Unlike an ordinary math class, which works sequentially through some sort of curriculum, in math team we work solely on problem solving. The problems we work on are unlike anything seen in an ordinary high school math classroom. They do not merely require solving for x and do not require knowledge of some grand theory to attempt. Yes, we do learn little tricks and techniques, allowing us to solve certain types of problems, but by and large the main skill we practice is creativity.
When you’re given some difficult math problem to solve, there is no single formula or equation to swiftly yield the solution. There are no rules for what steps may be allowed. A difficult math problem is instead like a blank canvas. There is/are a limitless number of possibilities for solving a given math problem, and it is up to you to figure out which one might lead you to a solution. Solving hard problems requires a mixture of creativity, intuition, and persistence.
During a recent practice, I gave my students a problem in which they were given three sides of a triangle and were asked to determine an altitude. One student used a brute force method: he set up a system of equations using the Pythagorean theorem applied to several right triangles. Another student astutely used Heron’s formula to first determine the area of the triangle before she found the altitude. A third student noticed that the given side lengths all appeared in various Pythagorean triples, and she could reverse engineer the solution from knowledge of these special triangles. A fourth student used his geometric intuition to approximate the length of the altitude, verifying whether his estimate was correct.
Watching all of these solutions unfold simultaneously, I was struck at the diversity our math team’s thought-process, how two students sitting next to each other might come up with two totally different ideas.
Every year, we travel to compete in two prestigious and difficult math contests: the Harvard-MIT math tournament (HMMT) and the Stanford math tournament. These contests bring in over 900 students from many of the elite high schools in the US and China. The problems at these contests are extremely demanding. They require intense concentration, novel solutions, and an open and agile mind. Both of these contests feature individual rounds in which students are working on problems individually, as well as team rounds in which students work in groups of 6 to 8. These team rounds are exciting to witness: students may be collaboratively writing mathematical proofs, animatedly debating the merits of a difficult problem solving process, or working on fast-paced “guts” problems. The “guts round” is the finale of the HMMT where students move through ten sets of problems as quickly as possible. Beginning with a set of three problems, students solve these as quickly as possible before running their solutions to the front of the auditorium to pick up another set of increasingly difficult problems. Their solutions are scored in real time and displayed on a large screen for the world to see.
Last year, our team did well at both Stanford and HMMT, and several of our students placed within the top 10th percentile. We prepared hard for the HMMT, which took place in mid November, as well as for a statewide contest in Colorado the week earlier. We brought 17 students to Colorado and 11 students to HMMT. While math contests are typically dominated by the male sex, it’s refreshing to note that our math team is evenly split between girls and boys.
The trip to Colorado was, for the most part, a big success. The most exciting part of the competition was the team round. In teams of three, we faced other schools in fast paced rounds consisting of 12 questions. The questions were read aloud, which was somewhat of a disadvantage to our students who are more comfortable with reading the question on their own. Our students had to buzz in the correct answers before their opponents in order to gain points. As a member in the crowd of spectators, I could see that it was clearly a nerve-wracking situation for the students at the tables on the stage.
Here are a few examples of the questions where students had at most 30 seconds to answer during the team rounds: How many diagonals does a hexagon have? What is the largest three-digit prime number? A dog walks two miles east, three miles south, then six miles west; how far is it from its starting point?
During the day of the Harvard-MIT Math Tournament, the high energy level of the students and teachers was palpable. In the morning, we arrived bright and early so we could talk about final changes and last minute strategies going into the competition. As we sat in a vacant classroom imagining we were MIT students, we could feel the excitement growing as more and more students from other schools showed up. In total, there were 135 teams and 800 students from the very best high schools in the US and China (the winning team came from Shanghai).
The first two rounds of the day were individual tests in which students had one hour to solve extremely difficult problems in algebra, combinatorics, probability, geometry, and number theory. Of the 800 students taking these tests, most were not able to solve any problems. Last year at HMMT, we had two seniors who were highly successful in the individual test. This year, our team had no stand-out mathlete. The problems we faced in these first two rounds were more difficult than anything from prior years. Even though we did not have high scores in the rankings during these early rounds, both Ronald Chiang and Chloe Jeon had some successes.
The third round of the day was a difficult team test. This was the first time our students were allowed to work together to solve problems. Although this test was much harder than the individual tests, our students tend to do better in a collaborative setting. Their problem solving skills complement each other extremely well and they do a great job of divvying up tasks between their areas of expertise. We did not find much improvement in our score on this test, but I could tell that the math team felt more comfortable and confident solving problems together.
After lunch came “guts”, the grand finale and most exciting round of the day. Guts is another collaborative round in which students work together in teams. The venue of this round is very different from previous rounds: instead of working in classrooms, a huge lecture hall is taken over by hundreds of students split up into teams. Each team strategically stakes out a small place in the lecture hall. Problems are given in sets of three–once a team solves all three, a student runs the problems up to a proctor who gives the student another set of three to solve. This repeats for 12 rounds. The problems require quick solutions; the more problems that you solve, the higher your score is. Meanwhile, a huge scoreboard is projected in the front of the lecture hall updated in real-time. As teams solve problems, they know exactly where they and their opponents stand based on the previous rounds.
The guts round is where we saw our largest improvement over last year. This round is incredibly exciting for our team, and it is well suited to our strengths. Round after round, our team was able to earn a perfect score. The problems required much creativity and ingenuity, and on top of this was the incredible stress of working in the lecture hall with hundreds of other students. We worked very well together, and when one of our students hit a roadblock, another was there to keep the momentum going. Our students came out of the guts round buzzing with excitement after having their best round at the end of a very long day.
Our two teams were ranked 92 and 123 out of 135 teams. These numbers were much higher in the guts round, but still we were far from the leaderboard. Though we still have a huge way to go if we ever want to win HMMT, our students felt very accomplished by the event. When I showed our team our results, they were ecstatic. We spent much of our plane ride home working on the problems which had stumped us in the tournament.