Holding Everything In Your Head
I’m always on the lookout for what makes a great explanation tick. If I don’t know anything about a topic, what’s the best way for me to accelerate the process of learning? Conversely, what are the signs I should be watching for that indicate I won’t get anything out of the material?
These two questions occupy a lot of my thinking because of my interest in education. Not only do I want to improve how I learn, I want to provide the best tools and resources for others who want to learn physics and mathematics. Being able to learn without experiencing frustration at every single turn is something I want to figure out.
I can only learn so much from my own struggles, but I think it’s good to acknowledge each of them as part of the puzzle which is education. One piece involves the idea of parsing the important details from the unimportant.
I see this all the time when working with students. They will read a problem, and it’s unclear to them which aspects are important and which can be disregarded. I’m not even talking about technical skill here. Rather, it’s about being able to sort the information into “useful” and “useless” bins.
What creates this struggle in a student? When I’m working with them, I’m tempted to give them the answer so we can move forward in a question, but I realize that their struggle is perhaps the most important step in the whole process.
In my own experience, I’ve noticed this when first learning a topic. As someone who by definition isn’t an expert, I’m entirely reliant on the teacher to give me the information that matters. The problem is that they will give a lot of information. There’s no way I can remember it all as distinct pieces. There’s little chance that they can remember it like this.
But that’s not how teachers think about it. The pieces of information aren’t totally independent. Knowing one will tell you something about the others.
If you know the equation for a line, you can tell me something about parallel lines and the equations that describe them. These two pieces of information aren’t distinct, they are connected.
Mathematics and physics is basically the study of connections such as these. Therefore, even if the individual pieces of information might seem important, the true purpose of the subjects is to give you the connections between ideas.
So there you go. Just understand the connections, and you’ll be fine.
If only it were that simple.
From my experience, it is very difficult for professors to simply tell students what those connections are. It’s like there’s a filter that stretches between the student and the professor, blocking the connection. The professor might think they are telling students something extremely important, but students won’t grasp the significance. The is a huge problem, because it means students then see the information as independent pieces. This makes it impossible to remember, when really an understanding of the connections between the ideas is what matters.
I can’t overstate how much of a difference understanding the structure behind a collection of ideas makes. It’s the difference between being aware and understanding. Both states of being have you knowing the information, but the latter is where you can grasp the significance of it. Not only that, but it’s easier to remember a larger quantity of information, since you simply need to understand the logic that brings you from here to there.
When I’m sitting in a new class, listening to the lecturer, everything is new. There’s no easy way for me to tell what is important and what is superfluous. The professor might not know anything is wrong. To them, it’s super clear which aspects are important. It’s reasonable to assume that the students are absorbing the important parts as well, particularly since they aren’t asking questions.
But I can unequivocally say that this is not the reality. It takes me a long time to go from seeing the information for the first time to actually organizing it in a way that emphasizes the connections. I would argue that this is the actual process of learning. Until this happens, all you have is a bunch of information stored in your head. Glimpsing the connections and being able to jump from one branch to another is the mark of learning and expertise.
Unfortunately, I feel like we don’t emphasize this enough when learning. We favour the absorption of information, as if knowing the material is the only thing that matters.
Don’t get me wrong, knowing pure information is important. It’s what lets us quickly jump between connections. I’m a fan of memorization to the point that it gives you a stronger instinct for your available tools. However, the issue is that there’s a limit to how much you can hold in your head at one time.
To see this in action, pick up a textbook on logic or number theory that is text-heavy and whose contents you are unfamiliar with. Try to follow a paragraph of argument without writing it out on paper. Just try to hold it all in your head.
It’s difficult, right?
I don’t think that’s an accident. Rather, it’s because you don’t know where to put all of the novel information being thrown at you! You end up trying to hold it all in your head at once, and this creates issues. Honestly, when I read sentences that mention more than three mathematical objects I’m unfamiliar with, I’ve probably gotten lost. I sometimes get lost when holding multiple objects in my head that I am individually familiar with.
This isn’t to say you can’t ever introduce new information. But you need to do it strategically. Presenting an impenetrable wall of text isn’t going to work well. On the other hand, interspersing sketches or diagrams with the text can help a ton. Don’t underestimate how much this matters. Even if your sketch is just writing down an equation, I would recommend doing it because it separates itself from the rest of the text.
Likewise, we need to be aware that the first thing we should be doing when learning a new topic is trying to organize the information we receive. Absorbing is part of the equation, of course, but stopping there isn’t going to lead to a lot of growth. It’s only when we take the information that we’ve learned and created the connections that we start seeing the subject as a whole.
Whether this means mind-mapping, discussing with friends, or teaching the ideas to others, it’s crucial to do this extra step of organization, or else the most you will accomplish is being able to restate some facts.