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).