Studying the Collective Development of Mathematical Knowledge for Teaching

In the past two decades, teachers' content knowledge has become a major focus in the mathematics education literature (Ball, Thames, & Phelps, 2008; Ma, 1999) and increasing teachers' mathematical knowledge continues to be a major focus in both education research and policy. Throughout the past three years, as part of an extended research and development project focusing on developing Mathematical Knowledge for Teaching (MKT) through online collaboration, we have considerable success documenting individual teachers' development of MKT and a significant connection between this development and specific online teacher professional development activities (Author, Date; Author, Date).  While this individual development is important, we also recognize that thinking and learning are social phenomena and that the development of MKT can also be described as appropriating the knowledge, skills, language, and practices of a particular community of mathematics educators. As such, we have begun to widen our focus to include studying the emergence of norms and collective mathematical practices and how these norms and practices support or constrain individual development.

The setting for this study was an online graduate mathematics education course that was designed to enhance teachers' understandings of the concept of function, specifically the ways the concept can serve as a unifying theme for a variety of school mathematics. While there is not sufficient space to describe it in detail, we have claimed elsewhere that such "unifying" conceptions are one important component of mathematical knowledge for teaching (Author, Date; Author, Date). The course was divided into four units: (1) introduction to functions and graphs, (2) trigonometric functions, (3) modeling, and (4) rate of change. The current study focuses on the three-week unit on understanding trigonometric functions. A total of 24 current middle and/or secondary mathematics teachers, each of whom were certified teachers in their home states, had completed a minimum of three mathematics courses beyond Calculus III and had an overall undergraduate grade point average of 3.0 or better as an undergraduate participated in the study.

In this session, we will present a brief overview of traditional methods for documenting norms and collective mathematical practices and specifically discuss our use of Rasmussen & Stephan's (2008) methodology for documenting collective activity online. Using an "argument" as the unit of analysis and Toulmin's (1969) model to describe the structure and function of various components of the argument, Rasmussen & Stephan (2008) describe a three-phase approach to documenting collective activity. Phase one involves the creation of an "argumentation log," where arguments are identified and documented through the claims, warrants, data, backing, conclusions, etc. Phase two involves looking across the entries in the argumentation log and identifying norms and mathematical practices that become part of the group's collective ways of reasoning. Evidence of this shift from individual to collective includes identifying instances when a particular way of reasoning or acting that initially has to be justified is itself later used to justify other ways of reasoning or acting (Rasmussen & Stephan, 2008) and instances when individuals appear to violate a proposed communal norm and whether this activity is constituted as legitimate or illegitimate by the group. In either case, an activity is more likely to be "collective" as more evidence of its collective nature is documented. Finally, phase three of the methodology involves organizing the identified norms and collective mathematical understandings around common themes, i.e. the collective mathematical practices. 

While data analysis is still ongoing, tentative results include three collective mathematical practices: quantifying angles in consistent ways, connecting triangle trigonometry and circular functions, and connecting analysis of trigonometric functions with other functions. The presentation will include detailed descriptions and examples of these collective mathematical practices and the normative ways of reasoning for each.

There are multiple levels of the significance of this work. First, as the popularity of online professional development grows, understanding the emergence of collective understandings in online settings is important for designing and improving high quality online professional development that emphasizes group work and collective activity. Additionally, we believe that identifying the specific collective mathematical practices that emerge in professional development is a necessary first step in studying and understanding the influence of discourse and conversation based professional development on teachers' mathematical development and instructional practices. One additional byproduct of this work is the observation that traditional methods for studying the emergence of collective understandings in face-to-face setting needed to be augmented and adapted for use with data from online professional development. These methodological issues will be discussed and the specific modifications needed to study collective understanding in online settings will be highlighted.

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59(5), 389(319).

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Rasmussen, C. L., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh & J. Y. Baek (Eds.), Handbook for innovative design research in science, technology, engineering, and mathematics (STEM) education (pp. 195-215). New York, NY: Taylor & Francis.

Toulmin, S. (1969). The uses of argument. Cambridge: Cambridge University Press.