Reading a Solution

When you have a lot of homework, it’s awfully tempting to just look up the answers. After all, you’re only doing it because you want to get other homework done, right? And if you take the time to read through the solutions, it’s good enough. You aren’t just copying down answers. You’re following the work that is done.

While the above situation sounds good in theory, in practice it is a terrible way to try to learn. It’s easy enough to find solutions to mathematics and science problems online, but finding them obviously diminishes the value of the problems.

If we compare the learning that happens due to working through a problem and the learning the happens due to reading a solution, I’m sure our intuitions will be correct. Working through a problem is a much better use of our time, even though it isn’t particularly efficient.

We’ve all been there: gazing intensely at a problem, frustrated that the answer won’t present itself to us. Then, usually just a few centimetres away lies a device that can find the solution for us. Why struggle when you can get through the small hurdle and continue on the problem?

My answer: tests.

Sure, it is nice to be able to look up just a “hint” to help you complete the problem that has been frustrating you for the last half an hour. However, the issue is that the “hint” most likely gave away the crux of the problem. Rarely are problems created only to test your ability to calculate one thing. Instead, they’re usually made to tackle specific concepts and make sure a student can recognize and deal with them. By turning to hints and help every time those points of confusion are reached, you’re robbing yourself of the skill needed to come up with an insight during a test.

Let’s consider an even more extreme idea: only reading a complete solution to a problem instead of even attempting it. While I am certain you will understand the solution (as you should, since that’s the point of a well-written solution), your ability to have gotten to the answer without any help has not improved. Instead, you’re fooling yourself into understanding how to do a problem, when really you only understand someone else’s solution to a problem. Therefore, reading a complete solution will make you think that you understand everything, but it’s more likely that you don’t know how to do it on your own.

Still, these online resources are good for something. I use them to verify my answers when I am done working on a problem. I rarely do what I described in the preceding paragraphs, because I know that when I reach a point of difficulty on a test, I can’t just magically skip to the next line of my solution like I can do when reading a person’s full solution. That blank space needs to be overcome first, and making the effort to practice writing full solutions by yourself without help is the key to being able to solve problems on your own.

Don’t just read solutions. Get good at making them.

Expertise Does Not Imply Teaching Skills

One of the incorrect assumptions I’ve long made is about teaching. Basically, I would get excited by the prospect of teaching some sort of concept to another person, and so I would work very hard on the presentation. However, there would inevitably come a moment when I’d realize that I wasn’t actually an expert in what I was talking about, so I would decide to stop the project.

My thinking went something like this: if I couldn’t be an absolute expert on the subject, then I couldn’t teach it. And obviously, since I am just a student and it is very difficult to be an expert in anything, who was I trying to explain concepts to people?

This crippled me for a while, and it still does (to a certain extent). The difference is that I’m now more aware of the truth, which is that teaching means bridging the gap between one level of expertise (a higher one) to the other (slightly lower, who is looking to learn). Therefore, the only requirement to be a teacher to someone is to know more than they do.

This is truly a new perspective on how to view teaching. It’s not about a bunch of students who know nothing that are taught by these experts who know everything. Instead, it’s about teachers sharing what they know in the hopes that the students can get to the same “level” as them.

Additionally, teachers don’t necessarily start as experts. Imagine you wanted to drive from your home to a new place. The first time you take the route, you’re constantly unsure of yourself, continually checking the roads to make sure you’re still on track. Now imagine trying to show that person the route after that first day. It will be somewhat difficult, right? However, if you’re asked to show them after doing the route a hundred times, there’s a good chance you’ll be able to show them the route with zero difficulty. Repetition breeds expertise.

The problem is that students don’t see this repetition. Unless they get a teacher who is just beginning, the teacher already knows what they are doing. Therefore, it appears to the student as if the teacher has no difficulty at all with the content. If a student then wants to teach (not necessarily as a profession, but perhaps just by writing articles, or tutoring, etc.), the illusion they’ve seen is that teachers are experts in their subjects, therefore they need to be an expert as well.

I can attest to this experience. I looked at my teachers, as well as other science educators online, and couldn’t imagine myself being in the same league as them. Looking at their content, I seemed so unsure in my information about science and mathematics, while those I looked to seemed confident. This stopped me from thinking I could be as good of a teacher.

If there’s a piece of advice I could offer, it’s this: don’t give up on teaching/education because you don’t feel like you have enough expertise. If you’re really passionate about teaching, the knowledge of the material will come in time. It’s a matter of repetition. Keep at it, and you will get better. As long as you’re teaching someone that is one “level” below you, consider yourself a teacher.

Repeating

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.

Fluency

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.