The study of pivotal teaching moments (PTMs) offers significant insights into mathematics classroom interactions. PTMs are student-generated instances within a lesson that provide opportunities for teachers to modify planned instruction (Stockero & Van Zoest, 2013). Given the importance of interactions and discourse in students' mathematical growth and understanding, we examine them in the context of "PTM episode triples." PTM episode triples consist of three main elements: (1) a PTM, (2) a teacher response to the PTM, and (3) the immediately ensuing student utterance. Analysis of data from nine elementary mathematics lessons shows that teacher responses that explicitly "pursued students' thinking" elicited significantly higher cognitive levels of student discourse than teachers who "ignored or dismissed" PTMs or simply "acknowledged [PTMs] but continued as planned." While many researchers have focused on PTM episode "doubles" (combinations of student-generated interruptions and the teacher's response), no studies to our knowledge have addressed PTM episode "triples" as we have. Implications of these findings prove helpful for identifying concrete ways that educators can increase the cognitive level of student discourse within their classrooms. Further, our integration and modification of existing theory and frameworks for analyzing PTM episode triples--incorporating the relationship between PTMs and teachers' responses to them, on the one hand, and the cognitive level of the ensuing student utterance, on the other hand--are a unique contribution to the field and provide a method by which researchers and educators can study classroom practice.