As a student, I’m used to diving right into the technical details of a topic. I don’t mind working through a wall of algebra, because that’s what I’m used to. If I wanted to describe how I learn in my classes, it would be: mathematics first, “high-level” understanding second. This isn’t a bad thing. I don’t mind going through the details first. Sure, I might not know how the concept relates to other ideas immediately, but I can learn that later.
The unfortunate thing is that this “high-level” understanding doesn’t usually come from teachers explaining ideas to you. Instead, I’ve found that it comes from observing how ideas are explained in class, and putting these observations together over a long period of time.
I’ll start with a simple example. In secondary school mathematics, one of the ideas is that of factoring quadratic expressions. Different methods are taught, including grouping, completing the square, and “filling in the blanks” by looking at the various coefficients. When you’re learning these techniques, they all seem different. You might even be told that you need to use a certain method for a certain kind of expression you see. As such, you follow what the teacher says. It’s only much later on that you realize that these different methods were essentially doing the same thing. They look different at first, but that’s only at the surface. Once you realize this, you also understand that calling them different “methods” is not helpful. Unfortunately, this moment of realization comes much later when you have completed the class.
This isn’t just an issue in mathematics. As a physics undergraduate, I’ve taken many physics classes over the years. Some of these classes seem distinct, but once you take a step back, you can’t escape the fact that they are all about the physical world. This means there are many connections between classes, waiting to be discovered.
If you have good physics professors, they won’t hesitate to point these out. That’s because understanding these connections can give you a better picture of the concept you’re studying. It removes the details and let’s you worry about the heart of the matter. While this might not help you calculate the electric field at a certain point, it will give you the ability to understand a subject at its broadest.
Grasping the conceptual
I’ve hinted at this before on the blog, but I’ve noticed that while I learn a lot of the technical details at school, I haven’t done a good job of grasping the overarching principles behind certain subjects. I think this conceptual understanding is quite important, and yet, due in part to the school system not rewarding this type of knowledge, I haven’t focused on it. This is something I want to work on, because it allows one to move fluidly within a conversation without having to pull up the technical details.
I’ve been thinking about ways this can be done. One method that I think holds promise is the idea of a “one-sentence summary”. It’s exactly what it sounds like. With any given topic, what is the one sentence that can describe its essence? Of course, I’m also barring myself from turning into a punctuation master and creating a run-on sentence that lasts forever. Instead, I’m thinking of something short and sweet, preferably memorable. The key here isn’t to find the perfect sentence, but one that is good enough.
Will it encapsulate everything about the subject? If the subject has any breadth, of course not. But that’s also not the point. Instead, I want to capture the bare minimum of the subject. If someone heard my sentence, would they agree with it? Perhaps they have other ideas that could take precedence over this one, but at least they agree that the one I gave is important.
In order to illustrate this, I’ll give you a few examples of one-sentence summaries that I’ve come up with.
- Entropy: A count of how many micro configurations give rise to the same macro properties.
- Metric: A tool to let you measure distances and angles in some geometry.
- Statistical mechanics: What properties appear when a lot of smaller parts form a system?
As you can see, these aren’t perfect. I’m sure that people with more expertise and experience with these ideas than I have can give better one-sentence summaries. However, the goal here isn’t to write the absolute best sentence for any topic. Rather, it’s to write the best sentence that I can think of at this moment in time. There’s no use in trying to achieve some absolute. What might be even more interesting is to update your one-sentence summaries whenever you think you have a better understanding, and keep the older ones. What you will end up with is a chronological record of your thoughts on a subject. It will let you see how your understanding has changed over the years.
These are just preliminary ideas to get you started. Adapt this as you see fit. The key is to try and gain some high-level understanding of a subject. The method itself doesn’t matter. I know that I’ve spent so much time in the weeds of subjects that it’s worth trying to take a broader perspective.
I want to be clear: I’m not looking to replace my usual method of learning. The details are important, and I don’t want to minimize that. If you want to learn about science or mathematics in particular, sooner or later you have to dig into those details and work with them. There’s no escaping that. Conceptual understanding alone won’t make you pass a test that deals with calculations.
That being said, I think we too often eschew conceptual understanding, as if it’s not as good as “the real thing” of calculation. I think this undersells the value of knowing a subject at a high-level. In my mind, they are both important, but distinct. If you just focus on the details, it will take a long time for the high-level understanding to come. One-sentence summaries could be a tool that can help you, but I don’t care if you use my method. Please, just find something that works for you. It’s so great to feel at ease with a concept to the point that you can talk about it at a high-level without worrying about the details (or that you’re butchering them).
When you first start solving a problem in mathematics, the goal is often to find a way to express the problem as some sort of differential equation. During this initial search, you don’t care how the equation looks. It’s more important to get it written down so that you can proceed.
However, once you do have an equation, the first step is not to try and solve it. That’s a rookie mistake. Instead, the question you should be asking yourself is, “Can I put this equation into a form I recognize?” Asking yourself this question can save a ton of time in solving a problem. After all, if you can recognize the form of the equation, then you know the answer without doing any more work.
This might seem like an edge case that never happens in practice, but that’s not true. In particular, mathematicians have studied the solutions of many ordinary and partial differential equations, and know the answers. Therefore, if you’re working with a differential equation (which is almost always the case in physics), you might be able to save yourself a lot of time if you recognize the form of the equation.
For example, any student in physics who has taken more than a few courses will recognize the differential equation representing simple or damped harmonic motion. Physics students come across it all the time. This equation comes up when considering swinging pendulums, motion of a spring, electrical circuits, stability of circular orbits, and even in the Schrödinger equation. It’s what you might call a pervasive equation.
I can guarantee that professors don’t go over the solution to this equation after perhaps the first semester of physics. The reason is that students learn how to solve this differential equation, so there’s no need to go through all the work again and again. Instead, they identify the equation, and then give the solution.
However, it might not always be obvious that an equation satisfies the differential equation for harmonic motion. If there are a bunch of constants littered everywhere in the equation (due to the physical situation), it can be difficult to see the underlying equation. How do we deal with this so that we can try and identify an equation?
The trick is to change variables and bundle up constants together as much as you can. If your equation has constants littered everywhere, see if you can divide the constants out so that you have less in total. In the same vein, if you can see a simple change of variables that will allow you to “absorb” some of your constants in the differential equation, that can also help in simplifying the equation.
The goal here is to try to make your equation as generic as possible. That’s often the best way to compare it to the known equations in mathematics which have solutions. When you look at solutions to differential equations, they won’t be given in terms of parameters like the mass of a particle. The constants will be generic. Therefore, it’s often in one’s best interest to “clean up” a differential equation as soon as possible in order to make it recognizable.
Remember, there’s nothing wrong with ploughing ahead and solving the equation right off. It can still work. It’s just that the constants present in an equation that are specific to the problem can muddy the waters of the solution. By dividing constants out and changing variables, the equation will shed its “particular” qualities, showing only the essence underneath. Then, one can save time by identifying it with a known differential equation.
The point when solving a physical problem isn’t to go through all of the mathematical detail for no reason. If a solution is already known, there’s no point to ignore that. Use the fact that you can recognize solutions to speed up your problem solving. In the end, it’s the physical solution itself that matters.
Snapping Into Focus
Learning new ideas in mathematics or science isn’t always easy. Heck, I would venture to say that most of the time it’s difficult. I imagine the experience is the same whether or not you consider yourself to be “good” in a given subject. That’s because, on some level, we are all in the same situation when it comes to learning. We need to figure out how to integrate new knowledge into our existing worldview.
In particular, I find that mathematical ideas and equations can be the toughest aspects of learning new material. The challenge for me always revolves around the question, “How can I restate these equations and expressions into words that I can understand?” (I’ve written about a similar idea of translating from words to equations before.) I find it helpful while trying to understand what’s going on within an equation. All equations have a story to tell.
I’ll be honest: even as someone who has seen a lot of mathematics, if you drop me inside a derivation without any background, the probability of having me understand what’s going on converges to zero. Mathematics requires context, and it requires focusing on a specific argument. Only once you’ve interacted with it will you start feeling comfortable with the specific equations and expressions.
It’s during the end of this period of struggle where something interesting happens. Just as you’re starting to to figure out what’s going on, things seem to “snap” into focus. The best way I can describe it is through an analogy with running in the fog. When you’re in the fog, you can’t see anything. The light attenuates quickly, and you end up seeing only twenty or so metres in front of you. However, if you climb a hill, there’s this moment where you break through the fog, get above it, and can see everything. While studying mathematics, this is where an idea clicks into place and everything makes sense. The great thing is that once you’ve gotten it, there’s no going back. The concept just makes sense now.
This moment is something I search for all the time, both in myself and in others. As a tutor, there’s nothing that makes me happier than seeing the student I’m working with suddenly exclaim that a concept is now clear to them. It’s the reason I tutor students in the first place. Sure, it’s a job, but it’s also rewarding to witness these moments where concepts snap into focus.
I love this feeling because it illustrates the difference between receiving information and internalizing it. As a student, I have many different classes, each with their own set of assignments, tests, and lectures. In an ideal world, I would be focused during each one. However, if you are (or were) a student, you know that this isn’t the case. Most of the time, we are distracted, not focused, or aren’t engaging with the material more than what is needed to pass the test. You might “understand” the material fine for the course, but I would argue that having this deep understanding where ideas snap into focus is a different situation. When this happened, it became so clear to me that I didn’t have to worry about forgetting it. The idea just made sense, and I felt like I could hold the idea in my head without effort. Contrast this to the feeling one gets when studying the day before an exam, and I think you will see what I mean.
Having this experience is great, but it’s also a lot of work. You need to engage with it, making sure each point makes sense.
Because of this, I can only reasonably commit to fully understanding a few ideas at a time. It depends on the number of ideas you can juggle in your head. Furthermore, I’ve found that engaging with the ideas from a class isn’t enough. In order to get the perfect alignment which is characteristic of something snapping into focus, I need to perform a deep dive. This can be done through writing or teaching.
This isn’t practical to everyone. We don’t all enjoy writing, and producing these pieces takes a lot of work. As such, there are other strategies you might want to employ. First, you can work through related problems that highlight this specific idea. An idea can seem fuzzy in the abstract but be clear when applied to a problem. As such, practice problems can be useful. Second, see if you can explain the concept without any extra help from a textbook to lecture notes. If you can do that, then there’s a good chance that the idea will snap into focus for you soon (or already has). Beware though: you need to make sure the explanation is clear to you. Often, we can be tempted to take the shortcut of merely parroting what is said by the teacher, but that won’t help here.
If you want to really understand an idea, at some point it will have to snap into focus. That’s non-negotiable. The act of snapping into focus is just a milestone in learning. As such, we should be thinking about how to get there, and the strategies we should use to do it. Like I’ve written above, going through problems and trying to explain the topic yourself are good strategies. Another one though that is important is asking someone else. Sometimes, it’s just a particular explanation that is holding you back from understanding. If you limit yourself to just what your teacher says, than you will be in trouble when they say something you can’t figure out. Finding an alternative explanation is the best way to go when this happens. This could be from a friend, from a textbook, or even from the teacher. The point is that sometimes we just don’t understand a particular route, and a different explanation is all it takes to snap into focus.
Most importantl of all, remember that learning is more than just showing up to class and getting a passing grade (or even a good grade!). It’s about struggling with a concept until finally the fog clears and everything falls into place.
Escaping the Path
There’s a lovely forest near my house. It’s a wonderful place that looks exceptional in the autumn, where the fallen leaves of the trees cover the path in a flurry of orange, red, and yellow. I love running there because it’s so peaceful.
Imagine that I told you I would show you this forest. After hearing me wax poetic about it, you’re excited to see it. We get to the forest, and I show you the path that goes through. We walk along it, and after a while you ask if we can get off the path to see the forest in its more “natural” state.
Puzzled, I ask, “But this path is the forest. There’s nothing else of interest other than what’s on the path anyway.”
We might not use the same words, but this is how a lot of us view mathematics. There’s a path (the curriculum), and following it is the only way to learn about mathematics. Forget about going off-path. That’s not even a thought that crosses your mind!
Unless you are really into mathematics, chances are you haven’t seen the wonderful little niches that the subject has to offer. This is unfortunate, but it’s a consequence of the fact that we tend to look at mathematics in terms of the path forged by the curriculum. It’s also not a problem which is limited to mathematics. Almost any subject will have this standard “path” that most people end up associating with the subject itself.
If I could send one message to my younger self, it would this. Don’t make the mistake of seeing the path as the subject itself. It’s only one particular way of looking at a subject, but there are so many more available. It just takes a willingness to look past the usual offering.
Unlike what we’re taught in school, mathematics isn’t a linear subject. Sure, it’s probably a good idea to learn about arithmetic before you learn algebra, but it’s not always as clear. The web of mathematics is thick and highly-connected, which means there are many paths you can take through the subject. Just because there’s a clear trail that has been created by countless curriculums does not mean you are forced to take that same path. In fact, I would encourage you to explore more. Look for those smaller connections. They can be as interesting as the regular path.
My hope here is to encourage you that mathematics is not only the curriculum you learn in school. It has so many other aspects that are off the path, if only you start exploring.
To me, this indicates two things. First, it means that we need to spread the message through our educational institutions, because it’s important that students see mathematics as more than only a curriculum. Second, it suggests that a way to get people interested in mathematics is to find something that they are attracted to. The key point is that this may not lie on the main path, but who cares? I’m more concerned with getting people to see mathematics as it is: an ensemble of many ideas, not just a linear path.
It’s worth wandering off the path every so often to see what else is on offer.