Relating

One of the most difficult things for me to do when I am learning is to make the conceptual leap from one idea to the next. Often, I’m not confused about how to do a problem. Rather, I’m stuck on an idea that preceded it which I am not fully on board with.

This is the reality that I’m frequently confronted with when I tutor students. Most of the time, the actual application of an equation to a problem isn’t an issue. Instead, the trouble comes from some conceptual piece of the puzzle that simply isn’t clicking into place. Usually, this comes from the teacher not fully explaining a concept, which then makes the student confused about how to jump from one idea to another. It’s also often a simple thing that manifests itself in the form of the student not being able to do any of the problem.

An example of a gap left in explanations to me is how students have to learn about various functions and what they look like. The list includes the following functions:

– Constant: $y = c$

– Linear or First-Degree: $y = mx + b$

– Quadratic or Second-Degree: $y = ax^2+bx+c$

– Inverse: $y = \frac{k}{x}$

– Exponential: $y = ab^x$

Ignoring the fact that the exponential function is thrown in there completely randomly, the other ones are given a bunch of different names. However, what is missing from this list is how the functions relate to one another. If we remover the random exponential function that was thrown in, a much more intuitive pattern can emerge from this list: each function is a successive power of $x$.

When I first showed this to the student I was tutoring, they didn’t even know what I was talking about when I used the idea of a negative exponent. But after walking through some examples with them, it was much clearer.

Here’s my modified list:

  • Inverse function (negative one degree): $y=kx^{-1}$
  • Constant function (zero degree): $y=cx^0=c$ (Ignoring the case of $x=0$)
  • Linear Function (first degree): $y=mx^1+b$
  • Quadratic function (second degree): $y=ax^2+bx+c$

If you look carefully, there are two important things I did here. First, I listed them in a logical order. They are ordered from the smallest power of $x$ to the largest power. This makes the functions easy to recognize. Count the powers and you’re good to go.

Second, I explicitly showed extra work in this list. I showed the powers of one and zero even though the final answer does not necessarily show them. The reason is simple: the students are learning and this is an easy foothold to take when learning about functions. When you’re suddenly bombarded with all these possibilities for functions, it can be difficult to keep them all straight in your mind. This way, you can easily see how the powers of $x$ are in each equation.

This is a simple point, but it’s something that is so important to get right in the beginning of teaching a concept. The longer this goes unnoticed, the more trouble the student will have since they won’t ever feel as if they have a comfortable foothold into the concept. Think about walking on ice. You’d feel a lot safer if you were wearing crampons or skates as opposed to summer shoes. In the same way, our goal is to give students these conceptual footholds, and the best way is usually through relating with something they already know.

Cementing

A common thread I see between many young students who don’t seem to “get” mathematics is that they aren’t told to look at the way they are taught mathematics as only that: a way. Unfortunately, the impression that is made on them is that mathematics is a strict set of rules that cannot be broken and must be followed every moment.

Imagine instead I was learning the English language, and that I wanted to express the emotion of anger. When I ask this, the teacher tells me, “You can say that you’re ‘angry’.”

Here’s my question: wouldn’t it be absurd if I went along my whole life using that one word to express how I was feeling? Instead of using the variety of synonyms that the English language provides in order to bring more nuance to my state of mind, I’d be repeating the same word over and over again, because I know that it will generally express what I want. In essence, it’s a “safe” option. It then gets cemented in my brain through repeated use, and I am stuck in this cycle forever.

I hope you agree that this wouldn’t be a very good thing to do to a student. Making them only see one way of looking at the world and not using any other words to describe anger robs them of expressing themselves, since it can never fully capture the particular shade of anger in that moment. However, we are doing the exact same thing to our students in mathematics!

Consider one of the earliest functions that are exposed to students: $y = f(x)$. This is taught to be a function of the independent variable $x$. After this is taught, example upon example is given using this form. However, students are then taught that functions don’t have to look like $f(x)$, but can be in the form of $f(y)$ as well, with an independent variable $y$. This isn’t necessarily a problem, but from the countless examples and repetition in the previous form, using this other method becomes confusing. Then, to make matters worse, they are taught that one can often switch between these forms by manipulating the variables.

As one can imagine, this confuses many people. How can $y$ turn from a dependent variable to an independent variable by just arranging the equation in a different manner? When do we have to use $f(x)$, and when should we use $f(y)$? These questions may be obvious to us, the tutors and teachers, but it isn’t so obvious to the students. I’ve seen it myself.

In my eyes, the solution is to make mathematics more dynamic. Instead of introducing the variables $x$ and $y$ as the de facto standard for the rest of their mathematical lives, show them that the variables we use are just their for us, the people doing the mathematics. It doesn’t really matter if something is called $x$, $y$, or $lambda$, they’re just symbols. From there, I believe students will have more confidence in using variables that aren’t your traditional $x$ and $y$. The way in which I try to help this with the students I tutor is to use the terms “horizontal” and “vertical” axis instead of the $x$ or $y$ axis. My hope is that this will get them out of their standard way of mathematics and make them realize that a lot of the things you do in mathematics is more out of convention than of need.

Obviously, the end goal is to get students using mainly just $x$ and $y$ for a lot of mathematics. However, the point is that these are just the traditional ways of doing things. There’s no requirement that the y-axis be the vertical axis. As such, I’d much rather change the names of variables and get them to think in ways that aren’t the usual, just to keep them aware that all of mathematics does not revolve around $x$ and $y$.

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.