Where Are Your Weaknesses?
During one of my calculus classes in university, we were running behind in terms of class content. Within the last few days of class, the professor announced that the topic of Taylor and McLaurin series and expansions weren’t going to be part of the final exam, but he would hold an optional “extra” class for those who were interested.
I was interested, but I also live far away from the university, which meant I didn’t want to drive a total of eighty minutes for fifty-five minutes of class. So I didn’t go. I figured it wouldn’t be a big deal, since it was just a small topic in my calculus class.
Unfortunately, I was dead-wrong. As any physics student knows, being able to write a function as a power series is a very useful technique to know, and is used all over physics. The reason is simple. It can be difficult to solve the differential equations that are encountered in physics, and using a power series expansion can allow us to solve problems to a great approximation. Knowing how to use this technique to express a function as a polynomial is powerful.
Because I skipped that class, I didn’t get to see this topic in detail. As such, I continued through my education with only a rough idea of how it worked. This meant that when professors would tell us to expand a function as a power series in order to solve a problem, I would always be slightly behind, not quite sure how to do it. I knew that it involved derivatives and factorials, but it was clear even to myself that I wouldn’t be able to do it on my own.
The problem here is one that many students face. They end up not really understanding a concept, or miss it for whatever reason during the semester, and then go on with their education with this missing gap in their knowledge. That’s fine, but these gaps do show up later on. At that point, it’s usually more annoying to go back and learn the concept, so students either try to fake their way through understanding, or fail.
It doesn’t have to be this way. I’ve decided to put in some time to look at expanding a function as a power series, because I know that it’s an important skill that I need in my toolbox. It won’t be something I’ll figure out in five seconds, but it’s a good investment of time.
I am sure these same kinds of weaknesses exist for you. Perhaps there was a concept that kind of “slipped through the cracks” for you when you first came across it, and you just haven’t thought about it in a while. These are your weak points, and it takes honesty to admit that they are there. Furthermore, it takes a certain amount of willingness to say, “I’m not satisfied with acting like I know this. I want to really understand it.” It’s not easy, but it’s important. I will keep on beating the same drum: in mathematics and science, concepts build on top of each other. If you don’t have a strong foundation, it is difficult to learn about new concepts. It’s possible, but your understanding will be riddled with holes. If you don’t believe me, find a topic that you know next to nothing about, and then find some lecture notes or a textbook aimed for an advanced audience. I’m willing to bet that almost none of it will make sense to you. That’s because you don’t have the foundational experience necessary to jump into these resources.
If you want to get better at anything, it’s crucial that you identify your weaknesses, and then work to improve them. This latter part is just as important as the former. Saying you have weaknesses is one thing, but working hard to address them is a different challenge. However, if you are willing to put the work in, it is doable.
Don’t do like I did. Don’t miss a concept and let it go ignored for years. It will come back to haunt you, so you might as well put in the work to understand it.
Pairing Simple Examples With Complex Machinery
Teaching a new concept within mathematics or science isn’t easy. It requires taking students outside of their comfort zones to try and understand how we model phenomena that is more complex than previously seen. This tends to require new tools and techniques, which means students have to shed their old tools in favour of these new ones. This can leave students disgruntled, particularly those who were attached to the old method of solving problems.
While teaching a new concept, one often revisits an older problem that didn’t require these new tools, and shows that the new tools can also solve the problem. This is done quite a bit, because there is a limited amount of time during a class, and older problems can be worked through more quickly. As such, they are good candidates for the first example of a new tool.
However, a problem can occur if a teacher only uses these older examples and doesn’t move on to newer, more difficult examples. The reason is that the students will see new machinery being introduced for no good reason. They will ask, “Why do I need this new machinery to solve a problem I can already do without these tools?” If the new tools aren’t vastly more efficient, this can be a valid point from a student, even if the teacher knows there are other reasons to learn this tool.
For example, consider classical mechanics. The first classes of classical mechanics involves vectors and calculus. One has to keep track of where the forces point at all times, and deal with each component separately. It works, but as the complexity of a problem increases, using this vector calculus method of solving classical mechanics problems becomes both tedious and somewhat intractable.
Then, one learns about a new technique to solve mechanics problems. This is the topic of Lagrangian mechanics, where vectors are nowhere in sight. Instead, generalized coordinates play a big role. The important point is that this method is much better than vector calculus in general. As such, it’s easy to showcase the power of Lagrangian mechanics when analyzing the motion of a bead on a spinning hoop (for example), which would be a messy affair using vectors. As a student, I thought, “Obviously, this is way better than what we were doing before. Lagrangian mechanics is a great tool to know how to use!”
But Lagrangian mechanics isn’t the endpoint. After learning about the Lagrangian and the Euler-Lagrange equations of motion, the Hamiltonian is used. For myself, this is where things got murkier. The Hamiltonian approach to mechanics was described to me as a more “sophisticated” approach to classical mechanics. I believe that, but in my eyes, the practicality of the Hamiltonian was lost on me. It was an alternative way to use the Lagrangian, but it didn’t feel any better. I wasn’t sold on why this new tool was useful.
I think the reason came down to not having any examples where I could say, “This is where I need to use the Hamiltonian1.” The Lagrangian method was adequate for me, so I stuck with it and paid the Hamiltonian method little attention.
This is an issue that can arise when teaching a lot of mathematics. We develop a tool with the students, they get better with that tool, and then we tell them, “Hey, there’s actually another tool you need to learn, and it will be useful.” Then we fail to deliver on this promise when we rely on simple examples and past problems to showcase the new tool. That’s not enough! If we want students to be on board with the new tools we teach them, then we have to push the students’ boundaries a bit. Put them into a situation where the old method fails, which will force them to try the new tool. Show them that this is what the tool really excels at, and that there isn’t a better way to solve the problem. Note here that I’m not suggesting that we tell students this is the way or the only way to solve a problem. Rather, we should be able to come up with examples where the “reasonable” strategy to solve a problem is with this new tool. If you don’t do this, you will get students who stick with the old method of solving a problem because it’s the method they are used to, and they won’t get to see how good another method can be. Your job is to at least give students a glimpse of that alternative way.
There’s one last important thing that I want to say: I despise test questions that “force” a method of solving on the student. There is only one good side to this, and it has to do with the fact that forcing students to use a certain method will give them a hint as to how they can solve the problem. Apart from that, I don’t think it’s productive to force or insist that students use a particular method to solve problems. Of course, there’s nothing wrong with having students solve problems in a particular way during an assignment (after all, that’s for practice). Tests, on the other hand, should be left wide-open in terms of solving strategies. A test should seek to answer the question, “Can a student use appropriate steps to get to the answers?” I realize that there’s the issue of covering the whole curriculum, but then I would suggest writing problems that steer students into a particular way of solving, but without forcing them to. This way, if the tool was introduced in a way that made it clear why it existed and what kind of problems it simplified, the students will be able to make that connection and solve the problem during tests.
I do realize that the Hamiltonian is used a lot in quantum mechanics, but the context of what I was learning in class wasn’t quite the same. The Hamiltonian in quantum mechanics is used to solve the Schrödinger equation, which isn’t quite like the procedures used in Lagrangian mechanics. ↩
One of the most important things you can do when learning a new subject is attempting it with your best effort. While this sounds simple, so many people think they are too good for this step and skip it.
I was reminded of this at the beginning of last semester. One of my professors said, “When you read the chapters in the book, do the examples that are there. It’s extremely important that you do. It might be tempting to go through and read the solution of a problem and then say, ‘Oh yes, that makes sense.’ However, doing this won’t help you out in the long run.”
I couldn’t have put it better myself. This is exactly the kind of issue I run into all the time, and I know that it can affect others just as much. We have this idea that reading implies learning, but there are many ways to read a textbook and learn next to nothing about solving problems. This includes skipping through the exercises and examples, preferring to just read the solutions. This is good up until the point that you are faced with a question on your own, such as on a test. At that point, you may find that the problems are not so easy.
It’s simple to say, “Oh yes, this solution makes sense. I get it.” It’s much more difficult to think of what that solution is on your own. But this is what doing the examples in a textbook is for! They are there to allow you to think of the solution without a helping hand. The solutions are for after you’ve done this step.
This is so important, because I see people all the time take ten seconds to read a problem before saying, “I don’t know what to do.” Well, of course you don’t know what to do! You’ve barely had time to think about it at all.
The purpose of doing all these exercises is to give you a sense of what works and what doesn’t. As a consequence, you start to gain an eye for what a solution might look like. When I tell the students I tutor how I solved the problem they are struggling with, they look up at me in disbelief, as if they would never be able to do what I did. But they’re wrong. I didn’t do anything special. In fact, I did something I’ve done a bunch of times. That doesn’t mean I was as good when I was in the same year as them. It just means I’ve been at it longer and have put the work in to make those problems trivial.
It seems like a waste of time, but the real key to learning and succeeding in your studies is to attempt to do problems on your own, without referring to your text. I’m not suggesting you memorize every single formula in physics. Rather, I’m talking about learning to come up with your own strategies to solve problems. From there, it’s just a lot of practice and refinement. If you prefer to skip the work and read the solutions, this skill won’t be developed. You’ll always be looking for the “trick” to solve problems, instead of applying techniques. Therefore, it’s in your best interest to work through as many problems as you can. It will pay off in your future.
Permission to Learn
When you were in elementary school, did you get a say in what you learned? How about in secondary school? Odds are, you were given a set of classes, and your time at school looked something like this:
We all had these instances when we wondered, “Why am I learning this?” We were then either told that we had no choice, or that that there were benefits to learning that material that we couldn’t see right now. One of these must have seemed reasonable, because you continued on.
We then get to university, where the classes are more varied, yet we still have to take certain ones. Hopefully by this point you like the program that you’re in, so this doesn’t seem as bad.
My question to you is this. Have you ever decided to learn something despite it not being a part of a class? Did you ever decide to learn something just because you were curious?
If the answer is “no”, I encourage you to ask yourself why you haven’t. I’m sure that there must have been something that piqued your interest at one point. Why didn’t you follow up on it?
The point I want to get at here is that, now more than ever, we don’t need permission to learn something new. Sure, what you’re curious about might not be on the test in class. It might be unrelated. But you’re curious, aren’t you? That’s enough, and it’s why you need to learn on your own.
This doesn’t tend to be an instinct, because our education system doesn’t prepare us for this. From the get-go, we’re told what to study. Forget about everything else, it’s not as important to you. Focus on what you need to know for the test.
That’s fine advice if you want to get good marks, but it doesn’t help you grow. We have so many resources available now that you can learn pretty much anything you want. Why not use that opportunity to move away from the curriculum you got from school and learn something that interests you?
We can all use the reminder. We don’t have to be in a class to learn something new. Giving yourself permission to learn something new is a powerful first step to growing much more than you thought possible.