"If Tim advances the ball four yards from the line of scrimmage, how many more yards are needed for a first down?"

"Sally has $25. Can she get herself and three kids into the movie and buy popcorn?"

It's Saturday afternoon, and all over America, adults are solving math problems like these. Meanwhile, at the University of Arizona math department, UA undergraduate and graduate students are teaching high-school kids about cryptography, relativity and the solution to Rubik's Cube.

Several times during the year, promising high-school math students — and sometimes their teachers — assemble to ponder fascinating math topics in a free workshop series "run entirely by graduate students," according to Katrina Piatek-Jiménez, who heads the program. Jiménez, a 1999 graduate of the University of California, Santa Barbara, expects to earn her doctorate from the UA in 2005 after "taking a year off" to teach high-school math.

Last year's workshop topics were cryptography, knot theory, and probability, says Piatek-Jiménez. In past years the series met on Saturdays, but the 2001 workshops were operated as weekday field trips for entire high-school classes. "That solved transportation problems that cropped up with weekend workshops run on weekends," she explains.

The cryptography workshop has featured "two employees of the National Security Agency who brought an Enigma machine — one of only about 10 still in existence — used in Nazi Germany during World War II," relates Alexander Perlis, one of the workshop instructors, who is working on his doctorate in math at the UA.

More than arithmetic
Students from any school can participate, says Perlis, adding, "many are from low-income neighborhoods."

Perlis sees two major benefits to the workshop program.

"High-school students are usually exposed only to math from 200 years ago ... 'arithmetic' ... but there are all these great ideas in modern math that ought to be presented earlier" in a student's math education, he says. Moreover, he adds, "the workshops make the grad students think about what they are doing and why they're doing it."

The hope is that the high-school student participants will enjoy the experience enough to study math in college and consider math careers. Organizers also hope high-school teachers who attend will get useful exposure to math topics and teaching strategies.

Turning people on to math
The high-school math workshops began in 1997 as a project of the Southwest Regional Institute for the Mathematical Sciences (SWRIMS), an NSF-funded program directed by UA math professor William Yslas Vélez. SWRIMS sponsors projects at the UA, Northern Arizona University and Utah State University.

At one time the high-school students were paid $35 per workshop, says Perlis, and their teachers also received what might be called a finder's fee. The high-school students are regarded as "very helpful subjects in an experiment who are helping us by coming to these workshops when they could be out doing whatever high-school students do," he says.

"Even after the experimental phase ended and the students were no longer paid to attend," he relates, "interest and participation in the workshops continued."

Activities such as the high-school workshops are among the reasons Piatek-Jiménez chose the UA math program — which she considers "unique" in that "you can do research in math education and choose to study many topics. I would like to further my study of math education," she says, "versus strict math research."

Perlis, too, enjoys the mix of research and teaching.

"I come from a family of mathematicians," says Perlis. "I've always been interested in cryptography. And while part of me would just like to do my own stuff, it feels good when you can turn people on to math."