYOT | II-JILD | 11-SON | 2025 \ 83 SCIENTIFIC REVIEW Article: “Integrative approach to teaching computer-based mathematical system” Author: Kayum Karimov, Ph.D., Acting Professor, Karshi State University, Uzbekistan. The article under review presents a comprehensive and well-structured study on the integration of computer mathematics systems and network technologies into the modern educational process. The topic is highly relevant in the context of global digital transformation, growing demands for technological competence in education, and ongoing reforms aimed at improving the quality of teaching and learning in higher education institutions. The author convincingly demonstrates that the global education system increasingly requires the incorporation of advanced information and communication technologies. The analysis of international experience – particularly that of Singapore, Russia, France, Japan, Korea and the United States – clearly illustrates the importance of adapting effective foreign practices to local educational contexts. The emphasis on developing the professional competencies of future teachers is fully consistent with national policy priorities and with the strategic goals articulated by the President of the Republic of Uzbekistan regarding the formation of the “Third Renaissance.” In this regard, the necessity of mastering modern computer programs and digital tools is substantiated not only by current educational challenges, but also by broader societal demands for highly qualified specialists capable of working confidently in a technologically enriched environment. The article makes several noteworthy scientific contributions. It identifies gaps in previous research related to students’ mastery of computer mathematics systems when these are used in combination with network technologies and interdisciplinary integration. It offers an in-depth analysis of the pedagogical potential of Maple, Mathcad, MATLAB and similar software packages for solving analytical, graphical and computational problems. The author proposes original methodological recommendations for the effective implementation of network-based learning environments, with particular attention to the use of NetSupport School. The study demonstrates how these technologies enhance students’ independent learning, creative thinking and analytical competencies, and it convincingly highlights the ongoing shift from a content-oriented educational paradigm to one that emphasizes the acquisition, application and independent construction of knowledge. These contributions are scientifically grounded and supported by both theoretical reflections and practical examples. The analysis of network technologies, and of NetSupport School in particular, is detailed and informative. The author carefully describes the program’s functions for classroom management, student monitoring, file transfer and interactive teaching, thus offering a valuable methodological tool for educators seeking to organize technology-rich learning environments. Overall, the article provides strong and well-argued justification for the practical importance of implementing computer mathematics systems and network technologies in university-level mathematics education. The findings demonstrate that such technologies deepen students’ understanding of mathematical concepts, strengthen their independence, motivation and responsibility for learning, enhance their digital literacy and communication skills, improve the organization and effectiveness of the teaching process and contribute to the preparation of specialists who are capable of meeting the demands of the modern labor market. The methodological recommendations put forward in the article can be directly applied in higher education institutions and may serve as a basis for the development of new curricula, training programmes and digital education strategies. In conclusion, the article is well written, logically structured and fully meets the scientific and methodological standards required for publication in a peer-reviewed academic journal. The research is original, relevant and firmly grounded in contemporary theoretical and practical frameworks. The conclusions drawn by the author are well supported by the analysis and are consistent with the stated aims and objectives of the study. On the basis of its scientific relevance, strong methodological foundation, thorough analysis and significant practical value, I recommend this article for publication in the journal without reservation. Disclaimer © This scientific review has been prepared by the editorial board of the “RESEARCH & DEVELOPMENT” journal and is intended solely for use within the journal’s internal expert evaluation process and editorial activities. This review is protected by copyright law, and its content may not be distributed, reproduced, or used for commercial purposes without the prior permission of the editorial board. The review has been prepared to assess the scientific quality, content, and methodological aspects of the author’s (authors’) work. It does not represent the personal opinion of the author(s) nor should it be interpreted as the official position of the journal. The editorial board bears no responsibility for the implementation, outcomes, or consequences of the recommendations, conclusions, or comments contained in this review. The review is provided to ensure transparency in the editorial process and to maintain quality control over scientific publications.