Assessing the Impact of Competency-Based Integration of Mathematical and Natural Science Knowledge on Critical Thinking, Analytical Skills, and Real-World Problem-Solving

Background: In modern education, there is a growing need to adopt a competency-based approach that supports the integration of knowledge across disciplines. One promising example is the use of Euler's formulas, which fosters comprehensive student development and the practical application of knowledge in real-world contexts. Geometry, particularly when dealing with lesser-known formulas, offers opportunities for creative, non-standardised instruction. Objective: This study aims to assess the effectiveness of implementing a competency-based approach that integrates mathematical and natural science knowledge, using Euler’s formulas, with a focus on developing students’ critical thinking, analytical skills, and real-world problem-solving abilities. Methodology: This study utilised a comparative theoretical analysis of classical proofs for Euler’s formulas to identify and extract their untapped pedagogical potential. Using geometric didactic modelling, the researcher developed enhanced, visually intuitive proofs designed to bridge the gap between abstract algebra and spatial geometry. These integrated tasks – combining geometry and physics – were piloted in the “Emotional Formula Geometry” project (Kyiv) and implemented during undergraduate seminars to assess their classroom efficacy. The qualitative validity of the tasks was established through expert peer review, and instructional reliability was ensured through consistent longitudinal use across multiple educational settings to assess student engagement and conceptual retention. Results: The reinterpreted proofs of Euler’s formulas facilitated a clearer understanding of complex geometric relationships. Their integration into practical construction tasks improved student engagement and interdisciplinary thinking. Conclusion: Tasks based on Euler’s formulas effectively promote interdisciplinary connections, enhance students’ spatial reasoning, and contribute to improved educational outcomes in mathematics and science. Unique Contribution: The study introduces didactically enhanced versions of Euler’s formulas and demonstrates their practical application in geometry-physics integration. This approach enriches teaching methodology and supports competency-based education. Key Recommendation: Further research should explore broader applications of classical mathematical relationships in cross-disciplinary learning scenarios and evaluate their long-term impact on student competency development.