When I’m learning a new topic, all of the details matter. If I want to understand what is going on, I need to have a firm handle on each detail. If I cannot imagine how each part operates and connects with the others, “getting” the concept is difficult. I suspect this happens for others, too. After all, if you want to understand without needing to take a result on faith, you have to know the concept as a whole.
This means learning takes a lot of effort up-front. I need to read the textbook, work through problems, and figure out what exactly is going on. If you imagine learning as a process that begins with a blank slate, anything that is introduced has equal weight. In other words, I cannot tell what new information is important and which is is not. Deciphering this requires extra effort and time, which is why it takes long to become comfortable with a new topic.
I have experienced this myself as I learned general relativity (and the associated mathematics) throughout my undergraduate degree. When I began, everything was new, which meant I had difficulty splitting what I was learning into a “useful” category and a “not-so-useful” category. This made it difficult to understand the big picture, because I could not determine the core ideas. Should I focus on the Einstein field equations, or was it better to become comfortable with the mathematical formalism first? Since I was learning the topic on my own, I did not have a bunch of support like you would have from a dedicated class. The result is that I spent a lot of time spinning my wheels and getting nowhere.
At least, this is how I perceived it. In reality, I was slowly gaining knowledge, figuring out what worked and what did not. However, there is no escaping the fact that it took an enormous amount of effort to get myself to the point where I feel comfortable.
As I right this, I am still not fully comfortable with the ideas. That being said, I worked with the concepts long enough to have some idea of how everything works. It is not perfect knowledge, but when I read about the geodesic equation, my mind does not offer up an empty box with a question mark inside. Instead, I have an idea of what the geodesic equation is. This is just one example, but it is the common thread I have seen while learning.
Familiarity breeds knowledge. Things might not make sense at first, but if you persevere and keep going, you will start to see progress. It might be slow, and it might take a lot of effort, but it will come. Since I needed to learn these ideas for the summer research I did as an undergraduate, I was given an incentive to keep going. This definitely helped, because I know it can be difficult to conjure up motivation out of thin air.
The reason I bring this up is that I always wanted to have the “effortless” knowledge that my professors have. Sure, I did well in school, but what I thought was amazing was how they seemed to know a bit about any piece of physics. Even if they were not an expert in that domain, they could quickly come up with an argument as to why the concept should be like this or that. How did they do it? I wanted to find out.
I think a big reason is that they have “offloaded” a lot of what they learn to the back of their minds. Essentially, they understand what parts of a concept are important, and they focus on those. The rest is “stored away” for retrieval if necessary. The reality is that a lot can be said by simply using the important parts, which is a lot easier to remember than everything. As such, they can explain the key features of a concept without too much difficulty.
Thinking back to material that I am comfortable with, I see a lot of agreement with this idea. As a tutor for secondary-level mathematics, I work with students solving basic linear equations, working through algebra, and solving geometry problems. These are ideas that I have so much experience with that I know what the important details in a problem are. While the student might read a problem and not know what “weight” to assign to each piece of information, it takes me seconds to figure it out. This is not a nod to my “brilliance”. Instead, it is the result of working on a lot of similar problems and understanding the key components. This lets me get to the core of the problem, without worrying about the details.
That is what I think learning really means. When you are presented with a concept, a problem, or a question, can you get to the heart of the matter, or do you get stuck in the details? As you learn a topic, I think you end up offloading a lot of the details to the back of your mind, allowing you to focus on what matters. Learning is offloading the non-essential details to the back of your mind.
I try to keep this in mind when I work with other students. If they are having trouble, it might not be because they do not understand. Instead, it could be a simple issue of not knowing what is important and what is not. In the case of mathematics, this can get tricky, especially with questions that are designed to fool you (hello, almost every word problem!). Giving a student the tools to let them abstract away from a particular problem and let them see the bigger picture is my mission, which means I want them to start offloading as much as possible.
I would compare this process to trail running. If you never ran on trails before, the key difference from regular running is that the terrain is uneven. Roots, rocks, mud, and a thousand other obstacles can hinder your progress, and “real” trails are not smooth. As such, you need to be careful when you run. However, if you try to focus on every single detail of the trail, you will often trip up on something. Instead, the best strategy is to take short and quick steps without thinking too much of where you are landing. In essence, you are offloading the task of obstacle avoidance to your subconscious.
In much the same way, I think it is critical to be able to grasp the core of a concept without worrying about the details. I am not saying that you should never worry about the details, but that they are not often useful in the beginning. Instead, the goal should be to get to the core, and then add in the details.