With a patch over one eye, titanium legs, and a bite to match his bark, math professor Paul Sally runs a tight ship.

Elements of Math Instruction, a graduate class in the Urban Teacher Education Program (UTEP), meets Tuesday nights in “The Barn,” otherwise known as Ryerson 352. The enormous room, with a high sloping ceiling and exposed rafters, was carved out of an even bigger undergraduate physics lab in the early 1990s. The north wall, which borders a hallway to Eckhart Hall, is part glass, revealing the rest of the original lab on the other side of the corridor.

At the helm of this hobbled-together space stands the man who nicknamed it: Paul J. Sally Jr., professor of mathematics and director of undergraduate studies in mathematics. Sally too has an affectionate nickname: the Math Pirate (sometimes Professor Pirate), for the black patch that covers his left eye. He lost the eye to diabetes in 1975; the vision in his right eye has degenerated to such a point that he’s now legally blind.

Yet Sally, who turned 75 in February, has never considered retiring. This course is one of three he’s teaching during winter quarter. Sally first walked into a classroom of high-school seniors in 1954, at the age of 21, “and I’ve been teaching in some school or another ever since.” He’s taught math at all levels, from PhDs to second graders. He won a Quantrell Award for Excellence in Undergraduate Teaching in 1967, his second year at Chicago, and he enjoys relating that Wayne Booth, AM’47, PhD’50, then dean of the College, told him that based on student recommendations, “I should have won it the first year, but they wouldn’t give it out to first-year teachers.”

Like many great teachers, Sally is a quirky despot in the classroom. According to one legend—repeated by students, his Wikiquote page, and an October 1, 2007, Boston Globe feature—he loathes cell phones so much that if one goes off during his class, the students are invited to line up and take turns stomping on it. Sally confirms that the legend is true, that he’s stomped on several over the years. In one class, he says, he found a forgotten cell phone on his desk; it wasn’t even ringing, but Sally threw it out the window anyway.

On this February night the 6-foot-3-inch Sally looks particularly menacing in all black: a turtleneck, Nike warm-up pants that don’t quite cover his titanium prosthetic legs (another diabetes loss), and New Balance walking shoes. “Yo, Sally,” shouts a student near the back. Students call out his name because he cannot see raised hands. “Can we go over question five on the homework?”

“Who’s asking?” he shouts back across the vast room.

“Briana.”

Sally works the word problem on the board: men sharing pineapples on a desert island, a typically farfetched excuse for multiplying fractions. His handwriting is clear enough to read from the back of the room. Suddenly, Briana’s cell phone rings. She silences it swiftly and gets away with it. During Sally’s lecture several of the teachers-in-training use their laptops to check e-mail or the Wisconsin primary results, and they get away with it too.

Laura Grandau, a UTEP mathematics-teacher educator, acts as an aide, writing the ever-growing list of assignments on the board. “Yo, Sally, is this more homework?” a student moans. Of course it is. From Sally’s Wikiquote page: “The problem with young people nowadays is that they actually don’t think that they should be doing math every day, all day.”

Sally is one of those rare academics who’s as content working on a basic level as on an advanced one. By trade he is a research mathematician who studies p-adic analysis and representation theory, work “at the intersection of algebra and analysis,” explains Peter Constantin, chair of the mathematics department. Sally, Constantin says, has been a “very important contributor.”

For a research mathematician, Sally’s interest in teaching basic math is “very unusual, but not completely unique,” says Diane Herrmann, SM’76, PhD’78, senior lecturer and associate director of undergraduate studies in math. “He was a teacher first, before he got his PhD, and that has continued to be a part of who he is.” Constantin places him within the Russian tradition, in which famous mathematicians wrote high-school–level problems. A “good comparison” for Sally, he says, is Andrey Kolmogorov, a prominent 20th-century Russian mathematician; another is William Thurston, a Cornell professor who won the 1982 Fields Medal. “Sally belongs to a small but very select group of people,” like Kolmogorov and Thurston, “who care about spreading knowledge beyond just the specialists.”

In 1983 Sally was appointed the first director of the University of Chicago School Mathematics Project (UCSMP), which developed the Chicago Math K–12 curriculum. The late Izaak Wirszup, PhD’55, who had received seed money from the Amoco Foundation, asked Sally, along with education professors Zalman Usiskin and Max Bell, AM’58, MAT’59, to help get the program under way. The project’s reforms—for example, letting very young students use calculators—have been so widely adopted that it’s hard to understand how revolutionary they were at the time, says Usiskin, who took over from Sally in 1987 and has been director ever since. An estimated 4 million U.S. students learn using Chicago Math.

 “Sally has a very long-standing interest in two kinds of students,” Usiskin says: “inner-city kids and talented kids.” Sally’s Young Scholars Program (YSP), cofounded with Herrmann in 1988, brings together both; it’s for mathematically gifted students, mainly from Chicago Public Schools, “with a few outliers,” says Herrmann. In the free summer-enrichment program, students learn about topics in number theory and geometry not covered in their regular curriculum. Initially funded by a National Science Foundation grant, YSP continues to be sustained by other grants. The program, which targets seventh through 12th graders, draws about three times more applicants than the 100 slots available.

In 1992 Sally—more interested in training teachers than in developing curricula as at UCSMP—founded his own program within the math department, Seminars for Elementary Specialists and Mathematics Educators (SESAME). To date 1,200 Chicago Public Schools teachers have taken SESAME classes. In 2001 Harvard University Extension began offering the program to public-school teachers in Boston and Cambridge.

Sally’s approach to teaching teachers is easy to describe, not so easy to achieve: math teachers should understand the material well enough to explain it in a few different ways. Teachers who can’t are doomed to the “slower and louder” approach, he says. “If the kids don’t get it the first time, you say it again slower and louder.”

The Urban Teacher Education Program class provides another opportunity for Sally to show young people how to teach math before they’re thrown into a classroom. Established in 2003, UTEP is a two-year graduate program open to fourth-year Chicago undergrads as well as college graduates. Many UTEP graduates go on to teach in Chicago Public Schools.

Tonight’s group of aspiring urban teachers has a certain urban style. Two women have nose rings, one also has tattoos, and one man wears a purple stocking cap. When Sally gets near enough, the cap catches his attention. “Take your hat off,” he barks. “No hats.” The student looks startled, not quite sure if Sally is serious. He is. The student whips off the hat.

“Yo, Sally,” calls a young man with dark hair.

“Is that Jason?”

“Yeah.” Jason’s question is not so much mathematical as pedagogical—or perhaps ethical. In his student-teaching position, he’s under pressure to teach third-grade geometry “to the test”: in this case, the upcoming Illinois Standards Achievement Test (ISAT).

“The teacher is on the guillotine,” says Sally. “As horrified as I might be, Jason, you’re going to have to be part of that. After the ISAT is over, you can have fun again and play games.”

Halfway through the three-hour class the students, who spent the day either teaching, observing, or assisting a class, break for dinner: boxed sandwiches that Sally pays for from a $5,000 Provost Fund Grant “slush fund.” (In future years, he says, he’ll pay out of his own pocket if necessary.)

Then it’s on to tessellation, an exercise that “works for kindergarten to graduate school, depending on your level of sophistication.” By now his all-black ensemble is lightly coated in chalk dust. He has dust around his mouth from chomping the chalk like a cigar. He writes a definition on the board: “Tessellation—to cover the plane with one or more shapes with no overlaps and no gaps.” One student confidently names a shape that tessellates: “A square.”

Sally asks another student for a shape that doesn’t tessellate.

She hesitates. “Circle?”

“There we go!” Sally says. “Beautiful!”

Sally has taught at Chicago for more than 40 years, but the decades of exposure to short Midwestern As and firmly pronounced Rs have made little impact on his Boston accent. Sally’s father, an Irish immigrant, was a mason and plasterer with an eighth-grade education; his mother was half Irish, half English. The family lived in the Roslindale neighborhood, later moving to the nearby suburb of Dedham.

At Boston College High School, Sally was a basketball star. Math came just as easily: good at it without having to try, he didn’t. While he was an undergraduate at Boston College, a professor explained that 75 points of the grade would be based on tests and 25 points on homework. “Guess what I got?” he asks. The correct answer, of course, is 75 points.

After college he enrolled at Brandeis University, which had opened its mathematics graduate program that year, 1957, and needed teaching assistants. There he finally started to work, but, typical of Sally, “I learned as much from teaching as from studying.”

Early the first year he met an Irish American woman named Judith Donovan, “the most brilliant student in the program by far.” They went out once in the fall, but over winter break she turned him down, using the old washing-my-hair excuse. He persevered, and on their third date, “We looked at each other and said, ‘This must be it,’ and decided to get married.”

The Sallys married in 1959, and as he struggled to finish his dissertation on representations of semisimple groups, the couple had three sons in three years. (The oldest, David, PhD’95, teaches business administration at Dartmouth College; Stephen, after earning a PhD in Slavic languages from Harvard University, became a corporate lawyer; the youngest, Paul III, is a “superb teacher” of math at New Trier High School in Chicago’s northern suburbs.)

Sally began teaching at Chicago in 1965, winning tenure in 1969. The same year, Judith Sally enrolled in Chicago’s graduate program. By 1971 she had her PhD and the next year was teaching math at Northwestern University. “One piece of advice if you want to have a happy marriage,” says Sally: “marry a woman who’s smarter than you are.”

The 11 undergrads in Analysis in Rn, which Sally team-teaches with a former student, Senior Lecturer John Boller, SM’91, PhD’99, sit in a thick clump at the front of the classroom. Sally, wearing a gray turtleneck and gray tweed sport coat with his usual black warm-ups, walks in a few minutes late. Boris, a young man with a soft Russian accent, is already demonstrating a proof on the board. While Boris narrates the complex proof, Sally looks in a completely different direction, as if following along on his own internal chalkboard. “Five or six years ago,” he says, “I developed the ability to remember a huge volume of math in my head.”

A student in an orange Chicago Bears cap takes issue with the proof and goes to the board to make corrections. Returning to his desk, he remembers the cap. “Thank you very much for taking your hat off, Dustin,” Sally says. Its removal leaves a sharp ridge in Dustin’s straight hair. “I gotta stay away from the orange hats,” he mutters.

The course is an experiment in inquiry-based learning, an approach developed by R. L. Moore, who earned his PhD at Chicago in 1905. Five years ago Sally was approached by the Educational Advancement Foundation in Texas, a group trying to spread IBL, a modified form of the Moore Method. The foundation funds IBL courses at Chicago and four other universities.

The idea is simple: students in IBL classes prepare and deliver the lectures. They use no textbooks but rather are given pages of theorems to prove. “It’s a spectacular way of learning,” says Sally—as long as the classes have 15 or fewer students. “The kids bring in beautiful lectures.” Sally first used IBL in his Honors Calculus course; this is the first year he’s tried it in Analysis. The class is “often very chaotic,” he says. “The teacher’s role is to prevent it from descending into anarchy.”

Student after student goes to the front. Sally sits off to the side, Boller and teaching assistant Michael Broshi, SM’04—who has quietly removed his baseball cap—in the back; the teachers never get near the board. The level of excitement in the classroom is palpable. At one point a student asks a question, and five of his classmates turn around to help explain.

“What does that function look like?” Sally asks. One-variable functions, as any algebra student knows, produce straight lines or curves when graphed. But with two variables, as in this case, the objects look like surfaces in three dimensions.

Dustin acknowledges sheepishly that he can’t visualize it in his head: he used Mathematica software to graph the function.

Sally is incredulous. “What, this thing here?” he shouts. “Just LOOK at it. You don’t need to use Mathematica, you use your BRAIN.” He and Boller talk through how the path should look. Following Sally’s description, the students slowly move their hands through the air as if doing some math-based form of Tai Chi. At the end of the 90-minute class the students loiter, talking math to the instructors and each other. They don’t want to leave.

Sally admits he saves most of his theatrics for his undergraduates, and that he uses different styles of lecturing to address different levels of expertise. At math conferences, with an already-interested audience, he presents the material in a straightforward way—though it’s hard to imagine he doesn’t slip in a joke or two. “The way he presents himself in the classroom has everyone paying attention, laughing, and thinking deeply,” says Scott Messick, a second-year math major who took one of Sally’s favorite courses, Honors Analysis, as a first-year. “He has an incredible sense of what people can do and calls on people for exactly the right questions.” When he gets in the classroom, says Herrmann, Sally becomes a “wild presence.”

Sally spreads his own lore as much as anyone. For a mathematician, such liberties are “actually more allowable,” says Boller. “We have to take the mathematics so seriously that there’s no room for opinion.” But when it comes to “nonmathematical thinking,” he says, “those stories do grow into tall tales.”

One story Sally likes to tell at the beginning of the quarter, when he’s trying to whip his undergrads into line, makes the point that exams can never, ever be rescheduled. As the tale goes, a student once claimed he had to miss a test because he had booked a flight home for Thanksgiving. Sally supposedly grabbed the kid’s cell phone and, as the entire class watched, called the airline and rebooked him on a later flight. True? Maybe. Sally, Herrmann, and an undergrad in that 2003 Honors Analysis class all remember it differently—and the student allegedly involved doesn’t remember it at all.

In the hallways Sally is just as theatrical, known for belting out “Danny Boy” and other Irish tenor tunes. (Both Ryerson and Eckhart have excellent acoustics, Sally says.) A music lover, he keeps a collection of cassettes neatly alphabetized in his office—mostly classical, but Whitney Houston, Carly Simon, and the Cowboy Junkies are there too—and works his way through them in order. Whenever Bach is playing, Beethoven will shortly follow.

The energy level in Advanced Algebra for Elementary Schools, part of his SESAME program, is about half that of Sally’s undergraduate Analysis course. At 4 p.m. everyone in the room has already put in a full day of teaching middle-grade students. They’re paying attention, but they aren’t exactly burning up the classroom.

Before Sally arrives, his teaching assistant, Mike Shulman, SM’05, goes over the homework. “Are we going to need this to pass the Algebra Initiative Exam?” asks an African American woman with a gray fedora and granny glasses. “Why are we doing it then? Why are we working just to be working?” The TA seems so flummoxed to hear a teacher ask the equivalent of, “Is this going to be on the test?” that he can’t summon an answer. If Sally were here, he certainly would have provided one.

Soon enough he appears, today in a white turtleneck and the warm-up pants. While lecturing, he occasionally grips a grab bar installed for him along the edge of the chalkboard. Sally knows the names and rough locations of all 50 students, who sit in assigned seats. The Eckhart lecture hall seems based on the mathematical game of penny packing: not one more desk could possibly fit.

“Is somebody sitting there?” Sally asks at one point.

“That’s my coat,” a student explains.

“That’s your coat,” Sally repeats. “Hello, coat!”

A cell phone rings. And rings. And rings. Its owner, an Asian American woman with highlights, is perhaps praying it will go to voice mail. “Smash it!” suggests a young man in a buffalo plaid jacket. After five rings, she gives up and shuts it off as surreptitiously as possible.

“Have I told you how much I hate cell phones?” Sally asks the class.

“Yessss.”

“We haven’t punished anybody severely yet,” he says. “The cold weather is enough punishment. As the weather warms up, we’re going to stomp on one.” Sally turns to the board and slowly explains how to factor polynomials. Some students are struggling. The pace might try the patience of a high-school algebra instructor, let alone a research mathematician. But Sally never loses his enthusiasm.

Chicago Public Schools teachers are often “woefully uninformed and assigned,” says Sally a few days later. “It’s not their fault.” Elementary-school teachers cover up to six subjects a day. “Nobody can do that. You can’t teach sixth-, seventh-, eighth-grade math without a specialist background.” And yet CPS is full of teachers doing just that.

Sally sees the overload partly as an unintended negative consequence of feminism. When other professions were closed to women, teaching was open. “So you had truly educated women teaching third graders,” he says. Today that’s not always the case. He’s pragmatic about how much effect SESAME can have. In public education in math, “progress has been very slow. The problem took a long while to be created. It’s going to take a long time to be fixed.”

Math teachers should know their stuff: that’s part one of the Sally approach. Part two is that students should be allowed to have fun. Fond of sports metaphors, Sally tells the story of a kid who starts working with a basketball coach. The kid takes 1,000 shots and runs five miles a day. After a while he gets bored and asks if he’ll ever be allowed to play basketball. The coach says, Nope, this is it. “That’s the way math is often taught,” says Sally. “Drill drill drill.”

In Sally’s method, students play math games to test their thought processes. “It’s an integral part of teaching,” he says. “Students need to be able to apply their skills to real, honest-to-goodness math problems. You have to give ’em a game.” Over the years Sally has developed a “truckload” of problems to give his students. Sometimes he likes to be sneaky, slipping in one that has never been solved with a bunch of easy ones. (An example of an unsolved problem: How many regular tetrahedra of edge length 1 can be packed inside a sphere of radius 1 if each tetrahedron has one vertex at the center of the sphere?)

Many of Sally’s games and problems can be found in the books he and Judith began writing when she retired from Northwestern in 2002. In Trimathlon: A Workout Beyond the School Curriculum (A. K. Peters Ltd.), chapter 7 details tessellation. Their second book, Roots to Research (American Mathematical Society), explores five mathematical problems that, like tessellation, span the range from elementary-school to graduate-level math. Chapters begin with a simple explanation of the problem and end with a discussion of current research. The book is based on another Sally theme: math is math, no matter the level of sophistication. The Sallys are now at work on Geometry for Teachers, a professional-development guide for middle- and high-school teachers. Sally is also finishing up a solo effort, Tools of the Trade: An Introduction to Advanced Mathematics, to be published by the American Mathematical Society this year.

But his true love remains the classroom. On his first day of teaching 54 years ago, Sally realized, “This is just where I belong. I just have this incredible desire to teach,” he says. “I’ve lost my legs, lost my eyes, lost my hair. But my brain is still sharp.”

Dedicated to outreach through writing and teaching, Sally “is also committed to not diluting the mathematics, and in some sense corrupting it. That is a legacy that is so important,” says department chair Constantin. “Not enough people take up this difficult fight. Somehow, we need to duplicate him.”