I think a lot about the process of learning, both in my experience and when I am teaching others. I find learning to be a fascinating topic, because it’s what lets us improve and become better at a variety of tasks, skills, and subjects. I’m also interested in the difficulties that are present when learning. In particular, are there things we should do or avoid while trying to learn?
One of the most frustrating things that can happen to me while I’m learning is to have just a tiny amount of confusion during an explanation. For example, a professor could be going through a derivation, and I’m nodding along as they write on the board. Then suddenly, a step is performed that I did not see coming, and doesn’t make sense to me. When that happens, I’ll often get sidetracked from the whole derivation, my mind always drifting back to that step of confusion. I’ll start asking myself why that step was allowed, and the longer this goes on and I don’t see the reason, the more distracted I become. In essence, this small confusion can derail my attention from the overall explanation.
Sometimes, the issue is a small one in which you just missed a part of the calculation, while others are due to larger ideas being swept under the rug. The problem is that you can’t always (or don’t feel comfortable enough to) ask the professor about what happened. The result is that you end up sitting there in confusion.
This happens to me all the time. However, most of them weren’t during lectures. They occurred when I was learning on my own through a textbook. I would be following a derivation, and suddenly a step didn’t make sense to me. The authors weren’t kind enough to include a few signposts in the text, so I was stuck on my own. Couple this with the fact that you can’t exactly ask a textbook a question and it’s clear that these little missteps can derail your learning.
The most frustrating part of these situations is that you know that you’re often just missing a small idea that you either can’t remember or aren’t considering. If someone would mention that idea, you would be well on your way. Without it though, the derivation is just confusing.
In order to combat this, you need to find someone who can answer your questions. The worst thing in my experience is to soldier on without resolving your confusion. I’ve tried that before, and the problem is that it builds a shaky foundation. Sure, you might be able to follow the rest of the steps, but if you can’t figure out why this one worked, you’re not fully comfortable with the derivation. I’ve also found that if you have one source of confusion and you skip by it, chances are there will be more confusion down the road. Sooner or later, skipping through various steps will come back to haunt you.
As such, you want to find someone who understands the material in order to nudge you back in the right direction when you’re lost or confused. When I find something confusing, I don’t even try to skip ahead anymore. I know that it doesn’t work in the long haul, so I take the time to step back, reassess my issue, and ask for help if necessary. That could include asking other students, professors, or even looking online for those with similar questions. The point is to avoid skipping past your confusion as if nothing was wrong. Most of the time it’s important to get all of the details right.
As I’ve studied in school for longer and longer, my approach to learning has changed. “Getting through” an explanation isn’t enough for me anymore. Learning isn’t a process of being given an explanation over and over again until you can repeat it without issue. For myself, learning is about feeling comfortable with whatever material I’m studying. I want to follow along each step in a derivation without feeling like anything is wrong or unreasonable. Of course, that doesn’t mean I won’t be surprised by the direction a proof takes. It’s just that I want to be able to say, “I understand how we go from this line to the next.” If I can do that, then I’m happy.
This doesn’t always happen on the first try. I often have to sit and think about a step that was done. That’s because people will skip steps, and depending on my level of knowledge in the subject, this can make it difficult to follow. But I’ve resolved to making sure that I can follow any explanation or derivation I come across. I know that I could get away with just knowing how to reproduce certain calculations for my classes, but that’s not enough for me.
On the other end of the equation, I try to be more mindful of how a student is feeling as I’m explaining a concept to them. Do they seem like everything is fine? Are they nodding along in a way that is engaged, or that’s merely going through the motions? I try to gauge this while teaching, because it’s so easy to get into a groove in which you assume the person understands, when really they don’t. I want to get at each and every source of confusion a student has, because it’s the only way to build a more solid mathematical foundation.
An explanation is only as good as each twist and turn. You could have the best explanation in the world, but if one part is confusing and difficult to follow, it harms the rest of the it. Therefore, it’s important that each part makes sense, or else the totality will be weaker than it could be. This is why I recommend pausing when you encounter a point of difficulty. Don’t forge ahead, but slow down and think about the issue. I guarantee that you will find it worth it by the time you understand the full explanation. Likewise, I recommend taking the time to think about how you craft your explanations. A single misplaced detail can derail the whole explanation, so build carefully. Anyone can throw together an explanation, but it takes thoughtfulness and care to do it in a way that a person can follow.