The Epiphanies of Math: How to get better at math
By: Sparsh Joshi
Tags: Mathematics, Experience, Math, Imagination, Storytelling

I was 15 in 2015, and all I wanted to be was a physicist. I had been on a learning spree of the keywords for years — from the standard model of particle physics to learning about the ergospheres of black holes.

I was reading some simple papers and their introduction — blogs, videos, podcasts. I was learning about things like quark-gluon plasma, quantum entanglement, and whatnot. All that seemed weird and wonderful, I had my eye on it.

It was my dream to solve the Hawking Information Paradox and AMPS paradox, both a blend of quantum mechanics and black holes. I remember reading something new in this area and immediately coming up with ideas that I thought were “revolutionary” or “the missing piece” to these problems. I would write doc files with catchy names like “Solving AMPS Paradox using Tensor Networks” or “Creating Chameleon-Electron pairs to Simulate Hawking Entanglement Pairs” with nothing but gibberish inside.

I would be super-excited to mail my hollow solutions to world-acclaimed physicists like Sean Carroll, Brian Greene, and Brian Schmidt without even consulting my local physics teachers. You might think they wouldn’t even consider having a look. Yet, it is the humbling greatness of them, that they would often respond to my emails! I suspect it was probably from the hopes of getting another teen in this endless endeavor of decoding the mysteries of the universe. But all their emails had a strange recurring feature, which almost felt like a grand conspiracy physicists made to not get me published. They wanted me to study more mathematics. To understand the basic mathematical tools that can help me make my argument more concise and meaningful.

Alas, if I had to pick a subject I was the worst at, it was mathematics.

Peer Pressure: A great math tool?

The 10th grade in India has an overrated role in a student’s life. S/He gets to decide which of the three streams they would want to opt for in the future and shape their careers.

The state education board conducts the exams in 6–7 subjects like science, social studies, and mathematics across the state. These tests are evaluated and scored in percentile — the higher, the better. Of course, everyone wanted to be in the top percentile. No doubt it is a huge industry for coaches and teachers to earn. And like most Indians, I went to a coaching class for studying all subjects. I remember that there were tests every Saturday by our Mathematics and Science teacher. I used to score impressively in science — almost always at the top of the class, with an 85 to 90% score, every time, impeccably.

With Mathematics, however, I could barely touch even the halfway mark.

Not that I was bad at Math that year — I had always been terrible with numbers, let alone variables and operations. I struggled with linear equations and geometry in my earlier grades. I could not understand why would we want to find x, and how do similar triangles even work. I could barely pass my exams and there was almost no scope of me doing any better in the finals of my 10th grade too. I struggled with math all throughout my “important” year. Even the last mathematics exam in our coaching, a week before my actual exam, gave me a dreadful 45 out of 100.

My parents, both great at math, were ready for some heart-breaking news, and I felt very depressed as the exam date approached. On the ultimate day before the math exam, I was almost ready to break down and beg for help — I could see that I was going to be left behind in this race because of one awful subject. It was like knowing the end before it began. Yet, I mustered the courage and kept solving my textbooks, worksheets and examples. I was making all the efforts to squeeze the water out of the stone and to create something that was good enough.

Then, something happened, something that completely changed the way I looked at Mathematics.

I remember that evening a day before the final exam. I was discussing some questions with my friends from the textbook, and I saw in it what could only be described as the absurdity of the world. A sudden moment made it clear to me why linear lines are of the form they are or why a quadratic equation forms a parabola. I could now understand why tangent segments to a circle from an external point are equal. I could understand everything the book told me. Overnight, I went from being depressed about the score to being confident. I had, a mathematical epiphany.

That day, I scored a staggering 92% — more than all of my scores in all other exams, even science!

That night marked my departure from the “un-learnablity” of mathematics. Sure, I have struggled with Mathematics in these 6 years, but it has always been about concepts and not about logic. I have never looked at mathematics as being “boring and hard”. I have seen it as a way nature manifests itself; and in this process, I have learned to appreciate the beauty of mathematics.

Math: Composition v/s Clarity

Photo by Tim Gouw on Unsplash

Somehow, I don't get math.

I have come across this statement multiple times, both from others and myself. And I am certain that it is not something that people say out of procrastination — I’ve seen people struggling with math even after putting in their best.

Of course, as a reader of this publication, you, my dear reader, might be good at mathematics. Some of you might have an innate intuition of mathematics, some learned it on the way. But all of us have seen people who struggle with it throughout their lives. People, who just don’t understand what is happening, even after much deliberation. Why is that?

After much profound thinking, I have realised that people learn mathematics on the basis of how they view mathematics: either by clarity or by composition.

To understand what I am trying to say, think of this equation:

The General solution to a quadratic equation

Most people who have graduated high school have heard of this formula. If you plug in the values correctly, you get the roots of a quadratic equation. All simple, nothing special.

Here is something interesting: Can you derive this formula? Given the general form of the quadratic equation, can you come up with a solution that lands to this result? In the presence of mathematical tools available at your disposal, can you create this formula on the fly?

This is where the Clarity v/s Composition view of mathematics comes into the picture.

The people with the vision of Composing mathematics ask nothing more than how. Their main concern is the composition, or the facade of the equation. It is what mathematics is for them — a useful tool that generates results. For them, it is enough to know how to generate results, and that’s it. You need not know how something works for you to use it. For them, just like a Toaster, mathematics is a tool that operates and gives an output. That’s it. Period. It is an instrumentalist approach to the world — the woven reality is shaped by the useability of the tool.

It is a very useful view, I must admit. It gets the job done without hassle in the intricacies. But be warned, for this vision rests on a very thin line. As soon as more and more formulas and theories come in, this view vanishes. Composers would get very confused, be in a frenzy and fail eventually.

The clarity-seekers not only focus on the how, but also think about the why. They seek to understand why something is the way it is, and exactly what something actually does. This means reading theory, deriving equations and solving examples — but more importantly, developing intuitions that are the building blocks of mathematics. It is a lot of work and a lot of brainstorming as you go further and further. It is a tough and absolutely tiring practice and philosophy.

However, this vision never fails you. You may find something hard, but nothing ever becomes “unsolvable”. This idea ensures that your basics and cores are solidified and you can develop better intuitions as a student.

You can assess easily which category you belong to. I, for one, transitioned from the former to the latter in a very short period of time. But we are yet to discuss the most important aspect of this duality: what makes a composer and a clarity seeker, what they are? I mean, no one filled a form to be either of them, yet here we are. What innate difference does make one a composer or a clarity-seeker? Well, it has nothing to do with mathematics, and more to do with your culture.

Becoming Better: How getting better at Math has nothing to do with Math

TIMMS, a series of international assessment exams, is taken all over the world to know the aptitude of people in the universe of mathematics and science. It's a tedious, hard and long exam — 120 questions of hard math problems for 4th and 8th graders of the world just to know more about the way their countries produce math and science-literate citizens. It is a tiring process, especially for kids so small. Naturally, a student would want to leave some blanks. And they do — some even leave as many as 15–20 questions. Yes, some leave one-sixth of the paper empty. Either because they don't know how to solve the question or are too tired of not knowing what to do.

But some don’t leave their sheets empty. And that helps us visualise the way mathematical ability is seen in the world. As it turns out, the average number of items answered on that questionnaire varies from country to country — but in a very specific order. And that order, when measured against the countries rankings on TIMMS, correlates perfectly. Meaning, students of a country who leave the test problems are also from the countries that are bad at math! You don't even need to check the papers anymore! You can check how many holes are unfilled, and you will get the results.

So, what is happening here? How can empty sheets be exactly related to how good a country is at math? Well, because getting good at math has nothing to do with math, but has a lot to do with perseverance. The students, who were on average ready to patiently sit and try to get the solution were much more likely to be from the country that had a great number of math prodigies. Just the ability to not give up, and see what is beyond the question in order to unravel the answer, makes the difference between a good and bad math student.

In Outliers, Malcolm Gladwell talks about Renee, a woman with no mathematical background who persisted in a test for over twenty-two minutes to evaluate the slope of the y-axis on a graph from scratch.

Would you ever try so hard to sit on a problem?

He writes:

We sometimes think of being good at mathematics as an innate ability. You either have “it” or you don’t. But…, it’s not so much ability as attitude. You master mathematics if you are willing to try...Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds. Put a bunch of Renees in a classroom, and give them the space and time to explore mathematics for themselves, and you could go a long way. Or imagine a country where Renee’s doggedness is not the exception, but a cultural trait, embedded as deeply as the culture of honor in the Cumberland Plateau. Now that would be a country good at math.

This perfectly captures the way mathematics works. It's not about your background or your vision, it is about your persistence, patience and hard work.