Ink Tea Stone Leaf

(This essay previously appeared on an earlier version of this blog)

When I was a junior in high school, I joined my school’s Academic Team. This was an activity that relied upon the ability of myself and my teammates to recall information from our classes faster than our competition could, and to buzz in like contestants on a nerdy teenage game show. These were two things, it turned out, that I was very good at. I ended up being the only junior on the varsity team that year, and when I came back for my senior year I was given the position of “captain.” I wore a fancy jacket and a tie to each of our matches. I even have one of those felt varsity letters in storage somewhere. This was my version of high school glory.

My grades didn’t always reflect my raw talent at remembering stuff quickly, due to any number of things that were the matter with my neurology. If I’m more proud of anything than being captain of the team, then it’s getting my shit together fast enough to graduate and head off to college. Regardless of my struggles, however, I was always absolutely confident in my ability to recall and understand academic information that I’d learned, in any subject. And I had a lot of fun doing it.

There was one significant exception to this, and that was math. When the answer to any question put to our team involved calculations of any complexity, I threw up my hands and deferred to my compatriots. I usually wasn’t confident that I understood the material, and even when I was confident, there was no way I could arrive at the right answer faster than they could. As it happens, we had some very talented mathematicians in our ranks, and my rather conspicuous disengagement from the subject didn’t seem to hurt our team. We were the second best in our league that year (undefeated save one), and we competed in a championship tournament (we did not win, but we had fun). There’s not much cause for regret as far as I can see – we all did the best we thought we could, and we did pretty well.

I’ve worked in education for a while now, and in spite of my coarser instincts I feel compelled by some sense of honor to affirm that all academic subjects are important and that none should be neglected. Certainly, I neglected math, and not always in the passive and indifferent sense. Sometimes that neglect was born of hostility and malice, or abiding resentment. Nowadays, my feelings have actually changed to the point where I regard the whole subject with curiosity and respect, even if some points remain obscure. Some of it is the inevitable mellowing of an opinionated teenager, but I also feel that circumstances have arranged themselves to show me the value in thinking about the processes and properties of numbers. I doubt I could be much more helpful on the team, but I’d like to think I would have tried a little harder.

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Despite my best intentions and the dictates of common sense, in my career I have occasionally found myself in a position to be teaching or tutoring students in mathematics. Fortunately for them, the math in question has seldom risen in complexity above what I managed to master in my high school days. I’ve gotten through it, sometimes by relearning the material on the spot, sometimes by hastily cramming a Wikipedia article.

These days, I’m working part time as a high school AVID tutor, acting as a facilitator and guiding hand in an otherwise student-directed process. In biweekly tutorial sessions, students are supposed to bring in a question they are stuck on, while the other students in their group help them work through it to gain understanding by asking constructive questions. It may come as little surprise that they mostly bring math problems, and often struggle to translate their specialized vocabulary into ordinary language.

As it happens, many of the groups I’m tutoring are studying right triangles and trigonometric functions: sine, cosine, and tangent. The realization that there would be a lot of these questions put before me in the past few weeks sent me into study mode, and I was pleased to find that, with a little review, these terms actually made a lot of sense to me. A lot of my math dread memories seemed difficult to explain, and the triangles themselves began to fascinate me in a way that I wish they had twenty years ago.

These tutorials I am participating in are, as I mentioned, designed to be student-led. An ideal tutorial would typically consist in my saying as little as possible, documenting healthy amounts of participation from all group members and providing assurances that they are moving along the right track (or putting up guardrails if they should begin to stray). But for any number of reasons, not all tutorials are ideal, and I must sometimes intervene to explain critical concepts that none of them can articulate, or that they haven’t properly learned.

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My brain has been steeping so long in right triangles and their properties that, in attempting to write a simple review of their properties for the benefit of anybody who can’t quite remember high school, I accidentally wrote an eight paragraph trigonometry lecture. The experience shocked and bewildered me, so for the sake of the reader’s sanity and mine, I’ll change tactics and try to focus instead on the right triangle’s more aesthetically satisfying qualities. This might be even weirder than the geometry lecture, but should be more fun at least.

You could say the 90 degree angle is the most classic of angles, sitting halfway between 0 degrees (which is not really an angle) and 180 degrees (which is also, arguably, not really an angle) to embody maximum angularity. Since the sum of all a triangle’s interior angles must be 180 degrees, the right triangle’s two smaller angles must add up to equal the third one; a + b = c. It’s a satisfying sequence, like executing a triple jump in Super Mario 64. Or, maybe something less niche.

Now consider the pythagorean theorem, concerning the length of the triangle’s sides. It’s definitely among my favorite mathematical theorems. Just as the 90 degree angle must be the triangle’s largest, so must the side opposite that angle be the longest. If each side of the right triangle is reimagined as the side of a square, then the area of the two smaller squares will add up to be as big as the third: it’s like Mario triple jumping exponentially into new dimensions.

Then there’s the trig functions – that is to say, the ratios of the lengths of the right triangle’s sides, as they correspond to the size of the non-right angles of the triangle. If one such angle has a sine of 3/5 and a tangent of 3/4, then these will be the cosine and cotangent, respectively, of the other such angle, and vice-versa. It’s an elegant symmetry, much like how a right triangle can be duplicated, mirrored, and then combined with the original to form a rectangle. I haven’t figured out how to extend the Mario metaphor here (and complete the rhetorical triple jump), but I’m sure the internet will have thoughts on the matter.

Incidentally, these relationships all stay exactly the same no matter how big the triangle in question is, or how it is morphed or distorted, proving the fundamental and consequential relationship between the angles and the sides in a logical and deterministic way. In other words, the measurements of a right triangle cannot be random. If all but one or two sides or angles are known, then the remaining cannot help but be known if one cares to know them.

Geometry, and mathematics in general, can be dazzlingly complex. People have extended the possibilities of what numbers can do and mean to the limits of the human imagination. But all that complexity is still built on these simplicities, and the willingness to follow all valid ideas to all their logical conclusions.

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Two decades ago, I would have been astonished at myself for holding forth on triangles, especially trig functions. Trigonometry was, in fact, the highest level of math I engaged with in my whole academic career; elaborating on the relationship between the magnitude of an angle and the arc of a circle, graphing waves, and a bunch of other things that I don’t properly remember whether they were a part of “Trigonometry” or not. My memories of Trig are so fuzzy that I’m not really sure how I gathered enough understanding at the time to pass it in my senior year. I don’t remember what my grade was and I don’t care to look it up.

To be sure, I’d come a long way by that point from the absolute nadir of my math education, eighth grade Algebra. In those days, I was so frustrated with anything having to do with calculation that I simply wrote “5” for most answers on homework and tests. The ramifications of this decision were not pleasant – my reasoning that 5, as the mean of all the numbers from 0 through 10, was likely to be the right answer to any problem did not impress my teacher or my parents. But I was at a conceptual brick wall, and being just a little too proud to look for a way around it, I doodled aimlessly on it instead.

I had to repeat Algebra I in 9th grade, which put me a year behind most of my friends in math and science classes. But with a year’s growth, a change of setting and teacher, and the application of some ADHD medication, the class actually started to make sense. It was not as though I were suddenly adept at the subject, but I was certainly no longer inclined to write “5” if I didn’t think it had more of a chance to be correct. I actually got an A that year, and I don’t need to look that up to be sure.

My capacity for the abstract thought necessary to do the math had certainly increased, to say nothing of my motivation to put in the work, and I had even begun to develop a personal philosophy of math. There were only four things, I reasoned, that one could do a number: add to it, subtract from it, multiply it, or divide it. All that mathematical notation I was being introduced to was just iterations on those basic operations, done in a particular order. Keeping it straight was a matter of realizing the relentless abstraction into something concrete and manipulable.

If math were something that could react with intention or malice, it would surely delight in putting my naïve philosophizing to the test. In any case, I’ve remained frustrated by some of the more abstract mathematical processes in the high school curriculum. Take, for example, the logarithm, a word that to this day connotes impenetrability to me. Even now, when I’m sitting in with students working through a logarithm problem, I am tempted to shut off my attention and await rescue from the responsibility of understanding it.

Maybe I lacked the patience to work through to the fundamental insight that would make logarithms transparent to me – maybe I just lacked the vocabulary to ask the right questions. They remain emblematic to me of a kind of intellectual black box – some kind of process occurs, which my calculator apparently understands but which I cannot reproduce with my hand, a pencil, and paper. I understand logs a little better now (at least at a basic level), but when I tried to explain them here, I couldn’t be sure I wasn’t talking nonsense.

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Theoretically, when I walk in to the tutorial classroom, I am taking part as a facilitator for motivated students with ambitions of higher education, who are aiming to develop the skills needed to contend with rigorous academics. Naturally, in reality the students come to the room with widely varying levels of preparedness and motivation. Some of them look like future college students; some of them look like teenagers with a lot on their minds and not much understanding of what they are doing or why they are there. For many, there is a noticeable gap between what we hope to inspire them to do, and what they are ready to achieve.

In a tutorial session, each student is expected to bring a question from their homework or classwork that is causing them difficulty(the initial question). From there, they are expected to explain what they already know about the question, describe what academic vocabulary is necessary to understand it, and show the process that leads to their point of confusion. They then formulate a tutorial question that expresses that confusion – it is the tutorial group’s task to help them resolve their point of confusion, so that the student can then solve the initial question.

This process goes wrong in many different ways. Often the tutorial question is substantially or even literally the same as the initial question – the student in effect declaring they don’t know the first thing about solving the problem. Too often, the student in the hot seat will quietly mumble their presentation as the others around the table sit in silence. Or, they’ll come with nothing to present at all.

There is no one reason that a tutorial presentation goes bad, but my observation is that many of the students are simply not prepared for abstract problem solving, and there are few operations in the process that do not seem like black boxes to them. They’re being led through an alien zoo and being asked to name the animals.

Thinking about abstract geometric concepts requires mastery of the basic vocabulary, and many students do not have this. Many times, I have heard them ask about an angle’s “sin,” as though the triangle had fallen short of the will of god, without realizing that “sin” is merely an abbreviation for “sine” to make it easier to fit on a calculator button or write into an equation. As they’ve moved through the weeks, I’ve watched students ask meekly for help in finding things like congruence, circumference, or area, then reveal under questioning that they simply do not know what these words mean.

Just this morning, I watched a kid struggle for several moments with a misfiring calculator, trying to find the supplementary angle of 120. As his frustration mounted, I chimed in to say “I think you can do this in your head. What is 180 minus 120?” He looked scared. I rephrased it to 80 minus 20. For a moment I was scared I’d have to bring it down to 8 minus 2, but he caught on.

I don’t mention this to mock this student or bemoan the state of our schools, but to say that I recognized his frustration and loss of wherewithal in myself. He’d been so worried about solving a complicated problem that he’d failed to recognize the simplicity of the piece of that problem he was directly engaged with. In a fundamental way, he was not connecting his present challenge with the applicability of a skill he’d learned a long time ago.

In my experience, as a student and an observer of students, there comes a point when the mind is willing and able to understand a concept, before which little progress is possible. Reaching this point is largely about first mastering more basic underlying concepts, but there is more to it than that. There is also something which is crudely a function of age, but more precisely a function of intellectual maturity – the recognition of the actual ability of one’s own brain, and the readiness to apply the brain in direct proportion to that ability. It is the lack of intellectual maturity that makes learning new material a drag, and it’s the largest obstacle to overcome for students, teachers, and educators in general.

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All 6 Trig Functions on the Unit Circle, by Beautiful Math

There are very good reasons to be intellectually mature, and to wish to be more intellectually mature than you currently are. The catch is that to the intellectually immature, these reasons may seem insufficient, incoherent, and irrelevant. This is why we so often sell children on education with the promise that it will convey social or economic advantage. Though it may, it’s very much not the point. I will forever insist to anybody who will listen that it is not the point.

Socrates’s well-known formula for wisdom is that the truly wise know that they know nothing.  To believe you understand anything with perfect certainty, is to invite a flood of questions that wash away the mud that conceals your ignorance. Our knowledge is always contingent, and what we believe to be case is always mediated by what seems to be the case. In a similar vein, to be intellectually mature requires humility as much as accomplishment. It’s a regard for the yet-to-be-known as sublime.

I am humbled by the realization that I don’t know how to teach intellectual maturity. My own sense and recollection is that I acquired whatever measure of it that I have as a result of particular life experiences, which cannot be duplicated. But in the opportunities I’ve had to revisit, as a more mature person, a subject that frequently bedeviled me in high school, I’ve reaped the benefits of my initial efforts to understand mathematics as a sincere appreciation for mathematical beauty. I can only hope that these students find their way to something similar as they grow.