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Lee Capristo, editor

The Mulberry Tree

Phone: (240) 895-4795

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St. Mary's City, MD 20686

By Lee Capristo

It’s the Putnam. The William Lowell Putnam Competition, a fiendishly difficult test of mathematics knowledge and creative problem-solving skill. Math junkies love it. Mr. Putnam conceived the competition to stimulate a healthful academic rivalry between colleges and universities back in the 1920s. The first competition, held in 1927, was actually in the subject of English, not mathematics. Since 1938, the competition has been run by the Mathematical Association of America. A three-professor committee spends four or five months devising 12 problems, worth 10 points each and arranged in order of increasing difficulty. “No calculators, no notes, no mercy,” says Lev Grossman, who wrote about the Putnam in Time (“Crunching the Numbers.” Time 16 Dec. 2002). Students compete both individually and as teams representing their colleges. Rivalries between colleges can be fierce or fun, as in the case of the Green Chicken Award shared between the colleges of Middlebury and Williams. The school that posts a higher team score (at a mini-Putnam they put together each fall) gets bragging rights with a large, green ceramic chicken for the year (Williams has it now).

The top five individual finishers, designated Putnam Fellows, get $2,500 each. One of them is also awarded the William Lowell Putnam Prize Scholarship of $12,000 plus tuition for graduate study at Harvard. The top five teams get between $5,000 and $25,000. Harvard, MIT, Stanford, Duke, and Princeton took the team prizes in 2007.

In the entire Putnam history, there have been three perfect scores of 120 points. In 2007, the top score was 110, but of the 3,753 students from 516 schools taking the exam, 1,596 (or 42% of test-takers) scored zero points. If you had scored just 2 points, you would have beaten half of your competitors.

Under the radar of the big names in this mathematical showdown are other colleges, like St. Mary’s. Associate (then Assistant) Professor David Kung convinced his five students in MATH 461 “Topology” to try it in 2000. They scored a total of 10 points. In addition to starting the Putnam tradition at St. Mary’s, Kung conceived the Putnam Lamp prize. “My grad school friend got a job at Gettysburg the same year I came to St. Mary’s. We heard that Williams and Middlebury had a competition and wanted to do something similar. We thought a strange, well-loved object should be the prize. I found the lamp at a thrift store. A student painted the lamp to give it its current, lovely look.” The lamp has gone to Gettysburg once for the higherscore, but has been held by St. Mary’s for the past five years, where it is proudly displayed in the math lounge.

In 2001, Kevin Beanland ’02 scored 11 points on the Putnam, beating the previous year’s total team score. In 2002, Kung turned his not-for-credit Putnam prep workshop into a one-credit class as student interest in the exam grew. That same year, Kristen Arrildt ’03 scored 20. In 2004, Peter Lo ’06 scored 22, the only year thus far that the St. Mary’s team has ranked in the national competition (St. Mary’s was 148th of all colleges and universities in the U.S. and Canada that year). Lisa Byrne ’06, who scored 10 points on the exam in 2004, went on to serve in the Peace Corps, where she taught physics and English (see page 17). Elizabeth Hargraves ’08 was high-scorer two years in a row (2006-2007) with a 10 and is looking for a job in mathematics. Daniel Tanner ’08 took the exam three times, scored twice, and is headed to Rice University for a Ph.D. in mathematics.

How does one prepare to take the hardest math test of all? In a class like MATH 391 “Putnam Seminar” or MATH 392 (General Problem Solving) the students learn problem- solving strategies and how different strategies are used across different areas of math. If they’re lucky, as St. Mary’s students are, they are taught by a talented teacher who motivates and inspires confidence. Keep in mind that it might be unsettling to be the professor who has to walk into a class like “Putnam Seminar” and lead students through problems for which he himself doesn’t know the solutions. Kung explains this dilemma: “What I’m really trying to do in the Putnam Seminar or in MATH 392 (General Problem Solving) is to help students see patterns in how to approach difficult problems (sometimes called problem-solving heuristics), and help them build better control mechanisms to use those heuristics.” As your skill with control mechanisms and heuristics improves, so does your skill in problem solving. Kung is known to interrupt his class while they’re working to ask, “What are you doing?” and “Why are you doing it?” and “What will it get you if it succeeds?” “Research shows,” he says, “that expert problem solvers know the answers to all three questions immediately at any point in the problem-solving process; novices can usually only answer the first.”

When Professor Ivan Sterling first taught “Putnam Seminar,” he went through the tables of all the Putnam problems and noted in reverse order the number of students nationally who solved them, to come up with “the easiest Putnam problems ever.” “I started with those for several reasons, the main one being that the more students that have actually solved a Putnam problem, the more students that will have at least some confidence during the exam. I think this paid off this year, when 15 students scored at least one point.” Another important tip that Sterling shares: whenever possible, draw a picture. “It can often help put things in the right place,” he says. Sterling rarely presents solutions himself in class. “I usually only do a problem myself when I have no idea how to solve it; that can give an accurate picture of what one mathematician’s approach might be.”

Kevin Beanland ’02, who’s now in his first tenure-track faculty position as assistant professor of mathematics at Virginia Commonwealth University, credits Sterling with nudging him toward a math career. “Getting a Ph.D. was not on my radar until Ivan suggested the possibility. He changed, for the better, the trajectory of my life.” While teaching as a graduate assistant, Beanland discovered that the top math students (as defined by GPA and test scores) were not always the most intuitive problem solvers.

Bruce Reznik, retired mathematician and former problem writer and grader for the Putnam, wrote an article about the skills that sharpen one’s likelihood of Putnam success in the book, Mathematical Thinking and Problem Solving (Ed., A.H. Schoenfield, Hillsdale, NJ: Lawrence Erlbaum Associates, 1994). He writes, “Success on the Competition requires mathematical ability and problem-solving experience, but these are not sufficient: a ‘Putnam’ temperament is also necessary. A contestant must be able to work quickly, independently, and without references and be willing to consider problems out of context.”

Peter Lo ’06, who scored St. Mary’s highest score yet in 2004, is a middleschool math teacher in Baltimore County. He appreciates what his St. Mary’s professors knew from their side of the desk now that he’s in the position of teacher rather than student: “One challenge I face daily is to teach these children at the level they can understand and learn. These 13- or 14- year-olds are developing and mastering abstract thinking, so it is difficult for them to learn algebra. I need to use the right instructional strategies (seat work vs. group work; think-pair-share) and tools (visual aids, technology) to enhance the lessons. When the students understand the concepts, it is, by far, the most rewarding part of my job. They usually just burst out, ‘I get this, this is easy.’”

Hanging out in the math lounge in Schaefer Hall, with the Putnam lamp and their own espresso bar, the St. Mary’s math students might be heard to shout the same thing on occasion.