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Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
431
NEGOTIATING THE ESSENTIAL TENSION OF TEACHER COMMUNITIES IN A
STATEWIDE MATH TEACHERS’ CIRCLE
Frederick A. Peck David Erickson Ricela Feliciano-Semidei
University of Montana University of Montana University of Montana
frederick.peck@umontana.edu david.erickson@umontana.edu Ricela.Feliciano-Semidei
@umontana.edu
Ian P. Renga Matt Roscoe Ke Wu
Western State Colorado University University of Montana University of Montana
irenga@western.edu matt.roscoe@umontana.edu ke.wu@umontana.edu
Math Teachers’ Circles (MTCs) bring math teachers and university mathematicians together to
engage in collaborative mathematical activity. Currently there are over 110 MTCs across 40 states.
A key claim is that MTCs are “communities of practice.” However, to date there has been no
research to substantiate this claim. In this paper, we explore the ways in which participants in an
MTC negotiate aspects of community formation.
Keywords: Teacher Education-Inservice/Professional Development, Teacher Beliefs
Founded in 2006 by the American Institute of Mathematics, Math Teachers’ Circles (MTCs;
www.mathteacherscircle.org) bring K-12 math teachers and research mathematicians together to
engage in collaborative mathematical activity. Unlike traditional professional development, which
tends to foreground pedagogical practice, MTCs focus on engaging participants in mathematical
activity. Notably, the model:
emphasizes developing teachers’ understanding of and ability to engage in the practice of
mathematics, particularly mathematical problem solving, in the context of significant
mathematical content. The core activity of MTCs is regular meetings focused on mathematical
exploration, led by mathematicians or co-led by mathematicians and teachers (White, Donaldson,
Hodge, & Ruff, 2013, pp. 3-4).
MTCs have expanded rapidly, and currently, there are over 110 MTCs in 40 states. As MTCs
have expanded across the country, a small amount of research has begun to explore MTCs as a form
of professional development for teachers. One significant finding is that MTCs can increase teachers’
mathematical knowledge for teaching (White et al., 2013). This is an important result, as
mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008; Hill & Ball, 2009) is
associated with effective math teaching (Hill, Rowan, & Ball, 2005). Further, surveys of MTC
participants have suggested that teachers who participate in MTCs begin to identify more strongly as
mathematicians (Fernandes, Koehler, & Reiter, 2011; White & Donaldson, 2011).
Finally, an often-stated claim is that MTCs are communities of practice that support sustained
teacher learning. For teachers, communities of practice help to support intellectual renewal and
provide a sustained venue for new learning (Grossman, Wineburg, & Woolworth, 2001). However,
there is currently no research-based evidence to support the claim that MTCs are—or develop into—
communities of practice. This is important because communities are not created by fiat, and not all
groups of teachers are communities of practice in the way that the term has been used in the
anthropological literature (e.g., Lave & Wenger, 1991; Wenger, 1998).
Given the importance of communities of practice to teacher professional development, it is
crucial to understand the ways in which MTCs are—or are not, or develop into—communities of
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
432
practice. In this paper, we explore the ways in which participants in an MTC negotiate aspects of
community formation.
Conceptual Framework
A community of practice is defined by three features: mutual engagement, joint enterprise, and
shared repertoire (Wenger, 1998). Mutual engagement refers to the requirement that participants
jointly participate in the practice(s) that binds and defines the community. Joint enterprise refers to
the purpose of the community. Shared repertoire refers to the objects that are naturalized in the
community—those objects that are so natural to members so as to be taken-for-granted, but which
may seem foreign or strange to outsiders (Bowker & Star, 1999).
In this paper, we pay particular attention to the negotiation of joint enterprise by focusing on the
essential tension in teacher communities: the tension between focusing on pedagogical practice on
the one hand, and engaging in subject-matter disciplinary practices on the other (Grossman et al.,
2001). This is an important consideration with respect to MTCs. Primarily, MTCs are meant to
engage participants in mathematical practice. The improvement of pedagogical practices is not a
“core” activity (White, et al., 2013). However, Grossman et al. (2001) contend that both foci are
essential elements in the joint enterprise of a teacher community:
We contend that these two foci of teacher learning must be “brought into relation” in any
successful attempt to create and sustain teacher intellectual community... Teacher community
must be equally concerned with student learning and with teacher learning. (p. 952)
Grossman et al. (2001) suggest that the negotiation of the essential tension will go through three
ordered stages as a group develops into a community. A “beginning” group demonstrates a lack of
agreement around whether the joint enterprise ought to be one focus or the other, and there is often
opposition tension between the two foci. An “evolving” group maintains the opposition between the
foci, but begins to demonstrate a willingness to allow different people to pursue different foci.
Finally, a “mature” community holds the two foci in productive relation, recognizing that “teacher
learning and student learning are fundamentally intertwined” (Grossman et al., 2001, p. 988).
Research Questions
On the one hand, pedagogical practice is officially backgrounded in MTCs so as to maintain a
focus on engagement in disciplinary practice. On the other hand, “for a group of teachers to emerge
as a professional community, the well-being of students must be central” (Grossman et al., 2001, p.
951). This makes us wonder, even if the stated goal of an MTC is to engage participants in
mathematical activity, what actually happens when a group of math teachers gets together to do
mathematics? Do teachers simply engage in mathematical activity? Do they focus on pedagogy? Or
some combination? Our study is the first to employ anthropological methods to answer
anthropological questions about math teachers’ circles—in particular, just what is the joint enterprise,
as it is negotiated by participants? Our research questions are:
1. In what ways, if at all, are the two foci of the essential tension—mathematical activity and
pedagogical practice— manifested in MTCs?
2. When pedagogical practice is invoked, how is it treated by participants?
Materials and Methods of Analysis
Our data come from the initial gatherings of a newly-inaugurated statewide MTC. The gatherings
include two after-school gatherings from each of five state-wide locations and a 3-day “summer
retreat.” These were the first gatherings for the statewide MTC, although two locations had
previously hosted MTC gatherings.
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
433
Local gatherings were facilitated by “lead teams” composed of 3-5 local teachers and university
mathematicians. These lead teams attended a group training session facilitated by the American
Institute of Mathematics, the organization that created and currently disseminates the MTC model.
The lead teams designed and conducted their gatherings independently of each other. The summer
retreat was organized and facilitated by the coordinators of the statewide MTC, four of whom are
also authors of this paper (Peck, Erickson, Roscoe, and Wu).
Gatherings were organized around “activities”—mathematical problems that participants worked
on groups of 3-6 people, followed by large group discussions of the problem. There was 1 activity in
each of the 10 local gatherings, and 9 activities in the summer retreat. In all, the data encompass 11
gatherings and 19 activities. Across all sites, there were 177 participants: approximately 80% were
practicing teachers (20% elementary, 30% middle school, 30% high school), approximately 10%
were post-secondary mathematics faculty, and approximately 10% were pre-service teachers (these
percentages are approximate because there are some participants for whom we do not have
demographic data).
Communities develop via engagement in joint activity. Participants interact with each other and
with artifacts, and through this interaction norms of engagement, joint practices, and a shared
repertoire emerges; a community develops and people become part of it (Bowker & Star, 1999;
Dean, 2005; Lave & Wenger, 1991). Because community development occurs in interaction, we used
video and audio recorders to capture the naturally-occurring interactions of participants as they
engaged in activity during the gatherings. For each of the 19 activities, we have video and audio
recordings of 2-6 problem-solving groups. Additional data include:
• Participants’ notebooks from the summer retreat. Participants used these notebooks for
jottings and work space during the retreat. They also used the notebooks to provide
written responses to a series of reflection prompts at the end of the retreat.
• Interviews with 10 participants from the summer retreat. This represents a selective
sample of all participants. We invited all participants to be interviewed. From the set who
agreed to be interviewed, we chose interviewees selectively in order to achieve a diverse
sample with respect to gender self-identification, level taught (elementary, middle, high),
and region of the state.
Our initial analysis focused on the recordings of MTC activities. We used a cyclical data analysis
method, which relied on both inductive and deductive approaches (Miles, Huberman, & Saldaña,
2014). First, we developed a list of deductive codes based on our conceptual framework. We then
engaged in the following process for each activity. Members of the research team each
watched/listened to a different group engaging in the same activity. The team member created a
content log (Maxwell, 2013) of the recording and coded the log according to the codebook, allowing
new codes to emerge from the data. We then met to discuss our observations and coding. We refined
our codebook and then used the refined codebook to code the next activity. We proceeded in this
fashion, with inductive codes emerging from the data and subsequently undergoing refinement, for
all 19 activities.
We used these coded content logs to identify key segments in which participants negotiated the
two foci of the essential tension (mathematical activity and pedagogical practice). We transcribed
these key segments and analyzed them using multi-modal interaction analysis (Erickson, 1992;
Goodwin & Heritage, 1990; Streeck, 2009).
We employed a similar procedure to analyze the participants’ notebooks and interviews.
Findings
We present our findings organized around our two research questions.
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
434
RQ 1: In What Ways, if at all, Are the Two Poles of the Essential Tension Manifested in
MTCs?
Perhaps unsurprisingly, we found that the majority of activity in MTCs involved engaging in
disciplinary (mathematical) practices. Pedagogical concerns occupied less than 5% of the “official”
activity. We gloss an activity as “official” if it was introduced by the facilitators as the focal activity
of the group.
Pedagogical concerns were sometimes explicitly backgrounded by facilitators. For example, in
the introduction to the one of the initial spring gatherings, a facilitator explained the goals for the
gathering:
You shouldn’t feel like there’s any expectation to be walking away this evening with anything
other than a good feeling, alright? We’re not trying to prove anything, this is just for us. We’re
not trying to say, “and now, fourth-grade math achievement will go up because-” ((laughter)).
That has nothing to do with it. You see, we’re just- what we’re trying to do is just, be a group that
likes mathematics.
In this turn, the facilitator invokes both pedagogical concerns (“fourth-grade math achievement”)
and disciplinary activity (“be a group that likes mathematics”). The turn is designed such that the two
foci are put into opposition with each other. This can be seen in the use of the adverbs “not” and
“just” to modify the verb “trying” in the second half of the turn (“not trying,” and “just trying”). In
particular, the use of the word “not” in “we’re not trying to say” negates the pedagogical focus. This
is reinforced with the exclusionary “just” in reference to mathematical activity.
This finding—that the primary activity of an MTC is mathematical, not pedagogical, activity—is
not surprising, given that engagement in mathematical activity is the explicit purpose of MTCs.
Furthermore, the finding that the two foci were treated oppositionally is also not surprising due to the
“beginning” nature of these teacher groups.
However, we also found that pedagogical concerns were invoked in multiple, interesting—and
sometimes surprising—ways during MTCs.
Pedagogy was rarely the official topic of activity. Most commonly, if pedagogy was the official
activity of an MTC gathering, it happened in the final phase of the gathering. This phase was framed
as a “reflection” time, and pedagogy was a topic for reflection. For example, at the end of the 3-day
summer retreat, the group met all together, and reflected on what it meant to do mathematics, based
on their experience in the summer retreat. After the group generated a list of attributes associated
with doing mathematics, the facilitator (Peck) referenced the list and said:
So if we think about all this stuff that doing mathematics is, um, take a moment and write just a
couple ideas about how you might incorporate some of this into your classroom.
Both the design of this turn, and the subsequent uptake by participants provide evidence that
pedagogy has become the official activity. By employing the imperative mood (take a moment and
write just a couple of ideas…), and applying it to pedagogical concerns (…about how you might
incorporate some of this into your classrooms), the facilitator signals that the official activity is now
related to pedagogy. Participants’ uptake confirms this. After the facilitator’s turn ended, participants
began writing and there is silence on the video and audio recordings. Analysis of participants’
notebooks confirms that each response involves pedagogy.
Notice also how this turn brings pedagogy into a productive relationship with mathematical
activity. Rather than treating pedagogy as separate from the mathematical activity of the retreat, the
facilitator, through the use of the word “incorporate,” suggests that the mathematical activity of the
retreat can productively be brought to bear on classroom pedagogy.
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
435
More commonly, pedagogy came up informally in relation to the concurrent mathematical
activity. As participants engaged in mathematical activity, they often related the activity to pedagogy.
For example, the strip of dialog in Table 1 occurred while participants were exploring the question,
“can any number be written using only powers of 2?” The three participants, Amy (5-6th grade
teacher), Diane (3rd grade teacher), and Patty (7-12th grade math teacher) discuss the mathematical
question in turns 1-17. In turns 18-20, they transition to a pedagogical discussion related to the
mathematical activity, which they continue for the remainder of the strip.
Table 1: Doing Mathematics and Talking Pedagogy
Turn
Speaker
Talk1
1
Amy:
I think, I mean-
2
Diane:
Well we have to be able to because how else- That’s how binary
works.
3
Amy:
How else could we-
4
Diane:
They have to-
5
Patty:
Make every number?
6
Diane:
Yeah, binary’s gonna work every time.
7
Patty:
mmm-hmm
((Amy looks at notebook, where she has written a list of powers of 2.))
8
Amy:
How do you get 127?
9
Diane:
((points to notebook)) There’s your two, 16 – ((moves finger along
notebook, where Amy has written successive powers of 2)) That’s
gonna be… one hundred… [twenty seven!
10
Amy:
[twenty seven! Okay…
((Talk continues in this fashion, for turns 11-17, with Amy suggesting an
number, and Diane showing how to make the number)).
18
Amy:
Two:::: f::::- ((smiling))
19
Diane:
You little pain in the butt! ((laughing))
20
Amy:
Hmmm… I’m trying to think like a f-
21
Diane:
Trying to think like a sixth-grader?
22
Amy:
Yes!
23
Diane:
They are difficult little critters, but they’re adorable!
24
Amy:
“well what if you want to do this? What if you want to do this?”
25
Diane:
“So tell me how. What’s the pattern you’re seeing?”
26
Patty:
So what grade do you start doing these problems?
27
Amy:
Binary?
28
Patty:
No-
29
Diane:
Exponents!
30
Amy:
[Oh exponents
31
Patty:
[Yeah, just- just exponents?
32
Amy:
Fifth- fifth grade.
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
436
33
Patty:
Well, we talk about squaring (crosstalk) not- not square roots-
34
Diane:
Just squared?
35
Patty:
Just squared. So I [sh-
36
Amy:
[I don’t even talk about square roots, until-
37
Patty:
So I show the kids that notation ((draws a “2” in the air)), and I
talk to them about how if it’s 54 square units, I show them how you
can write 54 units squared ((draws “2” in the air)) [with a-.
38
Amy:
[yeah
39
Patty:
with a- with an exponent two… so what grade do you start talking
about, “what does that exponent MEAN?” ((draws 2 in the air)) and-
40
Amy:
I do fifth and sixth grade, and in fifth grade I introduce it to
them-
41
Patty:
Okay.
1 In general, talk is transcribed using standard punctuation, so that a comma denotes a short pause, a period denotes a
longer pause after a falling intonation, and a question mark denotes a pause after a rising intonation. Ellipses ...
denote a long pause. Underline denotes vocal emphasis, co:::lons denote a drawn-out sound, and a hyphen- denotes
a restart. Vertically-aligned open brackets [denote overlapping speech. ((Double parentheses)) denote non-vocal
action.
Because participants often invoked pedagogy even when it was not the “official” topic, we found
that participants were engaged in pedagogical conversations or activity approximately 15% of the
time—3 times more than that which was accounted for in the official activity.
RQ 2: When Pedagogical Practice Is Invoked, How Is It Treated by Participants?
We found that pedagogical practice was treated as a normative topic of discussion in MTCs. The
strip in Table 1 is representative of this. Notice the framing of the turns where pedagogy is first
evoked, and the response to these turns:
20
Amy:
Hmmm… I’m trying to think like a f-
21
Diane:
Trying to think like a sixth-grader?
22
Amy:
Yes!
23
Diane:
They are difficult little critters, but they’re adorable!
24
Amy:
“well what if you want to do this? What if you want to do this?”
25
Diane:
“So tell me how. What’s the pattern you’re seeing?”
26
Patty:
So what grade do you start doing these problems?
27
Amy:
Binary?
28
Patty:
No-
29
Diane:
Exponents!
30
Amy:
[Oh exponents
In particular, notice the absence of an account for why pedagogy is being introduced. Neither
Amy nor Patty provides a rationale for why they are introducing pedagogy, and subsequent turns do
not hold them to account for such an introduction. Together the design of turns 20-30 can be taken as
evidence that for these participants, pedagogy is normative topic of discussion (consider how these
turns would be designed differently in a situation where pedagogy was not normative, say at an adult-
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
437
league softball game). This finding is somewhat surprising, considering the “official” framing of
MTCs as primarily focused on engaging in disciplinary practice.
Even though pedagogy was a normative topic, the way that it interacted with disciplinary
practices varied. In some cases, these two foci were treated oppositionally, as would be expected in a
“beginning” group like the ones that we studied. The first facilitator quote given above is one
example of this. A second example comes from the reflections of participants in the summer retreat,
one of whom wrote,
Some of the activities were good, but others were not helpful. I guess I was looking for more
options to take back to my classroom.
Using the word “but,” the participant contrasts “good” activities with those that were “not
helpful.” She goes on to identify “helpful” activities as those that could be used in the classroom.
This comes even after the participant discussed how much she had learned about doing mathematics
from the activities. This suggests that, for this participant, “engaging in disciplinary activity” and
“improving pedagogical practice” are two separate foci.
However, we also found multiple times where the two foci were held in “productive relation,”
which would be evidence of a more mature community. The strip of talk in Table 1 is one example of
this, where the content of the activity spurred a conversation about pedagogy related to that content.
We also see evidence for the “productive relation” in participants’ reflections. For example, a
different participant reflected:
Working together to solve problems reminds me how important and fun it is, and I need to do
that as much as possible in my class!
Here the participant uses a different conjunction: “and” instead of “but.” In doing so, she brings
disciplinary activity (“working together to solve problems”) into a productive relation with pedagogy
(“I need to do that… in my class”).
Conclusion and Significance
Math Teachers’ Circles have exploded in popularity since their introduction in 2006. A key claim
is that MTCs constitute communities of practice. However, this claim has not been subjected to
analytical scrutiny. In this paper, we take a step towards such scrutiny by employing anthropological
methods to analyze the ways that participants in MTCs negotiate their joint enterprise.
We found evidence that both disciplinary practice and pedagogical practice are part of the joint
enterprise. This is a surprising finding because the groups are “officially” framed around disciplinary
practice only, and are represented as such in the published literature (e.g., the excerpt from White et
al, 2013, in the introduction). This supports Grossman et al.’s contention that teacher communities
must include a focus on pedagogical practice.
We also found that, in negotiating the joint enterprise, MTC groups display hallmarks of both
“beginning groups” and “mature communities.” This complicates the claim that all MTCs are
communities of practice: our findings suggest that a beginning MTC, at least, may not be a mature
community of practice. Participants are still negotiating the essential tension, and some participants
struggle to hold the two foci in productive relation.
However, what is perhaps our most striking finding is that, most often, when pedagogy was
invoked, it was treated as normative by participants. Most of the time, when the “essential tension”
manifested itself, there was no tension at all. This finding complicates the model of Grossman et al.
(2001) in which communities must go through ordered stages of oppositional tension between
mathematical activity and pedagogical practice, before they can hold the two in productive relation.
This finding should be explored and elaborated in future research.
Inservice Teacher Education/ Professional Development
Galindo, E., & Newton, J., (Eds.). (2017). Proceedings of the 39th annual meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier
Association of Mathematics Teacher Educators.
438
Acknowledgements
The research reported in this paper was supported by a grant from the University of Montana.
References
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal
of Teacher Education, 59(5), 389–407.
Bowker, G.C., & Star, S.L. (1999). Sorting things out: Classification and its consequences. Cambridge, MA: MIT
Press.
Dean, C. (2005). An analysis of the emergence and concurrent learning of a professional teaching community.
Vanderbilt University.
Erickson, F. D. (1992). Ethnographic microanalysis of interaction. In M. D. Lecompte, W. L. Millroy, & J. Preissle
(Eds.), The handbook of qualitative research in education (pp. 201–225). San Diego, CA: Academic Press.
Fernandes, A., Koehler, J., & Reiter, H. (2011). Mathematics teachers circle around problem solving. Mathematics
Teaching in the Middle School, 17(2), 108–115.
Gee, J. P. (2011). How to do discourse analysis: A toolkit. Taylor & Francis.
Goodwin, C., & Heritage, J. (1990). Conversation analysis. Annual Review of Anthropology, 19, 283–307.
Grossman, P., Wineburg, S., & Woolworth, S. (2001). Toward a theory of teacher community. Teachers College
Record, 103(6), 942–1012. http://doi.org/10.1111/0161-4681.00140
Heritage, J., & Clayman, S. (2010). Talk in Action: Interactions, Identities, and Institutions. Chichester, UK: Wiley.
Hill, H. C., & Ball, D. L. (2009). The curious — and crucial — case of mathematical knowledge for teaching. Phi
Delta Kappan, 91(2), 68–72.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student
achievement. American Educational Research Journal, 42(2), 371–406.
Jaworski, A., & Coupand, N. (2006). Perspectives on discourse analysis. In A. Jaworski & N. Coupland (Eds.), The
Discourse Reader (2nd ed., pp. 1–37). New York: Routledge.
Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. Cambridge, UK: Cambridge
University Press.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge
University Press.
Maxwell, J. A. (2013). Qualitative research design: An interactive approach (3rd ed.). Thousand Oaks, CA: SAGE.
Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.).
Los Angeles: SAGE.
Packer, M. J. (2011). The science of qualitative research. Cambridge, UK: Cambridge University Press.
Streeck, J. (2009). Gesturecraft: The manu-facture of meaning. Amsterdam: John Benjamins.
Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge
University Press.
White, D., & Donaldson, B. (2011). How do Math Teachers’ Circles affect teachers? Themes from teacher surveys.
Joint Mathematical Meetings, New Orleans, LA.
White, D., Donaldson, B., Hodge, A., & Ruff, A. (2013). Examining the effects of math teachers’ circles on aspects
of teachers’ mathematical knowledge for teaching. International Journal for Mathematics Teaching and
Learning. Retrieved from http://www.cimt.org.uk/journal/white.pdf
