Educators, researchers, and policy makers increasingly recognize that participation in classroom mathematics discussions, especially engaging with others' ideas, can promote students' mathematics understanding. How teachers can promote students' high-level engagement with others' ideas, and the challenges teachers face when trying to do so, have not often been studied, however. Using coding of videotaped whole-class and small-group discussions in 12 elementary school classrooms, we analyzed the level at which students engaged with each other's mathematical ideas and the moves teachers used--both moves to invite student engagement and follow-up moves to encourage deeper engagement--to support student engagement. Teachers used a wide variety of invitation and follow-up moves to encourage student engagement and combined them in multiple ways in the moment to address the challenges students faced when trying to engage with others' ideas. We show the limitations of teachers' initial moves to stimulate engagement and the power of their follow-up moves to foster productive student struggle with the mathematics.