My Experiences With Math Class

It has only gotten worse since I left.

During my early years in elementary school, and beyond, I didn't care much for mathematics. Like most people, especially at that age, I had the misconception that math was just arithmetic, and I find arithmetic boring. When I was in second or third grade, I remember that we were told to memorize the multiplication tables. I never bothered, since I was lazy and didn't feel like it. On the quizzes, I did well enough just knowing a few and getting everything else by adding repeatedly. That's pretty much what I still do today. I also still add with my fingers pretty often, and I have no shame about it. So how come I ended up getting so into it?

In sixth grade, I got my first taste of algebra in the form of solving linear equations, like $5x + 3 = 7$. The teacher mentioned that we had probably done these things before in real life, in our heads, and just needed to get used to the symbols. At least for me, this was the case, so everything felt straightforward, and it was a nice change of pace from the horrible arithmetic of the past. I also remember that she once showed us an illustration of why the volume of some shape was what it was, which was very nice, but I don't remember which shape it was. I just remember it was some kind of pyramid.

In seventh grade, things continued to get better, at least with respect to math. Everything else got worse, since I was becoming a real edgelord asshole at the time: a trend that thankfully reversed itself, I think.

The problems were still really easy and not very annoying, and I ended up liking math enough to say it as my canned response when someone asked me my favorite subject. My real interest at the time, though, was programming, by which I mean stitching together incomplete, poorly coded, shitty games in GameMaker: Studio. If I wanted to do math, it was only because I was stuck in math class and was tired of hearing the teacher complain that everyone else keeps shutting down whenever a problem was slightly different from what they were taught to do in class.

I guess my teacher saw I wasn't like that, and decided I should try joining some math competition team my school had. When I did the exam to get in during lunch, though, I quickly found that I couldn't answer most of the problems, so I didn't get in. My teacher was convinced that I did it on purpose, since I wasn't very interested in joining in the first place. Like I've been saying, I didn't really care for math.

Regardless, something started happening around this time. By now, I had seen plenty of criticisms of the school system about how it is a place where creativity, natural curiousity, and independent thought go to die. As a result, I began to ask myself why it was that everyone else was shutting down whenever they saw a problem that was slightly different from what they were taught to do. I began to suspect that the problem was that since nothing was ever explained, but rather the students were just told what to do, it made all of math seem like arbitrary instructions with no rhyme or reason behind them. No wonder, then, that they couldn't do anything on their own: they don't even process that was an option, much less that there is a reason behind everything, because 90% of the time they don't even get the chance to figure out anything on their own, or see the reasons behind anything.

This is when I began to suspect that there was more to math than I was being told. I knew there was reason behind everything, but I didn't know what that actually looked like. Why didn't we ever get explanations? Why didn't we get the chance to make them? However, this never became anything more than a nagging thought, and I never bothered to investigate it because, as I said, I didn't care much for math.

In eighth grade I was placed in Algebra I. This is when things began to truly go downhill. At the start of the year, we got a weird "Algebra Nation" textbook, and we started doing weird things. For example, on some worksheets we were working on (not related to the textbook) our teacher told us to write down whatever these things were:

When we asked our teacher what this was and why they wanted us to do it since none of us knew, the teacher said he didn't know either. Now I understand that this "set-builder notation" is used to build sets, but back then it just seemed like a weird thing with a weird name. We also started solving equations for specific unknowns. Things like perhaps solving this for $w$:

Which, while I managed, I never trusted what I was doing, for the same reasons that others straight up shut down.

Early on in the year, I had to change schools. Now the weird textbook and the "$\in$" thing were gone, but in their place was something far more ridiculous. Each day started simply enough: the teacher would always seemingly be in a good mood, though I suspect it was a façade, and she gave us bellwork, which always pretty much consisted of either finding a slope of a line, and its associated equation, given two points. Sometimes there would be some problem about Yolanda and Xavier, but I don't remember what those were about. At some point, though, the whole class would get more and more rowdy, until our teacher couldn't take it anymore and ended up ranting for half of the class period about how difficult we were to deal with, furiously, inexplicably in an Italian accent. As far as I know, she was not Italian. I think my teacher got the impression that I was the "good kid," since I never actively caused much trouble. Secretly, though, I sympathized with the rowdy kids, simply because I didn't like her class. The previous teacher was cooler.

Through all of this, we got the garden variety Algebra I circus. Things like memorizing FOIL instead of understanding how the distribute property generalizes, being asked to shuffle equations of lines into "standard forms" for no particular reason, and of course, within the Cartesian framework, parabolas. Lots of parabolas. And throughout the whole course, I felt like I didn't understand parabolas or quadratics at all. All I knew was that there was this U-shaped curve which came with an imposing formula that said where it crossed the $x$-axis, which was important for some reason, apparently.

Not only did I have no clue why the weird letters of the quadratic formula were arranged the way they were (and I did wonder, but never looked into it), I also just didn't really have a good grasp of things, which became evident when after the final exam we did a Kahoot and I could barely answer any of the questions about parabolas or quadratics. I could see my teacher slowly become more and more concerned, ever so slightly.

Parabolas in general got me wondering. We had spent so much time finding slopes of straight lines, only to shift into something that wasn't a straight line at all. At one point, while the teacher was lecturing, I asked myself if it was possible to have the slope of a parabola, but quickly discarded the idea as it would have to be different in different parts of the curve, but slopes had to be the same everywhere, because that's what our teacher said about straight lines. How silly of me!

I remember we once went to the computer lab to learn about exponent rules of whatever. I didn't particularly care. I was more interested in playing around with Desmos. So, $y = x^2$ is a "parabola," huh? What about $y = x^3$? And what I saw left me with more questions than answers. It was a weird, crooked, skinny curve thing. Why was it like that? I felt like I had glitched out of bounds in a video game. Then I decided to do something fun: I would zoom into the curve as much as possible. As I did this, I noticed that the curve looked more and more like a line. I realized that it wasn't really straight, though, so I felt compelled to keep zooming in. I wondered if there would ever be a point where it really was straight, but understood that I would never reach it myself.

Sadly, I didn't put two and two together back then. Although I don't regret the way my mathematical journey has gone so far, I'd like to think that if I had known what math really was back then, that I could've discovered calculus on my own, and learned it way earlier than I did. But alas, back then all I knew was that if my teacher didn't tell us how to do something, it was probably impossible, so I certainly couldn't figure it out on my own.

As it happened, I didn't actually do the final exam for Algebra I, since my family decided to go on vacation on the same day for a few days, and I had to go since my mom didn't want to leave me alone in the house. I also missed the make-up because I wasn't told the exact date and I had another trip planned, and I didn't know who to ask, and frankly I just didn't really care all that much. I now know that this was a blessing in disguise, but to my past freshman self it was a source of consternation.

Not doing the Algebra I exam meant I had to take a remedial math class. Gasp! I was falling behind! A fate worse than death! The class was called "Liberal Arts Math" and it was insultingly easy, though I still had some doubts about things, like why "point," "line," and "plane" were called undefined when our worksheet had definitions right there, or why anyone would write $f(x)$ instead of $y$ when our teacher told us it was "just the same thing."

In that class, I once again found myself with a rowdy bunch of characters. Some of them couldn't even plot points in what I would now call $\mathbb{R}^2$. This meant I would sometimes find myself being asked for help. My most memorable exchange was when the kid that sat next to me asked me to calculate the area of a semicircle. It went something like this:

Me: What's the area of a circle?

Him: $\pi r^2$

Me: So what's the area of $half$ a circle?

Him: I don't know

I think he did figure it out in the end, though. I don't remember. In any case, while I was in this class I finally took the Algebra I exam and passed it, despite the fact that I had to hold my breath during every single question about parabolas and quadratics. This meant I got placed into Geometry for the next year.

When I learned that in Geometry, we would prove things, I was excited. Is this it? Will I finally see the deeper side of math I suspected must have existed but never got to explore until now?

Nope. Instead I got clumsy notations, a ridiculous, unreadable two-column format, and equally ridiculous, unreadable justifications like "Reflexive Property of Equality," "SAS Congruence," and my personal favorite, "CPCTC." I tried to like them. I really did. I understood that this must have been the deeper side of math I was looking for, right?

It didn't help that at one point our teacher gave us a cutesy little booklet of two-column proofs which had a predetermined number of rows for each problem. Because of this, I felt that any solution I came up with that didn't fill everything was somehow inherently wrong, rather than what it really might have been: a cleverer way to prove the theorem.

I still don't like two-column proofs. But what I now know that I didn't back then is that no one actually proves things like this. Not Euclid, who wrote Elements as prose and didn't even distinguish between congruence and equality, and who himself made appeals to intuition that did not come from his axioms despite the fact that he was essentially the father or rigorous mathematical proof, and certainly not modern mathematicians, who also write their proofs as prose in natural language, with notation only being used when it is useful, rather than gratuitously, and who sometimes use pictures to get their point across. The absurdity of what I was being exposed to was simply one of the many deranged insanities that only exist in K12 education, like "zero tolerance" bullying policies or putting kids in detention for not going to class.

Of course, I didn't know any of this back then, so I assumed this was how proofs had to be, and as a result I ended up with the false impression that math really was just vapid nonsense this whole time. The fact that I was forced to write proofs in such a ridiculous, artificially stunted way turned me away from the whole idea of proofs, and my distaste for math ended up rising again. No wonder the teachers didn't explain why things were true. Just imagine how ridiculously unreadable those explanations must be!

However, while my suspicion that there was a deeper meaning to mathematics had been successfully repressed, it was thankfully never destroyed. Though terribly tortured and on life support, it clinged on to me even as I went into Algebra II, and even as my Geometry teacher reviewed the so-called "trig functions."

This wasn't my first rodeo with the circular functions. My Algebra I teacher was the first teacher who told me what those curious buttons on the calculator did. Her case was special, though. Not only did she parrot SOHCAHTOA before actually telling us what the subject was about, but she also put lipstick on the pig by saying that it was the name of the Native American man who invented trigonometry, since I guess it sounded indigenous to her for some reason or another. I guess the actual history of the circular chords of the Ancient Greeks and the infinite sums of Indian astronomers simply didn't make good foil for a dumb mnemonic.

This time around, though, I had more questions. By now I had seen function notation being used to write things like:

Yet the definitions for the trig functions did not have $\theta$ anywhere on the right hand side! Sure, which side is "opposite" and which side is "adjacent" depend on which angle you're plugging in, but the fact that the functions were not written like those other ones made me wonder what was really going on. But again, back then I didn't know what math was, so I just swallowed my questions with the help of the knowledge that whatever the teacher said must surely be right.

Later on, in Algebra II, I was exposed to the unit circle definitions, and became even more confused. Now $\theta$ wasn't in degrees anymore, but rather the same as an arc length, and sine and cosine were now side lengths, not ratios. Why? Sure, just set $H = 1$, but this explanation felt hollow. And now I'm just supposed to accept that these things have a wavy graph, and that if you go a length $\theta$ in the $x$-axis, the corresponding height of the graph is Opposite over Hypotenuse? Where even is the triangle anymore?! To me, none of this made sense.

Of course, now I know that I was in good company. The mathematician G. H. Hardy raised attention to these issues in his Course in Pure Mathematics, in which he said: "The whole difficulty lies in the question, what is the $x$ which occurs in $\cos x$ and $\sin x$? To answer this question, we must define the measure of an angle." I also now have satisfying answers to all of these questions. I have decided to write them in my math notes, which I hope will be of use to someone else who has these questions.

Of course, I learned none of this in school. I learned all of this through 50% self-studying online resources and 50% figuring things out on my own (Which was necessary as most online resources out there are just as uninspired as the stupid classes on purpose since they're just meant to copy them). All school taught me was that I shouldn't worry about any of this, that proofs are nonsense, that it is far more important to know $\csc(-7\pi/4)$, and that I need to learn "Trig" before "Calc", despite the fact that any rigorous treatment of the measurement of circles and any rigorous definition of the circular functions requires the calculus or something like it as a foundation.

I'm not saying you're not allowed to learn about the circular functions before the calculus. My point is that you can learn math in any order you want, so long as you explore questions that arise naturally from your own exploration and curiosity, not just questions that show up on some curriculum. I'm just saying that I've come to the conclusion that people who tell you that knowledge of the circular functions is a must-have prerequisite for learning the calculus are either lying to STE students (not a typo) who just want to pass their boring Calculus classes, or they are so ignorant of the nature of mathematics that they genuinely believe the way Stewart, the College Board, or whoever else chooses to present the calculus in their classes, as well as which contrived exercises they choose to pull out of their ass, are just as fundamental to the calculus as the relationship between integrals and derivatives.

Tangents aside, why did I bother to figure out any of this? I've spent most of my time talking about how I didn't care much about math, and I just got done talking about how I ended up disliking proofs. What caused me to change course?

I started Algebra II the school year after everyone got locked in their houses because of the bug. Pretty much all real-life social interactions with anyone other than my mom got abruptly cut short. Now, in the new school year, pretty much everyone I knew got to be in the same classes, but I got separated from them. Most of them were in some honors magnet program, so now, in our junior year, our tracks had drifted.

Feeling alone and left behind, and still recovering from the effects "distance learning" had on me, as well as being equipped with a newfound disliking towards math (if you had asked me at the time, I would've told you my favorite subject was history. I know because I have a screenshot where I do exactly that), I began to flounder in Algebra II in ways I hadn't before.

I got a C on a quiz about factoring trinomials, which should've been an easy A. I'm still very slow at that. I didn't get polynomial long division or synthetic division. I still don't. I didn't get completing the square at all, and even now I still have to draw a picture to do it. The differences of squares and cubes felt arbitrary to me, and the properties of logarithms were not intuitive. The calculations had all become more tedious than ever. I began to feel like I was hitting the end of my mathematical ability.

I feared this day would come. I remember telling my Geometry teacher that I didn't want to catch up with my classes to take Calculus. Despite the fact that I had already asked myself the slope question that was one of the motivations of the subject, that I was fascinated by Zeno's paradoxes when I first heard about them on the internet in fifth grade, and despite the fact that I even since I was very young, I was entranced by the way each frame in a smooth animation is indistinguishable from the last, yet somehow, they come together to form motion, I didn't want to take Calculus. I didn't know what it was.

All I knew was that only depressed college students took Calculus, and only god-like geniuses could actually understand it. All I knew was that if I ever saw a $\int$ or a $\partial$ (whatever those mean), it meant I should run away as fast as possible. All I knew was that Calculus was to be feared. My experiences in Algebra II made it seem like this viewpoint was reasonable. I was already starting to struggle. Just imagine what Calculus would be like!