Brown and Edelson (Brown & Edelson, 2003; Brown, 2009) introduced pedagogical design capacity (PDC) as a way to understand how teachers perceive and mobilize existing resources to design instruction. Perceive indicates the ability to recognize, or notice, potential resources and mobilize highlights the importance of teachers' abilities to act on or with those resources (Remillard, 2005). The PDC construct is in its infancy. That is, the key dimensions of PDC have not been identified; and ways to measure teachers' PDC have not yet been developed (Brown, 2009). This dissertation study sought to add to the PDC construct by investigating the PDC for teaching elementary mathematics of four expert teachers. This dissertation is written as three articles - Chapters Three, Four, and Five.

Chapter Three describes how the teachers mobilized four different types of progressions: unit, series of instructional activities, number choices, and student solutions. Different curricular resources provided support for one or more progression types. For instance, Investigations (TERC, 2008) provided support for one teacher, Violet, for unit and series of instructional activities progressions, but not for number choice and student solution progressions. Violet was supported, however, by her CGI knowledge for these two progression types. These results add to the existing research in mathematics education around the notion of mobilizing hypothetical learning trajectories to provide rationales for designing instruction (e.g. Clements & Sarama, 2004; Fuson, Carroll, & Drueck, 2000; Simon, 1995).

In Chapter Four, I report findings on what kind of knowledge the teachers had of students as well as how the four teachers mobilized student resources to design instruction. Grounding that study in PDC and other studies focusing on teachers learning how to use children's mathematical thinking for instructional decisions, I found that teachers could detail strategies and were able to distinguish between details that speak to a child's conceptual understanding and details that speak to other mathematical practices. Furthermore, the teachers possessed individual knowledge of students including knowledge of the strategies students tended to use as well as individuals' dispositions, and mobilized individual knowledge of students to make instructional decisions. Those instructional decisions are based on moving students along a student solution progression. Student resources were also mobilized to introduce instructional topics, to develop PDC, and to take on roles traditionally reserved for teachers.

Chapter Five describes how the four teachers mobilized number choices - one type of progression identified in Chapter Three. Choosing number choices in problem posing is a knowledge base that has received little, if any, attention. By analyzing problems posed by four teachers, I found that the teachers mobilized number choices in seven different ways: to address mathematical content, to encourage a particular strategy, to provide differentiation, to develop relational thinking, to respond to children's mathematical thinking, for assessment, and to provide an entry point. The teachers mobilized number choices in these ways to move students along the four different types of progressions.