Circles with No Center
On the Benefits of Circles with No Facilitators
By Nick Grener, Fred Peck, and Matt Roscoe
By virtue of the fact that you are reading this, we assume that you hold a positive opinion of the phenomenon known as math circles. It is likely that you have experienced a math circle, either as a participant or perhaps as a facilitator. Furthermore, you probably enjoyed the experience and found it enriching, rewarding, or just plain fun.
Certainly, though, not every circle session generates the same experience for each participant and facilitator. For example, have you ever found yourself generating an interesting question about the topic at hand, but then feeling frustrated that you couldn’t pursue it because the facilitator is trying to direct the room in a different direction? Or experiencing stress while preparing to facilitate a circle where you did not know the audience well and were nervous about appearing sufficiently competent? Or even having to cancel a calendared math circle session for want of a willing facilitator?
In the years since we began to promote math circles in Montana for students, educators, and the general public, we have experienced all of these situations. Recently we have found a way to address some of these concerns that has fundamentally changed the nature of our gatherings. While this change is still playing out, we are excited to share here our new methodology, our observations on how we believe our Math Teachers’ Circle has improved as a result, and some resources that might help you implement it at your own math circle.
More specifically, over the years, our gatherings have evolved to adhere to a new organizing principle in which there is no facilitator or moderator, by which we mean that there is no “teacher,” “expert,” or “person who knows the answer and how to get it.” Instead, each month, a different member of our math circle community volunteers to bring a problem, question, or general setting that is mathematical in nature for the group to collectively consider. We call those who bring the problem “originators” of the session.
Importantly, for problems where an answer is pursued, the originator does not know the solution (or, at least, the full solution) to the problem. Oftentimes, the originator will steer away from the presentation of a specific question or set of questions altogether, and leave the question-generating aspect of the experience completely up to whomever shows up to the session. In any case, after the initial sharing of their seed idea, the originator steps back from the foreground and participates as an equal in the session with everybody else in the room.
The result looks something like this:
Perhaps the primary fear of running a math circle without a person at the “center” is that it will be too unfocused and will devolve into chaos, or — worse — disinterest. Our experience in Missoula, however, is that engagement is uniformly good, which we attribute to the fact that everyone in attendance is tacitly agreeing to be accountable for how the experience will play out. In other words, rather than one person organizing and structuring — and therefore shouldering all of the responsibility for — the gathering, the participants collectively take responsibility for engaging in sustained mathematical activity.
It also seems to us that the energy in the room is especially high in those cases where participants are invited to own the direction that the session takes by forming the questions that will be worked on. For example, a board game might be presented with the simple prompt of, “What math questions does this game make you think of?”, coupled with the freedom to actually explore the questions generated by the group.
Here are a few other advantages of unfacilitated math circles that we have experienced:
- They are stress-free to organize. It is far less imposing to originate a session than to facilitate a session, since the former merely requires sharing something of interest to you, whereas the latter imposes the expectations that you have thorough knowledge of the content of the session and keep the room directed during the meeting itself.
- They are more fun for the originator, since the originator gets to participate in the mathematical activity of the group, rather than having to orchestrate the activity. In fact, the circle is more exciting for all involved, since each discovery made by the group is an “aha” moment for every single person in the room.
- There is no clear ending to the session. In fact, our discussions sometimes continue well beyond the physical gathering as we continue to explore and make new discoveries and share them with one another via email. This, perhaps, is the single most important distinction from a traditional session where the facilitator feels pressure to direct the mathematical activity toward some predetermined end point, thereby discouraging the opportunity for further study.
So, how might you experiment with this method at your own circle? Here are some ways originators have recently begun meetings of our math circle that you might consider trying:
- Sharing a low-floor problem that (to the originator’s knowledge) has not yet been solved. A solid library of accessible unsolved problems from the history of mathematics can be found at MathPickle.
- Bringing an object or activity that can be mathematized. We have used games like Tenzi, Settlers of Catan, and Gobblet Gobblers to generate rich discussions. We have made and explored mathematical artwork including Celtic Knots and Spirographs. Finally, we have investigated interesting mathematical objects, such as 3D-printed Menger Sponges and sets of TANGLE toys.
- Providing participant handouts for one of the sessions in the Math Teachers’ Circle Network library. The session should look intriguing to the originator, but it should be something that they have not spent significant time working on themselves prior to participating in the circle. Often, the handouts themselves are quite structured. In this case, the originator might remove much of the structure while preserving the central question.
- Bringing a question that the originator is curious about. For example, in a recent gathering, a member of our group explained that they had, for a long time, been perplexed by the geometry of the formula for the tetrahedral numbers. The algebraic formula for the nth tetrahedral number suggests that 6 copies of the nth tetrahedron can be arranged to form an n × (n+1) × (n+2) rectangular prism. However, the originator could not figure out how to make such an arrangement, so they brought the question to our group. Originators have also brought problems that they saw on social media and found intriguing. James Tanton’s Twitter feed is a great source for these.
To be sure, the transition to a circle without a facilitator may take some adjustment, particularly if your sessions typically have a large number of folks attending. But if these ideas interest you, we encourage you to discuss them, give it a try, and see what happens. You might just find yourself renaming your “Math Circle” (the set of all points equidistant from a fixed center/leader) to a “Math…” — wait, what do you call a set of n points that are all equidistant from each other?
About the Authors
Nick Grener is a mathematics teacher at Hellgate High School in Missoula, Montana. Fred Peck and Matt Roscoe are mathematics professors at the University of Montana. All are leaders of the Montana Math Teachers’ Circle.
