### Can You Teach What You’ve Learnt?

Teachers are usually regarded as people who are experts at their subject. They may not be literal experts (such as those who teach scientific courses), but they have usually enough experience to make them experts. This expertise is usually forged through many years of teaching and working with students.

However, I’ve found that it has been extremely helpful for me to revisit content through the lens of a tutor. By switching roles from the person who is learning to the one who is explaining the concept, I’ve found that it helps reinforce the ideas in my own mind. When I can teach a concept to another student, it shows that I know what I’m talking about.

For me, the main reason I find this helpful is that teaching another student can be tricky. Often, the way you see something in your mind is not at all how another person sees it. Therefore, when teaching you have the challenge of not only transmitting an idea to the student, but of translating it as well.

This is particularly pertinent for the students I help in secondary school. Since that period of my life is several years old, the way in which I attack problems or situations demanding the skill set from that time has cemented in my mind. However, I need to sometimes put that aside when working with other students and focus on how they see the situation.

It’s a challenge that I think is worth thinking about. When learning a new concept, ask yourself: “Am I comfortable enough with this concept that I could actually explain it to someone else?” Trying to answer this question in the affirmative is a good way of assuring yourself that you’ve studied well.

It’s easy to have enough knowledge that you “pass” through a test without being noticed for your lack of knowledge. However, it’s much more difficult to do when you need to explain the entire concept to someone who has no experience. There, you can’t fall back on the chance that you won’t be asked a question on the one thing you aren’t sure about.

You can pass through a test without being discovered, but your goal should be to be capable of explaining a concept.

### Derivations and Feeding

When I do nearly any assignment for homework, I’ll make a rough copy of my work before writing the copy that I will hand in. I do it for a practical reason: while it may take extra time to write my work twice, the truth is that I often take a lot of detours on my first try tackling a problem. I go down dead-ends, make little mistakes here and there that need to be corrected, and generally do a lot of messy work. Once I get the correct answer, I can tidy all my work up in order to make my final copy as concise as possible.

This work well for handing in assignments, but unfortunately this strategy is often adopted too much while teaching students new concepts. Instead of giving students time to think about how to prove or derive a statement, teachers often give the instructions to the student at face value. Additionally, there’s no “rough” work given, making students sometimes wonder why a certain strategy is being used. I’ve heard many times the phrase, “Don’t worry, you’ll understand what I’m doing in just a second.” It’s a nice thought, but I think it creates an environment where students don’t have to think about what they need to do themselves. Instead, they just need to follow along as the teacher feeds them the steps and the answers. This might be easier, but I definitely don’t think it’s as good of a strategy in the long run as getting students to struggle and go down those wrong paths.

Along the same line, this tendency to always show the “final” work sends the message to students that this is the only way they should have seen the problem, and having any other ideas wasn’t going to work. However, that couldn’t be further from the truth! We need to send the message that rough work is essential in science, because no one is likely to get the answer right on the first try. You need to choose a direction and make some progress before you can decide if you’re going to reach your goal.

If we want students to be more engaged in the act of learning and taking notes while looking at a derivation, we can’t spoon-feed them the method to use. Sure, it will give them the way to do it in the future, but it’s not a good way to get them to learn the concept. What we need to be doing is encouraging them to try ideas, see if they work, and refine them. That’s the basis of intuition. You get it because you’ve seen many problems like this and you have a feel for what’s going to happen.

### Flashy

When I work with younger students in subjects like mathematics or physics, it doesn’t take much to impress them with my ability to quickly see through a problem and calculate things that would take them minutes. Just like any other student at my level, we often skip the use of calculators because it’s easier to just focus on the work we are doing and do the arithmetic in our head. The most prominent example of this, however, is in algebra.

I’m definitely no expert on the subject, but from my experience with other students, the thing that seems to confound them over and over is algebraic manipulation. Consequently, it takes longer for them to learn any other concept after, because they’re always using the algebra that was already tricky for them.

For example, I’ve worked with some students on the process of factoring expressions. In secondary school, this is usually done with trinomials such as $x^2+5x+6$. The goal is to get from that form to the factored form of $(x + 2)(x + 3)$.

To do this, there is a whole set of instructions that students copy down onto their memory aids for the test. When I worked with these students, the procedure is what they followed.

As they were doing that though, I was solving it in my head. We then compared answers, and I’m fairly sure it was surprising to them how fast I had gotten to the answer, as if I was able to pull it out from the page itself. They, on the other hand, were carefully going through each step of the procedure before getting the correct answer.

Both ways worked, but it seemed as if I had a faster way. I showed them how I thought of the procedure in my head, and I could tell that they probably weren’t confident in their own ability to do that in their heads. To them, they were only ever going to work it out on paper.

What I’m trying to illustrate here is the massive difference only a few years of practice can make. I wasn’t separated from them by that many years, and what was a whole new problem for them to solve was something that I could typically do in about ten seconds or less. It isn’t because I’m brilliant at arithmetic that I can do this. Rather, it’s because of all the work I had done for years to get the hang of it. I can do it all in my head now, but I remember doing countless examples when I was younger, trying to get the hang of it.

In most areas of physics and mathematics that you enter, talent isn’t a prerequisite. Instead, being willing to commit to hard work for a long time is what’s needed. Do that, and you’ll be able to eventually appear “flashy” in front of those who can’t do something as effortlessly as you can.

But be aware: this is true for virtually any domain in life. Athletes don’t pull off amazing plays out of only sheer talent. They have an enormous backlog of hours that were dedicated to improving in their sport, and which culminate to help the player make the amazing play. You will find few examples of people simply having so much talent that they can do everything with little effort. More common is that you will see the hard work that was done by just pulling back the curtain a little on that person and looking at their history.

### Integrals, Integrands, and Understanding the Notation

Even though mathematics is one of the subjects I enjoy studying, it’s not always easy. Nor does everything always make sense to me. One of these things was the notation for an integral.

If you’re reading this, you probably have an idea of what an integral looks like. The simplest one looks like this:

You recognize the $f(x)$ part, but the other two parts aren’t components you’ve seen before. When I first learned about integrals, I don’t think I really internalized the notions of the two other symbols, and it was to my detriment. In mathematics, if things look funny, it’s probably because you don’t fully understand them. The reason is that ideas are made to be as compact and helpful as possible in most mathematical concepts that you will see in your education. Said differently: complexity isn’t introduced for complexity’s sake.

The symbol on the left ($\int$) is just the sign of integration. Much like we write $\frac{d}{dx}$ or $f’(x)$ to denote derivatives, the sign of integration is there to tell you that an integration is being performed. However, the shape of the symbol isn’t arbitrary. The symbol is a sort of horizontally narrowed and vertically stretched out “S”, and it stands for the word “sum”. After all, integrals come from the idea of taking the sum of a bunch of tiny pieces of a function, and so the sign of integration is supposed to be a nod to that.

However, it’s the other symbol which has much more meaning. The $dx$ part of the integral is what signals to the reader what variable will be integrated. In my example, $x$ is the variable being integrated.

When I first learned about integrals, this was where I stopped digging deeper. I more or less just went ahead and calculated, only using the $dx$ as a small reminder that I was supposed to be interacting with respect to $x$. This approached worked for me, but it definitely wasn’t the best I could have done. Instead, here’s the much better way of thinking about the $dx$ part.

First of all, $dx$ is called the infinitesimal or differential of $x$. These are just long words to say that the portion of $x$ we are considering is really tiny.

You also probably know that a way to interpret a definite integral (one that has bounds of integration) is to think of it as the net area “under” a curve between two points. That’s the “physical” way to look at it, and it has a lot of meaning for our notation of an integral.

What kind of units does an area have? It has units of a length squared, such as $m^2$. Therefore, if our integral is just a bunch of addition of tiny “chunks” of area, each chunk needs to have units of area.

If we look back at our integral:

We can imagine the term $f(x)$ being considered as one of the lengths in question. So what’s the other one?

It’s the small portion $dx$.

The reason this was hard for me to grasp at first was that we were discussing $dx$ as though it was so small that it was basically not there. As such, I didn’t think of it as a “real” quantity. It was just a symbol that I put inside my integral, and it had no actual meaning. However, I couldn’t have been further from the truth. By taking $dx$ to have a geometrical meaning, the idea of an integral suddenly felt so much more grounded to me. The $dx$ wasn’t some ornament. It was an important part of the integral.

And that’s what I want you to get from this. The differential $dx$ is very important in concepts later on, so it’s a good idea to get a firm understanding of what that differential actually means.