Span and Deliver

Emma R. Carron, Amanda J. Guzman, and GARY L. NORRIS

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.


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.”


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