Mathematics competitions and contests

Summary: Mathematics competitions and contests help encourage students to practice and study mathematics and develop problem-solving abilities.

Well-designed mathematics contests provide excellent vehicles for students to hone their skills, expand their knowledge, develop their ability to focus, practice creative problem solving, and join a community of peers who love mathematical challenges. Mathematicians and educators organize competitions, help students prepare for them, participate on committees to grade the results, and assess contests’ long-term impact. Some mathematics competitors are known as “mathletes.”

MATHCOUNTS and USAMTS

There are a number of well-known mathematics competitions in the United States for middle school and high school students. MATHCOUNTS, a mathematics competition for sixth, seventh, and eighth graders, emphasizes problems from geometry, combinatorics, and algebra. The competition includes written and oral rounds with both individual and team competitions, and students advance from school, to chapter, to state, and to national levels. The USA Mathematical Talent Search (USAMTS) is an open mathematics competition for U.S. middle and high school students. USAMTS consists of two rounds of six problems per round and operates on the honor system, since participants are given a full month to work on the problems. The goal of USAMTS is to help students develop their proof writing ability, improve their technical writing abilities, and mature mathematically while having fun. The organizers strive to foster insight, ingenuity, creativity, and perseverance. The American Mathematics Competitions (AMCs) provide three levels of competitions. Students who perform well on the AMC 10 or AMC 12 exams, for students in grades 10 or 12 and below, respectively, are invited to participate in the American Invitational Mathematics Examination (AIME). Approximately the top 270 performers on the AIME and the AMC 12 advance to the United States of America Mathematical Olympiad (USAMO), which is the final round of the AMC series of contests. The top 230 AIME and AMC 10 only participants take part in the USA Junior Mathematical Olympiad (USAJMO). The top 30–40 performers on the USAMO, along with a dozen or so others from the USAJMO, attend the Mathematical Olympiad Summer Program, a training program from which the six members of the U.S. International Mathematical Olympiad (IMO) team are selected. Students who do well on the AIME typically receive scholarship offers from prestigious colleges and universities.

IMO

The International Mathematical Olympiad (IMO) is an annual two-day, six-problem, mathematical competition for pre-collegiate students that began in 1959. Approximately 100 countries send teams of up to six students. The problems are extremely difficult and involve ideas that are not usually encountered in high schools or colleges. Many IMO participants have become world-class research mathematicians, such as Noam Elkies, who eventually became the youngest full professor in Harvard University’s history at the time of his promotion. Filmmaker George Csicsery documented the 2006 U.S. IMO team in Slovenia. The documentary also included segments on families and schooling, girls, and the Olympiad, as well as the problems and their solutions. Melanie Wood, who was the first female to represent the United States in the IMO, noted: “Math competitions are great. They introduce all these new ideas and in particular give students who are at school the first chance to see how you can be creative in solving a problem.” She went on to obtain her Ph.D. in mathematics in 2009.

William Lowell Putnam Mathematical Competition

College students also participate in mathematical contests. The William Lowell Putnam Mathematical Competition is an annual mathematics competition for mathematically talented undergraduate college and university students in the United States and Canada administered by the Mathematical Association of America. The competition, in which both individuals and teams compete, consists of morning and afternoon three-hour exams, each with six problems. Although the problems are extraordinarily difficult and require highly creative thinking, they can typically be solved with only knowledge of college-level mathematics. The problems are so challenging that a median score for the 120-point exam is often 0 or 1. Of the more than 120,000 times the exams have been taken since the competition’s inception in 1938, there have been only three perfect scores as of 2010.

In recent years, about 4000 students and 400 teams have participated. The top five teams and individual scorers receive thousands of dollars in prize money. Many top five scorers, named as Putnam Fellows, have become distinguished researchers in mathematics and other fields, including Fields Medalists (the highest award in mathematics for people younger than 40) John Milnor, David Mumford, and Daniel Quillen, and Nobel laureates Richard Feynman and Kenneth G. Wilson. Several Putnam Fellows have been elected to the National Academy of Science. In 2010, Putman Fellow David Mumford received the National Medal of Science, bestowed by President Barack Obama.

MCM

Unlike other competitions, which place a premium on speed and individual performance, the Mathematical Contest in Modeling (MCM) contest rewards teamwork, research skills, programming skills, organizing ability, writing ability, and stamina. The MCM is a 96-hour mathematics competition held annually since 1985 by the Consortium for Mathematics and Its Applications and sponsored by the Society of Industrial and Applied Mathematicians, the National Security Agency, and the Institute for Operations Research and the Management Sciences. Approximately 1000 international teams of three undergraduates each produce original mathematical papers in response to one of two open-ended modeling problems. The students may use any references and the Internet but are not permitted to discuss their problem with anyone not on their team. Approximately 1% to 2% of the teams are designated Outstanding Winners. The skills required in the modeling contest are those typically most valued by employers. Many students who do not excel in problem-solving contests excel in the modeling competition.

Value and Benefits

The value of mathematics competitions is that they pique interest in mathematics and encourage students to pursue intellectual activities. The benefits of participating in mathematics competitions are very much like the benefits derived from athletic contests or becoming accomplished in playing a musical instrument. The intention is that those engaged in such activities develop a sense of accomplishment and a positive self-image. On the other hand, some object to mathematics being presented as a competition. While some students may thrive in a competitive environment, others may be discouraged. For some, the competitive environment highlights mutual interests, which can help create lasting bonds and friendships.

Like sports, participants in mathematics contests may learn to set goals and work toward them, be highly motivated, be able to focus, have self-discipline, perform under pressure, cope with success and failure, and have a competitive spirit. As in music, participants in mathematics contests must learn self-discipline, develop the ability to concentrate, pay attention to detail, and practice many hours. Perhaps the important lesson learned from participating in mathematics contests is that success is the fruit of effort.

Bibliography

Csicsery, George. Hard Problems: The Road to the World’s Toughest Math Contest. Washington, DC: Mathematical Association of America, 2008.

Flener, Frederick O. Mathematics Contests: A Guide for Involving Students and Schools. Restin, VA: National Council of Teachers of Mathematics, 1990.

Gallian, J. “The Putnam Competition from 1938–2009.” http://www.d.umn.edu/~jgallian/putnam06.pdf.

Rusczyk, Richard. “Pros and Cons of Math Competitions.” http://www.artofproblemsolving.com/Resources/articles.php?page=pc‗competitions.