During my science education at CÉGEP, there was a lot to learn. In two years, I took five physics classes, four mathematics classes, two chemistry classes, and a biology class. This was in addition to many other complementary and language classes I had, which meant there was a lot of content to get through over the years. Consequently, there was an impetus to prioritize work by looking at whatever was coming up in the next week. Once the material was covered, it could safely be forgotten.

I’ve written about this last week, but I want to discuss something a little different today: revisiting material. This is something I don’t see being done very often. Throughout all my years at school, I don’t know anyone in particular that would revisit their old material. Of course, I may just not have noticed (as others could make the same claim for myself), but it demonstrates how this isn’t necessarily an aspect of one’s education that frequently occurs.

You may have heard that people say that re-reading a book always gives them new information or a different perspective on the story. I believe learning is much the same. If we only go through material once, there’s a fair chance that many aspects of it will be lost on us. However, if we revisit a concept many times, there’s a better chance that we will learn to appreciate all of its nuances.

Have you ever wondered why a teacher is so good at their subject? It’s usually because they’ve worked on the concepts many times over. The best science and mathematics teachers I’ve had are the ones that have been teaching the material for years. Because of this experience, they are able to recognize and solve all sorts of problems, merely due to repetition. This is the exact same thing as teaching a small child the principles of addition, subtraction, multiplication, and division. Since these are the basic operations in mathematics that are learned at a very young age, it becomes basically second-nature once one has moved on to more advanced mathematics. Therefore, one’s experience is great enough doing these operations that there is seldom a problem.

For myself, I’ve recently been brushing up on my own skills in a subject that has been a little bit distant in my mind: linear algebra. I know that I did well in the subject, but there were always ideas in the class that made me a little confused. This list includes linear transformations, Markov chains, and working with the transpose of matrices. These caused me some trouble in my classes, and until now I’ve basically ignored them. I didn’t try to forget about them, necessarily, but I also didn’t try to understand these concepts.

Now though, I understand the value of revisiting ideas that gave you trouble. Even more, one should revisit ideas that one is already familiar with. This is important because it will help you remember ideas you may have forgotten or not learned well, helping your mathematical skills in the long run.

If you’re interested in re-examining some ideas from linear algebra, I recommend starting by Grant Sanderson’s new video series on linear algebra. He makes beautiful animated mathematics videos, and they’ve helped me solidify some of my understanding which was a bit hazy in linear algebra beforehand. I also like that they are brief yet still give deep insights into the intuitions behind different linear algebra concepts. I’ve found that this is a good method for me to revisit material because it isn’t as rigorous as the first time around, yet refreshes my brain enough that I can learn some more on the subject.

As science students, many of us dread to even think about the word “repeat”. This word has a certain implication of requiring one to re-do a class, which is something no student wants to do. Therefore, we focus on doing enough to get through a class once, and then not needing to know too much about it anymore. This can be done with both good and mediocre grades. I personally did well in linear algebra (in terms of grades), yet I still did well to revisit concepts in linear algebra. The urge to forget information in favour of new information is strong during our education, which means it’s imperative to go back and look at things you’ve learned a while back in order to keep it fresh.

Don’t do things once and then move on. Make sure you take the time to look back at the material from time to time in order to stay sharp.


As I’ve written about many times here before, I’m a big proponent of understanding why one is using a certain strategy or procedure during a problem. In my mind, understanding the essence of the process is a great way to learn. However, this comes with a huge caveat, which is rarely talked about. The piece that sparked my reflection is an article on Nautilus, where the author explains how she went from being a translator (and someone that wouldn’t even look at science and mathematics) to reinventing herself and becoming a professor of engineering.

This is obviously a radical transition. Think of someone you know that is in the arts and is definitely not a science person. Then, picture them becoming an expert in material that you’ve dealt with all your life. Undoubtedly, this would be a little unsettling. The mental model you’ve created for this person would essentially be broken, becoming a terrible approximation for that person.

The article is a great read (and I recommend you do read it in its entirety), but the crux of her explanation for how she was able to transform so well was through focusing on fluency rather than conceptual understanding. Repetition and examples were the elements of her training that were responsible for her improvement.

I’ll be the first to admit that my stance has been largely the opposite of hers. By knowing the concept well, I believed (and still somewhat do) that you can solve a wide variety of problems that you wouldn’t be prepared for if you only knew about the brute calculations. Therefore, repetition and examples were only useful insofar as the core idea is understood.

I think this is a good way to go, but I’ve accepted that repetition isn’t as bad of a teaching tool as it might seem. I shouldn’t be surprised, either. After all, the main thing that I preach when other runners ask me how I’ve gotten to be as fast as I am has been to say through many kilometres of training. That is repetition in action.

I suppose I come from a place that is more about getting an idea. If you can calculate something but you don’t know what it means, are you really learning? This is the kind of question I’d ask myself. And I’d always arrive to the same answer: conceptual understanding is always best.

However, I’m not so sure anymore. The best example I can think of is this: imagine you’re in class with an amazing lecturer. Everything he or she says is understood by you perfectly, and you’re hanging on to every word. At the end of the session, you’d be able to confidently say that you followed everything they said. Would this person be able to do the problems assigned?

As much as I want to say “yes”, I think I’d be deceiving myself. The truth is that this student would probably struggle on the problems. Not because they are incompetent, but because they aren’t fluent in the process. They weren’t lying when they said they followed every word the professor said. They simply couldn’t apply it to their actual work because they had never practiced it.

How many times do you listen to a teacher and think, “Oh yeah, I understand what they’re doing. This is easy.”? If you’re anything like myself, this happens a lot. However, the assignment will then come and I’ll be confused. It made so much sense when the teacher did it, but starting from scratch with no teacher is much more difficult.

The reason it’s difficult is that you’ve never done it before. Taking derivatives and integrals of most basic functions aren’t difficult, but they are if it’s the first time you’ve done them. Therefore, it is hardly surprising that practice and lots of repetition is the key to improving one’s skills in taking derivatives and integrals. You won’t necessarily gain conceptual understanding after the hundredth integral, but you’ll definitely get used to how they work and how you should approach solving them.

As the author points out, this applies more broadly then just mathematics and science. In general, one needs to practice over and over again to get better. It’s as simple as that. Understanding the concept is definitely an important part of one’s education, but it shouldn’t be the only feature of one’s curriculum. Instead, there should be a place for both conceptual understanding and lots of practice. Before this article, I’d perhaps be in favour of more conceptual understanding and less robotic procedures on how to calculate, but I think that’s doing a real disservice to students who are going to need those practical skills.

As usual, relying on one method of learning is recipe for failure. Mix both conceptual learning and practical learning in, and students will be better suited to understanding mathematics as well as knowing how to do the actual work.

Role Reversal

Like most people, I enjoy engaging in activities that I have a good time in. Seeing as many of my preferred activities are sports (though I do hold mathematics, science, reading, and writing in high regard as well), I like to be good at what I do as well. Therefore, the kinds of activities I do usually are ones I am good at. I’m good at basketball, so I play basketball with my friends. I’m good at running, so I enjoy running with others. I’m good at mathematics, so I’ll frequently help people out in their studies.

Because of this tendency to do things I’m good at, I’m not often shown the other side of the equation: the side where people aren’t good at the activity. I can only empathize so much when I talk with most of my friends about grades, because only a few get grades that are similar to mine. In this same way, I can hardly imagine what it’s like to be less than stellar at a sport, because I’m decent at most that I participate in. This isn’t to brag, it’s simply to say that I am usually towards the “stronger” side of the spectrum.

Of course, no one is good at everything, and this reality was brought into sharp focus once again while doing an activity with my friends. It was an activity that was enjoyable, but I simply am not great at it. Therefore, I was the weaker link on a team.

And honestly, it wasn’t fun. I didn’t enjoy being the worst on a team, but it was an interesting experience. It gave me the chance to see what a team or an activity looks like from the perspective of those who aren’t as good at it. It made me think about how I behave when I am in a position of being better than others at an activity. I hope to be more humble and treat those others not just as liabilities (such as in a team sport, perhaps), but as participants who want to succeed just as much as I do.


A refrain I often hear from people regarding my scientific and skeptic mindset is how cynical I’m being. They tell me I’m just shooting down anything that I cannot see proven in an equation or carried out in some sort of lab experiment. Since I can rarely get that kind of validation, I always seem to hold negative views of new ideas.

To a certain extent, I can see how others see this. My first response to new information is usually a way of invalidating it. Honestly, this does seem quite cynical. However, that’s because we often use terms interchangeably, and this is most definitely the case here.

According to, “cynical” is defined as: “distrusting or disparaging the motives of others, bitterly or sneeringly distrustful.”

This is obviously not my intent when responding to certain claims, though I’m sure it has occurred throughout my years of hearing crazy “science” from the media. The real word that should be used is skeptical.

According to, “skeptical” is defined as: “Having an attitude of, or showing, doubt.”

This is definitely the word I’m looking for.

In my eyes, the difference between these two words stems from the negativity the former expresses. When I hear a new scientific “fact”, I don’t have any negative view of it. I simply question its validity. That’s not particularly negative. In fact, I would argue it’s neutral, because questioning the fact simply means I want to know more about how this fact relates to other scientific ideas.

I don’t necessarily need _a controlled laboratory experiment to think something is true. I can make a certain subset of inferences about the world that are probably more or less accurate for my purposes. However, my skeptical side kicks in as soon as I hear something about a “wonder drug” or the “one simple thing you can do to be happier”. These statements are usually under the guise of being scientific, but in reality they are misguided at best and pseudoscientific at worse. When I hear _those kinds of statements, I try not to reason it out myself and instead ask for evidence. If something is as powerful or great as someone says, then surely there is a wealth of evidence to support it. If not, then there is something lacking in the statement.

A quote that I particularly like from Carl Sagan (at least, he is the prominent figure that I know of who said it) is, “Extraordinary claims require extraordinary evidence.” It encapsulates the idea that skepticism for something that is particularly incredible needs a lot of evidence. Otherwise, we could be swayed by anything just because someone is persuasive. With this mindset of skepticism, you give yourself a way to trust statements because of the evidence, not because it “sounds” good.

On a related note, I’ll often get accusations of being close-minded because of my skepticism. This usually results from some really enthusiastic person not understanding why I can’t seem to ever be convinced that something they say is true. What they won’t say is that the claims they always make to me are extraordinary, and come with little explanation or evidence. When I prod for evidence, they switch the onus onto me to go read and find out for myself because they didn’t go through the whole study or article.

Unfortunately, that’s not how good conversation goes (at least, in my experience). If one wants to state something, they’re going to need some evidence or else it will be dismissed pretty quickly. But even if I do accept to go read further about it, the conversation will continue as if the statement was true.

This is a problem, because the conversation has shifted dramatically from what we know (or are reasonably certain of), to what we don’t know. Furthermore, I will then be chastised for never keeping my mind open to new possibilities.

My response: I’m always open to something new, but I need to see some kind of credible evidence. And unlike what many people think, witnessing an event is not a great tool to make these kind of claims.

In the end, I’m skeptical because I understand the rigour needed to correctly do science. The scientific process is a process, and one of the steps in it is to repeat experiments. Therefore, having a one-off study isn’t necessarily convincing to me. Likewise, many studies saying the same thing won’t be convincing if they all have some kind of flaw in the procedure.

Being skeptical does not make you cynical. It forces you to look at evidence before you integrate that knowledge with the rest in your life. A skeptic will be open to change, but the change must be convincing.

And it just so happens that the best tool we have to make convincing statements about the world is science.