ZDM Mathematics Education (2011) 43:53–64 DOI 10.1007/s11858-010-0305-2

ORIGINAL ARTICLE

Co-facilitation of study groups around animated scenes: the discourse of a moderator and a researcher Talli Nachlieli

Accepted: 30 December 2010 / Published online: 28 January 2011  FIZ Karlsruhe 2011

Abstract This paper examines the methodological issues pertaining to study group facilitation assisted by representations of teaching. The study groups, which comprised high school geometry teachers, aimed at achieving two goals: enhancing professional development among teachers and studying the practical rationality of geometry teaching. Therefore, session facilitation involved playing two roles that were often contradictory. A specific co-facilitation model was developed to achieve both goals. This paper examines the characteristics of the discourse of each of the facilitators. The results of this study are used to explain how the facilitators’ discourse created a relevant environment for achieving both goals. Keywords Facilitation  Study groups  Professional development  Discourse  Routines of interaction

1 Introduction This paper examines study group facilitation assisted by representations of teaching among high school geometry teachers. The study groups were part of the Thought Experiments in Mathematics Teaching project (ThEMaT) directed by Herbst and Chazan. ThEMaT aims at collecting the practical rationality applied in teaching geometry and algebra in secondary schools. It seeks to identify tacit

T. Nachlieli (&) Levinsky College of Education, Tel Aviv, Israel e-mail: [email protected]

norms or tendencies around which practitioners make instructional decisions. Further, it examines how practitioners perceive and value those norms, as well as deviations from those norms, in selected instructional situations (Herbst & Chazan, 2003). For the purpose of eliciting the practical rationality of high school geometry teaching, teachers were recruited to form two study groups that met once a month after school hours. These study groups were aimed at achieving two goals: enhancing the professional development of the participating teachers and researching the practical rationality of geometry teaching. Therefore, session facilitation involved playing two roles that were often contradictory. On the one hand, the facilitators sought to provide the teachers with a secure space in which they could freely discuss issues relevant to their own teaching. On the other hand, the facilitators questioned the work of the teachers to test the boundaries of what they considered viable in specific situations. It is important to note that there was no clear boundary between the parts of the discussion aimed at professional development and those aimed at research. The project organizers handled this duality by assigning two staff members to co-facilitate the sessions—a moderator and a researcher. The moderator shared his experience as a geometry teacher with the participants and was responsible for creating a welcoming environment in which participants could exchange their thoughts and reflections. The researcher, who came from a different background and was eager to learn about the wisdom of practice, took on the role of a provocateur who questioned what often may be unnoticed by practitioners or is so obvious as to remain unquestioned. The goal of this study was to better understand how this dual facilitation was useful in accomplishing the goals of

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the study groups.1 Hence, in the paper I attempt to identify the routines of interaction in the discourses of the researcher and of the moderator by examining the characteristics of each facilitator’s discourse. Specifically, I considered the following questions: when did each of the facilitators choose to participate in the discussion? What type of reaction did each have to the conversation? How, if at all, did this reaction change the focus of the ongoing discussion? The results of this study are used to explain how the identified discursive sequences created a relevant environment for enhancing teachers’ professional development and for learning about the practical rationality of geometry teaching.

2 Theoretical background 2.1 Facilitating study groups The issue of facilitating focus groups for various purposes has been widely discussed in literature (e.g., Sherin et al., 2004; Stewart et al., 2007; Tochon, 1999). Despite differences in goals and contexts, there is general agreement regarding the centrality and complexity of the facilitator’s role in the success of study groups: ‘‘Mastering the technique of moderating a focus group is an art requiring a moderator to wear many hats and assume different roles…he or she has the unenviable task of balancing the requirements of sensitivity and empathy, on one hand, and objectivity and detachment, on the other’’ (Stewart et al., 2007, p. 69). Many studies have focused on the role of the facilitator in study groups in which representations of teaching, mainly video records of classroom observations, are used for teachers’ professional development (e.g., Levin, 1999; LeFevre, 2003; Sherin et al., 2004). In these studies, the facilitator is often seen as having a twofold role: to create a comfortable environment for the participants to engage in discussions and to make sure the discussions meet the goal of the study group with respect to the topics discussed and the level of discussion. These studies often address the effectiveness of enactment in professional development workshops and the var-

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This dual facilitation seems to work and is not questioned in this paper. From the research perspective, the data collected during the sessions are useful in learning about different aspects of the practical rationality of geometry teaching (e.g., Nachlieli et al., 2009; Herbst & Miyakawa, 2008). The effectiveness of the professional development enhancement was not thoroughly studied, but we relied on the fact that the practitioners participated in the sessions throughout the year. Indeed, some chose to attend the study groups for a second year and often noted how the discussions helped them in their instruction.

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ious relations between facilitators and participants with respect to designing professional development programs that can be implemented beyond a specific place and group of participants (Borko 2004). Some studies focus on the interactions that develop in focus groups. Some investigate the interaction among the participants (e.g., Myer, 1998), while others examine the relationship between the facilitator’s questions and remarks and the number and type of responses provided by the participants (e.g., Nemirovsky & Galvis, 2004). Yet, some others study the language used by the facilitators to achieve the specific purpose of the study group (e.g., LeFevre, 2003). These studies have concluded that not only the topics to be discussed, but also the types of responses desired must be considered when planning a study group meeting. Thus, the questions and remarks of the moderator must be planned accordingly (Nemirovsky & Galvis, 2004). The objective of the current study is to characterize the discourse of the facilitators by identifying unique interaction routines in each discourse. The results can be used to better understand how the role assigned to each facilitator was actually enacted, as well as to define a model of this type of facilitation. 2.2 Practical rationality and instructional situations The ThEMaT project focuses on learning about the practical rationality of mathematics teaching. Practical rationality is defined as a system of dispositions available to practitioners to construct reasonable ways to act while participating in an instructional situation instance. Dispositions are defined as ‘‘categories of perception and appreciation’’ (Bourdieu & Wacquant, 1992, p. 11). These categories of perception consist of the moments, actions, people and things perceivable by practitioners in the situations in which they practice. The categories of appreciation include the principles or values that members of a practice can use to justify or otherwise discard possible actions in an instructional situation (Herbst, Nachlieli & Chazan, 2011). Based on the work of Goffman (1974), Herbst (2006) proposed the notion of an instructional situation to describe the ‘‘frames that teachers and students use to exchange the work they do in and through their interaction for claims on having taught or learned the knowledge at stake.’’ We hypothesize that classroom life is organized into interaction segments, the goal of which is to produce mathematical work and exchange it for claims on the knowledge at stake. An instructional situation model should therefore include the following: what needs to be exchanged, who has to do what for that exchange to happen, and how the events unfold over time (Nachlieli, Herbst & Gonza´lez, 2009). For

Co-facilitation of study groups around animated scenes

example, for a teacher to believe that a theorem has been taught in class, various actions should be carried out, some by the teacher and others by the students (see Herbst, Nachlieli and Chazan, 2011, for further reading). In this study, we considered two instructional situations in high school geometry instruction—that of engaging students in proving, abbreviated as ESP (see Nachlieli, Herbst & Gonza´lez, 2009) and that of installing theorems2, abbreviated as ITH (see Herbst & Miyakawa, 2008).

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observing how teachers interact around these animations, we can learn about what they consider to be expected, what they consider to deviate from the expected and how they value these actions. The animation shown in the two sessions analyzed and discussed in this paper focused on the base angle theorem. The animation opened with the class going over several homework problems. The teacher designated one of these problems as ‘‘the important one’’: Given : D ABC AB ¼ AC:

3 Methodology

Prove : \B ¼ \C

Two means were designed to enable our study groups to accomplish their two goals—specially designed animations of classroom interactions and co-facilitation with a moderator and a researcher.

A relevant figure of an isosceles triangle was provided, including a hint: ‘‘draw a median to BC’’. A student proved the statement on the board. Later in the lesson, the teacher asked the students to solve the following problem:

3.1 Setting

Given: AB ¼ BC

3.1.1 Using animations as representations of teaching

D; E and F are midpoints: Prove: DDEF is isosceles

To elicit the practical rationality of mathematics teaching, the researchers designed a special type of representation of teaching: animated classroom scenarios depicting stories of classroom mathematical work, created to elicit specific hypothesized norms of teaching (see Herbst & Miyakawa, 2008; Herbst, Nachlieli, & Chazan, 2011, for more information on the animations). These animations represent mathematics classroom stories that in some ways resemble geometry lessons in the USA. Additionally, they intentionally provoke what ThEMaT researchers hypothesize as norms by showing actions that depart from those we assume are likely in such situations. We use the word ‘‘norm’’ to refer to those expectations about behavior that, if carried out, would go without saying (Nachlieli, Herbst & Gonza´lez, 2009). We assume that while viewing animated scenarios that in some ways resemble stories from their own classrooms yet in other ways differ, practitioners will be able to relate to and give meaning to the performed actions and react to those discrepancies that may seem unlikely. Like any other social activity, teaching is governed by patterns that enable us to communicate and attribute meaning to human actions. Therefore, we expect that in the discussions provoked by the animations, teachers will point out specific scenes or actions and provide the means for us to learn why these were deemed remarkable. That is, we expect that by

Figure 1 was provided. While this problem could have been solved using the theorem that was proved as a homework problem (which implies that the base angles of an isosceles triangle are congruent), the students re-proved the theorem while solving the problem. When the students noted that the proof was long, the teacher stated that it could have been shorter. The animation ended by the teacher, frustrated, asking a student about the problem that was proved earlier on the board, in an attempt to help students associate the two problems and base the proof of the second problem on the result of the first. The student could not remember this result (see Herbst, Nachlieli & Chazan, 2011, to read more about the animated story). 3.1.2 Co-facilitation of the study groups Managing study groups aimed both at professional development and at research may create tensions that can hinder a group in achieving its goals. On the one hand, it is important

Fig. 1 The second problem

B

D

E

2

We use the words ‘‘installing theorems’’ to refer to the work carried out by a teacher and her class to reach the conclusion that the theorems had been taught.

A

F

C

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to create a context in which participants feel secure and at ease, so they can reveal their thoughts and choice of actions regarding their classroom experience. On the other hand, researchers may question these choices to test the limits of the expressed actions. We handled these sensitive circumstances by assigning two or three project members to facilitate each session. One facilitator acted as moderator, while the others (1 or 2 facilitators) assumed the role of researchers. The moderator, who had experience teaching geometry, was able to empathize with the participating teachers, make participants feel comfortable about sharing their opinions and maintain the discussion’s continuity. Each researcher was introduced as someone with knowledge of teaching practices, but without the background or experience of the other participants, and therefore likely to provoke discussion (Nachlieli & Herbst, 2010). The moderator and one researcher facilitated the same group throughout the year. The sessions were planned jointly by all facilitators and often by other project members as well. 3.2 Participants We recruited high school teachers with three or more years of experience teaching geometry from a diverse set of school districts in the US Midwest. The participating teachers were organized into two study groups, each of which met once a month for 2 years to discuss issues related to geometry instruction. Each session lasted 3 h. Table 1 provides general information about the study group participants. The recruited teachers were informed that the goal of the study groups was to provide an environment in which high school geometry teachers could discuss issues related to their practice, so as to enhance their professional development as well as to offer researchers insight into the practice of teaching.

3.3 Data collection All of the sessions were video-recorded by two cameras that together kept track of the body gestures of all group members and recorded the spoken words of everyone at the table. In addition, two audio recorders were used to ensure the audibility of each of the participants. The video and audio records were transcribed to include all relevant data—everything that was said during the 3-h session (excluding conversations that took place during breaks) and relevant actions carried out. The data analyzed here comprise transcriptions of the verbal actions only. This study analyzed two study group meetings, one for each instructional situation: installing theorems (ITH111605) and engaging students in proving (ESP-011006). These two sessions were chosen because they took place after the groups had held several monthly meetings and the participants had become familiar with the general agenda of the sessions, as well as with each other. Also, the same animation was presented in both of them for the first time, allowing for less variation when comparing the discourses that developed in each session. The names of the sessions comprised the initials of the relevant instructional situation followed by its date. The study groups met monthly, beginning in August 2005. The analyzed meetings took place in November 2005 and January 2006. ITH-111605 was led by a moderator and two researchers. ESP-011006 was led by a different moderator and by one of the researchers from ITH-111605. Although the differences between these groups as far as the type of instructional situation discussed are not relevant to this paper, I have chosen to maintain the names of the sessions for coherency with other papers written about the ThEMaT project. Table 2 shows the distribution of turns for the moderators and researchers analyzed in each session. The results in the table indicate that in each session, about 15% were moderator turns, whereas only about 7% were researcher turns.

Table 1 Analyzed sessions

3.4 Description of the analyzed study groups

Session name

Moderator

Researchers

Participants

ITH-111605

Mod1

Res1, Res2

Denise, Penelope, Megan, Tina, Cynthia, Edwin

ESP-011006

Mod2

Res1

Carl, Esther, James, Megan, Karen

The study groups were designed to enable participants to discuss issues of their choice relevant to their current work, as well as to consider selected classroom circumstances. These circumstances were often presented by means of representations of teaching.

Table 2 Distribution of moderator and researcher turns Session name

No. of turns

No. of Mod turns

No. of Res1 turns

ESP-011006

971

142

59

ITH-111605

943

147

74

123

No. of Res2 turns

13

% Mod turns

% Res turns

14.6

6.08

15.59

7.84

Co-facilitation of study groups around animated scenes

Each of the two analyzed study groups had a different agenda, yet the same animation was presented for the first time in both. Session ITH-111605 opened with a ‘‘show and tell’’ activity, in which participants were encouraged to present personal narratives about their geometry classrooms. They were then shown an animation, introduced as being about installing the theorem that the base angles of an isosceles triangle are congruent (see Herbst, Nachlieli & Chazan, 2011, to read more about the animated story). After the animation was shown in full, parts of it were shown again during the ensuing discussion, as requested by the participants. ESP-011006 opened by introducing the group to a new participant who had moved over from the other group. Then the group viewed an animation, the same as the one presented in the session ITH-111605. This time, however, the animation was introduced as an enactment of students working on proof problems. After discussing the animation and re-viewing different parts as requested, the participants were asked to write a continuation for the animated story. 3.5 Studying researcher and moderator discourse In all the study group meetings, we found that the cofacilitation and the use of animated representations of teaching allowed for an interesting interplay between three different classrooms throughout the discussions: (1) the participating teachers’ classrooms, comprising specific students and a teacher; (2) a general classroom, comprising a ‘‘modal’’ teacher and ‘‘modal’’ students created and referred to by practitioners according to their experience; and (3) an animated classroom, in which participants do not actually exist and who act under circumstances that may not necessarily have occurred. The interplay among these three classrooms allowed for shifts in the discussion. For example, the discussion was likely to shift from speaking about the participants’ students (‘‘I have kids who will not use the angle symbol’’) to speaking about modal students created by the speaker according to her practical experience (‘‘but kids don’t understand that that’s not how proofs are done’’). The discussion also might include statements such as ‘‘he [the animated teacher] should have written that on the board’’ when referring to hypothetical circumstances provoked by the animation. I chose to focus on identifying patterns in the discourse of each of the facilitators with respect to this interplay, as it seems central to eliciting the practical rationality of geometry teaching as well as to teachers’ professional development. The teachers were provided an opportunity to consider actions that would not normally occur in class. Circumstances of this type are liable to raise considerations that might otherwise be overlooked in everyday practice

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and therefore allow participants to view their instructional choices in a different light. By focusing on such hypothetical actions, the teachers discussed what they expected the modal teacher or the modal students to do, as well as what they themselves or their students would do in class. Furthermore, those shifts allow participants to express why certain actions are not likely to take place, or why they should take place. Such discussions may be helpful in testing the boundaries of the practice of teaching as considered by teachers. Therefore, the suggested analysis of the facilitators’ discourse focused on identifying discursive routines related to shifts between these three contexts. To this end, I adopted Sfard’s notions of discourse and discursive routines (Sfard, 2008). Discourse is used as an instance of communication made distinct by its repertoire of admissible actions and by the way these actions are paired with reactions. Discursive routines are defined as repetitive patterns that can be spotted in discursive sequences produced by interlocutors in response to certain familiar types of situations. These patterns are described using meta-discursive rules (Sfard, 2008). The set of meta-rules that defines a routine specifies both the ‘‘when’’ and the ‘‘how’’ of a discursive action and can be divided into three subsets: (1) the opening conditions, which delineate the circumstances in which the procedure is likely to be invoked; (2) the routine procedure, which defines or constrains the actions of the performer; and (3) the closure conditions, which describe the circumstances that signal the completion of the routine. The first and third sets of meta-rules are the ‘‘when’’ of the routine, and the second is the ‘‘how.’’ The ‘‘when’’ of the routines I sought refers to the times during the session when the facilitators chose to react, while the ‘‘how’’ refers to the type of their linguistic reaction. All of the facilitator turns were analyzed, turn by turn, as were the turns of other participants found to be related to the specific facilitator’s turns. The initial step in the analysis was to identify the categories that seemed to describe the facilitators’ discursive routines. The following questions regarding the ‘‘when’’ and the ‘‘how’’ of the interaction routine were then posed for each facilitator turn in the session. 1.

When did the facilitator choose to respond verbally? Specifically, what were the practitioners discussing prior to the facilitator’s response: the animated classroom? a general classroom?, their own classrooms? Who was the facilitator reacting to: the previous speaker? several previous speakers?, a specific speaker who had spoken at an earlier stage? Who was the facilitator addressing: the speaker he or she was reacting to?, the entire group? What were the practitioners discussing after the facilitator’s reaction: the

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animated classroom?, a general classroom? their own classrooms? What type of reaction did the facilitator have to the conversation? The facilitator statements and questions were categorized as: (1) revoicing, including statements that repeat, replace, summarize, elaborate and translate what participants say (O’Connor & Michaels, 1996; Forman & Ansell, 2002); (2) orienting, including statements that encourage participants to listen to and attend to the ideas of others (O’Connor & Michaels, 1996) or to a particular contribution of an outsider, thus redrawing the classroom geography; (3) offering or requesting an elaboration on something previously discussed; (4) clarifying something previously discussed; (5) offering alternatives; (6) proposing reasons for a suggested action; (7) confronting or comparing ideas, actions or situations (see also Nachlieli & Herbst, 2010). This categorization is based on Ghousseini’s (2008) review of the literature on teachers’ moves while managing classroom discourse and has been adapted for examining facilitator discourse. I added two categories specific to this analysis of facilitator discourse: (8) concreteness, referring to narrowing an idea voiced about a hypothesized or general class to the specific teachers’ class; and (9)

Investigating the ‘‘when’’ of the discursive routine (question 1) may enable us to learn about the direction of the shifts in conversation related to each of the facilitators. It may also shed light on differences in participants’ reactions to each of the facilitators. The ‘‘how’’ of the routine (question 2) may allow us to learn about the various linguistic strategies used by facilitators in general, and about those strategies that seem related to shifts in the conversation in particular. 3.6 Examples of analyzed data The following examples are presented to clarify the methodology used to analyze the transcripts. 3.6.1 Example 1: shifts in the discourse The following excerpt is taken from a discussion held in session ITH-111605, after participants viewed an animation about students solving a problem, the proof of which relied on the result of a problem proved earlier (see Herbst, Nachlieli & Chazan, 2011, for further reading). Turn 530 is analyzed here. Since the suggested analysis focuses on the opening and closing of the interaction routine, as well as on the routine procedure, the preceding and following turns were also considered.

525 Res1 So they [the animated students] did connect to the problem. 526 Tina Right, but they [the animated students] never connected to the… 527 Megan But they [the animated students] didn't just skip, they [the animated students] redid the 528 Tina Yeah they [the animated students] redid the whole five steps that she had plus the other problem. 529 Edwin Oh that's what I mean. 530 Mod1 We'll watch this in just a minute, ag ain, just, just this part. But I have a question, if this is, this is the argument they made. Okay, this is what they made and they were saying here one of the students said oh it's too long. And what if, you know, in your class you had this proof on the board and someone says it's too long and some kid says well yeah, it could be much shorter, or how could it be much shorter. What would you do? Would you try to, what would you expect your students to do? Would you let your students try and find a shorter way? 531 Denise Yeah. 532 Mod1 How 533 Tina I would ask them how to shorten it. I would tell them it could be shortened. How do you think you can shorten it?

detachment, involving adopting the voice of an observer to detach from the group. I used these questions to code the different turns in the transcripts of the two sessions. All the codes for each of the facilitators were analyzed, including the relevant participant turns.

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Words written in square brackets provide added information regarding the deictics used. Learning about the ‘‘when’’ of the interaction routine requires examining the turns prior to turn 530 that appear to have prompted the moderator’s reaction. In turns 525 through 529, the participants discussed and interpreted

Co-facilitation of study groups around animated scenes

actions performed (or not) by the animated students (‘‘they never connected’’). In turn 530, the moderator reacted to the previous speakers and addressed the entire group. She referred to the animated students and then shifted the focus of the discussion from the animated classroom to the participants’ own classrooms. The story remained the same, but the context changed. One example is the statement: ‘‘in your class you had this proof on the board and someone says it’s too long and some kid says well yeah, it could be much shorter.’’ This is a description from the animated story that the participants were discussing at the time. The moderator then brought up a hypothetical set of circumstances, ‘‘what if, you know, in your class you had this proof…’’ and placed it within a concrete setting (their classrooms). That is, the facilitator explicitly changed the context of the discussion. In turn 533, Tina projected herself as a teacher in her own class: ‘‘I would tell them it could be shortened. How do you think you can shorten it?’’ That is, she joined the moderator’s discourse about the specific class and suggested alternatives to the animated teacher’s actions in the animation. 643 Tina

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Each facilitator turn was coded. Table 3 shows the coding for turn 530. The columns ‘‘Disc. Prior’’ and ‘‘Disc. After’’ (discussion prior to and after turn 530) are coded as follows: AC, animated class; GC, general class; SC, specific class. The columns ‘‘React to’’ and ‘‘Address’’ refer to the individual that the facilitator reacted to and addressed. These columns include the codes: G, group; PS, previous speaker; SS, specific speaker. The codes in the column ‘‘Type of turn’’ refer to the categories outlined in Sect. 3.5. In this example, the type of turn was coded ‘‘8’’ (concreteness), as the moderator shifted the conversation from speaking about the general class to relating to a specific class. This column could be categorized using more than one code. The column ‘‘Shift?’’ is a yes/no column addressing whether or not there was a shift in conversation regarding the type of class discussed. 3.6.2 Example 2: the researcher’s discourse The following excerpt is taken from session ITH-111605.

Right, and then do this one and just go and say does everyone understand this one? … I would have taken the D off and put an M, and just kind of drawn the other one right from that one. …

644 Mod1 645 Tina

So you'd just help them do the connection. Yeah, just make more of a connection there. Especially since I've got a limited amount of time. 646 Res1 Do others agree with that? That this, if you would draw the new problem on top of this, this figure? 647 Tina I don't think I'd draw it on top of that one, I would draw it right next to it. 648 Res1 You’d draw it next to it 649 Tina I would make ABC the same way, and I think in the new one he draws he puts B at the top though doesn't he? 650 Megan Yeah it changes. 651 Tina He [the animated teacher] changes the order of the letters. I would have drawn the exact ABC … Half the kids wouldn't make the connection but at least this is the triangle they saw. 652 Mod1 So to have both triangles in front of them and have the formal proof. 653 Tina Yeah. 654 Res1 One thing that Edwin mentioned earlier, that I think it makes me think why one might not do what you're suggesting, is that he said students know that B is congruent to C, but they have not connected that with base angles of the isosceles triangles, something like that. And it seems to me that like I draw the new triangle with the letters in the same position and all that, they might just think oh it's B equals C, but continue not making the connection to the idea of the base angles. [Long pause] Do you see my point? 655 Tina Oh yeah. Well they [the animated students] didn't make the connection when they did the problem with the base under. 656 Megan Some [of the animated students] did. Some thought that, they were like 657 Cynthia They wanted to use it. 658 Tina But they [animated students] couldn't use it because they haven't done that problem. That was that taboo area you know? 659 Res1 So what I was thinking when you guys were saying about drawing next to it or drawing it on top was well yeah you get them to actually solve the problem but do they know why they can solve it?

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Table 3 Coding a moderator’s turn No.

Facilitator

Disc. prior

React to

Address

Disc. after

Type of turn

Shift?

530

Mod

AC

G

G

SC

8

Yes

In turn 643, Tina suggested an alternative to the animated teacher’s actions. She then spoke about what she would do under such circumstances: ‘‘I would have taken the D off and put an M.’’ That is, the location shifted from the animated classroom to her class. The moderator then turned to the previous speaker and generalized Tina’s suggested actions while offering an interpretation of those actions (‘‘you’d just help them do the connection’’). Tina’s response (turn 645) suggests that the focus was still her own classroom. She accepted the moderator’s statement and elaborated, focusing on herself as the teacher while detailing the constraints explaining her choice (‘‘a limited amount of time’’). The researcher then reacted and turned to others in the group, seeking their opinions regarding Tina’s suggestion. Their reactions can help test the limits of the teacher’s role in the discussed issue. Specifically, these reactions can test whether participants considered the suggested action to be viable or desirable. They can also help shed light on how teachers may or may not justify the actions of another teacher. To address the researcher’s question, Tina compared the animated teacher’s action to the alternative action (turns 649, 651). In turn 654, the researcher challenged Tina’s idea with a suggestion that had been made earlier in the session by another participant. I call this type of reaction ‘‘confrontation’’. The participants reacted by suggesting interpretations of the animated students’ actions. That is, the researcher’s reaction (confrontation) relates here to a shift in the discourse from speaking of specific classes to discussing the animated class. In turn 659, the researcher used another strategy, detaching himself from the group and making himself an observer: ‘‘So what I was thinking when you guys were saying.’’ In addition, he questioned the benefit of the participant’s idea. In this example, one moderator turn and four researcher turns were analyzed. Coding of the researcher’s turns is shown in Table 4. The discussion focuses on the actions of the participating teachers as alternatives to those of the animated teacher. All turns use hypothetical speech (e.g.,

‘‘I would draw,’’ turn 643) as they refer to the hypothetical circumstances presented in the animation. In this excerpt, two linguistic strategies can be identified in the researcher’s discourse: contrasting two ideas voiced by different group participants, and presenting the researcher as an observer of the group. Applying these strategies leads to shifting the focus of conversation back to the animated students’ actions and the interpretation of those actions (‘‘that was a taboo area,’’ turn 658).

4 Findings: moderator and researcher discourse characteristics This section examines the initial research questions about the discursive routines that may characterize the discourse of each facilitator: When did each of the facilitators choose to participate in the discussion? What type of reaction did they have? How (if at all) did this reaction change the focus of the ongoing discussion? In the following section, the results are analyzed to explain how these discursive segments created a relevant environment for teachers’ professional development and for learning about the practical rationality of geometry teaching. 4.1 The when of the routines of interaction The following describes what appears to prompt the routine of interaction for each of the facilitators. In general, the moderator in his role of session manager sat at the head of the table facing the computer screen, enabling him to operate a slideshow, project the animation and orient the discussion. The researcher, in contrast, sat among the participants. The moderator was found to engage in all parts of the session, and was the one who opened and adjourned the meeting as well as ran the main activities within each meeting. He responded to the participants more often than did the researcher (see Table 2 for the number of

Table 4 Example of coding a researcher’s turns No.

Facilitator

Disc. prior

React to

Address

Disc. after

Type of turn

Shift?

646

Res

SC

PS

G

SC

1, 3

No

648

Res

SC

PS

PS

SC

1

No

654

Res

SC

PS

G

AC

7

Yes

659

Res

AC

G

G

GC

7, 9

Yes

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Co-facilitation of study groups around animated scenes

turns for each facilitator in each session). The researcher seldom participated in the introductory and closing segments and rarely engaged in activities aimed at sharing teachers’ classroom experiences that were not necessarily related to study group issues (‘‘show and tell’’). About one-quarter of the moderator turns focused on session management, such as setting the day’s agenda and meeting participants’ needs (e.g., pens and paper, food). The researcher seldom participated in management issues. In identifying the contexts of conversation during which each of the facilitators chose to react and the context of their reactions (participants, general, and animated classrooms), we were able to learn about the continuity and shifts in their discourses. Both facilitators seemed to react to each context and to maintain a certain degree of continuity regarding the context of the discussion. Nevertheless, about 20% of the researcher turns in both sessions included a shift in conversation, whereas less than 13% of the moderator turns included such shifts. That is, the moderators seemed to maintain more continuity when reacting to the participating teachers than did the researchers. It is important to note that while viewing the animation, the participants were encouraged to bring up topics they wanted to discuss, and they shifted the discussion between the various contexts more than the facilitators did. The direction of the facilitators’ shifts also differed, as shown in Tables 5 and 6. In both sessions, the moderator’s turns involved shifting the discussion toward speaking about the participants’ specific classes (47% of the shifts in both sessions, as shown in Table 6). Fewer shifts were away from speaking about specific classes (12% and 33% of the turns, as shown in Table 5). Tables 5 and 6 reveal a different picture with respect to the researchers. Most of the shifts were away from specific classes (50 and 58% respectively, Table 5), while fewer shifts were toward discussing specific classes (6 and 33%, Table 6). A chi-square test shows that the patterns of interaction of the moderator and that of the researcher differ significantly in the ITH-111605 session for both types of shifts: away and toward a specific context (a = 0.05 and a = 0.016, respectively).3 The changes were not found to be significant with regard to differences between the moderator and researcher in the ESP-011006 session.

61 Table 5 Shifts in discourse away from a specific context From the general class

From the specific class

P

12% (2)

76% (13)

12% (2)

a = 0.05

31% (4)

19% (3)

50% (8)

ITH-111605 Mod (n = 17) Res (n = 15) ESP-011006 Mod (n = 15)

27% (4)

40% (6)

33% (5)

Res (n = 12)

8% (1)

33% (4)

58% (7)

NS

Table 6 Shifts in discourse toward a specific context To the animated class

To the general class

To the specific class

P

Mod (n = 17)

41% (7)

12% (2)

47% (8)

a = 0.016

Res (n = 16)

63% (10)

31% (5)

6% (1)

Mod (n = 15)

13% (2)

40% (6)

47% (7)

Res (n = 12)

33% (4)

33% (4)

33% (4)

ITH-111605

ESP-011006 NS

address, the context to which they referred and the linguistic type of the response. The facilitators had the option of addressing one or more of the participants: the previous speaker, that is, the speaker who prompted the reaction, the entire group, or a specific participant of their choosing. The findings suggest a difference between the two sessions in how the facilitators interacted (see Table 7). In ITH-111605, both the moderator and the researcher addressed the previous speaker on

Table 7 The addressees of each facilitator Group

4.2 The how of the interaction routine This section describes the type of reaction of each facilitator. The description includes who the facilitators chose to

From the animated class

Former speaker Specific speaker P

ITH-111605 Mod (n = 111) 43% (48) 56% (62)

– (1)

Res (n = 72)

5% (4)

43% (31) 52% (37)

NS

ESP-011006 3

As the samples tested are small, a Mann–Whitney test was also used to test for significant changes. The results were found to be similar.

Mod (n = 124) 43% (53) 57% (71)

– (0)

Res (n = 54)

3% (2)

58% (31) 39% (21)

NS

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62

T. Nachlieli

Table 8 The discussion context of each facilitator Animated class

General class

Specific class

P

Mod (n = 95)

66% (63)

4% (3)

29% (28)

a = 0.05

Res (n = 65)

65% (42)

12% (8)

23% (15)

Mod (n = 111)

29% (32)

20% (22)

51% (57)

Res (n = 58)

26% (15)

33% (19)

41% (24)

ITH-111605

ESP-011006 NS

most turns. In contrast, in ESP-011006, the researcher tended to address the group, whereas the moderator addressed the previous speaker. Changes between patterns of interaction of the moderator and the researcher were not found to be significant. The context to which the facilitators referred also appeared to be related to the specific study group. Whereas in ITH-111605 both facilitators seemed to focus more on the animated class, in ESP-011006 the specific class was the focus (see Table 8). While in both sessions the participants viewed the same animation for the first time, the difference in focus may be explained as follows. In the ITH-111605 group sessions, participants seemed to have more trouble following the mathematics statements in the story. Therefore, they spent more time in interpreting the story and spoke more about the animated class. In the ESP-011006 group sessions, participants had the opportunity to choose topics for discussion while reacting to the animation. Hence, such differences between the groups are to be expected. A chi-square test shows a significant difference between the pattern of interaction of the moderator and that of the researcher in session ITH-111605. Such a change was not found for session ESP-011006. The main linguistic types characterizing the facilitators’ reactions included revoicing, requesting clarification, requesting or proposing alternatives, confrontation and detachment. Differences in linguistic type were found between the two sessions. In the ITH session, the moderator’s primary type of response was revoicing, while the researcher’s was confrontation. In contrast, in the ESP session the moderator made extensive use of confrontation, whereas the researcher provided and requested alternatives to discussed actions. Confrontations took the form of opposition to participants’ ideas or actions. In turn 654 of Example 2 in the previous section, the researcher generated confrontation between ideas suggested by two participants at different times during the session. The findings suggest that when the moderator used confrontation, the

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participants usually addressed it in the context of their own experience, whereas when the researcher used confrontation, participants tended to speak in general terms in the context of a general class or of the animated story. The strategy of detachment was practiced solely by the researcher. While detachment was not used frequently, it seems to be a key characteristic in the researcher’s discourse. Detachment was used by the researcher in 5 of the 59 turns in the ESP-011006 group session (8%) and in 4 of the 87 turns (5%) in the ITH-111605 group session. In turn 659 of Example 2, the researcher used several strategies that could be interpreted as detachment: explicitly placing himself as an observer to the discussion and making a distinction between the group (you) and himself (I) (‘‘what I was thinking when you guys were saying’’). 4.3 A descriptive model of moderator and researcher co-facilitation The following model (Fig. 2) describes the findings suggested by this study with respect to co-facilitation. This model describes the shifts in conversation between discussing the animated classroom (AC), the general classroom (GC) and the specific classroom (SC) with respect to the participants in the conversation and the facilitators who seemed to provoke these shifts. This model shows that shifts in conversation toward speaking about a specific class were found mainly in moderator turns, whereas shifts away from a specific class were carried out mainly by the researcher. No differences were found with regard to the shifts between the animated class and the general class. The type of turn that appears to be related to such shifts was mainly confrontation (con). However, as noted earlier, it is important to note that this same strategy resulted in a different type of response when used by the researcher and when used by the moderator. This may suggest that the participants were engaged in different discourses when addressing each of the moderators.

⋅

AC Mod (con)

Res (con)

Participants

⋅

SC

Res (con) Mod (con)

⋅

GC

Fig. 2 A descriptive model of shifts in conversation

Co-facilitation of study groups around animated scenes

5 Discussion: how do these discursive segments create a relevant environment for teachers’ professional development and for learning about the practical rationality of geometry teaching? The study group meetings analyzed in this paper were designed to achieve two goals: promoting professional development and researching the practical rationality of geometry teaching. Specific animations were designed to achieve these two goals, and two project members were chosen to facilitate each session, a moderator and a researcher. The intention was that the co-facilitation would encourage discussions to promote both goals rather than for each facilitator to be responsible for a specific goal. Videos from all the sessions, as well as analyses of the data collected (e.g., Nachlieli, Herbst with Gonza´lez, 2009), suggest that the discussions promoted by the representations of teaching included shifts among three types of settings: the animated class, the general class and the actual classes taught by the participants. Those parts of the discussions differed in their focus. References to the animated class were mainly hypothetical, while discussions of the actual class focused on concrete actions that happened or could happen, and discussions of the general class included statements about what was expected in general. Those shifts were supported by the specially designed representations of teaching and by the co-facilitation of the sessions. In this paper, I have proposed a way to characterize the discourses of each of the facilitators with respect to their routines of interaction. Using Sfard’s (2008) notion of routines, I addressed both the ‘‘when’’ and the ‘‘how’’ of these routines by asking the following questions: When did the facilitators choose to participate in the discussions, and what type of reaction did they have to the conversation? The findings about shifts in conversation paint a different picture regarding the two sessions examined. While patterns in the discourse of the moderator and those of the researcher were found to be significantly different in the ITH-111605 session, they were not found to be significant in the ESP-011006 session. That is, although the direction of the shifts identified in the moderators’ turns in both sessions, as well as the direction of the shifts identified in the researchers’turns was found to be the same, the pattern of interaction regarding those shifts was significantly different only regarding ITH-111605. The findings suggest that the discourse of the moderator, which seemed to maintain continuity of interaction, encouraged participants to voice their thoughts and concerns. Shifts in conversation were mainly toward the participants’ own teaching experience. Such shifts can help practitioners test alternative actions in light of their actual practice and create an environment for them to become more aware of their own choices in class. The researcher,

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in contrast, seemed to draw the discussions toward the general and hypothetical, thus enabling participants to see beyond the particular and the situational. The use of linguistic choices such as confrontation obliged participants to consider boundaries that may seem viable or desirable in light of alternative ideas or interpretations to actions, thus enabling them to modify their discourse about teaching and learning. This interplay can raise practitioners’ awareness regarding gaps between what they consider possible in teaching and learning, on the one hand, and the choices available to them in their classroom on the other. Such discussions can also help them become more aware of the type of efforts that should be invested to narrow this gap. It is important to note that only two sessions were analyzed for this study and that the discourses of each of the moderators in the two sessions analyzed were not similar. This could suggest that while there may be specific ways of behaving in the session according to the assumed role, the differences could also be explained by the personal style of each of the facilitators. Eliciting elements of practical rationality involves identifying what practitioners consider to be norms and what categories they use to relate and evaluate those norms and deviations from the norms. During the study group sessions, the facilitators were attuned to hearing the stories of the participants and thus were able to help them reveal the spontaneous narratives related to their teaching, as well as general and hypothetical narratives about teaching. In the interplay between the various contexts co-played by moderator and researcher, the slight tension between the specific, on the one hand, and the general or hypothetical on the other could broaden the range of possible actions from which the participating practitioners can choose to act, as well as inform researchers about the limitations and possibilities of the practice of teaching, as it is experienced by teachers. Acknowledgments The work reported in this paper was supported by the US National Science Foundation (NSF), Grant ESI-0353285 to Patricio Herbst and Daniel Chazan. The opinions expressed are those of the author and do not necessarily represent the views of the foundation.

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