Students' opportunities to persevere in making sense of mathematical ideas have long been considered significant to learning. Building on existing literature and a case study of video-based teacher collaborative sensemaking, we propose a conceptual framework for bridging perseverance and sensemaking. This framework synthesizes dispositional, metacognitive, and contextual-interactive theoretical perspectives on perseverance. Informed by these three research perspectives, the conceptual framework brings forth three interrelated mediators for bolstering perseverance practices and dispositions towards mathematical sensemaking: students' positions as capable sensemakers, explicit problem-solving heuristics, and facilitation of student participation within their collective Zone of Proximal Development (ZPD). We argue that the three mediators, when brought together, provide a holistic and generative lens for teaching and teacher learning. To illustrate the framework and its utility, we build on a case study featuring a veteran middle-school mathematics teacher across his classroom facilitation of students' engagement with a classical mathematical task, the Tower of Hanoi, and a subsequent video-based debrief about the lesson with his colleague and our research team. We first frame the analysis around classroom events, and then investigate teacher learning opportunities in the lesson debrief. By making explicit the complex work of directing perseverance towards sensemaking, this study provides a more nuanced understanding of perseverance for teaching and teacher learning. Moreover, developing clarity around notions of perseverance in mathematics classrooms helps mitigate the potential dangers of the term being taken up in ineffective or even harmful ways.