Mathematical representation has an essential role in solving mathematical problems. However, there are still many mathematics education students who have difficulty in representing ill-structured problems. Even though the ill-structured-problem-solving tasks designed to help mathematics education students understand the relevance and meaningfulness of what they learn, they also are connected with their prior knowledge. The focus of this research is exploring the used of mathematical representations in solving ill-structured problems involving quadratic functions. The topic of quadratic functions is considered necessary in mathematics teaching and learning in higher education. It's because many mathematics education students have difficulty in understanding these matters, and they also didn’t appreciate their advantage and application in daily life. The researchers' explored mathematical representation as used by two subjects from fifty-four mathematics education students at the University of Nusantara PGRI Kediri by using a qualitative approach. We were selected due to their completed all steps for solving the ill-structured problem, and there have different ways of solving these problems. Mathematical representation explored through an analytical framework of solving ill-structured issues such as representing problems, developing alternative solutions, creating solution justifications, monitoring, and evaluating. The data analysis used technique triangulation. The results show that verbal and symbolic representations used both subjects to calculate, detect, correct errors, and justify their answers. However, the visual representation used only by the first subject to detect and correct errors.

Mathematical representation is an important aspect of mathematical problem-solving. But prospective student teachers’ ability of a mathematical representation in ill-structured problem solving is still very limited compared to that in well-structured problem-solving. However, ill-structured problems supported mathematical abstraction used in mathematical concept understanding. This study described the characteristics of prospective student teachers’ representations and translation among representation in ill-structured and well-structured social algebraic problem-solving. Thirty Indonesian prospective students teachers have to solve the ill-well algebraic problems by using think-aloud while recorded using a video recorder and the researchers observed by writing on a field note to record the important events. Data were analyzed using a comparative method so that it was obtained the different characteristics of representations between solving ill-structured and well-structured problems. The result showed that verbal and symbolic representations used by subjects to compute, detect and correct an error, and justify their answers in ill-structured problem solving but visual representation only used by the first subject to detect and correct error. The subjects lack to uncover necessary information to solve the ill-Structured problem compared to that of a well-structured problem. It means that representation is important to be studied further because it affected to ill-structured problem ability.