High school life, research topics, and advice for the young aspiring mathematician. (Part 3 END)

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This will be the final post in my three part saga about mathematics and high school. You can check out parts one and two here and here respectively if you haven’t already read them. In this post I want to end off by offering the final part that was mentioned in the title, research topics (I have decided to write about advice for the upcoming mathematics undergraduate in a different series some time far in the future).

If you have been reading any of my previous posts, you might remember me talking about how a critical part of learning mathematics is knowing how to explore, be creative, and think about problems outside of the classroom. It is better to sow your tiny seeds interests early than later, because as many of you find out from learning maths, many of the areas  interconnect beautifully if you just take a while at the subtleties.

Thus, below, I have provided some topics of interest that can be researched at the high school level but delve deeply into undergraduate topics; all they need is a bit of perseverance, time, and creativity to get started. You can probably even get your teacher in on this. Ask him/her if instead of completing droll textbook problems or monotonous exercises, you could start a research topic instead! No more spending one hour every day copying down exactly what the book asks you to do or sketching and drawing precise margins (at the university level, precisely drawn graphs is one of the lesser important aspects of mathematics). Instead, mathematics turns out to be a continuous process that you can do every day, without the stress.

... unless of course, your teacher wants you know the critical skills of copying and pasting. Then, it is advisable to keep at those 'exercises'. {MS Clipart}

So without further ado, here they are (with difficulty rankings relative to high school content):

Population Dynamics

Difficulty: 7/10
Mathematics Topics: Differential Equations, Chaos Theory
About: Students can examine the different  population models such as Malthusian, logistic, and island biogeography. Investigations can be about the limitations of each model, the strengths and the ideal conditions for the model to work.
Bonus: Students can talk about how current day population models are more similar to chaotic systems than deterministic ones

 Poker, Probability, and Game Theory

Difficulty: 6/10
Mathematics Topics: Game Theory, Probability, Statistics, Simulations
About: Students can investigate the best course of action to take with various poker hands, categorized by winning strategy. Approaches can be from a game theoretical, probabilistic or statistics side or all three.
Bonus: Create a computer simulation to accept or deny your theory through a null hypothesis and Monte Carlo simulations.

Rings, Fields, Groups, Vector Spaces, and Algebras

Difficulty: 9.5/10
Mathematics Topics:
Abstract Algebra, Linear Algebra
About: Students can discuss the differences and similarities between the above algebraic objects and how to construct them. Investigations can be made about various isomorphisms between algebraic objects and about interesting objects such as the power set of the real numbers being an algebra under the union and intersection operators.
Bonus: Construct your own simple algebraic object (maybe the set of all watermelons that taste like oranges; something silly) and see if you can go from a magma all the way to an algebra.

Fractals and Iteration

Difficulty: 9/10
Mathematics topics: Fractals, Chaos Theory
About: Students could take a look at the definition of a fractal and what exactly is fractal dimension. Investigations can be made about the construction of fractals under various domains, their applications (e.g. coastlines), and their self-similarity properties. Students could try to generate fractals through freeware found online.
Bonus: Students could investigate what higher dimensional fractals would look like if projected on to the 3D or 2D plane. An example of a 3D projection into the 2D plane would be a contour map.

Methods for Computing Mathematical Constants

Difficulty: 8/10
Mathematics Topics: Computer Science, Simulations, Calculus
About: Students can investigate the different methods for computing mathematical constants such as \pi, e, and \phi the golden ratio. Some investigation could be made about the complexity of the algorithms and how fast they converge to the constants.
Bonus: Come up with your own unique constant and an algorithm used to compute it. Talk about complexity and rate of convergence as well as if it can be represented by current day mathematical constants

Cardinality of Various Sets

Difficulty: 8.5/10
Mathematics Topics: Set Theory, Ordinals
About: Students can investigate the various cardinalities contained in sets such as the natural numbers, integers, irrational numbers, rational numbers, real numbers, and algebraic numbers. Students could also define their own set and try to determine the cardinality of that set.
Bonus: Students can devise a model for classifying the cardinalities of different sets based  on certain characteristics contained in each set.

If you have read Part 2, you might recall that I did write a research paper on a topic of my choice. From the above topics, my essay was actually about the last one, cardinalities, which you can access a copy, here. Some time in the future, I will be restoring that essay to a cleaner LaTeX typeset document. You may use it as a guide, but I highly encourage that readers take the time to learn about the above topics themselves. After all, to quote Georg Cantor:

 

English: Georg Cantor

"The essence of mathematics is in its freedom." {Image via Wikipedia}

 

 

I hope you all enjoyed my little mini-series rant about my high school experiences and if you did, I would love to hear about your experiences in the comments below. In the upcoming months, stay posted for even more content as I begin to dive straight into some serious mathematics this term and start to bring you my quality typsetted notes.

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.

High school life, research topics, and advice for the young aspiring mathematician. (Part 1)

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Brace yourselves as this is going to be a very long post – split into two to three parts – of myself reminiscing back to my high school years.

Specifically it will cover my experiences in:

  • Grade 9,10 and part of 11
  • My developing interest in mathematics
  • The beginning of my Extended Essay on set theoretical topics

[Part 1]

To recount my experiences in high school, I can honestly say that for my first two years doing mathematics at the ninth and early tenth grade level that I really hated it, or at least hated how it was taught. Many of the problems were built around repetition and memorization and the “interesting” problems that were brought up in class were nothing more than algebra problems that were sugar-coated with words.

Problems like “if Alice is twice Bob’s age fifteen years from now and she is twenty right now, how old is Bob?” populated this category endlessly. I would often solve such problems and then write something to the tune of “Above is an algebra problem. To find Bob’s age, ask Bob for his health card.” just for kicks… and then proceed to lose marks because of it.

Because of the monotony of class, I eventually developed a system wherein I would be able to learn the material in the shortest amount of time while having free time to work on other subjects. Sort of a pseudo-form of procrastination if you will. I would sometimes even spend time reading ahead to the grade eleven and twelve books. Now, while I would always receive 95%+ on the tests, because I did not do any homework or in-class assignments, my marks would never peak beyond 85%. This frustrated me quite a bit.


"You may have solved the Riemann hypothesis, but you haven't completed your exercises on exponents yet. Therefore, you get a B- at best." {freedigitalphotos.net}

By the middle of the tenth grade my system still worked, but besides working on other class material during maths classes I was still bored out of my mind. It was only on a whim that I entered the Grade 10 Waterloo Cayley mathematics contest, or maybe it was out of boredom (I can’t recall). But taking that maths contest was possibly the best decision that I made up to that point in my life. This is because after writing it, not only did I win the contest and become the school district champion, the experience lit a flame within me whose goal was only to discover how far into mathematical ocean could I dive. All of this because the contest really challenged me and provoked me to improve myself to sheer effort.


"And also to allow for another chance to get back at those pretentious math club members and their non-mathy maths games." {freedigitalphotos.net}

In grade eleven, when I began driving straight into learning rigourous mathematics, I also began my two-year journey through the arduous IB (International Baccalaureate) program. One of parts of the program that I genuinely enjoyed was writing the Extended Essay, and in my case it was on Mathematics, specifically Cantor and his classification of the different cardinalities and properties of the cardinality of the continuum.


This was what essentially shaped my two years writing that essay... what else could be more fun? {wikipedia.org}

Now during the introduction of the essay and its components, which were taught through a series of lectures, my advisers made it very clear that no one chooses a maths topic unless they wanted a low grade or wanted to write about poker (which is what the last guy did… two years ago). Throughout the lectures, the content mainly emphasized skills that would be helpful for those writing history or English papers. None of it was particularly helpful for researching and formatting maths papers.

When it came time to select subjects and topics, sure enough, many of my classmates fell into either English, History or the odd Science related topic. When it came to my turn though, I shyly squeaked out my interest in developing a maths paper and instead of an expected “That’s a great choice.” or “What an interesting subject.” like all my other peers, I received a response to the tone of “Hah. Good luck with that. Nobody ever chooses math.” and yes those were exact words. No else in my class decided to do a maths paper after that.


"He wants to learn math? Hah! Next he'll be saying he wants to become a mathematician." {freedigitalphotos.net}

Undeterred though, I was hopeful that my personal faculty adviser could provide input on my thesis topic. Unfortunately, my maths adviser had never done any set theory during his undergraduate career and while that did surprise me a little I later found out, a week into researching my topic, that his majored subject was in physics and not mathematics.

I knew, then, that I had a lot of work ahead of me.

On the Irrationality of Pi.

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One of the couple of mathematics related details that really bugged me during high school (other than the fact that we constantly had to put those annoying arrows in front of ALL of our “vectors”), was the irrationality of \pi.

There was no easy proof that I could use, with my current knowledge at the time, that could prove it was irrational, unlike simpler numbers such as \sqrt{2}. I could not even find any teacher at the time who could explain to me using basic calculus why the ratio between the diameter of a circle and its radius was never rational.

Skip many years ahead, and I came upon this interesting, albeit “short” proof:

http://www.lrz.de/~hr/numb/pi-irr.html

Although the proof can be summarized through eight remarks and a few properties about integrals, trying to prove each remark can be a bit challenging. I would estimate that the average high school grade 11 or 12 student would not be able to get past the second remark without great difficulty.

That is why I compiled this, a fairly easy to follow guide detailing each of the remarks and their derivations. In the .pdf, I also left a couple exercises for the reader to do just for fun (but if really need to know the answer, don’t hesitate to private message me). However, if you would like to do the proofs yourself, send me a message or an email and I would be happy to post your proof up on the site.

If you want to know how I made the article, please check out the link in the “Resources” page under “Typesetting”.