Not by the numbers

It’s nearly noon on a damp September day as uppers and seniors spill into Panama Geer’s second-floor classroom in the Academy Building. The students park backpacks beside their chairs and immediately head to the whiteboards sandwiching the century-old room.

In pairs, they talk through solutions to the advanced math problems assigned for the day. The hieroglyphics that take shape in dry-erase marker would be ciphers to many observers, but the students in Math 400-B seem comfortable with this code.

“OK, guys, we have to start talking about the problems pretty soon,” Geer announces as progress slows on a few outstanding problems.

“I see three hard problems still out there — well, there are more than three hard problems — but … let’s get going.”

The problems the class tackles on this day and every day are Exeter’s own. They are a handful of the more than 3,700 problems the Math Department has developed through the years to spiral concepts throughout the curriculum and build problem-solving skills that emphasize not the answer but rather the winding path to get there.

“I think there’s a general feeling in the way that we do mathematics that being wrong is part of the process here — and a good part of the process, actually,” says Geer, in her eighth year as a math instructor at PEA after years teaching internationally and at the college level. “And, that wrong answers are often more informative than correct ones.”

Recurring themes — symmetry, optimization, vectors, graphing, rates of change — are woven throughout the sets, so a student will encounter them repeatedly, from term to term, year to year.

“If you open up an algebra book, the problems at the end of each section expect the students to basically only use the algebra techniques of that chapter to solve any of the problems,” says Gwyn Coogan ’83, Math Department chair. “You’ll never have to recall ideas from geometry and you’ll never have to draw a picture. Same thing with your geometry class. Your geometry book doesn’t expect you to be able to do any algebra.”

If silos stifle understanding, so, too, does stagnancy. That’s why every summer, the math faculty gathers to make sure their custom curriculum is ship-shape. For a week in late June, instructors converge on a classroom for an exercise in mathematics democracy.

They debate alternative approaches to solving the problems — tapping into their own experiences with the curriculum and feedback from their students — and try to ensure that the content is current and relevant.

“The body of knowledge of math is getting bigger and bigger,” Coogan says, quoting her Math Department colleague Xitai Chen. “We want to make sure that what we show our students is as good as it can be.”

The new math

Exonians have been studying math since John and Elizabeth Phillips signed the Deed of Gift. Longtime teachers like Don Dunbar, Bill Campbell and Frank Gutmann steered generations of Exeter math classes with aplomb. Many studied math texts written by PEA instructors such as Dick Brown and Mary Spruill Kilgore.

When Rick Parris joined the Math Department in 1978, a new idea took shape. Parris began developing supplementary math problems to the approved texts for his classes. As he wrote in 1984 in a summary of grant work: “My interest in such problems is in part due to the pleasure I get from working them myself, but it also stems from my belief that the only students who really learn mathematics well are the ones who develop the staying power and imagination to be problem-solvers.”

By the early 1990s, Parris was teaching his classes almost exclusively using his problem-based materials. Parris, who died after a short illness in 2012, chafed against the routine of teaching material from a book in class, then assigning homework on that topic. He and some of his colleagues wanted to offer materials that fostered more agency.

With the support of Math Department Chair Anja Greer, Paris and fellow instructors such as Tom Seidenberg began writing a custom curriculum for 200-level courses. The teachers would gather at Parris’ house in the evening, writing problems for the following day and generally making them up on the fly. Over time, problems for the higher levels were added to the catalog. All emphasized discovery over destination. Rather than handing someone a hammer and nails and saying, “Build me a house,” the idea was to simply point them toward the toolshed.

Without the contextual clues, students must call on all the math they’ve learned through the term and over the years, not just from that day’s lesson and not just to heed a teacher’s prompt. The idea echoes “Pascal’s Method.” The 17th-century French philosopher argued that “people are generally better persuaded by the reasons which they have themselves discovered than by those which have come into the mind of others.”

The Harkness table has proved to be the perfect arena for this approach. Students dig into the material before class, then sort through their work together, bouncing ideas off one another and often winding up in a very different place from where they began.

“When I do math now, I do it without any prejudice,” says Jack Liu ’20, who is early into a two-term sequence of Math 610: Multivariable Calculus. “I don’t really think about whether this is a hard problem or an easy problem. I think about attacking it holistically, trying not to make any presumptions, trying not to jump to the answer. To really feel out the math, and be creative with it.”

That is precisely the point. Math is like an art form, and just as in art class, the idea is to develop an environment in which students’ creativity is fostered. Students figure out the rules on their own and apply those rules as they see fit.

Work in progress

The problem sets have become Exeter vernacular. Every student studies them. By the end of prep year, students are as familiar with the recurring character “Alex the geologist” as they are with their roommates.

But the sets are not dogma. The work Parris and company spearheaded nearly three decades ago was the foundation, but the house is never finished. Problems evolve as new paths to an outcome are discovered. Problem No. 155 in the Math 2 sets is a good example. Through the years, students from the classes of 2005, 2016 and 2019 have come up with viable solutions, as has a math teacher in Cleveland and a student in Chicago (the sets are published online). Each of these approaches was vetted by the math faculty at the conclusion of the school year and, once it passed muster, was added to the “commentary,” or teachers’ notes.

Last summer, several members of the math faculty gathered in Room 108 — Geer’s classroom — in the Academy Building to vet the recommendations and make the updates. Visitors to Room 108 are greeted by a cartoon from The New Yorker on the classroom door. In it, a group of lab-coated academics cower from a colleague gone mad above the caption, “Give him whatever he wants! He’s threatening to divide by zero!” It is an early giveaway that math happens there.

Geer guides the annual debate — which is limited to one week — in a process she calls collegial if not always smooth. “There are a lot of different types of mathematicians, and we all bring different strengths to the department, but it’s one of the few times when we, with all of our varied strengths, come together and really talk math,” she says. “It’s not necessarily about classroom techniques. … We’re really diving into the layers of mathematics. And because we are writing it ourselves, there’s a freedom associated with that. We’re uncovering the ideas, or trying to lead students to uncover ideas in ways that we think are the most fulfilling for them mathematically.”

That the discoveries often come from students makes the work even more rewarding, Geer says.

“We might have viewed a problem as, ‘OK, this is a kind of problem students really need to do algebra for. Or they might need to do a lot of pencil calculations for.’ And then there’ll be a student who does it completely differently. Like they’ll just completely tackle it from some obscure picture, or make a connection to something that they did in a science class and they’ll say, ‘Well, if you think about it this way instead…’ That’ll peel the layers off the onion and suddenly open up a new way of visualizing, a new way of allowing the students to uncover the ideas.

“Once that seed has come up in a classroom, a colleague will say, ‘Hey, my students did this. Isn’t that cool?’”

Technology has forced the problem sets to evolve, too. When Parris began writing problems, Bill Gates had yet to launch Windows and the internet was years away from popular embrace. The tools available to students have changed, and asking them to do certain things they were asked to do 25 years ago is akin to a mandatory course in cursive writing.

“If the curriculum that most kids are learning before they get to our school has changed, certainly we want to change to reflect that, to move them from wherever they are to wherever they think they should go,” Coogan says. “You have to move kids closer to the edge of mathematics.”

Back to class

The seniors and uppers in Geer’s Math 400-B class get stuck on No. 714, a three-part problem about how lottery commissioners should invest their revenue. Bona Hong ’21 rattles off her work for parts A and B to consensus, but Part C remains elusive.

“I guess I’m not really understanding the question,” someone says.

“It’s a geometric series,” Cheikh Fiteni ’21 submits.

“Let’s write it out, so we can all see what you’re suggesting,” Geer says.

As Fiteni scrawls an equation on the whiteboard, his classmates start to plug in the numbers to find the solution.

“The initial investment must be $363,344.40,” announces Jackson Carlberg ’21. The other students agree. No. 714 is solved. Next problem.

“That was a pretty good collaborative endeavor, everyone,” Fiteni declares. “Teamwork!”

The class works through more problems, with the discussion punctuated with words of encouragement from Geer: “Perfect!” and “You guys are gooood.”

B block concludes with Problems Nos. 722 and 723 left unresolved. They’ll be first up the next time the class meets.

As the seniors file out, Geer congratulates them on their effort.

“Super productive today, guys,” the teacher says.

“Thank you, Ms. Geer,” shout back several of the students. “See you tomorrow.”