There is beauty in fundamentals.
Unlike most of my posts, this will be more retrospective than speculative. One of the questions I’m asked with unfortunate frequency is, “Lily, how do I learn X?” or some variation thereof. By my own admission, I am hardly an expert; I joke quite candidly that the COVID-19 lockdown and this past year has been effectively a hyperbolic time chamber of mathematics for me. However, it’s hard to argue that in this weird mess I’ve created I didn’t do at least a few things correctly, and I’d love to talk a bit meta about my methods for approaching learning and problems in general.
For me, there were two major impetuses for my dive off the deep end into mathematics this past year. As a Ph.D. student in bioinformatics, of course, there’s some pretty solid grounding needed in quantitative methods to apply it properly to biology, but especially in the design of my program, most students started off in the wet lab (a.k.a. lab-based biological research) rather than in the quantitative sorts, and our class curricula kind of reflect that. In the interdisciplinary fields, especially bridging a more-or-less completely empirical field (biology) with a largely theoretical one (computer science), you tend to see specialization in most folks on one end or the other. The jury is still out whether for a bioinformatician, for example, one side is more important than the other, but you kind of get a hodge-podge type learning, and I wouldn’t necessarily say a deep level of mathematical insight is required to excel in the field (although compared to an average person, perhaps it is).
I came into my undergraduate studies looking to do physics, because I had machinations of studying quantum mechanics and being the next Stephen Hawking (except for the whole ALS thing, hopefully). I quickly hit a wall. One of the hidden secrets of the world simply is there isn’t “one” math brain, there’s several types:
1) Spatial-brain — These folks tend to be great navigators, puzzle solvers, and tend to breeze through the high-school odyssey of geometry all the way to calculus and, if they choose, topology and the like at the graduate levels.
2) Logic-brain — These folks tend to implicitly see order and groupings, and tend to gravitate towards logic-heavy fields like computer science or conversely law. Usually subjects like algebra up to discrete mathematics go really well for them.
3) Intuits — This is the most enigmatic type, folks who essentially have an unexplained mastery of mathematics without any clear process or grounding. The intuit path is fascinating because it tends to go hand-in-hand with perceived giftedness in youngsters, and has a high probability of ruin for the same reason (if everything comes easy to you, you do not learn how to learn, essentially).
This isn’t to say each person belongs to one of those types like a weird caste system. In general every person, no matter how mathy they think they are, will be some combination of all three categories. The bigger issue to bring up is the impact of standardized education here.
I am not a geometer. When I was in middle school, I fancied myself a genius at mathematics, and perhaps I was. I can still do 3 by 2 digit multiplication in my head, and to most non-mathematical people you’re basically a witch if you can do that. I had no issue in any type of algebra, really. But then I met geometry.
I have the spatial ability of a lobotomized monkey. I’m not exaggerating here; when I was younger, I took a giftedness test for the Johns Hopkins Center for Talented Youth search, and got an 11% on the visuospatial part of the exam (I qualified based on other factors, but ouch). And it showed in geometry, badly. I went from a straight-A math student to getting a C in honors geometry first semester of ninth grade. I ended up having to downgrade to regular math classes, and I stayed in that track throughout high school. I never ended up taking calculus in high school at all.
The issue is how we educate students on math, much as we do other subjects. I am not a geometer, but in college I taught myself calculus I and II over a summertime, and ended up getting nearly straight As throughout a top computer science undergraduate program. I similarly hit a wall when I went into physics the first semester of college, however, thanks to — you guessed it — kinematics. But the problem is for most of us, we look at mathematics education as a linear track, rather than a radiating umbrella.
For most students, especially up until the later years of college, we treat mathematics education as a track, starting from basic arithmetic and ending for most around pre-calculus or calculus I. However, even for the more gifted or quantitatively focused, this track largely continues in sequence through college up to about differential equations. This largely works, in the sense it may be close to the optimal fit for the largest group of people (variance-bias tradeoff, lol), but when it fails, it fails terribly.
For me, I came from a family mixture of mathematicians and non-mathematicians. My grandfathers were excellent at math; my parents, much less so (they went into medicine). I thought when I hit the wall in geometry it was the family curse catching up to me, and instead of reflecting on why I was having issues, it seemed more or less the natural order of things. However, this was incorrect, and is a trap a lot of potentially gifted mathematicians fall into.
First and foremost, we can reflect on the idea of the splitting of types of math brains. The way standardized education and the linear track works is that once you’re off the gifted track, you’re off. Many people get traumatized somewhere on the track when running into difficulty they don’t even think there’s a potential path to get back on. They give up, and instead resign themselves to “not being a math person”. In my case, I still cannot properly visualize shapes and rotations. I will likely never be a gifted physicist, especially in theoretical branches, because I am not a spatial-brain. However, discrete mathematics comes close to naturally for me. I can visualize large scale systems and how they operate, and love understanding the fundamentals of why.
The second and more painful reason why many resign themselves out of mathematics is poor educational habits. This can take a few forms, but the most common are:
1) One or more “bad” teachers — This doesn’t mean a wicked or incompetent teacher. We all learn differently. For example, I likely have some sort of attention deficit, and learn without question better from doing than from lecture. However, there’s many who learn optimally the opposite way. That said, we have one professor. Who gets to be the mathematician? The one who matches the professor’s teaching style. The other one is simply “bad at math”.
2) Running out of time — In most forms of education, there is an implied time limit and structure in how one must learn. In grade school of course this takes the form of the academic calendar and the lesson plan. In general, the lesson plan is a function of two factors, neither of which are optimized to you — the approved school curriculum (designated by state and local agencies, usually) and the teacher’s preference. The bigger issue in mathematics comes from the fact that mathematics, to varying degrees, does follow a compounding track — without proper knowledge of arithmetic, for instance, algebra is perhaps impossible. Without probability theory, statistics is impossible (to do well). So when you take a structured course, you — and all other students — are continually racing against the Clock of Normal Learning. If you fail to master the material in time, you have two options — fail out (usually not something that happens, and when it does, it comes with harsh penalty) or take shortcuts. In many ways, taking shortcuts is worse for you than failing out. Let’s explore why.
You can usually tell a math geek from everyone else simply by the reverence at which they explain the material. For many if not most, we look at the rules and tools of mathematics as simply tools. For a salient example, let’s consider linear regression. The basic concept is stupidly simple — we fit a line (and sometimes intercept) to a group of points to try to predict a trend. With basic algebra I (notionally, y = mx+b) we have the intellectual tools to understand how to use linear regression.
The danger is most of us stop intellectually there. It’s quite easy to see linear regression as a black box tool under our control, with the mantra to use it as the basic predictor before moving onto more exacting methods. But the problem most people ignore is — how, or rather why, does it work? Trivially, you could say that the intrinsic value of fully understanding linear regression is knowing when it doesn’t work, but this can be more abstractly understood as extrapolation.
Many of us who hate math, if you drill down to it, see math as a morass of formulae to be memorized. This works as long as you haven’t hit the limits of your memorization ability, to some degree. This also works fairly well during standardized education, simply because the measurables of mathematics education are the formulas and how to use them. But we’re humans, we’re not machines. The beauty of mathematics isn’t to re-invent what is already known, but to understand the reasoning and logic behind it. We learn mathematics to solve problems, and in general the problems we encounter will not be the same as in the books. They will be little variations thereof. But without a proper grounding in why things work, how can you hope to extrapolate it outwards? Or to put it in statistical parlance, the way most of us learn mathematics is to overfit because we optimize for GPA. In the process, many of us develop a deep mathphobia, and our brains instinctively shutoff when we see equations and curly brackets.
But there is beauty in the fundamentals. Understanding the fundamental reasons why will provide you the foundations you need to grow your knowledge, helping you conquer the morass of equations and extrapolate your knowledge into something both useful and beautiful. I personally believe everyone has the ability to learn mathematics, and I hope along the way to teach myself and others more about the field.
Ciao for now,