Forms of Knowledge of Advanced Mathematics for Teaching

Abstract

1.IntroductionTeachers, particularly at the middle and secondary level, are frequently required to take advanced mathematics courses (e.g., real analysis, abstract algebra, etc.) as part of their teacher preparation program. This is based on a general sense of the importance of strong content knowledge for teachers. However, many teachers at some point question the relevance of these advanced courses for their future teaching careers (e.g., Zazkis & Leikin, 2010). Indeed, as Monk (1994) noted, the quantity of advanced mathematics preparation does not guarantee teaching quality; that is, one's own mathematical understanding does not necessarily translate into an ability to enhance the understanding of others. With the recent focus on practice-based approaches to conceptualizing teacher's mathematical knowledge (e.g., Ball, Thames, & Phelps, 2008; Petrou & Goulding, 2011) - which has led to documenting specific ways teachers use their knowledge of mathematics in their professional work - much more needs to be done to identify, define, and document how advanced mathematics may inform the actual work of teaching. Although this problem has been observed for a long time - Felix Klein (1932) made comments about this in the early 20th century - little progress has been made in this regard.In contrast to defining specific content areas in advanced mathematics or undergraduate course recommendations (e.g., CBMS 2001; 2012), this paper focuses instead on more general forms of knowledge of advanced mathematics that may be productive for the teaching of school mathematics. Following a grounded theory approach (e.g., Strauss & Corbin, 1990), we developed a framework of five forms of knowledge of advanced mathematics for teaching that, while not necessarily an exhaustive or exclusive list, moves the discussion of mathematical knowledge for teaching - particularly of advanced mathematics - beyond just listing what content teachers need to know, and toward a more general conception of how knowing advanced mathematics could positively interact with the work of teaching. We begin by reviewing related literature and situating this work within it.2.Literature2.1.Teachers' Knowledge of advanced MathematicsRecent efforts to conceptualize the mathematical knowledge required for teaching have incorporated practice-based approaches to teacher knowledge - that is, the content knowledge teachers need should be relevant for the practices and work of teaching. In particular, researchers have used this perspective to conceptualize different "domains" of knowledge, at both the elementary and secondary levels (e.g., Ball, Thames, & Phelps, 2008; McCrory, et al., 2012; Heid, Wilson, & Blume, 2015). Notably, many of these - and others' (e.g., Zazkis & Leikin, 2010)- allude to the importance of knowing advanced mathematics. For example, the Mathematical Knowledge for Teaching (MKT) framework (Ball, Thames, & Phelps, 2008) included the domain Horizon Content Knowledge (HCK), which is related the conversation about advanced mathematics (e.g., Wasserman & Stockton, 2013; Jakobsen, Thames, & Ribeiro, 2013; Zazkis & Mamolo, 2011); and in translating to secondary mathematics teaching, McCrory, et al. (2012) included knowledge of advanced mathematics as a domain of their Knowledge of Algebra for Teaching (KAT) framework. Heid, Wilson, and Blume (2015) described Mathematical Understandings for Secondary Teaching (MUST), which, although not connected explicitly to advanced courses, have connections to larger mathematical practices in the discipline which are often honed in courses such as abstract algebra, real analysis, or an introduction to proofs course. Thus, many regard advanced mathematics as important for teaching.However, finding explicit connections to practice has been more difficult. This fact is perhaps captured best by the "provisional" nature of the HCK domain - it was left lessdeveloped and less-defined than other parts of the MKT framework. …