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Not by the numbers

Exeter’s custom-made math curriculum fosters understanding through problem-solving. (Try your hand with our embedded math quiz.)

By
Patrick Garrity
October 23, 2019
Students work on a math problem together around a Harkness table.

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 are 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.”

Students in a math class at Phillips Exeter Academy.

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.”

Being wrong is part of the process here — and a good part of the process.”
Panama Geer

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, and newly arrived colleagues Jerzy Kaminski and Tom Seidenberg wanted in. Kaminski says 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. All three teachers wanted to offer materials that fostered more agency and what Kaminski called “intellectual courage.”

The trio, with the support of Math Department Chair Anja Greer, began writing a custom curriculum for 200-level courses. The three teachers would convene 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.

Students huddle around calculators to collaborate on a math problem at Exeter.

“Let’s contrast two situations,” Kaminski says. “One, you came to class and the teacher says, ‘Today we are going to learn about vectors.’ So, we all know that the class would be about vectors, and no matter what happens, you better use vectors, one way or another. Or at least say something intelligent about vectors, one way or another.

“The second approach, we want to solve some problems and I am not saying we are going to use vectors or if we are going to use derivatives or something else. You decide what is the best way to do it, based, obviously, on your previous knowledge.”

Without the contextual clues, students must call on all the math they’ve learned through the years, not just from that day’s lesson and not just to heed a teacher’s prompt. Kaminski calls the approach “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, Kaminski says, is precisely the point. Math is like an art form. “You want an environment in which students’ creativity is fostered. You don’t just go by rules you impose; you want students to figure out the rules on their own and apply their own rules and be aware that this is what they are doing.”

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, Kaminski and Seidenberg 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.

I don’t really think about whether this is a hard problem or an easy problem. I think about attacking it holistically.”
Jack Liu ’20

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.”

Two students look at a math problem on the board.

 

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.”

How’s your math?

A selection of problems from Exeter’s math sets

(Answers at the bottom of the page.)

The evolution of a solution

Image of two mapth diagramsExeter’s math problem sets are never complete. Each year, the math faculty reviews the 3,700-plus problems and makes modifications as needed. We asked Math Instructor Panama Geer to walk us through how a problem evolved over time:

In a series of problems, students discover the formula for the volume of a sphere. In the original sequence, students considered an inverted cone whose radius and height are equal and calculated the area of circular slices of the cone (see the right side of Figure1). Next, they considered a hemispherical dome inscribed in a cylinder  (with the same radius) and again calculated the area of slices of this shape, but this time slicing the shape formed by the space outside the dome but within the cylinder. These new slices form rings (see the left side of Figure 1) which, when sliced at the same height, have the same area as the slices of the cone.

Putting this together, they realized that the volume of the negative space outside the hemisphere but inside the cylinder was exactly the same as the volume of the cone. Because they already knew how to find the volume of a cylinder and the volume of a cone, they were able to deduce that the volume of a hemisphere was the difference between the cylinder volume and the cone volume, and consequently derive a general formula for the volume of a sphere. The problem had been set up this way for many years.

During the 2012-13 academic year, puzzled students in one of Greg Spanier’s classes were asking questions and wrestling with visualizing the shapes involved. Inspired by their questions, Greg wondered if student understanding might be improved by adjusting the setup of the problem. Greg suggested the diagram in Figure 2. Here students compare the area of slices of a hemisphere to the area of rings around a cone inside a cylinder.

The result is the same; each area, measured at the same height, is equal (assuming all the radii and heights are equivalent), thus showing the volume inside the hemispherical bowl is the same as the difference in volume between the cylinder and cone. Student questions inspired Greg, and in turn our editing group, to think of the sequence of problems from a different perspective, thus enhancing the understanding for both faculty and students alike.

Answers to the math quiz

Problem 1: 4 + 3 . 2 = 10; (4 + 3) . 2 = 14

Problem 2: Every parent knows an algorithm to solve this problem: One person divides the Snickers bar and the other person chooses. This is called the divider-chooser method

Problem 3: One way of coloring the graph is shown below. The colors are represented by the numbers 1, 2, 3, and 4.

Problem 4:  It is impossible to color the graph if only three colors are available. To see why, you could look at the state of Nevada and its neighbors.

Problem 5: 

Problem 6: 546 sec

Problem 7: (a) 53.9 min (b) yes, 50 min (c) 46 min; drive to a point that is 7.5 km from N

 

Editor's note: This article first appeared in the fall 2019 issue of The Exeter Bulletin.

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