Cultivating learning opportunities for mathematical practices in urban middle and high school mathematics classes [electronic resource]

International comparisons of student mathematics achievement (OECD, 2014) continue to show that US students are lagging significantly behind many other countries both in math content areas (e.g., quantity, space and shape) and in mathematical practices (e.g., formulating, interpreting). Mathematical practices, such as constructing arguments or representing mathematical ideas, are complex disciplinary ways of working that are central to both doing and learning mathematics (Bass, 2011; RAND, 2003). Studies have shown that math instruction that includes opportunities to engage in mathematical practices leads to improved student learning (e.g., Boaler & Staples, 2008). The Common Core State Standards (2010) are organized around a set of core mathematical practices that delineate the ways students should be able to engage with mathematics across all grade levels and content domains. Soon teachers will be expected to help all students engage in these practices, but few US math classrooms currently offer students any opportunities to engage in practices like constructing arguments or making sense of complex problems (Stigler & Hiebert, 2009; Litke, 2014). Furthermore, although we have evidence that participating in mathematical practices ultimately supports student mathematics learning (Boaler & Staples, 2008), we know much less about how teachers can support students in learning to engage in these practices. To inform efforts to help teachers and students in meeting the rigorous demands of the standards, my dissertation seeks to define the detailed work of teaching that supports learning opportunities for mathematical practices in urban secondary math classes. I conducted parallel case studies of four middle and high school mathematics classes in an urban public charter school in an under-resourced community. Since opportunities to engage in mathematical practices are so rare in math classrooms, I purposively sampled teachers based on prior research on what supports student engagement in mathematical practices, such as the use of cognitively demanding tasks (e.g., Bieda, 2010; Stein et al., 2000) and class discussions (e.g., Chapin et al., 2009; Ellis, 2011). The data collection occurred during the first month of school, since the beginning of school is particularly important for understanding how normatively was of doing mathematics are established. I collected all mathematical tasks and video records of class discussions in all four case study classes, as well as interviews with teachers and students. The findings from my dissertation can be organized in three major themes. First, in Chapter 4, I focus on a number of methodological challenges involved in identifying and describing mathematical practices in discourse data and detail potential strategies to address these challenges. Then in Chapter 5, I examine what this analysis technique allowed me to observe about student engagement in mathematical practices in the case study discussions across the first month of school. I found that mathematical practices frequently co-emerged in classroom discourse. A second theme is taken up in Chapter 6, which focuses on how teachers cultivated opportunities for student engagement in mathematical practices in discussions of mathematical tasks. I found that, although the role of the written math tasks was important, ultimately how teachers enacted those tasks with students in class discussions mattered more with respect to supporting student engagement in mathematical practices. The third theme, which is the focus of Chapter 7, explores how teachers made mathematical practices explicit as they emerged in these class discussions. I offer a framework of reprising moves that differentiates between different types of teacher moves that make aspects of mathematical practices explicit. I show how teachers were able to make these practices visible to students without teaching them prescriptively or in a teacher directed way. The findings from this study have key implications for both research and teacher education. First, the evidence of how closely connected mathematical practices are in activity suggests that future research on mathematical practices might need to consider how to examine multiple practices simultaneously. The finding that, while tasks matter, teaching matters more for supporting student engagement in mathematical practices, highlights that implementing a "Common Core aligned curriculum" is important but insufficient; teachers will need support in how to enact curriculum with students and lead discussions to foster mathematical practices. Finally, the framework of reprising moves that make mathematical practices explicit has implications for how teachers might support equitable learning opportunities for mathematical practices for all students while still providing authentic opportunities for students to participate in these central disciplinary ways of working.

Keywords: Education.
Advisor: Boaler Jo
Aukerman Maren (Maren Songmy)
Bass Hyman
Grossman Pamela L. (Pamela Lynn)