Among mathematics teacher educators, a consensus has emerged that exemplary teaching involves attention to students' thinking. This consensus stems, in part, from theoretical and empirical work highlighting the importance of teachers' being able to make thoughtful in-the-moment decisions, building on students' ideas to adjust the ongoing lesson. Since this in-the-moment flexibility draws in large part on teachers' noticing capabilities, mathematics education researchers have paid increasing attention over the past decade to key characteristics of teacher noticing and to ways to help teachers develop these skills and inclinations. In this paper, we explore how engaging teachers in annotating videos of mathematics classrooms can provide important insights into teacher noticing of students' mathematical thinking. We argue that, when used in particular ways, video tagging affords researchers and teacher educators the ability to examine the relationship between what grabs a teacher's attention and how teachers interpret the thinking displayed in those moments, and instructionally actionable feedback about how teachers are progressing in their attending to and interpreting of students' thinking, two related skills that may or may not develop in sync.