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.