AN ANALYSIS OF A CASE STUDY IN SUPPORTING ELEMENTARY
PRESERVICE TEACHERS’ PROFESSIONAL GROWTH
Kay McClain Maggie McGatha
Vanderbilt University Northern Kentucky University
kay.mcclain@vanderbilt.edu mcgatham@nku.edu
Abstract: The purpose of this paper is to document the
effectiveness of a CD-ROM based case in supporting a group of preservice
elementary teachers’ developing understandings of what it means to teach
mathematics for understanding. In doing
so we will provide a context in which to examine the use of cases in supporting
elementary preservice teachers’ professional development, including their
understandings of issues of both content and pedagogy. Our analysis will document the issues that
emerged as the preservice teachers analyzed and critiqued the classroom episode
highlighted on the CD-ROM. In
particular, we will point to significant pedagogical and content issues that
emerged as a result of the preservice teachers’ investigations.
The purpose of this paper is to document the effectiveness of a CD-ROM based case in supporting a group of preservice teachers’ developing understandings of what it means to teach mathematics for understanding. In doing so we provide a context in which to examine the use of cases in supporting elementary preservice teachers’ professional growth, including their understandings of issues of both content and pedagogy (cf. Barnett, 1998; Barnett, 1991; Copeland & Decker, 1996; Fennema, et al, 1996; Harrington, 1995; Lampert & Ball, 1990; Merseth & Lacey, 1993; Schifter, 1990; Stein, et al, 2000; Sykes & Bird, 1992). As part of our analysis we present excerpts taken from an elementary mathematics methods course that incorporated a series of three CD-ROM based cases in the coursework. In particular, our analysis provides a detailed account of the use of one of the cases1. The case is developed around a lesson that focuses on ranking lists, totals, and average which was taught in a seventh-grade classroom. Some of the resources on the CD include edited classroom video with scrolling transcript, pre- and post-interviews with the teacher with scrolling transcript, copies of all the student work, the teachers’ lesson plan, and a seating chart for ease of identification of individual students. The focus of the analysis is on the issues that emerged as the preservice teachers explored the case and how those issues supported the instructors’ agenda.
Setting
The methods course that is the focus of this analysis is the first in a two-course sequence at a private university and is intended for elementary preservice teachers in their sophomore year. There is no associated field experience. As a result, the instructor2 uses a series of three CD-ROM based cases to create classroom contexts that can be explored by the preservice teachers. A primary goal of the course is to support the preservice teachers as they begin to tease out the complexities involved in teaching mathematics at the elementary-school level. As such, the National Council of Teachers of Mathematics Standards [NCTM] documents, numerous articles from the NCTM practitioner journals, Making Sense: Teaching and Learning Mathematics for Understanding and three CD-ROM based cases comprise the basis of the resources.
Data
Data for this study were collected during the fall semester of 1999 and consist of daily videotape-recordings from one camera while the case was being utilized. Additional documentation consists of copies of all the preservice teachers’ written work, surveys completed by the preservice teachers, a survey completed by the instructor, daily field notes that summarize classroom events, notes from the teacher's daily planning, and taped interviews conducted with the instructor after each class.
Methodology
The general methodology falls under the heading of a teacher development experiment (TDE) (cf. Simon, 2000). This methodology is derived from the constructivist teaching experiment (cf. Cobb & Steffe, 1983; Steffe & Thompson, 2000) and whole-class teaching experiments (cf. Cobb, 2000) by acknowledging that a team of “knowledgeable and skillful researchers can study development by engaging in fostering development through a continuous cycle of analysis and intervention” (Simon, p.336, 2000). The distinction between the TDE and the teaching experiment is that the TDE is concerned not only with the mathematical development of the participants (i.e. the preservice teachers), but also their professional development. In this way, the TDE can be characterized as a “whole-class teaching experiment in the context of teacher development” (p. 345).
Results of Analysis
During the exploration of both the second and the third CD-ROM based case, the instructor focused the preservice teachers’ explorations on four main themes: (1) mathematical content (2) planning for instruction, (3) facilitating the lesson, and (4) understanding students’ thinking. These themes were chosen in support of the instructor’s goal of concurrent development of the preservice teachers’ understanding of (1) the mathematics they will teach and (2) effective strategies for teaching mathematics for understanding. Both the text resources and the CD-ROM based cases highlighted these aspects of teaching. (For example, in the text The Nature of Classroom Tasks coordinates with planning for instruction, understanding students' thinking, and math content. On the CD-ROM the issues matrix highlights excerpts from the classroom focused on each of the four themes.) The issues that emerged around these themes as the preservice teachers investigated the case provided a basis for significant discussion of issues of both content and pedagogy.
Issues surrounding the mathematical content were addressed as the preservice teachers engaged in the same mathematical tasks in which the seventh-grade students participated. In particular, the students were asked to first generate a list of features that they consider when purchasing sneakers. They were then asked to work in groups to rank order the list from least to most important. The final aspect of the task was to take the six ranked lists that had been generated by the groups and compile them into one list that summarized the data in the individual lists.
The preservice teachers’ solution methods included (1) placing each quality in the rank in which it appeared most often, (2) adding the ranks of each quality across the lists and re-ranking according to the sums, and (3) finding the average rank for each quality and then re-ranking by average. The preservice teachers then compared and contrasted the solutions as the instructor attempted to help them tease out the mathematical differences in each method. Questions from the instructor such as, “What if you hadn’t found the average and you’d just taken your sum. Do you think it would have changed the order any?” offered opportunities for the preservice teachers to clarify their understandings of the mathematics of the lesson as they worked to understand the differences in the solution methods.
In discussing planning for instruction, the preservice teachers noted the importance that the video teacher attributed to the anticipation of student responses as she planned her lesson. For the preservice teachers, this was an aspect of the classroom with which they had no experience. In addition, they had no basis for making these anticipations. They therefore questioned how they could incorporate this strategy into their own planning. However, as they reflected back on the mathematical exploration from the lesson, they realized that their own activity could provide a basis for these anticipations. The instructor facilitated this process by making it an explicit topic of conversation.
Instructor: Okay, so here’s what I want you to do. I want you to take your student hat off a minute and put your teacher hat on. And I want you to think like teachers now for a minute. If we gave this task to seventh-grade students, all right, seventh-grade students, what do you think they would do?
As the preservice teachers discussed facilitating the lesson, they experienced a perturbation in that they believed all students’ solutions should be valued equally. The teacher on the CD-ROM explained in her interview that she selected and sequenced students to share their solutions in order to advance her mathematical agenda. This caused a discussion about the role of student solutions in whole-class discussions. The preservice teachers had not reasoned about how to effectively build from their students’ contributions. They had only considered that they should ensure that all students have a chance to share.
This appeared to be a pivotal episode for the preservice teachers as they acknowledged that they might need to make judgments about the worth of a student’s contribution. This highlights a tension in teaching by making explicit the importance of the teacher’s understanding of (1) individual student’s offered solutions and (2) the relation of the offered solution to the overall mathematical agenda. The importance of the teacher’s attention to both of these aspects of a student’s contribution brings to the fore the complex nature of the practice.
A second issue that emerged while discussing facilitating the lesson concerned the classroom norms and student participation in whole-class discussions.
Nance: She asked “why” a lot just to have the students reflect on the like it’s kind of along with classroom norms, like justifying why they came up with their solutions.
Steph: And after a while they started doing that on their own. There was evidence that they, before she needed to ask could you explain that, they would just go ahead and someone would raise their hand and question one of the students.
The preservice teachers noted that the process of negotiating the classroom norms evolved in the course of this one lesson. They saw evidence of the importance of making the norms explicit and then following through in action. They judged the results of such norms as positive and something that they would want to work to achieve in their own classrooms.
A third issue that emerged concerned the fact that at the end of the lesson, the teacher did not identify the “best” method for solving the task. Some of the preservice teachers repeatedly referred to this as giving the lesson “lack of conclusion” and “uncertainty.” However, other preservice teachers disagreed. In the course of the ensuing discussion, the preservice teachers noted the similarities in the method involving sums and the method involving averages. They pointed to the mathematical significance of the difference and discussed possible goals that could be achieved by allowing students to solve the task with either method. Through this discussion, they were beginning to find merit in highlighting the diversity in students’ ways of reasoning.
The fourth theme, understanding students’ thinking, was especially problematic in that the preservice teachers questioned the video teacher’s ability to adequately assess the students since all of their activity was conducted in groups. The preservice teachers felt that without an individual written assignment, there was no way to effectively assess. They viewed this as a weakness of the lesson. In discussion, the instructor pointed to the problems inherent in their suppositions. They were implying that a written assignment had to be given every day in order for a teacher to be able to make a valid assessment. The preservice teachers had difficulty reasoning about how a teacher might use the results of interacting with students to conduct formative assessments that would then inform instructional next steps.
In contrast, the preservice teachers commented favorably on the teacher’s interactions with the students as they worked in groups and discussed in a whole-class setting.
Rosa: She’d just ask questions that would make them, to see if they really understood and she would ask a lot of different kids so she’d make sure that the whole class understood, not just the vocal ones.
Here it appears that Rosa is acknowledging the importance of using classroom discussions as a means of assessing students’ understandings. However, the preservice teachers characterized this aspect of the teacher’s practice as focusing on “classroom norms.” They were unable to make the link between supporting the development of a classroom participation structure that valued students’ thinking and how that would contribute to ongoing assessment of students’ understanding.
Conclusion
In this paper, we have highlighted the opportunities that emerged for the instructor as she built on the issues raised by the preservice teachers while still working to advance her agenda. In particular, the instructor noted that the case provided an exemplary way for her to focus on mathematical content in the context of teaching for understanding. As preservice teachers begin to tease out the complexities of teaching, it is important to situate that development in a deep understanding of the mathematics that they will teach. This implies a concurrent development of (1) an understanding of mathematical content and (2) an understanding of effective pedagogical strategies for the classroom. Cases which provide the resources necessary to focus on both of these aspects can play a pivotal role in preservice teachers’ development by creating opportunities for grounded discussions of issues of both content and pedagogy. It is important to note, however, that we are not attributing the professional growth and development of the preservice teachers to the case described. What we do claim is that the CD-based case provided the instructor with the means of support necessary to achieve her goals for the course in the context of deliberately facilitated discussions.
Notes
1. The CD-ROM based case that is the focus of this study was developed by the first author in collaboration with Janet Bowers of San Diego State University, and Helen Doerr and Joanna Masingila of Syracuse University with funding from the National Science Foundation under Grant No. REPP 9725512.
2. The second author was the instructor of record in the methods class that is the focus of this study. The first author is the teacher on the video in the CD-ROM case.
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