Building on MOSTs: Investigating Productive Use of High-Leverage Student Mathematical Thinking was a collaborative project among researchers at Brigham Young University, Michigan Technological University and Western Michigan University that focused on improving the teaching of secondary school mathematics by exploring the teaching practices that allow teachers to elicit and take advantage of MOSTs.

MOSTs are high-leverage student contributions ("teachable moments") that occur at the intersection of three critical characteristics of classroom instances: student mathematical thinking, significant mathematics, and pedagogical opportunity. In essence, MOSTs are student mathematical contributions that provide an in-the-moment opportunity to engage students in joint sense making (Leatham et al., 2015). The purpose of this project was to develop a theoretical conceptualzation of a productive way teachers can coordinate their work around MOSTs—a teaching practice we refer to as building. Building on a MOST is a teaching practice that takes advantage of the opportunity that a MOST provides (Leatham et al., 2021) in a way that adheres to principles of effective teaching of mathematics (NCTM, 2014). We formally define building on a MOST as engaging the class in making sense of the MOST to better understand the mathematics of the MOST. Unpacking that definition led us to theorize that building is comprised of four elements:

1. Establish the student mathematics of the MOST as the object to be discussed;
2. Grapple Toss that object in a way that positions the class to make sense of it;
3. Conduct a whole-class discussion that supports the students in making sense of the student mathematics of the MOST;
4. Make Explicit the important mathematical takeaways from the discussion.

Although we began with an initial conceptualization of building and its constituent parts that made sense in theory, our experience studying MOSTs suggested that the practice is seldom enacted in classrooms. We thus engaged in studying the practice of building on MOSTs by working with teacher-researchers to generate and then analyze instantiations of the practice.

During the first year of the project, we developed and piloted MOST-Eliciting Prompts (MEPs)—brief mathematical undertakings that have a high likelihood of surfacing particular MOSTs as students share their thinking in response to the prompt. These MEPs were developed by drawing on our prior work studying MOSTs, the literature, and our own teaching experiences. MEPs were constructed around broad mathematical ideas so they could be "dropped into" different 6-12 grade classes at almost any time during the year.

After the MEPs were developed, we held a teacher-researcher retreat with a group of 13 teacher-researchers (TRs) in summer of 2018 and taught them about our theorized building practice so they could create instantiations around the MEPs in their classrooms. During the second year of the project, the TRs engaged in cycles of (a) implementing a MEP, (b) providing initial reactions and feedback, (c) participating in a small-group debriefing meeting, and (d) participating in a large-group research meeting. The goal of these cycles was to capture increasingly better enactments of the building practice.

During the third year of the project (2019-2020), we analyzed the building enactments that were generated by the TRs during the second year of the project (2018-2019). Our original plans to use this analysis to create and present refinements of the theorized building practice to the TRs before a second round of enactments during 2020–2021 were postponed due to the COVID-19 pandemic. Instead, we spent our fourth year (2020-2021) doing a deeper analysis of the 2018-2019 data. This resulted in a further refinement of our theorized building practice.

2021-2022 was the fifth (no-cost extension) year of our project. We met with the TRs in the summer of 2021 to update them on our current theorization of the building practice so that they could repeat the cycles from the second year of the project (see above) and provide more refined enactments of the MEPs to test out our current conceptualization of the building practice. We then analyzed these enactments in order to further refine our conceptualization of the teaching practice of building on MOSTs.

2022-2023 was the second no-cost extension year of our project. We began this sixth and final year of the project by finalizing our conceptualization of the teaching practice of building so that it can be utilized by teachers to help them take advantage of the MOSTs that occur in their classrooms. We then shifted to writing about and sharing the refined conceptualization of building on MOSTs with teachers, teacher educators, and researchers.

The core research questions of the project are:

• What does it look like to build on MOSTs in a way that coordinates the core principles underlying their productive use?

  Our work to answer this research question will involve analysis of records of the teacher-researchers' attempts to implement the building practice during six MEP enactment cycles. Drawing heavily on the procedures we have established for analyzing MOSTs and for analyzing teacher responses to student thinking, this analysis will have two main foci. The first focus is understanding whether our current building prototype—including the four building subpractices—satisfies core principles of quality mathematics instruction that we have gleaned from mathematics education literature. These principles are that student mathematics is at the forefront, students are positioned as legitimate mathematical thinkers, students are engaged in sense-making, and students work collaboratively. The second focus is understanding whether there are other collections of subpractices that also satisfy the core principles. In answering this research question, we hope to better understand how the coordination of subpractices produces the teaching practice of building.

• What are variations in how enacting the building subpractices coordinate the core principles?

  Our work to answer this research question, like the previous research question, will involve the retrospective analysis of records of the teacher-researchers' attempts to implement the building practice during the six MEP enactment cycles. However, to answer this particular question we will focus specifically on the enactment of each of the subpractices of the teaching practice of building on MOSTs. By focusing on these subpractices, we hope to better understand what the individual subpractices look like, and which enactments of these subpractices do and do not work well.

• What are teachers' experiences in attempting to build on MOSTs?

  Our work to answer this question will involve the ongoing analysis of the teacher-researchers' attempts to implement the building practice and their reflections on these attempts during the six MEP enactment cycles. This work will include watching MEP enactment videos and reviewing teacher feedback to select segments or collections of segments that will be the focus of group discussion among the project PIs and the teacher-researchers. Subsequent analysis of teacher feedback and meeting discussions, as well as the video of the MEP enactments, will allow us to ascertain what aspects of building are particularly accessible or challenging, and consider how teachers can be supported in their enactment of building on MOSTs. This analysis will also inform the ongoing planning of our work, including modifications to the building prototype and the work we do with the teacher-researchers to prepare them for future MEP enactments.

This project is funded by the National Science Foundation, grant #'s:

WMU DRL-1720613
MTU DRL-1720566
BYU DRL-1720410

Any opinions, findings, and conclusions or recommendations expressed in these materials are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.