This Chapter reports the findings for each of the four cases studied. Chapter 5 is dedicated to the cross-case analysis that was conducted on all four cases. The section that follows describes the participants and the format used to report the case findings. I have dedicated a separate section for each case within this chapter. I begin the cases with what I consider the most complex and detailed case, Beth.
Within Case Analysis: Participants
There were four cases corresponding to each teacher participant. I list the four
participant teachers in table 4.1 below, along with the name of their institution7 and mathematics course where their observation period took place. I include additional descriptions about the institutions and the teachers in the Methods Chapter (Chapter 3).
Table 4.1. Teacher Cases
Teacher Case Institution and Course from the Observation Period Beth Sundryville High School: Algebra 2
Shannon Sundryville High School: Algebra 1 in a Sheltered Language Instruction Program
Eddy Mixville High School: Algebra 1
Dena Beacon Community College: Quantitative Analysis Course
Within Case Analysis: Findings Format
The findings for each case were divided into two main sections. The first section provides necessary background information to situate the findings in their particular institution, the mathematics course under study and the students that composed each class. Table 4.2 provides a description of the type of content that was reported in each subsection.
92 Table 4.2. Description of Case Background Subsections
Background Subsection Content Description 1 Course and their students’
learning paths
This subsection provides information about the course in relation to the curricular or programmatic needs that it meets at its hosting institution. It also describes the math learning paths that the students in these courses would have followed and/or have access to.
2 Specifics on the Teacher’s course
This subsection provides particular information about the design of the course as implemented by the teacher. It also includes details about the particular unit that was studied during the observation period.
3 Students and their Classroom This subsection provides additional information about the class make-up and a description of the classroom
environment.
The second section reports the findings for each case in response to the research questions. I revisit the research questions below:
• What forms of knowledge of the student (e.g., mathematically foundational, identity, community, etc.) do practicing algebra teachers use to leverage the learning of algebra for students from underserved populations?
• How are these multiple and diverse forms of knowledge of the student applied in practice to support the teachers’ algebra learning goals for their students?
• In what ways do these teachers perceive their use of these forms of knowledge as helpful in supporting their students’ learning of algebra?
• What models can be developed to understand these teachers’ practice as they attempt to advance their students’ learning within their situated context of instruction?
The format for this findings section used four primary categories (i.e., teacher’s learning goals, forms of knowledge of the student, teaching interventions, and central phenomena). I also describe each of these categories (i.e. subsections) of the findings section in table 4.3
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Table 4.3. Findings Subsections and their Content in Response to the Research Questions
Findings Subsection Content Description 1 Teacher’s Teaching goals
RQ1
I triangulated the teaching goals from the full case study data for each teacher. I used their teaching goals to further understand the ways that they considered important to advance their students’ learning.
I found that their use of the forms of knowledge of the student depended on these goals.
2 Forms of knowledge of the student found in a Teacher’s case
RQ1
The forms of knowledge of the student were reported in table form. Although their use was not quantified, they were listed in an order that reflects relevance in use.
I also denoted for each form of knowledge if it was found to have a direct association to any from the original categories in the start list (denoted with an O), or if the form of knowledge was not from the start list (denoted as N).
3 Teaching interventions used by the Teacher based on forms of knowledge of the student
RQ2
Each teacher exhibited central behaviors in how they used forms of knowledge of their students. I have called these central behaviors: interventions.
This subsection includes:
a) A definition for each intervention situated in the context of each case and generated using the full case study data. b) A description of how the forms of knowledge of the
students were used to inform each intervention.
4 Central phenomena in the Teacher’s practice
RQ3 & RQ4
Findings from the first three subsections above were used to understand the central learning phenomena taking place in each case.
In this subsection I describe each teacher’s perspective on their students’ learning, derived from the full case data and in association with their interventions. I also describe the interplay between the teacher’s interventions, situated in the teacher’s perceptions on how their students learn.
I end the findings section with an analytical model that I used to study the central phenomena describing how the teacher’s used forms of knowledge of the student and the resulting learning experience for their students.
The teachers’ learning goals for their students elucidated on each teacher’s conception of what it meant to advance their students’ learning within their practice. The teachers’ learning goals in combination with the forms of knowledge of the student that were found to be used by the
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teachers, together, respond to research question number one (RQ.1). The teaching
interventions, which were informed by forms of knowledge of the student and used to advance
their students’ learning respond to research question two (RQ2). The central phenomena grew out teachers’ perspectives on their students and their learning, thus responding to research question three (RQ3). Analytic models were constructed for each teacher based on all of the case data, describing the central learning phenomena originating from teachers’ use of
knowledge of their students, responding to research question four (RQ4). Research questions three and four are further addressed within the cross-case analysis (Chapter 5) and the
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