Imagine I had a block of aluminium and I asked you to describe it to me.
Perhaps you start by describing it qualitatively. Maybe it’s a cube of side length 0.10 m. If there’s a light in the room, perhaps you’d note how reflective the cube is, suggesting that it’s some sort of metal. You might even pick it up and note that it isn’t heavy, suggesting one of the lighter elements.
But this isn’t the only way to describe the block. If you took a more “complete” approach, you might do some tests on the block to determine its composition, concluding that it is indeed aluminium. Then, you could mathematically describe (at least, in principle) every particle that makes up the aluminium block.
What I’m trying to illustrate here is that there are different ways of looking at a block of aluminium, and they can both be right in their own domain. Additionally, it can be good to look at the block as a whole “thing”, instead of a bunch of particles. In other scenarios, the opposite may be true.
However, this is a bit too much for the purposes I want to talk about now. Where I get a lot of use out of different ways of looking at something is for a mathematical or physical concept. When learning, a teacher tends to only do one derivation (if they do one at all) for an idea. That’s great, but there’s always a chance that the way the teacher understands the concept is difficult or not in line with the way you think. Consequently, the proof can seem complicated even though it isn’t, just because you’re not comfortable with the tools involved. It’s kind of like always using traditional running shoes while training, and then suddenly being given racing flats for your race. The racing flats will mostly likely be a lot better than your training shoes, but you won’t reap the benefits because they will feel odd to your feet, making it difficult to run fast. Likewise, a teacher may select a method that is the most efficient, but you can’t appreciate the gains because you’ve suddenly been thrown into using methods you aren’t comfortable with. The solution to this is therefore to seek alternative methods to derive said equation, hopefully in a way that you are familiar with.
That is all well and good, but I think there’s use to doing this even when you’ve understood the first way of deriving a result. By looking at different ways to get to the same answer, you’re effectively giving yourself different ways to “look” at the concept. For example, my professor for waves and optics actually went through several different ways of calculating the intensity of a diffraction pattern. Some used algebra, others used vector addition, but the end result was the same.
What this did was give me a firmer grasp of the ideas at hand. I could explain them in different ways and still get to the same answer, which is both a good sign and makes me comfortable with the concept. Obviously, this might not work for every mathematical or physical idea, but it is a good way to help strengthen your understanding of a subject. I’ve said this before, and I’ll hammer it home again: a strong foundation is the most crucial part of learning.
So go build that foundation.