Prospective and in-service teachers' use of pedagogical mathematical practices in approximations of teaching practice

Researchers have identified ways to support mathematics teachers in connecting advanced mathematics to their teaching of secondary mathematics, particularly through the use of Approximations of Practice (AoPs), which provide opportunities for teachers to engage in simulations of teaching practices without the complexities associated with being in a real classroom (Grossman et al. in Teachers College Record 111(9):2055-2100, 2009a). In this study, we designed AoPs that approximated the teaching practices of attending to, interpreting, and deciding how to respond to students' mathematical thinking (Jacobs et al. in J Res Math Educ 41(2):169-202, 2010), as well as leading a class discussion about the students' ideas (Smith and Stein in 5 Practices for orchestrating productive mathematics discussion, National Council of Teachers of Mathematics, Reston, 2018). We conducted interviews with teachers enrolled in a master's-level Mathematics for Teachers course. The curriculum and interviews focused on three mathematical content domains: probability, abstract algebra, and real analysis. We administered AoPs prompting the teachers to interpret and respond to student thinking and structure whole class discussions about students' strategies. Through our analysis of their responses, we identified the teachers' pedagogical mathematical practices (PMPs) (Wasserman in Learn Math 42(3):28-33, 2022), i.e., practices and habits shared by mathematicians and mathematics teachers, used as they responded to these AoPs. We also identified design features of the AoPs that are conducive to teachers' enactment of certain PMPs. We contribute new PMPs to the literature and provide implications for the advanced mathematical preparation of mathematics teachers.