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Winter 2008 Contents

Bill McCallum, UA Distinguished Professor of Mathematics, is known among mathematicians for his work in number theory.

The University of Arizona Alumnus / Winter 2008

MATH: Growing in Circles
How putting university math researchers, education professors, and K-12 math teachers together can help bring U.S. mathematics up to speed.

by Steve Cox
Jacob Chinn photos

Beyond the glass wall of Bill McCallum’s office, eight stories above the UA campus, a panorama of the brown Santa Catalina Mountains spreads beneath a deep blue sky. Blackboards covered with mathematical notations fill two walls, and on the fourth wall hangs a Peruvian folk picture, in cloth, of a mathematics classroom.

It’s like the set for a television series starring a fast-talking, bearded mathematician who solves dauntingly complex problems of increasing urgency for the United States and, indeed, for the world.

In fact, McCallum does sport a beard, and he is a mathematician — a distinguished professor of mathematics. He does indeed talk very fast, and enthusiastically, about his solution to a single dauntingly complex and increasingly urgent problem.

Through a new Institute for Mathematics and Education he established at the UA with the National Science Foundation support, McCallum is undertaking to bring mathematical teaching and learning in the United States up to speed.

“The attitude that says teaching is an art, some people are good at it, some not, is a coded way of not taking it seriously,” he says. “Wrong. We have to take teaching more seriously than anything else. We’re not going to get by with a few great teachers. The scale of the problem in mathematics education is such that we need lots of decent teachers.

“You can learn how to teach better than you are doing now.”

McCallum was named a Distinguished Teaching Scholar by the National Science Foundation in 2005, and his Institute is doing everything in its power to draw UA faculty and K-12 math teachers together. It sponsors a master’s degree in teaching middle-school mathematics, for example, and a math circle for middle-school teachers.

The UA math department has a history of linking math professors with K-12 students and their teachers, from a high-school research project on killer bees to the University of Arizona Summer Mathematics Camp for students of promise. But McCallum’s new institute is drawing national attention.

“Mathematicians have important contributions to make to assuring the quality of school education,” says Hyman Bass, professor of math and math education at the University of Michigan. And university mathematicians are most effective, he says, when they collaborate with professional educators, an effort in which McCallum “is exercising national-level leadership.”

McCallum’s institute “is a small organization with modest resources,” Bass continues, “but it can have high-leverage impact on the field.”

Schoolteachers can get practical lesson plans from the professors — as the math circles demonstrate — but they also benefit from professors’ store of pure knowledge.

“The more deeply teachers understand the theory behind middle-school mathematics,” McCallum points out, “the better they can teach it. That’s where research mathematicians can help them out.”

The Queen of Sciences

A Harvard-educated Australian, McCallum is known among mathematicians for his work in number theory —“the queen of mathematics,” in the opinion of Carl Friedrich Gauss, a 19th-century scientist who called mathematics “the queen of the sciences.”

Number theory tackles the properties of numbers in general, and its purity was what, until recently, gave number theory its royal allure.

As G. H. Hardy, a British prince of number theory, said proudly at the dawn of the 20th century, "I have never done anything ‘useful.’ No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

The distributive law lies behind just about every step you perform in solving an algebra problem. It says that, if you add two numbers and multiply them by a third, it’s the same as multiplying each number by the third and then adding.

Hardy might be dismayed to learn that, in the 21st century, number theory is no longer pure, nor useless — it plays a central role in encryption, the array of computations that protect credit card numbers, bank transactions, e-mail addresses, and national security.

McCallum himself prefers purity. He rarely watches TV, but once, exhausted from talking number theory all day at a math conference, he retired to his hotel room and clicked on the TV. To his astonishment, he was confronted with a crime show on which a detective mentioned the Riemann hypothesis, a problem that even the highest-flown mathematicians have not solved in nearly 150 years.

“I thought I was hallucinating,” says McCallum.

(Not that the TV detective made much use of the Riemann hypothesis. TV writers retain math consultants, McCallum says, to feed them such lines.)

The Riemann hypothesis is so difficult that a million dollars has been offered for the first correct proof. And number theory, if only in its billion-dollar effect on worldwide commerce, is increasingly urgent.

But the problem that has captured McCallum’s attention — improving math education — is much more difficult than winning a million-dollar prize and even more urgent than the fate of global commerce.

Ever since 1957, when Soviet rocketeers surprised their U.S. competitors by launching Sputnik 1 — the first manmade satellite to orbit the Earth — Americans have worried about keeping up with the rest of the world scientifically. Concerned citizens have looked to American schools to do a superior job of teaching science and mathematics. By the 1990s, it seemed clear that the schools were not measuring up.

In September 2000, John Glenn, the former astronaut and U.S. Senator, chaired a national commission that declared that 21st century America would need a citizenry educated in math and science to address genetics, health, the impact of computers, global change, and other pressing public issues. Yet the commission found alarming scores among American youngsters.

“Among 20 nations assessed in advanced mathematics and physics ... none scored significantly lower than the United States in advanced mathematics, and only one scored lower in physics,” the Glenn Commission found. American math and science students were not “world-class.” But the commissioners offered a solution: “The most direct route to improving mathematics and science achievement for all students is better mathematics and science teaching.”

As the 21st century got under way, schools made an effort to strengthen their math and science curricula. By the year 2005, though, two-thirds of American fourth-graders and eighth-graders were still not even proficient in math. What was holding them back?

The American students seemed to be competent.

“Everyone is capable of learning substantial mathematics,” argues Bass, the Michigan professor.

Sheila Tobias, author of Overcoming Math Anxiety, agrees. “When students do fail at math, it’s not because they lack competence,” she says, “but because they lack confidence.” As Tobias told Education World, girls may feel that “math is for boys,” and members of minorities may feel that math is for “people unlike me.”

A whole range of variables might impact students’ performance in math, from gender, ethnicity, and economic class, to computer games and IM-ing. Plus, K-12 math itself is actually getting more difficult.

“Current middle-school math,” McCallum explains, “used to be high-school or college-level math. It involves lots of algebra, solid geometry, and trigonometry” — the killer class that once stopped high-school seniors in their tracks.

And as math gets more difficult, it’s getting more important.

The changing economy demands that “more of our work [be] done using sophisticated math,” McCallum says. “Although computer programs will ‘do the math,’ we still need to know math in order to have a sense of the correct answer.”

The problem seems to be how to teach that difficult math to those competent kids.

To put American math students at the top of the heap, says Bass, “the most critical and most difficult challenge is improving the quality of instruction.”

Global Competition

The wider world is full of math and science whizzes, and nowadays they work cheap. Not only that, they’re staying home. For a hundred years, the United States has depended on attracting foreign scientists and mathematicians, from Albert Einstein to Bill McCallum, to build a robust economy and to teach.

Now that more countries are industrialized, they’re building universities where their own scientists and mathematicians teach their own students.

“I was just in China for a weekend conference on teaching math,” says Roger Howe, William T. Kenan Professor of Mathematics at Yale University. “The Chinese system of higher education is four or five times as large as it was in 2002, and the Chinese want to be sure that their college teaching meets international standards.

“They brought in mathematicians from the U.S. and the former U.S.S.R. to talk about teaching key subjects in math. We had an audience of 500 university math instructors. It was quite a show.”

Elsewhere in the world, students join math circles to prepare for competitions. “Russia built an extremely strong math community,” Howe says, “and many of the members got their start in math circles, where they saw math they wouldn’t normally see in schools.”

Professor McCallum in his classroom overlooking the UA campus with the Catalina mountains in the background.

More recently, some American students have followed suit, Howe says. Students from math circles in Boston and the Bay Area have done very well in the International Mathematics Olympiad. The U.S. team almost always places among the top three in this competition, along with China and Russia.

The Chinese strategy — involving university mathematicians in promoting math education — “is crucial,” Howe says. Perhaps recalling the dissension between American university professors and public-school math teachers during the “math wars” of the 1990s, he adds, “It must be done in a cooperative manner. That is exactly what McCallum’s institute is doing.”

Full Circle

The UA twist is a circle of middle-school math teachers, not students.

“Middle-school math teachers often start out teaching elementary school, not specializing in any subject,” McCallum says. “They move up to middle school without having been trained to teach math specifically.”

Under the auspices of McCallum’s Institute, middle-school teachers from Tucson gather each month, with a different UA math professor each time, to talk over one teaching topic, to share classroom experiences, and (the hook) to have dinner together.

Ginny Bohme, who taught math at Tucson High Magnet School for 30 years, leads the teachers’ math circle. Energized by an initial workshop at Stanford in the summer of 2007, she immediately recruited a circle of about 25 middle-school math teachers from Tucson and its suburbs.

At 5:30 one fall evening, Bohme and Nate Carlson, a teaching postdoc who has come to the Institute from the University of Kansas, arrange nametags, water bottles, and snacks in a classroom on campus. Twelve teachers drift in and greet this evening’s leader, an affable retired UA professor named David A. Gay.

Tall, white-haired, wearing a retiree’s polo shirt, Gay stands behind a bare lab table. During the next two hours, he will clutter it with wooden cones, dissected pyramids, and clear plastic cylinders and spheres.

His topic: using physical, three-dimensional models to teach solid geometry to middle-school students.

“They have a hard time grasping this idea of volumes, anyway,” says Gay. “Flat drawings of volume lack reality. People think that math all has to be done in your head — that you can’t use real things. But that’s the way Archimedes did it — by measuring and comparing physical objects: How much water can I pour into this sphere? This cone?”

It’s an entertaining and instructive reminder that the senses — seeing and touching — are useful tools in math.

The middle-school teachers are fully engaged with Gay’s hands-on demonstration, and they are happy, also, to break for the trays of steaming Chinese food that Bohme has set before them.

After dinner, Gay brings out his tour de force. He holds up a plastic globe the size of a cantaloupe and asks, “How can you find the surface area of a sphere?” He studies his globe. “Well, maybe you could wrap a rope, a piece of clothesline, around it. You would know the width of the rope, and its length, and you could get the area that way.”

From beneath his bench Gay produces a sphere already covered with spiraling cotton clothesline.

“And then if you could unwrap the clothesline and lay it flat…“Gay produces a square of cardboard with clothesline, spiraled to fill a tight circle, glued to it.

“I’ve measured the clothesline,” he says confidentially. “It’s the same length as the one on the sphere.”

Like walking them through a miracle, step by step, Gay shows the middle school math teachers how you can use the sphere and the flat circle of clothesline to write the mathematical equations to calculate the surface area of a sphere.

The teachers plunge into working, hands on, with Gay’s models. It’s a real way that these math teachers can help bring their students up to speed.

And a piece of Bill McCallum’s dauntingly-complex puzzle of increasing urgency drops into place.

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