#### by Doug O’Roark

Executive Director, Math Circles of Chicago

The New York *Times* recently published an article entitled “The Right Answer? 8,186,699,633,530,061 (An Abacus Makes It Look Almost Easy)”. Its lead photograph features over 100 children seated at desks, facing forward, working individually. This is yet another in a long series of public relations disasters for mathematics. This depiction of mathematics is nothing new, and I most suspect readers experienced no cognitive dissonance in seeing mathematics represented this way.

*Traditional Forms of Math Enrichment — and the Problem with Contests*

Mathematicians collaborate to explore exciting open-ended questions. Unfortunately, this may be the world’s best kept secret. The problem? We have two major gateways to participation in the community of mathematics: the classroom and the contest. The classroom, of course, is universally familiar, and innumerable efforts have been made to improve the student experience in the classroom.

Here I focus on the second gateway, the contest. When I say ‘contest’ I refer to the kind that looks like that Abacus competition pictured in the newspaper. These are quite common. For the majority of children competing for the first time, contests have a number of common features:

- A large group of students gather in one location. They do not talk to each other.
- Students complete closed-ended problems with unique correct answers, which is necessary so that responses can be judged impartially.
- Contest problems are generally predictable. This allows students to memorize formulas and tricks to save time during the contest. Frequently students prepare for contests through practice problems that turn contest
*problems*into*exercises*—known results that can be completed efficiently to save time during the contest, preserving time for checking answers, since partial credit for thinking is not given. Speed is prized. - At the end, four classes of prizes are awarded: 1st place, 2nd place, 3rd place, and no place — with this last group the mode by a wide margin. One can argue that this is a feature of any competition, but the ramifications of competition in a subject as fundamental as mathematics are quite serious. Giving up on math is not same thing as giving up on chess or basketball.

Again, there are many exceptions to this general description of contests, but those exceptions are usually experienced by those who have previously been successful in close-ended contests. One can argue students can get exposed to beautiful problems that they themselves might extend. A colleague once told me that the best part of the contest was after papers were submitted and kids started to “sing the contest” — eagerly talking to their friends about the most interesting problems.

I’ve seen students benefit from contests. But I do think it’s time to reflect on some of the downsides of this kind of enrichment. As we seek to expand access to the world of mathematics, do we really want our main form of math enrichment to narrow the gateway?

*More Subtle Problems*

I’ve known many high school students who have done well in contests and were subsequently motivated to major in mathematics. In turn, I suspect there is a selection bias that supports math contests as the main form of math enrichment. If you didn’t do well in contests, you didn’t major in math, and therefore you haven’t stuck around later on to question the practice.

For others, contests have a negative impact on identity. As I mentioned, “no place” is the most common ranking for the large majority of contest participants. **Contests send and reinforce a fixed mindset message.** Children often compete as part of a team from their school, and schools with more resources tend to perform better. But when less-prepared children from underresourced schools compete, they may see other children achieving at a higher level, and may be led to believe that they just aren’t as good as others out in the wider world. This is particularly pernicious when it comes to underserved communities and communities of color.

Teachers are often inspired by contests, benefit from writing them, and gain personal connections with students that they might not otherwise have formed. But, as I mentioned, preparing for contests prioritizes efficiency over depth. Few contests reward alternative solutions or depth of thinking. Speed matters; hence, game theory dictates that we ought to teach procedures.

Finally, and perhaps most subtly, the world of math enrichment largely embodies a “Field of Dreams” approach: “If you build it, they will come.” My question: *who* comes? If you look at the participants in existing contests, you are more likely to get boys, children from more affluent schools, and people who have the social capital to know where to find the contests. And there is a ‘Matthew Effect’–minor advantages accrue early in life, so that by the time children are participating in contests, what seems like a fair assessment of talent is really just a piling on of advantage.

The math enrichment gateway needs to change, or the usual suspects will be the only ones making it through the gate. We will continue to lose underdeveloped talent—children with latent ability who will never reach their mathematical potential. Career choice, economic mobility, and civic engagement will continue to be unnecessarily limited for many.

*What Might We Do Instead? The Case for Math Circles*

In the last five years, the single greatest impact on my thinking about teaching has come from Alan Schoenfeld’s Teaching for Robust Understanding (TRU) Framework. TRU asserts that to create powerful math learning environments, we need to attend to five crucial dimensions: mathematical connections, cognitive demand, access, agency/identity, and formative assessment. The ongoing program of research and practice of Schoenfeld and his team is working to show that these five conditions are both necessary and sufficient for robust learning to occur, and to explore effective, efficient ways to make them happen.

TRU is being applied in classrooms and it applies equally well to math enrichment settings. Math circles, festivals, and summer camps can be designed to be equitably accessible in a way that contests simply cannot be. Students exploring open ended problems for a math symposium have an opportunity to experience agency at a level that neither the classroom and the contest rarely provide.

Let’s consider math circles. Math circles usually occur outside of regular school hours, where interested children investigate novel mathematics in sessions led by an adult with a strong affinity for math.

Like math contests, math circles can be cognitively demanding. But the phrase ‘cognitive demand’ can be deceptive. Sometimes it’s read simply as ‘hard’. But — ironically — making mathematics hard is not difficult. TRU describes cognitive demand as: “The extent to which classroom interactions create and maintain an environment of productive intellectual challenge conducive to students’ mathematical development. There is a happy medium between spoon-feeding mathematics in bite-sized pieces and having the challenges so large that students are lost at sea.”

Contests rarely provide appropriate levels of cognitive demand to a broad range of students. A math circle, with a more classroom-like environment can be designed to provide that “happy medium” for individual students. Students can work at their own pace and have a personally rewarding experience.

The advantages of a math circle become fully clear in other dimensions of the TRU framework. Because they do not center around competition, math circle sessions are more welcoming spaces for a broader range of students. We can make our sessions accessible. We can improve the likelihood that more diverse students identify with the subject, consider it to be fun and worthy of long-term pursuit. Problems can be fine-tuned to become easier or harder based on the pace at which a given student digests the new material.

And, let’s not forget the mathematics itself. According to the TRU Framework, in order for a classroom to be mathematically powerful, “The mathematics discussed is focused and coherent, …[and] connections between procedures, concepts, and contexts… are addressed and explained.” Typical contests throw disparate problems at students, where mathematical connections between problems can be non-existent. Compare that to a math circle—based, perhaps, in one main rich activity, or a chain of problems that feature a strong connections that can reveal surprising and beautiful results.

*What’s next for math circles?*

While I see the potential of math circles, I also recognize that if we do not implement them thoughtfully, we can end up reproducing many of the shortcomings of contests. The field of dreams approach can still limit who attends, and we may end up serving the same children we served before with contests. Perhaps those children will have a better experience than in competition, but the audience might still be made of the usual suspects. We need to consider where (and when) math circle meetings are held; whether we can hold them free of charge; and how we can forge partnerships with community leaders to thoughtfully recruit children in underserved communities.

Moreover, as we consider serving a more diverse group of children, we need to consider how to serve those children effectively. A math circle leader needs to know how to teach well. They should also know how to build connections with and between students, while also having a deep understanding of the mathematical connections at hand. (I admit this is a significant topic unto itself, and that teaching well is an enormous challenge—but it’s absolutely essential that it be addressed for this enrichment based in student collaboration to be successful.)

Four years ago, I became the Executive Director of the Math Circles of Chicago (MC2). We aim to provide more equitable access to high-quality math enrichment. Since I began, we’ve quadrupled in size and now serve over 800 students.

We strive to make math enrichment accessible by reducing economic hurdles, fighting barriers introduced by geographic and school segregation, attending to student identity during sessions, and promoting interpersonal connections.

In 2015 MC2 had three sites, two on Chicago’s north side (the wealthier and whiter part of the city) and one on the near south side. Four years later we have circles at eight sites that are geographically dispersed around the city. This fall we opened a new site meeting on Saturday mornings in Back of the Yards, one of Chicago’s poorest communities, with an enrollment of over 60 students.

All of these programs are free to families.

Our population of teachers has diversified as we have grown. Initially many of our leading teachers were in graduate programs at universities like UIC, UChicago, and DePaul. While we continue to add more such teachers, we now have many more classroom math teachers, both from middle schools and high schools around the city.

The inclusion of teachers, particularly those teaching middle school, means that as a community that we have much more institutional knowledge of working with younger children (we serve kids in 5th to 12th grades, and in practice more than 80% of those served are middle schoolers).

We also provide workshops for our teachers—from 2015-2018 Dolciani Math Enrichment grants funded the development and implementation of workshops like, “What is a Math Circle?”, “Math Circle Teaching Basics”, and “Intermediate Problem Solving”, along with workshops built around group observations of math circle sessions. These workshops build connections between our teachers and build an esprit de coeur.

MC2 is still very much a work in progress. We need to improve in the evaluation of the work we are doing, particularly to measure whether it’s effective in meeting the needs of the varied children we are trying to serve. It takes time to build relationships within a community, with teachers who can both lead sessions and who can help advertise this opportunity effectively.

Messaging matters, and over time we’ve found that expressing clear core values helps us find the right teachers, community members, and family members.

- Math should be fun and empowering.
- Every child can do and can enjoy rich mathematics.
- Every child deserves equitable access to rich mathematics.
- Students should be agents of their own learning.
- Math can and should be collaborative.

This stance has made us discourage the use of the word ‘gifted’. We certainly draw many well-prepared students (among others), but we are careful to signal that the math experiences we offer are for everyone.

I started by talking about the popular image of mathematics, but changing that image is not enough. The substance of the experiences we provide—Competitive or collaborative? Cooperative or individual? Mathematically connected or disconnected?—will determine whether we can attract a broader audience to the intellectual joy that we know mathematics can offer.

The mathematical community is powerful. High stakes tests make mathematics a pathway to many opportunities, in college, in careers, and in our ability to influence our communities. Many of us spend a significant amount of time writing, coaching, and judging contests. I think it’s time to question how much time is spent this way, and how, proportionally, we might spend more time involved in collaborative, cooperative, connected mathematical experiences that provide access to the many, and a deeper experience for all.

Excellent article, Doug. You have explained the math circles concept very well. I hope to see more Math Circles both in Chicago and the rest of the country.

I remember being discouraged by math contests because they seemed to be built on this idea of competition: beat everyone else to reach the top. While the original intention was to stir curiosity and a love for math in the participants, the contests ended up doing the exact opposite for the vast majority; instead of kids that want to learn more because they’re just so fascinated, we have kids that want to memorize more tactics and drill on more problems to win contests. Of course this is a generalization, but it’s basically inevitable in a competition environment, where inquiry and understanding is only secondary concerns and beating everyone else is primary.

Only by moving away from contests was I able to view math as fun, interesting, and full of exploration. I imagine many other students feel the same.