In mathematics, engaging in reasoning-and-proving (RP) includes investigating mathematical relationships, formulating conjectures, evaluating others' conjectures or arguments, generating arguments, and communicating mathematical knowledge. These mathematical processes are key habits of mind that help learners think critically within and outside mathematics classrooms. Researchers and policymakers have reflected the centrality of RP by emphasizing the importance of helping students in all mathematical content areas and in all grade levels to engage in rich proving activities and become competent at generating and evaluating mathematical arguments (NCTM, 2000, 2009; Yackel & Hanna, 2003). In contrast to these recommendations, however, K-12 students' initial experiences with formal proof traditionally occur in high school geometry classes, where students often experience only one slice of the range of RP processes with a focus on proving given statements using a formal two-column proof format (Harel & Sowder, 1998; Herbst, 2002). Prospective teachers of elementary grades (PTEs) also typically have limited experiences developing and using proof, which could hinder their future elementary students' engagement in these RP processes (Balacheff, 1988; Martin & Harel, 1989). How are PTEs currently being prepared to provide rich opportunities for their future students experience and understand RP?Mathematics for elementary teachers courses are the primary site for supporting this development of PTEs' mathematical knowledge for teaching. Geometry and Measurement is a common content slice for these courses, with explicit attention to the work of proof (Cannata & McCrory, 2007; McCrory, Siedel, & Stylianides, 2008). This attention on RP is important for developing PTEs' ability to explain why mathematical relationships are true, as well as for preparing them to help their students develop RP-abilities. By examining how graduate teaching assistants (TAs) implemented RP mathematical tasks when teaching a Geometry and Measurement course for PTEs, this study investigated the opportunities for PTEs to learn about RP in such courses.Specifically, six TAs, each assigned to teach their own sections of the course, participated in the study. Through classroom observations and interviews, the way these TAs engaged PTEs in RP and how TAs' conceptions of RP illuminate their instructional decisions around RP tasks was examined. RP tasks were defined as tasks with potential to engage PTEs in RP processes. Findings indicate that TAs engaged PTEs in a range of RP processes. For a plurality of observed tasks, however, opportunities for PTEs to engage in RP were decreased. TAs often decreased RP opportunities by stating a conjecture or proof before PTEs had a chance to generate them. There are multiple factors (e.g., TAs' conceptions of the purposes for teaching about RP and how to facilitate class discussions) that influenced their instructional decisions. This research has implications for professional development to support college mathematics instructors' teaching.
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