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The Durability and Invisibility of Practice Fields: Insights from
Math Teachers Doing Math
Frederick A. Peck
a
, Ian Parker Renga
b
,KeWu
a
, and David Erickson
c
a
Department of Mathematical Sciences, University of Montana, Missoula, MT, USA;
b
Education Department,
Western Colorado University, Gunnison, CO, USA;
c
Department of Teaching and Learning, University of
Montana, Missoula, MT, USA
ABSTRACT
In this paper, we revisit a long-running conversation about situated learning and the design of
environments for disciplinary engagement. Throughout the 1970s and 1980s, scholars advanced an
anthropological critique of the then-dominant acquisitionist paradigm of formal schooling with a
situated view focused on membership in communities and participation in practices. The critique
led to a practice turn in education and a consensus model for reform-oriented school classrooms
as orchestrated practice fields where students engage in disciplinary practices within a structured
environment. Questions remain, however, about the nature of the practices and communities that
this model engenders. We join this conversation through an anthropological investigation of a self-
organized group of teachers who gather outside of school hours to engage in collaborative math-
ematical activity. Participants have the flexibility to conduct their mathematical activity however
they want; yet as we show, they tend to reproduce a practice field resembling a reform-oriented
school mathematics classroom. This may seem unremarkable, even desirable for many reformers.
However, assuming that teachers can or should only replicate practice fields when doing mathem-
atics may be selling them short. Our findings suggest a durability and invisibility to practice fields
that may be limiting the possibilities for the production of novel learning communities
within schools.
Through the 1970s and 1980s, Jean Lave and others advanced an anthropological critique of the
then-dominant acquisitionist paradigm of learning in which knowledge is understood as a decon-
textualized commodity that can be possessed by individuals in one place and time, transported to
a new place and time, and then deployed there. Within this paradigm, schools serve the world
outside their walls by ensuring students receive relevant knowledge for future use. Through stud-
ies of learning and thinking in the wild (e.g. navy ships, Hutchins, 1995; tailor’s shops, Lave,
1977; and dairies, Scribner, 1985), anthropologists advanced a radically different perspective on
knowledge. Instead of an acquired commodity, they argued, knowledge was interwoven with
knowing—a situated activity always bound to the material and social world in which it emerges.
This upended the core logic of schooling, because, as Lave (1996) explained, “if there is no other
kind of activity except situated activity, then there is no kind of learning that can be distinguished
theoretically by its ‘de-contextualization,’as rhetoric pertaining to schooling and school practices
so often insists”(p. 155).
A situated perspective, Lave and Wenger (1991) demonstrated, renders visible the participa-
tory, inherently social process of enculturation and becoming while entering into community.
Their reframing of schools as communities with particular norms, practices, and roles revealed
how school participation is another form of community enculturation. Within classrooms, it was
CONTACT Frederick A. Peck frederick.peck@umontana.edu Department of Mathematical Sciences, University of
Montana, Missoula, MT, USA.
ß2021 Taylor & Francis Group, LLC
COGNITION AND INSTRUCTION
https://doi.org/10.1080/07370008.2021.1983577
noted, the school community (re)produces and naturalizes peculiar artifacts called knowledge and
associates them with an understanding of learning tightly coupled with teaching (Brown, Collins,
& Duguid, 1989). Indeed, though schools may appear to be dealing with knowledge in the form
of decontextualized abstractions, from a situated perspective “these abstractions cannot exist in
their own right …[and] must be continually reproduced in the practices of the community”
(Packer, 2001, p. 509). Participation in the learning practices of school communities thus (re)cre-
ates peoples’perception of academic knowledge as a distinct thing that can be given
and received.
More recently, this line of critique has contributed to a practice turn in the field away from
how teaching facilitates the transmission of knowledge toward how it might immerse students in
desired epistemic practices (Ford & Forman, 2006; Forman, 2018). The aim has been to engage
learners in the activities of professionals and disciplinary experts such that “children might learn,
by becoming apprentice mathematicians, to do what master mathematicians and scientists do”
(Lave, Smith, & Butler, 1988, p. 62). Students can then “grasp”how a disciplinary practice works
by taking on the “various roles that exist in that practice”(Ford & Forman, 2006, p. 8). The
prevalence of the practice turn within the literature and recent reports (cf. National Research
Council, 2012) and standards documents (Common Core State Standards for Mathematics,
National Governors’Association Center for Best Practices, Council of Chief State School Officers,
2010; Next Generation Science Standards, National Research Council, 2013) suggests scholarly
agreement around the key takeaways of the anthropological critique, but different interpretations
emerged and complications persist. Assessing the field following the ascendance of situated learn-
ing theory, Barab and Duffy (2000) identified two general ways it was taken up to transform
learning experiences within schools: what they termed the “psychological approach”and the
“anthropological approach”(p. 26). Although both approaches take apprenticeship as the model
for learning, they end up with very different foci and prescriptions for the design of learning
environments.
The psychological approach takes practice as its point of departure. The goal is to engage stu-
dents in disciplinary practices via problem situations designed to be authentic in two ways. First,
they should mimic the sort of ill-structured problems that might be encountered outside of the
classroom. Second, they should require the same demands of engaging in disciplinary practice,
including, for example, taking on disciplinary roles (Ford & Forman, 2006). The focus on authen-
ticity ensures that concepts and procedures are “situated in the contexts of their use”(Collins,
Brown, & Newman, 1987, p. 457). Classrooms thus become a kind of highly-designed arena, or
practice field (Senge, 1994), distinct from the “real field”as “contexts in which learners …can
practice the kinds of activities they will encounter outside of schools”(Barab & Duffy, 2000,p.
30). Collins et al. (1987) imagined these school-based simulacrums as “cognitive apprenticeships”
(p. 3). Since then, a number of design principles have been devised for creating practice fields in
schools, including authenticity in context, complexity, collaboration, roles, and practice; student
ownership of the problem and the inquiry; and opportunities for reflection (Barab & Duffy, 2000;
Engle & Conant, 2002; Ford & Forman, 2006; Stein, Smith, Henningsen, & Silver, 2009). A key
aspect of a practice field is the presence of someone—e.g. a teacher—who does not so much
engage in disciplinary practices to address problems in the field but rather orchestrates disciplin-
ary activity such that the learners can remain productively engaged. Thus, practice fields have a
hierarchical participant structure, with a clear division of labor between those who design and
implement the practice field and those doing the practicing (Barab & Duffy, 2000).
By comparison, the anthropological approach takes community as its point of departure. From
this perspective, practices and communities exist in a reciprocal relationship, captured in the oft-
cited notion of a community of practice (Lave & Wenger, 1991; Wenger, 1998). The reciprocality
is important as communities are defined, in part, by their practices, but also, crucially, practices
are defined and legitimized within communities (Barab, Warren, del Valle, & Fang, 2006). As
2 F. A. PECK ET AL.
such, practices are imprinted with social and historical dimensions as they are passed on and
refined by community members over time through their mutual engagement in a joint enterprise.
Furthermore, communities of practice do not emerge fully formed by fiat of a designer
(Grossman, Wineburg, & Woolworth, 2001). Instead, practices and communities co-emerge over
time, each providing structure and meaning for the other. Temporality matters, with individuals
developing identities as they progress from newcomer to old timer, taking on (or taking a stand
on) the shared heritage of the community as they participate in its practices. From the anthropo-
logical perspective, learning is thus much more than an epistemic process—it is an ontological
process in which members, practices, and communities exist in productive tension, evolving and
becoming together (Packer & Goicoechea, 2000; Penuel & Wertsch, 1995).
Unpacking this tension, researchers (Barab & Duffy, 2000; Wenger, 1998) have identified the
following defining features of a community of practice: (1) a joint enterprise, which describes the
purpose of the community; (2) a shared repertoire of cultural resources; (3) mutual engagement
in the practices that bind the community; and (4) a history, with a reproduction cycle though
which new members are added and the community is (re)produced and changed. Schools exhibit
all four features (Packer, 2001), which complicates the idea that schools can be bracketed off
from yet still replicate the dynamics of disciplinary communities and other “real world”scenarios
(Barab et al., 1999). Schools are arguably no less “real”than these communities, and “attempts to
bring ‘real–world’problems into the school classroom will founder on the fact that the tasks can-
not remain the same. Because the social relations and cultural resources of the classroom are
inevitably different from those in the real world, the tasks are always transformed”(Packer, 2001,
p. 500).
This poses an acute problem for designing disciplinary engagement in classrooms. Although
practice fields involve a good faith attempt to mimic authentic conditions, neither practices nor
problems can be transported, unscathed, out of one community and into another. The activities
take place in a separate community and thus differ in meaningful ways from the conditions they
are meant to simulate. Using physics as an example, Lave and Wenger (1991) observed how stu-
dents and physicists engage the discipline and understand it differently within their respective
communities. Crucially, they note how the “actual reproducing community of practice, within
which schoolchildren learn about physics, is not the community of physicists but the community
of schooled adults”(pp. 99–100).
In many ways, the field remains reluctant to take the anthropological perspective and wrestle
with the implications of teachers as “schooled adults”whose instructional practices define and are
defined within school communities. Doing so reveals how the prerogatives of professionalized
teaching—i.e. planning, management, clearly defined outcomes—shape what it means to do and
learn math, science, or other disciplines in ways often taken for granted (Lave, 1988; Penuel,
2016). Complicating things further, as Lave (1996) reiterated decades after her initial anthropo-
logical work, teaching is not a necessary precondition for learning to occur; people can and do
learn without explicit direction or the orchestration we typically associate with classroom teaching
(see also Barab et al., 1999). This is not to say that such orchestration or well-designed practice
fields do not produce learning; rather, classroom learning is interwoven with the teaching practi-
ces that produce it and distinguish schools from other communities (O’Connor, Wortham, &
Rymes, 2003; Penuel, 2016).
In mathematics education, our field of study, ample research and significant reforms have
challenged the acquisitionist assumption and led to novel designs around practice, identity, and
community in classrooms. As we discuss below, these reforms have largely followed the psycho-
logical approach and established practice fields as the consensus model for designing learning
experiences within mathematics classrooms. It is a model that we as faculty employ in structuring
our classrooms and guide pre- and in-service teachers toward producing in their own. The mod-
el’s logic is arguably so uncontested as to be largely invisible to scholars and practitioners. Even
COGNITION AND INSTRUCTION 3
so, the idea of a practice field became visible and strange for our team as we revisited the
anthropological critique while investigating groups of math teachers who regularly gather within
Math Teachers’Circles (MTC) to engage in mathematical practice. Whereas many professional
teacher groups such as professional learning communities (PLC) and teacher teams are externally
organized by administrators and compulsory, MTCs are self-organized groups who meet volun-
tarily on their own time. They are also organized primarily around doing mathematics problems
rather than addressing pedagogy or other attributes of professional practice. Participants thus
have the flexibility to conduct their mathematical activity however they want. Yet as we show,
they tend to reproduce a practice field resembling a reform-oriented school mathematics class-
room. This may seem unremarkable, even desirable for many reformers; from an anthropological
perspective, however, this finding suggests a durability and invisibility to practice fields that mer-
its further interrogation.
Reform-oriented mathematics classrooms as practice fields
The aforementioned practice turn has enabled researchers and educators to take significant strides
in challenging the acquisitionist learning paradigm and teacher-directed model that remains per-
sistent in classrooms, particularly in mathematics education in the United States (Hiebert, 2013;
Litke, 2020). Within the acquisitionist paradigm, typical lessons begin with a lecture from the
teacher in which students are taught procedures and concepts, often supported by worked exam-
ples. Students are then given time to practice the new material on routine exercises largely mir-
roring the teacher’s examples. Discourse usually follows a script in which the teacher initiates a
known-answer question, the student responds to the question, and the teacher evaluates the cor-
rectness of the response (Mehan, 1979). In rejecting this model, the mathematics education com-
munity has developed innovative and culturally responsive ways to structure classrooms so all
students have opportunities to engage in mathematical practices. Generally, these reforms, as
codified in guidelines (CBMS, 2016; NCTM, 2014), consensus documents (Kilpatrick, Swafford,
Findell, & National Research Council, 2001), observation protocols (Gleason, Livers, & Zelkowski,
2017; Rogers, Petrulis, Yee, & Deshler, 2020; Schoenfeld, 2014), and influential publications
(Cobb, Stephan, McClain, & Gravemeijer, 2001; Horn, 2012; Smith & Stein, 2018; Stein et al.,
2009), reinforce a consensus model of classrooms as practice fields where students engage in col-
laborative mathematical inquiry within an environment designed to support such engagement.
Central to the endeavor are problem solving and argumentation practices (Hiebert et al., 1996;
Liljedahl, Santos-Trigo, Malaspina, & Bruder, 2016; Schoenfeld, 1994; Yackel & Cobb, 1996).
Rather than having concepts and procedures introduced through a lecture, conceptual learning is
developed by doing and discussing complex problems, or tasks, that students may not know how
to solve (Stein et al., 2009). Problems are selected by teachers to fit certain criteria. First, they
should have a high cognitive demand by prompting complex thinking and unstructured inquiry
(Stein et al., 2009). Second, they should be group worthy (Horn, 2012) by requiring multiple kinds
of mathematical competence. Third, they should draw on students’prior knowledge (Bransford,
Brown, & Cocking, 2000). Finally good problems should be aligned with a mathematical goal
(Marcus & Fey, 2003) such that they leave a “residue”in the form of a new understanding that
remains after the problem is solved (Hiebert et al., 1996).
Teachers cannot simply drop these problems into a classroom and expect meaningful engage-
ment. Rather, the learning experience is carefully orchestrated, with distinct phases demanding
particular teacher actions (Jackson, Garrison, Wilson, Gibbons, & Shahan, 2013; Stein, Engle,
Smith, & Hughes, 2008; Stein et al., 2009). First is the problem setup or “launch”phase to intro-
duce students to the problem and initiate productive inquiry. Successful launches, according to
Jackson and colleagues (2013), begin with familiarizing students with key contextual features of
the problem. For example, making sure students know what sub sandwiches are before doing a
4 F. A. PECK ET AL.
problem describing a scenario with subs. The launch should also activate prior knowledge and
reintroduce key mathematical ideas necessary to venture into new mathematical terrain while
doing the problem. A major challenge for teachers is maintaining the problem’s cognitive demand
while getting students started on it (Stein et al., 2009). If teachers provide too much support, they
may unwittingly lower the cognitive demand of the task (e.g. by explaining or suggesting a solu-
tion strategy).
With the problem solving launched, the implementation phase begins as students work in
small groups to solve the problem (Smith & Stein, 2018). Students should have agency to pursue
strategies that make sense to them but teacher support in doing so. During this phase, teachers
circulate through the room, structuring the inquiry based on student needs. The goal is to keep
things problematic enough to be productive but not so problematic that the inquiry devolves into
frustration (Hiebert & Wearne, 2003; Liljedahl, 2017). If the activity is too problematic, the
teacher may provide scaffolding or feedback to support students’inquiry without reducing
the cognitive demand (Anghileri, 2006; Barlow et al., 2018; Engle & Conant, 2002). In addition,
the teacher often presses students to justify their approach and articulate the meaning behind the
mathematics (NCTM, 2014; Stein et al., 2009).
Monitoring students’emerging responses prepares the teacher for the third and final phase in
which students share strategies and justify their solutions to classmates. While this can sometimes
feel like mathematical show-and-tell, skilled teachers leverage students’work as the grist for
sophisticated mathematical discussions and collective learning (Stein et al., 2008). Effective
orchestration of discussions engages students in the practices of justification and argumentation
such that sanctioned disciplinary knowledge emerges (Yackel, 2002). Such orchestration starts
with teachers selecting students to share work based on mathematical goals. Doing so ensures that
particular strategies or content understandings are made available to the entire class. Teachers
may highlight right answers or sophisticated strategies; but they may also have students share
opposing conceptions, or a proof and refutation, or strategies at a variety of levels of sophistica-
tion. Shaping students’work into a cohesive mathematical story requires purposeful sequencing of
presentations. For example, teachers might have students present contrasting conceptions back-
to-back to seed debate; or they may have students present in order of increasing sophistication to
build understanding. Once the selected mathematical ideas have been shared, teachers conclude
the lesson by drawing connections between the ideas to clarify important distinctions and establish
the big idea across strategies (Smith & Stein, 2018).
Taken together, these phases and the actions within them constitute a practice field for math-
ematics, with a hierarchical participant structure and a clear division of labor between teachers
and students. Students’engagement in complex mathematical inquiry is orchestrated by an
expert—not necessarily a professional mathematician, but rather an expert in facilitating disciplin-
ary engagement to produce desired conceptual learning (Ball, Thames, & Phelps, 2008). Over
time, classrooms can develop into a community, with teachers and students mutually engaged in
the joint enterprise of developing a shared repertoire of collective practices and participation
norms (Cobb & Hodge, 2011; Yackel & Cobb, 1996). Critically, though, the hierarchical teacher/
student participant structure plays an outsized role in defining the classroom community, includ-
ing the source of the mathematical problems, the nature of the problem solving, and the structure
of the classroom activity. As such, the joint enterprise of a reform-oriented mathematics class-
room is less akin to, say, a community of professional mathematicians than to the production
and maintenance of a practice field for mathematics (a contrast we briefly revisit later).
The practice field model described above is admittedly an ideal type, and there are many varia-
tions in schools and nuances raised in the literature (e.g. techniques for forming groups, Cohen
& Lotan, 2014; Liljedahl, 2014; strategies for productive scaffolding, Gonz
alez & DeJarnette, 2015;
van de Pol, Mercer, & Volman, 2019). However, given its prominence, it is reasonable to suggest
that the model reflects the general form of the field’s desired mathematics classroom, and a
COGNITION AND INSTRUCTION 5
shared interest among scholars and educators on incorporating or at least adapting practices
from professional communities to develop young people’s conceptual knowledge and procedural
skills for eventual entry into a “real world”beyond school. Such assumptions and concerns are
hallmarks of the psychological approach to addressing situated learning, the tenets of which
seemed reasonable yet also unusual in the mathematical activity we observed during
MTC gatherings.
Research methods
We employed a field-based ethnographic methodology in an effort to describe and thus fur-
ther understand (Eisenhart, 2006) the ways participants in MTCs engaged in mathematical
activity. Our data came from 19 gatherings of a statewide MTC in the western United States.
The gatherings took place in 5 locations across the state. Across all locations, there were 192
unique participants: approximately 85% were practicing teachers (30% elementary, 25% middle
school, 30% high school), 5% were post-secondary mathematics faculty, and 10% were pre-ser-
vice teachers. Within a location, the participants were relatively stable, with an average gather-
ing having approximately two-thirds returning participants. There was very little overlap in
participation between locations, with only 6 of the 192 participants attending gatherings in
multiple locations.
MTC gatherings were locally organized by different lead teams composed of 3–5 local
teachers and university faculty, all of whom had participated in at least one training work-
shop on how to run a MTC, facilitated by the American Institute of Mathematics (AIM).
Lead teams organized their gatherings independently of each other but were loosely coordi-
nated by a central university-based group that maintains a statewide website and mailing list
along with convening the AIM training workshop. Three authors of this paper (Peck,
Erickson, and Wu) are members of the central coordinating group, and, as such, are active
participants and promoters of MTCs. During the analysis and interpretation of data, these
individuals saw themselves as researchers and actively endeavored to set aside their positive
feelings toward MTCs to take a more neutral position, following the data to conjectures and
conclusions, even those that challenged beliefs and assumptions. This critical orientation was
supported by one member of the research team (Renga), who is not involved in implement-
ing MTCs and provided an outsider perspective.
We used video to capture the naturally occurring interactions of participants during the 19
local gatherings. Most gatherings lasted about two hours. Participants in most gatherings broke
into smaller problem-solving groups, and we recorded 2–4 of these groups for each gathering in
addition to recording whole group discussions. Content logs were produced for all videos to iden-
tify areas of conceptual interest for further transcription and fine-grained analysis (Erickson,
2006; Jordan & Henderson, 1995). We then used a cyclical, constant comparison method to
develop theory from our data (Lincoln & Guba, 1985; Maykut & Morehouse, 2002). First, mul-
tiple members of the research team independently watched and coded the same video. To develop
the initial codes, we asked ourselves, “What is going on here?”We met to discuss our coding and
resolve disagreements; we then used these consensus codes to code the next video, allowing new
codes to emerge inductively.
Through several iterations of this cyclical process, we developed a set of codes for the
activity—what the participants were doing—in each gathering. The types of activity included:
organizational activities, introductions, socializing, problem setup, problem solving, participant
sharing and summarizing, lecture, and pedagogy discussion (Appendix A for the complete
codebook). Not all activity types were observed in every gathering. Even so, iterative passes
through the data revealed patterns in the activity, and the structure of a typical gathering
emerged along with interesting variations. To facilitate this analysis, we created data displays
6 F. A. PECK ET AL.
(Miles, Huberman, & Salda~
na, 2014), including timeline maps that showed the temporal
unfolding of activity in each gathering and weighted directed graphs that showed transitions
among activities.
We then engaged in axial coding to relate the codes to each other. Using the data displays, it
became clear that gatherings often followed a similar sequence of activities, which could be
grouped into three macro-level phases. First, gatherings typically started with a preliminary phase,
including personal introductions and other organizational work such as explanations of MTCs
aims. This was followed by engagement in a mathematical activity phase, including setting up
mathematical problems, engaging in problem solving, and sharing and summarizing results.
Finally, the gatherings often concluded with a wrap-up phase that sometimes included brief dis-
cussion relating the math activity to teaching and learning, as well as more organizational work
such as giving door prizes or discussing the next gathering. Our focus in this paper is largely on
the gatherings’second phase devoted to doing math, though we do consider how participants
framed their activity in the preliminary phase.
Two other prominent themes emerged in the axial coding: (1) gatherings seemed to have a
particular participant structure and (2) problems and problem solving seemed to play an import-
ant role in gatherings. To analyze these themes further, we asked the following questions:
Regarding the participant structure, what are the different roles in gatherings and what are the
rights and responsibilities for each role? With respect to problems, what sort of problems are
addressed in gatherings and where do these problems come from? Finally, with respect to prob-
lem solving, what is the nature of problem solving in gatherings?
We investigated these questions by analyzing transcripts of participants’talk in conjunction
with our observations of non-discursive aspects of the social organization of the activity (e.g.
standing, sitting, circulating around the room, etc.). In analyzing the discourse, we attended to
the design of turns at talk, as well as the ways in which these turns were taken up and sequenced
in interactive sequences (Heritage & Clayman, 2010; Packer, 2011). For example, to investigate
the roles and responsibilities in gatherings, we examined transitions between phases, which were
often initiated through turns at talk. We paid particular attention to the design of these turns,
noting how they commonly involved the imperative and subjunctive moods (“So, get into groups
of like four or five people …”). We analyzed which participants produced these turns, and we
investigated the effect of these turns by analyzing the ways they were taken up by the other par-
ticipants in the gatherings.
Finally, we triangulated our findings through semi-structured interviews (Galletta, 2013)of
nine participants from the MTC gatherings we studied for this paper. All participants were
invited to be interviewed; from those accepting the invitation, we chose interviewees selectively in
order to achieve a representative sample with respect to grade level taught and MTC group. We
did not have access to how participants self-identified, but they seemed to skew white and female.
Participants were asked about their history with mathematics and teaching, their reasons for join-
ing MTCs, and their thoughts about the gatherings. Interviews were audio recorded and
transcribed.
Findings
In studying the mathematical activity, our main finding is that MTC gatherings typically resembled
a practice field for mathematics similar to a reform-oriented classroom, which is evident in (1) the
structuring of the activity in phases, (2) the high degree of orchestration and participant struc-
ture, and (3) the nature of the problems and mathematical activity. We elaborate each of these
in turn.
COGNITION AND INSTRUCTION 7
Structuring of mathematical activity in phases
We found that gatherings were highly structured during math time. On average, one hour and
20 minutes of the approximately two-hour running time of each gathering was spent doing math-
ematics. During this time, there were on average between seven and eight (SD 3.5) distinctive
segments of activity, with certain activity types repeated. As shown by a weighted directed graph
of activity transitions across all gatherings (Figure 1), the dominant activity types during math
time were problem setup (13% of time), problem solving (63% of time), and participant sharing
and summarizing (15% of time). Also, as demonstrated by the two largest arrows in Figure 1, the
activities during math time followed a particular temporal sequence as problem setup was imme-
diately followed by problem solving, which was then followed by participant sharing and
summarizing.
There was some variation in how these segments of activity emerged, illustrated in the activity
timelines shown in Figure 2. The most basic sequence is exemplified by Gathering L (top panel),
which included only three segments of activity following the dominant sequencing of [problem
setup] ![problem solving] ![participant sharing and summarizing]. These phases mirror the
phases of reform-oriented math classrooms, with a distinctive launch, opportunity for problem
solving, and orchestrated discussion.
Although Gathering L provides a basic template, only 2 gatherings had only one sequence of
setup, solving, and sharing. In the remaining 14 gatherings, we observed three variations. The
first variation was to repeat the sequence multiple times. In Gathering B (middle panel), for
example, there were three successive iterations of the sequence (the “other”activity that occurred
in the middle involved giving door prizes). The second variation, exemplified in sequence 1 of
Gathering A (bottom panel), involved a lecture immediately following problem solving in the
place of participant sharing and summarizing. During these lectures, a facilitator simply gave out
the answer(s). The third variation involved breaking-up sustained periods of problem solving
with instances of participants sharing and summarizing. As evident in sequence 2 of Gathering
A, problem setup was followed by a sustained period of problem solving, which was peppered
with instances of participant sharing and summarizing. In these instances, participants often
shared partial strategies with the group.
Orchestration of mathematical activity
Along with the phased structuring of the mathematical activity, we found that this structure was
orchestrated by a single individual or a small number of individuals, whom we call “facilitators.”
Figure 1. A weighted directed graph of activity types within the mathematical activity phase across all gatherings. Nodes are
sized in proportion to the amount of time spent on that activity type, and edges are sized in proportion to the frequency of the
transition they represent. For clarity, the following activity types are depicted as “other:”organizational work, personal introduc-
tions, and discussions of teaching and learning (together, these activities account for 2% of the time).
8 F. A. PECK ET AL.
Our analysis suggests that facilitators were established as programmatic experts within gatherings,
initiated transitions to structure the activity, and provided support, scaffolds, and feedback to
other participants. Altogether, this produced a hierarchical participant structure similar to what is
seen within classrooms (teacher/student) and necessary for producing a practice field for
mathematics.
Facilitators are established as experts
In the beginning of the gatherings, facilitators often established their expertise with MTCs by pro-
viding their credentials while introducing themselves to the assembled group. For example, in
one gathering the facilitator introduced himself as a local teacher, and then stated:
I had the pleasure of joining the local team up in [local city] for a conference a few weeks ago, and I’m
gonna show- I’m gonna- I used to live in [a city on the West Coast], and I’ve been a part of the circle
movement, uh, for a while [ …] so uh, I’ll share with you an activity I learned in the Bay Area math circle.
1
Figure 2. Timeline maps of three gatherings. The x-axis represents time, as a percent of the total time. The categories on the
y-axis are activity types. The activity segments in each gathering are represented with black horizontal rectangles, with vertical
lines representing transitions between segments. The maps show common sequencing of the activity types.
1
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. Underline denotes vocal emphasis,
and a hyphen- denotes a restart. [Brackets] denote text that has been added to, or modified from, the spoken words.
Bracketed ellipses [ …] denote spoken words that have been excluded from the transcription to preserve analytic focus.
((Double parentheses)) are used to provide contextual and other non-spoken elements that assist in interpreting the
transcribed text.
COGNITION AND INSTRUCTION 9
Here, he established himself as an MTC expert by providing his credentials, which included
his history with MTCs, participation in a training workshop, and prior experience with the prob-
lem under consideration in the gathering.
Facilitators were also established as experts by other participants in the gatherings. A common
means for doing so involved asking the facilitator questions about a problem or its solution.
After spending 6 minutes exploring a problem that involved moving wooden pegs along a linear
board (Problem A in Appendix B), for example, two participants looked to the facilitator for
clarification:
1. Participant X: ((addressed to Facilitator)) Oh, you can’t go back?
2. Participant Y: ((addressed to Facilitator)) You can’t go backwards?
3. Facilitator: You cannot go backwards.
4. Participant Y: Oh! Well then, never mind, because I went backwards.
By asking the facilitator a question about the rules of the mathematical problem (turns 1-2),
the participants established the facilitator as a person who had expertise in the problem that they
did not have. The disparity in expertise was reinforced in turn 4 as Participant Y accepted the
facilitator’s claim as a truth about the problem.
Facilitators initiated transitions to structure the activity
Transitions between activity segments were almost always prompted by a facilitator. Across all of
the MTC gatherings we examined, 128 (91%) of the 141 transitions between activity segments
were prompted by facilitators. For example, as one gathering started their mathematical activity,
the first sequence involved a problem called “friends and enemies”(Problem B in Appendix B).
During problem setup, the participants enacted a scenario in which they all moved around
according to a set of rules. They then broke into small groups to consider some mathematical
questions related to the problem. The transition from whole group to small group was occasioned
by the facilitator stating:
So, get into groups of like four or five people, find some space. Somebody right here, four or five. We got
two here. Just get where you have some space to work. ((Participants move into open spaces and get into
groups)). Okay, so what I want you to do is just spend a few minutes thinking about the possibilities for
what could or could not happen, depending on how you- so I want to just analyze the problem a bit.
In the first part of this quote, the facilitator used the imperative mood to issue a command
(“get into groups”). This was followed by a desire, expressed in the subjunctive mood (“what I
want you to do is”). This last part was taken by participants as a request, which then followed by
working in small groups to analyze the problem. Taken together, this talk worked to structure the
participants’activity, shifting them from the activity of problem setup to that of problem solving.
Facilitators across gatherings similarly issued commands (imperative) and expressed desires for
action (subjunctive) to initiate transitions between activities. To prompt problem solving after
setup, for example, one facilitator stated, “Try these, they’re a bit unexpected.”At another gather-
ing, a facilitator prompted participant sharing and summarizing after problem solving activity by
proclaiming, “Alright, let’s uh, let’s go ahead and stop what you’re doing and come on and come
back together a little bit here.”Being more suggestive, a facilitator initiated problem setup by
imploring the participants, “Okay, I would like you guys to bear with us for a second and just try
something.”Likewise, another facilitator at a different gathering moved participants from sharing
insights on a problem to starting a new one by stating, “So, one of the things that I want to do
[…] I want you to take your towers and set them on the appropriate place on the mat.”Such
statements served to structure—or orchestrate—the mathematical activity.
10 F. A. PECK ET AL.
Another common prompt for such structuring involved the facilitator asking questions.
During one gathering, for example, the participants worked together to create and undo tangles
in a pair of ropes using particular rules (Problem C in Appendix B). At one point, the facilitator
prompted a transition from problem solving to participant sharing and summarizing by asking
the whole group, “So does anyone have a strategy for solving this, that they would like to show?”
Although this is technically a yes/no question, in the gathering it functioned as a prompt for
groups to share their strategies. Similarly, later in the gathering, a second facilitator prompted a
transition from problem setup to problem solving by asking the group, “For practice, do you
guys want to try, how do I do five twists? What if you start with five twists?”Subsequently, the
participants began exploring the facilitator’s question, indicating that the question worked as a
prompt to structure the mathematical activity.
Facilitators provided support, scaffolds, and feedback
During problem solving, facilitators tended to circulate around the room asking questions and
providing participants with various kinds of support, scaffolding, and feedback. Sometimes they
confirmed whether or not an answer was correct. In one gathering, for example, after a small
group completed a logic puzzle (Problem D in Appendix B), they shared their solution—it took
them thirteen moves—with the facilitator, who confirmed, “that’s the lowest.”Facilitators also
offered support by asking pointed questions, such as when a facilitator explained to a small
group, “you just have to maximize your time,”and then asked, “What’s the best way to maximize
your time?”Facilitators often used these pointed questions to suggest particular strategies.
Recommending that participants simplify a complex problem (Problem E in Appendix B) for
example, a facilitator suggested that the group, “think smaller [ …] what if there is only one
human on the island, one human and one zombie, and then from one to two to three?”
Similarly, in another gathering (as participants explored Problem F in Appendix B), a facilitator
addressed the entire group and asked, “In math, when you want to find a pattern, what do you
look for? You look for the extreme. And what is the extreme in these number strips?”Facilitators
were thus overseeing participants’work and intervening to move it forward.
A less-common means of support involved providing material artifacts—including manipula-
tives—to help structure participants’activity. In one gathering, participants explored a problem
that involved using polyominos to “stomp”gophers on a grid (Problem G in Appendix B). The
facilitator provided an erasable whiteboard with a rectangular grid printed on it, along with poly-
omino manipulatives that were sized according to the grid squares on the whiteboard and sup-
ported participants in empirically testing different conjectures. Likewise, in another gathering, the
facilitator provided cups and M&Ms for participants to simulate sending a message through a
coded medium, which could then be physically passed through a group of participants who acted
as the medium (as described in Bachman, Brown, & Norton, 2010).
Facilitators also guided the sharing and summarizing of solutions after problem solving. At a
minimum, facilitators structured the order of the presentations by calling on participants to pre-
sent. They also provided evaluative feedback in 39 (80%) of the 49 segments in which participants
shared with the whole group; we observed this behavior in all but two gatherings. In a typical
example, after participants had been exploring a problem for 15 minutes, a facilitator called on a
participant to “share what they got.”The participant presented a strategy using a whiteboard at
the front of the room, after which the facilitator provided evaluative feedback saying, “I think
that’s great.”In another instance, at the end of a gathering, a facilitator posed quick questions
while lecturing on the solution to a problem that involved lying about numbers written on a strip
of paper (Problem F in Appendix B). “So what did we lie about?,”he asked. Multiple participants
responded, “Zero.”He then echoed and affirmed the answer: “Zero.”These evaluations worked to
convey that problems had fixed solutions that were known to facilitators, and they worked to
structure participants’activity in the direction of those known answers. In this way, facilitators’
COGNITION AND INSTRUCTION 11
evaluations, coupled with other forms of support, further demonstrated how facilitators were
actively orchestrating participation in the mathematical activity.
Gatherings had a hierarchical participant structure
The evidence we have presented thus far supports a broader observation that there were two
kinds of people at gatherings, facilitators and participants, each with different responsibilities.
Across MTC gatherings, we noted how facilitators were generally standing while participants
were generally seated at tables. Facilitators provided the problems and knew the answers while
participants worked on the problems. They addressed the whole group at will, while participants
only addressed the whole group when invited. Facilitators also structured the mathematical activ-
ity by providing instructions to participants, who then followed those instructions. During prob-
lem solving, facilitators circulated freely, stopping at tables to interact with participants, who
generally stayed seated and only interacted with groupmates. Finally, facilitators called on partici-
pants to share solution strategies, and participants obliged the request. Taken together, this par-
ticipant structure largely mirrors that of a practice field, with some participants facilitating, or
orchestrating the activity like teachers, to promote meaningful disciplinary engagement among
the other participants.
Non-routine, complex problems with known solutions
As noted above, the activity in reform-oriented classrooms is organized primarily around rich
problems and student engagement in problem solving. We observed such engagement in nearly
every MTC gathering in our data as participants engaged in problems for which they did not
seem to have a ready-made method to solve (see Appendix B for sample problems). Evidence
that these problems were actually problematic for participants comes from analysis of their uptake
in the form of engaging in problem solving activity. In all but one case, problem setup was fol-
lowed by problem solving.
Similar to analyses of mathematicians’problem solving in the literature, participants’activity
was both agentive and disciplined (Livingston, 2015). In particular, activity tended to proceed in
a sequence of “free”and “fixed”moves (Pickering, 1995), in which the participants exercised
agency over the direction of their work, and then followed through with the implications of those
choices. The excerpt below is a representative example of how the sequence of free and fixed
moves emerged during most gatherings. In this instance, the participants played a game called
Stomp (see Problem G in Appendix B). Participants were given grids with gophers drawn in cer-
tain squares. They were to use a given polyomino to stomp the gophers, such that wherever the
polymino was stomped, the squares would change state: if a square had a gopher, it would be
stomped out, but if a square was empty, it would acquire a gopher. The excerpt is taken after
participants had been working on the problem for about 30 minutes. In what follows, two partici-
pants discuss possible moves while drawing trominoes onto grids on their whiteboards:
1. Participant P: So, see if we do this one- ((draws tromino))
2. Participant Q: Okay.
3. Participant P: Then we can, if we want, we can add three, just by stomping here. ((gestures
to a location on her grid)) And those two will be gone. So if we can eliminate, basically, the
top row, we can have one, two, three. ((pointing to grid squares)) Right? If we stomp here,
those can be gone. And can we repeat this in the bottom row?
4. Participant Q: You can clear the outside that way. But I keep getting left with those two in
the middle.
12 F. A. PECK ET AL.
5. Participant P: Okay, so say if we do next, if we do this one ((draws tromino on board)),
then we would get this ((adds color on board)) and not here ((erases color from board)).
And you’re saying, if we go this way again [ …] If we want to, we can do this way-
6. Participant Q: Yeah, and get them out of the middle.
7. Participant P: So now the middle ones are not out, and we have left, those two. ((points to
grid squares))
8. Participant Q: But since there’s two, there’s never anything we can do with them, even if
they were next to each other.
9. Participant P: But we can add more.
10. Participant Q: Yeah we would have to add more.
Here, we see the participants narrating a series of free and fixed moves. They were free to
“stomp”their tromino anywhere inside the grid. This freedom was reflected in phrases such as
“we can, if we want”(turn 3) and “if we want to, we can”(turn 5). However, the free moves had
consequences. Following a series of free moves in turns 1–3, Participant Q noted the consequen-
ces in turn 4 by observing how they had “clear[ed] the outside”but are “left with those two in
the middle.”It was not up to the participants to decide which squares were colored following
their free moves. They could, however, make another free move, which Participant P articulated
in turn 5. This, in turn, led to the consequences narrated in turn 7 and a new free move in turn
9, and so on such that free and fixed moves followed from each other. We documented similar
sequences of free and fixed moves in all gatherings that involved problem solving.
At the same time, as documented earlier, participants’engagement in this kind of mathemat-
ical practice during gatherings was highly orchestrated by facilitators who knew the answers. It
was common for facilitators to suggest strategies during problem solving, ask pointed questions,
and provide artifacts and feedback to help guide participants toward achieving those answers. In
some cases, facilitators even segmented tasks into smaller chunks, which were sequenced in order
to lead participants to an answer (e.g. Problems B and F in Appendix B). Although participants
usually spent about an hour engaged in problem solving activity (mean 52.5 minutes, SD
24 minutes), there were only 2 gatherings in which problem solving was completely uninterrupted
(as in Gathering L in Figure 2). In the remaining gatherings, problem-solving was interspersed
with interruptions and the resulting average length of uninterrupted problem solving amounted
to 18 minutes (SD 17 minutes).
Furthermore, the problems were fixed and bounded in ways that were similar to problems in
reform-oriented math classrooms. The gathering length bounded them temporally, as problems
started when the gatherings started and ended when the gatherings ended. In all but 2 gatherings,
problems were provided by facilitators and were not subject to change during gatherings (see
Appendix B for examples). Each problem had a fixed solution, and solutions were provided at
the end of gatherings if participants had not solved them. So while problem solving involved free
moves, neither the problem nor the solution were subject to change in the course of prob-
lem solving.
Disconfirming evidence
Sixteen of the 19 gatherings followed the template that we articulated above, in terms of the
structure of the mathematical activity, the orchestration by facilitators, and the nature of the
problem. This suggests a stability and consistency to the findings. However, we also noted some
disconfirming evidence, in the form of three anomalies. At one extreme, we documented one
instance in which the facilitator lectured for the gathering’s duration. At the other end were two
gatherings in which the math activity was barely orchestrated. In these latter gatherings, the
group collectively developed questions to which no one—including the facilitator—knew the
COGNITION AND INSTRUCTION 13
answer. This led to a less overtly hierarchical participant structure with considerably less
facilitation.
Other possibilities for “doing mathematics”
Again, it may seem unremarkable that groups comprised of mostly mathematics teachers would
pursue doing mathematics together through the logic and structure of a practice field that
resembled a reform-oriented math classroom. In fact, in light of the progressive reform goals
described earlier, their instantiation of mathematics activity through effectively organized, orches-
trated, collaborative, and cognitively demanding problem-solving is a noteworthy accomplish-
ment. Without diminishing this accomplishment, we want to highlight that practicing teaching or
reproducing a reform-oriented classroom were generally absent from participants’stated goals.
Although facilitators sometimes recognized that problems might be useful in classrooms (espe-
cially for use on special days, like half-days or days with assemblies), only one gathering was
framed by participants as primarily about improving teaching. Participants in all the others expli-
citly framed gatherings as about doing mathematics. In fact, they made pains to distance their
activity in MTCs from the work and concerns of teaching, focusing instead on doing mathematics
for its own sake. “We are not trying to teach you something here to use in your classroom”,
explained one participant to her colleagues as she opened their gathering. Likewise, another par-
ticipant stated to his group:
We don’t have any expectations except you walking away with a good feeling. 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—” It has nothing to do with it. What we’re trying to do is just, be a group that likes mathematics.
The expressed aims of doing mathematics for its own sake, for pleasure, and for personal
enrichment were echoed during interviews of MTC participants. When asked to share their expe-
riences within gatherings, all but one of our nine interviewees explained that the primary activity
in MTC gatherings was to do math. As one participant explained, “We get together to do math,
and to just hang out and have fun doing math.”Another characterized the MTC activity by stat-
ing: “[We] did math. We didn’t focus on standards, we didn’t focus on strategies, we didn’t focus
on teaching, but we did math.”
This framing of the primary purpose—to do mathematics—is shared by the larger MTC com-
munity. On its website, for example, the Philadelphia Area Math Teachers’Circle (n.d.) promises
“a space for math teachers to work as mathematicians do!”Donaldson, Nakamaye, Umland, and
White (2014) similarly indicate that the major goal of MTCs is “to encourage teachers to develop
as mathematicians by engaging in the process of doing mathematics”(p. 1336). Teaching con-
cerns and connections did surface during gatherings (Renga, Peck, Feliciano-Semidei, Erickson, &
Wu, 2020), but the participants appeared determined to set them aside to create a community
whose joint enterprise was doing mathematics. Even so, as we have shown, the resulting activity
produced what resembled a classroom community, with teachers treating other teachers as stu-
dents—albeit, where the students engaged in complex, non-routine tasks as advocated by the lit-
erature on practice fields.
Perhaps this was an inevitable outcome, especially for mathematics teachers. Teachers and
many other schooled adults (Lave & Wenger, 1991) are likely to associate mathematical practice
with the formalized version they experienced in schools (Lave, 1988). But an expansive view of
mathematics reveals a richer picture. Researchers have shown how numeracy, algorithmic think-
ing, and other forms of mathematical practice and learning emerge in non-school settings, includ-
ing in the community of professional mathematicians (Barany & MacKenzie, 2011; MacKenzie,
1999; Pickering, 2006), in workplaces (Gainsburg, 2007; Masingila, 1994; Nunes, Schliemann, &
Carraher, 1993; Scribner, 1985), in “everyday”settings in Western societies (Lave, 1988; Nasir,
14 F. A. PECK ET AL.
2000; Taylor, 2013), and in non-Western societies (Ascher, 2002; Saxe & Esmonde, 2012). This
work draws attention to notable differences in the doing of mathematics in and out of classrooms
and underscores the cultural dimensions of practice (Nasir, Hand, & Taylor, 2008). A group that
sets out to do mathematics could thus draw inspiration from any number of communities for
their practice; they do not have to recapitulate a practice field that resembles a reform-oriented
classroom. In the case of MTCs, it is reasonable to infer that participants set out to do mathemat-
ics in ways that resemble the community of professional mathematicians. However, there are
important distinctions in the approach to collaborative problem solving between a reform-ori-
ented math classroom and the community of professional mathematicians (Table 1).
As with reform-oriented mathematics classrooms, problems play a central role within the com-
munity of professional mathematicians (Halmos, 1980). For both students and mathematicians,
Table 1. Key differences between “doing mathematics”in a community of professional mathematicians versus in a reform-ori-
ented classroom.
Doing mathematics …
…in a reform-oriented classroom
…in a community of professional
mathematicians
Joint enterprise Producing and sustaining a practice
field for mathematics
Doing mathematics
Participant structure Two kinds of people: teachers and
students, with a very explicit division
of labor. Teachers structure, support,
scaffold, and give feedback on the
problem solving activity, while
students engage in problem
solving activity.
Members have a variety of expertise
and status, which affect their specific
roles in the community. However,
nearly all members contribute to
collective problem-solving activity
that is directed toward finding
unknown results.
Source of problems Problems are provided by the teacher,
fully-formed and well-posed.
Students work on the problems that
they are given.
“Natural questions”arise from an
individual’s work. Mathematicians
pose problems to the community.
Certain problems gain importance
within the community, including
those which are posed by senior
mathematicians. Problems may start
out as ill-posed explorations of
intriguing phenomena and part of
the work of a mathematician is to
develop well-posed versions of a
problem. Mathematicians exercise
choice in the problems that
they pursue.
Nature of problems and problem solving Problems are non-routine and complex.
The solution and solution strategy
are not known by the problem
solvers but are known by the wider
community and the teacher. Problem
solving may involve a dance of
agency, but the problem and
solution are fixed in advance.
Problems are resolved and wrapped-
up by the end of the session.
Problems are non-routine and complex.
The desired result (solution, proof,
etc.) is not known by anyone,
including the problem solver and
other members of the community.
Problem solving involves a dance of
agency in which both problems and
solutions may be mangled. Problems
are rarely resolved in a
bounded time.
Means of structuring the activity Activity is structured by the teacher,
who structures the activity into
distinct segments (e.g. problem
setup, problem solving, sharing and
summarizing).
Activity is structured by individual
creativity and norms of
the discipline.
Consequences of engagement in activity Individual success or failure, preparation
for future activity.
Preservation and extension of the
shared heritage of the community,
identity development as a member
of a community.
COGNITION AND INSTRUCTION 15
the problems are non-routine and complex, and solving them requires sustained inquiry.
However, there are important distinctions. Instead of solving problems provided by teachers,
mathematicians exercise choice in the problems that they pursue. The problems come from a var-
iety of sources. Often, mathematician pursue problems that arise “naturally”in their work (Bass,
2011). Or, mathematicians may choose to pursue problems that are deemed important by the
community, including those which are posed by senior mathematicians (Halmos, 1980; Lockhart,
2009; Zeitz, 2017). In a practice field, problems are well-posed, while mathematicians sometimes
work on ill-posed problems related to intriguing phenomena, and part of the work involves deter-
mining a well-posed version of the problem. Whereas teachers tend to know the answer to the
given problem, mathematicians work on problems for which no one in the community knows
the desired result (including the solution to a problem, or a proof of a conjecture, or a novel
proof of a theorem;Bass, 2011). Furthermore, as discussed earlier, teachers structure the mathem-
atical activity for the problem solvers (students) into distinct phases and anticipate and plan out
the process. For mathematicians, the process is bounded by disciplinary norms and historical
constraints that are enforced by community members. Problem solving is often temporally emer-
gent in the give-and-take between creativity on the one hand, and accountability to norms and
community members on the other (Pickering, 2006). Although the community often holds each
other accountable to solving problems as they are stated, it is also possible that problems can be
abandoned or modified in the course of solving them (Pickering, 2006). Taken together, the
potential malleability of problems and the temporally emergent nature of solutions form a con-
trast with the activity in reform-oriented classrooms where problems and solutions are usually
fixed a priori for students.
There are other notable differences (e.g. participant roles, consequences of engaging in activ-
ity), which, taken together, establish a community of professional mathematicians that is very dif-
ferent from the communities created in practice fields. The contrast serves the point that the
mathematical activity in MTC gatherings, like the activity in classrooms, is a particular form of
practice. From the anthropological perspective, a group could arguably not “do what mathemati-
cians do”in the fullest sense of the phrase unless they were members of the community of math-
ematicians. However, MTCs could reduce the orchestration of their activity to offer participants
greater agency and creativity in ways similar to professional mathematical practice.
We are not suggesting MTCs should strive to situate themselves within the professional math-
ematics community. Such a goal is no more necessary for exploring mathematics than yoking the
activity to professional teaching through an explicit goal of improving classroom instruction.
Indeed, assuming that teachers can or should only replicate practice fields when doing mathemat-
ics may be selling them short and limiting their imagination (and the field’s) for disciplinary
engagement. If a group of basketball coaches got together to play basketball, it would be odd if
one or two of them started coaching the others and structuring the activity like a team practice.
Basketball coaches, like the general public, understand the difference between practicing basket-
ball and playing basketball. The fact that teachers gathering to do mathematics structure their
“play”like a practice raises questions about whether the same can be said for mathematics or
other academic disciplines.
Revisiting the anthropological critique
In advancing a philosophical argument for social practice theory, Barab et al. (1999) contend that
the field of education tends to hold on tightly to particular assumptions about teaching and learn-
ing in an effort to reduce complexity while making targeted improvements. Letting go, they sug-
gest, is the only way to radically reimagine learning and schooling. Of course, fear of the
unknown boosts the appeal of holding on to status quo understandings. The anthropological cri-
tique of schooling levied by Lave, Barab, Packer, and many others helped the field confront the
16 F. A. PECK ET AL.
grip of acquisitionist theory on schooling, with its dualistic views of learners’minds, knowledge,
and abstract skills as separated from their embodied, socially-situated participation in the world.
The orientation toward community offered a corrective to the individualism undergirding acquisi-
tionist theories of learning. As the basis for classroom practice, Lave (1996) observed, such theo-
ries can lead educators to unjustly “prescribe ideals and paths to excellence and identify the kinds
of individuals [ …] who should arrive”(p. 149).
Since then, many education scholars have embraced situated understandings of learning, with
a practice turn drawing attention to the knowledge producing activities within disciplines and the
design of immersive, authentic experiences in the classroom (Forman, 2018; see also Herrenkohl,
Tasker, & White, 2011; in science education, Manz, 2015). Likewise, there is an emerging consen-
sus that community matters. In mathematics education, fostering a community of practice within
classrooms has moved from the literature (Rogoff, 1994) into popular textbooks as an explicit
goal of impactful teaching (Huinker & Bill, 2017)
2
. This strikes us as significant progress toward
a more accurate and just way of understanding learning by expanding the frame of inquiry and
concern (Hand, Penuel, & Guti
errez, 2012). Even so, the field’s use of “community”may still be
too narrowly conceived as (1) a vehicle for controlled disciplinary engagement and (2) a support
for individual learning, thus constricting our collective vision for what a community of practice is
and could become in schools.
Studying how communities emerge through their joint activity reveals the tensions participants
negotiate while deciding What are we doing? and Who are we here? (Hand & Gresalfi, 2015;
Renga, Peck, Feliciano-Semidei, et al., 2020). As noted earlier, people quantify and mathematize
the world in all sorts of ways, though schools tend to be most people’s primary contact with the
more formalized, academic mathematics discipline (Lave, 1988; Nunes et al., 1993). We see the
residues of this contact and the school context in the MTC participants’reproducing a particular
kind of classroom environment while doing mathematics, what we have called, following Barab
and Duffy (2000), a practice field. That this reproduction happened outside of school, in the
absence of students or professional directives, suggests a durability and invisibility to practice
fields in ways that reform-oriented educators may not intend. Practice fields are durable, in that
they permeate out-of-school spaces where they arguably are not appropriate, while at the same
time remaining invisible as the unquestioned, common-sensical means by which a group of
schooled adults goes about “doing mathematics.”This, in turn, suggests that practices fields in
schools may not leave learners (or at least, teachers) with the grasp of a disciplinary practice in
ways that advocates of practices fields desire (Ford & Forman, 2006; Forman, 2018). Indeed, in
the groups that we studied, it seemed that disciplinary practice was equated with school practice,
with its attendant tight orchestration of activity and bounding of problems and solutions.
Watching adults “play school”while seeking to distance themselves from it underscores how the
reproduction of disciplinary practices beyond the school context requires not only (re)cognizing
their appropriateness for novel situations (Packer, 2001), but also seeing the instructional dimen-
sions of practice as strange so new possibilities for practice might emerge.
This paper contributes to these efforts. The unique circumstances of MTCs made the familiar
dynamics of practice fields strange, revealing the grip of the psychological approach to situated
cognition both within and outside of schools, and demonstrating that practice fields are not sim-
ply neutral venues for engagement in “authentic”practice but rather that they produce a peculiar
form of engagement in disciplinary practice that is different in important ways from that of pro-
fessionals. Making this visible is an important contribution to our field’s discussions around
2
Although the persistence of an idealized subject narrative of the gritty student (Kirchgasler, 2018) and the use of
individualistic techniques such as data walls to motivate student performance (Harris, Wyatt-Smith, & Adie, 2020) suggest that
individualist theories continue to hold sway over teaching practice.
COGNITION AND INSTRUCTION 17
disciplinary practice, especially insomuch as it may be the first step in loosening the grip of the
psychological approach.
In this regard, the history and grassroots nature of the MTC movement gives us hope that
they could be a space for experimentation with alternatives. MTCs are an outgrowth of Math
Circles and other extracurricular experiences that originated a century ago in Eastern Europe and
eventually came to the United States in the 1990s. Wiegers and White (2016) explain how the
first gatherings in Bulgaria and Russia grew out of the expert community’s“higher emphasis
[compared to their U.S. peers] on the vertical integration of mathematicians sharing their math-
ematical passion with younger students”(p. 238). They note how gatherings operated independ-
ently of formal schools, with university faculty and graduate students meeting with school-aged
children to collaborate on complex problems. Concerns over access meant gatherings were typic-
ally free, open to all, and held in convenient locations that could include schools but not always.
The first Math Circle in Boston followed a similar model by enabling young people to do math-
ematics with passionate experts and peers in an informal setting. Often the experts introduced
non-routine problems, but unlike traditional classrooms, novices had more leeway in the math-
ematics practices employed and the direction of the inquiry. The young people could thus step
outside the bounds of formal schooling to enculturate more directly into the disciplinary commu-
nity (Lave & Wenger, 1991).
Altering or softening those bounds remains difficult. Doing so arguably requires educational
designers and teachers who see beyond the bounds and are willing to challenge them. MTCs are
promising in this regard, as teachers come together of their own volition to explore disciplinary
practice decoupled from school so they might prioritize doing mathematics for reasons other
than promoting students’disciplinary learning (e.g. doing math for personal pleasure; Renga,
Peck, Feliciano-Semidei, et al., 2020). Leaning into the radical potential of MTCs could support
efforts to take the field in unusual directions. Notably, it could push STEM scholars beyond a
preoccupation with integrating disciplinary practices into classrooms in more “authentic”(and
inevitably imperfect) ways to consider more ambitious transformations of schooling (Penuel,
2016). To this end, we are currently studying the anomalous MTC gatherings with the less hier-
archical participant structure; early analysis suggests these gatherings bear a closer resemblance to
the early Math Circles or a coffee shop salon than a reform-oriented classroom.
Framed anthropologically rather than psychologically, MTC participants’efforts at decoupling
mathematics from teaching and reprioritizing their aims has important community building
implications. MTCs arguably do more than give teachers opportunities to experience novel prob-
lems and approaches to mathematics practice; they offer teachers opportunities to (re)produce
novel kinds of community through their practices. Indeed, one of the implications of Lave and
Wenger’s(1991) claim about the production of “schooled adults”is the possibility of alternative
trajectories and outcomes through the production of people who understand themselves as being
learners in unconventional, non-school, or unschooled ways. This, though, may require awareness
of alternatives—an awareness that may not be possible as long as practice fields remain durable
and invisible. Echoing Lave (1996), we see no way of realizing such alternatives in schools with-
out further interrogating the tight linking of teaching and learning associated with them. Doing
so would make the school enculturation dynamics of practice fields less invisible and, perhaps, as
strange in K-12 classrooms as they were in MTC gatherings.
Acknowledgements
We are grateful to the MTC participants and facilitators who graciously opened their gatherings to our study. We
thank Ricela Feliciano-Semidei for her assistance with data analysis, and Carrie Allen, Sara Heredia, Joanna
Weidler-Lewis, and Kathryn Wiley for providing valuable feedback on the manuscript.
18 F. A. PECK ET AL.
Funding
This work was supported by grants from The American Institute of Mathematics, The Montana Office of the
Commissioner of Higher Education [grant number S367B140023-14A], The National Security Agency [grant num-
ber H98230-17-1-0299], and The University of Montana.
ORCID
Frederick A. Peck http://orcid.org/0000-0002-2212-0535
Ian Parker Renga http://orcid.org/0000-0001-7467-1120
Ke Wu http://orcid.org/0000-0001-7645-9051
David Erickson http://orcid.org/0000-0001-7247-9039
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COGNITION AND INSTRUCTION 23
Appendix A.
Codebook
Appendix B
Sample problems
Problem A:
This problem is sometimes known as “The leaping frog puzzle”or “n-in-a-row.”It is described in many books,
including Ball and Coxeter (1987). In the gathering, participants were given a board with 9 holes in a line. The
left-most 4 holes were filled by white pegs, and the right-most 4 holes by brown pegs. The instructions were pre-
sented orally by facilitators as follows:
1. Facilitator 1: The idea here is, we want to get all the white where the brown are, and the brown where the
white are. And you can move over one or you can jump.
2. Facilitator 2: And you can only jump over opposite colors.
3. Facilitator 1: Right. And so the idea is, is that, how many moves would it take?
Problem B:
This problem is known as “friends and enemies.”We are not aware of the originator of the activity, nor could
we find a published source. It was shared with us by Bob Klein of the Alliance for Indigenous Math Circles. In the
gathering, the activity was presented in stages. Each stage was presented orally by one facilitator, as follows:
Stage 1:
1. Facilitator: Here’s what we’re going to do.