Conference PaperPDF Available

"It's a Different Mindset Here": Facilitation Challenges in a Practice-Based Professional Development

Authors:
“It’s a Different Mindset Here”: Facilitation Challenges in a Practice-Based Professional
Development
Gil Schwarts
University of Michigan
Irma E. Stevens
University of Michigan
Patricio Herbst
University of Michigan
Amanda Brown
University of Michigan
Schwarts, G., Stevens, I. E., Herbst, P., & Brown, A. (2022). “It’s a different mindset here”:
Facilitation challenges in a practice-based professional development. Proceedings of the
forty-fourth annual meeting of the North American Chapter of the International Group
for the Psychology of Mathematics Education. Nashville, TN. (pp. 14791487)
Available at: http://www.pmena.org/proceedings/
“IT'S A DIFFERENT MINDSET HERE”: FACILITATION CHALLENGES IN A
PRACTICE-BASED PROFESSIONAL DEVELOPMENT
Gil Schwarts
University of Michigan
gils@umich.edu
Irma E. Stevens
University of Michigan
istevens@umich.edu
Patricio Herbst
University of Michigan
pgherbst@umich.edu
Amanda Brown
University of Michigan
amilewsk@umich.edu
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Keywords: Problem-Based Learning; Professional Development; Teacher Educators.
Background and Theoretical Framing
In recent years, research on the facilitation of professional development (PD) programs has
received growing attention due to facilitators’ key role in the implementation of educational
initiatives (Rösken-Winter et al., 2021). Within this emerging field, scholars have discussed the
preparation of facilitators (Lesseig et al., 2017), their knowledge and practices (Borko et al.,
2014; Karsenty et al., 2021), and facilitators’ professionalization processes (Schwarts et al.,
2021). When referring to novice facilitators who are also practicing mathematics teachers, the
main difficulties described in the literature are related to their limited capacity to lead in-depth
discussions with teachers (Borko et al., 2014) and to the complex navigation between their
multiple identities as facilitators, teachers, and colleagues (Knapp, 2017). PD facilitators find it
difficult to gauge the extent of their involvement in managing participant discussions (Lewis,
2016) and they struggle to enact their ambitious goals when designing and implementing
activities (Jackson et al., 2015). Overall, these results support the argument that “being a good
teacher does not necessarily imply the ability to help others develop their teaching” (Even, 2005,
p. 334). In spite of this work, there is still much to learn regarding the underlying factors that
constrain facilitators working in the context of practice-based PD. In particular, it is important to
understand in what ways complexities of the activity of facilitation rather than deficits of the
facilitators themselves, might account for difficulties observed.
This paper seeks to frame the challenges of facilitation as rooted in facilitators’ implicit
expectations about learning in practice-based PD settings. For that purpose, we refer to
facilitators’ challenges in the context of the PD triangle offered by Carroll and Mumme (2007,
see Figure 1). This triangle embeds the well-known instructional triangle (Cohen et al., 2003),
including its claims about the interconnections between the different components of instruction,
into the facilitators’ level. The PD triangle includes the three vertices: the facilitator, the teachers
(hereafter, referred to as participants or practitioners), and the practice of teaching and learning
mathematics in the place of the content (see Figure 1). As in the case of the instructional triangle,
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1479
we can hypothesize this PD triangle to be situated in environments, particularly the institutional
environments that enable practitioners to participate. This framing allows us to situate
facilitators’ expectations with respect to the other components of the PD environment. The
expectations we are interested in refer to how and what participants are supposed to learn
(participants-practice arrow), and the facilitator’s role with respect to this learning (how the
facilitator relates to practice, or the facilitator-practice arrow, and how the facilitator relates to
the participants, or the facilitator-participants arrow). We hypothesize that facilitators’ years-long
experiences as practicing mathematics teachers who have also participated in PD inform these
expectations and influence facilitators’ practices in ways that are not always discernible for them
and for developers of PD environments.
Figure 1. The PD triangle (adapted from Carroll & Mumme, 2007, p. 11)
This paper focuses on facilitation in a practice-based PD, where practitioners “learn in and
from practice” (Cohen & Ball, 1999, p.18) by collaboratively inquiring on artifacts of practice,
analyzing them and arguing about them. In such settings (i.e., ones that center on practitioners’
reflections and where there is no specific content to be taught), the facilitator’s role as a
moderator of practitioners’ discussions is even more complex (Schwarts et al., 2021). In this
context we ask, how do facilitators’ expectations based on their prior experiences shape their
work facilitating a practice-based PD?
Context: The Storyprocess
The context of this study is Story (Herbst & Milewski, 2018)a process of teacher
collaborationthat aims to engage practitioners in collective scripting, visualizing of, and arguing
about a problem-based lesson. Storyhas evolved over its various iterations, but it has
consistently maintained the goal of having practitioners represent their practice through
storyboarding a collective lesson (see Brown et al., 2021; Milewski et al., 2018, 2020 for
examples). Inspired by Japanese lesson study (see Herbst & Milewski, 2018), the main design
concepts of Storyinclude opportunities for rapid prototyping of lessons and a user-
centered design (Herbst & Milewski, 2020). Thus, the discussions that are at the core of
Story position practitioners as experts and support them in talking with one another about
their rationale for making certain, sometimes competing, decisions in the classroom.
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1480
Figure 2. Scenes at the beginning and the end of the Tangent Circle lesson
(© 2021, The Regents of the University of Michigan, used with permission)
In the iteration of Storyconsidered in this report, thattook place between February
and April 2021,secondary geometry teachers engaged over six weeks in discussions focused on
a problem-based lesson aimed at introducing the tangent segments theorem. This lesson starts
with the posing of a problem (see Figure 2, left) and ends with the statement of the instructional
goal of the lesson (Figure 2, right). The lesson is illustrated in a storyboard using cartoon
characters to represent a teacher and their students. Given the problem and the instructional goal
of the lesson and one potential instantiation of the lesson provided to them in advance,
participants started this Story at the visualization phase, annotating what the provided
version of the lesson looked and felt like to them and inserting comments where they thought the
teacher might have done something differently. During six weeks they prototyped alternatives to
those moments including alternative moves to manage whole class discussions and review
students’ work. Participants were expected to collectively argue about alternative ways in which
the teacher could have handled events in the lesson, scripting the alternative scenes, which would
also be storyboarded and visualized. As they scripted potential instantiations of how the lesson
might unfold, it was expected they would engage in argument about decisions the teacher would
make in the lesson. Connecting it to the role that experimentation plays in Clarke and
Hollingsworth’s (2002) model of teachers’ professional growth, Milewski et al. (2020) have
described Story as “virtual professional experimentation” (p. 624). During Story,
practitioners worked on the different segments of the lesson in a combination of synchronous and
asynchronous activities. These different forms of interaction permit practitioners to dedicate time
to tinker with particular moments in a lesson and consider alternative ways of handling various
contingencies that might arise in an instance of the lesson.
The avowed goal of each cycle of Story is to produce a representation of a
collaboratively developed lesson. Although the particulars of the entire design of Story
are beyond the scope of this paper, we describe three different design features unique to the
present iteration of the Storydesign. First, participants were not expected to attend every
synchronous meeting – each participant was scheduled to attend only three or four out of six
such meetings, so every synchronous meeting included a subset of participants (and not always
the same people in each subset). Second, the meetings were not intended to build on one another;
instead, each of them focused on a specific segment of the lesson. Third, by design, the facilitator
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1481
is not responsible to use the software to prototype the lesson scenes the participants propose;
rather two assistants (undergraduates skilled in the storyboarding software) are present in every
synchronous meeting. These storyboarders represented suggestions participants made in real
time during discussions. The resulting storyboards served as collective artifacts the participants
could use to visualize what the lesson looked and felt like so as to argue about decisions the
virtual teacher had made later on and possibly occasioning new scripting or revisions.
Method
This study is part of a five-year research project focused on teachers’ learning about
discussion-based teaching of Algebra and Geometry lessons. The facilitators for the 2021
iteration of Storywere practicing secondary school teachers in their content area. For
Geometry, the facilitator was Quincy, an experienced teacher who had been a participant in a
previous iteration of Story, and this was her first time facilitating the workFourteen
participants took part in this cycle.
Data collection and analysis
To identify how the facilitator’s prior experiences in teaching and professional development
framed her management of the Story process, we searched in our records for evidence
that her expectations were not being met. This approach builds on the idea that individuals’ tacit
expectations can be revealed through their reactions to deviations from customary practices
(Herbst et al., 2011). The data corpus for the 6-week cycle includes, among other things,
recordings and transcriptions of six synchronous meetings and follow-up debriefs (6*90
minutes). Using thematic analysis (Braun & Clarke, 2006), records were inspected for evidence
that the facilitator’s expectations were not fulfilled. This evidence included moments during
facilitation or in the follow-up debriefs in which the facilitator expressed (using statements or
gestures) surprise, confusion, conflict, discomfort, or puzzlement. We also searched for the
sources of these expressions as could be inferred from the facilitator’s expression (e.g., when she
said “It’s a different mindset here, having help”, we inferred that her expectation to work
individually, rather than have storyboarders representing the lesson that participants script, is
related to her classroom experience in which she does not have assistants). After the
identification of these moments, they were organized according to themes and were mapped to
highlight the interrelations between the facilitator’s expectations, the experiences that were likely
to shape those expectations, and the design principles of Story.
Findings
The following are two prominent themes from the analysis that are representative of the
facilitator’s expectations.
Theme 1: The traditional turn taking that animates the PD triangle
The first theme illustrates the facilitator’s difficulty in adjusting herself to work in
collaboration with the storyboarders and to manage time. To illustrate this difficulty, we provide
some more details on the work of storyboarders during PD discussions. Story
synchronous meetings usually open with a question that the facilitator poses about a scene in the
lesson at hand (for example, “Are there things that the teacher might say in their introduction of
that student work?”, Turn 242, second meeting). While participants raise alternatives, the
storyboarders create images that represent the participants’ suggestions and incorporate them
into the representation of the lesson (see example in Figure 3). These images are an essential
component of the argumentative process in Story since they provide a shared
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1482
representation on which participants can argue. However, after Quincy’s facilitation of the first
meeting, she reflected that the use of these new images was in fact overwhelming for her:
So I guess the part that I felt most awkward about was like transitioning between that, and
then like jumping down to the depictions [...]. Partly I was worried like I… did I give them
[the storyboarders] enough time to depict before jumping down? so I was stalling on that a
little bit (Turns 546-547, first meeting debrief).
This quote reveals the facilitator's difficulties in managing a discussion while simultaneously
considering the materials the storyboarders were creating and the time they needed to create
them. In addition, she mentions that moving between slides and referring to ad-hoc ideas was
“awkward”. She expressed similar feelings two weeks later, after the third meeting:
Not sure how to like really utilize what you guys [talking to the storyboarders] are doing on
the depicting, like I can see you're like going crazy, I don't know if I need to like just dive
into the depiction earlier (Turn 601, third meeting debrief).
Figure 3. Image created by the storyboarders during the third synchronous meeting
(© 2021, The Regents of the University of Michigan, used with permission)
Beyond knowing whento use the images, another source of tension for Quincy was howto
use them: In this meeting, the storyboarders represented almost every comment made by
participants. Doing so resulted in many new images, illustrating different pathways in the lesson,
that Quincy felt obliged to “utilize”. All of the above evidence points to Quincy's expectations of
having control, both on the timing and on the mediation and interpretation of participants’
contributions. This claim is supported by another comment made in the debrief of the third
meeting, where Quincy compared facilitation to teaching:
When I'm teaching off of the Google slides right now, like every day, I’m in control of them
or I’m having the students interact with them, so it's very [...] It's a different mindset here,
having help (Turn 681, third meeting debrief).
These moments suggest that Quincy’s expectation of having full control is informed by her
experiences as a mathematics teacher, used to working individually (“It's a different mindset
here, having help”, Turn 681). That is, “having help” is something she is unaccustomed to, and
perhaps, at that moment, she felt that the work of the storyboarders did not help her at all. This
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1483
challenge points to a disruption in the traditional turn taking between teachers and students (e.g.,
McHoul, 1978), or between facilitators and participants. In the latter case, the facilitator is
commonly the only one who is responsible for the interpretations of the participants’
contributions and their mediations with the PD content. Even if participants raise unexpected
ideas, the facilitator knows (even if only tacitly) that it is her job to address them. However, in
Story, the participants’ ideas are also interpreted by the storyboarders. The expectation
that her facilitation role is similar to that of a teacher seemed to activate in Quincy the
expectation that her job included managing more interactions—namely ensuring that: (1)
participants’ ideas are explicit enough for the storyboarders to depict, (2) the storyboarders have
accurately captured participants’ ideas. In addition, as described above, she appeared to take
responsibility for pacing the discussion according to the time depicting took, even though
storyboarders were not part of the conversation. It follows that even for practitioners who are
familiar with managing discussions and allocating turns of talk, facilitating a Story
discussion is a challenging task as it may involve a decreased sense of control. We posit that
such disruption, although challenging for facilitators, is an important feature of the learning
environment offered in Story: It requires participants to explicate their arguments, specify
them, and explain how they relate to previous comments. That is, because the participants are
able to see their own and others’ contributions represented visually in real time, the participants
are in an environment in which they have greater opportunities, and perhaps feel more
accountable, to engage with and reflect on one another’s comments. This, along with the
facilitator’s prompting, recruit participants’ attention to others’ arguments, and sharpens the
ways they communicate and argue about their practice. For the facilitator, however, it adds
layers of complexity.
Theme 2: The agreed-upon goal of the meetings
The second theme alludes to the ways the facilitator envisioned the goals of Story
which were sometimes in conflict with the program design and the participants’ goalsThis
tension was manifested in Quincy’s goal to improve the storyboarded lesson in a certain way,
including her expectation to build on previous meetings while doing so. As mentioned above, a
main design feature of Storyis that only subsets of the enrolled participants engage in
each synchronous meeting. Accordingly, all but one of the participants who attended the second
synchronousmeeting had not been present in the previous meeting. Quincy planned to work in
this meeting on part of the lesson where the students were stuck and the teacher redirected them
by discussing with them pre-selected students’ work, as had been decided by the group who
participated in the first meeting. However, when she asked the participants which pieces among
the pre-selected student work they would like to present, they did not answer her question, but
instead wanted to work on improving the beginning of the lesson. The facilitator, while surprised
by this initiative, followed the participants’ lead. Nonetheless, in the end of the meeting, she told
the participants:
I sort of… I mentally… I guess [I] was expecting to pick up the conversation where we left
off with a totally different group of people, and maybe that wasn't the most realistic
expectation. You guys brought different ideas to tonight, and so we had a little different
conversation (Turns 540-541, second meeting).
The facilitator’s expectation was to continue visiting the lesson chronologically picking up
where it had been left by participants the prior week, however, the participants in this second
week might not be ready to deliver to that expectation inasmuch as they had not necessarily
reviewed what the prior week’s group had done. Quincy followed the participants’ ideas,
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1484
although her expectation had been that the participants would go along with her goal, and
contribute to the group’s common attempt to improve the lesson. When reflecting on this
meeting, she noted that, “I just didn't expect it with adults, for some reason” (second meeting
debrief, Turn 703), “it” refers to the derailing from the original plan. This expectation is also
evident in the following reflection, taken from the debrief of the fourth meeting:
Project leader: So you started saying that you were worried at the beginning, what do you
mean? (Turn 553, fourth meeting debrief).
Quincy: Well, when they were like “we don't do proofs”.[...] Oh well, that's the
premise of this lesson. If we're not gonna play that game I don't know
where we're going. But they came around, I think it was nice that Quintin
[one of the participants] could help bring everybody in (Turns 554-557,
fourth meeting debrief).
This statement indicates that Quincy was disrupted by the participants’ divergence from the
path she envisioned (i.e., discussing proving), which suggests that she had in mind the
expectation that their scripting of the lesson during the meeting should contribute to the direction
that had been set in the prior meeting (“If we're not gonna play that game I don't know where
we're going”). This highlights a tension present in Story and related to the role of the
lesson as a motif for the work (representing the lesson is the avowed goal of the activity) but not
the outcome of activity (learning with and from colleagues is the outcome of the activity). Along
those lines, the lesson serves as a resource that enables participants to learn how to work together
and communicate on their practice, under thepremise that the lesson itself can evolve in multiple
ways, each having its own merits, and participants can ponder on their decisions without being
encumbered by the expectation to enact best practices. The facilitator’s uneasiness with the
takeover by the participants showed that she expected to be able to maintain alignment between
the avowed goal and the expected outcome, as a teacher usually does. Interpreting this tension
with the PD triangle, the facilitator seemed to take for granted the equivalence between avowed
goal (which refers to content) and expected outcome (which refers to having conversations and
arguments about practice among colleagues) and found that equivalence disrupted by the
participants’ desire to construct yet an alternative lesson. We hypothesize that her expectations
may stem from her experience as a teacher who is used to having students agree with her on the
purposes of lessons.
Discussion
Above we described two implicit expectations a facilitator had when leading Story for
the first time. These results corroborate previous findings about facilitators’ difficulties to lead
PD activities (Borko et al., 2014; Jackson et al., 2015; Jacobs et al., 2017). The method used in
this study shows that such challenges can stem from the expectations facilitators bring with them
from their prior experiences as mathematics teachers in the classroom and as participants in other
PD programs. Although some of the disruptions described above (such as the presence of
storyboarders) represent idiosyncratic characteristics of Story, situating them in the PD
triangle (Carroll & Mumme, 2007) allows us to generalize into broader themes. That is, the first
theme illustrated a disruption in the communication among participants and facilitator about
practice, while the second showed disruptions in the facilitator-practice edge of the triangle. We
highlight that both disruptions were features of the original design, aiming at defamiliarizing
practice in a way that would encourage practitioners to collaborate (Herbst & Milewski, 2020).
StoryCircles is not purposefully designed to disrupt facilitators, yet the observation that
Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (2022). Proceedings of the forty-fourth annual meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education. Middle Tennessee
State University.
1485
disruptions happened offers insights into how the innovative nature of practice-based PD
programs is complicated and cannot be simplified to a train-the-trainer model. The analysis
above leads us to suggest that the implicit expectations the facilitator held, which surfaced only
when she facilitated Story for the first time, point to implicit norms that shape facilitating
and participating in PD settings. These norms are related to the instructional norms that are
obtained in mathematics classrooms. The analysis suggests that teachers-become-facilitators
carry with them the norms of the instructional triangle as they strive to make sense of the activity
of running professional development. The facilitator’s implicit expectations seem to align to the
expectations instruction imposes on classroom teachers and that have been described using the
theory of the didactical contract (Brousseau, 1997; Herbst, 2003).
This framing can help situate and explain the challenges of novice facilitators, by revealing
the complexity involved in the work of facilitation. In the same way that improvement or change
in teaching is bounded by regularities that are difficult to depart from (Herbst, 2003), we showed
how facilitators’ practices are constrained by implicit expectations that exist even if they are at
odds with the explicit design of the intervention. These results can contribute to the current
discussions on issues of fidelity, integrity, scaling up, and implementation (e.g., Jacobs et al.,
2017; Karsenty, 2021): Rather than assuming that the PD design principles and goals are
transparent for facilitators, that all they need is training and good will to implement PD programs
with fidelity, this study shows that considerations of implementation require attending to the
background expectations teachers-become-facilitators bring with them to the job. The same
capacities and experiences that give them street credibility to lead practice-based professional
development can hamper their capacity to manage practitioners’ learning of practice.
Acknowledgment
This work has been done with the support of grant 220020524 from the James S. McDonnell
Foundation. All opinions are those of the authors and do not necessarily represent the views of
the Foundation.
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... In this paper, we investigate an on-demand, asynchronous, adidactic simulation model, which is an adaptation of the facilitated practice-based professional learning model StoryCircles where teachers collectively design a problem-based lesson by creating a storyboard. In StoryCircles, peerinteraction about mathematics instruction is an essential resource for participants' learning Schwarts et al., 2022Schwarts et al., , 2023. Therefore, this adaptation has led us to explore the following overarching question, guiding our research project: In the absence of peers, MTE, or didactic feedback, what mechanisms can practice-based digital simulations rely on to promote mathematics teachers' learning in and from practice? ...
... These goals are achieved through carefully designed structures that support collaboration, coupled with skillful facilitation. The role of the MTE or facilitator in such contexts is to create a supportive and inclusive atmosphere and lead discussions that amplify teachers' voices and promote reflection (Schwarts et al., 2021(Schwarts et al., , 2022. ...
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This chapter conceptualizes and illustrates StoryCircles, a form of professional education that builds on the knowledge of practitioners and engages them in collective, iterative scripting, visualization of, and argumentation about mathematics lessons using multimedia. The drive to invent and study new forms of professional education for mathematics teachers, such as StoryCircles, is predicated on the need to improve mathematics instruction. While many such efforts aim to support teachers to make broad sweeping changes, few take into account the actual predicaments of practice that make such changes difficult. StoryCircles aims to support teachers in making incremental improvements to practice by eliciting teachers’ practical wisdom and enabling participants to use each other’s knowledge and experience as resources for professional learning. In this chapter we outline critical characteristics of the StoryCircles interaction and illustrate how they are connected to seminal anchors in the professional development literature. We also illustrate those features with examples from various instantiations of StoryCircles. We close by providing some considerations for the affordances we see for the model both for the profession and for individual groups of teachers.
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Many believe teacher leaders can play a central role in meeting the needs of students in schools, but it is presumptuous to think that teachers intuitively know how to lead their colleagues or schools without any focused support. This paper describes how one middle school mathematics teacher learned to enact leadership through an informal role as a teacher leader. The results of this autoethnographic study also reveal how the teacher’s view of leadership changed during the year of the study. As that leadership identity transitioned, the researcher found that adopting a lead-by-example and lead-learner stance supported her work with colleagues. A presentation of findings related factors that supported and hindered the transition from mathematics teacher to that of teacher leader are shared. Factors that supported teacher leadership included maintaining a disposition of continuous learning, developing a community of practice with colleagues, and developing a systems view of leading. The experiences and factors that hindered the process of becoming a teacher leader included confusion about one’s leadership role, navigating the middle ground between colleague and leader, and the lack of communication with administration. The findings in this paper suggest several courses of action for supporting emergent teacher leaders.
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This article presents research on how teacher developers in the United States learn to conduct lesson study. Although the practice of lesson study is expanding rapidly in the US, high-quality implementation requires skilled facilitation. In contexts such as the United States where this form of professional development is relatively novel, few teachers have participated in lesson study, so leaders of lesson study groups do not have that prior experience to draw upon for facilitation. To establish lesson study groups, teacher developers are therefore needed in the US context, but we know little about how leaders who are new to lesson study learn to do this work. To investigate this, two novice teacher developers were followed for a period of eighteen months, from their first exposure to the literature on lesson study, through their participation in lesson study conferences, apprenticeship with an experienced lesson study leader, and into their independent conduct of lesson study groups. Data show that the facilitators learned to contend with such issues as teacher resistance, the use of time, and the shifting imperatives of directing teachers’ work versus stepping back to give teachers autonomy in determining their collective work. The article concludes by suggesting that lesson study functions as a countercultural bulwark in the field of teacher learning by promoting a participant-driven, time-intensive form of professional development, and that, despite its novelty and complexity, teacher developers with strong mathematical and pedagogical backgrounds become reasonably skillful facilitators in a surprisingly short span of time.
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Novel (as opposed to familiar) tasks can be contexts for students’ development of new knowledge. But managing such development is a complex activity for a teacher. The actions that a teacher took in managing the development of the mathematical concept of area in the context of a task comparing cardstock triangles are examined. The observation is made that some of the teacher’s actions shaped the mathematics at play in ways that seemed to counter the goals of the task. This article seeks to explain a possible rationality behind those contradictory actions. The hypothesis is presented that in managing task completion and knowledge development, a teacher has to cope with three subject-specific tensions related to direction of activity, representation of mathematical objects, and elicitation of students’ conceptual actions.