Building on MOSTs was a collaborative project among researchers at Brigham Young University, Michigan Technological University, and Western Michigan University that explored how teachers can productively take advantage of Mathematical Opportunities in Student Thinking (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 conceptualization of a productive way teachers can coordinate their work around MOSTs?a teaching practice we refer to as building.

We initially conceptualized building on a MOST (hereafter referred to as building) as engaging the class in making sense of the MOST to better understand the mathematics of the MOST. That is, teachers engage in the teaching practice of building when they take actions to support the class's co-construction of an argument about the MOST that results in their better understanding the mathematical point of the MOST.

To study the practice of building we worked with 13 teacher-researchers (middle and high school mathematics teachers) to generate and then analyze instantiations of the practice. Through multiple cycles of enactments and analysis we refined the conceptualization of building described below.

Building is comprised of four elements (see Figure 1): (a) Establish the student mathematics of the MOST as the object to be discussed; (b) Grapple Toss that object in a way that positions the class to make sense of it; (c) Conduct a whole-class discussion that supports the students in making sense of the student mathematics of the MOST; and (d) Make Explicit the important mathematical takeaways from the discussion.

The first element of building is to Establish the student mathematics of the MOST as the object to be discussed. This element consists of four aspects (see Figure 2). The teacher establishes precision by ensuring that the MOST is clear, complete, and concise. Establishing an object makes the MOST a "thing" that can be tossed to the whole class. The teacher establishes intellectual need by ensuring that the class recognizes the need to make sense of the contribution. Finally, the teacher establishes a conversational bubble, signaling to the class that the focus of the discussion is shifting toward making sense of the MOST.

Having established the MOST as a precise object that has intellectual need, the second element of building is to Grapple Toss that object in a way that positions the class to make sense of it in the established conversational bubble. There are two aspects of the Grapple Toss: the object and the sense-making action (see Figure 3). The Established MOST will be the object of the sense-making action. An effective Grapple Toss question is as specific as possible while maintaining the cognitive demand of the sense-making activity.

Once a teacher has Grapple Tossed the Established MOST, the third element of building is to Conduct a whole-class discussion that supports students in making sense of the student mathematics of the MOST. The general structure of the Conduct element can be summarized as follows: When a student contributes an idea during Conduct, the teacher needs to determine whether the idea contributes to the joint sense-making argument around the MOST (see Figure 4). The subsequent sequence of teacher actions depends on their determination of whether the contribution is argument-related. If a teacher determines that the contribution is not argument-related, they put aside the idea and redirect the class to the sense-making discussion (wherever they happen to be on the arc of the argument). If, on the other hand, the teacher determines that the contribution is in fact argument-related, they need to make a move that invites students to use this new idea to help make sense of the MOST. In essence, the flow of the Conduct element is a repetition of this pattern of weaving an argument-related contribution into the argument or putting aside contributions that are not argument related and setting students up to contribute yet another new idea. 

After the teacher infers that students have made sense of the MOST during the Conduct discussion, the final element of building, Make Explicit, entails making explicit for students the mathematical takeaways of the foregoing sense-making discussion. There are three aspects of Make Explicit (see Figure 5). Resolution entails ensuring that students have made sense of the MOST?that, as a class, they have resolved the intellectual need the MOST gave rise to and are aware of having done so. Generalization requires the teacher ensure that the resolution is connected to the broader mathematical point afforded by the MOST. Transition involves making explicit the relative importance of the MP to the lesson and then moving on with the lesson. 

Last Modified: 07/13/2023
Modified by: Shari Stockero