In this paper, using written responses of 37 PSTs preparing to teach grades 1-8 mathematics, we examined explanations they constructed to support their problem solutions and explanations they provided in support of their critiques of student-generated explanations. We also examined features of explanations on which PSTs drew in their critiques of mathematical explanations of students. Our results draw attention to the importance of helping PSTs develop competencies in constructing and critiquing mathematical explanations concurrently. Although explanations PSTs generated for their critiques of student explanations were weaker compared to the explanations PSTs formulated for their own problem solutions, PSTs proficient in generating mathematical explanations were also more proficient in analyzing and critiquing mathematical explanations. We identified seven criteria PSTs used while analyzing and critiquing student-explanations. These criteria reveal what PSTs might value, or pay attention to, as they critique student-explanations. We share implications for mathematics teacher educators to consider and suggest directions for further research.