Story and photos by DENISE JARRETT

Jackson Middle School students (clockwise from left: Miranda White, Jake Purcell, Banah Graf, Annie New) strike a pose that teaches them about geometry, movement, and teamwork.

y keenly confronting the enigmas that surround us … I ended up in the domain of mathematics. I often seem to have more in common with mathematicians than with my fellow artists.

-Maurits Cornelis Escher

PORTLAND, Oregon-

She's a head taller than he is. They eye each other dubiously, then turn back-to-back. The boy plants his feet firmly as the girl leans into him, her shoulder blades pressing into his. Trusting their weight to each other, they strike a counterbalance and sink into a deep knee-bend.

Dancer Keith Goodman taps out an accompanying tempo with a clave, then pauses as mathematics teacher Michael Lang steps forward. Lang asks the other assembled eighth-graders, "When Melinda and Terrance were standing back-to-back, what kind of symmetry was that?"

A reflection!

"Okay. And when they lowered to the floor, what kind of transformation occurred?"

A translation!

Lang asks them to explain. A translation is when an object is moved to a new position without changing its orientation. He probes further: So, what would a glide reflection be? A glide reflection combines a reflection with a translation in which only one element slides, or glides, to a new position. Lang nods in agreement.

This is math class, but unlike any these students have experienced before. They are midway through a three-week unit integrating geometry and art. Right now, they're learning how choreographers incorporate tessellations and symmetry into dance. Goodman directs the students to break into groups. The youngsters, a combined mathematics class of about 60 students at Jackson Middle School in Portland, Oregon, disperse across the auditorium floor in boisterous but intent clusters.

As a choreographer and dancer with Conduit, a Portland contemporary dance studio, Goodman often incorporates tessellations-arrangements of shapes into symmetrical and repeating patterns- into his works. His dances influenced by the African Diaspora, for instance, resonate with images, patterns, and rhythms found in the dances of Africans who were displaced around the world by the slave trade. But it wasn't until Goodman began coteaching with mathematics teachers Lang and Ken Reiner in this geometry project that he recognized the extent to which he relies on mathematics when choreographing. He's eager to share this revelation with these students.

"Okay," Goodman calls above the din. "You're going to create a movement tessellation that lasts from eight to 24 counts that you mirror, or reflect, on either side of a line. Now, at some point in that movement, you're going to rotate one of the sides 180 degrees, keeping the movement going, then you're going to do a glide reflection of it."

The students begin striking poses, mirroring each other's gestures and measuring dance steps- exploring their creative powers at the same time that they are developing their understanding of geometric principles.

Symmetry is part of the weave of the world. It's in the inkblot patterns of butterfly wings and in crystals of snow. It underlies the beauty and function of suspension bridges and the flying buttresses of cathedrals. It is apparent in computer programming and DNA sequences. From artists and musicians to architects and mathematicians, people devise patterns and symmetries to express human experience and to reshape environments. The language of mathematics governs these enterprises, and mastering this language can powerfully affirm one's place in the society of human beings.

For teachers like Ken Reiner, 50, and Michael Lang, 43, empowering young people to take their place in society is a potent motivation. They teach eighth-grade mathematics to students of varying academic abilities. Both men are highly effective teachers. They keep abreast of mathematics standards and best practices. They design active and challenging lessons, and question with skill and enthusiasm. And yet, they worry that they aren't reaching every student.

An opportunity to try something new arose when Jackson Middle School received a Bernstein Education Through the Arts (BETA) grant from the Leonard Bernstein Center. The center awarded the school $30,000 a year for three years to integrate arts into its core curriculum. Lang and Reiner subsequently designed a geometry unit on tessellations involving mathematics, visual art, and dance. Neither had done this kind of thing before, and planning the new unit took a considerable amount of their time-about 20 hours stolen from the school day, afternoons, and weekends. Nevertheless, they shared an emerging conviction that art could be a key to mathematical understanding for many students.

For Reiner, watching his students work with a choreographer on their movement tessellations was a memorable moment in his career as a mathematics teacher.

"Keith [Goodman] asked the kids to come up with some movement pattern using certain mathematical concepts, and every one of those 16 groups of students immediately began working," Reiner says. "I was amazed. I never thought eighth-grade students would be so uninhibited and would come up with such concise and amazing movement patterns."

Anna Hermann's regular tessellation features fish and quadrilaterals.

Banah Graf arranged turtle shapes into repetitive patterns in this regular tessellation.

The teachers' primary goals for the project were that students would understand the geometry of polygons-angle measurements, regular and semiregular polygons, and how polygons tessellate-and that students would make a connection between these mathematical concepts and the real world.

Reiner and Lang introduced students to the masterworks of the Dutch graphic artist Maurits Cornelis Escher and the American dance company Pilobolus, based in rural Connecticut but known to audiences worldwide. These artists create geometric patterns in highly innovative and dynamic ways. Students examined Escher's precise and fantastic worlds, delighting his engravings of morphing triangles, lizard mosaics, and staircases that trick the eye. They were intrigued by the ways in which the artist plays with natural phenomena -gravity, optics, and spatial dimensions.

They watched on videotape how Pilobolus undertakes similar explorations in dance. The leotard-clad dancers create geometric shapes and patterns with their bodies, joining hands and feet, or entwining in improbable clusters of torsos and appendages, only to metamorphose into new forms and patterns.

Early in the project, students completed worksheets on lines of symmetry and practiced arranging polygons into regular and semiregular tessellations. They drew their own tessellations by combining polygons into imaginative and colorful images-such as goldfish, foxes, and sunbursts-and arranging them into repetitive patterns. When Goodman, an accomplished professional dancer and seasoned instructor at Buckman Elementary in Portland, agreed to collaborate with Lang and Reiner on their project, he sat in on several classroom activities to familiarize himself with the concepts students were learning. He then began meeting with students in the school's auditorium.

Goodman captured the students' attention and imagination, and they worked with high energy and focus. He taught them to choreograph movement tessellations-brief dances or drill routines that combine the geometric principles students had been studying in the classroom. The auditorium echoed with young voices quickly mastering and mingling the two vocabularies of dance and mathematics.

"The arts are an integral part of mathematics, especially in the field of geometry," says Reiner. "If you don't relate mathematics to the real world, to such things as dance, you're doing a disservice to students. There is so much that you can get out of studying the arts that will help you to understand mathematics, it's incredible."

Lang agrees: "By bringing in the arts, students see that mathematics is not just present when they're sitting in their chairs in the classroom. It's all around-schools of fish, patterns in the sky, the flow of traffic. It all has a mathematical pattern."

The two men arrived at this conviction by way of different life experiences. Art has played a steady role in Lang's life. As a child he learned piano, and now he plays guitar every night for his own children. But it wasn't until he saw an exhibit of Escher prints at the National Gallery of Art in Washington, D.C., that he began to perceive how art might give rise to "teachable moments" in mathematics. The prints depicted mathematically conceived images based on geometry and symmetry. Lang could see how students, likely to be intrigued by the fanciful images, might also be drawn to the mathematical concepts the artist employed in his craft.

"That was a pivotal moment for me," he remembers. It led him to think about other areas of art that might facilitate students' understanding of mathematics, such as weaving, music, and dance.

For Reiner, athletics have been at the center of his creative life. Basketball and baseball as a youth, and now golf, have taught him to develop his own physical skills and also perform in front of spectators. He also knows what it is to be a good team player. In this instance, he listened to his "teammate," Lang, when he suggested that they collaborate on an interdisciplinary unit in which students would learn the concept of tessellating polygons by choreographing their own dances.

"I said, 'You want to do what?'" recalls Reiner with a laugh. "It was so far from traditional teaching-the dance part was just not me! But the more we talked about it, I began to think, why not? Middle school students need movement, and this seemed just perfect."

Nevertheless, Lang and Reiner experienced some anxiety about doing the unit. They worried about the extra planning time it would require. They worried about preparing students to meet the state's benchmarks in mathematics. And they worried what parents would think. To ensure that students would gain important mathematical knowledge, the teachers incorporated concepts and activities recommended by the Oregon state standards for mathematics and the National Council of Teachers of Mathematics.

"I had to justify it for myself by going to NCTM and the state standards and making sure that there were direct links between this project and the standards," says Lang. "You're always worried about the community's reaction to what you do in the classroom. I was worried that parents would think I was supposed to be teaching 'math,' not dance."

The teachers also had to overcome a mild case of stage fright. Teachers involved in Bernstein-funded projects receive training in how to integrate arts into other disciplines. One training activity required Lang and Reiner and other teachers to perform a skit for each other. Although they stand before an audience of students every school day, the teachers found performing in front of their colleagues to be unnerving. But they summoned their courage and, in the end, the skit turned out to be quite a romp, with extravagant costumes borrowed from the Portland Opera. For a day, the adults played like children.

"We really became middle school kids. And now we have a lot of empathy for the students, because it's hard to be up in front of your peers," says Reiner.

But for their young students, performing for peers was only one challenge. Facing the bewildering and timeworn trials of adolescence presented a host of others. Boys and girls had to touch and trust each other as dance partners. They had to muster their still-developing powers of coordination and rhythm. And the choreography required them to think both concretely and abstractly. The adults stood by as the youngsters persevered with surprising maturity.

Forming movement tessellations with their own bodies helped many students to grasp the differences between a regular tessellation and a semiregular tessellation. It also helped them to understand how to accomplish symmetry in patterns by rotating, reflecting, translating, and glide reflecting. The project earned high marks, not only from students but also from parents who remarked on their children's surging enthusiasm for math class.

"All my anxieties were worthless!" says Lang, "It worked out really well."

It is now several weeks after the project, and students sit on the floor, reflecting on the experience. They're eager to share their thoughts, and their words tumble over each other's in an ebullient rush.

"It was easier translating a glide reflection in the dance. I never understood it when we were just moving shapes around on paper."

"I had to look at it on paper first before I could do it in the dance."

"I didn't like math, it was boring, and I didn't want to come to class and do nothing. Now, I kind of got into it."

"Dance is filled with mathematics, you just have to find it."

In addition to learning mathematics, it's evident that students also learned a little more about themselves.

"I learn better in groups, where you can learn from each other."

"Some people have individual talents, and they can bring that to the group."

"We had to cooperate. If both people weren't cooperating in a counterbalance, you'd fall over."

So, what role can art play in mathematics?

A quiet voice answers from the back: "It cleans the mind and the soul."

The students laugh at their classmate's poetics, but they nod, eyes bright.

Denise Jarrett is a writer with NWREL's Mathematics and Science Education Center.

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