I had a hell of a breakthrough in learning some math this morning…and I wish it had happened ten years ago. But first a little background:
For most of my life I’ve thought I was bad at math, which really sucked because I was otherwise a relatively bright kid. When I was really young I had a talent for memorizing things and it helped me coast through tests to the delight of my parents and teachers. It was also a huge help at the easy stuff such as basic arithmetic and multiplication, but it when the time came to hit up algebra I hit a brick wall. I could not for the life of me make the uber-abstract world of algebra work inside my head and it made me feel stupid for the first time in my life. In middle school and high school, algebra study sessions at the dinner table frequently left me in tears after spending hours and hours not understanding what was going on.
It’s not like math didn’t work for me at all…I just didn’t realize the things I was also interested in counted as math. Geometry was easy for me and the proofs for it made complete sense to me. In one middle school math class I lost focus in, I spent one long afternoon deducing what a 4-dimensional hypercube must look like. My teacher upon seeing the work was out of her depth and tried to redirect my attention to my class work. I found out five years later I’d gotten it right. Likewise, by high school I’d immersed myself in the world of comic book art and fancied a career for myself as a penciler. In learning this art, I’d assimilated the canon proportions of the human body and face, learned rules of perspective and how it can make the sizes of things shift to an observer. I’ve also had a lifelong fascination with patterns–finding arrangements of things like the tiles on the ground in the subway, arrangements of rivets on merry-go-rounds, etc, and working to find symmetrical arrangements of these patterns that can be infinitely repeatable. During phone conversations in my parents’ home I took to finding an algorithm for which of the kitchen tiles to step on such that I could navigate around an island countertop and always perfectly arrive back at the original tile (it took a couple years and countless phone calls, but I finally got it). But I never, ever thought of any of these things as math.
Math in my mind was abstract, it was something minds more nimble than myself did. For the life of me, I could see no beauty in the arrangements of numbers and variables. They were just things…staid, boring facts that were presented as sacrosanct rules which I must repeat the usage of in problem sets over and over again to prove I’d learned them (I frankly never did learn most of them). Calculus in particular was a word that filled my mind with dread. Calculus was something the smartest of the smart did…it was math akin to doing magic that only the sharpest of kids got to do in high school and which I never had a desire to work toward or do myself. It was something that you had to get through algebra to get to, after all, and if it required more of that unholy work, I wanted no part of it. Quite frankly, I didn’t even know what it was until my late 20s.
In college, I’d already decided math wasn’t for me and had figured out a way to game the requirements of my major to avoid taking any math classes. At CU Boulder, the college of arts and sciences would accept any engineering course as a math course, assuming that math was all that engineers actually did. I found an obscure freshman level engineering class called Telecommunications I to take. The course was on how to write HTML–something I’d already been doing for 4 years. It was the easiest “A” I got in college and with it I managed to skip Statistics, Differential Equations, and the host of other crazy-sounding classes my peers constantly complained about. Success! Or at least I thought…
Then in 2006 I went to Burning Man for the first time and it introduced me to fire dancing. More specifically, poi spinning. Poi is a really fascinating art because it inspires such incredibly different reactions from the people who practice it. At one end of the spectrum are people to treat it as ornamentation for movement and dance and at the other end of the spectrum are people for whom the analysis of how the tool moves is a seriously and decidedly complex pursuit. This latter end of the spectrum tends to attract people with backgrounds or interests in mathematics, physics, and programming. I picked up the tool myself in the spring of 2007 and was immediately hooked. My inclination toward pattern recognition and reorganization had found a very comfortable hook to hang its hat on and I found the practice of the art taking up more and more of my time.
As I assimilated the tool more and more, I began thinking of its movement not in terms of the arrangements of disparate movements into recognizable patterns, more colloquially known as “tricks,” but as complex combinations of simpler basic movement elements and it was these that fascinated me the most. By controlling and rearranging these elements in different fashions, I found I could for the first time create movements and tricks that no one else had yet performed or recorded. It didn’t happen overnight and to be honest I couldn’t tell you when I crossed over the line into doing something original, but what it did do was get me hooked on understanding the mechanics of the tool. At one point in the pursuit of this knowledge, I became obsessed with finding the distance the poi head would travel in all the different shapes I could make.
A good deal of poi movement bears significant similarity to a type of geometry called roulettes or trochoids, which are complex curves that tend to overlap with themselves and cycle in such a way that they produce flower-like patterns much like a spirograph. Mathematically, these patterns can be described using parametric or polar equations. To find the distance traveled by the poi head inside these shapes, it meant that I had to not only learn this type of math, but also the dreaded calculus. For, as I eventually learned, calculus was the mathematical study of curves and now everything I was doing relied on them.
And this is where things got really hard…math is traditionally taught in such a way that properties of it–say, the distributive property of multiplication, the quadratic formula, etc, are presented almost as canonical law. Students are then given problems that require these properties to solve and when the produce enough correct answers, they are thought to have learned the property successfully. This was always my issue with math: I understood the properties as presented, but I never understood WHY they worked. This is something that no math class I’ve ever taken has bothered to do–there seems to be an assumption that, this property having been proven, there is no need to understand how it was proven or why it works. Ironically, given that students are always required to show their work on a problem, we were never shown the work of how the elements we were working with worked themselves.
This became a problem in tackling my poi problem: I could find many, many references online as to what the parametric equations I needed to know to draw out poi patterns looked like, but none on why those equations worked and how they could be altered to produce different patterns entirely. I got very, very lucky in that at a fire festival two years ago I had a chance encounter with a juggler named Adam Dipert who took me through the process of creating the equations I needed. I do not think it’s a stretch to say that the ten minutes I spent with Adam that day taught me more about math than the entirety of high school did, nor that Adam is easily the best math teacher I’ve ever had. Far from just giving me the equation, Adam started by showing me how an equation could describe the movement of the poi head around then hand, and how the properties of that equation could be used to extrapolate a way of describing the hand’s movement as well. I won’t claim that instantly all was revealed, but I did now have the tools to figure out everything I wanted to know. It was as close to eureka as I could hope to get. I began creating problems for myself, not just of the flower-like roulette patterns, but also of three-dimensional poi moves that knotted and bent their way through space like corkscrews and doughnuts. And this was when math switched over from being a set of staid, lifeless facts and became a living, breathing thing.
Part of the problem, at least in my case, is that I am a hacker by nature. I see systems and patterns and upon figuring out the rules that govern these systems and patterns I want to find ways to recombine those patterns into new ones. It’s little wonder poi appealed to me, then, but this is a type of thinking that seems altogether alien to the way math is taught in most public schools. As students we are presented with the dry properties and problem sets, but rarely if ever presented with real-world problems for which the math we are being presented is the answer. We quickly forget all the knowledge our parents and the other tax payers of our school districts have paid handsomely for because it seems unrelated to the things we spend our time doing. When math had an application for something I was interested in: drawing human bodies in a way that seemed proportionally correct, finding the correct number of tiles to skip to navigate the kitchen in an easy-to-repeat pattern, I was fascinated. When math was presented as an isolated dalliance with inscrutably abstract numbers and figures I was lost and convinced I was stupid.
I’m still working my way toward an answer to my problem on poi distances. A friend wrote a computer program for me a couple years ago that took parametric equations and measured the distances traveled by the curve, but it’s only accurate to four decimal places. As repeated patterns have emerged from the data, I need to know how to write the equations that produce these distances themselves so I can isolate mathematically what these proportions are instead of just knowing what integer they are. To do this I’ve finally had to step up and teach myself calculus. I’ve had some wonderful help: Khan Academy has been hugely helpful for some of the basic step-by-step knowledge. Some of the a ha moments have come from a wonderful book that was a Christmas present from my girlfriend’s family: “The Calculus Diaries”, a wonderful book that is short on explanation of how calculus works but wonderfully detailed in all the problems it can solve. But by far the best tool I’ve had to work with is the brain of a hacker.
Many of the tools I need to simplify the equations I work have long since been lost to the annals of time. I’d love to claim I held onto all the quadratic, binomial, and trinomial properties I was supposed to have learned in high school, but it would be a lie. Now, however, I’m able to look at the numbers as yet one more thing I can hack. After tackling basic derivatives in Khan Academy’s hugely helpful video, I decided I wanted to take some of the solutions presented at the end of the last video, showing the derivatives for f(x)=x^whatever and rather than just take them at face value, create my own problems to test them and find out why they are what they are. It took a long time…I’m positive that if I’d remembered more of my high school algebra there are many steps I could have skipped or simplified, but in all honesty the extra work made it that much more rewarding when I finally got the answers myself. I tried here and there to find guides online for some of the work I was doing, but I still lack the vocabulary to describe most of this work. When it came right down to it, the solution I found was satisfying not just for reaching the answer, but knowing WHY the answer worked and thus how it can be obtained in situations that veer wildly off the grid.
As a quick note, I will say that there is another disincentive to learning math in this fashion and it has more to do with the attitudes of people who don’t consider themselves math people. People who are fascinated by math growing up may be marginalized as nerds or geeks at an age when children are known to be cruel by nature, but it’s quite another to encounter it adulthood. More times than I’d like to count, I’ve tried to shared some of the breakthroughs I’ve had with close friends only to find them recoiling at the idea of having to comprehend any complex math as an adult. The marginalization that kids who excel at math find isn’t just limited to childhood and it’s very disheartening to find it rearing it’s head even among my 30-something friends. It’s hard to take seriously the claim of friends who consider themselves open minded and whole-heartedly support their friends exploring art, music, and other means of personal expression but somehow find expression through math abhorrent.
At 31, I’m a decade past when most people who are going to learn calculus have learned it (and probably a few years after they’ve already forgotten it). There are countless lessons on it I’ve missed and I can’t help but feeling a great sense of regret that an education of this type wasn’t available to me as a child when my self-image was being ravaged by the onset of algebra. But now, being a math hacker, I know that the knowledge of it is going to be far more ingrained than anything I would have learned in high school or college. The value of the thing, after all, is the use we accrue from it. So, parents with kids who are having math problems, see if you can’t get your child more engaged by finding the real world problems that they solve. See if you can’t find how to make math applicable to the things they care about. Math isn’t hard…it’s just very good at convincing people it is.