Unveiling Mathematical Elegance: Exploring The Synergy Between Algebra and Geometry

In today’s dynamic and complex society, the ability to represent mathematical concepts and solve real-world problems is essential for mathematics education students. This study explores how algebraic and geometric representations can be synergized to solve environmental problems commonly faced in everyday life, particularly through the “builder’s problem” involving quadrilaterals. Using a qualitative, exploratory case study approach, one student with high mathematical ability was selected through a 10-item Mathematical Ability Test (TKM). Data collection included a problem-solving task and semi-structured interviews to examine the student’s reasoning processes. Data were analyzed using Miles and Huberman’s model: data condensation, data display, and conclusion drawing. Triangulation of test results, written work, and interview data ensured the validity of findings. Results indicate that the student was able to construct a mathematical model of the problem algebraically and then reinterpret it geometrically, revealing an equivalent but more conceptually elegant solution. Specifically, solving for the yard boundary of a rectangular plot algebraically led to the same value as determining the radius of an incircle within a right triangle geometrically. However, while the algebraic representation was more precise, the geometric approach offered deeper visual insight into the structure of the problem. This study highlights the importance of dual representations in enhancing mathematical understanding and problem-solving. It suggests the integration of real-world contexts and multiple solution strategies in teacher education programs, and recommends further research with diverse participants to deepen insights into algebra-geometry interactions in problem solving.