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How to make a mathematician
In a classroom on the tough South Side of Chicago, a math teacher cuts a daunting figure, his head rising above a six-foot-high, freestanding chalkboard, a black cloth patch over his left eye. He begins the lesson abruptly, calling out to a class of eighth, ninth, and 10th graders, “What’s the sum of numbers from 1 to 111?” A boy wearing a University of Chicago sweatshirt begins to speak up in the back, and the teacher interrupts loudly, “Who’s that?” He lurches toward the youth. His gait is a testament to six decades of battle with Type 1 diabetes that has left him with titanium legs, today tucked in black sneakers. Both arms are out in front of him, because even with his “good eye” he can’t really see. “Is that Dylan?” In a sharp Boston accent, he orders Dylan to hustle up to the front of the classroom. “You’ll be my victim today,” he informs the young man, and then he identifies a second victim.
It might have been an intimidating moment, if the students weren’t all taking it in stride, chuckling or plowing ahead with their problem-solving. The setting wasn’t a public school: It was a spacious room with a high arched ceiling and rows of windows looking out on the quadrangle of the University of Chicago, in the South Side’s Hyde Park enclave. Twenty-five students, mostly from Chicago public schools, turned up on a Saturday morning in February for the math equivalent of a weekly pickup game, open to any youngsters who, for the span of three hours, want to take a swing at solving math problems that would vex many of their teachers. They were there because they chose to be there, for what is called the Young Scholars Program. It’s one of the many math-education ventures of Paul Sally ’54, MA’56, a celebrated University of Chicago professor whom students, from pre-teens through graduate school, affectionately call “the math pirate.”
Sally is a rare beast in the mathematical jungle. On the one hand, he stands atop the food chain, among leading research mathematicians, who are known generally as a reclusive species not highly disposed to the teaching life. And yet, he takes great pleasure in standing in front of a classroom of middle and high school students, giving out fist bumps when they reason their way to what he often styles a “beautiful proof.” As Marek Dobrenko, an eighth-grade math whiz at O.A. Thorp Scholastic Academy, a magnet public school 19 miles away, remarked while helping himself to chocolate chip cookies during a break at the Young Scholars class, “It’s just a lot of fun spending your Saturday morning learning the logic.”
Sally devotes as much passion to working with teachers as with students, acting on his precept, “To teach mathematics, you have to know some.” One of the boldest iterations of that principle is a project he spearheaded seven years ago in Chicago called the Algebra Initiative, a program of rigorous study that qualifies teachers to offer that critical subject in middle schools. Certification from the Algebra Initiative, now available at three universities around the city, is required by the Chicago Public Schools before an instructor can teach the subject at the middle school level.
“He’s really well known among research mathematicians, and he’s equally well known in education circles,” from elementary to university level, says Albert Cuoco, who directs the nonprofit Center for Mathematics Education, in Newton, Massachusetts. Sally’s research area is number theory, a branch of pure mathematics, which is concerned less with the utility of mathematics than with the beauty of its abstractions. (One way that Sally tries to capture this beauty for students at all levels is by introducing them to the Platonic Solids, five perfectly symmetrical shapes that Plato wrote about in his dialogue Timaeus, and which were further developed in Euclid’s Elements.) His mission as an educator, however, appears all encompassing: “to spread mathematics far and wide,” says mathematician Ben Howard, who discovered his love of the discipline as a high school student in the Young Scholars Program (which Sally founded in 1988 with colleague Diane Herrmann). “That’s what he lives for,” says Howard, who is now an associate professor of mathematics at Boston College.
At 77, Sally the math evangelist is renewing his mission, aiming to spread the Algebra Initiative beyond Chicago (his best bet is Boston), even as some observers see much of Sally and his ways that can’t be replicated or scaled up.
Ramping up the quality of math and science education is widely considered a pressing national priority. A sign of this urgency is the Obama administration’s “Educate to Innovate” campaign, which looks to link government agencies, corporations, and universities in efforts to bolster teaching in those subjects and make American students more competitive globally in the technology and science arenas. Sally has sat on his share of advisory panels and steering committees that address atmospheric issues such as curriculum and appropriate standards of math achievement. His signature approach, however, is unstintingly personal.
He goes to students. And he goes to their teachers, through such projects as the Algebra Initiative and SESAME (Seminars for Elementary Specialists and Mathematics Educators). Launched in 1992 and still directed by Sally, SESAME offers a breadth of courses in pedagogy as well as in content areas such as probability and number theory, focusing especially on the conceptual foundations. It is one of a number of programs run by various institutions that lead to the state of Illinois’s certification to teach math and science in middle schools. Aside from the enterprises run from his bustling office at the University of Chicago, Sally’s footprints are found in settings ranging from his alma mater, where he has intervened at pivotal times in the evolution of Boston College’s math department (see sidebar, below) and where he has personally endowed the Sally Award for math achievement by graduating seniors, to the K–8 Thorp Academy set in a working-class neighborhood on Chicago’s northwest side. Sally has “essentially adopted” this magnet school, serving as generous donor and star attraction at special events with titles like “Math Stories: An Evening of Reasoning and Proofs,” says Linda Hunt, who teaches and coordinates math studies at Thorp.
At the University of Chicago, Sally is the sort of professor whose reputation among faculty extends well beyond his department. Colleagues more versed in Coleridge than calculus know enough not to tell him, “I can’t do math.” When he hears the confession, whether at a party or on a plane, Sally responds, “That’s funny, I can’t read.”
Among the university’s students, “Everyone hears about him as soon as you get on campus,” says Eleanor Brush, a fourth-year math major from Hartford, Connecticut. Brush is sitting at the front desk of Sally’s office on the second floor of the gothic-style Eckhart Hall, answering the phone and playing with a Rubik’s cube. One legend students relate has to do with Sally’s classroom policy on cell phones, which is reputedly to invite students to line up and take turns stomping on any phone that rings during class. Sally himself will only say that he follows through on this threat if a phone goes off five or six times in the same class meeting. With clearer conviction, he tells of instances in class when he’s borrowed a cell phone (he doesn’t own one) after learning of a student’s plan to miss math class for an early flight out at Thanksgiving or spring break. Sally says emphatically that he has rescheduled flights in class “more than once.”
“They think there are things in life that take precedence over mathematics,” he says, tilting his six-foot-three-inch frame forward during an interview in his office, “and I try to disabuse them of that notion.” This is part of Sally’s shtick, his nothing-really-matters-but-math routine. His students are quick to note that on the first day of class, Sally informs them that the only excuse for missing an exam is “if you died and your funeral’s that day.” He also instructs them in the proper way to address him in class: “Yo, Sally.” This, because he can’t see their raised hands.
Bracing classroom encounters aside, Sally has a devoted cadre of students, former students, and other coconspirators who proudly identify themselves as members of “Sally’s gang.” The inner ring includes roughly a dozen students who work for him, reading aloud his e-mails, sorting through the milk-crate file boxes that crowd his office, helping out with research tasks, and otherwise lending a hand on his projects (they’re paid mostly with grant money obtained by Sally). The outer rings extend to about 50 others, including city of Chicago teachers and administrators.
“He’s really a great and cool guy. He’s 77, but I like hanging out with him,” says University of Chicago second-year student Jonathan Gleason, who came to Sally asking if the professor knew of any jobs available on campus and quickly found himself driving Sally around and helping to manage the Young Scholars Program. One of many students and allies to course through the office on that Friday in February was Christina Pei, who graduated from Chicago in 2006 and is now teaching at an experimental middle and high school in Flushing, New York. Pei says she flew in for the purpose of participating in Sally’s gang activities, her chief destination being the Young Scholars session the next morning, in which she would be assisting as a counselor, someone who helps facilitate student discussion of math problems in small groups. After the Saturday session, Pei said, “Everything he does, I want to be part of.”
By all accounts, Sally’s generosity is as outsized as his humor. The story is told of a farm girl from Italy who dreamed of pursuing graduate study in math at a great American institution, and was accepted by the University of Chicago. All she had to do was take the Test of English as a Foreign Language, or TOEFL, which she elected not to do, knowing she would almost certainly fail.
The determined student journeyed to Chicago anyway (having mistakenly gone first to Minneapolis) and hung around the math department until she was allowed to begin sitting in on classes informally. School officials fully expected that within weeks, she’d pack it in—though not for lack of math prowess. And they were right.
One fall day in 1990, she sat at a pay phone in Eckhart Hall, tearing up as she booked a flight back to Italy. Sally walked past the pay phone at that moment. He signaled her to come to his office, and she left a short time later with a personal check for $1,000 and a plan to get squared away with the admissions office so she could begin drawing income as a graduate teaching assistant.
That farm girl was Gigliola Staffilani, now Abby Rockefeller Mauze Professor at Massachusetts Institute of Technology and a leader in the field of pure mathematics. Staffilani says that whenever she sees Sally, she tells him that he made all the difference in her life and career. He says that’s nonsense: It was her talent and drive that put her where she is today.
“I don’t agree with that. I would have never made it, without his help,” Staffilani says in her hesitant English. “If I had gone back to Italy then, I would have never returned.” The TOEFL test ceased to be an issue. Sally worked behind the scenes, and the next thing Staffilani knew, the requirement had been waived in her case.
Paul J. Sally, Jr., was born in Boston’s Roslindale section on January 29, 1933, the son of a “jobber” who ran a one-man contracting business—plastering, roofing, and bricklaying—out of the trunk of his car. Sally earned a half-tuition scholarship to attend Boston College High School, where he was an all-star basketball player, and was lured with another half-tuition scholarship to Boston College. Both scholarships were for academic promise, and when asked about his scholarly feats at the University, Sally, who majored in math, responded with a long, loud laugh. “I muddled my way through,” he says. “My grades were mediocre at best.”
At times too clever for his own good, he never saw the point of doing math homework as long as he could tackle the assigned problems in class. Sally relates that homework counted for 25 percent of the grade in one class; he scored 100 on all the tests but racked up a final grade of 75, owing to his personal homework policy.
John Ford, a retired social-work manager, was a Boston College classmate of Sally’s and is now alumni chairman of the Class of ’54. He went to hear his old friend speak in McGuinn Hall last December about the Algebra Initiative (Sally is a frequent visitor to Chestnut Hill). “He was just as brash as he was when he was an undergraduate. He was as much of a wise guy as ever,” Ford says of the presentation in which Sally, among other zingers, dared a well-known textbook company to sue him for proclaiming loudly that its Algebra 1 text is an “absolute mathematical disgrace.” This is part of the funny, earthy, tough-guy image that the Roslindale boy cultivates, but Ford and other friends say that Sally’s true character is defined by his loyalty and grit. “He’s there for people,” is how Ford puts it.
Shortly after graduating from Boston College, Ford was in Army basic training at Fort Dix, New Jersey. Sally was dating Ford’s sister in Somerville, Massachusetts, and Ford’s girlfriend was back there, too. The soldier was homesick, and Sally promised to drive down with Ford’s sister and girlfriend. “Paul had an old, beat-up, mid-’30s car, and most of the floorboards were gone,” the friend recalls. As luck would have it, on the weekend of the trip, the Northeast was battered by a hurricane. “He drove through the rain and wind . . . just so I could see my girlfriend,” Ford says. “I knew he would get there in spite of the storm. It was sort of his character.”
Sally says he got a late start as a mathematician, by professional standards. He likes to quip that mathematicians do their best work when they’re 21 years old (which is not so wild an exaggeration) and that at that age, he was dribbling a basketball. Sally was also hitchhiking regularly the 10 miles to the Heights from Dedham, where his family had moved. After graduating from Boston College, he stayed on to get a master’s in math, in what was to him a “dream” situation. He’d been offered an Intramural Fellowship, which involved assisting in intramural sports while pursuing graduate study. “They were paying my way to go to school, and I spent all my time in the gym,” he remembers fondly; he also substitute-taught in Dedham public schools. “I hadn’t yet caught the drift that you have to work hard at mathematics.”
The second year out of college, he taught full-time at Boston College High School while finishing his master’s and moonlighting as a cab driver. Third year out, he worked at a mathematical consulting company in Carlisle, deploying the theorems for such functions as air traffic control, while teaching part-time at Regis College.
Then, as he turned 25, came his day of decision. He was among the first Ph.D. math students admitted to Brandeis University, and he recalls that soon after arriving there, the chairman of the department pulled him aside and said, “Sally, we know about you. You’re bright, but you’re a screw-off. If you don’t get to work, we’ll throw you out on the street next week.” Sally got serious and in addition to earning regard for his studies became a popular instructor on campus. Abram Sachar, the renowned first president of Brandeis, was once quoted in the student newspaper as complaining that the best teacher in the math department was a graduate student (Sally).
Sally speaks of several moments in his young-adult life when friends nudged him to his next level of ambition and performance. These friends include the truck driver with whom he worked part-time delivering furniture during college, who told the young man he would not be happy for long in such an occupation (Sally at the time wasn’t so sure); and an older, fellow math teacher at BC High who said teaching alone would never satisfy Sally’s mathematical curiosities. Sally is not one to air reflections on how mathematics became his calling or when he knew he had a gift for it. As he often does when asked a personal question, he answers with a question—”How do you know you want to play for the Boston Celtics? It’s a dream.”
At Brandeis Sally met a woman named Judith Flanagan Donovan, whom he married in 1959; the couple had three children, all boys, while he was still in graduate school. Two of his sons are in education—one as a curriculum director at a public high school with two campuses in Chicago suburbs, the other as a management professor at Dartmouth College. The third son is a partner in a Boston law firm. Judith Sally retired in 2002 as a math professor at Northwestern University and has coauthored two books with her husband, including most recently, Roots to Research: A Vertical Development of Mathematical Problems, published by the American Mathematical Society in 2007. Sally is currently working on two more books, one with Diane Herrmann and one on his own.
Sally went to the University of Chicago to teach in 1965, and never left. But he made his name as a research mathematician during two visiting stints, in the late 1960s and early 1970s, at the Institute for Advanced Study in Princeton. There, with his colleague J.A. Shalika, and under the influence of the famed Indian mathematician Harish-Chandra, he cranked out a series of acclaimed papers in the area of representation theory, notably in harmonic analysis, which involves the examination of basic waves in physics. Sally received particular recognition for his work on the related topic of p-adic analysis, which the average intelligent person “has no business knowing about,” as he told an interviewer. (Staffilani says that Sally’s lecture on that subject at a 2007 MIT symposium is generally remembered as the most impressive mathematics presentation there in recent years.)
All the while, Sally kept a hand in math education at the lower levels, teaching weekly classes and making special presentations at elementary and middle schools in Chicago. He made a grander entrance into that arena in 1983 as founding director of the University of Chicago School Mathematics Project, better known as “Chicago Math,” a curriculum designed mainly for grade schools. Still used in educating an estimated four million children nationwide, its teaching materials and textbooks tie math to everyday activities such as counting money and playing cards.
Four years into the Chicago Math experiment, Sally departed as director, pointedly. “I got fed up with the educational bureaucracy,” he recalls, expressing the view that school leaders generally felt the best way to engage students in math was to make the math easier. He wanted to make it more challenging, in part by teaching the concepts behind simple mathematical operations—why any number multiplied by zero equals zero, or why the product of two negative numbers is always positive.
Though one of the most illustrious, Sally is not the first mathematician to throw up his hands after seeking to elevate math instruction in America’s schools from outside the classroom. Staffilani of MIT says that at professional meetings, mathematicians often vent about the halting pace of progress toward raising standards and reshaping curriculum. As Staffilani relates, Sally decided to take his effort “in a completely different direction.” He went inside the Chicago school system and straight to the teachers, “straight to the fact that if people know more, they can teach better.” One has to know a lot of mathematics to teach the “simple things” with confidence, says Staffilani.
This idea that if you want to teach math well, it helps to know math well, is behind all of Sally’s outreach to teachers. And when he speaks of knowing math, he’s not talking about calculation skills. In Sally’s class, the best students of any age aren’t necessarily the ones who tally up the answer first. They’re the ones who use logic and reason and imagination to devise a way, preferably an elegant way, of getting there. They enjoy the journey.
Sally would want middle school teachers who attend his weekly SESAME classes to do more or less as he does on Saturday mornings at sessions of the Young Scholars Program. When he asked the two dozen teenage students that morning in February for the sum of all numbers between 1 and 111, he wasn’t looking for a single number. And he didn’t drill his students on a rule for arriving at the answer.
After offering clues, he had the students break into discussion groups, as they might at a college-level seminar on Chaucer. The students sat at rectangular tables and were assisted by six college students and one alumna, all belonging to “Sally’s gang.”
At one table, University of Chicago third-year Kris Harper engaged two African-American and two Hispanic students (two boys, two girls) in conversation about ways of scaling the problem. Harper mentioned something called the “reverse-and-add-trick,” and two of the young scholars interjected almost in unison, “Pascal’s Triangle!” referring to a geometrical arrangement of numbers named for the 17th-century French philosopher and mathematician Blaise Pascal. After five or 10 minutes, Sally gathered the counselors to get reports on how the pupils were faring. He listened, shuffled to the blackboard and, with his uncovered eye an inch from the surface, chalked out the conclusion:
|n (n + 1)|
(The top part of the formula equals the bottom part, if “n” equals 111.) Sally’s conclusion did not, of course, offer a bottom-line number, but took the students to the point where the rest was mere calculation. Sally then introduced the next of several problems he would raise that morning (he has a desk drawer full of suitable challenges collected over the years).
Sally says too few eighth graders are taking algebra in Chicago and in inner cities elsewhere because their instructors can’t teach the subject. This matters, because a solid course of algebra puts eighth graders on track for calculus in 12th grade, which is, for one thing, a factor in admission to many colleges and universities. The picture is getting brighter in Chicago. Since it began in 2004, the Algebra Initiative has credentialed 311 middle-school teachers after running them through a challenging course of study and administering a qualifying exam so rigorous that many teachers fail it on their first attempt. The end result is that more Chicago eighth graders are being steered into algebra—12 percent last year, up from 7 percent in 2003, according to David Jabon, a math professor at DePaul University (another of the initiative’s local university sponsors) and a former student of Sally’s who is conducting an evaluation of the project.
Sally likes to send his “young scholars” to the blackboard and otherwise put them on the hot seat, but he’s gentler when teaching teachers, who tend to know less than those students. “He’s infinitely patient with them,” and careful, says John Boller, his unofficial right-hand man and a senior lecturer in math at the University of Chicago. Sally’s goal is to win their trust, which he does, not only in the classroom but also by his presence in the schools and his skill at working with the school system. “He goes to bat for teachers like no one I’ve ever seen,” says Hunt of Thorp Academy, who took Sally’s classes in SESAME and now teaches SESAME classes in curriculum and instructional methods. She cites his advocacy of professional development for teachers, which includes lining up financial help for those who can’t come up with the tuition for SESAME.
At the national level, the question Sally often gets from funders and policy makers is: When are you going to “scale up?” On the one hand, he’s dismissive of the suggestion. “If I could do eighth-grade algebra in Chicago schools, it’s already scaled up,” he says, alluding to the size of that school system, with 409,000 students. On the other hand, he is looking to expand, evangelizing far and wide about the Algebra Initiative, with presentations in San Francisco last January at the annual joint meeting of the American Mathematical Society of America and the Mathematical Association of America, as well as articles in preparation for professional journals. Cuoco of the Education Development Center says Sally is involved in a push to bring the Algebra Initiative to his native Boston through a partnership with the Hub’s public schools, Northeastern University, and the center. “Boston’s interested,” Cuoco says, adding that foundation funding will be the key. Meanwhile, across the Charles River, in Cambridge, the Harvard Extension School has been replicating the SESAME formula since 2001.
Solomon Friedberg, chairman of the math department at Boston College and a former student of Sally’s at the University of Chicago, is among the research mathematicians who tell of how they were drawn to math education largely because of Sally. “For me, it was inspirational” to see a top researcher in the field involved in K–8 education, he says. Using almost the same words, Boston University professor Glenn Stevens, who is not a former Sally student, recalls that as a young mathematician in the mid-to-late 1980s, “I was always inspired by Sally’s presence at meetings.” Stevens is himself a number theorist. He also directs PROMYS (Program in Mathematics for Young Scientists), and he serves as principal investigator for Focus on Mathematics, two Boston-area programs that foster professional development among teachers and math ability in children and teens. He says Sally and the “warmth of his community” of research mathematicians was an encouraging sign to him that one could succeed as a scholar while forging links with schoolteachers and young students. Sally’s full effect in this regard “can’t be measured, but it’s been really significant,” Stevens observes.
The math pirate shoves aside questions about why he’s committed to math at every level. When pressed, he says he “sees a need” and goes from there. The 77-year-old does say he intends to keep on teaching—”for as long as I can find the blackboard.”
Mathematics at Boston College has journeyed far since the days when proofs and problems were at times taught by Jesuits who doubled as French or philosophy instructors. And the latest testament to that headway is the launching of a new doctoral program in mathematics at the University.
The math Ph.D. is set to begin in the fall after getting the Board of Trustees imprimatur. It will focus primarily on two research areas, according to math department chair Solomon Friedberg: number theory/representation theory, which has applications for the secure transmittal of information on the Internet, for example; and geometry/typology, the far-flung applications of which include the study of curved space in the cosmos.
Friedberg says the advanced program will draw top scholars of these pursuits and will help attract students from the next generation of research mathematicians to Boston College. It will be the first new Ph.D. in the College of Arts and Sciences since sociology, political science, and theology were added 40 years ago. The School of Nursing accepted its first doctoral students in 1988.
The administration has sponsored an extensive academic review of the math department over the past couple of years, conducted by three outside mathematicians from institutions prominent in the field. The evaluation team reported, among other findings, that the department exhibits an “outstanding” level of purpose and collegiality; that the University “now has research faculty in mathematics that one would expect to see at institutions like Michigan or Columbia,” and that the establishment of a Ph.D. program would etch Boston College onto “the national map” in this discipline.
Paul Sally ’54, MA’56, a leading research mathematician and a professor at the University of Chicago (see story), underscores Friedberg’s role in advancing research and other aspects of the department at Boston College. “He’s turned the place on its ear. It’s really abuzz now,” Sally says of his former graduate student at Chicago, who arrived in Chestnut Hill in 1996 and became chairman three years ago.
The chairman pushes some of the credit back to Sally, who, for one thing, urged Boston College to recruit Friedberg as a professor and then persuaded him to accept an appointment. (Sally has made similar interventions with regard to other new professors at Boston College, where the math department has hired seven ladder-rank faculty over the past 15 years.) “I’ve certainly called him up a number of times, asking advice” about departmental matters, Friedberg notes, pointing also to Sally’s frequent presentations at Boston College.
A half dozen or so doctoral students will enter the new program in the fall. A notice recently posted by the department on the American Mathematical Society’s website says, “Our program will be small, thereby facilitating close interaction between students and faculty.” In addition to the research priorities, Friedberg says, the doctoral program will put emphasis on training “effective communicators of mathematics” for teaching and writing.
Although Latin, Greek, and religion were privileged subjects at Boston College during the early years, mathematics was part of a cluster of other subjects (including history and geography) taught from the beginning. During the 1950s, the School of Education gained repute as a leader in the preparation of math teachers for primary and secondary schools, at least partly through the efforts of Stanley Bezuska, SJ, and his Boston College Institute of Modern Mathematics, now the Mathematics Institute. The math department has long offered master’s degrees through the Lynch School of Education—including a master’s of science in teaching—and through the Carroll School of Management. Two years ago, the department began partnering with MIT in creation of the BC-MIT Number Theory Seminar, a series of high-profile lectures focusing on developments in that field. On Boston College’s end, the seminar is organized by Friedberg and associate professor Ben Howard.
At the undergraduate level, the University will institute next year a bachelor of science degree in math to complement the bachelor of arts. New, yearlong courses in algebra and analysis will allow BS students to “learn these key areas of math in greater depth at the upper-division level,” says Friedberg. He adds that the BA program will remain strong and especially suited for those who are preparing to teach math at the pre-collegiate level or are double-majoring in math and a non-quantitative field. —William Bole
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