A mathematical task is defined “as a classroom activity, the purpose of which is to focus students’ attention on a particular mathematical idea” (Stein, Grover, & Henningsen, 1996, p. 460). A task can take on different forms and be designed with a wide variety of goals in mind. It can vary from being one problem to consisting of several connected problems, from focusing on learning the steps of a particular algorithm to making sense of a mathematical concept. Since the choice of mathematical task largely determines the opportunities students have to learn mathematics (Stein, Smith, Henningsen, & Silver, 2009), the importance of selecting, designing, and enacting high quality mathematical tasks has never been clearer. Nevertheless, there is much to learn about their design and use. As noted by Watson and her colleagues (2013), research reports rarely give sufficient detail about tasks for them to be used by someone else in the same way. Few studies justify task choice or identify what features of a task are essential and what features are irrelevant to the study. (p. 9).

In this chapter, we attempt to fill these gaps by providing an illustrative example of a process for selecting, modifying, implementing, and redesigning a mathematical task in a mathematics content course for prospective elementary teachers (PSTs).

This process, a task design cycle, was born out of an ongoing collaboration among six mathematics teacher educators whose goal was to learn about effective ways to design, implement, and modify mathematical tasks for use in mathematics content courses for PSTs. The purpose of this chapter is to share our task design cycle with instructors who teach these courses. We describe how we went about selecting a task designed for elementary school students and modifying it for use in our mathematics content courses for PSTs. Once modified, the task was enacted, and data on the enactment (e.g., student work samples) were collected and analyzed. Subsequently, the task was modified again based on instructors’ reflections and data analysis.

This chapter is divided into three parts: (1) a description of the final version of the mathematical task that resulted from our task design cycle; (2) a description of the task design cycle itself (selection, modification, implementation, redesign) that led to this task; and (3) recommendations for instructors of PSTs based on what we learned from our work.