FEATURES OF INTRODUCING GENERAL SECONDARY EDUCATION STUDENTS TO THE GENERALIZED CANTOR SET USING NUMERATION SYSTEMS - Malyshenko, Kateryna (Physical and Mathematical Education, 2025)
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".