A COMPREHENSIVE METHODOLOGY FOR DEVELOPING MATHEMATICAL MODELING AND ALGORITHMIC THINKING THROUGH OPTIMIZATION AND COMBINATORIAL ANALYSIS

This article proposes a comprehensive methodology for developing mathematical modeling and algorithmic thinking skills through optimization problems and combinatorial analysis. The approach integrates algebraic structures, elementary calculus, and discrete mathematical methods to enhance students’ ability to analyze complex systems and construct optimal solutions. Special emphasis is placed on step-by-step modeling, functional analysis, derivative-based optimization, and combinatorial enumeration without the use of graphical tools. The methodology provides a structured framework for teaching, including algorithmic decomposition, recursive reasoning, and logical generalization. The effectiveness of the approach is demonstrated through theoretical explanations and practical applications in education.本文提出一套综合方法论，旨在通过优化问题与组合分析，培养学习者的数学建模与算法思维能力。该方法整合代数结构、初等微积分与离散数学方法，以提升学习者分析复杂系统、构建最优解的能力。本方法特别侧重无需借助图形工具的分步建模、泛函分析、基于导数的优化及组合枚举环节。该方法论为教学构建了结构化框架，涵盖算法拆解、递归推理与逻辑泛化等内容。本文通过理论阐释与教育领域的实际应用，验证了该方法的有效性。