...1999). If one accepts the premise that research holds value for educational practice (Margolinas, 1998; NCTM Research Committee, 2006; Silver, 1990), it is important for teacher educators to develop instructional experiences that bring teachers into the discourse surrounding educational research. At the present time, communities of teachers and researchers are often largely separated by communication-related barriers (Sowder, 2000; Silver, 2003).

Lesh and Lovitts (2000) observed the following about the relationship between research and practice:

In mathematics and science education, the flow of information between researchers and practitioners is not the kind of one-way process that is suggested by such terms as information dissemination. Instead, to be effective, the flow of information usually must be cyclic, iterative, and interactive (p. 53).

An implication of this statement is that "transmission" view of familiarizing teachers with research is naive, just as mathematical pedagogy based on such a view is misguided (Kline, 1977). Lesh and Lovitts (2000) went on to state, "Although simpleminded, 'delivery-and-reception' metaphors are recognized widely now as being inappropriate for describing the development of students, teachers, or other complex systems, these same machine-based metaphors continue to be applied to the development of programs of instruction" (p. 57).

Teachers' Conversations as Complex Systems

Complexity science provides a framework for designing and analyzing the types of complex systems for the development of teachers mentioned by Lesh and Lovitts (2000). Davis and Simmt (2003) provided a discussion of the implications of complexity science for mathematics education. They defined complex phenomena in the following manner:

First, each of these phenomena is adaptive. That is, a complex system can change its own structure ... Second, a complex phenomenon is emergent, meaning that it is composed of and arises in the complicated activities of individual agents. In effect, a complex system is not just the sum of its parts, but the product of the parts and their interactions (Davis & Simmt, 2003, p. 138).

These two defining characteristics also apply to complex systems that arise in other disciplines, such as cells, bodily organs, cultures, economics, and ecosystems (Johnson, 2001).

Literature pertaining to mathematics teacher education contains empirical examples of complex systems emerging among teachers as they converse with one another. Davis and Simmt (2003) characterized a study group of teachers trying to solve mathematics problems as a complex system. The structure of the conversations among the study group changed as individuals each brought unique contributions to the solutions of problems to the conversation. Smitherman (2005) described similar dynamics among a group of pre-service teachers discussing fraction concepts. She described how she conducted a classroom conversation among pre-service teachers by asking them to share their thoughts on the fraction one-third. By the end of the conversation, many of the aspects in the NCTM (2000) standards connected to fractions had been considered by the group. In each complex system, knowledge was constructed in a non-linear fashion as individuals contributed their perspectives on the objects of study at hand to the conversation.

In studying teachers' interactions within complex systems, it is important to keep in mind that complexity science provides a framework for analyzing human interactions, which are never devoid of contextual peculiarities (Stacey, 2003). Accordingly, the goal is not to strip away contextual factors in order to discover abstract principles that will invariably transfer to other systems, but instead to model the interrelationships, connections, similarities, and differences among components in the given system (Smitherman, 2005). This mode of inquiry is in some ways a more authentic approach to educational research, since, "Teachers know that knowledge is context-laden, slippery, and changeable but that the current analytic has insisted that good knowledge is solid, unchanging, and context-free" (St. Julien, 2005, p. 113). St. Julien (2005) went on to state that although complexity science research does not have the abstraction of general principles as its guiding focus, the system models it produces can help sharpen and refine the intuitions of readers with similar goals working in similar settings.

Purpose of the Study

The purpose of the present study was to construct a model of the complex system that emerged among a group of practitioners discussing a statistics education research report related to concepts they were responsible for teaching. Consistent with the lens of complexity science, the goal was to construct one possible model of the observed system rather than a singular, definitive account (Davis, 2005). The primary reason for constructing the model is to help teacher educators begin to understand and shape similar conversations among the practitioners with whom they interact.

Methodology

This report can be described as an intrinsic case study (Stake, 2000), because the goal was to understand and model the dynamics of interaction among a particular group of practitioners rather than to make broad generalizations. Nevertheless, it is likely that readers will draw personal generalizations from the case study by recognizing common empirical ground with their own experiences. In essence, the goal of the case study is to facilitate the formation of such naturalistic generalizations (Stake & Trumbull, 1982).

Participants

Nine practitioners from a school district in the Mid-Atlantic U.S. participated in the study. Three taught mathematics at one middle school in the district, two at another district middle school, and one taught mathematics part-time in after-school and summer programs. The remaining three played supporting roles for teachers, as one was a district-wide resource teacher, one was a new teachers mentor, and another was the curriculum coordinator for the district. A summary of some of the characteristics of the participants is provided in Table 1. All names in Table 1 are pseudonyms.

Procedure

Participants took part in the conversation described in this study as part of an ongoing professional development program I had designed and implemented (Groth & Bergner, 2007). This placed me in the position of a participant-observer (Glesne, 1999) rather than that of a detached researcher. The conversation took place online in an asynchronous learning network (ALN). Harism's (1990) definition captures the key characteristics of an ALN: "(a) Many-to-many communication; (b) place independence; (c) time independence; (that is, time-flexible, not a temporal); (d) text-based; and (e) computer-mediated interaction" (p. 43). Data gathering was facilitated by the fact that the text of the discourse was captured immediately when entered online. The ALN conversation described in this study took place over a one-week span.

In addition to facilitating data collection, there were pedagogical reasons for having the conversation in an ALN. Shotsberger (1999) and Newell, Wilsman, Langenfeld, and McIntosh (2002) each reported using ALN environments to promote peer discourse and reflection among mathematics teachers about pedagogical issues. The many-to-many feature of an ALN allows individuals to engage in a number of threads of discourse simultaneously, therefore encouraging the emergence of a complex system. The place and time independence features allow individuals to participate where and when suits them. This was important for the participants in the present study because their professional lives took place in different buildings within the school district.

Since participants received credit toward recertification for participation in the online discussion, I set parameters for taking part in it. All participants were to make at least four posts to the discussion board each week. At least three of the four posts were to be responses to others. This parameter was set in order to help the group move toward collective knowledge construction, which can be a positive outcome of participation in an ALN (Salmon, 2004). However, no guidelines were given for the specific content of the posts, since such moderator-imposed restrictions had been counterproductive to reflection and peer discourse in previous studies (Dysthe, 2002; Wickstron, 2003).

The ALN conversation described in this report focused upon an article entitled "Mean and median: Are they really so easy?" (Zawojewsky & Shaugnessy, 2000). The article gave examples of tasks used on the National Assessment of Educational Progress (NAEP) to assess students' understanding of measures of central tendency. It also provided details about students' responses to the tasks. Important findings from the NAEP data included the fact that many students did not understand the relative advantages of the mean and median as measures of center. The authors of the article recommended that readers use the NAEP tasks to probe their own students' thinking about the mean and median in order to gather information to design instruction. I encouraged the study participants to point out perceived strengths and weaknesses of the article and to respond to comments made by their colleagues during ALN interaction.

Data Analysis

A total of 50 messages were posted to the online discussion board during the conversation of the article. As the ALN discourse unfolded, I read each one of them. This analysis led to posts designed to extend participants' thinking. The specific nature of my posts is described along with those of the participants in the next section. My ongoing analysis provided context for a retrospective analysis after the ALN interaction had concluded.

Retrospective data analysis began with the construction of thread response trees (Aviv, Erlich, Ravid, & Geva, 2003) to help model the structure of the ALN discourse. A sample thread response tree is given in Figure 1. Each node in the tree represents a message posted to a discussion board. Nodes are labeled with the initials of the individual making the post. Arrows in the tree point from a post toward the message to which the post replied. The numerals on the arrows correspond to the chronological order of each post in...

NOTE: All illustrations and photos have been removed from this article.



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