CASCADE APPLIED PROBLEMS IN MATHEMATICS AS A MEANS OF DEVELOPING KEY COMPETENCIES OF STUDENTS

Abstract

Problem statement. Contemporary educational reforms in Ukraine emphasize the development of key competencies as mandatory learning outcomes at all levels of secondary education. State regulatory documents — the New Ukrainian School (2016), the State Standard of Basic Secondary Education (2020), and the State Standard of Profile Secondary Education (2024) — identify a broad spectrum of competencies that students are expected to acquire, including mathematical, civic, environmental, entrepreneurial, and digital competencies, among others. The school mathematics course has significant potential for integrated development, particularly through purposefully designed systems of applied problems that embed mathematical content in authentic real-world contexts. However, existing mathematics textbooks rarely include problems of this kind, creating a clear methodological gap between normative expectations and classroom practice. This article addresses that gap by focusing on a specific type of applied problem — the cascade applied problem — and on approaches to its effective use in teaching students at the basic and profile secondary education levels.

Materials and methods. The study employs theoretical research methods: analysis of regulatory documents; reference and educational literature on the theory and methodology of mathematics teaching; and synthesis, comparison, and generalization of the findings. The theoretical foundations of the study draw on international research in mathematical modeling and competency-based education.

Results. The article introduces and substantiates the concept of applied problems in mathematics as a cascade, an original didactic construct. A cascade-applied problem is defined as a problem that combines an authentic real-world context with a cascade of interrelated requirements of increasing complexity, mandatory interdisciplinary connections, a discussion question requiring reasoned judgment, and a purposeful orientation towards specific key competencies. The article proposes four sample cascade problems with detailed solutions, methodological commentary, and analysis of the competencies developed through each problem.

Conclusions. Cascade applied problems in mathematics represent a promising means of developing students' key competencies in the school mathematics course. Their cascade structure naturally provides for differentiation, enabling students of varying levels of preparation to engage meaningfully with the same problem. Solving such problems may contribute to the development of both mathematical and other key competencies stipulated by educational standards. A perspective for further research is the experimental verification of the effectiveness of cascade applied problems in real school practice, including an investigation of their impact on students' mathematical literacy across different age groups.