The core research questions of the project are: (1) What is the nature of high-leverage student thinking that teachers have available to them in their classrooms? (2) How do teachers use student thinking during instruction and what goals, orientations and resources underlie that use? (3) What is the learning trajectory for the teaching practice of productively using student thinking? and (4) What supports can be provided to move teachers along that learning trajectory?

Leveraging MOSTs (Mathematically Significant Pedagogical Openings to build on Student Thinking) is a collaborative project among Brigham Young University, Michigan Technological University and Western Michigan University that focuses on improving the teaching of secondary school mathematics by improving teachers' abilities to use student thinking during instruction to develop mathematical concepts. The core research questions of the project are: (1) What is the nature of high-leverage student thinking that teachers have available to them in their classrooms? (2) How do teachers use student thinking during instruction and what goals, orientations and resources underlie that use? (3) What is the learning trajectory for the teaching practice of productively using student thinking? and (4) What supports can be provided to move teachers along that learning trajectory? The project is developing a theory of Productive Use of Student Mathematical Thinking (PUMT Theory) that articulates what the practice of productively using student mathematical thinking looks like, how one develops this practice, and how that development can be facilitated.

Design research methodology underlies the work of four interrelated phases: (1) Student thinking - testing and refining a preliminary framework by expanding an existing data set of classroom discourse video to include more diverse teacher and student populations; (2) Teachers' interactions with student thinking - assessing teachers' perceptions of using student thinking and how they make decisions about which thinking to pursue; (3) Teachers' learning about student thinking - using a series of teacher development experiments to improve teachers' abilities to productively use student mathematical thinking during instruction; and (4) Shareable products - creating useful products that are in forms that encourage feedback for further refinement. Data include video recordings of classroom instruction (to identify MOSTs and teachers' responses to them), teacher interviews (to understand their decisions in response to instances of student thinking), and records of teacher development sessions and the researchers' discussions about the teachers' development (to inform the teacher development experiments and future professional development activities). Project evaluation includes both formative and summative components that focus on the quality of the project's process for developing a PUMT Theory and associated tools and professional development, as well as the quality of the resulting products.

Leveraging MOSTs provides critical resources - including a theory, framework, and hypothetical learning trajectory - for teachers, teacher educators, and researchers that make more tangible the often abstract but fundamental goal of productively using students' mathematical thinking. The project enhances the field's understanding of (1) the MOSTs that teachers have available to them in their classrooms, and how they vary in diverse contexts; (2) teachers' perceptions and productive use of student thinking during instruction; and (3) the trajectory of teachers' learning about student thinking and how best to support movement along that trajectory. Using student thinking productively is a cornerstone of effective teaching, thus the PUMT Theory and associated supports produced by the project are valuable resources for those involved in mathematics education as well as other fields.