# A Foray into Math Circles

This year I’ve started facilitating math circles in Portland. For those who don’t know, a math circle is a small, extracurricular gathering of similarly-aged children to explore math topics. Usually an hour long, a math circle serves simultaneously as a means to introduce kids to math topics outside the standard curriculum, to encourage critical thinking, and as a social space for kids who like math to enjoy it without a social stigma. Math circles typically have a single adult facilitator, whose goal is to present an “accessible mystery,” and then do their best to help the kids navigate their own thoughts and questions about it.

I attempted three math circles this year—that is, three different groups of students, each with multiple sessions—and I only really consider the last one a success. This was also prompted by a short online course/discussion group run by The Global Math Circle which asked each member to run a real math circle so they could discuss it. And I read Math from Three to Seven.

My first group of kids was my next door neighbor’s 8-year-old daughter and two of her friends. Our first circle was on the introductory muffin problem from Bill Gasarch’s book Mathematical Muffin Morsels: how can you evenly split 5 muffins among 3 students so that no student gets a smaller piece than any other? it was mostly a flop. The girls didn’t find the problem very interesting, and it was simply too advanced. To wit, they quickly figured out that you can give each student a full muffin and split the other two muffins pieces of size `{2/3, 1/3}`

, so that one kid gets two 1/3-size pieces. And they agreed that you couldn’t possibly make it work if every muffin piece had to be size 1/2. But then they were unable to think of any fractions that were between 1/2 and 1/3. We tried talking about that, and while one girl was keen on it, the other two decided it was time to goof off. The second and third circles I did with that group weren’t much better, and then Summer and friend-group dynamics mixed up everyone’s schedules and preferences and it fell apart.

A second group was four 6-year old boys and girls in the neighborhood my parents live in. There was a 6-year old boy whose family lived down the street from my parents, and one day when I was walking my toddler down the street with my family, we met them and I mentioned I’m a mathematician. The boy piped up that he loved math. A few days later I brought him a 15 sliding-block number puzzle, and later a graph coloring puzzle booklet. Then his mom found some neighbor kids who wanted to try a math circle. This group was a bit more successful. We did some combinatorics puzzles (counting ways to stack 3 blue and 2 yellow blocks), and basic versions of Nim. And some activities that didn’t work out so well, like the fold-and-cut theorem—which was hard because they didn’t have the dexterity to fold and cut paper precisely enough. After four sessions, still, enough students didn’t want to return that it was just down to the original 6-year-old boy, and we couldn’t really have a math circle with one kid—part of the point is to get the kids to teach and learn from each other.

The third and most successful group consisted of 7-year-old siblings of the kids at my son’s preschool, which he started going to after the other two circles ended. We did counting puzzles, different flavors of Nim, Knights and Knaves puzzles, fold-and-cut theorem, the “no three in a line” puzzle, and more. And most of the kids were pretty interested in the topics. Some of the kids were *really* interested in the topics, as much as a 7-year-old can be, I think. And I’m hoping to start with this group again in 2023.

Facilitating math circles has been a lot of fun, despite regular issues with “classroom management” (i.e., stopping them from roughhousing or distracting each other too much from the math). I wanted to share a few reflections, aside from the typical thoughts on which topics worked and which didn’t.

The first is that it’s hard to find willing participants! If I didn’t happen to run into neighbors with willing kids and friends, or I didn’t happen to have a kid myself, I probably would have to resort to awkward things like advertising? Or contacting a local school? But even with a kid of my own, it’s hard to say, “please send your child to my house once a week so we can do math in my basement (without you in the room)!” Having even a brief prior relationship with the family of at least one kid really helped.

Second, explaining the point of our activities to the parents seemed to help motivate the parents to encourage their kids to keep coming. In my first group the parents weren’t there (and most of the activities flopped), and in the second group one or two of the parents sat in, but for the parents who didn’t, their kids seemed to peter out. Often the kids will say things like, “how is this math?” which is of course expected, but I can only imagine their failures to explain to their parents what they did (especially the younger kids). For the third group, after each circle I sent a pretty detailed email to all the parents with:

- A description of the activities we did and why they were mathematical.
- Interesting ideas that each specific kid had (if I could remember them).
- A list of extensions/tweaks they could do with their kids at home, if desired.

And the parents did continue to engage them between sessions, at least for the kids who really enjoyed it.

Third, I was continually surprised by which activities the kids enjoyed and didn’t enjoy. And as a result, I had to maintain a significant number of “backup” activities in case my choices flopped. One problem I thought would be a shoe-in was the game of Set, but it flopped immediately, and was worse because they couldn’t help but shout over each other and grab all the cards whenever they thought they saw a set.

One activity I thought would flop, but ended up being a huge hit, was Knights and Knaves. In that activity, Alice and Bob are either always-truthful (knights) or always-liars (knaves). They each say one statement, from which you have to ascertain their identities from the four options. In the first puzzle I gave them, Alice says, “Bob is a liar!” and Bob says, “Neither of us are liars.” The kids immediately thought Alice was a liar because she was accusing Bob, and they all declared she was the most “sus.” But after exploring it, they found Alice was the truth-teller. I think it also helped that I wrote Alice and Bob’s statements on physical paper signs and taped them to chopsticks so I could hold them up when I posed the problem. Physical manifestations of problems always seem to help, even in this superficial way.

Fourth, I struggled to find the right balance of admitting some fun and goofing off, while still having it be focused enough on math. I don’t want math circle to be strict. If kids are bored and goofing off, that’s a signal that we need to switch activities, and acting the taskmaster will probably ruin their attitudes for math. And sometimes their goofing off is an expression of creativity, and I can channel that to math if I’m quick on my feet. As one example, with my third group I tried a (failed) activity about standing waves in a slightly-taut rope. They were going way to wild waving the ropes around. And then one kid said something like, “look, I can send secret messages!” (by “pulsing” transverse waves up/down or left/right). I quickly pivoted to an activity where two kids held each end of a rope. One kid asked a question, and the other kid answered it only by sending waves down the rope. They were asking all counting questions, and it got interesting when they asked me “how old are you?” and I ‘lost count’ (then they invented a scheme to communicate in base-10). And then non-counting questions (“what’s your favorite color?” or “what’s your favorite food?”). That turned out to be a really neat activity, and all it took was to identify a clever thought, and push it farther so that they had to start to think critically about it.

The hard part with kids goofing off is when it’s *just one* student that is bored and the rest are really invested. Invariably, boredom manifests as distracting the other kids. And the kids who were interested would even snap back, “I’m trying to think about this puzzle!” With one egregious case I even had to ask one kid not to come back. I did this by talking to her parents and explaining that she doesn’t seem interested, describing what her distracting behavior was, and explaining that it was hurting the experience for the other kids and that making her come back would likely be worse in the long run than stopping and trying again in a year or two.

In the end, I think the most important lessons I learned about running math circles from this is to never trust your intuition about what the kids will latch on to, to bias yourself toward easier activities with a high ceiling, and to have lots of backup activities. So if you know of any fun activities that would make a good circle, please let me know! You can reply to this email or hit me up @j2kun@mathstodon.xyz.