<![CDATA[Mathematical representation is a critical competency that prospective mathematics teachers must possess in order to understand concepts and solve problems effectively. However, previous studies still leave conceptual gaps in explaining how prospective mathematics teachers construct and coordinate representations as cognitive guidance mechanisms, particularly when calculating the area of a triangle. This study aims to uncover the involvement of visual representations and cognitive guidance during the process of solving problems involving the area of a triangle. This study employs an exploratory qualitative approach. Data were collected through problem-solving tests and semi-structured interviews with five subjects selected from 93 prospective participants. The data were analyzed based on the use of visual representations, the concept of coordinate transformation, plotting accuracy, and indications of cognitive guidance. The results revealed four characteristics of representation use: (1) no visual representation, incorrect coordinate transformation, failure to plot coordinate points, and no cognitive guidance; (2) visual representation, incorrect coordinate transformation, incorrect plotting of coordinate points, and cognitive guidance; (3) does not involve visual representations, correct coordinate transformations, does not plot coordinate points, and is not cognitively guided; and (4) involves visual representations, correct coordinate transformations, plots coordinate points accurately, and is cognitively guided. Visual representations help understand geometric structures and spatial relationships, while symbolic representations formalize computational procedures.]]>
