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.

Alternative Perspectives

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.

Interconnecting Histories

Through secondary school, I had to take history classes. During them, I learned the history of Canada, and how there were a bunch of important wars, conflicts, and political moves that brought us to where we are today. I took classes on this for four years, and throughout all of it, the theme was the same: war between the French, British, and the native Americans, and how both the French and British flourished while taking over from the native Americans (something that is often only brushed upon in school, which is odd since it really is a place for ripe ethical debate).

What didn’t happen was a discussion on the scientific breakthroughs of the centuries we studied (roughly 1600-1980). We didn’t learn how the ingenious development of electrical power spread out through the world, nor did we learn how the Copernican heliocentric view of the solar system changed the way people looked at Earth with respect to the rest of the cosmos. We didn’t study the engineering breakthroughs of telecommunications, and how they changed the way information is transmitted around the world. We also didn’t study anything about ancient history, by which I refer to the history of life itself. None of this was studied. It was only war, trade, and politics, and for just one small part of the world.

Of course, I realize that there is only so much material that can fit into a course, and that the total history of our species is long. However, I feel like we do ourselves a real disservice to only focus on a couple of points like we do in secondary school. I fear we give students the impression that what is taught is our (local) history, and we fail to give them the bigger picture. Those other connections I mentioned above are definitely important ones, and I feel like at least a couple deserve to be up there with the material that is taught.

When I reflect back on my experience in history classes, I remember a lot of dates, names, and places that had to be memorized for tests. I also know that the idea of the course was to give students an understanding of the big picture through essay questions, but often they didn’t capture how wonderfully exciting history could be. I think learning about a much more global history that incorporates science into the fold would have generated that excitement more than the order of the provinces of Canada joining the official country.

I don’t know if it is realistic to expect the curriculum to be updated to fill this function. Perhaps this sort of thing could take over one of the optional classes in later years. I feel like I’d be able to do something with this idea, because showing the wonders of science (particularly, when you only have to dive into the qualitative aspects), students will be captivated. There’s always more history behind the broad strokes that are made by curriculum history, and I think many interesting science stories lurk behind them, just waiting to be presented.

Personally, I’ve only begun on this journey a bit. I’ve read some history books on science, but not many. I know that there are many more things left for me to learn, and that’s why I’m going to discover them. I want to make sure that other students think about this opportunity as well, and not like they know their history because they passed four history courses in secondary school.


It’s no secret that if you walk into a classroom at random (but more specifically, a science or mathematics classroom), you will see a lecturer up front, with a bunch of students listening and taking notes. Sometimes, it might even seem like the students are barely paying attention to the words of the teacher, preferring to just copy what is on the board or screen.

I’ve heard many people criticize this aspect of school, saying that educational institutions have transformed learning into a state of copying what’s up front and not even thinking during lectures. I’m sure you’ve also seen the odd student in a classroom who isn’t taking notes. Often, they just don’t care about paying attention in class, but other times, they feel like they learn better from only listening.

I’ve thought about both of these approaches a lot, and I can’t completely come up with a perfect answer to which one I think is better. It’s probably tradition more than anything that made me take notes through each year of school. I still do, in almost every class.

Critics of taking notes might say, “But you can get all the stuff the professor is saying from books! Lectures are for listening and absorbing the way people think.” (I’ve gotten this line of reasoning as I read Richard Dawkins’ An Appetite For Wonder.)

I agree with the general idea, but I think this misses a crucial aspect of what writing notes does, at least for me. When I write notes, I don’t necessarily write everything down from what the teacher does, because some of it isn’t important. What I do write down is usually different laws or equations, and their derivations. To me, this is useful to have. Yes, I could get it all from a textbook, but writing it down in my own notebook allows me to make the information even more dense. I don’t have to go about explaining all the things that are obvious to me. I can jump right ahead to the heart of the matter. This way, I rarely have to flip through a textbook to find what I want, because I have it in my own writing.

The added benefit of doing this is that I can write down explanations that I need while looking at a concept, which others might not find useful. Since I’m writing notes while the lecture goes on, I can give myself little reminders here and there to help me out for later. This may seem trivial, but it can be a big help for when I look at my notes later (which I do, though Dawkins says he never did).

Additionally, I think there’s something to be said about remembering things you’ve written down. I forget things easily, but it seems like just the act of writing them down helps me remember. I often don’t even need to look at the reminders I write for myself because I remember writing them. I think it’s very possible that this same sort of thing happens when writing notes in class. I don’t have research on this, but writing the ideas down seems to help me internalize them, much like actually doing problems is more useful to me than simply skimming through them. Therefore, I think taking notes isn’t quite as bad as it may seem at first glance.

The way I take notes is simple: I write down equations and lines of arguments. I usually end up writing definitions, but only insofar that they help me understand what is next. Crucially, I try to participate often in class, which means I’m not just writing down notes and staying consumed in the act of writing all the time. I try to stay connected to questions being posed, and I also try and think of my own questions.

My advice is this: if you feel that you’re doing fine without taking notes, there’s absolutely no need for you to start. Personally, I find taking notes in class allows me to skip having to go and look at all the equations and theory later while I’m doing homework. I have it all in front of me in my notes, which are usually easier to read at a glance. If you do take notes, keep in mind that you don’t have to write down everything that is said and put up front in class. A lot of it is context, and the key is to give yourself enough context to remember the content of class, but you don’t have to recreate it. Notes are fine, just don’t go overboard.