Number-Crunching

I find it incredibly disappointing that so many people in the general public seem to regard people who use mathematics in their profession as “number-crunchers”. Each time I hear someone say it, I die a little on the inside (even though I know they mean it in a good way). It’s as if the only notion of mathematics that these people have is that one does arithmetic. In my mind, it’s like saying that all a photographer does is take photographs or that a businessperson only makes calls all day for deals. It’s a narrow-minded view of any of those disciplines, and it gets a lot of it wrong.

First of all, our minds are brilliant things. The human mind has come up with descriptions of the universe that are much more accurate than any other story we’ve ever told about the universe. Therefore, why would we want to waste our time on performing a bunch of calculations using our minds or by inputting them into a calculator? The answer is that we don’t. We use computers for this because they are both better than us at it, and we can then use our minds for something else. Number-crunching may have once been what a lot of scientists do, but now we’ve got computers for the job.

(Of course, we still do have to statistically analyze data, but the point I’m trying to make is that a lot of the heavy lifting has now been shifted to computers.)

Second, this view of mathematics fails to see how much of an art form it is. Yes, I said it. Go out to any artist, and I’m sure they’ll soon start telling you about how a blank canvas is often paralyzing, and making art becomes much easier when there are constraints involved. Does that remind you of something? Oh yeah, that’s virtually the job of the engineer.

Moreover, mathematics at its purest is just that: logical implications following from certain constraints. Notice that I didn’t say anything about numbers or formulas. While these do have their place within mathematics, they are only there because of their utility.

So why do so many people seem to believe mathematics is just number-crunching? The answer is fairly obvious: elementary and secondary education. There are two things at play here: necessity, and algorithms.

Mathematics is not optional when you are in elementary or secondary school. You take it every year, without any say. In secondary school you are usually introduced to optional courses. These include things like fine arts, drama, dancing, fitness, wood and metal working, and so on. Notice that none of these “options” are mathematics. Why? Because you’re still forced to take the mathematics course, year after year. This is fine for those who enjoy mathematics (as I do), but it can be incredibly frustrating for those who don’t. Then, when students either enter the workforce or pursue more education, most of them won’t get to have any more mathematics education, only having learned the basics. Therefore, they will only see mathematics as remembering how to do very “set up” problems and knowing which formula to use.

This brings us to the other issue: the curriculum. Unfortunately, the curriculum isn’t exactly made to stimulate interest in the students with respect to mathematics. Instead, it’s mostly about knowing how to use different formulas and to recognize that this kind of question will lead to this kind of answer. It’s basically a lot of plug-and-play, which is why I’m not surprised that many people see mathematics as simply performing an algorithm or “number-crunching”.

The only solution I can see is to radically change the curriculum for elementary and secondary education. At the moment, I don’t think many students are ever excited to go to mathematics class, which is a shame because a lot of the ideas are wonderful and interesting. I know that mathematics has the content to interest young student, but the onus is on those who design curriculum to do so in a way that brings this

Simple Harmonic Motion

If you’ve ever taken a physics class on waves, the first type of mathematical wave you learn is the one due to what is called simple harmonic motion. The idea is pretty simple, so I’ll go through a rough derivation here.

First, we imagine what the situation looks like. It’s usually shown as a mass attached to a spring moving along an idealized (read: frictionless) surface, as shown below:

Idealized situation with only the force of the spring coming into play.

As we can see, there is only one force acting on it: the force of the spring, $\textbf{F}_s$. This force is proportional to the displacement $x$ of the mass attached to the spring and is known as Hooke’s Law. Therefore, if we write out the forces that act on this body using Newton’s second law, we get (along the x direction):

Once we’re at this point, we need to figure out what the solution to this second order differential equation is. I won’t go into full detail here, but you can use a bunch of different methods to solve for the function x we are looking for. You won’t need to solve the equation each time, since doing it once will give us the pattern we need in order to describe simple harmonic motion for all situations.

I’ll use the method of writing down our characteristic equation and solving.

We give the quantity $\sqrt{\frac{k}{m}}$ another variable name: $\omega$. This is called the angular frequency, and is measured in radians per second.

From here, our solution is given by:

For most applications that we use in physics, B ends up being zero. Additionally, $\phi$, which is just the phase constant and adjusts where the function begins, is often zero too. This means we are left with a rather simple equation that accurately describes how an idealized version of a mass moving on a spring would work.

But I don’t want to stop there. Instead, I want to show you another way of looking at how we describe simple harmonic motion. In particular, look at the form of our solution to the differential equation. It’s a combination of trigonometric functions, which we know can describe something else: a circle.

Simple Harmonic Motion through Uniform Circular Motion

At first glance, these two ideas might seem to have nothing to do with each other. Circular motion goes round and round while our simple harmonic motion situation only goes up and down or side to side. What can they possibly have in common?

(Before I dive into it, I just want to note that this is one of the coolest things I saw in my waves and optics class when I had to take it.)

First, we will build up our experiment before I show you the mathematics of it. Imagine you have a circular disc that is rotating at a constant rate on a table. Now, attach a small object somewhere along the circumference of the disk so that it spins with the disk. This means that the object on the disk is exhibiting uniform circular motion. Next, place a screen behind the rotating disk. Finally, we shine a light edge on to the disk so that its shadow is displayed on the screen behind it.

What do you think you will see? What will the object that is rotating on the disk look like on the screen?

The answer isn’t that the object will move from one side to the other at a constant rate. Instead, it will move faster in the centre and slower on the ends before it turns around. Effectively, it will look just like simple harmonic motion!

Because I know you are just craving to see what this looks like outside of your mind’s eye, here you go:

(A brief note: I did not make this myself. I found it on the Desmos site, and I unfortunately could not find the author. If you know who made it or if it is yours, please reach out to me and I’ll add your name to it. Also, press the “play” button on the ‘a’ in the calculator for it to start.)

The red function you see moving from left to right is the function given by simple harmonic motion that we saw above. It’s moving simply to show how the function lines up with the circular motion.

In the centre, we can see a point on the circle moving round and round through uniform circular motion. At the same time, there are two vertical lines in yellow acting as “screens” on each side. Notice the horizontal line connecting the point on the circle with the point on the screen. This is exactly what would happen if we shone light onto an object and have projected it onto the screen.

That’s a very nice way of viewing it physically, but what about mathematically? What is that screen? In essence, it’s the projection of the circle on the vertical axis. Imagine squishing the circle such that it creates a vertical line. That’s the role of the screen. Put another way, if we define the circle parametrically as $(cos\theta, sin\theta)$, we get the projection on the vertical axis by only looking at what happens to the y-component of the parametric curve and setting the x-component to zero.

And now, we have two ways of looking at simple harmonic motion. We can either view it as the physical motion, or as a projection of uniform circular motion onto a straight line (and since we’re dealing with a circle, you could project the circle onto any line you wanted).


Hopefully, this gives you a better idea of the basics of simple harmonic motion and how it relates to uniform circular motion. In the future, I’m going to go into detail on what happens when we add more forces to the system, since physical systems are never (or rarely ever) idealized.

Struggle

One of the unfortunate side effects of having a curriculum and set schedule in mathematics is that one never gets to think about concepts for too long. Instead, the goal of a class is to simply throw a bunch of ideas to students and let them “ponder” the ideas on their own time. This is seriously backwards, and it’s at the heart of what is wrong with a lot of mathematics education today.

Think back to your time as a student in mathematics. When was the last time you were presented with a problem, worked on it, and then had to stop and mull over the idea for a few days? If you’re education has been anything like my own, the answer is nearly never. That’s simply not how school works. You never hear the teacher give you a problem that you can think about and come to a solution on. What happens is that a teacher will explain the problem, and then jump right into the solution, with no explanation in between. And that’s if you’re lucky. Sometimes, an answer will be given to you without there even being a question!

I’m willing to bet that if you’ve ever been asked a mathematics puzzle, you usually think about it for a few seconds, decide that you may as well just listen to the explanation, and jump right ahead. It’s what I do, because I know I will understand the explanation, and I don’t want to pause the video or stop reading because thinking about the problem will be difficult.

However, I’ve realized that this is a way to make me think that I know a lot about mathematics, but it’s a sort of pseudo-knowledge. After all, why do we engage in mathematics if not to try and solve problems and puzzles? Once the answer is given, there’s no more fun to it anymore. It’s just an algorithm or step-by-step process to be applied.

That is why I’m in favour of learning how to struggle in mathematics more. And I don’t mean the kind of struggling that occurs because you don’t know what formula to use. I see this so often when I tutor students, and I’m starting to harbour the belief that it’s because schools are just presenting them with a vast amount of information without any use for it. They don’t know what to do in problems because they don’t even understand what the problems are about. The reason: they have no link in their minds between the material they learn in class and the problems presented to them. And since it doesn’t match up, they are at a loss for what to do.

In place of just giving them a bunch of theory on different concepts that they will have to remember how to apply an algorithm, we should have them struggling on the problems that led to the various equations and concepts that are learned. By doing this, it gives the student an idea what it is they are learning about. These aren’t just abstract symbols on a piece of paper. They are ideas about objects, and not giving students this information and not allowing them to struggle on these sorts of problems before giving them the answer as class notes is doing more harm than is worth the productivity boost.

Starting Simple

If you want to get a concept across, the best thing to do is to start simple. Learning can be difficult and many parts initially might not make sense, so it’s important to make the “jump” to that knowledge in a way that we can follow. If not, it will simply be too difficult to make that conceptual leap.

This is incredibly important when being introduced to a formula of some kind. The thing about formulas is that they can be used with little understanding of what is actually going on. It’s definitely possible to do well in a physics or chemistry or even mathematics class by only knowing formulas without actually knowing what is happening within the formulas.

For example, I was recently learning about the concept of curl and divergence in multivariable calculus. Without actually knowing the idea behind curl and divergence, I could simply used the formulas and respectively without any problems. Since I know how to compute partial derivatives, finding the value for curl and divergence is relatively trivial.

However, the deeper understanding comes from when a formula is explained. I hate having formulas simply presented to me. What do they mean? How did certain terms come into play for calculating this certain thing? These are questions that interest me, because they give a reason for a formula. Without a reason, I’m just blindly computing.

This is why I was pleasantly surprised to view these videos from Khan Academy, because the author (Grant Sanderson) always gave intuitions before presenting the formulas. This way, there wasn’t befuddlement as to why a certain formula was supposed to represent curl or divergence. Instead, it made sense because of the intuitions and explanations beforehand.

How did he do this? By constructing the simplest case possible, and building the formula up from there. Because he approached the formula like this, it made sense as to why a certain operation was being done. Consequently, the explanation was easy to follow, and the formula wasn’t surprising once it was presented.


Let’s face it: mathematics can be difficult and abstract at times, making it difficult to make sense of formulas. Often, it can seem like formulas pop out from nowhere, which make them look mysterious and exotic. However, by exploring the simplest cases possible for the situation and slowly working up the intuitions, it makes it much easier to understand the origin of a formula.

This idea can just as well be extended for ideas in physics or other sciences, and I firmly believe it should be. I’ve been in classes in which time didn’t allow for us to derive various formulas, so they were simply presented to us as correct. While efficient, this made it difficult to understand how each component of a formula worked to calculate the thing we were searching for.

Therefore, whenever you get a chance to understand how a formula came about, take it. Do not simply absorb the final answer. In the long run, it will be more productive to figure out how a formula works instead of just knowing it.