Different types of ambiguous student contributions occur in mathematics classrooms. In this study I identified (1) different types of ambiguous student contributions and (2) the different ways teachers respond to one particular kind of ambiguous contribution, clarifiable ambiguity. Note that clarifiable ambiguity is ambiguity that stems from a student who uses an unclear referent in their contribution and can be clarified in the moment by the particular student. Literature has focused only on ambiguity that has the potential to further the development of mathematical concepts and has only theorized about teacher responses to this specific type of ambiguity. This study identified an additional three types of ambiguous student contributions: Student Appropriation of Teacher Ambiguity, Irrelevant Ambiguity, and General Ambiguity. It was important to identify all the different types of ambiguous student contributions because teacher responses should likely be different to the different types of ambiguity. In addition, through analyzing the teacher responses to the clarifiably ambiguous student contributions, this study found that teachers addressed the clarifiably ambiguous student contributions about half the time. When the teachers did address the clarifiable ambiguity, the majority of the time the teacher clarified the ambiguity themselves with the most common response being the teacher honed in on the clarifiably ambiguity and asked for confirmation from the student on the accuracy of the clarification.

While the use of student thinking to help build mathematical understandings in a classroom has been emphasized in best teaching practices, teachers still struggle with this practice and research still lacks a full understanding of how such learning can and should occur. To help understand this complex practice, I analyzed every instance of student thinking and every teacher response to that thinking during a high school geometry teacher's whole class discussion and used these codes as evidence of alignment or misalignment with principles of effective use of student mathematical thinking. I explored the teacher's practice both in small and large grains by considering each of her responses to student thinking, and then considered the larger practice through multiple teacher responses unified under a single topic or theme in the class discussion. From these codes, I moved to an even larger grain to consider how the teacher's practice in general aligned with the principles. These combined coding schemes proved effective in providing a lens to both view and make sense of the complex practice of teachers responding to student thinking. I found that when responding to student thinking the teacher tended to not allow student thinking to be at the forefront of classroom discussion because of misinterpretation of the student thinking or only using the student thinking in a local sense to help advance the discussion as framed by the teacher's thinking. The results showed that allowing student thinking to be at the forefront of classroom discussion is one way to position students as legitimate mathematical thinkers, though this position can be weakened if the teacher makes a move to correct inaccurate or incorrect student thinking. Furthermore, when teachers respond to student thinking students are only able to be involved in sense making if the teacher turns the ideas back to the students in such a way that positions them to make sense of the mathematics. Finally, in order to allow students to collaborate a teacher must turn the mathematics to the students with time and space for them to meaningfully discuss the mathematics. I conclude that the teacher's practice that I analyzed is somewhat aligned with honoring student mathematical thinking and allowing student thinking to be at the forefront of class discussion. On the other hand, the teacher's practice was strongly misaligned with collaboration and sense making. In this teacher's class, then, students were rarely engaged in sense making or collaborating in their mathematical work.