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1. What StoryCircles Can Do for Mathematics Teaching and Teacher Education
The chapter aims to outline the critical characteristics of the StoryCircles practice and illustrate
how they are connected to seminal anchors in the professional development literature, with
examples, ending by providing some considerations for the affordances we see for the practice.
• Definition: 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 environments. The process is
based on the interplay among phases of scripting, phases of visualization, and phases of
argumentation about the visualized lesson. Over time the group of teachers engaged in
StoryCircles produces a storyboard that could be seen both as an artifact that represents shared
knowledge of how a lesson might flow and as the trace of a discussion featuring bits of teaching
knowledge anchored in the work of a specific lesson. Thus, the StoryCircles is an activity in
which practitioners canlearn from each other.
• StoryCircles gather a group to produce and consume a story; the group contrasts that story with
other stories to enhance its face value, critiquing one another’s pedagogical ideas in order to
improve the story. That is, merely putting colleagues together would not necessarily create a
StoryCircle.
• Aim: 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.
• The StoryCircles practice does not guarantee change in a given direction; it creates conditions
in which incremental improvements in instruction can be generated, visualized, and compared
with existing instructional practices, so practitioners can move beyond abstract consideration
of the visions for change expressed in standards and other documents, to inspect concrete,
practical entailments of those visions, voicing their perspectives through such inspection.
• Teachers’ opportunities to learn can be located in the consideration of alternatives they had not
considered before and in the possible arguments that teachers might have with each other as
they decide which alternatives satisfy criteria (e.g., desirability, feasibility, plausibility,
probability, usability, efficiency) that the group develops for the lesson they are creating.
• The conjecture we make with StoryCircles: If we can increase the chances for colleagues to
summon what they know about mathematics instruction while they are together, their
knowledge can be shared, questioned, and expanded.
• The view of instructional improvement in the context of storycircles
In our view, for instructional improvement to succeed, it is essential that it anchor its design in
actual teaching practice to co-opt rather than combat existing capacity. Rather than treating
practitioners as lacking knowledge or skill to implement the changes that policymakers desire,
efforts at instructional improvement need to attend to and build on what teachers already know
and can do. When we allude to what teachers know we not only include their explicit knowledge
of subject matter and pedagogy, but also, and more centrally, we point to the tacit knowledge
(relational- how teachers will talk to students-, somatic- to anticipate their management of space
and proxemics-, collective- what shared expectations they should be aware of-) that teachers may
have acquired through participation in instruction. Thus, we conjecture such tacit knowledge might
be shared, challenged, bolstered, refuted, and possibly also learned.

2. StoryCircles’Anchors in the Literature
1- The professional development schools (PDS) started by the Holmes Group (1986, 1990)
It reated conditions for the collaboration of practitioners with researchers that included
research on practice, professional development, and teacher education. Building on the PDS
philosophy, it becomes crucial as a principle for facilitators not to direct the work of a
StoryCircle but to let it take shape from the interactions among participants.
2- The Japanese lesson study (LS).
It highlights the importance of a lesson as a possible motif or center for such focus on practice to
precipitate collaboration and learning by a group of practitioners. That is, these lessons are key
representations of practitioners’ knowledge (Morris & Hiebert, 2011). LS also shows how the
research literature on the topic of the lesson might inform the development of the lesson, and how
the successive trials of the lesson in various classrooms, observed by the group, provide a
mechanism for visualization through enactment that can inform improvements to the lesson. This
visualization through enactments illustrates what a natural feedback mechanism could look like
for teacher learning and has stimulated us to think about other ways teachers could visualize the
lesson they are co-developing.
3- The Professional learning communities (PLC) – Communities of practice
PLC focuses on student learning, which is a key element to helping motivate and develop teachers’
instructional practices to better meet students’ needs. Likewise, the practice of teacher study
groups has been used for professional development; it has expanded consideration of the
representations of practice that teachers could use to learn from each other (Little, 2003), for
example bringing attention to samples of student work and narratives of practice. Representations
of students’ thinking in the form of their written work can play a significant role for the
transformation teachers’ participation in study groups as well as a key for the groups’ professional
learning trajectories (Kazemi & Franke, 2004). From this approach, we retrieve the notion that the
formation of a CoP takes more than simply putting people together in professional development.
Rather the identification of a common, real problem and the development of activities to work on
that problem, using artifacts of practice such as student work may all be key factors to the
intentional development of a CoP.
4- Practical argument
Practical arguments are ways to elicit the possible premises for a plausible or actual action. They
are important research and development tools because they allow an observer’s access to the
practitioners’ relevant rationales for action and because they create foci for practitioners’ reflection
about their practice. The creation of graphical resources for representing teaching using
storyboards and animation built on that work with video, inasmuch as animations of cartoon
characters enabled researchers to represent alternatives in teaching that could be offered to teachers
for consideration and elicitation of practical argument (Herbst & Chazan, 2006; Herbst, Nachlieli,
& Chazan, 2011). From this work on practical argument and its successors, we draw the idea that
the consideration of alternatives can lead to discussion of reasons for action, and such reasons can
enable the bolstering and challenging of arguments among colleagues.

3. 5- Practice-based teacher education
In particular, the notion of approximation of practice, or that environments of reduced complexity
can be used to engage novices in practicing the work of teaching, supports not only the use of
rehearsals in teacher education (Lampert et al., 2013) but also the use of scripting and
storyboarding (see Zazkis, Liljedahl, & Sinclair, 2009). That is, storyboarding adds more learning
opportunity to that afforded by scripting (Herbst et al., 2016).
6- The engagement of teachers in action research
Corey (1953) was one of the first champions for action research in education, advocating for
empowering teachers to identify problems of their own practice rather than depending on
generalizations. Stenhouse’s argument is that teachers should be involved in the research process
to strengthen their own judgment as well as to emancipate themselves from justifying their practice
to researchers (as cited in Rudduck & Hopkins, 1985). While the philosophical underpinnings of
action research might recommend more idiosyncrasy in design and individual sense making,
teachers have also gathered in collaborative support communities of like-minded individuals who
have shared their action research projects to engage in school improvement. We rescue from the
action research approach the possibility that joint activity in StoryCircles might lead to divergent
outcomes, for example represented in the form of different lessons being created or motive for
different kinds of actions being generated.
The Design Features of StoryCircles
[Present incarnations of StoryCircles]
1- The Group of Practitioners at the Center
Figure 1 represents the group of practitioners at the center of the StoryCircles activity; this is
indeed the most fundamental element of StoryCircles. A group of practitioners is essential for
StoryCircles to contain the expectation that teachers might learn from each other. Each individual
brings their own experience and knowledge of practice to serve as background for the
visualization, and their own sensibilities and work ethics to serve as resources for scripting. Each
participant can adapt their ways of thinking and knowledge by contrasting them with those of
others in the group. It is important to consider whether group composition is a variable (makes a
difference for participants’ opportunities to learn) that needs to be studied. For example, we have

4. run StoryCircles for in-service and for preservice teachers, for in-service teachers from the same
school and for teachers from different schools. Furthermore, even if the knowledge to be transacted
will come from the participants, a facilitator is needed to keep a StoryCircle functioning as such
and not devolve into a different kind of group. The facilitator’s role includes his or her work
helping participants bring their best selves to the collaboration and maintaining the collaboration
going through and around the circle.
2- The Phase of scripting (what is involved in scripting)
In a sense, StoryCircles is very different than a lesson plan. Lesson plan formats usually ask for
sections like goals, objectives, activities, materials, assessment, and so on. But lesson plans do not
usually call for the anticipation of what will be said and done by the teacher or the students. In
contrast, StoryCircles aim at producing a script that eventually would include anticipations of what
will be said and done. The activity of scripting will require other work, including the figuring out
of what the teacher has to say and do, how and when to say and do it, and what students might do
in the context of those actions. Yet, components of a lesson plan can be useful resources in
StoryCircles. The goals, activities, or student misconceptions one might find addressed in a lesson
plan can be important resources for supporting the scripting of a lesson.
The strategic goals of a StoryCircle (targets), might be chosen either by the facilitator or by the
participants. A facilitator is needed for the specific work of scripting to manage social interactions,
specifically by encouraging participants to script rather than to comment, to work on the specifics
of actions and sayings rather than to stay for too long elaborating on intention or examining
meaning. As the participants propose what they would like to ask the students to do, the facilitator
will ask how they would pose the question; as the participants wonder whether a particular question
might elicit a student’s specific conception, the facilitator might ask what the student with such a
conception could say that would provide evidence of the conception, etc. Thus, scripting involves
proposing what to say and do, using as resources some targets, or strategic elements that usually
belong in a lesson plan.
3- The phase of visualizing
Our take on visualization consists of representing what participants script in the form of a
storyboard, with cartoon characters playing the roles of teacher and students. Participants get to
visualize the script when they peruse the storyboard. Our realization of the need for this kind of
visualization is connected to the earlier observation that much of what teachers know of practice
is tacit—while some of that tacit knowledge can be elicited through the tactical demands of
scripting, other tacit knowledge, particularly that which concerns the management of space,
multimodality, and multivocality, can use visual representations that are less transient than
enactment. When teachers and students do things in the classroom, they do so in a particular
physical place; they also do so use a combination of communication modalities, certainly speech
but also, gesture, facial expression, body movement and proprioception, inscription, and material
resources; they also orient themselves to and address some or all of the other classroom
participants. Many of those things are done without explicit decision-making; the choices of which
student to address and how to do so are made naturally.
Two important resources are provided in StoryCircles to facilitate this storyboard representation.
One is a semiotic system that adds to language other graphics (icons and indices) to support visual
representation of the classroom environment. Icons such as classroom furniture, teacher and
student characters, sheets of paper, manipulatives, calculators, a whiteboard, an overhead
projector, etc. Indices such as speech bubbles to contain speech lines by participants, caption boxes
to complement the narrative, or boxes for whiteboard content are also included.

5. The other resource is a web software tool, which allows users to manage the layout of those
graphics on a canvas and the creation of a set of related frames for the storyboard. For this, we
have used the Depict (Herbst & Chieu, 2011) software tool with StoryCircles.
While the work of storyboarding could conceivably be done by the participants themselves or by
the facilitator, we have been exploring the potential of employing an individual dedicated to
storyboarding (hereafter, the storyboarder). This storyboarder is usually an experienced user of
Depict who can fluently use the graphics and the software to represent what the participants are
saying. The presence of the storyboarder allows us to run StoryCircles with minimal delays
resulting from technoliteracy limitations, though it also constrains the extent to which alternatives
can be shared.
4- The phase of Arguing (eliciting and arguing about alternatives)
The activity of eliciting and arguing about alternatives needs to happen in some space where
practitioners can interact with each other. One of those spaces is, obviously, the face-to-face group,
and some of our StoryCircles happen in that context, taking a couple of hours in the same room,
assisted by the storyboarding technology and a computer projector, to work on scripting,
visualizing, and arguing about a lesson. But to only consider face-to-face environments for
StoryCircles can be limiting, both regarding who can participate and regarding how much
interaction can be expected about a given lesson. The Internet provides us with resources to think
about how else StoryCircles could run. Multimedia based online forums and chats could be seen
as a precursor of StoryCircles. Chieu, Herbst, and Weiss (2011) examined how practitioners and
novices interacted about an animated lesson in those online chats and forums, showing potential
for these discussions to permit noticing of important aspects of practice.
As we developed the practice of StoryCircles, we anticipated they could run in an online mode as
well as in a face-to-face mode. The online mode might be in a forum mode, for example with
participants contributing lines of scripting as forum entries and a storyboarder posting versions of
a storyboard that incorporate those contributions; this mode makes it possible to take more time to
develop lessons and to allow participants more time to think about the script they produce. But
also, the availability of sophisticated software for online videoconferencing has been useful to
think about StoryCircles to operate at distance and synchronously so as to create more of a sense
of a real conversation, similar to the face-to-face case. It has also been helpful for us to experiment
with a mix of synchronous online videoconferencing and asynchronous forum to put together the
two sorts of benefits.
How StoryCircles’ Features Work in Practice
By providing these examples we suggest what we think are best practices and what we think are
desirable outcomes. The following examples illustrate the central role played by the StoryCircles
participants in the achievement of the goals of this practice.
• 1: To Tell a Classroom Story, a Group Is Better Than an Isolated Individual
In our experience the collective activity is beneficial. The teachers’ collective efforts resulted in a
lesson that was much more attentive to diverse learners.
• 2: Using a Facilitator
Looking across the interactions, some of the more productive roles for the facilitator include
ensuring everyone’s voice is included in the conversation, maintaining a focus on the production

6. of the storyboard, and helping participants reach consensus about contested aspects of the
storyboard. Two other important and related roles of the facilitator are keeping the focus of the
StoryCircle on the production of the story and ensuring that participants reach consensus on
contested issues. Further, by effectively managing the flow of ideas, there seems to be no need to
make ad hoc efforts to bring the group back to focus on the lesson. Those moments where multiple
alternatives are under consideration by the group are critical for the facilitator to act to support the
group’s learning.
During a focus group interview about a recent StoryCircle, one participant said, “[The facilitators]
are great but sometimes they come in with their experience and we don’t have that. And we don’t
get to really hash out what we think so sometimes I feel like we’re led in a direction that might not
have been where we would have gone. We may have gotten there eventually but we just didn’t
have enough time to think about it.” Building on that comment, another participant added that he
wasn’t sure if the purpose of the StoryCircle was to capture the participants’ instructional practice
or the facilitators. That sort of confusion of purpose seems important to avoid.
• 3: Scripting
- Scripting as the Interplay of Consideration of Targets and Development of Tactical Moves
By targets we mean all sorts of strategic (goal-oriented) considerations for a lesson to be scripted;
those targets are usually included in a lesson plan (e.g., the lesson’s objectives, or the activities to
use). The targets of a lesson can help orient its scripting, but they are not enough. It is not enough
to know the lesson’s objectives to visualize the lesson, as objectives don’t determine what the
lesson will be like. The moment-to-moment, tactical moves of a teacher adapting to their students’
responses to the tasks presented creates a dynamic of their own. Yet sometimes the lesson
objectives are not met because tactical needs end up taking the lesson to other places.
Example: During the StoryCircles interaction, the participants worked hard to ensure the depicted
teacher’s responses to students’ mathematical contributions were comprehensible to the students.
The participants devoted most of their time and energy within the StoryCircle towards the
development of these tactical moves. As they did this, the script they designed drifted from their
goal to develop a lesson that focused on multiple methods.
- Choosing Good Tasks Can Support Learning Through StoryCircles
For example, for practices such as engaging students in tasks that promote reasoning and problem
solving, as sketched out in the National Council of Teachers of Mathematics (NCTM)’s (2014)
Principles to Actions? Capacity for teaching with tasks that engage students in what Stein, Grover,
and Henningsen (1996) called “doing mathematics” might be enhanced through engagement in
StoryCircles by including the use of such tasks as targets of the scripting— taking tasks from
curriculum materials designed to engage students in reasoning and problem solving. Research
suggests teachers resist the introduction of novel tasks for many reasons, including the difficulties
associated with managing various kinds of tensions associated with novel tasks (see Herbst, 2003).
StoryCircles can be a way to support teachers in practicing launching such tasks, anticipating the
variety of students’ approaches along with the difficulties that students might have, and exploring
different ways of supporting students and handling their contributions. Collective storyboarding
enables teachers to set aside what they are comfortable doing and try something more adventurous
(e.g., tasks that students could attack in different ways), enabling their colleagues to anticipate
what might follow (viz., diverse ways in which students might attack the task), and having a group
with which to discuss and explore how to address those.
This example helps to illustrate how the need to make some changes in practice may initially be
ignored by teachers but can become more palatable to teachers as they are able to visualize those

7. changes in StoryCircles. Thus, teachers can become more open to the possibility of maintaining
the cognitive demands of tasks as they discuss possible student responses with colleagues.
- Adding Records of Student Work to StoryCircles
An important goal of professional development is to increase capacity to notice and understand
students’ thinking. StoryCircles can help in that direction, particularly when supported by records
of students’ work. Records of student work can prove useful for focusing teachers’ attention on
student thinking (or intervene to promote students’ learning) and thereby supporting teachers’
learning. We have found that bringing student work that participants must use as targets in the
development of a storyboard is a useful strategy for helping ensure teachers deal with particulars
of students’ ideas that might not otherwise come up.
• 4: Visualizing (features of visualizing)
- Graphic Semiotic System for Visualizing Lessons
Discussions of practice in typical forms of professional development often privilege language-
based representations of practice and do not make use of other semiotic resources to represent tacit
or nonverbal issues. When narrating orally stories about events in their classroom, teachers often
leave out such details as where they were standing and what sort of facial expressions they or their
students used during an exchange.
The storyboarding element of script visualization in StoryCircles can help bring up these often-
neglected details to the fore. This helped us realize how the visual classroom representation-
oriented teachers to nonverbal aspects of classroom practice. While the teachers quickly agreed
that a particular student’s work would be displayed on the board, once it came time to create the
storyboard frame, the participants seemed quite surprised that there was less consensus about
whether the student herself would be allowed to accompany her work to the board or whether the
teacher would present the work on the student’s behalf. This illustrates how fleshing out details of
multimodal communication might also promote consideration of additional alternatives and
argument about their value.
- Using a Storyboarder
We involved a skilled user of the Depict software, playing the role of dedicated storyboarder. What
sort of difference does the storyboarder stand to make in such an interaction? Are there ways the
presence of a storyboarder supports or hinders participants’ work? On the one hand, when the
storyboarder’s work lags the participants in documenting alternatives, the group gets very quickly
lost in the conversation. On the other hand, the storyboarder can be too quick in documenting
participants’ ideas. When the storyboarder represents a suggestion offered as “rough draft
thinking”, we have seen participants surprised at seeing their own thinking materialize so quickly.
Similarly, we have seen participants react with statements like “Well that’s not quite what I meant,
but it’s okay.” So, it seems that the storyboarder needs to aim to be neither too fast, nor too slow,
but right on time—possibly not being too precise initially. One way for the storyboarder to handle
the expectation of precision is by pairing early depictions with clarifying questions such as “Did I
get this right?” to send the message that the storyboarder is not interested in taking over the story
but giving the participants an opportunity to adjust the storyboard.
Aside from the storyboarder’s timing and precision, we have also seen evidence that the
storyboarder’s actions can make a difference in the interaction. For one, the storyboarder can, at
times, do too much. In an occasion when the storyboarder put effort in non-substantive aesthetic
aspects of the storyboard, like lining up the desks or adjusting location of the character’s eyes, we
noted distraction in participants’ attention from discussing how the lesson could unfold. For this

8. reason, we find it beneficial if storyboarders wait until a natural break in the StoryCircle to worry
about aesthetic elements.
Taken together these examples illustrate the added benefit of using the resources associated with
the visualizing phase of StoryCircles. These include the central feature of a graphic language and
software with which to make explicit elements of instructional practice often implicit in scripts
and the ancillary feature of a storyboarder to help participants derive benefits from visualization
without being encumbered by technology.
• 5: Arguing (StoryCircles’ arguing phase)
- StoryCircles Engage Teachers in Arguments About Practice
By engaging in argumentation regarding practice, teachers have opportunities to learn a wider
variety of alternative methods for solving common practice dilemmas and such variety may be
useful for developing flexibility within their own teaching. Argumentation may also support
teachers in better understanding the reasons behind various alternatives and prompt them to reflect
about how their own instructional choices may or may not reflect particular professional
obligations.
To move forward in the story, the participants needed to hear one another’s competing ideas
regarding how to handle the same instructional moment and provide justification for their ideas.
We have observed this sort of confrontation of core elements of the lesson in other StoryCircles
involving teachers from different schools. Because these arguments take time, groups of teachers
from different schools may take longer to construct a single representation of a lesson.
- Overcoming the Limitations of Place and Time with Online StoryCircles
Our initial pilots of StoryCircles took place in face-to-face settings, with participants, a facilitator,
and a storyboarder sitting around a table with the storyboard projected on a screen. But face-to-
face StoryCircle sessions provide little time for argumentation and for iterative revisions; while
one might be able to run regular StoryCircle meetings in the context of a university class, this is
less feasible when working with practicing teachers taking professional development at the same
time as they teach, each in their own location.
We have also explored two variants with potential to overcome those limitations:
Videoconferencing and forum exchanges. Geographic location can limit who can participate in a
face-to-face StoryCircle, and the extent to which a lesson can be scripted, visualized, and argued
extensively is limited by the time available for face-to-face interaction. Online professional
development presents an opportunity to address such challenges.
In the project Embracing Mathematics, Assessment & Technology in High Schools through
LessonSketch StoryCircles we engage participants from distant locations in StoryCircles using
both video conferencing and online forums. After that meeting, participants used a multimedia
forum tocontinue contributing to and revising the storyboard (see Fig. 6), and to discuss others’
contributions. The storyboarder attended to the forum content, depicted new contributions weekly,
and uploaded revised storyboards to the forum.
Online participants report enjoying the hour-long video conferencing sessions, and those who had
previously participated in face-to-face sessions report experiencing the videoconferences as
similar, though shorter than face-to-face sessions.
• 6: Reiterating the Process (iterative nature of StoryCircles)
StoryCircles is not meant to be a linear sequence of phases of scripting, visualizing, and arguing.
Rather, the phases of StoryCircles are meant to iterate continuously, with visualization providing
grist for argumentation, and argumentation providing suggestions for new or revised scripting.

9. - Iterations Around the Circle
The work of StoryCircles involves teachers in iteratively scripting, visualizing, and arguing about
a lesson. In actual teaching, while reflection could suggest what to do differently at another time,
what one has done in a classroom cannot be modified. But in StoryCircles, if a group does not feel
good about a lesson when visualized, they can argue for alternatives, altering the script
accordingly, and visualizing the alternative. Engagement in these iterations can make room for
learning. For example, in the context of a StoryCircle with preservice teachers, one participant
suggested that the teacher posed the question “What might be mixed up?” to the class regarding a
piece of student work. Once the storyboarder added that very question to the storyboard, the
participant who had suggested the question looked at the storyboard and said, “That’s a vague
question. Yeah, that’s not a good question.” Then the participants engaged in revising the question,
which ultimately underwent two more phases of comments and revisions before the participants
were satisfied. Surely, to ask the same question to students in eight different ways is quite atypical
in actual practice. But the chance to do this revising work in the context of StoryCircles helps
participants hone their practice of composing questions in ways that can develop skill for
questioning in real practice.
- Enabling Participants to Work Independently of Storyboarders or Facilitators
There is a possibility that the group of participants may see the facilitator or the storyboarder as
ancillary, seeking more autonomy for themselves. We view this desire for autonomy positively, as
validation that scripting, visualizing, and arguing about alternatives for a lesson can be legitimate
learning activities within a community of practice, and as validation that participants can learn
from each other using this practice. However, that in the context of the longer terms project some
bids for autonomy were coupled with a desire to keep the storyboarder. We suspect this continued
appreciation of the work of the storyboarder stems from the rapid nature of the online conversation.
While participants are, indeed, able to carry out technical work like the storyboarder, participants’
comments during the online meetings indicate an appreciation for the speed with which the
storyboarder can represent their thinking. Distinct from their ability to work at the same speed, we
think the participants do not fully perceive other aspects of the storyboarder’s work as critical for
enabling the work.
About the facilitator, we have evidence that participants can and will participate in StoryCircles in
the absence of a facilitator, if they are already used to working with each other. For example, in
face-to-face StoryCircles among teachers with pre-existing work relationships (e.g., colleagues in
the same school), the facilitator’s role was reduced to initially framing of the interaction as, for the
remainder of the session, participants took over the role more typically associated with the
facilitator.
What Learning Can We Anticipate for Participants of StoryCircles?
- Students’ side, learn about the multimodality and diversity of students’ work.
Participants in StoryCircles can become more aware of the multimodality and multivocality of
classroom interaction by deliberately considering how they represent instruction. Storyboarding,
more than scripting, can reveal the fallacy of considering students as just a single counterpoint
voice to the teachers. The realizations that students communicate also with their facial expressions,
gestures, and body language and that there is more than one message possibly being transacted can
become more apparent as participants hear how their colleagues attend to their students and what
they make use of when they represent those anticipations in a storyboard.

10. While multimodality and multivocality describe characteristics of classroom communication a
teacher needs to address when teaching, they also contain the traces of larger scale cultural changes
happening as individual teachers develop their careers. In a StoryCircle, particularly one with
teachers who entered their profession at different moments in time, there is the chance for learning
about student identity and culture, by hearing how colleagues relate to students’ actions, and to
possibly generationally different ways of learning mathematics.
- Teachers’ side, learn about teaching and increase their knowledge of the mathematics
needed to teach
We expect StoryCircles participants will become aware of options available for them to do things
that are often tacit. For example, in assigning their students a task, some teachers might state it
verbally, others might distribute it in written handouts only, yet others might combine the two
approaches.
Furthermore, the literature on teachers’ mathematical knowledge for teaching has demonstrated
teachers’ experiences make a difference for knowledge they have about teaching mathematics.
StoryCircles gives teachers an opportunity to share with others how that knowledge impacts their
decision making for teaching a particular lesson. For example, before embarking on the storyboard
creation, one group of teachers was deeply embroiled in a discussion about which geometric solid
to begin with for a lesson about finding the surface area of pyramids. One set of members thought
starting with a regular triangular pyramid made sense but offered little justification. Another
member argued for starting with an oblique triangular pyramid with right triangles for two of the
three lateral faces. This person justified their suggestion on account of a typical student error that
the “height” needed to calculate the surface area is the height of the pyramid rather than the height
of the lateral face. This kind of knowledge about how to consider students’ common errors to select
tasks for students is typically attributed to teachers’ knowledge of content and students (KCS).
- Teachers’ side, increase the repertoire for doing various things.
As participants visualize scripted lessons, we expect they will increase their repertoire for doing
various things. While it may be obvious there is more than one way to do things, people can take
stock of specific alternatives by considering how their colleagues do things. Sometimes one of
those alternatives may feel more compelling; but even if they appear just as matters of taste, to be
aware of them as options is an example of incremental growth in teaching capacity that might
translate into incremental improvement in instruction (e.g., more variation in the activity structures
that a teacher uses in class).
- Teachers’ side, promote argumentation about practice and deliberateness in one’s own
practice
The consideration of alternatives promotes argumentation about practice and deliberateness in
one’s own practice. The confrontation with alternatives in StoryCircles is likely to require, at the
very least, the development of rationales for such routines. Yet we contend the StoryCircles work
may serve to question the rationale and eventually also the routine itself. Because such questioning
is done in the context of scripting a story, by colleagues who appreciate each other’s autonomy,
we also expect the questioning not to be overly intrusive or require a complete divestment from
elements of one’s rationale (as could be the case if participants were engaged in a professional
education experience with a facilitator intent on changing how they teach).
- help practitioners see the commonalities undergirding each of their professional practices,
and to embrace accountability to the same obligations
More importantly, StoryCircles may help practitioners see the commonalities undergirding each
of their professional practices, and to embrace accountability to the same obligations. In this

11. matter, it is also important that StoryCircles do not privilege the voice or the concerns of the
facilitator so much as to alienate the concerns of practitioners. For example, concerns of an
institutional nature, often minimized by mathematics educators, such as the time activities take or
the extent to which they cover the curriculum, are less likely to be ignored by practitioners engaged
in these discussions; rather they are likely to be incorporated into collective arguments that might
also seek to upgrade the quality of the mathematical work or give more agency to students.
How Could StoryCircles Be Enhanced?
- A first place is in the formation of the group.
While thus far we include only practitioners and a facilitator, assisted by a storyboarder, it is
conceivable that other types of participants might be brought in, at least virtually. The voice of
organizations like NCTM could be brought in in the form of quotes from important documents
such as Principles to Actions (NCTM, 2014); the voice of research on students’ learning could be
brought in in the form of readings from literature in which researchers speak to practitioners (e.g.,
Clements & Sarama, 2014). Nachlieli and Herbst (2010) describe study groups structured with two
extra roles, a facilitator and a provocateur: While the facilitator maintained a stance of deference
to practitioners and managed the conversation, the provocateur was free to question participants
and present alternatives that might come from the literature.
- A second place made is about the targets used for scripting.
As we noted, these targets might include particular mathematical goals or particular mathematical
tasks. It is quite reasonable that the tasks chosen might be ones that can be used to engage a class
in what Stein et al. (1996) call “doing mathematics.” We also noted that among the resources used
as targets, one might include records of students’ work; if those records included particular errors
students make, the facilitator could orient participants to the lesson to be scripted as one which
shows how the teacher could help students reason their way out of those errors.
- The visualization phase is another one in which the work could be customized to enhance
the chances that capacity for desirable practices be developed.
For example, the storyboarding environment could be seeded with some givens that steer the
storyboard in certain directions. Profiles of individual students to be found in the storyboarded
classroom could be developed in advance—describing aspects of an individual student’s
background, prior knowledge, or school history. Those profiles could be made available for
participants to use as they think of the lesson, to visualize the class and hopefully use them, thereby
bringing up for discussion issues of differentiation and equity.
- Enhancing StoryCircles Through Connections to Actual Practice
One important possibility for enhancing StoryCircles is through making an explicit connection
between the storyboard (being) developed and participants’ classroom practice. In our own work,
we have seen synergistic potential for engaging teachers in cycles of collectively scripting and
implementing lessons through StoryCircles. In the EMATHS project we have engaged teachers in
StoryCircles across several months to script lessons with the idea that the lessons would eventually
be implemented in each of their classrooms. The expectation to eventually implement the lessons
in their classroom has grounded the conversation in practice in some interesting ways. Without the
eventual implementation, it seems the temptation to view the scripting as merely a thought
experiment would have been very real. However, the coming implementation seemed to ground
this conversation in a discussion about the practical implications of such a lesson.
In any case, these kinds of moments leave us with questions about how the complexities of the
classroom can be represented within a StoryCircle to help members better understand and leverage

12. the wisdom of practice that develops across a variety of teaching contexts. In the next phases of
the EMATHS through StoryCircles project (currently in progress) we explore those questions as
teachers contribute to the revision of the storyboards by providing artifacts collected during the
implementation of the scripted lessons. Specifically, we engage teachers in this work by asking
each to take the lead on implementing a particular lesson by: (1) collecting and sharing
implementation artifacts such as video or audio recording as well as students’ work; (2) selecting
a segment of the storyboard to improve based on those artifacts. The implementation artifacts have
the potential to help members make practical arguments about the existence of critical differences
in their context that make for important differences in teaching.
In this way, the iterative cycles of scripting, visualizing, and arguing of StoryCircles can be
combined with another cycle of StoryCircle, implementation, and sharing which can be something
of a proving ground for teachers to engage in developing knowledge about teaching that is then
tested and validated across various practical contexts.
Conclusion:
In addition to the benefits that StoryCircles can provide to teachers, the outcomes of StoryCircles
can serve to create a professional knowledge base that responds to what Morris and Hiebert (2011)
advocated for: A knowledge base for teaching in the form of lessons. This brings us back to the
thus far ignored part of the title of this paper, what StoryCircles can do for mathematics teaching.
If the process of StoryCircles may be useful for practitioners to learn from each other, the artifacts
produced in a StoryCircle, the storyboards of lessons, including the arrays of alternatives they
include, are a representation of the knowledge of the profession. Furthermore, the lessons that get
scripted through the StoryCircles process can embody the knowledge base of teaching and thus
benefit the profession by creating resources that go beyond the curriculum or the lesson plan, and
actually preserve practices in the form of a story that others can use and learn from.

13. The Role of Simulations for Supporting Professional Growth: Teachers’ Engagement in Virtual
Professional Experimentation
• Aim: It examines the StoryCircles model of professional development (Herbst & Milewski,
2018), which guides teachers into a simulated type of professional experimentation to support
teacher growth through the design and improvement of lessons using storyboards. Focusing on
the experiences of two secondary mathematics teachers, we illustrate how the StoryCircles
processes of scripting and argumentation were associated with teacher growth.
• It investigates the extent to which Clarke and Hollingsworth’s Interconnected Model of
Professional Growth can be useful in investigating the learning that may be accomplished in
the context of StoryCircles as a type of simulated professional experimentation.
• The study context
Emerging digital technologies allow for the possibility of supporting teachers’ engagement with
practice-based pedagogies in online or blended settings. Herbst and colleagues (2016) explored
the affordances of digital technologies for supporting practice-based pedagogies in teacher
education, including describing how various uses of technology can help address the problems and
support the practices of teacher education. But what mechanisms might support individual growth
as teachers engage with those technologically enhanced pedagogies? In this paper, we consider
how a virtual environment—StoryCircles—can support teacher learning from practice through
professional experimentation (Clarke and Hollingsworth, 2002).
Based on Clarke and Hollingsworth (2002) model of teachers’ professional growth, particularly
the domain of practice (teachers’ attempts to try new activities within that broad domain), and
considering the potential of technologically mediated teacher education, we wonder whether and
how their model could accommodate teacher growth through professional experimentation in
simulated settings, which we exemplify in this paper. As a contribution to an investigation of this
question, we examine teachers’ interactions within a particular intervention, StoryCircles (Herbst
& Milewski, 2018), which we claim is a virtual case of professional experimentation.
• Assumption of using StoryCircles as an intervention
We posit that a StoryCircles interaction embodies some of the characteristics of professional
experimentation. First, it creates a space for teachers to try out new instructional practices. Second,
it provides an opportunity for teachers to script storyboards rather than teach actual lessons, those
opportunities are virtual. Third, It involves a group of practitioners reacting to each other’s ideas,
their virtual nature does not necessarily deprive the participants from what they would get in
classroom experimentation: participating teachers still have to risk doing things that (other teachers
might think) do not quite work and have to cope with responses and reactions from students in the
virtual classroom (whose voices are animated by colleagues who bring in their knowledge of
students to respond to experimentation moves).
• Definition of StoryCircles in this context
StoryCircles (Herbst & Milewski, 2018) is a form of online professional development that gathers
teachers (using video conferencing and asynchronous forum software) to collectively represent
how a lesson, built around a particular mathematical task, might unfold. It is considered to be a
virtual case of experimentation to further define and explore how this type of experimentation may

14. support teacher growth. As teachers script aspects of the lesson, a storyboarder works in the
background, using online software to represent the scripted lesson in the form of a storyboard. The
storyboarder shares and displays the storyboarded representation of the ideas for the teachers to
visualize the lesson and argue about alternatives.
StoryCircles as a simulation of practice
Practice-based approaches aim to support teacher learning by centering teachers’ experiences “in
the tasks, questions, and problems of practice” using records and artifacts of practice such as
student work, video records of classroom instruction, or lesson plans (Ball & Cohen, 1999). With
the increased interest in developing practice-based approaches, scholars have begun to describe a
variety of ways that such pedagogies might be used to develop common instructional activities
and curricular materials for teacher education (Ball, Sleep, Boerst, & Bass, 2009).
Some scholars have taken a different tack, investigating ways to engage teachers in practice-based
pedagogies using simulated environments (e.g., Brown, Davis, & Kulm, 2011).
We contend that StoryCircles share many of the features ascribed to other forms of simulated
practice. StoryCircles is 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 environments (Herbst & Milewski,
2018).
In StoryCircles, teachers are provided with a mathematical task or instructional goal and asked to
collectively create a storyboard representation of how such a lesson would unfold. To do this, the
teachers engage in cycles of scripting events in the lesson, sharing the kinds of actions they
envision the teacher to take as well as anticipating the ways in which students might respond.
Each StoryCircles participant scripts actions for both the teacher and students in a storyboard
where cartoon characters in classroom settings play the role of teachers and students. Teachers’
contributions are depicted in a storyboard and displayed or participants to view. As a form of
simulated professional experimentation, StoryCircles differs from other kinds of simulations of
practice because the various alternatives under consideration can be captured in storyboarded
representations of practice that are both durable (i.e., unlike discussions which are ephemeral) and
malleable (i.e., unlike video records of teaching).
A central component of the StoryCircles process is the technology mediated visualization of the
lesson which is done with the Depict (Herbst & Chieu, 2011) storyboarding software. Depict is
part of a suite of tools in the LessonSketch platform (www.lessonsketch.org) in which users drag
and drop customizable graphic elements (e.g., users can select from a suite of backgrounds for
representing K-12 classrooms, move and change the orientation of furniture, or select characters’
facial expressions) from a library onto a canvas to create representations of classroom practice.
While some of our StoryCircles have happened in face-to-face settings, with the storyboarder
displaying his or her screen with a digital projector, the StoryCircles interaction described in this
paper took place in synchronous meetings using video conferencing and screen sharing software
with follow-up synchronous discussions that happened in online forums.
In this project we supported participants by employing a storyboarder (a non-participant who had
experience using the software) to represent participants’ contributions. The storyboarder usually
asking clarifying questions of participants (e.g., where the teacher to be standing during a particular
portion of the lesson). On other occasions, the storyboarder stayed intentionally quiet, representing
only a minimum of what they heard and waiting for participants to request more details.
Participants’ contributions and the subsequent depiction of those contributions are sometimes met
with alternatives by other participants. To collectively decide on which alternatives should be

15. included in the common storyboard, the group shifts into a cycle of argumentation by offering
various forms of justifications for the given alternatives. Once the group resolves the argument,
the group moves back into scripting the next bit of the lesson or revising the segment they had just
discussed.
• The theoretical framework of teachers’ professional growth
Clarke and Hollingsworth (2002) propose a model for professional growth that frames teachers’
growth as the result of the interaction between four separate domains of teachers’ professional
worlds. The external domain encompasses factors outside the teacher, such as information
presented during a professional development event, or the curriculum adopted by the school. The
personal domain includes the knowledge, beliefs, and dispositions an individual teacher has. The
domain of practice encompasses all facets of a teacher’s professional activity, including the
instructional activities that the teacher and their students engage in daily. Lastly, the domain of
consequence includes academic, socio-emotional, and other outcomes teachers attend to during
the school day (p. 951). These separate domains are connected through the processes of enactment
and reflection. Clarke and Hollingsworth (2002) claim that each of these individual domains can
impact the other domains as teachers engage I n new kinds of enactment or reflection on teaching,
eventually enabling teachers’ professional growth.
For example, the adoption of a new textbook (i.e., change in the external domain) may influence
the content that a teacher covers (i.e., change in the practice domain through enactment) which
may help a teacher develop a new understanding of a mathematical idea (i.e., change in the
personal domain). Conversely, a teacher’s experimentation with new instructional practices (i.e.,
change in the domain of practice) may have a positive impact on students’ motivation to learn (i.e.,
change in the domain of consequences) and through reflection on these changes, teachers’
knowledge about teaching students may shift as well (i.e., change in the personal domain).
We define professional experimentation in simulated environments to be any kind of
experimentation that takes place in settings distinct from an actual classroom involving actual
students. Professional experimentation in simulated settings engages teachers in approximating
(Grossman et al., 2009) elements of practice in environments that are mediated by either
individuals playing the role of students, or virtual representations of classroom settings. We see
these simulated forms of professional experimentation as offering immersive spaces for teachers

16. to: (1) experiment with new kinds of instructional practices and (2) pause an experimentation to
reconsider decisions considering numerous alternatives. They can do that without having to
simultaneously face the risks that accompany experimentation in actual classrooms (the domain of
consequence).
• The intervention: Storycircles in relation to the theoretical framework
To make use of the Interconnected Model of Professional Growth, however, we need to first
identify where professional experimentation in a simulated setting might fit in the Interconnected
Model of Professional Growth. From our perspective, there are two reasonable places such an
activity might fit. The first is in the external domain, where such activities might act as a stimulus
for changes in teachers’ knowledge, beliefs, and dispositions or changes in the domain of
classroom practice. On the other hand, as teachers engage in scripting how the task would be
handled in the classroom, they can experiment with new instructional practice in the simulated
setting, moving it into the domain of practice. Thus, it is also reasonable to argue that the scripting,
visualizing, and arguing that constitute StoryCircles can serve as professional experimentation
(albeit in a simulated setting) just as much as classroom practice does in the Interconnected Model
of Professional Growth. For our purposes, we elect to frame simulated professional
experimentation as a specific kind of professional experimentation achieved within the domain of
practice.
• Research question
We examine the StoryCircles model of professional development as a kind of virtual professional
experimentation ushering teachers and facilitators into a simulated space to support teacher growth
through the design and improvement of lessons. In that context, we ask what kinds of evidence of
change in the personal domain we can see in instances of experimentation in StoryCircles.
• Methods
The StoryCircles interactions we describe in this article took place among four secondary inservice
geometry teachers, along with a facilitator and a storyboarder, between the months of January and
March of 2016. The data selected for the analysis comes from a series of interactions focused on
the collective construction of a storyboard depicting how a lesson focused on a Unit Circle task
from the EMATHS curriculum might unfold in a 10th grade geometry class. These interactions
took place across two synchronous video conference meetings and seven follow-up forum
discussions. Within that larger group, we elected to focus our analysis on the interactions of two
participants, Dana and Terrie, with similar contexts. Further, they both spent considerably more
time than others in the forums. The data gathered include video and audio recordings of the
synchronous sessions, forum entries, the collectively created storyboard, and summary comments
provided by the participants about the professional development and the lessons they created.
Our analysis of StoryCircles interactions began with a creation of field notes, recorded by
researchers on the project. Next, we segmented the field notes using interactional analysis,
identifying changes in participants’ focus (Lemke, 1990; Jordan & Henderson, 1995). Our
analysis of the storyboards was limited to the initial storyboard the group considered during the
synchronous meeting and the final storyboard the group completed, where we compared the two
artifacts for changes across the entire storyboard (such as additional frames being added) as well
as changes within particular frames (such as revisions to work on the board).
• Preliminary results
- Evidence about teachers’ growth by examining participants’ artifacts and interactions.
- Scripting as a means for professional growth
- Arguing as a means for professional growth

17. - Various ways that teachers might engage in simulated professional experimentation.
One of the ways that we can understand teacher growth in the context of a StoryCircle is by
considering changes in the interaction among and artifacts produced by the participants.
the intention of the task was that students would engage in an exploration of the Unit Circle without
relying on previously developed trigonometric ratios. In the two months that followed that
meeting, participants spent a total of two hours together in synchronous video conferencing
meetings and several hours connecting across a series of asynchronous forums. During that time,
the participants collectively produced 7 additional versions of the story (16 of those new versions
were submitted by Dana). The difference between the final version of the Unit Circle Lesson and
the one Dana had originally represented can also be understood by examining a handful of the
storyboard frames. That is, the final lesson is quite different than the one Dana and others had
originally supposed they would be representing where knowledge of special right triangles was
taken as prerequisite.
In this section, we examine Dana’s experiences, using the Interconnected Model of Professional
Growth to focus on how professional growth can result from engagement in the phases of scripting
from the StoryCircles model.
• Implications
One of the implications of this work is that the Interconnected Model of Professional Growth can
be useful for understanding how teachers grow professionally while engaged in simulated
professional experimentation. The characteristics of the Interconnected Model of Professional
Growth, including its non-linear, flexible structure, make it a particularly useful tool for identifying
various kinds of change sequences. It could also be useful for drawing comparisons across different
forms of simulated professional experimentation. That said, our use of the Interconnected Model
of Professional Growth was not without its challenges. Perhaps the most salient was the difficulty
we experienced in placing the activity of simulated professional experimentation in just one of the
domains. This difficulty points to some uneasiness assimilating simulated professional
experimentation within the domain of practice, perhaps suggesting that a modified version of the
Interconnected Model of Professional Growth might better account for the multiple resources that
play a role in supporting teachers’ professional growth.
Second implication of the work is the potential of the StoryCircles approach (Herbst & Milewski,
2018) for supporting growth across teachers taking different kinds of stances towards the external
domain. For some time now, there has been a growing consensus (see Richardson, 1994) that the
field needs to develop new ideas about teacher change that avoid the two extremes of a top-down
perspective on change (as something coming from the external domain) and the bottom-up
perspective on change (as something that stems naturally out of teachers’ own personal domains).
As exemplified across these two very different kinds of participants (one an enthusiastic scripter,
one a respectful dissenter), we see the StoryCircles approach as creating space for a middle ground.
StoryCircles creates a space where teachers are supported to grow professionally in two distinct
ways: by experimenting with ideas from the external domain, and by bringing those ideas under
the scrutiny of their own knowledge and experiences stemming from the personal domain.

18. Questions:
• What is the critical of the storyboarding within the visualization phase?
• What if there are no alternatives proposed by the practitioners, how the argumentation phase will be
meditated?
• How many cycles should be done to finish the process of storycircles?
• To what extent does the storycircles two modes differ from each other related to teachers’ professional
growth (the interconnected model of teachers’ growth)? If we could see the difference between both
models, we can give rationale to using the simulated form.
It is important to know as background that there have been studies that have compared face-to-face with
online professional development and found worthwhile programs delivered online can be just as effective
for teacher learning as when delivered face-to-face (Fishman et al., 2013). Similarly, forum-based
discussions have been found capable of supporting discussions regarding dilemmas of practice (McConnell,
Parker, Eberhardt, Koehler, & Lundeberg, 2013; see also Chieu et al., 2011, 2015).
For example, what are the differential effects on teachers’ engagement or on the quality of the represented
lessons
of online versus face-to-face StoryCircles? Or, what are the differential effects of visualizing by reading a
script versus visualizing by reading a storyboard? The internal study of StoryCircles will support its
design and help decide what counts as an implementation, or as Shadish, Cook, and Campbell (2002)
would put it, would
help understand the molar treatment we call StoryCircles in terms of its molecular composition, or how its
various features interact with each other, or even what needs too be observed when gauging the fidelity of
implementation. But in addition to those effects, we should also be interested in the effects StoryCircles
may produce in
instructional improvement—how does it increase the resources teachers have available when they teach
or even the qualities of their instruction?
• For our purposes, we elect to frame simulated professional experimentation as a specific kind of
professional experimentation achieved within the domain of practice?
• Within the interconnected model, knowledge means explicit of tacit knowledge as well?
Points to investigate:
- Clarke and Hollingsworth (2002) propose a model for describing teachers’ professional growth
- Tacit knowledge as described by Collins (2010): Relational (it enables participants of a practice to
relate to each other without completely explicating what they mean; it allows teachers to transact ideas
with students in ways that might seem ambiguous to others), Somatic (it is inscribed in bodily skill
acquired with practice; how individuals manage their bodies. E.g., how to look over students’ shoulders
while one circulates the classroom), and Collective (Knowing how social organization and culture
interpenetrate behavior and communication. E.g., what is acceptable to do in a given culture might not
be the same in another country- it might be illustrated with the differences in teaching scripts identified
by scholars investigating teaching across cultures.
- The professional development schools (PDS) started by the Holmes Group (1986, 1990) created
conditions for the collaboration of practitioners with researchers that included research on practice,
professional development, and teacher education.