past. This research study identified some instances of this dynamic relationship between the written and enacted curriculum in college-level mathematics classes designed for elementary teachers.
Section II: Conceptual Framework
This research used a modified version of Brown's (2012) framework to conceptually frame the study. Brown proposed the Design Capacity for Enactment Framework (Figure 2.1) to explain how the enacted curriculum is a result of the interplay between a teacher and a written curriculum. This framework highlighted both the
resources of the teacher and the resources of the curriculum. Teachers’ resources included their goals and beliefs. Teachers’ beliefs included beliefs about the subject matter, teaching and learning the subject matter, about curriculum, and beliefs about their own particular students. Teachers’ resources also included teachers’ subject matter knowledge and their pedagogical content knowledge.
Curriculum resources in Brown’s model included “physical objects,”
“procedures,” and “domain representations” (p. 26). Brown studied teachers’ use of science curricula, so physical objects, such as laboratory equipment, were particularly important to him. In mathematics classes, while physical objects such as rulers, calculators, manipulative materials, and graph paper are used during instruction, they may not rise to the same level of importance or saliency as a component of curricula. In Brown’s explanation, procedures referred to directions for teachers, scripts of enactment,
instructions, or problems for students to solve. Domain representations referred to explanations or representation of concepts.
Figure 2.1. Brown's (2012) Design Capacity for Enactment Framework
Because Brown used this framework to describe the curriculum use of science teachers, further articulation of each of the curriculum resources in the context of
mathematics courses for pre-service teachers is helpful. Domain representations referred to explanations of mathematical ideas, models, and representations. Because these courses concern mathematics for teaching, representations of elementary students’ mathematical thinking would also be a domain representation. These could be found in the student editions of textbooks or in the Instructor’s Guides. Some of these domain representations are on Power Point slides. Procedures include problems and activities that students engage with. They also include instructions to the instructor about how to
facilitate the class. In this study, two of the curricula also had videos of the curriculum being enacted in other college courses, with commentary. These videos are also a
component of the procedures. Physical objects relevant to this study include graph paper and three-dimensional solid blocks. Some of these physical objects were paper shapes that pre-service teachers cut out. The categorization of these different resources is less important than the specific resource being studied. For example, one problem (or procedure) resulted in pre-service teachers creating trapezoid cut outs (physical objects) in order to create a model that explained a mathematical formula (domain representation). Within the Instructor’s Guide, there were sample questions to ask pre-service teachers (procedures) and different ways pre-service teachers might justify the formula (domain representations). This research study holistically analyzed the participants’ use of this set of curriculum resources around this mathematical idea and problem and ones like it, rather than identify the category of the different resources. Similarly, this study
investigated how instructors use curriculum resources and what elements supported them in their instruction. It did not label the curriculum usage as offloading, adapting, or
improvising.
In this study, some adjustments to this model were used. First, “instructional outcomes,” for specificity, were referred to as “instructional practices.” This change clarified that it was the practices that were impacted, and not necessarily student learning outcomes, which involved other variables not present in the framework. Second, for clarity, “teacher” was changed to “instructor” to represent college faculty. Furthermore, mathematical knowledge for teaching teachers (MKTT, Olanoff, 2011; Superfine & Li,
2014; Superfine & Wenjuan Li, 2014) was added to Teacher Resources. Though it is not yet completely clear whether this knowledge is an extension, subset, or altogether different construct from subject matter knowledge (SMK) and pedagogical content knowledge (PCK), it is clear that instructors need and have knowledge of mathematics, pedagogy, and elementary contexts that goes beyond the knowledge of that needed by elementary teachers.
In addition, the framework was adapted to show the dynamic nature of instructor’s use of curriculum. Curriculum materials are tools, and tools are used for enacting a practice. Just as the musician learns by listening to himself play sheet music, or a pole vaulter learns by reflecting on his performance, instructors learn by enacting a curriculum (Choppin, 2009, 2011; Remillard, 1999; Remillard & Bryans, 2004). Their knowledge, goals, and beliefs are changed or reinforced after noticing student responses to the enactment of curriculum. Therefore, after each iteration of enactment, what instructors notice or attend to in curriculum materials changes. While the published curriculum materials themselves do not change, the instructor’s interpretation of them may. This phenomenon is made more explicit in Figure 2.2.
Figure 2.2. Dynamic Adapted Conceptual Framework
According to this dynamic framework, while curriculum resources have the potential to impact instructional practices, instructional practices also impact how instructors view curriculum materials. For example, instructors who establish norms where students explain their thinking have different opportunities to learn about their students and about the affordances of particular aspects of curriculum materials than instructors who do not use this instructional practice. This learning in turn impacts how instructors interpret, modify, or adapt curriculum materials in subsequent enactments.
wide range of backgrounds. Some have K–12 teaching experience while others do not; some have advanced degrees in mathematics while others have advanced degrees in education, psychology, or other fields (Masingila et al., 2012). Based on their past experiences, they come with different knowledge bases and different beliefs about what high-quality mathematics instruction looks like. Therefore, they may notice different aspects of curriculum materials in their planning and interpret them in different ways. Additionally, the curriculum materials themselves can also influence instructors’ actions, beliefs, and knowledge. While curriculum materials can influence instructional practice, the instructor also has an existing repertoire of practices and works within a professional community that has existing norms and standards for teaching practices. Instructors with K–12 teaching experience may be more likely to have been socialized into teaching practices that actively engage students during class time, while mathematicians may have been socialized into teaching practices that focus on precise, well-articulated delivery of information (see also Schoenfeld, Thomas, & Barton, 2016). This study seeks to describe how curriculum materials might interact with these other factors to support powerful instructional practices.
Different instructional practices, in addition to the instructors’ beliefs, knowledge, and goals, then impact what instructors notice from student responses to curricular
enactment. Some instructors may attend to student affect, while others may attend to conceptual understanding, while still others may attend to students’ developing abilities to engage in mathematical practices like justification. While goals and beliefs influence whether instructors notice these issues, these issues also require instructional practices
that allow them to surface. For example, some questions posed to students may elicit evidence of conceptual understanding, but if instructors answer all false claims themselves, rather than giving the authority to the students, they may not have an opportunity to notice whether students improved their abilities to make sense of ideas. The next section, therefore, reviews the literature on instructors’ learning from and use of curriculum materials.
Section III: Research on Instructors’ Learning from