On Saturday, April 14, more than 100 volunteers in matching T-shirts converged on the copper-domed kiosk of the Smithsonian Institution’s Ripley Center. They brought squares of colored paper with them into the bowels of the building, along with bags of macaroni and pinto beans. They set up tables in the exhibit concourse and hung hand-written signs above them. “Instant Insanity,” read one.

For all that phrase’s menace, the volunteers had not come to the nation’s capital to drive residents or tourists crazy. They traveled from all across the United States to stage a Julia Robinson Mathematics Festival to encourage visitors of all ages to engage with, delight in, and be challenged by mathematics.

In one of the classrooms opening off the main concourse, David W. Brown of San Diego used “operations” on pairs of hands to introduce tweens to a mathematical object called the Klein Four Group.

Across the hall, Olga Radko (UCLA) explored with elementary and middle school students the ramifications of measuring distance not as the crow flies, but as the taxi drives.

As Brown coaxed his audience to “liberate” themselves “from thinking about math as being just about numbers,” Radko walked hers to the realization that, in “taxicab geometry,” circles are square.

Circles of a different type turn out to play a big role in events like the April 14 festival. Brown and Radko and most of the other math enthusiasts on hand belong to the National Association of math Circles (NAMC). Inspired by an Eastern European method of teaching math through the discussion of problems, math circles are enrichment programs that put mathematicians and mathematical scientists in direct contact with precollege students. Together they prepare for mathematics competitions or investigate interesting mathematical topics neglected by school curricula or learn math through games and hands-on activities.

“One thing all math circles have in common,” explains the NAMC Wiki, “is that the students enjoy learning mathematics, and the circle gives them a social context in which to do so.”

Saturday’s festival was part of the three-day Circles on the Road 2012 workshop that brought math circle directors and instructors from across the country together to share strategies and success stories. Sandwiched between two days of panel discussions and activity demonstrations, the festival gave attendees the opportunity to put discussion about outreach into practice.

Twice during the day, Bill Mueller of the Massachusetts Math Circle got a crowd of elementary students excited about ideas usually limited to a computer programming class. Mueller enlisted “robot” assistants to order five volunteers from shortest to tallest, first using the pairwise comparison and swapping operations of bubble sort and then switching to the more robot-intensive parallel sort. Audience participation never flagged.

“You’re going to jump out of your pants if I don’t call on you,” Mueller said to a boy so eager to evaluate the relative efficiency of sorting algorithms that he couldn’t sit still.

Julia Robinson Mathematics Festivals exist to generate such enthusiasm. “Our hope,” wrote Dave Auckly of the Mathematical Sciences Research Institute (MSRI)—one of the event’s sponsors [and the Festival series is a program of MSRI]—in an email, “is that many of the participants will look for other mathematical opportunities after the festival.”

The first festival was held in the San Francisco Bay area in March 2007. Hosted by Google, the festival grew out of software consultant Nancy Blachman’s fond memories of a high school math contest. Blachman, with her husband, David desJardins, and Palo Alto math teacher Joshua Zucker, organized a day of puzzles, games, and collaborative problem-solving that emphasized fun and cooperation.

“We envisioned having so many problems that not even a mathematical genius could solve all of them during the festival,” Blachman wrote in an online history of the festivals. Organizers “hoped that each attendee would be able to find something rewarding at his or her level.”

Inclusiveness lies at the heart of the festival; its namesake suffered discrimination because of her gender. Although Julia Robinson was every bit the mathematical equal to her husband Raphael Robinson—it was Julia who contributed to the solution of Hilbert’s tenth problem, not Raphael—the University of California, Berkeley, awarded him a professorship while leaving her a lowly instructor. Nonetheless, Julia became the first female mathematician elected to the National Academy of Sciences in 1975.

Festival developer Blachman credits the “feeling that the event would encourage girls and minorities—groups underrepresented in the company’s workforce—to go further in mathematics” as part of Google’s motivation for backing her effort.

Since that inaugural event in 2007, more than two dozen festivals have been held across the country, from Arizona and Wyoming to North Carolina and Virginia. An estimated 350 visitors attended the festival at the Smithsonian Institution. The Fairfax Math Circle was out in force, its students and chaperones distinctive in turquoise “The Essence of Mathematics Is Its Freedom” T-shirts. Virginia lawyer Neena Scaria, who showed up at the Ripley Center with her husband and young son, said she learned of the festival from the buzz about it on a Fairfax County Association for the Gifted blog.

Other attendees, though, stumbled across the event by accident.

“What are you guys doing here?” asked a woman who had wandered from the crowded National Mall into the relative calm and quiet of the Ripley Center.

“Math!” answered algebraist Monika Kiss (Saint Leo University) from behind the registration table.

The walk-in looked doubtful, but she drifted toward the origami station nonetheless.

In a festival so free form and unscripted, one activity stood out: the math wrangle. Another Eastern European import, a math wrangle pits two teams of three students against each other in a sort of mathematical debate. Teams take turns challenging each other to present solutions to difficult problems drawn from a preset pool they have had time to work on in advance. Explanations are open to critique from the opposing team, which can earn points by noting flaws or omissions or by indicating a way to increase a solution’s elegance.

Steve Dunbar, director of the MAA’s American Mathematics Competitions, hopes to see math wrangles catch on. “They have a place anywhere people ought to be getting up and talking about mathematics,” he said on Saturday as the first of the day’s two wrangles got under way.

And what a wrangle! The two teams, both representing the Fairfax Math Circle, did their coach, Bob Sachs, proud, showing off their ability to both do and articulate mathematics in what Dunbar deemed the “smoothest-running wrangle” he had ever witnessed.

Before the 2011 Circles on the Road workshop, there were 80 math circles registered with the NAMC; by the following fall, the number had jumped to 120.

“We hope to see similar growth when we look at the list in the fall of 2012,” Auckly said.

Katharine Merow is a Washington, D.C., writer.