A Mind For Numbers

Fahim sank further into his seat, in a classroom with fellow STEM majors sitting upright and diligently taking notes. He doodled drawings on the margins of his notes, away from the disarray of calculus equations on limits and derivatives. His professor turned towards the board and wrote a problem describing a function as the limit approached zero.

He tapped the zero twice with the chalk.

“Now– this part matters,” he says. “When we say as x approaches zero, we’re not asking what happens when x is at zero. That’s a different question entirely.”

Some students nod, others keep writing.

“We’re asking what happens when x gets very close to zero. Closer than close. Infinitesimally close. In fact, some functions may not be defined at zero. There could just be a hole there for all we know. Doesn’t matter. We want to know the rate of change at a certain point. Just like you all— you may or may not know where you are headed. But, you can see how you are doing right now.”

Fahim’s gaze was fixated at the chalkboard— as the numbers, arrows, and graph lines molded into a gibberish amalgam.

“Some of you need to pay attention to this, your quiz grades last week speak for themselves,” he said as he scanned the classroom and locked eyes with Fahim momentarily. “We’ve gone over limits as they approach zero with trigonometric functions several times to prepare for this quiz.”

Fahim looked away out of guilt, back at his doodles on the outskirts of his notebook. His last quiz was graded a four out of ten, written in a bold red ink that demurred a sense of urgency.

Please see me after class— was the last thing he saw written on the top of his quiz before he crumpled and shoved it into his bag. His heart sank in his chest further than he sank in his seat. He was familiar with these conversations, especially with his parents— disciplinary in nature, hopelessly optimistic at best.

What is this, baba? Fahim’s father would ask, usually holding a failed math exam. His glare would pierce his wire frames, enough to shorten Fahim’s breath and tighten his chest.

Baba, you need to learn this. You learn math, you can do anything. You can be engineer. Maybe doctor, even better. The family needs a doctor. We bought this house for you- so you can have a room to study. Now you want to study English in college? How will we pay this mortgage for you?

He’s had many iterations of these conversations with his father throughout the years, oftentimes citing his aptitude in writing with hopes it would eclipse his faults in math. Fahim was more than capable of writing essays with creative ideas, bulletproof arguments, and cited sources. However, according to his father, his biddings in creative writing could never lapse his frailties in mathematics.

Fahim, you already speak English. Last twenty years, your mother and I cannot find a good job because our English is no good. You ever think about this? Needing to learn English for a job? You don’t need to worry about it, you’re born knowing it. This is a luxury, baba. Now you want to study English— in college? You want to pay to study English? What’s there to study- when you know it already?

Fahim would often bury his face in his hands, already knowing whatever argument he had would be rendered inefficacious.

Everybody knows math, baba. Same language, no matter you’re Bengali, American, or anything.

After class was dismissed, Fahim anxiously approached his professor’s desk.

“Hi Professor,” Fahim approached cautiously, “you wanted to see me?”

“Yes, Fahim, do you have your quiz?” his professor asked inquisitively.

Fahim pursed his lips and reluctantly pulled out the crumpled quiz from the bottom of his bag. The questions were barely legible between the creases of the crinkled paper he gently unraveled.

“Not even a folder, huh?” his professor asked facetiously. “Fahim, I was hoping to review some of the material with you. Our exam is coming up in two weeks, and I want you to avoid failing or dropping this course. Is there something I should know about?”

Fahim was taken aback. He assumed he would get yelled at, or reprimanded with some form of monologue mixture of guilt and fear. It was enough to elicit a glimmer of vulnerability he would never share with his father.

“Professor… I don’t know what to tell you. I’m just not a math person. My goal is to pass this pre-requisite, and finish my degree. I’m an English major. Once this class is over with, I’m over it. I don’t see math the way Steve & Rajinder see it— I mean, they have a reason to learn it. Data Science, Biology, Engineering– this is for them and I don’t understand why I am paying $900 for a course I’ll never use.”

Fahim paused and flinched— expecting to be interrupted and unheard. His professor looked on intently, with a relaxed gaze and furrowed brows. He took a deep breath and continued.

“Professor, I’m sorry. I’ll try to memorize all our notes and I will do my best to pass the exam. I just need a C+ to graduate, and I’ll be out of your hair for good,” Fahim confessed.

His professor’s gaze lingered for a few moments longer until it shifted to his desk, at the crinkled paper.

“Fahim, you like books?” the professor asked gently.

This time, his gaze was gentle and his eyes lifted from above his glasses as it sat just on the bridge of his nose. Fahim responded skeptically with a gentle nod. His professor slipped out a book from his messenger bag.

“I want you to try and read this before our exam,” he said, handing a book to Fahim. The title in bold letters on the cover of the book: A Mind for Numbers: How to Excel At Math And Science (Even If You Flunked Algebra).

Fahim examined the cover of the book and figured exactly why his professor decided to recommend this to him. He lightly skimmed the blurb of the book, the back contained text on the author Barbara Oakley. A Ph.D recipient who struggled with mathematics in school, until she returned to it many years later with a new approach and mindset.

“Fahim, I don’t think you’re bad at math. I just think someone needs to speak it in your language. Let me know how it goes,” his professor said with a soft smile.

This book is for high school students who love art and English classes but loathe math.

That was all it took for Fahim to be glued to this book over the next week. He read it religiously, scouring the pages and finding himself between the stacks the way he never would in a mathematics textbook. Barbara Oakley described her meticulous journey of flunking mathematics throughout high school, simply labeling it as something that wasn’t for her. Something she couldn’t grasp, and didn’t merit the effort that easily came to her when it came to learning languages. That is, until about a decade later, when she decided to re-approach mathematics with a new mindset.

Fahim never thought he’d see himself in a book that had any inkling of mathematical notations or complicated equations. He’d always felt as if his brain was simply not computable with numbers, and was hard-wired for feelings, emotions, and prose. The mathematics that he was taught began to leave the backlogs of his brain from memorization sequences to a universal fervor that stretched throughout everyday things. The time it took to defrost chicken breast became a line graph over an x-axis scaling the number of hours before his mother returned home. The electricity bill his father complained about every month became an equation of different components and factors. The mini-map that guided him in his racing game became an array of cells that computed orthogonal values, resulting in the ideal route to complete the latest mission.

He began to find the universe in specks, and beauty in the mundane.

Fahim learned several new techniques to studying and approaching math, including chunking, recall testing, and diffuse mode thinking. Allowing his brain to approach problems even when he wasn’t actively trying to solve it. He’d study in bits instead of powering through hours of deep studying typically followed by periods of burnout.

Those long periods of deep, overpowering studies were remnants of his father’s approach to mathematics. To simply digest the material by any means necessary.

One more hour, Fahim, then you can go to sleep.

His father had a no holds barred approach when it came to maintaining a standard in school. He’d sit with Fahim under a dim light, with his scribbled notes scattered around the table.

“Baba, I’m tired. Can we do this again tomorrow?” Fahim would cry at nine years old.

“Do you think Amit waits until tomorrow to learn this? He’s your age and has a perfect grade in this class. Why don’t you? Do you want to be a ‘B’ student forever?” his father reigned.

“Baba…” Fahim would faintly whine.

“Amit’s parents work hard. They came from Bangladesh just two years ago, and he’s already doing better than you. You were born here. Did you know some people come all the way from Bangladesh just to study— and here you are, cannot learn your times table,” his father presided.

Fahim would often attempt to win his father’s approval by other means.

“Baba, I finished Magic Tree House today. It’s the third one this week, and I wrote two book reports,” Fahim said proudly.

His father was not easily impressed.

“Fahim, good job. But this is not enough. Reading books won’t make you money. Don’t you want video games? Do you like the mice that sneak around at night? Look at your Amma- do you think she works hard for you to not do well in school? Keep going.”

They’d stay up hours on end throughout the night, until Fahim managed to finally burn the times table in his mind.

Barbara’s book offered an alternative to this approach. The gritty, hard-willed approach to learning by any means necessary. Exacerbated by immigrant parents and an educational system that laid heavy emphasis on annual exams.

He decided to test Barbara’s study techniques with a practice mock exam his professor sent out last week. Fahim powered through the exam until a question began to stump him:

Find the limit of x approaches zero for the function x-squared multiplied by the cosine of one over x.

His thoughts quieted as his mind froze. He completely forgot how to approach this problem, given that it is impossible to simplify the cosine of one over x directly. His mind began to wander as he defaulted to doodling within the blank margins of the mock exam. He drew random shapes and scribbled various lines stretching and shooting in different directions. That is, until the lecture returned to him.

One day, his professor drew three curves on the board. Two bounding curves– one above, one below, and a wild oscillating function sandwiched in between. He turned to the class.

Alright, so here’s the idea. Sometimes a function is too unpredictable to figure out directly. It could be bouncing all over the place, like this one.

He tapped the middle curve, the one darting between the others like a trapped bird.

But if you can trap that function between two others– ones you do understand– and they’re both approaching the same value… then you got it. The middle one is being squeezed, and it has to go where the others are going.

He underlined the limit in the equation.

“It’s like you’re being pulled from two sides. But that pressure? It gives you clarity. It forces you to settle.”

He paused.

That’s the Squeeze Theorem. Pressure from both sides, bringing order to the chaos.

Fahim jolted up in his seat when he observed the connection— his scribbled drawings were chaotic but found order within the margins. The limit of the function for cosine of one over x in his problem was zero. It was intrinsic, logical, and poetic at the same time.

He circled the answer from the multiple of choices and finished the mock exam that night, feeling brisk and enamored seeing math the way Barbara had intended.

Over the next week, Fahim studied his notes diligently. He did not push himself for hours on end, but instead, approached mathematics like a form of art. An expressive, complex, yet logical language whose ethos was embedded in different facets of the universe.

On the day of his final exam, he sat in his seat as the professor walked around distributing the exam. He leaned over to Fahim:

“Hope you studied hard. Good luck!”

Fahim, with a distinctly optimistic attitude compared to a few weeks ago, replied upbeat:

“Yes Professor, I did. And I finished the book by Barbara Oakley, by the way. Thank you!”

The professor nodded in amusement as Fahim began taking his exam.

Find the limit of x approaches zero when the sine of x is over the cosine of x.

“Okay. This isn’t too bad. The sine of x over the cosine of x is just the tangent of x. As x approaches zero, the tangent of x also approaches zero, since both sine and cosine are continuous and well-behaved near zero. So the answer…is just zero.”

What is the limit as x approaches zero with the function one minus cosine of x divided by x-squared?

“This tripped me up before. I can’t just plug-in directly– it would just be zero over zero. I can use L’Hopital’s Rule for this, taking the derivative of the numerator and denominator separately gives me sine of x over two-x, which just becomes one-half. Easy.”

Find the derivative of f of x equals x-squared at x equals three using the definition of the derivative.

“Alright. This is just finding the rate of change at x equals three. And then using f of three plus h eventually gives us the limit of h approaches zero of six h plus h-squared results in six.”

He powered through the exam solving each question handily. However, one seemingly simple question stopped his momentum.

Find the limit of x approaches zero when the sine of x is divided by x.

His mind drew a blank:

“Sine of x over x as the limit of x approaches zero. This was one of the trigonometric identities. Professor drew them out for us, but everyone told me to just memorize them. What was it…”

Fahim defaulted to doodling within the blank margins of the exam. He drew trees, faces, and inanimate objects to keep his mind afloat. He thought to himself:

A mind for numbers…

He decided to step away from the margins and draw within the gridded lines of the graph paper the exam was printed on. Starting with a line for the next doodle, he imagined a pair of axes within the gridded cells. The line began at a convex between the cells and uniformly dipped below the imaginary axis line.

Barbara said to rely on visualization, he thought to himself.

He suddenly remembered the sine wave his professor drew in class, as it flattened near one.

That’s it! The limit of sine of x is equal to one because it approaches one as x approaches zero.

Fahim eagerly stood up and submitted his exam. He left the class and stepped out onto the campus lawn. He sat alone on a bench, relieved to see his only math course come to a close. He watched as students flocked away from the campus, cheery, and celebrating the end of final exams. Some tired, others energized.

He couldn’t help but recall a lesson from his professor earlier in the semester.

“Now this right here,” he tapped the chalkboard, “is the heart of it all. The Fundamental Theorem of Calculus.”

Fahim paid attention earnestly.

“It’s very profound, you see. It tells us that the rate of change of everything you’ve accumulated…is where you are right now.”

He paused.

“That’s not even math right there, that’s life.”

A few students looked up.

“Everything you’ve built up– every hour of studying, every book you’ve read, every step taken to campus— it adds up. That’s an integral of all your hard work. But if you want to know who you are right now— you take the derivative. The rate of change of where you are now.”

Fahim smiled as he rested his back against the bench.

His feet laid on a curb overseeing the grassy lawn on campus. A sprinkler head rotated in waves back and forth— stopping just before it reached a full rotation. The shadow of the flagpole shortened slowly as the sun came down- matching the derived rate as the sun finished its daily orbit around the earth. The ripple of a breeze on the grass— a slow, continuous change.