Can You Explain It?
Despite needing to compute many quantities in my various physics and mathematics classes, I like to take a slightly different approach to studying for a test. First, I will do a bunch of practice problems in order to strengthen the “muscle memory” of how to do certain types of questions. This usually means going over assignments and doing examples. This particularly helps with refreshing my mind for questions I haven’t done in a few weeks.
However, this isn’t the complete extent to which I prepare for my tests. When it is the night before my test, I make sure to review concepts instead of specific problems. What this means is that I won’t be doing actual calculations. Instead, I’ll make sure I can explain the steps that I need to do in order to solve certain questions. What this does is refresh my mind to the types of questions that may be asked during the test, and I find this gives me the confidence that I know what I am talking about. While some may simply do the exact homework problems again, I want to make sure that I can adapt to something different on the test. My hope is that doing this will get me thinking about the concepts in general.
As such, I’d suggest trying the same during one of your tests. Don’t just do the practice problems and call it a day. Make an effort to be able to explain what you’re doing and the things you should be looking for in order to solve certain questions.
What is the one thing in mathematics or physics you feel completely comfortable with? In other words, which concept do you find you have grasped to such a degree that you’re able to get a good grasp on any problem concerning it? For myself, the mathematic concept that I feel pretty comfortable with is the idea of tangent lines and planes to certain functions (multivariable or not). I’m not saying that I understand a problem within three seconds, but I’m generally able to figure out the solution without too much difficulty. In physics, the concept I’m comfortable with is waves. I’m able to write equations for the wave-like motion of various objects and phenomena, and I’m good at extracting information out of them. Therefore, that would be my most comfortable area at the moment.
(Of course, it may come as no surprise that I’m comfortable with these two areas because I’ve recently taken a class on them. As such, it’s not exactly surprising that I’m used to them.)
However, there are plenty of concepts which I am not familiar with. In mathematics, I always get caught up when I have to deal with chords in a circle or if I have to use hyperbolas and other not-frequently used functions. In physics, the concepts of energy, torque, moments of inertia, and other physics related to rotating bodies can often throw me for a loop.
The reason, I think, is that I never had a full, thorough explanation of most of these concepts. As such, I’m always operating on a level that includes a bit of knowledge, but not necessarily one that makes me confident that I know what to do. In other words, I have a shaky foundation. And, as we all know, a shaky foundation has consequences for later on. When we aren’t comfortable with a subject, the problems will only resurface in other concepts later on.
To combat this, I’m setting myself a goal of filling the holes of my knowledge and finding those full explanations that I might have missed at different points in my education. That way, I’ll be improving my foundation so that I can actually move forward and not be so confused with concepts.
If I can offer one piece of advice, it is this: don’t settle for half-explanations. When learning, make sure that you understand the concepts to a point that you are completely comfortable with the ideas. If you have questions, make sure you ask them. Don’t worry about seeming “dumb” or not knowing enough. What’s more important is that you understand, and nothing else.
Taking Out The Pencil
I’ve already mentioned this before, but there are more and more high-quality shows and sites on science and mathematics out there. Whether it’s popular science like what you’d find on Nautilus, or mathematics channels like 3Blue1Brown, there’s a lot of potential to learn science and mathematics in a way that is both informative and beautiful.
(I say beautiful because I am a person who absolutely detests reading text or watching video with low production value. That’s why I try to make my site as pleasing as possible to read, and is also why I tend to read most of my feeds on my chosen RSS app. I like looking at content in a pleasing way.)
With that being said, I’ve noticed a disappointing trend in the way I read and watch new content in science and mathematics. Instead of taking out a piece of paper when some kind of equation or relationship is being explored, I’ll tend to take what the person is explaining at face value and continue watching. I do this primarily because it’s much easier to just listen and not work through the conclusions myself. In essence, I tend to be lazy.
However, I’m not completely blind to this. I know that it’s not the best way to learn new topics. In my experience, the best way to retain new information I learn is to actually work with it. It’s not enough to passively absorb it. The real learning occurs when you work out the relationships for yourself on a piece of paper, sometimes struggling to get the answer, but learning all the while.
Science can be great to consume as just a qualitative affair. That’s what you’ll get from reading popular books on science. Usually, a bit of history is mixed in with the author waxing poetic about science. These stories are usually some of my favourite, but they can also be misleading, because they only show you the surface of the science. Therefore, one might mistakenly think they understand a particular bit of science, when really they only understand the outcome or result, as opposed to how it actually works.
I want to try and push back on that a bit. Now, I do my best to not be a passive consumer of science, but someone who is engaged as the author goes about explaining. When I watch a video on mathematics and I see that it is moving too fast for me, I stop the video and work out what I’m missing. I don’t simply move on and tell myself that I’ll figure it out. I’m sure I could do that and be able to follow much of the rest of the video, but the consequence is that I’m not actually working with the idea.
Next time I encounter something I’m not entirely comfortable with, I’m pausing and actually attempting to work it out, because I know that this is how I’ll learn.
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.