A complex system analysis of practitioners' discourse about research.

The recent past has seen an increasing frequency of calls for teachers to implement "evidence-based" practices (Davies, 1999; U.S. Department of Education, 2002). At the same time, it has been noted that teachers see research as largely irrelevant to practice (Lester & William, 2002; Steen, 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 a "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 the overall discourse for the week in which the thread was constructed. Given these labeling conventions, the first post in a thread is attached to an arrow that does not connect to any other post. Therefore, in Figure 1, the node labeled AB signifies the first post in a thread but the second overall post to the discussion board that week. The node connected to it signifies that CD replied to AB's post with the fourth overall post for the week, and so forth.

[FIGURE 1 OMITTED]

Once response trees had been constructed for each thread of discourse taking place during the study, retrospective analysis continued through the construction of narratives describing the nature of each strand (Glesne, 1999). Narratives are valuable for modeling complex systems because they help tell the story of the system as it evolves in a way that quantitative tools cannot (Kauffman, 2000). In constructing the narratives, chronological order was generally followed to preserve a sense of how each thread unfolded. The thread response trees and their accompanying narratives are presented together in the next section of this report.

Results

During the study, five conversation threads were formed by participants. Three of the threads were quite extensive, whereas the other two consisted of only a few messages. The threads that formed are presented in chronological order in this section according to when the first message to the thread was posted. This convention was followed in order to preserve a sense of the role each thread played in the overall discourse. The subsection headings are the titles that were given to the first post in each respective thread by the author of that post.

Thread 1: Are They Ready?

Maura L. made the first post to the discussion board. Her post started a lengthy discussion thread, as shown in Figure 2. She wrote,

    It was interesting to note that a small percentage of students in
    middle school really understand the mean and median nor know which
    measure is the best to use for a set of data. I was surprised when
    I had to begin teaching this because I never heard of statistics
    until college and even having had 3 statistics classes in college, I
    am still confused as to what is best. Oh I know what they all are
    and I understand how to find them, but middle schoolers just have
    not developed an analytic mind. If I tell them what to do they can
    do it but leave them to find them on their own, they are confused. I
    even give them pneumonic [sic] devices and they still can not
    remember. Maybe we are expecting too much?

With this post, Maura suggested that the reason why middle school students exhibited poor understanding of mean, median, and mode was that the topics themselves were developmentally inappropriate for middle school students. She also implicitly suggested that the concepts might be taught by rote, through mnemonic devices.

[FIGURE 2 OMITTED]

With the second post to the discussion board, Rhonda W. expressed agreement with Maura's through that middle school students often show poor understanding of mean, median, and mode. She wrote, "They can define the terms and tell you how to find each, but not much beyond that. They also have a difficult time deciding which best represents the given data." Maura replied back to her with more observations about students' difficulties in understanding. She noted, "I just would like them to understand that the very highs and very lows in using the mean can skew the data. I have showed them many times how this works with grades and most just do not get it." At this point, Terri K intervened to shift the discourse of the thread away from discussion of student deficiencies. With the fourth overall post to the discussion board, she wrote,

    I think that measures of central tendency need to be personalized to
    accomplish student buy-in. So I wonder if personalizing the question
    might not help? Which would they like to use for their grades, the
    mean or the median? Allow them to work it out together? Seriously, I
    wonder what would happen if we would use the higher of the two
    measures in determining their grades. Do you think that the median
    could show more of what a student has learned at times?

With this post, Terri shifted the focus of the discourse from generating a list of students' difficulties to proposing an alternative teaching method for introducing the concepts.

With the eighth overall post to the discussion board, Barb V. expressed agreement with Terri's proposed teaching method. She stated,

    Many of the students love to argue, lots of times with me. I agree
    with Terri, what a great thing it would be to empower students to
    logically discuss the grade they think they deserve and be able to
    communicate why they deserve it. Students need to be able to have
    control of something, this would give them a great opportunity to
    have the last word.

This post by Barb became a new hub of conversation within the thread. As shown in figure 2, it elicited four direct responses, and three of those responses generated further strings of conversation.

The first direct response to Barb's post came from Sarah G. She expressed further enthusiasm for the idea, mentioning, "Even students who claim to not care about their grades would be motivated to argue their way to a higher grade." In reply to Sarah, Yvonne P. agreed that the teaching method could be effective, yet said she would not be likely to do it in her own classroom. Yvonne, however, changed her mind after Rhonda W. posted a response to her message and stated she planned to try to have students calculate their own grades. After reading Rhonda's post, Yvonne stated,

    I know that in my previous post I said that I didn't think that I
    would actually do this--but the more I think about it--the better it
    sounds. I think that this would be most helpful for students in the
    beginning of the school year so that they may be able to use that
    line of thinking as the year progresses and will hopefully consider
    how a zero brings down an average.

Sarah G. used the forty-fifth post to the discussion board to support Yvonne's decision and urge others to adopt her position as well, stating,

    We all know that learning is often very different from the effort
    put forth. Either way, grades are so subjective: weighted, total
    points; credit/no credit assignments; the number and value of
    assignments; variations in the use of rubies and the rubrics used.
    No one way will ever be a perfect reflection of what they have
    completed or what they have learned. Why not take advantage of this
    subjectivity and make them think a little about the grade that they
    want?

Kristin S. expressed support for Sarah's position by mentioning in the thirty-forth post, "The combination of being in control of their grade--and the opportunity to argue a point--wow, the students would be in hog heaven."

A shorter sub-thread of conversation was sparked by Maura L.'s response to Barb V.'s post. As shown in Figure 2, the fourteenth overall post to the discussion board was made by Maura in response to Barb's expression of enthusiasm for the teaching method proposed by Terri K. Maura re-expressed her earlier pessimism about the idea, stating,

    One problem with students arguing their own grades is that at least
    in my room grades are weighted. The other problem is that having
    them figure their own grades at grade time would be a problem for
    lack of time ... As I said before, I am not so sure statistics
    should be taught when students are at an age when their minds become
    analytic.

Despite her expressed pessimism about the idea, Maura did mention wanting to teach the concepts differently the next school year, stating she wanted to "have alot more discussion among the students and do more small group discussion." Barb V.'s response to Maura, which was the thirty-seventh post for the week, expressed support for Maura's proposed shift in the form of her instruction by agreeing with the position that having students talk with each other was an important aspect of mathematics instruction. At the same time, Barb V.'s post was silent on Maura's contention about developmental appropriateness.

Greg Z. started a third sub-thread of conversation with his response to Barb V.'s expression of enthusiasm for Terri K.'s proposal of letting students calculate their own grades. With the thirtieth overall post to the discussion board, he stated,

    There was an activity in the old math curriculum that was great for
    this debate (argument). It had three different basketball players'
    scores from several games. Each had an advantage. One with the
    highest mean, one with the highest median, and one with the highest
    score in any game. I would ask them which player they would want on
    their team and why. The problem was that many of them would pick the
    player that just had the highest score in one game even though they
    were terrible in the other games. Rarely did that matter to them! I
    tried to argue the point of consistency, but they were not so
    concerned with this point. But then again, we are arguing with
    someone who is always right. What do we expect?

Here, Greg contributed to the collective conversation about helping students understand mean, median, and mode, while also expressing a degree of pessimism about students' abilities to learn the measures. In replying to Greg, Barb V. focused on the positive aspects of his post, stating, "That sounds like an awesome lesson, could I have a copy of it to implement in my classes?" Terri K. also chose to focus on the positive aspects of Greg's post in response to him. With the forty-seventh post of the discussion board, she suggested possible enhancements to the lesson, such as using famous players' names and statistics and incorporating group work in the activity.

The fourth and final spoke on the hub of the conversation started by Barb V. did not generate a sub-thread of conversation. Instead, it was a single message posted by Kristin S. in response to Barb. With the thirty-fourth post to the discussion board, Kristin simply expressed agreement with Barb's enthusiasm for Terri's idea of having students calculating their own grades. Kristin wrote, "I agree that students love to be in control. Having control over some portion of the day is a great motivator for the students." This was almost identical to Kristin's post expressing agreement with Sarah G.'s enthusiasm for the same idea.

A short portion of the first thread stemmed from a reply Greg Z. made to Maura L.'s initial post. With the twenty-ninth post to the discussion board, Greg tried to further delineate the issues involved in choosing the best measure of center, stating,

    The debate over which measure presents the data most accurately is
    where we want the students to decide. It is a debatable point and
    one without an answer. We had many lengthy discussions on this
    question and there is no one-fit solution in this case.

Barb V.'s reply to this post embraced the fact that the point was "debatable." In it, she mentioned,

    What a great lesson it would be to allow them to hold a debate over
    which they believe to be the best way. You could allow them time to
    prepare their 'case' and then allow them to debate. Kids love to
    argue their point.

This post implicitly linked this short thread of conversation with the rest of the discussion thread, which focused upon a discussion of the merits of allowing students to determine which measure of center should be used to determine their grade. By the time Barb made the post, which was the thirty-ninth overall for the week, she had already made similar posts at other points in the thread.

Thread 2: What Should we be Teaching our Students?

The second thread of the week, shown in Figure 3, consisted of a single message. With the fifth overall post to the discussion board, Terri K. wrote,

    Do we settle for students who can perform mathematical computations
    correctly, or is being able to decide correctly which solution fits
    the situation? I think we have read a number of articles that
    illustrated the importance of student mathematical thinking and
    decision-making. Here is another article that illustrates that there
    is more than computation to making sense of a problem. Students see
    averages all the time--from grades--sports. When do they see
    examples of mode and median? Do we clearly demonstrate when those
    central tendencies would yield more useful data? Which measure shows
    what the students know? Which would give the fairest grade? Which
    measure would we use in looking at the endurance of a pitcher or a
    runner? We can look at monthly budgets--At what point during the
    month do we need to have the most cash ready for bills? The public
    transportation is affected by ridership--How would they decide which
    routes to cut or increase? The cafeteria looks at meal participation
    to plan the ordering for the meals--How might they use alternative
    measures of central tendency to help them with their ordering and
    meal planning? How would you tie instruction of measures of central
    tendency into your instruction? Is this something we should discuss
    when we get together?

Even though this post came early in the discourse and asked a number of questions directly of other participants, it did not elicit any responses.

[FIGURE 3 OMITTED]

Thread 3: Subtle but Important Differences

The first message in thread 3 was intended to extend the conversation that had begun in thread 1. I sought to prompt the participants to reflect on the extent to which their statistics instruction consisted of rote teaching of procedures as opposed to engaging students in authentic data analysis as envisioned by NCTM (2000). The hope was that this sort of reflection would move some participants beyond "blaming the students" for lack of understanding of mean, median, and mode. With the sixth post to the discussion board, I wrote,

    Consider the following two word problems. They are similar, but the
    first one is a typical textbook problem, while the second one is
    not:
    (1) Seven 100-point tests were given during the Fall Semester.
    Erika's scores on the tests were: 76, 82, 82, 79, 85, 25, 83. Find
    Erika's mean score.
    (2) Seven 100-point tests were given during the Fall Semester.
    Erika's scores on the tests were: 76, 82, 82, 79, 85, 25, 83. What
    grade should Erika receive for the semester? There is a world of
    difference between these two tasks. Which type of the two tasks is
    more prevalent in your instruction? How do you suppose the thinking
    that is triggered by the second task differs from the thinking
    triggered by the first task?

I hypothesized that this post would help prompt participants to re-examine their teaching practices and beliefs about the field of data analysis, since it had previously been used successfully with pre-service teachers for these purposes (Groth, 2006). It generated some sub-threads of conversation and some individual responses, as illustrated in Figure 4.

[FIGURE 4 OMITTED]

Maura promptly replied to my message with an observation about her current practices that led to a short exchange between her and Sarah G. With the seventh overall post to the board, Maura wrote,

    I have presented the second problem to students which makes them
    think about a grade. I have also given them a problem with a list of
    grades and asked which grade would you rather have; one figured by
    the mean or median and why. What I need to do is to have more
    classroom discussion as one of the earlier articles suggests. I need
    to have a classroom driven by student discussion and not teacher
    lecture. I am looking forward to trying this. Hopefully, meaningful
    discussion will lead to more understanding of the concept.

Sarah's reply to this message expressed agreement with Maura's decision to try to implement more discussion in her class. Maura's reply back to her simply re-affirmed their stated agreement. Hence, this short sub-thread of discourse did not push either individual to re-consider existing positions.

In addition to engaging in discourse with Maura L., Sarah G. responded directly to my post during this thread of conversation. With the eleventh overall post to the board, she answered one of the questions I posed in the first post in the thread, stating,

    The first task does not require higher-level thinking, but the
    second task does. In order to answer the second task, a student
    would have to know how to compute (at least) the mean and median,
    compare the varying results, and then justify his or her choice of
    the more appropriate measure of central tendency.

This post by Sarah did not ask for input from any other study participants and did not receive any further responses.

A post made by Terri K. in response to my original post did provoke a small sub-thread of discussion, although the sub-thread had little statistical or pedagogical substance. With the twelfth post to the discussion board, Terri wrote,

    Based on experience, I think that the first example is what has
    occurred in our classes. We have demonstrated to students what that
    25 does to the mean. (Oh, why obfuscate? We refer to it as an
    average.) ... What would you say if a student said that because both
    the median and the mode are 82, that grade reflects more of the
    student understanding than the mean? Anyone can have a bad day. How
    would you probe the student's thinking?

Kristin's response to this post simply expressed that she would be surprised to hear a student make such an argument about their grade, and also joked about Terri's use of the word "obfuscate." The responses of Yvonne and Terri back to Kristin simply acknowledged the joke.

The final two responses that were made directly to my initial post to start the thread did not lead to any sub-threads of conversation. Greg Z. and Charles M. each posted personal reflections on the questions I posed in the initial post. With the twenty-eighth overall post, Greg reflected,

    The second question obviously reflects higher level thinking when
    the student is posed with a choice, then you should discuss if you
    would determine the two answers differently and why. However, an
    average is an average. Should all students' work count? Or, should
    we just throw out the bad days? At my job, we are still accountable
    for the bad days.

In the fiftieth overall post, Charles stated,

    I would use the second example, because it is more of a real life
    problem and requires the student to do more than crunch some
    numbers. I think it is also important that the student know not just
    how to find the mean and median, but why we might need to know about
    the concept of central tendency.

Even though these posts did not generate further sub-threads of conversation, they did demonstrate that each individual had reflected to a degree on the difference between rote and reform-based conceptions of teaching data analysis.

Thread 4: Use of the Word "Average"

The fourth thread of conversation again began with a moderator post. With the thirteenth post to the discussion board, I wrote,

    Some textbooks use the word "average" exclusively for "arithmetic
    mean" (the "add-and-divide" algorithm). Others use the word
    "average" in connection with the mean, median, and mode. Which usage
    do you prefer? Why? Are there advantages or disadvantages in terms
    of students' learning?

The intent of this post was again to push participants to consider how their teaching practices might influence the degree of conceptual understanding exhibited by students. I hypothesized that this issue would be important for participants to consider in light of research illustrating the role that informal conceptions of the word "average" play in students' understanding (Watson & Moritz, 2000). The post generated a fairly lengthy thread, as shown in Figure 5.

[FIGURE 5 OMITTED]

Terri K. made the first response to my post at the beginning of thread 4. In the sixteenth post to the discussion board, she wrote,

    I think that in terms of measures of central tendency we do students
    a disservice if we teach children to average a list of numbers
    without tying it to the specific reason we are investigating the
    numbers. If all the students know is how to average, that knowledge
    goes nowhere; it is just computation. As far as using the term
    average to refer to mean, median, and mode would seem to confuse the
    issue. What is the thinking behind referring to all of the measures
    of central tendency as the average?

Here, Terri set forth her pedagogical views about teaching mean, median, and mode while also wondering aloud why one would refer to all three as "averages." Rhonda W. responded to Terri's post by agreeing with her thought that the teaching of measures of central tendency should go beyond mere computation.

Rhonda W. also replied directly to my post at the origin of the discussion thread. She agreed with my observation that textbooks made different uses of the word "average." In the seventeenth post to the discussion board, Rhonda wondered,

    For so many years all they've heard is the word average: grades,
    sporting statistics ... Can we reprogram them to think of average as
    mean, median and mode? I'm not sure, but I'm willing to give it a
    try if others agree that we should.

Yvonne P. then replied directly to Rhonda stating,

    In my opinion one of the reasons why students (and sometimes us
    teachers) get confused by the "labeling" for measures of central
    tendency is that when we use real world applications for finding
    measures of central tendencies, in the real world--they ARE called
    "averages." For example--we use grades to explain the concept of
    mean and the students have heard for years-your "average" in math
    class is ______."

Each individual agreed that the word "average" was often used in connection with the arithmetic mean. This was also a key observation of the article (Zawojewski & Shaugnessy, 2000), since it discussed the fact that students tend to prefer the arithmetic mean even if the median or mode are more centrally located in a data set. However, neither put forth pedagogical suggestions based on this observation.

At this point in the thread, I sought to steer the discourse toward a discussion of the pedagogical implications arising from the possible multiple meanings of the word "average." In order to do this, I replied to Rhonda W.'s questions about whether or not it was worthwhile to have students think of the median and mode as types of "averages." In the twentieth message on the discussion board, I posted an excerpt from a book entitled How to Lie with Statistics (Huff, 1954) that illustrated how individuals often chose the measure of center (mean, median, or mode) that best suits the argument they want to make. I went on to say that it might be considered a basic statistical literacy issue for students to understand the multiple common uses of the word "average." This post generated a new sub-thread of conversation, shown in Figure 5.

One branch of the sub-thread stemming from my message containing an excerpt from How to Lie with Statistics was started by Rhonda W. She connected the excerpt to a conversation she had recently had with another teacher about how individuals use the word "average." In the twenty-second post to the discussion board, she stated,

    If you go into an ice cream store and ask about the most popular
    flavor, probably the clerk would respond by saying something to the
    effect that on average, vanilla is the favorite flavor. Vanilla is
    not a number and in their minds, they are giving an average, but to
    us they are giving the mode.

Terri K. responded to Rhonda's observation to note that her earlier confusion about why we would want to make students aware of multiple meanings for the word average had been cleared up by this example.

The second branch of the sub-thread stemming from my message containing the How to Lie with Statistics excerpt began with a post from Sarah G. In the twenty-third post to the discussion board, she wrote,

    Wow, I remember learning the difference between mean, median, and
    mode; although I had known the average for years before this. I
    honestly never realized that the word average is used to refer to
    the median and mode, in addition to the mean. This will definitely
    make me be more critical when I encounter the word "average." As I
    stated earlier, these discussions have been equally as beneficial
    for me as the actual articles.

Terri K's response to this message expressed that she also first became aware of the same issues related to "average" during the conversation. She also affirmed Sarah's observation about the value of the discussion board conversations. Barb V.'s response to Sarah in this thread also focused on expressing common ground with her, as Barb stated with the thirty-fifth post to the board, "I am glad I wasn't the only one who just realized that fact. I could give you the mean, median and mode for questions asked of me but I guess I never really thought about what it meant." The last two messages by Terri and Barb marked the conclusion of the How to Lie With Statistics sub-thread.

In addition to the How to Lie with Statistics sub-thread, another sub-thread of moderate length branched off of Rhonda W.'s post containing questions about how to go about teaching students to think about the mean, median, and mode. As illustrated in Figure 5, Greg Z. started this new sub-thread with the twenty-seventh post to the discussion board. In it, he stated, "I definitely agree that we must associate and relate with what the students are bringing with them and build off of it. How they think of the word average, as opposed to how we think of it." The subsequent posts by Kristin S. and Barb B. in this sub-thread simply expressed agreement with the principle of building on students' prior knowledge. No further connections to the research article under consideration or to pedagogy were made.

The very last message in thread 4 was authored by Charles M. With the forty-ninth post to the discussion board, he replied to my initial post to the thread by writing,

    I think the kids are more comfortable with "average" because it's
    always used in their daily lives. For the most part mean and median
    are used for about 3-4 weeks out of the year. I think if we use
    these concepts more throughout the year the students would be able
    to interchange them better.

Given its temporal order in the discourse and the fact that it was not situated within any of the other sub-threads of conversation, this post did not lead to any further discourse among participants in the conversation.

Thread 5: Easy Difference, No Consistency

A small thread of conversation, shown in Figure 6, was generated by a message from by Greg Z. With the twenty-sixth post to the discussion board, he wrote,

    In my opinion, the difference between each of these two measures is
    not only easy, but is easy to remember for my students. Finding them
    accurately has proved to be some sort of a challenge, but my
    students can tell you what mean and median are. All I ask them is
    what is in the middle of a dual highway (interstate, freeway). They
    respond the median. Then I ask them again where it is in relation to
    the two roads (in the middle). Then they usually will remember that
    the mean is also the average (add, count, divide). For most of my
    students, it is consistency they lack rather than the difference
    between them or what they mean. Then, if we need to continue on to
    mode, I repeat consistently mode/most, and point out they begin the
    same way. I will agree that the two can be confused and difficult to
    find when given other tasks associated with them like in the
    examples in the article.

The early portion of Greg's post expressed disagreement with the claim of the week's article that mean and median were difficult for students to understand. However, near the end of the post, he recognized a difference between the procedural tasks he had given students in the past and the conceptual nature of the tasks used in the research described in the article.

[FIGURE 6 OMITTED]

Greg Z.'s reflection on his past practices sparked similar reflections on the part of Rhonda W. and Yvonne P. The structure of the rest of the sub-thread sparked by Greg's reflection is shown in Figure 6, and it proceeded in the following manner:

    Rhonda: Greg, my students and I believe most students can define all
    three and tell you how to arrive at each, but I'm not sure if they
    can truly tell me how each relates to the data. I know that they
    have a difficult time deciding which measure best accurately
    describes the data.
    Yvonne (in response to Rhonda): Rhonda--I totally agree. My students
    do fairly well when it comes to determining the mean, median and
    mode but they are not able to explain what those measures tell about
    a set of data or why it is important to be able to determine the
    measures of central tendency. I would like to have my students to
    be better able to do that this year. I have a goal!!
    Terri K.: Rhonda, you have voiced the important idea, and Yvonne,
    hooray for your determination. You two have brightened my week. The
    computation and the definitions mean nothing without knowing when
    and why to use them. It is the capacity to make those decisions that
    we want to develop in our students.

In sum, this final thread began with Greg's reflections on his practice prompting others to notice similar aspects of their own practice, and it ended with Terri affirming the value of the reflections by Rhonda and Yvonne.

Discussion

The thread response trees and the narratives in the previous section illustrated that a complex system had emerged among the participants in the ALN conversation. In examining thread response trees, it is apparent that the structure of the discussion changed itself continuously as new posts accumulated, and hence the first defining criterion for a complex system specified by Davis and Simmt (2003) was met. It is also apparent that the structures of the threads were determined not only by the accumulating posts but also by the interactions (or lack thereof) among participants, thus satisfying the second defining criterion

The value of identifying the conversation as a complex system is to provide a frame of reference through which the conversation can be further analyzed. Davis and Simmt (2003) identified five conditions that help sustain a complex system: internal diversity, redundancy, decentralized control, organized randomness, and neighbor interactions. In the following, a brief definition of each of the five conditions will be given and their relationships to the conversation modeled in the present study will be discussed. Possible implications for teacher education and further research will also be offered, with the recognition that readers are likely to glean their own implications from reading the narrative of the case.

Internal Diversity

Internal diversity is concerned with the differences in the nature of the contributions individuals make to a complex system. It is an important factor because "A system's range of possibilities--its intelligence--is thus dependent on, but not necessarily determined by, the variation among and the mutability of its parts" (Davis & Simmt, 2003, p. 148). Peer discourse is a particularly effective mode of teaching when individuals are pushed to justify opinions in the face of conflicting views (Manouchehri, 2002). Lack of diversity in a learning system such as an ALN can result in trivial agreements among teachers where individuals are not challenged to reconsider existing conceptions. Disagreements are important for fostering ALN discourse (Matusov, Hayes, & Pluta, 2005), and these can come about only through diverse participant contributions.

The ALN in the present study did have a degree of internal diversity. In the first thread, for example, a diversity of views on what the research study under consideration implied for classroom instruction was represented. Maura L. started the thread by stating that the article implied that mean, median, and mode shouldn't be taught to middle school students, because she perceived the article to demonstrate that students were not developmentally ready for the concepts. Terri K. disagreed with this perspective, stating that the article should really push teachers to reconsider their approaches to teaching the concepts. By shifting the focus of the discourse, Terri prompted a variety of posts about possible teaching approaches to the concepts.

On the other hand, a variety of discussion threads came to an end when one participant simply identified an area of agreement with another. Some of Barb V.'s posts fit this category, particularly her contributions to the third conversation thread (Figure 5), which were the thirty-fifth and thirty-sixth overall posts to the discussion board. The former post stated a simple agreement with a point raised by Sarah G., whereas the latter did so with a point raised by Kristin S. An implication for those facilitating similar ALN conversations is that participants should be helped to understand how disagreements can be viewed as learning sites and possibilities for extended conversations rather than nuisances and obstacles (Matusov, 1996). In discussions of research reports in particular, facilitators should guard against the notion that such reports contain absolute "answers" to pedagogical issues. Well-argued disagreements with others and with the author of the research report under consideration are vital in helping teachers reflect upon how research translates to their own professional contexts (Sparks-Langer et al., 1990).

Redundancy

Although diverse viewpoints are necessary to sustain a complex system, a degree of similarity among agents in the system is also necessary. Davis and Simmt (2003) noted, "Sameness among agents--in background, purpose, and so on--is essential in triggering a transition from a collection of me's to a collective of us" (p. 150). From a complexity science perspective, the term "redundancy" refers to similarities among agents that are helpful in system-building rather than its more negative connotation of unnecessary duplication.

Redundancy played a role in sustaining the ALN discourse in the present study. Participants shared a common background in that they are worked within the same school district. As members of the same district, they had common curricular responsibilities, which included teaching the concepts of mean, median, and mode. Given this common background, they were able to draw upon their past experiences to describe how they had previously taught the concepts. For example, Greg Z. drew upon this background with the twenty-sixth and thirtieth posts to the discussion board. In the former post, he identified difficulties students had with the concepts, and in the latter he posted an activity he used to partially overcome student difficulties. Each of these posts sparked further discussion among participants. Without a common background and concern for teaching the concepts discussed in the given research article, it seems likely that conversation would have been irrelevant to individual participants' needs and those of the collective. An immediate implication for ALN discussions of research articles is that the moderator of the conversation should become familiar with various aspects of the school contexts of the participants. Even if participants do not share the same curriculum, it is likely that other relevant similarities can be identified through careful examination and inventory of the professional concerns of the participants involved.

Decentralized Control

In discussing the notion of decentralized control in complex systems, Davis and Simmt (2003) stated,

    Within a complex system, appropriate action can only be conditioned
    by external authorities, not imposed. The system itself "decides"
    what is and is not acceptable ... a key element in effective
    teaching is not maintaining control over ideas and correctness, but
    the capacity to disperse control. (p. 153).

This view proposes an alternative to both the "teacher-centered" and "learner-centered" paradigms for instruction by suggesting that "control" is distributed among various members of the system itself. It is not simply in the hands of either the learner or the teacher.

In the present study, decentralized control did play a role in sustaining ALN conversation in some respects. First, it was apparent with the very first message posted to the board by Maura L. that teachers did not regard the research article under consideration as an absolute external authority dictating pedagogical practices. Her skepticism about the advisability of teaching mean, median, and mode to middle school students eventually led to an extensive discourse thread. Second, it was apparent that the participants did not view the facilitator as an absolute external authority. Although moderator posts led to the formation of two threads of discussion, control of the course of the conversation was dispersed over several participants. In examining the various thread response trees, it is apparent that various participants' messages became hubs for conversation. For example, Figure 2 shows that a message posted by Barb V. became a hub of conversation, and Figure 5 illustrates the same for a message posted by Rhonda W. In addition, the threads shown in Figures 2, 3, and 6 involved no moderator participation at all.

Despite the instances of decentralized control, the discussion threads shown in Figures 4 and 5 have the moderator at the center. It seems reasonable to assume that at least some of the messages in these threads were posted in an attempt to please the moderator, perhaps because I had set the parameters for awarding credit for participation. This phenomenon seems most pronounced in Figure 4, where the various posts made in response to the moderator's initial post did not lead to lengthy strands of discourse. Even though no restraints were put on the content of the posts (Dysthe, 2002; Wickstrom, 2003), participants may have assumed that I expected a number of replies to my posts. This suggests that in situations where ALN discourse is facilitated by a university instructor or another external entity, it may be important to emphasize the notion of decentralized control with participants. The purpose of the discourse should be seen to be advancing knowledge construction rather than trying to "please" the moderator.

Organized Randomness

Davis and Simmt (2003) noted, "The structures that define complex systems ... maintain a delicate balance between sufficient organization to orient agents' actions and sufficient randomness to allow for flexible and varied response" (p. 155). If too many restrictions are placed on the activity of a system, the restrictions will choke out activity by encouraging too much redundancy. If too few restrictions are placed on it, an overabundance of diversity can lead to lack of focus and meaningful knowledge construction. In formal educational settings, whether they are online or face-to-face, the instructor plays a significant role in placing constraints on the system. Expectations for participation in discourse that are (or are not) communicated to students unavoidably help shape the structure the system ultimately takes. Given this situation, it becomes important to examine the potential impact of moderator-imposed constraints on an environment similar to that considered in the present study.

One moderator-imposed constraint in the present study was that at least three posts to the discussion board were to be responses to posts made by others. The intent of this constraint was to communicate to participants that collective knowledge construction was a primary goal of the ALN. As one examines the ends of the discussion threads in each figure and reads through the accompanying narratives, it is apparent in some cases that this constraint may have encouraged some posts that lacked substance because they were made simply to satisfy the constraint. On the other hand, it is impossible to say what this particular complex system would have looked like without the constraint in place. It may be that without it, the extended strands of conversation observed would not have developed. The key for those moderating in an online environment seems to be to communicate the importance of the goal of collective knowledge construction. If one moderates a group that already understands this principle thoroughly, it may be unnecessary and even harmful to impose a constraint on how many messages must be made in reply to others. On the other hand, if one moderates a group that is relatively new to online learning, the constraint on how many messages must be made in reply to other may play an important role in communicating expected norms within the system.

As noted earlier, in the threads shown in Figures 4 and 5, conversation began with a moderator post. While this could be due to participants wishing to please the moderator, another contributing factor to this phenomenon seems to be that the participants were implicitly seeking out constraints on the content of the posts. By waiting for a moderator post to the discussion board, they could perhaps further discern what would be "acceptable" to post. This suggests that imposing absolutely no restrictions on the content of the posts made to the board, as in the present study, may allow for an uncomfortable degree of diversity for some participants. A mid-road between not imposing any restrictions on the content of posts and prescribing precise content could be to provide a set of heuristics for participants to consult in deciding upon what to contribute to the discussion. For ALNs focused on the study of educational research, a partial list of these can be gleaned from posts that helped build threads of conversation in the present study. They include: (a) Questioning how the research study may be applicable/relevant to classroom practice; (b) Affirming the practices/beliefs of another and providing a detailed explanation for agreement; (c) Expressing disagreement with the content of the post of another and providing a detailed explanation for the disagreement; (d) Expressing a classroom dilemma and asking for input from others; (e) Posting excerpts from relevant pedagogical activities and inviting others to do the same; and (f) Proposing a shift in the direction of conversation within a given thread.

Neighbor Interactions

On the surface, the phrase, "neighbor interactions" seems to suggest the need for physical interactions among participants. However, Davis and Simmt (2003) contend, "These neighbors that must bump up against one another are ideas, hunches, queries, and other manners of representation" (p. 156). From this perspective, ALNs appear to have an advantage over synchronous or face-to-face instruction, since they allow extended periods of time for "neighbors" to bump against one another, and they allow for multiple simultaneous strands of discourse containing different ideas, hunches, and queries rather than just a single strand.

Participants in the ALN in the present study contributed to neighbor interactions that would not have taken place otherwise. Sarah G. and Terri K. each explicitly noted this in thread shown in Figure 5. Sarah expressed the value of encountering various different ideas encountered during discussion board interaction, and Terri affirmed her statement. Despite the fact that all participants worked in the same school district, opportunities to meet to discuss teaching and exchange ideas were sparse. Without the ALN, the neighbor interactions vital to sustaining a learning community among teachers were very difficult to come by due to the structure of the teachers' professional setting, typical of others in the U.S. (Stigler & Stevenson, 1991).

Although the thread response trees and the accompanying narratives suggest that most participants took advantage of the opportunity for neighbor interactions afforded by the ALN, they also show that some did not. In particular, Charles M. placed himself on the periphery of the discourse by not contributing until the end. Kristin S. also placed herself near the periphery in the same manner. Although each of these individuals may have benefitted from reading the neighbor interactions taking place on the discussion board, they did not make substantive contributions to sustaining the life of the discourse by interjecting elements of diversity or redundancy. To remedy this situation, one might drawn the conclusion that further constraints should have been required to make a post a day over a given period of time. Before taking such a step, however, consideration must be given to the principle of organized randomness. The proposed constraint may do more harm than good to the discussion board if it excludes individuals or ideas from the conversation or prompts individuals to make trivial postings to the board. Ultimately, the decision about whether or not to impose such a constraint must be made by the moderator based on knowledge of participants' characteristics. In any event, the moderator must strive to make participants aware of the need for active exchange of ideas, and that such exchange cannot take place if individuals adopt parasitic orientations toward the discussion.

Conclusion

There appears to be a natural match between teachers' study of educational research and the use of an ALN discourse environment. Just as research "dissemination" is properly understood as a non-linear process (Lesh & Lovitts, 2000), ALN discourse can be non-linear. In the present study, the ALN allowed teachers to exchange ideas related to a research report describing students' understanding of statistical concepts. Rather than a top-down approach of having an "expert" explain the research and its implications, ideas were built from the ground up through online discourse.

Although the practitioners' discourse did not deal exhaustively with the research report, the conversation contained elements necessary to sustain a complex system, including diversity, redundancy, decentralized control, organized randomness, and neighbor interactions. It should be noted, therefore, that there can be a distinction between a "complex system" and "high-quality discourse" about an issue. A complex system can succeed or fail to contain high-quality discourse. It seems likely that a blend of success and failure will be present in most situations. This report provides an example of a system containing both successes and failures. Instances of quality discourse were apparent, as when Maura's assumption that mean, median, and mode were too difficult for students was challenged by Terri. Instances of missed opportunities for quality discourse were also apparent, as evidenced by an absence of practitioners' discussion about trying the NAEP tasks with their own students. The value of a complex system model of ALN discourse is that it can help moderators understand factors including the quality of conversations and become conscious of opportunities to help shape the discourse.

The complex system model described in the present study highlights specific issues to which other potential ALN moderators can attend. In particular, moderators should seek out opportunities to help participants understand the components of a complex system, as detailed in the discussion section, so that all participants actively support the discourse. Helping mathematics teachers understand the dynamics of such an environment may well be a challenge, particularly if they subscribe to the "transmission" paradigm of instruction (Kline, 1977) in their own practice. Although this paper highlights some possible inroads in this endeavor, it, like any other research report, does not provide final answers. It is hoped, however, that the complex system model presented this paper will help sharpen and refine the intuitions (St. Julien, 2005) of others seeking to bring practitioners into the discourse surrounding educational research.

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Randall E. Groth

Salisbury University 

Table 1. Study Participants

                           Number of
                 Number    years teaching
                 of years  middle school
Participant      teaching  mathematics     Role in school district

Yvonne P. (YP)   12         6              Full-time mathematics teacher
                                           at middle school A
Charles M. (CM)   3         3              Full-time mathematics teacher
                                           at middle school A
Maura L. (ML)    16        15              Full-time mathematics teacher
                                           at middle school B
Rhonda W. (RW)   27         8              Full-time mathematics teacher
                                           at middle school B
Greg Z. (GZ)      4         3              Full-time mathematics teacher
                                           at middle school B
Terri K. (TK)    27         1              School district curriculum
                                           supervisor
Sarah G. (SG)     5         0 (full-time)  School district resource
                                           teacher
Kristin S. (KS)  17        10              School district new teacher
                                           mentor
Barb V. (BV)     23         0 (full-time)  After-school and summer
                                           middle school mathematics
                                           teacher