Penn professor brings calculus to the masses through Coursera

Posted: February 12, 2013

Do you remember that the secant function has vertical asymptotes?

Neither did I, but it came right back to me when I listened to the soothing tones of Robert Ghrist.

He is a mathematics professor at the University of Pennsylvania, but since January, his audience has grown exponentially: 48,000 people in more than 62 countries.

Ghrist is spreading the gospel of calculus through an online education service called Coursera, and he scored a big vote of confidence last week when a higher-education umbrella group said the course deserved official college credit.

As a former math teacher who switched to journalism more than 20 years ago, I am inclined to agree, though it is not for everyone. Derivatives. Integrals. Taylor series. L'Hopital's rule. All this after a speedy review of functions and trigonometry (hence the secant asymptotes - more on that later).

Ghrist assumes his students - who hail from Santa Barbara to Sri Lanka - have a strong math background.

"You're about to dive into one of the loftiest achievements of human thought," Ghrist says in his opening remarks for the course in the warm, measured cadence of someone on public radio.

He is not wrong. Calculus is a vital tool in too many realms of human endeavor to count: building bridges, snapping digital photos, measuring which medicines cure the sick, sending rockets into space. But most people never take it, and when they do, Ghrist fears they are not taught in a way that allows them to grasp the underlying concepts.

He aims to change that with his videos, which feature colorful, animated images, diagrams, and equations, along with clear explanations. Didn't catch it the first time around? Simply pause and rewind.

And don't skip around.

"Calculus is a story, like an epic story, with grand characters, lofty themes, struggle, and eventual victory," he tells students. "I want you to live that story from beginning to end."

Of the 48,000 who have signed up for the 13-week course, just 8,000 have stuck with it as active participants. That is still a vast number - "every math teacher's dream," Ghrist tells me - and too many for him and his teaching assistants to help everyone who has questions.

So they help one another. The course website is a hotbed of international cooperation, with David from Slovakia explaining trigonometry to Valeriia of Ukraine, and Pilar from Spain giving tips to William from Oregon. Ghrist and his staff monitor the students' online discussion groups, giving them a helpful nudge if they veer astray.

The reason the secant has vertical asymptotes, by the way, is that it is defined as one divided by the cosine, which ranges from -1 to 1. So whenever the cosine is zero, the secant is 1 divided by zero - a mathematical no-no. In Ghrist's graph of the secant, the line periodically shoots off toward infinity.

Later, as the lectures dive into actual calculus, Ghrist takes issue with how many high school students are taught that the derivative of a function is the same as its slope. That is a narrow definition, and not very useful with functions that have multiple inputs and outputs, he says.

"That is an interpretation of the derivative, not a definition," he says. On the screen appear the words "Slope? Nope!"

He then goes on to sketch out several broader explanations, including that the derivative is "the limit as the change in input goes to zero of the change in the output over the change in the input."

True, but pay attention! Ghrist moves on pretty quickly to the next concept.

I decide to try my hand at some of the homework. On the first problem set, a review of functions, trigonometry and logarithms, I score just 8 out of 12 - ouch.

But cut me some slack. I've been in the word business for more than 20 years. In addition, the instructions did not point out that some of the multiple-choice questions called for more than one answer, which Ghrist acknowledged when I asked him later.

Had I known that, I would have gotten at least 9 of 12 - honest! Then I moved on to one of the actual calculus problems, and got it right. Haven't done calculus since studying engineering at Dartmouth College. Phew!

It is fun to feel part of a huge international experiment. The students clearly agree. Discussion forums indicate they range in age from nine to their mid-70s, from professionals seeking an edge to those who find math fun.

The recent endorsement by the American Council on Education means some colleges will accept Ghrist's course for credit, provided students pay for a third-party service to verify, identity, and proctor the exam. The fee is still being worked out, but it will be less than $200, of which Penn will get a share.

Current students signed up before they knew of that arrangement, clearly out of a pure thirst for knowledge.

"It's fantastic," says Stefania, a math teacher in northern Italy who seeks to hone her skills.

Then there's Kathy from Michigan, whose goal is to be able to help her son with his AP calculus class.

Maybe I'll be dropping her a line.

Contact Tom Avril at 215-854-2430 or