Mathematical Contests¹

The Hungarian Eötvös Competition, first held in 1894, is credited as one of the first mathematical competitions for high school students. To this date (now named Kürschák Competition), the contest features problems that require the students to be creative and reward mathematical thinking, rather than simply check acquired technical skills.

Also in Hungary, the mathematical journal KöMaL started in the same year, aimed to prepare students and teachers for those kind of problems; and also still runs today. The journal invites students and teachers to write and send solutions to their problems, and features them in later editions. Paul Erdős and many more Hungarian mathematicians were among the high school students sending solutions to KöMAL.

Around the same time, Gazeta Matematică was born in Romania, with a similar purpose to promote the exploration of mathematics to middle and high school students.

International Mathematical Olympiad (IMO)

The first IMO was held in Romania, and was disputed among seven Soviet Bloc countries. It is considered today the most prestigious mathematical contest for high school students.

Yearly, each of the around 100 countries send six of their students to participate in the contest. There, each mathlete takes two 4.5-hour exams containing only three problems each; although the prompts do not require extensive knowledge to be understood, they are very hard to be solved.

To select which mathletes join the USA IMO Team in each year, the Mathematical Association of America holds a series of contests, AMC, AIME, USAMO, TSTST, TST; for which many teachers, institutes, summer camps, and universities prepare their students.

My path in Mathematical Olympiads

During middle school in Greater Rio de Janeiro, a new math teacher in my school held extra classes off-shift: the students learned about logic, calculus, and programming. He registered the school for the Olimpíada Brasileira de Matemática, prepared the students — including me — for the contest, and because of it, I was able to earn a scholarship in a private school in Rio de Janeiro. In the new school, I continued studying and competing in mathematical Olympiads and was twice part of the Brazilian IMO Team.

I certainly enjoyed taking these competitions, but for me the really cool part of being a mathlete was attending the classes that prepared for them. These classes were very distinct from the other classes I had.

In my usual high school classes, the teacher would be the one primarily talking, the students would copy what they said, read the textbook chapter related to the lecture, and solve the exercises there to prepare for the similar exercises on the test.

My Olympiad classes usually started with a handout consisting only of problems, many with very simple statements; then, the students would try to solve the one they found more interesting; when someone had an idea for a problem, they would silently write and develop their idea on the board; other students could decide to read the idea (and potentially collaborate, ask questions, etc.) or could decide not to read the spoiler (maybe because they hadn’t even started thinking about that problem, or because they thought they were close to a solution by themselves). When the board had many ideas about a problem and the problem was close to being solved, the teacher would gather the attention of all students to work, as a group, towards putting together those ideas into a full solution. At the end of most classes, the teacher barely touched the boards; nevertheless it was usually filled with much mathematical content.

I pretty much felt that my folks and I were developing mathematics in order to solve those mathematical problems.

Shift to a Brazilian University

In my first semester as a undergraduate student, in a Brazilian university, my classes looked a lot more like the former experience than the latter, and that made me sad.

During the lectures, the instructor would exclusively follow the textbook, make definitions, claim and prove theorems. There was little interaction between the instructor and students, and the students with each other. I didn’t feel connected to the material, to my professor, or to my peers.

Arriving at Haverford College

At Haverford, I feel like I am somewhere between those scenarios. Although my classes employ some ideas that deviate from the more straightforward style of teaching, things still work the same in the big picture: I feel like I am absorbing mathematics, rather than recreating it.

To be fair with Math Department at Haverford, this is not exclusive of here. On the contrary, most of those behaviors are standard in universities and are even imposed by rules around math academia.

My Math Project

One of my experiences at Haverford that helped me to explore mathematics was writing an expository paper for Haverford’s Algebra course. I enjoyed it very much. I was able to merge a problem that I find very interesting (how to send messages prone to misinterpretations) with the concepts I was learning in class (like manifolds) in a way that I could not have foreseen; which made me very interested.

Motivation and Ownership

To me, one of the key problems in my negative experiences was the lack of motivation originated by me and my peers.

Here is an extreme example on how things can quickly get lost:²

• The natural numbers from a monoid, described by the Peano axioms.
• $\mathbb{N}$ can be extended to a commutative group, $\mathbb{Z}$.
• Lo and behold, $\mathbb{Z}$ carries a ring structure.
• Rings are nice, but fields are much better: localize to get $\mathbb{Q}$.
• Rationals are great, but there are lots of gaps: the Cauchy completion has none: voila $\mathbb{R}$.
• …
• Now let’s prove Stokes’ theorem on $\mathcal{C}^1$ hypersurfaces in $n$-space.

In such a straightforward presentation of mathematics, the question “Why are we even doing this?” is usually answered with trust on the system, with something along the lines of “It’s gonna be useful to us to define things this way.” Rather than trusting on the instructor, for example, a good set of challenging problems can lead the students to want to define things certain ways — sometimes equivalent to standard ones, sometimes distinct — and make students actively look forward creating mathematics.

Sadly, I wasn’t able to find historical resources that could explain how the standard of mathematical teaching for undergraduate students settled as a linear presentation of ‘definition–theorem–proof.’ From my experience, it seems like teaching math classes like this is simply easier for the instructor. Since books and articles are written linearly, following the textbook material is not hard at all; since the whole class is following the same chapters and content, mathematical exploration outside the boundaries of the textbook is not encouraged at all, which makes the class more manageable for the instructor.

It makes sense to make the instructor’s job easier — it is already pretty challenging. However, the actual learning happens inside the head of the students! The goal when designing a class should be to make it easier for students to learn — as opposed to make it easier for the instructor to ’teach’. There is some but not much research in this area. For example, studies show that combining some form of visual representation (a figure, a diagram, or an animation) with text representation (a paragraph, or an audio clip) is more effective than the text representation alone; while the change from an image to an animation does not seem to interfere with the student’s learning. While the scientific findings aren’t clear and strong enough to heavily guide the design of courses, instructors should get help from their students — the real protagonists of the classroom — to help in organizing the subsequent courses and the curriculum itself.

The scenario at Haverford

Haverford already puts in practice some attempts to deviate from the standard way of teaching mathematics. There are some inquiry-based courses, that There are also group activities that happen during some classes. Additionally, all classes have student feedback forms — a (small) way in which students can help subsequent courses to improve and promote more learning. I still wish more though. I would really love if a course centered about problem-solving and student exploration of mathematics existed, offering another path in which Haverford students could interact with mathematics.