High school life, research topics, and advice for the young aspiring mathematician. (Part 2)

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This post will be a continuation of the content found in Part 1 which can be accessed here. In Part 3, I plan to end with a list of topics for those who are interested in writing an Extended essay in mathematics as well as some information for those who wish to study maths in university.

Specifically, this post will cover:

  • The process of writing a maths essay versus a maths contest
  • The most important characteristics of an aspiring mathematician

Recapping from part one, I was just taking my first couple steps into rigourous mathematics through an extended that was to be written on a mathematical topic independently for the IB program diploma. The only trouble was that I had no idea what to choose for my thesis and I could not depend on my adviser to guide me through picking one because he mainly worked in physics.

The only out-of-high-school mathematics topic that really stuck in my head during that time was Cantor’s set theory and cardinality concepts, which was fresh in my mind because I recently attended a workshop on them during my visit to CMS maths camp. Mainly his diagonal argument bijecting the natural numbers to the rational number was what really interested me.

... and Rubiks cubes. We had a lot of talented Rubiks cube players. But who would ever write an essay on Rubiks cubes anyways? {acobox.com}

Going on this small hunch that this topic could lead to a potential thesis question, I headed down to the library, read set theory books and browsed the internet hoping to come upon something interesting. However, most of the books and content that I did get a hand on (about 80%) was far beyond the scope of what was taught in the high school curriculum, so I did what any reasonable young mathematician would have done. I borrowed some books about how to read mathematics books.

This took about a week to really get the notation and symbols such as \forall, \exists, \bigoplus, and \ vs / which you normally would see in only first to second year university mathematics.¬†Once I got up to speed reading the set theory related books though, I decided to choose the broadest thesis that would accompany all the knowledge that I gathered so far. Something to the tune of “How could one construct a system for classifying the cardinality of any given set based on how the set is constructed?” what was immediately came up.

What followed was an entire year to construct an answer to my thesis, all under the constant pressure of course work and mandatory community service. But to me this didn’t matter, as long I spent more than a couple of minutes each day pondering about the solution, this kept me going and interested. New ideas would spring up constantly and unexpectedly because of this and sometimes even in my dreams! What kept me moving was living the experience and looking at what would eventually be there at the end.

Kind of like running a marathon, except without the burning sensation in your legs and lungs {acobox.com}.

4000 words and 21 pages of mathematics later, I completed my first draft of my paper in July 2009. I brought it to my adviser who was amazed at all the new concepts that he had never seen before and praised me for going the extra mile for keeping the maths simple and succinct.

Here I would like thank Mr. Vaithilingham (my adviser) for his guidance and support, offering advice about the format and alternate languages I could use to write my paper (this was where I had my first exposure to LaTeX). Although he did not go into the subject matter, he was kind enough to provide me with sample papers and offer me advice about what criteria my paper should fulfill.

In September 2009, I submitted my paper and in January 2010, I received the results. I received an A, the first one ever in the history of my high school. But while the result certainly made me happy, I felt that the effort and time that I put into writing my paper were the true rewards.

So what advice could gained from my long-winded story for the young mathematician? Well for that, we have to visit another one of my experiences in high school (last one, I promise).

Before, in the first part of this series, I mentioned that I wrote the Grade 10 Cayley Waterloo contest in grade 10. Well, before ending my high school career, I also took three other contests, the Fermat (for Grade 11) in grade 11 and the Euclid contest (for Grade 12) twice in grade 11 and 12. I can say that all four experiences were many distances apart from the process of writing a maths paper.

For example, I can guarantee you that there is no one way to effectively study for these contests other than perhaps reviewing your grade 11 formulas (no calculus was necessary). But with only pure memory skills, one can expect to get about 40% on the Euclid or 80% on the Fermat at best.

What is really necessary in these contests is a sharp intuition and the ability to apply shortcuts, shortcuts and more shortcuts. This is because contests are structured to test how quickly a contestant can blitz through the first half and apply critical thinking skills and deduction to the later half. This is especially important in the Euclid, where the general consensus (among my peers who have written Euclid) is that the first six questions should be written within the first 30 minutes while questions seven to ten should be done within the next two and a half hours; they’re that challenging.

"I knew I shouldn't have spent those thirty minutes trying to remember the area of a tetrahedron!" {CSULA}

This is where you can see the line drawn between the two activities. While writing a paper requires endurance and perseverance, a contest is more based on quick intuition, spontaneity and a sharp creative mind. But which of the two is more important? While having all these traits is necessary to become a successful mathematician, I personally believe that the former list outweighs the latter in terms of importance.

Why, you ask? Well, the hard fact is that mathematics is not easy. In fact, many of the problems that you will encounter in your graduate or even undergraduate career will not be solved within a day or even two. As an example, Andrew Wiles, the famous mathematician who proved Fermat’s Last Theorem, spent nearly ten years on that proof alone, which is a great testament to his commitment.

I found out about this first hand when I first entered into the “Advanced” mathematics classes in the University of Waterloo. Suddenly, problems on assignment because mini projects often taking several days to solve. It was very different to the finish-in-one-day assignments that I found in high school and part of first year university.

As further reinforcement, the author of “A Mathematician’s Survival Guide: Graduate School and Early Career Development” (a book that I would highly recommend) echoes these sentiments. In one chapter he describes a student, a young prodigy, who excelled in all his upper year maths courses, aced Putnam contests (the university version of the Euclid) regularly and was accepted to all the top graduate schools in the country. Yet, when he finished his graduate thesis, earning his Ph.D. the young student was

"so disgusted with mathematics that he quit and went to work in a
genomics laboratory"

So what happened? Well the author goes to explain that the student, during his undergraduate years, was always on the “Putnam Exam mindset”, always expecting to receive quick results without any hard work. He just didn’t have the tenacity or discipline required to do graduate studies.

Trying hard, persevering, these are the necessary things required to succeed in not just mathematics but in any subject that you study.