(Page 2 of 8) - Experiential models of students' geometry thinking: Case studies of two secondary mathematics teachers authored by Rhodes, Ginger.

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Methods

This research paper details case studies of two high school geometry teachers. These teachers,

Judy and Barbara, were selected because they were experienced teachers with advanced college
degrees who were willing to discuss their instructional practices and attend working group
sessions. Data were collected for both teachers during one semester in one of their semester long
geometry classes. Over the course of the semester there were 13–16 classroom observations and
six working group sessions. Field notes were taken for each observation. For six of the classroom
observations there were 30–60 minute pre-observation and post-observation interviews. The focus
of the interviews was the teachers’ descriptions of the components of the learning trajectories.
When possible, the classes were observed the day before and the day after these six observations.
The six lessons for each teacher and all of the working group sessions were videotaped. In
addition, all teaching artifacts for the observations and the working group sessions were collected
and analyzed.

The complexities of the classroom make it difficult for teachers to focus specifically on

students’ mathematics. With that in mind, there were six working group sessions that occurred
approximately every two weeks. The purpose of the sessions was not to change teachers’
instructional practices, but to support the two teachers in their efforts to discuss their students’
mathematical thinking. The tasks in these sessions included solving a mathematics tasks, watching
videos of students solving tasks, creating tasks to use in their own classrooms, and discussing their
student solutions to their developed tasks. In two of the sessions the teachers created or modified a
task to use in their own classrooms. The two lessons that included those tasks were part of the six
lessons that pre- and post-interviews were conducted.

Data was analyzed based on the learning trajectory framework and using an interpretative

stance in order to examine “how individuals experience and interact with their social world, [and]
the meaning it has for them” (Merriam & Associates, 2002, p. 4). I began initial analysis during
data collection through a research journal. The contents of this journal included my reactions to
lessons, ideas about possible answers to the research questions, and themes that were emerging.
After data collection I began a retrospective analysis with four stages. During the first stage I
coded field notes and interviews. Next, using interview data, field notes, and teaching artifacts I
created the teachers’ learning trajectories. In order to distinguish between what the teacher planned
to happen during the class and what the teacher noted about what actually happened during the
class, I created two learning trajectories for each of the six lessons—the projected learning
trajectory (PLT) and the enacted learning trajectory (ELT). When necessary, I used video data to
fill in missing components of field notes. Third, I used the coding scheme to revisit the data by
coding the created learning trajectories. In the process of coding the learning trajectories, I
compared the PLTs and ELTs for individual lessons. Finally, I looked for confirming and
disconfirming evidence for each of the themes that emerged.

Results

The learning trajectories that I created for both teachers’ six lessons were complex due to the

factors that influenced their development. I want to acknowledge that I am adapting Simon’s
(1995) notion of hypothetical learning trajectories to create the teachers’ learning trajectories. The
learning trajectories were based on the teachers’ thoughts about the initial learning goal(s), the
students’ current mathematical understandings, the learning activities, and the changes that

Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29

annual meeting of the North

American Chapter of the International Group for the Psychology of Mathematics Education,
Stateline (Lake Tahoe), NV: University of Nevada, Reno.