By Jake Wong
Runner-up of 2019 Winter Prose Competition
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A fraction can mean anything by itself. One-half is fifty percent. A glass of water is half empty or half full. Half of all marriages end up lasting forever. Half of my face has more freckles than the other. A halfby itself is confused: it does not know which way to turn or what to describe or where to place itself. But numbers find meaning together. If we take ahalfand add afourth, and then an eighth, and then asixteenth, and keep adding numbers in that pattern forever, the sum gets closer and closer to 1.
(1/2) + (1/4) + (1/8) + (1/16) + (...) = 1
When we say that a series converges to a limit, what we mean is that a list of fractions, which go on according to some pattern, will forever approach a number—in this case, that number is 1. A limit is the collective ambition of this series: something it will never attain but will work forever towards. These numbers have a cohesive stride, an order, and a destination. I like patterns that don’t have an ending but have an answer anyways.
~
When I was eleven or twelve years old, my mother and I would sit for hours at the dining table trying to solve math competition puzzles. They usually came in a set of fifteen, printed neatly on a single sheet of paper. When we were lucky, we could solve half of them. More often, we’d solve only one or two. After working for hours on a set of seemingly short and simple questions, we’d both end up frustrated. I knew my mother had cracked a puzzle when she’d take out a fresh sheet of paper, start writing equations, and seek my gaze. But even when I understood her ideas, I didn’t have the technical skills to back them up. I was usually lost in the details she glazed over: factoring, condensing square roots, simplifying fractions. That’s when she’d start lecturing, and I’d start crying, and my father would come over and intervene. Ok, enough, he’d plead with us, no more math.
Those puzzles weren’t easy, and I wasn’t considered incompetent—but certainly not talented, either. A few of my friends, Dan and Ben, had a sharp talent for competition math. In half the time I spent on a single question, they’d consistently solve eight or nine. Every puzzle had a soft spot—an Achilles’ heel, so to speak—that could easily reveal the solution. Though my mother and I were unaccustomed to such mathematical tricks, it was still disheartening to think that those puzzles we agonized over were so easily solvable. They required a special kind of insight, the shrewdness to know where to aim. Dan and Ben knew. They picked up a pencil and wrote confidently, one line after another following logic: a direct path to the answer. My mother and I didn’t know where to aim—we flung ourselves around the page, trying our luck with vague polynomials and double-digit guesses, reaching for anything, feeling our way through, desperately hoping to find a solution.
~
In math, an integral is the total change over time. To integrate is to combine pieces together. If we integrate a curve, it means we can add up all the directions it went, up and down over a certain period. We add up all the changes, bit by tiny bit, and in the end, we get the total change. We integrate to ask: how much did something ultimately change from start to finish?
~
In middle school, when I was doing math competitions, math was a measure of intelligence and academic success. One student, Andrew, traveled to a high school every day to take math three years above his current grade level. He’d leave on a school bus all by himself in the middle of the day, and when he returned, we’d praise his intellectual gifts and call him professor. We conflated equations and numbers, no matter the context, into a single topic representing elite knowledge. Unintelligible equations and mathematical symbols came to represent the world of high-level wisdom, an awareness about the world we might someday attain if we were good enough. Perhaps it was the thought that Albert Einstein, our archetypal genius, might not be the best history or English student, but he would certainly ace a math test. If we wanted to be as wise as Einstein then, we ought to be good with numbers.
We admired math in those days—but with a notion of wisdom that was tenuous, at best. We stumbled wide-eyed through long, shiny hallways carrying piles of binders in our clumsy arms. We jammed our lockers for the first time. We complained about dirty bathrooms, gossiped in them, and separated ourselves into social cliques. We raised our hands during lunch to buy gummies. And we did math. Not as a personal endeavor, but as a way of belonging—to the group of good kids, the group who had a shot at being Einstein-level smart, the group of future rocket scientists and Star Wars engineers.
~
If an integral is the total change over time, then perhaps we want a starting point and an ending point to measure from. We might ask: how many inches did we grow from ten to twenty years old? How many miles did we travel to get from home to here? How much can we learn in four years of college?
~
Three times a week at 10:20 AM, I walk into a class known notoriously around Duke as math-two-twelve. I sit in the back corner, right side, next to the dirty window that overlooks a giant cherry tree. Five-petaled pink flowers spread over its delicate branches in mid-April. When the windows are open, it smells crisp in here with the scent of newly blossomed plants and fresh morning dew.
I glance quickly around the room. Mostly everyone in here is a college freshman. I wonder if they’ll become engineers, mathematicians, physicists, statisticians, or somebody else entirely. I’m a sophomore. I wonder about myself too.
Unlike other math professors, Dr. Bray doesn’t use the chalkboard. Instead, he connects his tablet to the projector screen. He prewrites all the lecture material, and we print them out. During class, he scrolls through his notes, emphasizing certain points by coloring them with an electronic pen. I bring my own set of colored pencils and try to match his color scheme.
This is a delicate, delicate theorem, he says while coloring it green. Some students don’t like the way they say he dramatizes math, but I find his quotes meaningful. I have a special red pen reserved for writing down the ones I like. Imagine air, he says, circulating around in tiny circles. Use your geometric intuition. I urge you to convince yourself that this theorem is natural. This is the moral equivalent to our original idea. I write them all down, quote after quote. Red scribbles of ink everywhere. I like the way he describes things, the way he uses words like delicate and moral and natural. He imposes meaning on formulae, rather than simply stating them. We both sense that math is meant to be deeply understood, not simply used. I feel comfortable in this small acknowledgement I share with him.
As I’m looking out the window, his words linger in my mind. What does it mean for a theorem to be natural? To be delicate? What if we could understand equations the same way we understand language? Maybe I’m just fooling myself. Is it even possible to find math—the same way we find sentences, people, or cherry trees—beautiful? A heavy gust of wind blows through, tugging at the branches and making petals flutter and fly swirling away in the air.
~
If we want to know an integral, the total change over time, we might want a starting and ending point. But sometimes, we don’t have fixed years to measure from. We might ask: how many inches could we grow in a lifetime? How much could we learn? How many times would we change our minds? How much do people change?
~
Over winter break, I reunited with Andrew, the star student who took high school math in middle school, and who had become, it turns out, one of the few friends from home I kept in touch with. We ate at a small café and went to Starbucks together. It was mid-afternoon on a chilly January day just after New Year’s. We sat outside under the cloudless sky and shining sun.
“If I had gone to Georgia Tech with you all, I’d probably be studying computers,” I said. “Instead, I went to Duke and now I’m considering things horrible like math. Remember how bad I was at math competitions?”
I asked him whether I was making a big mistake. Whether striving for beauty and a deeper understanding in math was worth all the arduous work of homework sets, more hard puzzles, and long hours. I’d always felt that Andrew was more practical and logical than I was—and that my decisions, which often seemed to stem from some mysterious intuition, warranted his advice.
“But no matter what you choose, you’ll still be you, don’t you think?”
It was reassuring to hear that word, you, spoken from a friend like it was something tangible, something rooted far deeper down than what a person studies, knows, or thinks in school.
“I’m sure there are many things you’d be happy doing,” he continued. “And maybe in some other multiverse, you might pick another major, and maybe you’d be a tiny bit happier. But who really knows, right?”
His answer didn’t fill me with confidence. But I nodded anyways, trying my best to take his advice to heart.
~
A few months later, at the start of spring, I walk into the Academic Advising Center to declare math as my major. I expect there to be a long line of people. There are none. Instead, I’m greeted by two whiteboard signs that read, “Major Declaration Here!” in big and bubbly lavender letters.
I carry a thin stack of paperwork. It contains a schedule of the classes I want to take, and my short “Why Math” essay. Math, as I am learning, requires only the persistence to think deeply and imaginatively about simple things. I mouth my own words as I read them over and over, convincing myself that they’re true. There’s nothing in the essay about middle school puzzles, nothing about studying math for a chance at feeling and beauty and language. It’s too much to write and I can’t write it well right now because I can’t understand it all.
I enter the office to hand everything in. The woman sitting at the front desk smiles at me but doesn’t ask my name. I assume she knows it from the paperwork. She doesn’t read the essay either. Every page is filed neatly into a folder before she congratulates me. Right outside the office, I take a picture with a life-sized cardboard replica of Vincent Price, our university president. I take some complimentary cookies, a tangerine, and a bottle of water as I exit. The whole process is over within five minutes of entering the building.
The same day I declare my major, I stay up late with math homework. The questions are harder this week. I sit in the kitchen with my textbook open, papers flung across the table, a mug of water half-empty.
An integral is the total change over time. Sometimes I think things have changed a lot, that instead of doing math to prove myself, I’m doing it for myself. Instead of crying during my mother’s lectures, I go to math-two-twelve and stare out the window at cherry blossoms swirling in the wind. Instead of stumbling around with binders in my hands and buying gummies for lunch, I carry around a backpack filled with math and snacks and a thin black laptop.
I always do this—I see an integral and my mind wanders off about how things change, or things don’t change, or how little pieces of change add up over time and maybe that’s how people change.
But I force myself back. Tonight, the homework questions are short and simple: only a few lines long at most. They remind me of old math puzzles, the ones I never had a blazing talent for. I know the homework can be solved quickly, but I still don’t know where to aim, so I sit there struggling, hoping for a solution, feeling my way through as the clock ticks by. I wonder if it’s foolish or imaginative to think of math as something we can feel.
It’s 1 o’clock in the morning, then 2, and by 3 A.M. everyone else is long asleep when I glance at my reflection in the window. On the kitchen’s thin, dark windowpanes, the reflection materializes like someone in a dream. I tilt my head and blink my eyes. The reflection follows. I study him in detail. Half of his face has more freckles than the other. Half of his homework is done. Half is not. Why did he pick such a hard subject to study? He’s grateful to be able to pick anything he loves to strive for. Half of him is feeling slightly lost. Half of him is happy.
But a half is just a half. Numbers find meaning together, remember? A half plus a fourth plus an eighth plus a sixteenth, and so on forever, approaches a whole. Some piece of me is always dreaming, feeling my way through when I can’t think of what else to do. A part of me misses home. Another part wants to stay and learn, wants to understand. Wants gummies. Wants a cherry tree.
Critique
Thank you so much for your submission to our Winter Story Contest! First of all, your connection of math and writing was super creative and we loved how you discussed summation of fractions in your introduction and conclusion. Before I get into the nitpicky edits I’m going to start with some broader critique we picked up. No question, this was well-written, but we’re hoping this feedback can make it even better.
As a whole the piece read a bit more like a personal statement than a story. Your anecdotes and description were there, but we think that the root of this issue was probably the length. Could you have cut one or two of those anecdotes to make the story more concise? Instead of focusing on every up and down related to your math career, could you have highlighted just a few of those standout moments? Deleting some portions would give you the space to really build up the elements you did choose to keep: you could integrate even richer imagery (pun? no? sorry) and really pull us into the moment.
Personally, I really enjoyed the whole flower-petals theme, and how it popped up a few times in the story. You wove it perfectly into your beautiful conclusion. In fact, your conclusion as a whole—connecting his half reflection with the fractions from the beginning—was fantastic. Reflections can be difficult to play with and are often cliché, but you pulled it off here.
Here’s some more detailed feedback:
“I wasn’t considered incompetent—but certainly not talented, either.” (pg. 1) Be careful with your use of the passive voice! Switching it to active may but us more in the scene and make this even more of a story. Instead of “I wasn’t considered incompetent”, try “My teachers didn’t think…”
“We admired math in those days—but with a notion of wisdom that was tenuous, at best. We stumbled wide-eyed through long, shiny hallways…” This paragraph is great. The kind of built-up detail I think you could pull into the rest of the piece by deleting a few paragraphs here and there.
“He prewrites all the lecture material, and we print them out.” (pg. 3) This sentence didn’t feel necessary to me.
“Instead, I went to Duke and now I’m considering things horrible like math.” (pg. 3) Is this a typo? I think you meant “horrible things like math”.
Overall, we really enjoyed this piece! By considering the changes above we’re confident it can go even further. Thank you again for your submission!