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Interview Prompts to Uncover Mathematical Knowledge for Teaching: Focus on Providing Written Feedback

IntroductionMathematical knowledge that is specifically connected to the work of teaching has been investigated empirically and theoretically, leading to a significant progression of its conceptualization. In particular, Ball, Thames, and Phelps (2008) developed a practice-based theory of content knowledge for teaching and introduced mathematical knowledge for teaching (MKT) as the mathematical knowledge needed to carry out the work of teaching mathematics. Such knowledge has been studied by examining teaching practice, such as job analysis (Ball & Bass, 2003) and by developing its measurement (Hill, Schilling, & Ball, 2004). MKT has been identified as important in teaching mathematics (Lewis & Blunk, 2012) and in student achievement (Hill, Rowan, & Ball, 2005; Rockoff, Jacob, Kane, & Staiger, 2011).Research on MKT has been conducted with practices of teaching that are prominent in mathematics classrooms, though not all practices. Ma (1999), for example, used four items with some exceptional tasks of teaching and mathematical demands, but did not include all topics and practices in and from teaching. Items developed by the University of Michigan include substantial tasks of teaching, but not a sufficiently organized approach to mathematical topics and teaching practice.1 Ball et al. (2008) emphasized a practice-based approach to study content knowledge for teaching. While introducing their conceptualization about such special knowledge, they focused on several and seminal practices of teaching, such as presenting mathematical ideas, responding to students' "why" questions, choosing and developing useable definitions. Nonetheless, the conceptualization has not yet been broadened to explore all practices or all topics in terms of MKT. In other words, these studies do not cover the extensive terrain of mathematical demands in teaching across contexts. For sustainable and elaborated development of the study of MKT, the mathematical demands entailed in teaching now needs to be explored systemically using a clear and comprehensive map of the practice of teaching, mathematics, features of learners, and national or international curriculum or grade levels.How MKT is studied has yet to receive sufficient attention. Ball and her colleagues studied MKT with a set of analytic tools they developed, using their wide range of experiences and disciplinary backgrounds, for coordinating mathematical and pedagogical perspectives (Ball et al., 2008; Thames, 2009). However, their experiences have not been shared in terms of researching MKT. A robust, reliable, and consistent study of MKT can be expected with an appropriate and good method. Substantial areas of mathematical demands in different practices of teaching still call for investigations from many researchers. Like the collaborative work on the Human Genome Project (International Human Genome Sequencing Consortium, 2001; Naidoo, Pawitan, Soong, Cooper, & Ku, 2011), research on MKT would need cooperative work for elaborate and systematic conceptualization. A major prerequisite for such collective work is the identification of a method to research MKT. If relevant methods of studying MKT are specified, research of MKT will be powerfully advanced. To systemically research MKT, a comprehensive method needs to be specified.To address the problems, the current study focuses on both using interviews as a comprehensive way to study and measure MKT and researching mathematical demands in providing written feedback, which is an unexamined teaching practice. It does so by building on lessons from the interview prompts used in Ball (1988) and Ma (1999) and on items used to measure MKT (Ball, Bass, & Hill, 2004). Specifically, this paper explores the ways in which interview prompts are developed and used to provide the content and character of MKT, and what, through the use of such interview prompts, might be learned. Particularly, what is entailed mathematically and pedagogically in providing written feedback? …