Establishing Problem Spaces and Boundaries during Small Group Work on a Contextualized Problem

In this study, we examine how United States (U.S.) grades 6-12 teachers working together on a contextualized mathematics problem negotiate issues arising from the ill-structured nature of the problem such as what assumptions one may make, what real-world considerations should be taken into account, and what constitutes a satisfactory solution. We conceptualize the process of negotiating these questions as the construction of a "problem space," characterized by the boundary between considerations deemed relevant or essential to the problem and ones thought to be beyond the scope of the problem. We use data from group discussions of the problem to consider ways in which problem spaces are co-constructed by learners, instructors, and problem authors and how these problem spaces evolve over time. Using Palm's authenticity framework (2002; 2006; 2008), we analyze and describe ways in which more or less authentic aspects of the problem intersected with the construction of problem spaces and boundaries. Lastly, we conclude by discussing implications of these findings for the design and implementation of contextualized mathematics problems with middle and high school teachers, and, potentially, with middle and high school students.