This study focused on the practice of selecting student strategies for whole-class discussions with teachers working toward a vision of instruction that is responsive to students' mathematical thinking. Selecting strategies is one of the "5 practices" identified in Smith and Stein's (5 practices for orchestrating productive mathematics discussions, National Council of Teachers of Mathematics, 2018) seminal work on orchestrating productive mathematics discussions, but I argue that this practice has been under-appreciated and under-researched, and implementation remains challenging for many teachers. To address these issues, I explored the selecting of student strategies with two sets of upper elementary school teachers engaged in professional development focused on the teaching and learning of fractions. One set consisted of 3 teachers who had demonstrated expertise in teaching that is responsive to students' mathematical thinking, and I explored the selecting of strategies in these teachers' classrooms. The other set consisted of 30 teachers with varying levels of expertise in this type of teaching, and I examined their selecting of strategies during professional development activities. Findings led to the creation of a two-level framework that showcases teachers' criteria for selecting strategies for whole-class mathematics discussions. The framework, which extends and adds a coherent structure to selection criteria currently identified in the literature, highlights considerations related to the mathematics of the strategy, the author of the strategy, and the class engagement with the strategy. The framework, which is presented through a case, was designed to not only provide guidance for the purposeful selection of strategies but also to be accessible and useful to teachers at any phase of their development in being responsive to students' mathematical thinking.