What does it mean to think like a mathematician? One of the great paradoxes of mathematics education is that, although mathematics teachers are immersed in mathematical work every day of their professional lives, most of them nevertheless have little experience with the kind of work that research mathematicians do. Their ideas of what doing mathematics looks like are based mainly on their own experiences as students. Creating opportunities for students to engage in the kind of flexible thinking that is characteristic of mathematicians' practice can be a challenge for all teachers. For their classrooms to become venues where students experience mathematical activities that foster flexible thinking, teachers need to change how they think about and do mathematics. Teachers at all levels should make the mechanisms of problem posing explicit and draw attention to how they can be used to navigate through open-ended problems. Through engagement in such mathematical activities, teachers and their students might come to view mathematics differently. By extension, these teachers might come to teach mathematics in a manner that is more authentic to actual mathematical practice. (Contains 2 figures.)