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"Some students ask me, 'Are we really doing math? Where are the equations?' That’s the point. It's to give a different perspective of what math is."

Maggie Regan, Duke Assistant Research Professor of Mathematics

The exercise is more than a rope trick, explained Regan, who is an assistant research professor in the math department. Each twist and turn in the ropes represents a calculation with positive or negative fractions. The idea is to come up with a sequence of moves that will untangle the mess and get them back to zero where they started.

The students were part of a “,” an informal math club where students explore open-ended puzzles and riddles under the guidance of practicing mathematicians.

Similar clubs have been popular in Eastern Europe for more than a century, but math circles didn’t start popping up in the U.S. until the 1990s. Today, there are more than 300 such groups nationwide. They take different forms, but all of them have a common aim: to help people experience math as mathematicians do.

There are no ready-made formulas to memorize. No grades, tests or homework. Rather, the focus is on imagining, spotting patterns, persevering. Exploring ideas and asking difficult questions.

The meets once a week during the school day. Each session, the students think their way through different problems. One time, the kids had to color maps using as few colors as possible, and so that no two neighboring countries or states were the same color. Another time they had to place pretend “sheep” on a game board so there was no straight path for a wolf to get to them and eat them.

It’s common for students to say they don’t feel like they’re doing math.

“Which is kind of the point,” said Regan, who runs the program with help from other volunteer instructors -- faculty, postdocs and graduate students -- from the department. “It's to try and change the view of what it means to do math, and who is a mathematician.”

“And it's just fun.”

The sixth-graders playing with ropes were abuzz as everyone talked simultaneously, coming up with ideas and trying out different strategies.

“That was weird,” one student said, stumped by a set of ropes failing to straighten out after multiple moves.

“Maybe if you can’t undo it by yourself, you can just use scissors,” another boy joked.

“Okay, so that doesn't work,” said Nelson, an assistant research professor of mathematics. “Let’s try another option.”

The students paused, took stock, tried a different approach. Suddenly there was a yell from the back of the room, then a chorus of whoops and claps as a team found a solution.

The hour-long sessions are focused on two age groups: one is for fifth- and sixth-graders, and the other is for second-graders.

“Especially for younger students, a lot of it is about strategy, problem-solving, and seeing how many solutions they can come up with,” Regan said.

One Monday morning last November, small groups of second-graders sat in circles on the floor of the school gym, contemplating a row of upside-down paper cups.

“Are you ready to play a game?” Regan asked.

They had to figure out how to get the cups in a single stack without breaking the rules. The exercise is related to an area of math called combinatorics, Regan said.

The students took turns shouting out ideas and trying them out.

“I have a good one,” said a 7-year-old in a pom-pom hat as she grabbed a stack of cups and jumped it two spaces to the left.

Another student piped up: “What if you try the other direction?”

Duke assistant research professor and graduate students and coached them through their options.

“Do you guys think that’s all the ways you can go?” Peart said.

“I have an idea,” said a girl in polka dots. “What if you moved that one to this one?”

“I have a new challenge for you,” Regan said. “We’re going to add one more cup and see how it goes. Do you see a solution?”

“This is hard,” someone grumbled.

In a flash of insight, one of the students figured out a way to stack all five cups on top of each other in a single tower. “She got it!” her team yelled.

“I want a high five from everybody,” Regan said. “Good job today.”