FEATURES OF INTRODUCING GENERAL SECONDARY EDUCATION STUDENTS TO THE GENERALIZED CANTOR SET USING NUMERATION SYSTEMS
DOI:
https://doi.org/10.31110/fmo2025.v40i2-03Keywords:
Cantor perfect set, Cantor function, numeral systems, development of research skills, STEM-educationAbstract
Formulation of the Problem. Ukraine is currently undergoing an active reform of basic secondary education. In our opinion, mathematical education should be structured in such a way that, upon completing general secondary education, students not only acquire the ability to perform basic mathematical computations but also develop skills in solving problems involving integrated topics, identifying various applications of studied concepts, forecasting and analyzing different mathematical models, formulating hypotheses, and proving or refuting them. Achieving these goals significantly depends on students' interest and intrinsic motivation to study mathematics. We propose organizing the learning process to ensure that students not only study the provided material but also actively engage as researchers.
Materials and Methods. A combination of theoretical and empirical methods was employed, including an analysis of scientific literature on the identified issue and observation of the mathematics curriculum in general secondary schools to explore possibilities for integrating the topic "Cantor Set and Numeral Systems".
Results. The study demonstrated that it is possible to construct Cantor perfect sets for numeral systems with a base of 2n+1, where n is a natural number. Specifically, the quinary (base-5) and septenary (base-7) numeral systems were considered, enabling further generalizations. As a result, perfect sets were obtained, and their perfection was proven. Cantor functions for the quinary and septenary numeral systems were constructed, and patterns were identified, leading to the generalization of the Cantor function for all numeral systems with a base of 2n+1.
Conclusion. It was determined that for any natural number n, it is possible to construct a Cantor perfect set with a base of 2n+1, for which a Cantor function can also be developed. The analysis of regulatory documents, mathematical, instructional, and psycho-pedagogical literature highlights the relevance of fostering research competencies among secondary school students in Ukraine’s current educational development phase. In our study, this is achieved through solving an integrated problem on the topic "Cantor Set and Numeral Systems".
Downloads
References
Bios, Dzh. E. (2022). Mathematics: Textbook for 5th grade general secondary education institutions. Kyiv: "Formula".(in Ukrainian)
Bios, Dzh. E. (2023). Mathematics: Textbook for 6th grade general secondary education institutions. Kyiv: "Formula".
Borysenko, O. A. (1995). Differential geometry and topology. Osnova. (in Ukrainian)
Hrynevych, L. M., Elkin, O., Kalashnikova, S., Kobernyk, I., Kovtunets, V., Makarenko, O., ... & Shyyan, R. (2016). The New Ukrainian School: Conceptual foundations of secondary school reform. (in Ukrainian)
Zhaldak, M. I., & Mykhalin, H. O. (2011). Elementary facts of set theory in the school mathematics course. Mathematics in School, (3), 12-23. (in Ukrainian)
Zhaldak, M. I., Mykhalin, H. O., & Dekanov, S. Ya. (2007). Mathematical analysis. Functions of several variables. NPU named after
M. P. Dragomanov. (in Ukrainian)
Merzlyak, A. H., Polonskyi, V. B., & Yakir, M. S. (2021). Algebra 8. Textbook for classes with an in-depth study of mathematics. Kharkiv: Gymnaziia. (in Ukrainian)
Model Curriculum "Mathematics. Grades 5-6" for institutions of general secondary education (authors: Vasylyshyn M. S., Mylyanyk A. I., Pratsovytyi M. V., Prostakova Y. S., Shkolnyi O. V.).
Retrieved from https://drive.google.com/file/d/1YMPwWKLNmdHTQ6wj4_5aUH0sPafkCBqX/view. (in Ukrainian)
Model Curriculum "Mathematics. Grades 7-9" for institutions of general secondary education (authors: Vasylyshyn M. S., Mylyanyk A. I., Pratsovytyi M. V., Prostakova Y. S., Shkolnyi O. V.). Retrieved from https://drive.google.com/file/d/1hxfR8CXPRbsZ16yos4CykfiJ-K5U-cKu/view. (in Ukrainian)
Onopriienko, O. V. (2023). The competency potential of learning activities in mathematics lessons in the New Ukrainian School. In Primary education in the paradigm of the New Ukrainian School: Challenges of the time: Collection of materials of the All-Ukrainian scientific and practical conference (April 27, 2023, Hlukhiv) (pp. 104-107). Hlukhiv National Pedagogical University named after Oleksandr Dovzhenko. (in Ukrainian)
On some issues of state standards for complete general secondary education, Resolution of the Cabinet of Ministers of Ukraine No. 898 (2022). URL: https://zakon.rada.gov.ua/laws/show/898-2020-п#Text. (in Ukrainian)
Approval of the Concept for the Development of Natural-Mathematical Education (STEM Education), Order of the Cabinet of Ministers of Ukraine No. 960-р (2020). URL: https://zakon.rada.gov.ua/laws/show/960-2020-р#Text. (in Ukrainian)
Tarasenkova, N. A. (2016). Competency-based principles of ensuring continuity in mathematics education at different levels of education. In Implementation of continuity in mathematical education: Realities and prospects: Materials of the All-Ukrainian scientific and practical conference (September 15-16, 2016, Odesa) (pp. 108-111). Odesa: South Ukrainian National Pedagogical University named
after K. D. Ushynskyi. (in Ukrainian)
Shkolnyi, O. (2024). Methodological features of studying the logical foundations of mathematics in the integrated course "Mathematics" for 7th-grade students of the New Ukrainian School. Didactics of Mathematics: Theory, Experience, Innovations, (2), 20-28. (in Ukrainian)
Casola, L., & Taylor, T. E. (Eds.). (2019). Increasing Student Success in Developmental Mathematics. National Academies Press. https://doi.org/10.17226/25547
Hayes, E. (2011). Science teachers take to the stage. Science in School, (19), 6-9.
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools. (2002). National Academies Press. https://doi.org/10.17226/10129
Rizayeva, L. (2022). Formation of research skills of students through solving problems in teaching mathematics in primary classes. Cypriot Journal of Educational Sciences, 17(8), 2567-2579.
Tandeep, K., McLoughlin, E., & Paul, G. (2022). Mathematics and science across the transition from primary to secondary school: a systematic literature review. International Journal of STEM Education, (9).
Tifi, A., Natale, N., & Lombardi, A. (2006). Scientists at play: teaching science process skills. Science in School, (1), 37-40.
Downloads
Published
Issue
Section
Categories
How to Cite
License
Copyright (c) 2025 Катерина Малишенко
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
- Authors grant the journal a right of the first publication of the work under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0)that allows others freely to use (read, copy and print) submissions, search content and link to published articles, disseminate their full text and use them for any legitimate non-commercial purposes (i.e. educational or scientific) with the mandatory reference to the article’s authors and initial publication in this journal.
- Original published articles cannot be used by users (exept authors) for commercial purposes or distributed by third-party intermediary organizations for a fee.
