Reflections on Honours Linear Algebra

A week or so after my exams finish each term, I find it helpful to write up a brief after action report of sorts for each of my courses, in which I reflect on how effective my various studying practices were. They’re usually each just a couple of pages of bullet points or, at best, disconnected paragraphs that I use when planning out how I will spend my time in future terms. However, in addition to all the expected ups and downs of online learning, my most recent term included my largest upward mark swing in a specific subject so far at UBC. Ergo, I thought it appropriate to formalize my notes on the factors contributing to my performance in MATH 223: Honours Linear Algebra, in an attempt to learn all available lessons from my experiences this term and formalize a strategy to utilize them going forward.

Background

Before discussing MATH 223, some context is necessary. 223 was not my first experience in an honours mathematics course. Before beginning post-secondary, I was among the strongest math students in my high school. In my enriched pre-calculus and AP Calculus classes, I routinely earned marks in the high nineties and even the occasional hundred, consistently placed among the top scorers in my school in the University of Waterloo math contests, and occasionally helped coach the Math Challengers team in problem-solving techniques. In short, as far as I knew, I was rather strong in mathematics. Consequently, when picking my courses for my first year at UBC, I elected to take MATH 120: Honours Differential Calculus, thinking that it would be a natural progression from high school and that, although it would be a noticeable step up, I would be able to handle it with a moderate level of effort. This presumption proved to be inaccurate.

The term started rather well. I found the topics covered challenging but fun. It was a new kind of math and a welcome change of pace from the computational grind of AP Calculus. Going into the first midterm, I had an average of 96% on the first three assignments. I believed that I was quite familiar with the material and would perform relatively well. I wrote the exam and felt my predictions had been accurate. I wasn’t expecting a hundred, but I thought I understood all the questions and gave reasonable answers.

I was walking across campus one afternoon when my phone buzzed with a notification. It was an email notification from Canvas stating that marks had been posted for the midterm. I clicked on the link, logged in, and saw the first and only C+ that I ever obtained on an exam. The F average was of little consolation. It was blindsiding, it was embarrassing, and it left a scar of self-doubt that would last for the next fourteen months. I had thought that I was adept, if not strong, with the course’s material thus far. When I saw my graded exam, I didn’t see a glut of clerical errors; I saw a series of misunderstandings of concepts, which further compounded my disgust with myself. Every point lost was a massive failure of my studying policies. I knew that major reforms were in order. However, it was too late to fully recover my MATH 120 grade.

I was able to reverse course over the next month and a half, but that exam dropped my course grade enough to result in my only A- in a course at UBC, and my only non-A+ in a course in what I would consider one of my core subjects (computer science, physics, and mathematics).

Despite this experience, I signed up for MATH 223: Honours Linear Algebra the following summer. I had been told that the course was more or less necessary if one wished to have a thorough understanding of concepts in upper-year courses, particularly quantum mechanics and machine learning. Deep down, I probably also wanted to prove to myself that the barely-mitigated disaster of MATH 120 was behind me. I knew that it would not be an easy four months. However, this time, I would be prepared.

That summer, I began reading ahead and formulating plans on what my studying and homework habits would look like come September. MATH 120 had been my impetus for beginning to write after-action reports like this one, which meant that I had a wealth of notes on and analyses of my study habits from first year. With them, I devised a list of failings that would have to be corrected going forward.

The four great mistakes that I took note of were that I had relied too much on my peers, been convinced by my own faulty arguments, lacked a single resource to consult when studying, and overfocused on practice problems to the detriment of general understanding. To amend each of these issues in my study habits, I had to reflect on how I went wrong in MATH 120 and how I would fix things in MATH 223.

Go it alone.

In the first lecture of MATH 120 and more than a few of my other first-year courses, the professor made a point of encouraging us to discuss homework problems with each other (with the obvious caveat that our work had to be our own). In hindsight, this advice was damaging to my final grade and devastating to many of my classmates’. Particularly in honours mathematics, one of homework assignments’ primary purposes is building an intuition for the sort of problem-solving that students will be expected to do in a given course. When a student works with classmates to come up with the key insight for a problem and then writes a full proof alone, he or she only serves to spoil the solution and evade a good exercise. Additionally, due to a selection bias in the problems students work together on, the hardest problems are disproportionately avoided. This leads to crippling deficiencies manifesting in an exam setting.

I was by no means the largest victim of this approach in MATH 120, but it did still negatively affect my learning. With this in mind, I set a new policy for myself. No longer would I work on homework with classmates (or at least not with people in the classes whose homework I was working on). There would be no hints from Piazza or discussions in group chats. Studying with others would be permissible, but every homework problem would have to be solved alone.

Because of this rule, the weekly office hours on Wednesday and Thursday served as a hard deadline for a rough draft (more on this later) of my solution for the week’s problem set (otherwise I would hear about other students’ solutions during office hours), which would be due Friday night. This early deadline had the benefit of shielding me from most of the chatter from classmates asking each other about the homework that would come closer to the actual deadline.

One obvious drawback of this approach was that it made working through problem sets and studying much more solitary. Fortunately, through a chance occurrence at the start of the term, I began to have study calls over Zoom with an acquaintance (who became a rather close friend over the course of the term) from my design team. Despite not sharing any courses with her, it was extremely helpful to have someone to encourage me to stay accountable (both in my studying and my then-ongoing search for my first co-op job) and, of course, to chat with on occasion.

Preventing myself from contacting my classmates until I completed the week’s homework was quite possibly the hardest change I had to make to my workflow. While I believe it contributed in no small manner to my success in the course, I found it to be rather isolating and draining. The shared sense of struggle on the assignments back in MATH 120 led to me developing fairly close relations with many of my classmates, something on which I feel I missed out, especially in this online year. I think it may be a better idea to allow myself some more homework-related discussions with my classmates, so long as I try to come up with the main ideas myself beforehand.

Solve before you explain.

In several of my first-year courses, we were encouraged early on to learn LaTeX and use it for our homeworks. In hindsight, I was slightly overenthusiastic in my adoption of it. Instead of using LaTeX for a “good copy”, I would start my homework by copying the provided .tex file, adding a section for solutions, and solving as I typeset. This worked (and, in fact, aligns with how I still write prose), but it was suboptimal. By solving the problems in my LaTeX editor, I was hampering my ability to properly build up a solution that I could be fully confident in.

In MATH 223, I devised a new strategy, one which transformed completing an assignment from a single task that is completed over the course of one or two days to a three-step process that plays out over the course of a week. The problem with my old approach was that it lent itself well to a sort of confirmation bias or self-deception. By writing the good copy as I was solving the problem, I was thinking of and writing down arguments for the validity of my solution without having verified it. This made it easier for mistakes to slip through. An improved process would have to decouple the mathematics from the argument, To do this, I “solved” each homework problem three times.

First, I tried to write down a rough sketch of a proof with just the key insights. I looked at the week’s assignment the night it was posted and two or three times each subsequent day until it was solved. Whenever I had a novel idea, I jotted it down. This was usually done in pencil on a sheet of scrap paper, or, in one instance, on Google Keep while on a ride back from the orthodontist. Once I had that sketch, I put it away for at least a few hours. After some time had passed, I came back to my note and, if I still thought it was sound (or at least was likely sound), I began the second phase.

Once I had an idea of which direction to take my proof, I needed to make it rigorous. I also did this step on paper. The goal here was to write out a full solution and completely convince myself that it was correct before I tried to convince anyone else. As such, I kept my writing as symbolic and terse as possible, while maintaining a high level of detail. Again, after writing out this draft, I left it for some time before coming back and reading through it again with fresh eyes.

Finally, after I was confident that I had a full solution that was logical and reasoned, I wrote a good copy in LaTeX. Since first year, I have consistently found that typesetting my final homework submissions in LaTeX encourages me to write more precisely and descriptively. I often liken it to the correlation that some of my peers find between dressing more formally and performing better on assessments. The medium is cleaner; there’s less potential to hide shoddy work on a typed page (unintentionally, at least). For MATH 223, I typically typeset my assignments during the weekly office hours on Wednesdays and Thursdays, which gave me a chance to ask clarifying questions and finally listen to others discuss their solutions. Note, however, that this was not some baked-in editing stage. If I found a mistake while typesetting (fortunately, this only happened twice), I considered it to be as grave an error as submitting that mistake and dedicated extensive further review to whatever concept I got wrong. As usual, once the typeset assignment was finished, I put it aside before giving it one more read, in which I compared it to my rough draft.

Naturally, this process had its downsides. It was time-consuming, it often felt tedious when applied to simpler problems, and it took some of the fun out of problem-solving. That said, it never failed me. In the ten homework assignments that I handed in, not a single error slipped through the series of filters that my process had created. This was the most effective change I made in my study habits and I intend to continue it well into the future.

Find a single source of truth or make one.

Another failing I found in my study habits from first year was that I was easily distracted by all the learning resources available in my courses. I would read the textbook, go through slide decks, and browse my handwritten notes in equal parts. However, these resources were never of equal value.

In MATH 120, we nominally had Tom Apostol’s Calculus as our textbook. Book of Proof, by Richard Hammack, was also recommended as a supplement for proof techniques. However, neither of these books covered the full scope of the course. There was no pre-existing single source of truth. Instead, I was responsible for making one with my lecture notes, though I did not realize it at the time.

The notion of single sources of truth for courses first dawned on me when I was writing my initial batch of reflections for the courses I took in my first term. I noticed that in courses like chemistry and computer science (CHEM 121 and CPSC 110, for those interested), where the course had a custom textbook (technically an edX course in the case of computer science), studying from the textbook paid much larger dividends than courses that lacked such resources. This realization changed my strategy for future courses, including MATH 223. If there was no single source of truth, one would have to be made.

I have always taken essentially verbatim notes in lectures, both in my math courses and elsewhere. That remained true in 223, but I did change how I used them. For me, someone with a computer science and physics background, the lecture notes I accumulate over the course of a term are just that: notes, a summary of the material covered. Writing them helps me with retention, and reviewing them occasionally helps while studying. In honours math, however, I have come to believe that notes are not a summary; they are one’s closest thing to a single source of truth and should be treated as such. Even when math courses list a required textbook (and especially when they list multiple), no external resource is going to match the course’s content exactly. Because of this, I had to construct a single source of truth for MATH 223 using my lecture notes.

I had never previously been one to copy out neat, formatted versions of my notes after I take them, but I was forced to change that. MATH 223 had a rather bizarre schedule for a linear algebra course. We first covered essentially all of the possible linear algebra one could do on 2x2 matrices in a week and a half; then we moved to arbitrary-sized matrices and covered essentially all of MATH 221, UBC’s non-honours linear algebra course, in the next couple weeks; and finally, we went through everything again with full generality, using fields, abstract vector spaces, inner product spaces, and spectral theory and some new applications, like systems of differential equations and singular value decomposition, along the way. It was unorthodox, not very axiomatic, and yet an extremely effective way to learn abstract linear algebra that I have not seen replicated in any textbook. The primary drawback for me was that it made a mess of my chronological notes, so at the end of each lecture day, I transcribed and reformatted my verbatim notes into a OneNote notebook, organizing everything in what I found to be a more logical order. This notebook served as my single source of truth for the course and saved me a great deal of time while working on homework and studying.

In hindsight, OneNote was probably a suboptimal choice for how to store my notes. I knew my “good copy” would have to be digital, due to the constant modifications to and expansions of material from weeks prior (as well as my rather horrid handwriting). However, I probably should have simply committed to writing a proper LaTeX document to save myself all the formatting troubles. This would have had the added benefit of being easier to share with others once I was done with it. Regardless of format though, keeping everything I needed to understand anything in the course in one place was a major contribution to my later success.

Diversify modes of learning.

If I were to sum up everything I heard at all of the “learning to learn at university” workshops I attended during orientation in one sentence, it would be “Learn by doing.” However, the past year and a half have taught me that drilling with every available practice problem whose likeness may appear on the exam is not enough to gain a full understanding of a topic. Obviously, actively reading through notes and textbooks is important when reviewing for a course. Many of my classmates in MATH 120 attempted hundreds of quite similar practice problems and ended up learning algorithms, not concepts. I did not make this mistake, but I did notice that I had trouble generalizing many of the ideas in the course. I could explain a formal definition using predicate logic, but turning that into a picture would take me a few minutes (and not just because of my lacklustre drawing abilities). Completing a fairly wide range of practice problems had still left me unable to fully understand some content. After considering this problem for a while, the realization I came to was that the problems I had been practicing with were only providing one perspective on the material. To remedy this in MATH 223, I would need to practice using more diverse methods, so I could ensure I was learning concepts, not recipes.

One way I sought to put this idea into practice was by trying to solve various classes of common problems computationally. This was only possible because of the fortunate coincidence that I was taking a linear algebra course, where computational methods for most problems (eigenvalue computations notwithstanding) are generally quite analogous to how one would solve those problems on paper. Because of this coincidence, I was able to learn a great deal from building up basic libraries for all the usual computations I found myself doing, like matrix multiplication, Gauss-Jordan elimination, and Gram-Schmidt orthogonalization. I first built these libraries using Python using NumPy arrays but as few other non-primitive functions as possible. Then, as a fun challenge for myself, I re-implemented many of the algorithms in Racket using a matrix data structure I had created. Personally, I found these exercises to be extremely helpful in building a general understanding of mathematical concepts. Solving computationally allowed me to look at a common problem in its most general form, and recognize any interesting edge cases that may have been lurking.

To be clear, I am under no delusions that my epiphany of “just make a computer do it” will be of any use to someone trudging through Real Analysis II. However, in certain contexts, learning the numerical methods for a problem can be quite valuable, and it is certainly something I intend to do in my future math and physics courses. More generally, one should try to diversify how they learn as much as possible, not just by both reading notes and doing practice problems, but by doing different types of practice problems, and inventing one’s own if need be.

Conclusion

I want to emphasize that none of this is prescriptive or even intended as a recipe for anyone else to follow. These are simply policies I adopted in one course, in one term, which led to one final mark of 100% (albeit one in the hardest course of my degree thus far). The true test of their efficacy will come in a little under a year, once I conclude my work term and am once again taking classes full-time. Once back, I will put the lessons of MATH 223 into practice and see whether they continue to hold.

Putting plans for the future aside, one question remains unanswered: is it all worth it? Each novel strategy I devised for myself came with a trade-off. Working alone meant forgoing much of the camaraderie I had in first year with the other students in MATH 120, many of whom remain close friends. My homework process kept linear algebra problems buzzing around in the back of my head during what little time I had for recreation. Maintaining a single source of truth for the course meant working late into the night reading, writing, and organizing notes. Forcing myself to practice linear algebra in as many different ways as possible kept my notes, not my books, at my bedside for much of the term and tapped some of my limited energy for programming in the term when I was taking my first real software engineering course and had just begun work as the lead server TA for UBC’s introductory computer science course. Indeed, it is extremely easy to look back and think about how I could have made my life a good deal simpler this term, potentially even obtaining the same mark. In the end, however, when I look at my final mark and think about all the trials I overcame and all the progress I made since first year to reach it, it would be a lie to say I have many doubts that my time was well-spent.