Making math fun by prepping for friendly competition
Mark Saengrungkongka, a first-year MIT undergraduate scholar, stood on the blackboard and defined his answer to a math downside just like those that may seem on the William Lowell Putnam Mathematics Competition, a prestigious annual math competitors for faculty college students in america and Canada administered by the Mathematical Affiliation of America (MAA). After he completed presenting his proof, the category gave a spherical of applause. “That was a really good answer,” math professor Yufei Zhao instructed the category.
It was a number of weeks into the semester, and whereas there have been a number of latecomers, college students in Class 18.A34 (Mathematical Drawback Fixing) paid shut consideration to the intricate proofs introduced by their fellow college students on the blackboard. The viewers requested probing questions and identified gaps within the arguments.
This undergraduate seminar, higher often called the Putnam Seminar, brings collectively first-year college students who’re within the annual competitors. Lately, MIT has completed first within the December examination, and all 5 top-scoring college students, often called Putnam Fellows, have been from MIT the previous two years.
However the Putnam examination can also be designed to only instill a love of math to all who try the insanely onerous downside units. One of many objectives of Zhao’s class is to make fixing these issues extra like collaborating in a enjoyable group puzzle somewhat than a worrying competitors. For a lot of first-year college students, it’s additionally a pleasant strategy to ease into MIT life.
“The scholars within the seminar often are available with a robust preparation from math competitions,” says Zhao. “However faculty is somewhat totally different from highschool math Olympiads. A aim of the seminar is to assist them transition from a highschool math Olympian to a profitable faculty scholar and past.”
Every week, Zhao begins his seminar gently, with an off-the-cuff dialogue. He asks the lessons how they’re feeling in regards to the semester and talks about life as a university math scholar. Discussions embody class choices, coping with setbacks, and profession paths. One scholar asks about the best way to discover analysis alternatives; one other scholar asks about suggestion letters.
Zhao is aware of that his college students are already considering forward about upper-level math lessons, however he hopes to gradual them down a little bit in order that they will take their time to essentially perceive and respect what they’re studying. “There’s an inclination for these college students to do an excessive amount of,” he says.
About 10 minutes after the beginning of the category, Zhao wraps up the dialogue and begins the coed shows.
Mohit Hulse introduced an answer to a combinatorics downside from the 2018 Putnam Competitors. When he realized that he made a small mistake in the midst of his presentation, there was some supportive laughter, and he confidently continued on. The viewers was respectful, and his classmates usually helped with an thought.
When he was completed, Zhao added compliments and ideas. “I recommend trying up the proof of the Chernoff certain, which is kind of comparable,” he says. “That was a pleasant answer.” He then pointed to the board and gave some strategies. “I assumed this step might be omitted.”
Along with discussions and scholar shows, the seminar additionally options weekly lectures by upper-year college students, together with veterans of the Putnam Seminar. The lectures spotlight math problem-solving strategies helpful for the Putnam Competitors in addition to present a lens into superior arithmetic.
This 12 months, roughly 60 incoming first-year college students utilized to the Putnam Seminar, amongst which 21 have been chosen. The seminar is internationally numerous, with college students from 10 international locations exterior america: Australia, Armenia, Canada, China, Georgia, India, Korea, Portugal, Singapore, and Thailand.
“This class builds connections,” says Zhao. “All of them are new to MIT, and plenty of are arriving within the U.S. for his or her first time. They’re all fascinated by arithmetic. I hope that the seminar will assist them meet different college students and type a supportive group.”
Any MIT scholar is welcome to attend the lectures, though the presentation classes are restricted to the seminar college students. These downside units are additionally made accessible through MIT OpenCourseWare for different college students and lecturers.
Working towards shows
The seminar is designed to offer a uncommon probability for first-years to develop their mathematical communication expertise, together with blackboard presentation and proof writing. Zhao says he has been innovating on the format of the seminar lately, with ever-increasing emphasis on oral presentation observe and suggestions.
“Undergraduates don’t get a variety of presentation alternatives, particularly blackboard shows,” says Zhao. “We hear rather a lot from MIT alumni that they want that they had obtained extra coaching in communication expertise at MIT.”
Two earlier seminar college students, senior Dain Kim and sophomore Tomasz Ślusarczyk, assist the category as undergraduate assistants. Final 12 months, Kim ranked sixth place within the competitors, and was awarded the Elizabeth Lowell Putnam Prize for being the highest feminine scorer, and Ślusarczyk earned an honorable point out. As undergraduate assistants, Kim and Ślusarczyk maintain common workplace hours the place college students come to observe shows to a small viewers of some different college students, with out the professor current. These workplace hours began final 12 months in response to college students in search of extra presentation alternatives exterior classroom hours.
Kim says that she benefited rather a lot from taking the seminar as a first-year scholar.
“Particularly in math lessons at MIT, it’s onerous to get an opportunity of giving a presentation to different college students, until it’s a CI-M [Communication Intensive in the Major] class, as a result of most lessons are lecture-based,” Kim says. “I might hear from different college students how they approached the issues that I couldn’t clear up, and I might additionally observe math shows.”
Ślusarczyk, who took the seminar final 12 months, credited the category with transitioning a few of his math contest approaches and mindset to research-oriented arithmetic. “Combining the problem-solving focus with a better degree of mathematical maturity was positively an ideal instructional expertise and improved my Putnam expertise rather a lot,” he says. “The talents developed within the seminar have been invaluable throughout problem-set classes, workplace hours, or analysis conferences. The category positively helped me rather a lot with my profession plans — I discovered rather a lot about research-oriented math and determined that I wish to pursue analysis in a math PhD program after commencement.”
The competitors
The Putnam examination was based in 1927 by Elizabeth Lowell Putnam in reminiscence of her husband William Lowell Putnam, and has been provided yearly since 1938, administered by the Mathematical Affiliation of America.
Final 12 months’s grueling six-hour examination featured 12 proof-based math issues, every price 10 factors, drawn from calculus, algebra, geometry, combinatorics, quantity principle, and extra. The 2021 examination was taken by 2,975 undergraduates from 427 establishments, 150 of them from MIT.
The highest rating was 119 out of 120 factors, with the median rating a mere 4 — which meant that almost all college students didn’t absolutely clear up a single downside. However of the highest 105 scorers who completed with honorable point out rankings or larger, 63 have been MIT college students.
The highest 5 scorers obtain the distinguished title of Putnam Fellow. For the second time within the competitors’s historical past, all 5 Putnam Fellows got here from MIT, and so they have been all Putnam Seminar alumni. Within the greater than 80 years of the Putnam Competitors, solely eight rivals achieved the rarified four-time Putnam Fellow standing, together with three from MIT. MIT math professor (and former MIT Putnam coach) Bjorn Poonen was one such four-time Putnam Fellow when he was an undergraduate at Harvard College.
Again in 1973, when MIT math professor Richard Stanley began instructing at MIT, he observed that MIT had numerous college students taking the Putnam. He had taken the examination 3 times throughout his Caltech years.
“My finest end result was ninth general,” he recollects. “At the moment, Caltech was the dominant Putnam college. I can bear in mind my freshman 12 months one of many individuals in my scholar home complaining that though he was seventh within the nation, he was solely fourth at Caltech and second in his hallway!”
When he checked out how MIT college students fared on the examination, he thought that there was room for enchancment.
“I assumed {that a} seminar on problem-solving would possibly fire up some curiosity and profit the scholars,” he recollects.
Stanley’s undergraduate seminar on mathematical problem-solving featured his weekly lecture dedicated to a Putnam-related matter reminiscent of quantity principle, linear algebra, producing features, or inequalities. “These are big topics, however the lectures have been very narrowly centered on offering background and examples for doing properly on the Putnam,” says Stanley.
He would then assign two downside units every week, one based mostly on the lecture, and one other devoted to “enjoyable” issues. The second class that week had college students talk about their options.
“The category’s second focus was simply the enjoyment of issues,” he says. “Many math college students get pleasure from problem-solving and the chance to be taught fascinating tidbits from all areas of math that aren’t more likely to be discovered in normal lessons. Additionally they just like the camaraderie of pleasant competitors.”
Stanley was additionally the Putnam coach for about 35 years, later joined by Professor Hartley Rogers. The Putnam class appeared to drag MIT college students into many first-place staff victories. Eight alumni from that seminar ultimately grew to become Stanley’s PhD college students, and he wrote a ebook, “Conversational Problem Solving,” based mostly on his seminar.
Above all, Stanley emphasizes placing the enjoyable in Putnam. Moreover, he says, there’s a draw back to all of the successful.
“It’s as if the Crimson Sox dominated all different groups yearly and received each World Sequence in 4 video games,” says Stanley. “Nice for Boston followers, however not for baseball in basic.”
As a first-year scholar in 2006, Zhao attended the Putnam seminar taught by Stanley and Rogers and went on to earn three Putnam Fellow spots; he missed the fourth by a single level. When he returned to MIT in 2017 to affix the school, he started instructing the Putnam Seminar.
Zhao credit Stanley for creating high-quality downside units, that are nonetheless getting used within the seminar. And Zhao can also be persevering with Stanley’s aim to steer college students away from seeing the seminar as merely coaching for Putnam.
“The category helped me to take the Putnam examination as a by-product,” says Kim. “Each week I solved an issue set, and I loved attempting some onerous issues. It helped me to get used to Putnam-type issues. However this isn’t only a ‘Putnam preparation class.’ Though we’re speaking about Putnam issues and a few ideas seem there, the aim of the category just isn’t ‘elevating the scores of contributors.’”
In the beginning of a late-November seminar, Zhao requested his college students how they felt in regards to the upcoming examination in lower than two weeks, and some talked about the phrase “pressured.” However others mentioned how a lot they simply loved engaged on the issues. “I wish to take it with out a lot strain,” mentioned one scholar. Added one other, “What works for me is that I’m competing with myself, and never others.”
Zhao then gave the category his Most worthy recommendation: “Ensure you get plenty of sleep the times main as much as the examination.”