My first semester studying mathematics at a university has officially come to a close, and I would like to take a moment to reflect upon the journey. While this was neither my first time at university nor my first time studying math, it was my first time doing both at the same time.
I took five courses this semester, plus an extracurricular study group of sorts with a professor and a few other students. Two of the courses were general education courses I needed to take and the other three were math courses. I took number theory, introductory proofs, and an advanced engineering mathematics course (AEM for short). The study group is focused (I use the present tense because the group is ongoing) on algebraic topology. The courses were challenging, but served as a smooth introduction to upper division math courses. I was fortunate enough to have excellent professors in all of my classes.
Of the three math courses, the proofs course was no doubt the most demanding. It wasn’t the amount of work or the difficulty of the work that made it challenging so much as it was how different the style was from other math courses I’ve taken before. Most math classes before this one have focused on finding the correct answer(s) to a problem while writing proofs is all about showing, rigorously, why something is true. To use the analogy of mathematics as a language: previous classes were about learning how to spell. Proofs class is about writing paragraphs. Taking this class made me thankful that I had previously taken a symbolic logic course which exposed me to the inner workings of proof techniques. The logic course was also difficult, so by taking that one I was able to spread the effort of learning proofs out over a bit longer time frame.
My time self-studying mathematics certainly helped me in proofs, but I wouldn’t say that the self-study made the class easier. What the self-study really did was change the way I engaged with the class. My head was filled with math language which I could regurgitate, but I had little control over it. My early proofs in the class were not elegant. In fact they were more like word vomit. They were long, convoluted, and sometimes missed the point they were trying to prove. So my early difficulty in proof writing was mostly working on slimming down the unnecessary parts of my proofs and making them more elegant.
My mentor back at community college had recommended that taking number theory alongside the proofs class would be a good idea, and he was correct. The number theory course does not have the proofs class as a prerequisite so it also serves as a gentle introduction to proof writing. Although there was some overlap between the two courses, the extra coverage felt more helpful than tedious. Additionally, number theory gave me an entirely different perspective on what numbers are and how to work with them.
Advanced engineering mathematics was a interesting course. On the one hand, there was not a lot of ‘new’ material in it. We mostly covered topics from linear algebra, differential equations, and vector calculus, which all had their own courses beforehand. However, the way the material was covered helped me to gain a much deeper understanding of the material and how it can be applied in the real world. There was also some nice coverage of Markov chains, which are amazingly useful and deserving of their own post some time.
Beyond the coursework, the semester also brought about many social changes for me. At the start of the semester I did not really know anyone. I was a bit nervous, to be honest. Luckily, I met so many great people and made some wonderful friends. Joining study groups with people no doubt improved my understanding of the material and encouraged me to study more so that I would not be a burden on the group. The times where I was able to contribute by helping someone else to understand something felt rewarding and fulfilling. It was also a lot of fun to hang out with groups of people who share a similar passion for mathematics. I am extremely glad that I sought out the math club on campus and got involved with them.
The grades I received this semester are an encouraging sign that all of my hard work is paying off. Overall, I earned three A’s and two A-‘s, something which would have been inconceivable to me when I first started college. Grades are not the only measure of progress. My knowledge has expanded and my confidence improved.
The main downside to studying math in school again is that I have had much less time and energy to spend exploring it on my own time. This is both a good and bad thing. On the plus side, by being forced to focus in on specific topics I gained a deeper understanding of them then I might have if I had merely explored them for fun. On the negative side, being restricted to a small set of topics can sometimes feel boring and restrictive. Building that discipline to spend time learning things properly rather than chasing every flashy new topic is an important part of the education.
My course load next semester is quite varied and sounds a bit more interesting than this past semester’s content. I will be in four math courses, plus the algebraic topology group and a GE course. The math courses are abstract algebra, complex analysis, numerical methods, and the second semester of the advanced engineering mathematics course. I am a bit nervous taking this heavy of a load, since I know that it means I will absolutely have to keep up with my studying habits or I will fall behind fast. However, after this previous semester I am confident that I am up to the challenge and the work put in will pay off.
Over the break I had intended to spend my time on a dynamical systems book which I checked out from the library, but I honestly have not touched it too much. Instead, I have been working through a book on complex analysis and some material on machine learning/numerical methods. It has been nice to take a break and rest my mind. Days on the beach and other reading has cleared my head and recharged my energy reserves. All in all, this was a good first semester and I am excited to take the next steps on my journey.