EDIT 1: There really should be a +1 to the CO section. I just forgot to add it, when I first made the poll. Sorry!
EDIT 2: +1 to Topology as well. Please let me know if you guys have any more areas of mathematics that I’m missing!
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.
So without further ado, here they are (with difficulty rankings relative to high school content):
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
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.
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.
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.
Mathematics Topics: Computer Science, Simulations, Calculus
About: Students can investigate the different methods for computing mathematical constants such as , , and 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
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:
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.
To all my readers and friends out there, I just want to start off by wishing everyone a happy new year! While you may not have heard from for a quite a while now, I assure you that I have been working on quite a few other projects. Here are a few that you look forward to:
I hope to see old and new readers coming in for the new year because now is where the content will be really flowing in!
While I am still quite busy preparing for Exam FM (completing 10 mock exams), I do have quite a few short-term and long-term plans for this blog.
For the sake of being concise, here is a short list of what I have planned; many of the projects will only be started after exam FM which is on the 10th of December. However, for some of these project, I might have some to time to start on them this week:
In addition, I will be posting random tidbits of interesting maths finds here and there, but just don’t expect them to come that often during these coming couple of weeks.
For my first official content related post, I thought I would start by giving out something to you, the readers. Specifically, something that’s usually marketed on many financial websites and is often used to save money and carefully plan out investments and expenses. What I am talking about is an Excel budget workbook , and it’s free of charge, of course!