Elegance

Ask any scientist or mathematician, and this is the quality that they would love their solution to have. They want the result to be elegant, simple, and intuitive.

To give you an example, I remember doing a problem in my calculus class which involved using a bunch of trigonometric functions. Naturally, the integral kind of exploded as I worked on it, and the result was super-complicated. However, after applying a bunch of different identities and swapping sines and cosines, the answer came back as simply tangent of theta.

When I got this result, I immediately knew I was right. The result was just too perfect after all that work for it not to be true (of course, this is a bias). Additionally, the answer made me feel good. It was a nice answer to look at, particularly after all the work required to get there.

This underscores our tendency in science and mathematics to revere simple answer. Consequently, we tend to “dress up” our equations and concepts in order to make them much more compact than they are in reality. I have two examples to illustrate the point.

First, in physics (particularly, wave motion), there’s the notion of forced oscillations for a spring or other kind of object feeling some sort of oscillation. The illusion of “dressing up” the equation was so strong in this sense that I felt moved enough to create a small comic of it:

ForcedOscillation1ForcedOscillation2

Even as my teacher talked about this equation, she looked sheepish. As soon as we saw the whole equation, we could see why (and this was only the steady state solution).

The second example comes from the recent World Science Festival, where I watched the panel on gravitational waves. During this panel at around the thirty minute mark, the moderator (Brian Greene) walked through some of the equations of general relativity, and showed just how complicated these equations can be. Despite looking relatively (sorry!) simple, the equations are just being dressed up to cover their complexities. There’s nothing necessarily wrong about this, but it does illustrate how equations in science and mathematics can be a bit more challenging than they appear. This is all done in the name of elegance. If we can make an equation more compact, we will do it.

Often, we seek the elegant answer, wanting to have something simple after working through a bunch of mathematics. This leads us to covering up the complexities of many equations, which make them difficult to understand while looking in from the outside.

Perhaps we should embrace a little more complexity?

Are You Willing to be Mediocre?

At first glance, the answer would be, “no”. However, giving that answer would be missing the point.

We all want to be experts. It doesn’t even matter what the expertise is in. If you ask most people, they’ll gladly accept being an expert at nearly anything. This is because achieving a state of expertise means you are wise and went to the trials of becoming an expert.

What is missed, however, is the fact that expertise arises from mediocrity. To be an expert means to have been an amateur. It’s virtually impossible to leapfrog from not knowing anything about a subject to being an expert. This simply does not happen.

Instead, expertise occurs as a result of a lot of practice. And, more importantly, being wrong.

We don’t like to be wrong. Often, being wrong feels like a personal attack on one’s character, as if we don’t feel as intellectual as the rest of our peers. We don’t like being wrong because we do not enjoy displaying a weakness to those in our social circles.

However, the reality is that expertise requires you to be mediocre. There’s no shortcut. In a way, being an expert means you’ve once been an amateur and have learned from all your mistakes. We don’t like to think of it like this, though, because experts are thought to be people who don’t get anything wrong.

What we need to realize, then, is that those who become experts have become experts because they’ve accepted the journey to get there. Mainly, they’ve accepted that being mediocre is just a phase in a process. Being only a phase, they realize that it will come to an end, and eventually lead to being an expert.

It’s a comforting myth to spread, but it’s wrong. Experts become experts as a result of being mediocre, not in spite of. Therefore, if you want to be an expert at what you do, embrace the phase of mediocrity. Through hard work, you’ll find that it is only a phase and does get better.

The path to expertise always includes mediocrity.

Slowly Chipping Away

Trying to solve the problem all at once is complicated, and usually too messy. When trying to solve something in one step, it’s easy to make mistakes or otherwise fail. It’s the nature of trying to figure out faster ways to perform tasks.

It’s the same way for your goals. On the one hand, you can attempt to take on too much, too soon. You can fill your schedule up with your goal to such an extent that everything becomes overwhelming. At that point, odds are you will stumble backwards from the pressure of it all.

If you want to look at the masters for inspiration, you will see this strategy all the time. Rarely will a master attack a problem head on, because they know it won’t work. Instead, they develop a methodical strategy to slowly get part of the problem solved, one step at a time. This is because it is much easier to handle a small part of the problem than trying to deal with the whole thing at one time.

When you have a choice for how to tackle a goal or situation, the smart choice will usually be to slowly chip away at it, instead of trying to get it done in one session. This way, you can create momentum for yourself, as well as stopping yourself from being paralyzed out of overwhelm. And, chances are, a big, ambitious goal will be tough to handle in one go.

Chipping away at your goal may be slower, but it ensures that you don’t get stopped by being too big.

These Rules Are Stupid

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As I was tutoring one of my students about algebra, I could see that he was struggling with making sense of the rules of how to manipulate equations in order to solve for a given variable. The equation involved fractions and like terms, and looked something like this:

He was having trouble with doing his steps in order. For example, he was dividing to solve for “x” before he placed all the other terms on the other side.

To be honest, it was a bit of a challenge for me to address, because I had so much experience working with basic equations such as these that the process was basically automatic. I knew the steps I took to get there, though, so that’s what I told him.

“You know how put the like terms together on one side, so take all the “x” terms and put them on one side of the equation, and then divide by that factor. You can’t divide before putting everything on the other side. And when terms switch sides, you have to switch signs, unless you’re trying to solve for “x” and you have a factor in front of it, in which case you have to divide by that factor.”

Okay, I didn’t really say all of that. But I did try and get the gist of the message to him.

Once I corrected the order of the steps he was taking, he muttered to me, “These rules are stupid.”

I’ve thought about this phrase for a while now, because it indicates a certain view of mathematics. Mainly, that it’s a bunch of rules that need to be followed. However, I strongly oppose this view. Instead, I think it would do us well to remember what a lot of mathematics demonstrates: relationships. Furthermore, it shows how these relationships are necessary if the original assumptions are true.

Let’s take something absurdly simple. Imagine you have two points in a flat space of $R^2$ (a sheet of paper, basically). Now, if I asked you to find a straight line that connected these two points, you would quickly find that there is only one way for you to do this. It’s inescapable. This is what is called an axiom (in this case specifically, an axiom for Euclidean geometry).

From this foundation (with other axioms), we then can figure out a whole bunch of things about lines, shapes, and the angles between them. We aren’t exactly inventing rules. A better way to think of it is that we are following the logical conclusions of things we’ve already accepted as true.

If we agree that this and this are true, then by extension, that must be true.

This kind of thinking can be applied to equations that need solving. For this situation, the idea is that an equation represents a sort of balance. Therefore, if we introduce something on one side of the equation, the only way we could conceivably keep this state of balance is to introduce the exact same something on the other side of the equation.

From this realization, we can then figure out how to solve the original equation shown up above. To do so, we’ll bring all of the “x” terms on the left side, and the rest of the elements on the right.

From here, we can put the terms together using common denominators, which yields:

Lastly, to solve for “x” we need to bring the factor on the “x” term to the denominator of the right side. In terms of “balancing” the equation, we need to divide by $-\frac{7}{5}$ on both sides of the equation in order to solve for “x”. Doing so yields:

Now, finding this wasn’t done by applying rules, per se. Instead, it was about following the logical end of our equation. Given our initial equation, we could only modify it in identical ways on both sides, eventually finding our answer.


In secondary school mathematics, the emphasis is on learning theorems and formulas, which are often regarded as “rules”. This is unfortunate, because it then makes the student worry about remembering how to use every single rule, leading to these proclamations of the rules in mathematics being stupid.

I believe a way to combat this is to get students to see that mathematics isn’t governed by rules, but that these are necessary conclusions that we have to make if we accept more basic axioms of mathematics. They aren’t negotiable, just as it makes no sense to say that breathing is a “rule” for humans. It’s not that breathing is a rule, it’s that there’s simply no other way to power our cells for the long term without some kind of aerobic capacity.

My wish is that this view of mathematics is pushed forward. Personally, I don’t remember ever hearing about axioms in secondary school, and these are supposed to be the bedrock of how we formulate most mathematical ideas (perhaps not inductive proofs, but those aren’t usually covered in secondary school). Instead of thinking about how mathematics just has these rules which work, it’s much more gratifying to think about them as a collection of necessary conclusions given certain information.

If we did this, perhaps I wouldn’t hear my students I tutor bemoaning how the rules of mathematics are stupid. They would understand that the theorem has to be true, just as surely as any triangle will have inside angles adding up to 180 degrees.

Let’s avoid dumping rules and formulas onto students, and start showing them how and why these exist.