Sunday, July 9, 2006
Practicing Math May Sound Like Detention, but to These Students, It's Play
By John Fleck
Of the Journal
Jack Ingalls jumped up to the whiteboard and grabbed a fist full of colored pens.
"There's a really elegant solution," the Sandia Prep eighth-grader said as he began sketching out the answer to a math problem.
Switching between red, green and blue, Ingalls drew a series of circles, squares and triangles and explained the simple way out of what sounded like an impossible tangle of a problem:
You've got a deck of cards with a triangle, circle or square on each card, and each one is red, green or blue, and the colors are either light, medium or dark, so how many possible combinations of three cards are there which ...
The problem went on, and to a visitor unfamiliar with the terrain, Ingalls' solution sounded every bit as tangled as the problem.
But not to the students around the table. They grasped its elegance.
"Rule number one," said Bill Cordwell, a Sandia Labs physicist and the informal math coach for the Saturday morning gathering in the Manzano High School library: "Don't get intimidated by an impossible-sounding problem."
Welcome to the world of teenage math competitions, a subculture where elegance is a great virtue, and where problem-solving is play.
Math's competitive edge
Some 200 students from around the state, mostly high school-aged, gathered on the first Saturday morning in February for the annual UNM/PNM competition, which brings out the state's best.
Lined up in the front row were most of the kids from Cordwell's Saturday practice session, their brows furrowed, their eyes sometimes darting toward the ceiling of the University of New Mexico Lecture hall.
Find all positive integers "n" less than 1,000 with the following properties: the remainder when "n" is divided by 25 is 1, the remainder when "n" is divided by 7 is 1, and the remainder when "n" is divided by 4 is 1.
This is not the math we learned in school, the math of times tables and long division, cookbook solutions where the teacher explains how to deal with the problem.
In competition math, the problems are tricky, and you have to figure out the approach for yourself, explained Manzano High School sophomore Chen Zhao.
"Classroom math is fun," explained Ingalls in an interview, "but the competitions are special."
Cordwell looked out at the students gathered in the Manzano High School library one Saturday last fall in bemused wonder.
Fifteen of them were gathered around a cluster of tables, penciling furiously, poking at their calculators, chattering about the problems they were stuck on.
"There's a lot of brainpower here," Cordwell said.
If this were basketball and you were a sports fan, the name "Cordwell" would almost certainly ring a bell. The oldest of the Cordwell siblings, Bob, was a member of the U.S. team at last year's International Mathematical Olympiad and was one of USA Today's top 20 "All USA" students.
While he's attending the California Institute of Technology, his younger sister Kristin, who just finished her freshman year at Manzano High School, is taking up where he left off. She and teammate Zhao won this year's New Mexico Supercomputing Challenge.
The Cordwell's younger sister Katherine this year was the youngest student ever to make the finals of the UNM/PNM contest, competing as a 10-year-old fifth-grader. The previous youngest? Kristin Cordwell.
One part informal coach, one part teacher and one part enthusiastic cheerleader, Cordwell has become a catalyst for the math contest world, inviting some of the top math students in the city to play.
Think of a pickup game of basketball at a city park where all the best kids show their stuff, and you'll have some idea.
They gathered under the informal banner of "The New Mexico Math League." One wore a sweatshirt that read: "We are geniuses, not just barking mad."
Students surpass teachers
UNM math professor Cristina Pereyra, who wrote the problems for this year's UNM/PNM competition, said she has the same experience every year realizing the students are more clever than she is.
Writing the test problems is tricky. She wants them to be difficult enough to challenge students but not based on arcane mathematics that the students will not have studied.
Some are two-part problems something simple that leads to something a bit more challenging. "Give 'em a problem that's a bit easier," she said, "then a stretch."
And invariably, on some of the problems, the students will come up with solutions much more clever and "elegant" than her own.
You are given 13 points on a circle, equally spaced. Suppose each point is colored either red or blue. Can you always find three points of the same color that are vertices of an isosceles triangle?
The solution she had in mind was a bit laborious, involving drawing circles with complex Christmas trees of triangles inside them, trying to narrow down their options. But a few of the kids found a better way three simple steps to prove that the answer was "yes."
When the math kids talk about "an elegant solution," that is what they mean a simple path through the thicket to the answer lying on the other side.
To Pereyra, the elegance is a delight. Sitting in a UNM-area coffee shop a few days after the test, Pereyra pulled out a piece of paper and sketched out the solution.
"Isn't that beautiful?" she said, still excited at what the students had done.
A week later, at one of Cordwell's Saturday practice sessions, the students were discussing the problem and the various Christmas tree approaches, when Kristin pointed out the three simple steps. Her father looked at the board for moment, and then a smile came across his face as he saw how simple it actually was.
No brute force necessary
If you have enough time or computing power, you can solve a lot of very large mathematical problems by what the math crowd calls "brute force" trying huge numbers combinations of things to see where the answer might lie.
But in competitions, the best solutions involve clever insights that narrow down the possibilities, so you only have to look at a few things.
Ingalls said he enjoys playing golf for the same reason.
"I like it," he said, "because it's a game for which brute force is typically not going to get you anywhere."