This study is an examination of 79 elementary prospective teachers' (PSTs') capacity for recognizing the core ideas involved in modeling fraction addition problems and their difficulties in solving and presenting the process of fraction addition using area, length, and set models. PSTs completed a written task in which they represented the process of solving a set of fraction addition problems of varying complexity in terms of the sizes of the addends, the sums, and the denominators using the three models. The PSTs' responses to the task were analyzed and clustered to identify the most salient and recurring patterns of erroneous approaches to modeling. While the PSTs recognized the core ideas involved in modeling fraction addition problems, their actual work samples demonstrated several areas for improvement. Some PSTs had invalid answers and representations that were indicative of their misunderstanding of key concepts. Some PSTs had correct answers, but their model did not support sense-making. Additionally, some PSTs' representations did not clearly incorporate the salient properties of the area, length, and set models. Our analysis generated implications for mathematics teacher educators regarding what PSTs need to learn, develop further, unlearn, and refine in order to effectively teach uses of representation in elementary classroom.