abstract: This dissertation reports three studies of the relationships between meanings teachers hold and meanings their students construct. The first paper reports meanings held by U.S. and Korean secondary mathematics teachers for teaching function notation. This study focuses on what teachers in U.S. and Korean are revealing their thinking from their written responses to the MMTsm (Mathematical Meanings for Teaching secondary mathematics) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. This paper then discusses how the MMTsm serves as a diagnostic instrument by sharing a teacher’s story. The data indicates that many teachers name rules instead of constructing representations of functions through function notation. The second paper reports the conveyance of meaning with eight Korean teachers who took the MMTsm. The data that I gathered was their responses to the MMTsm, what they said and did in the classroom lessons I observed, pre- and post-lesson interviews with them and their students. This paper focuses on the relationships between teachers’ mathematical meanings and their instructional actions as well as the relationships between teachers’ instructional actions and meanings that their students construct. The data suggests that holding productive meanings is a necessary condition to convey productive meanings to students, but not a sufficient condition. The third paper investigates the conveyance of meaning with one U.S. teacher. This study explores how a teacher’s image of student thinking influenced her instructional decisions and meanings she conveyed to students. I observed 15 lessons taught by a calculus teacher and interviewed the teacher and her students at multiple points. The results suggest that teachers must think about how students might understand their instructional actions in order to better convey what they intend to their students. The studies show a breakdown in the conveyance of meaning from teacher to student when the teacher has no image of how students might understand his or her statements and actions. This suggests that it is crucial to help teachers improve what they are capable of conveying to students and their images of what they hope to convey to future students. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2019