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Article Excerpt
Like many teacher educators, we have used video with teachers in a variety of contexts, from methods classes to invited workshops to action research groups. Moreover, the videos have come from a variety of sources, including the teachers' own classrooms, our prior research, and published materials. In some cases, we found that we could count on certain video excerpts to promote substantive discussion among participants time and time again. In other cases, however, excerpts that we thought of as stimulating did not promote the kind of teacher discourse we envisioned. In discussions with each other and with other educators about these experiences, we found ourselves asking, "What makes a video clip interesting?" In particular, we wanted to understand what features of a video clip make it a useful resource for having mathematics teachers examine student mathematical thinking.
To investigate this issue, we draw on our work with teachers in video clubs in which teachers meet with colleagues to view excerpts of their teaching (Sherin & Hart, 2004; van Es & Sherin, 2008). Certainly, teachers' discussions in a video club are shaped by a myriad of factors: the teachers' relationships with each other (van Es, 2009), the role of the facilitator (LeFevre, 2004), the school and district context (Gamoran et al., 2003), and more. Here, we examine specifically the nature of the video clips viewed and their influence on the conversations that take place.
In brief, we find that we can characterize a range of video clips of student mathematical thinking using three criteria: (a) the extent to which a video clip provides windows into student thinking, (b) the depth of student mathematical thinking shown in the video, and (c) the clarity of the student thinking shown in the video. Furthermore, rather than claim that a particular level of windows, depth, and clarity is required for productive discussions of student thinking, we identify key relationships between these three criteria that we believe influence teachers' discussions in important ways.
A Situative Perspective on Teacher Learning
Our research draws on a situative view of learning that emphasizes that learning is a social process that takes place as individuals participate in a community (Greeno, Collins, & Resnick, 1996). Furthermore, the activities, discourses, and tools used by the community are understood to be central influences on the learning that takes place (Lave & Wenger, 1991). In particular, the use of tools and artifacts by community members contributes to the development of shared goals and understandings (Engestrom, 1999).
The situative perspective offers important implications for the design of teacher professional development. For example, close attention should be paid to how teachers are organized for learning and to the resources that are used in such contexts (Gamoran et al., 2003; Horn, 2005). Putnam and Borko (2000) argue in particular that the situative perspective highlights the need to ground teacher learning in the practices of teaching. Like others, they recommend using artifacts of practice such as curriculum materials, student work, and classroom video to situate professional development in the context of teachers' work (Ball & Cohen, 1999; Kazemi & Franke, 2004; Nikula, Goldsmith, Blasi, & Seago, 2006; Smith, 2002). In our research, we rely on the use of video as a way for teachers to explore issues of teaching and learning. Moreover, we do so in the context of video clubs to position the activity of viewing video within a professional community.
Characteristics of Video for Teacher Learning
Video is generally thought to be a valuable medium for exploring teaching and learning because it captures much of the richness of the classroom setting. Furthermore, in contrast to the fast-paced nature of instruction, watching video can provide teachers with the time and space needed to reflect on classroom interactions. In considering how to design video-based artifacts for teacher learning, some researchers discuss the technical quality of the video (Goldman-Segall, 1998; LeFevre, 2004; Roschelle, 2000; Towers, 1998). They emphasize the importance of high sound quality and sufficient lighting and discuss the optimum length of time for a video clip and the benefits of various recording formats.
In other work, researchers focus on the content of the video excerpts--that is, on who and what is shown. Three issues are particularly relevant for the current study. First, there is widespread agreement that teachers will gain more from watching authentic, realistic classrooms than from watching staged interactions (Brophy, 2004; Merseth, 1996). In part, this claim is based on the idea that teachers would recognize staged lessons and not take them seriously as examples of instruction. As a result, developers of video cases often try to ensure that the classrooms shown are representative of the types of classrooms with which the user is familiar (e.g., Seago, 2004). As Brophy (2004) explains:
ideal videos show teachers with whom viewers can identify implementing a curriculum similar to the one they use or will use, in a classroom similar in appearance and student composition to the classroom in which they teach or will teach. (p. 289)
A second issue concerns the availability of information about the context in which a video excerpt is derived. Some teacher educators find that when background information is not provided, participating teachers ask for it (Webb, Diana, Luft, Brooks, & Brennan, 1997). "What happened just before the clip began?.... How often does Derrick speak up in class?.... Have students done this kind of activity before?" Clearly, understanding the context of a teaching-learning interaction is critical if we want to truly understand what is happening (Lampert & Ball, 1998). The real question, then, is how much information is necessary and in what form. In some programs, a series of video excerpts from a single classroom are provided in an attempt to offer more information about the context than would be possible from a single excerpt (Boaler & Humphries, 2005). In other cases, professional development designers provide supplemental information (video or text-based), such as a commentary from the teacher, background on the students, or detailed lesson plans (Seago, Mumme, & Branca, 2004). At the same time, researchers caution that if teachers know too much about the context, they may favor using that contextual knowledge, rather than evidence from the artifact they are examining, to reach their conclusions (Kagan & Tippins, 1991; Seidel, 1998).
Third, research presents varied perspectives as to whether video excerpts should illustrate exemplary practices or teaching dilemmas (Brophy, 2004; Merseth, 1996; Wang & Hartley, 2003). A number of programs use video as a context to show "best practices" around a particular teaching domain (Oonk, Goffree, & Verloop, 2004; Rosaen, Degnan, VanStratt, & Zietlow, 2004). The idea is that video can be used to help teachers develop a vision of what is possible, a vision that may often be quite different from the kinds of teaching practices teachers typically have the opportunity to see. In contrast, other researchers suggest that video be used as a context for reflecting on practice. Rather than illustrating model implementations of new teaching techniques, video for the purposes of reflection should portray the problematics of teaching. As Seago (2004) explains, "we found that the most useful video clips were based on situations where there was some element of confusion (either the students' or the teachers') that typically arises in classrooms" (p. 267). This sentiment is echoed by Shulman (1996) in discussing the development of narrative cases; he claims that moments of "failure or surprise" are particularly educative for teachers. Central to this perspective is the idea that video clips should provide something for teachers to puzzle over or speculate about--and that it is through this process of inquiry that teacher learning will likely occur.
Using Video to Examine Student Mathematical Thinking
As stated earlier, our goal is to use video as a vehicle through which teachers explore students' mathematical thinking. The need for teachers to attend closely to student thinking has been emphasized by the National Council of Teachers of Mathematics (NCTM). Specifically, the Principles and Standards (NCTM, 2000) states that "teachers need to move beyond a superficial 'right or wrong' analysis of tasks to a focus on how students are thinking about the tasks" (p. 24). Furthermore, teachers need to be able to do this at specific moments in time with respect to specific students' ideas (Ball, 1997).
Learning how to listen to and interpret students' mathematical ideas, however, is not a simple task (Chamberlin, 2005). Cohen (2004) explains that one challenge teachers face is recognizing that students have interesting, substantively rich, mathematical ideas. A second challenge concerns the ability to interpret these ideas. Research has shown that in many cases, U.S. teachers do not have well-organized schema for interpreting student thinking of particular mathematical concepts (Carpenter, Fennema, Peterson, & Carey, 1988; Ma, 1999). Yet teachers can benefit from the opportunity to develop new analytic approaches for analyzing the mathematical ideas that students raise during instruction. Cohen (2004) illustrates both advances in teachers' own understandings of mathematics as a result of examining student mathematical thinking and changes in instruction in ways that provide increased support for student learning.
Three Dimensions of Video Clips of Student Mathematical Thinking
With the previous discussion of video and teacher learning in mind, we now turn to the central question of this article: What features of a video clip make it a useful resource for having mathematics teachers examine student mathematical thinking? To explore this question, we focus our investigation on three dimensions of video excerpts of student mathematical thinking: (a) the extent to which a video clip provides windows into student thinking, (b) the depth of student mathematical thinking shown in the video, and (c) the clarity of the student thinking shown in the video. We hypothesize that all three dimensions play a key role in establishing a video clip that promotes teacher discussion of student mathematical thinking. Furthermore, we suggest that different video clips may display different degrees of windows, depth, and clarity of student thinking. In what follows, we discuss the basis for our claims about each criterion. The reader may want to refer to Table 1, which provides detailed information about classifying a video clip across each dimension.
The first criteria concerns the extent to which a video clip provides windows into student thinking--that is, ways of "seeing" what a student is thinking. Lave and Wenger (1991) discuss the importance of access to community practices if one is to learn to participate in such practices. Similarly, we expected that for video to be a means for teachers to explore student mathematical thinking, student thinking would need to be clearly visible in the video. Student thinking is frequently observed through students' verbal explanations and written work. In addition, teachers use nonverbal cues, such as gestures, facial expressions, and hesitation or confidence in answering, when making assessments of student understanding (Goldin-Meadow, 2004; Kagan & Tippins, 1991; Webb et al., 1997). As shown in Table 1, a video excerpt considered to be low on the windows dimension provides little evidence of student thinking from any source. A clip considered to be medium on this dimension provides some evidence of student thinking but with little detail. Finally, a video rated as high in windows provides detailed information from one or more sources (e.g., student drawing on board and student verbal description).
The second criterion refers to the depth of student mathematical thinking--in other words, the extent to which the mathematical ideas that students consider in the video are substantive in nature. Several research studies demonstrate that as student responses move beyond a focus on correctness and rote use of algorithms, teachers begin to develop a deeper appreciation for the complexity of student mathematical thinking (Schifler, 1998; Smith, 2000). Therefore, we hypothesized that video clips illustrating substantive student thinking would most effectively promote teacher learning. A video clip considered "low depth" is one in which students complete a task that is routine for them. In a "medium depth" clip, students continue to work primarily on tasks that are routine for them, but some reasoning is evident. In contrast, a "high depth" clip is one in which students engage in mathematical reasoning and problem solving. Note that a judgment of depth is a judgment not of the potential cognitive demands of a mathematical task itself but rather of students' engagement with the task (Stein, Grovel & Henningsen, 1996). (1)
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