# Span and Deliver

There’s a hand on the screen. To say it’s larger than life is to say nothing. It is the main ...

There’s a hand on the screen. To say it’s larger than life is to say nothing. It is the main focus on the screen that is movie-theater width and height, its area an integral proportional to how long it extends, taller than a person, wide as the classroom, where nearly every seat is taken.

Paul Bamberg’s hand on the screen is paused before writing. In real life, he’s leaning over the projector in front of his class in Science Center A. Paul keeps a pen in his front shirt pocket and wears a microphone on his second shirt-button. There is a videographer in the back of the room and her camera has remained motionless, trained on Paul in the center of the classroom, the cinematic focus of the lights hung on the walls. Paul’s hand turns side to side slightly, waiting for the ink or brain waves to flow. He uses a blue Bic pen that’s coming perpendicularly out of the paper, like his thumb does when checking for the direction of magnetic fields.

Paul moves away from the projector and takes three quick steps to the first row of students. He swipes his glasses away from his eyes before saying quickly, “The row reduction algorithm converts a matrix to echelon form. A pivotal one in the last column is the kiss of death.”

As he walks back to the projector, a student in the back raises two fingers.

“Paul,” he says.

“Yes,” Paul says, while swiping off his glasses.

“I’m a little bit confused.”

So Paul starts again, explaining the strategic placement of zeros. Row reduction is a technique that lets you solve systems of equations. It’s a powerful tool, and once everyone gets it, they’re nodding their heads in agreement, looking up between their papers and the giant screen. This is the part of math that can be gotten. You sit, you stare, eventually things make sense.

The thing about the real stuff of pure math, however—the flashes of brilliance, the lucid path through tangled equations—is that it seems like you either have it or you don’t. Like the level of musical adeptness that goes beyond hours and hours of practice, or perfect pitch: natural ability, that most feared and lauded thing. What can we call it except a thing? Like pornography, we know it when we see it. Genius, prodigy.

Which is what’s so strange about this class, Mathematics 23: “Linear Algebra and Real Analysis.” It’s not Math 55, which has its own Wikipedia page—where near-genius mathematicians converge to blaze through four years of math in a single year. It’s not Math 19 or 21 either, generally operating under the assumption that you just need certain practical equations for your science experiments, or your summer internship on Wall Street. Instead, Math 23 promises that genius-prodigy-material, theoretical math, can be taught along with the more practical material—that you too can learn about groups and topology or what it means to span a space. “It spans the subspace,” Paul says. He gestures into the air. The ceiling is blue. There is wood paneling on the walls.

It’s what explains the lack of empty seats in Science Center A, today and every Tuesday and Thursday afternoon; the fact that students of all ages sit and listen, students from New York and California and Uganda. There are doctoral candidates in social sciences, freshmen, humanities concentrators, a much more equal number of males and females than Larry Summers’ infamous declaration would suggest. There is a 10-year-old who sits somewhere in the middle. All here, taking notes, watching Paul trace symbols on the screen.

Paul

One of the first things that Paul G. Bamberg ’63 tells his students is that he wants them to call him Paul. He doesn’t have tenure, so technically it would have to be Dr. Bamberg—and that’s just rubbing it in, so Paul is much better, he likes to say. It’s also a practical thing; if you go up to him and say, “Hey, Paul,” he knows you’ve taken a class with him.

Today a senior lecturer in the mathematics department, Paul has been a math person since very young, teaching himself when necessary. In high school, his Westinghouse Science Talent Search project was a statistical paper involving baseball, developing the conceit—re-invented to popular fame 25 years later—of adding the on-base percentage to the slugging percentage as a good indicator of the successful hitter (concerning baseball, and Markov processes: “Your actual lineup, of course, is a product of nine matrices”). Paul was a physics concentrator at Harvard: “When I was an undergrad,” he says, “my main extracurricular activity was grading homework and writing homework solutions for the introductory physics courses.”

After Harvard, Paul went to Oxford as a Rhodes Scholar, where he worked on elementary particles. This was 1964; post-atom bomb, post-relativity, a time when physicists everywhere were searching for an elegant framework for the universe. “The longed-for Theory of Everything,” Cambridge physicist John D. Barrow once wrote, “promises to provide the final discovery after which all physics will become the refinement of its content, the simplification of its explanation.”

At the time, there were two basic visions of what it was exactly that made up the universe. The first, quark theory, was that there were a handful of elementary particles. The second, the theory supported by Paul and his thesis advisor, was that there were an infinite number of these elementary particles. This was a so-called non-compact theory of elementary particles—non-compact for not closed, not bounded, not finite. Paul worked out an entirely new structure of hyper-geometrized functions to rigorously support this hypothesis.

In his fourth year of graduate school, however, particle accelerators were being built that were powerful enough to look at the atom in a more experimental way, pointing towards a less-than-infinite set of underlying particles. “They were providing what was looking like fairly compelling theoretical experimental evidence for quarks,” says Paul. “When one of my classmates got a job with the title of quark physicist at a university in England, I figured that I had backed the wrong horse.” This is how theoretical physics is: a theory can be genius, elegant, sophisticated, and as right as a thing can be in its own innate sense—but sometimes, reality simply turns out to be different.

He never returned to particle research again: “It was a field that was developing so fast that unless you spent all your time on research you got scooped.” The new mathematics he discovered about generalized hyper-geometrized functions was interesting and valid, but the physical applications didn’t work out as well.

“That particular branch of thesis research was a dead end,” he says, “but what I really wanted to do was teach anyway.”

The Class

Paul has been teaching for 43 years. After returning to Harvard, he was involved with Physics 1—the introductory physics class for pre-meds—for a quarter century, and he’s been in charge of Math 23 for the past five years.

He’s won multiple awards for his teaching, including the Petra T. Shattuck Excellence in Teaching Award and the Dean’s Distinguished Service Award. But as is his custom, he’s not particularly verbose as to why he loves to teach (“I think what I like the very most is interacting with students”).

His passion only appears in action. When asked what he’d teach if he could give a lesson to everyone in Massachusetts, he gets visibly excited. “This is crazy,” he says. “I would teach everyone the small affine faculty senate. Either that or I’d teach everyone to solve quadratic equations with compass and straight edge.” Then he jumps up to the blackboard and starts drawing perpendicular lines. “It’s absolutely elementary.”

Central to the ethos of Paul’s class is the proof system. Mathematical proofs are strings of logical statements, like stepping stones leading from an extremely basic starting principle to an interesting or useful result. In Math 23, students are responsible for about 20 proofs. Manjinder S. Kandola ’14, who had no experience with proofs in high school, says, “I actually talked to an M.D./Ph.D. student last week who had taken 23, and asked, ‘Are you ever going to need proofs later?’ And he said, ‘The actual proofs, you don’t. But what’s interesting is that the kind of logic and way of thinking you develop from having to do those proofs is very important.’”

Paul would agree in part, but for him it goes deeper: it’s simply as important to Paul that you know why something works as how it works. He is, at times, disdainful of “plug-and-chug” methodology, just barging forward without understanding what you’re doing. Translating this philosophy into a grading rubric, however, proved more difficult. Students are responsible for knowing all 20 proofs: for their final grade, they receive .9 points for presenting one of these proofs to another student, and .1 points for listening to one (all catalogued carefully in one’s Proof Log). This direct transaction, however, leaves out an important step. “I started thinking about the logistics,” said Paul, “and I had to invent proof parties.”

Proof parties are publicized student gatherings at which at least 10 proofs are presented. Twenty-dollar refunds are available for refreshments (“We got nachos,” Kandola grins). Kandola relates a recent Harvard FML entry: “It said, ‘Proof loggers have become my primary social networking site.’ And then someone commented, ‘You seriously need to go to a proof party.’”

But the proof parties are emblematic of perhaps Math 23’s most astonishing trait: its social aspect. “It’s established a social schedule,” says Dylan C. Nagler ’14. “Sunday night is in Leverett, Monday night is in Grays, in between is office hours, you talk to people, you present proofs to each other.”

And while Paul might have the look of the traditional math guru (there’s the running joke: How do you know Paul’s feeling social? He’ll look at your shoes rather than his own when he’s talking to you), he is known for his accessibility, setting up extra one-on-one classes with students who have partial time conflicts with Math 23, for example. According to Gurbir S. Dhillon ’13, a current Course Assistant in Math 23 who took the class last year, Paul sent out an e-mail two days before last year’s final, inviting his students over to play an online video game with him. The game was called “Guild Wars.” “He’s quite a formidable player, yeah,” Dhillon says, “He had a whole personal island and all this stuff.”

Paul is known, too, for his anecdotes and shenanigans during class, with which he punctuates the material. “When we start talking about neighborhoods and open sets,” Nagler says, “one year he wore a red cardigan to class, pulled out his glasses and said, ‘Welcome to Mr. Rogers’ open neighborhood.’”

Keeping the math interesting and engaging is important to Paul, as is the fact that his class is a social experience. Everyone works together in the absence of a curve. You’re supposed to make friends, and everyone’s varied experiences are supposed to help you understand the material. This year, however, Math 23 put its open-neighborhood environment to the test, with the inclusion of a new type of student.

Youth And Beauty

It’s nine minutes after in the fourth floor Science Center classroom when Kevin walks in. He’s not quite late: the CA, Anji Tang ’13, follows closely on his heels, apologizing and saying that they really don’t have to get there right on time, she has a bit of a ways to walk. One kid in the back cracks, “Where else are we gonna go?” Math 23 lecture has just ended and it’s a quick elevator ride up to section. Kevin is grinning and taking his seat.

He has a blue binder, grey sweatpants, and a sweatshirt tied around his waist, holding his “Star Trek” T-shirt down like a belt. He’s a little smaller than his other classmates, and, at 10, a little younger, and he sits near the front with an almost-clear view of the board. Today’s proof concerns the mechanics behind differentiability. While one student presents, Tang passes back last week’s p-sets, and when Kevin gets his, he pumps his fist and says, “Yes,” softly to his desk.

Proofs can be tiresome. This one will go on for exactly 20 minutes, neat handwriting stretching from one side of the board to the other. Some students are lazing back in their seats as if waiting for the main event (which they are—homework help). Somebody takes a quick look at freestyle bike tricks on their laptop. Kevin reads the emergency-escape-plan sign on the wall next to him, but he nods his head vigorously in time with every “such that,” and “there exists.” He has an empty Poland Spring bottle that he fools with, and there are six yellow mechanical pencils fastened to the inside of his binder, five together, one apart. He shakes the bottle with his hand, unconsciously, as if trying to get the last drop out.

Kevin, whose name has been changed as per his parents’ request, is the youngest member of Math 23 and is generating a fair amount of excitement. He has been homeschooled up until now, although he did take online courses through a Stanford program until this year. “But he’s getting to an age,” says his father, “where I think it’s a positive thing for him to interact with other people, not just him and his computer.”

Kevin enjoys programming, Legos, and videogames, and the first time his parents knew that there was something wonderful and unusual was when Kevin was 4, on a beach in Jamaica. “We were sitting on the sand under an umbrella, watching the waves,” says his father, “and he was drawing in the sand with a stick. He then commented, out of the blue, how it was interesting that sqrt (5) + sqrt (5) = sqrt (20), and not sqrt (5+5) as one might naively suppose.” When asked how he knew this, Kevin drew a right triangle in the sand, specified a base of length two, a height of length one, and a hypotenuse of length sqrt (2x2+1x1)=sqrt (5). He said this was due to the Pythagorean theorem, which he’d just learned about. “He then noted that if you double the length of all the sides of the triangle you get a hypotenuse of length sqrt (4x4+2x2)=sqrt (20).” His father pauses. “This was when we first realized that Kevin had a special affinity for mathematics.”

At the end of section Kevin stays seated and asks the student next to him if he can do a proof for him. One of his legs is tucked under the other. While he explains the proof, the other student nods and smiles encouragingly, not needing to stop for corrections. “A lot of people tend to assume that he’s some sort of crazy calculator, but that’s not it at all,” Kevin’s father says. “We had to drill him on his times tables, he always forgets stuff like that. But he’s really good at the abstract.”

Office Hours

Paul himself is pleased to have Kevin in the course. “Some years ago the top student in Math 138 was a 13-year-old homeschooled student,” he says. “I myself had tried to learn calculus when I was 9 and had to give up because I couldn’t find anyone who knew any calculus and could help me.” He smiles when discussing his youngest pupil, his hand fiddling with his glasses. “I’m not at all surprised that there’s a youngster who can handle this course. And he seems pretty with it.”

For Paul’s office hours the door is always open. There are five folding chairs across from the blackboard that stretches from one end of the room to the other, and then the red armchair where Paul sits, holding court. Students go in and out to present proofs, talk about theory, discuss homework questions.

It’s here that Paul is in his element, correcting pluses and minuses in long mathematical sentences, coaxing students through the Bolzano-Weierstrass theorem, writing symbols on the walls. One student, who is heading out of a chat with Paul, lingers in the doorframe while another student begins a proof.

“Is it OK if I sit and watch?” he asks.

“Of course, of course!” says Paul, and the student takes a seat and gazes at the numbers and letters on the blackboard.

About Paul, Nagler says, “You know, no surprise, he’s also amazing at computer science.” Which is perhaps an understatement: Paul took a few years off from teaching in the late ’80s and ’90s to develop speech recognition software. The software relies on the fact that you can model speech as a so-called hidden Markov process, and some little mathematical tricks of replacing probabilities that are conveniently close to zero. Today it’s known as Dragon NaturallySpeaking: “You can buy it in Best Buy,” Paul says. It’s why he gets his own Wikipedia entry, along with his Rhodes Scholarship.

Modeling speech is a different type of research from his work in particle physics, but it has in it certain echoes: words, the building blocks, a universal theory. As a teacher, his work is partly in translation too. In class, Paul often insists that the mathematical symbology ∀, ∈, ∃, is the hardest part of math, and that it’s easy to impress your roommate when you know little things: “∀” means “for all,” “∃” is “there exists.” And then of course, there are the things beyond the symbols—the true thrill and genius of math, the part that Paul hopes his class can impart.

There are moments when the light breaks, even briefly, like it did for Kevin on a beach in Jamaica playing in the sand. “It happened last week,” Kandola says. “Paul was writing something. And he actually made a computational mistake, when multiplying a matrix. So he went on to the next part, and I was just looking over it like, ‘This should be something else.’ And then someone pointed it out, and I was like ‘Yes!’ Like, I’m not a moron. I actually picked up on this.”

“There was one specific proof that I sat through and just completely understood,” says Nagler. “Sometimes it all clicks and I really appreciate that.”

“It’s a bright bunch of kids,” Dhillon says about the students in his section. “It’s all pretty new to them, still retains that certain sense of wonder.” He snaps his fingers searching for an example. “Like, I don’t know, like definitely once you hear about the fact that there’s different orders of infinity the first time. It’s just absolutely mind-blowing, until you get gradually acclimated to seeing these unbelievable things on a frequent basis,” he says.

“It’s kind of cool seeing the initial blush of excitement.”