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## Description

The idea that problems and tasks play a pivotal role in a mathematics lesson has a long standing in mathematics education research. Recent calls for teaching reform appeal for training teachers to better understand how students learn mathematics and to employ students' mathematical thinking as the basis for pedagogy (CCSSM, 2010; NCTM, 2000; NRC 1999). The teaching practices of (a) developing a task for a mathematics lesson and, (b) modifying the task for students while enacting the lesson fit within the scope of supporting students' mathematical thinking. Surprisingly, an extensive search of the literature did not yield any research aimed to identify and refine the constituent parts of the aforementioned teaching practices in the manner called for by Grossman and xiii colleagues (2009). Consequently, my research addresses the two questions: (a) what factors do exemplary elementary teachers consider when developing a task for a mathematics lesson? (b) what factors do they consider when they modify a task for a student when enacting a lesson? I conducted a multiple case study involving three elementary teachers, each with extensive training in the area of Cognitively Guided Instruction (CGI), as well as several years experience teaching mathematics following the principles of CGI (Carpenter et al., 1999). I recorded video of three mathematics lessons with each participant and after each lesson I conducted a semi-structured stimulated recall interview. A subsequent follow-up clinical interview was conducted soon thereafter to further explore the teacher's thoughts (Ginsberg, 1997). In addition, my methodology included interjecting myself at select times during a lesson to ask the teacher to explain her reasoning. Qualitative analysis led to a framework that identified four categories of influencing factors and seven categories of supporting objectives for the development of a task. Subsets of these factors and objectives emerged as particularly relevant when the teachers decided to modify a task. Moreover, relationships between and among the various factors were identified. The emergent framework from this study offers insight into decompositions of the two teaching practices of interest, and, in particular, the utility of the number choices made by the teachers.