Building a Community of Professional Learners
Boundary-Spanning Mathematical Play
By Teresa K. Dunleavy, Jennifer N. Lovett, Elizabeth K. Metts, and Shannon N. Reider
Overview: In this piece, we share the boundary-spanning ways we engage our work with the Math Teachers’ Circle of Middle Tennessee. We conceptualize mathematical play within and beyond its typical contexts — across a spectrum of higher education and PK-12 classroom professionals.
In our work as mathematics professionals for the Math Teachers’ Circle of Middle Tennessee (MTCMidTN)¹, we are deeply connected to engaging in mathematical play. As teachers and teacher educators, our hope for the MTCMidTN is to engage participants as mathematicians who both enjoy mathematical exploration and who learn from and with one another. We identify as boundary spanners: We are professors, teachers, and university students who engage in doing mathematics together, hoping to disrupt typical notions of what it means to engage in mathematics across and between higher education and PK-12 contexts.
The MTCMidTN was designed as a collaboration between in-service teachers who were graduates of the Vanderbilt University M.Ed. Programs in Secondary Education and M.Ed. in Teaching & Learning in Urban Settings, along with mathematics teacher education faculty from Vanderbilt University and Middle Tennessee State University. The first author is an assistant professor of math education and co-director of secondary education at Vanderbilt; the second author is an assistant professor of mathematics at Middle Tennessee State; the third taught high school mathematics for eight years and is now a doctoral student; the fourth taught middle school mathematics for eight years and was a numeracy coach during the 2020–2021 school year, when this article was written.
We center MTCMidTN around mathematical play, aiming to engage in complex mathematics for the sake of joy. We specifically use the term “mathematical play” to describe engaging in math exploration based on enjoyment and curiosity, as driven by doing mathematics. Mathematical play also implies that, against the typical norm, the purpose of engaging with a given math idea is intentionally not answer-oriented (see, e.g., Dunleavy, 2018). Instead, we are curious to hear from one another about creative ways to solve problems.
In this article, we examine our specific boundary-spanning learning context, in order to illustrate the potential ways that mathematics professionals can learn in and through practice, with and from one another (Hundley et al., 2018; Lambert, 2010). We also seek to illustrate how different kinds of boundary-spanning experiences can nurture the growth and development of mathematics professionals.
Across our work, we engage in mathematical play in a way that intentionally weaves mathematical pedagogy and practice, so as to support the ongoing development of who we each are as mathematicians and how we advance our own mathematical thinking, alongside the mathematical thinking of those we teach and with whom we interact. As mathematics teacher educators, we bring a situated view of learning (Lave & Wenger, 1991), such that professional learners develop through changing participation while engaging in our community. Due to the lack of joy and narrow success in mathematics nationally, we also center research on equitable teaching and learning (e.g., Aguirre et al., 2013; Cohen & Lotan, 2014; Dunleavy, 2018).
One key role for us as boundary spanners involves mediating the power structures of the participants, so that not all of the knowledge is perceived to be with the people who either have the most experience or the highest status. We work to establish a community where the developing expertise is distributed and valued across participants from different backgrounds and who bring different experiences. Fundamentally, our work involves unpacking who mathematics professionals are, where boundary-spanning mathematical play takes place, and what kinds of opportunities for mathematical play are bridged between individuals.
Design Principles for Boundary-Spanning Mathematical Play
Acknowledging the daily pressure and stresses that often prevent teachers’ enjoyment of mathematics in and of itself, we have conceived of the MTCMidTN as focused on three design principles that shape how we engage in mathematical play together:
- First, we are driven by the notion that mathematics teachers from different professional backgrounds have something to learn from one another.
- Second, MTCMidTN must exist outside university coursework and required professional development meetings.
- Third, mathematical play intentionally bridges coursework and traditional fieldwork.
In the following section, we unpack each of these design principles.
Design Principle #1: Who.
Boundary-spanning mathematical play includes professionals from higher education and PK-12.
One of the key features of a Math Teachers’ Circle, as defined by the national Math Teachers’ Circle Network (2019), is that we are made of professionals from different professional spheres. Powerful learning happens in the classroom in differentiated groups, when folks come from different backgrounds and have different kinds of previous experiences (Cohen & Lotan, 2014). This idea is applied to our intentional, random, heterogeneous grouping when we engage in work together. A heterogeneous group of professionals affords opportunities for individuals from different professional homes to learn from one another’s diverse experiences in the same way that students learn from one another, addressing status hierarchies that exist across the spheres of our professional work (Aguirre et al., 2013; Cohen & Lotan, 2014; Dunleavy, 2018). We have found that mathematical play across professional contexts is critical to our foundational design.
We have also witnessed how our diversity of expertise results in the most powerful kinds of opportunities for mathematical play. Across four years of working together, we have engaged over 175 individuals, spanning professional identities across university faculty, university students, school teachers, administrators, and district leaders.
One example of the power of boundary spanning across professionals happened during the first meeting of our second year. We passed out the paper-folding puzzle below as a warm-up:
The warm-up was a puzzle: Participants were invited to cut out the figure along the outer border, so that the squares remained connected, and to fold the paper such that the pictured eight letters ended up in alphabetical order on top of one another. During this time, participants asked questions about whether one could cut in between the letters in order to complete the task. When we replied no, that cuts in between letters were not allowed, two professors — a mathematics professor and a mathematics education professor — insisted that some cuts in between letters needed to be made in order to complete the task. Simultaneously, two middle school teachers and a teacher candidate solved the task without the requested “in-between cuts” that the professors insisted were necessary. After this MTCMidTN meeting, one of the mathematics professors reported walking around with the paper in his pocket for the next week, working to solve it every so often without being successful.
What we find particularly powerful about this example is that whereas mathematics professors might typically be perceived to be as or more capable to complete a given math task than middle or high school teachers, this task flipped that potentially predicted outcome. The fact that MTCMidTN was able to offer a task requiring mathematical thinking that featured middle school teachers’ success alongside mathematics professors’ struggles illustrates our potential for distributing mathematical knowledge across professionals. In this context, the power of who can do interesting and challenging mathematics has been expanded.
Design Principle #2: Where.
Boundary-spanning mathematical play exists outside of required coursework and professional development meetings.
MTCs bring together professionals across the spectrum of PK-12 and higher education mathematics contexts. We designed the MTCMidTN specifically to offer opportunities for participants to capitalize on math play that can happen outside of traditional higher education classrooms and professional development settings. Pragmatically, the varied professionals who participate in MTCMidTN work in contexts with different hours of operation. Philosophically, learning within a new structural space allows for new norms that characterize our respective boundary-spanning learning contexts and break down typical power dynamics, thus supporting learning through distributed expertise. We attribute a part of the ongoing success of the MTCMidTN to supporting the development of a positive identity for these communities.
By intentionally scheduling MTCs outside of required coursework and professional development, we place agency front and center. Part of the power of enacting agency is participation based on curiosity and choice rather than on professional responsibility, capitalizing on (and hoping for!) the potential for an individual’s positive association with and joy in mathematics. Reaching out to teachers across our local school districts and alumni networks, our per-event attendance has ranged from 16 to 45 people. Across the last 4 years we have had a typical attendance of around 25, about a third of whom attend regularly. Our participation, particularly because it includes people from across the spectrum, illustrates that professionals enjoy coming together to engage in math play.
Design Principle #3: What.
Boundary-spanning mathematical play intentionally bridges what happens in higher education math with what happens in school math.
Many times, math teachers do not have significant opportunities to discuss math outside of either teaching or engaging in a mandated course or professional development time (Campbell & Dunleavy, 2016). This lack of conversation outside of professional environments implies a lack of opportunities to build the connective tissue between what happens in higher education coursework and what happens in teaching and learning. Our boundary spanning exists in the way that we use our contexts to build bridges between what happens in universities and what happens in PK-12 classrooms. We promote this bridge both because we value the varying expertise that exists in these separate spaces, and because we know that bringing the expertise of these professionals together can only strengthen respect between and across these spaces.
During each meeting, we support participants to work in heterogeneous, randomly assigned groups on interesting mathematics problems. Groups engage in low-floor, high-ceiling problems that have an easy way to get started, offer multiple solution paths, and are often still yet unsolved by the time we wrap up. We act as boundary spanners for how professors, mathematics teachers, teacher candidates, and undergraduate and graduate students engage in math problems together.
Our hope is to enjoy mathematical problem solving while engaging in discussions about how to approach and how to think about mathematics in multiple ways. We aim to use mathematical play to bridge math knowledge with pedagogical content knowledge. Further, we assert that new contexts like MTCMidTN are needed, in order to bring varied professionals together in productive conversation. In other words, we use mathematical play to mediate the theory upon which university work is based with the practical knowledge of teaching, upon which classroom work is based.
The MTCMidTN addresses the development of mathematics professionals’ capacity to engage in mathematical play.
The work of the MTCMidTN invites participants to engage in mathematics in the ways that we would envision wanting our students and ourselves to engage in mathematical play. By simultaneously engaging faculty, teachers, teacher candidates, and undergraduate and graduate students in the same mathematics task, we intentionally work to engage in boundary-spanning work that redefines how, where, and with whom mathematical play can happen. Further, because our participants are randomly assigned to work with and meet new colleagues during each visit, there is great potential to bridge professional learning across mathematics professionals with different backgrounds. This heterogeneous, ever-evolving, and growing group affords us the opportunity to build a community that develops care, respect, and a desire to learn about mathematics from new perspectives that might not otherwise have been considered.
Growing our Capacity for Boundary-Spanning Learning Opportunities
Consistent with strong school-university partnerships, we utilize boundary spanning in order to engage “who, where, and what” mathematical play looks like.
Our commitment across university and school contexts pushes us to conceptualize opportunities for mathematical play that exist within and beyond the spaces where coursework and fieldwork typically happen. Specifically, we have sought to design our context so that we focus on supporting mathematical play such that it exists:
- across the spectrum of mathematics professionals,
- outside of traditional coursework and PD, developing capacity for content expertise alongside collaboration and innovative problem solving, and
- to bridge university mathematics with school mathematics.
As we considered these design principles, we created the MTCMidTN such that educators have opportunities to learn alongside professionals across the educational spectrum and to increase their capacity for content expertise through collaborative and innovative problem solving, as well as to disrupt traditional status hierarchies for who is perceived as engaging in meaningful mathematics. As such, we intentionally work to weave pedagogy and practice in order to support a professional community committed to lifelong mathematical play. Boundary-spanning learning and teaching requires the simultaneous integration of both knowing and doing. We believe such learning contexts support and bind together the development of professional knowledge and skill, contributing to our success as a community.
Acknowledgments
The first author would like to thank Jeanne Peter and Amy Palmeri, who participated in early conversations about the concepts developed in this work.
About the Authors
Teresa K. Dunleavy is Assistant Professor of the Practice of Mathematics Education at Vanderbilt University. Jennifer N. Lovett is Assistant Professor of Mathematics at Middle Tennessee State University. Elizabeth K. Metts is a doctoral student at Vanderbilt University. At the time of writing this article, Shannon N. Reider was a numeracy coach at Metro Nashville Public Schools. All are co-founders of the Middle Tennessee Math Teachers’ Circle (MTCMidTN), located in Nashville, TN.
Footnote
[1] When we refer to the MTCMidTN participants as mathematics professionals, we mean that we are a cross-section of professionals that include university faculty from mathematics education and mathematics departments, in-service teachers, mathematics coaches, teacher leaders, and administration, and university students who identify either as future teachers or future teacher educators (including B.S., M.Ed., Ed.D., Ph.D., and postdoctoral researchers).References
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