By Convention

It’s game seven of the Stanley Cup Finals, and both teams are about to get on the ice. The camera hovers around them, and you notice that everyone has a complex array of fist bumps, arm movements, and rituals. Some even have smelling salts that they wave in front of them before the match starts. Superstition is rampant, and you roll your eyes as a scientist, knowing that it’s all nonsense.

Except that it isn’t complete nonsense. Sure, there’s no supernatural effect from any of these actions, but they do serve a purpose: convention. By doing the same thing every game, the players are able to calm themselves and focus on the match ahead. By repeating the same actions all the time, it becomes a new normal.

This is hardly a phenomenon of professional hockey, nor is it exclusive to sports. Convention is present in all aspects of our lives, and science is not immune to it. In particular, the field of physics is filled with conventions. You can barely wade into a few pages of a textbook without encountering some (often contrived) convention.

In and of itself, there’s nothing bad with convention in science. It makes things standard so that everyone has an understanding of how things are done. Unfortunately, physics is riddled with historical conventions. The problem with history is that a lot of it was wrong or misguided. As such, we end up having terminology that doesn’t even reflect the actual phenomena, such as cosmic rays, which aren’t photons but are particles.

This can be confusing enough, but there are also historical conventions in terms of notation, even when legibility is reduced. By honouring historical convention, you end up with ridiculous situations like this:

I take no responsibility for the accuracy of this comic.

The sane response would be to say, “Alright, let’s get rid of all this bad notation and terminology. Let’s update to the twenty-first century!”

However, there’s an additional problem. You notice these crazy conventions when you’re a student. It’s obvious then, but your professors say that you just have to suck it up and get used to it. Not having much of a choice, you accept it. Fast forward ten years, and you will end up telling your students that they need to get used to convention. After working with the convention for years, you won’t find that there’s anything strange with it, while the young students will find it weird.

Just like that, the cycle continues.

Breaking these historical conventions isn’t easy. It’s a mixture of not wanting to ruin the old notation in the literature and being “used” to the normal way to do things.

I have a different view. We are scientists, which means we shouldn’t be bogged down by historical baggage that has pervaded the field in the past. If a new convention is needed, we should step up to make that change. It might be slow, it might not work, but it’s worth trying. Not for us, but for our students later on.

I know that I won’t be writing those Greek indices on my tensors anytime soon.

Where's the Surprise?

Do anything enough and you will get used to it.

This is an unfortunate truth in the realm of mathematics and science education. As teachers and tutors, we know our subjects well. We know the punchlines and the proofs. We know the end result of any lesson. Through years of working within a subject, there is little surprise about how things fit together. Everything is planned out, which means there is a lack of novelty with respect to the subject.

This is inevitable. Ideas won’t be new and shiny forever. Instead, they become like old acquaintances, familiar players in your classroom. Because the ideas are less “fresh” in our minds, we can start to become disillusioned with the results. Of course the Pythagorean theorem holds, how could it not? It’s obvious that the shortest path between two points in a plane is line! These are just two examples, but they illustrate the fact that the novelty wears off.

Triviality implies lack of novelty?

When there’s a lack of novelty, slipping into a routine is easy. I’ve seen it with many teachers. They can teach their subject just fine, but the enthusiasm is gone. Classes go by monotonically, and students can feel the lack of energy.

We need to do better. We need to bring the element of surprise back into the classroom.

When is the last time you were surprised by something you learned? Simply the fact that a thought jumped out at you as you are reading this shows how great a surprise can be while learning. It creates an experience that sticks in the mind of the student, and they can remember it for a long time. I think we can agree that this is preferable to taking down notes day after day.

We need to instill more elements of surprise in our teaching of mathematics and science. A byproduct of surprise is delight, which will make students both enjoy and remember your classes more than others.

“Wait,” you might say. “That sounds great, but there’s nothing surprising about the topics I teach. They’re all basic!”

If that’s your response, you aren’t working hard enough.

I didn’t say that crafting surprise was easy. I’m arguing that it’s necessary. As teachers, we have the creative control over how material is going to be presented (to a certain extent). This means we are in the possession of the idea, the punchline, and the formal result. It’s up to us to mix these ingredients together in the right proportions to create lessons that are surprising. Yes, we can just give the results to the students one after another, show them example problems where they calculate a number, and finish the topic with a quiz. Or, we can work harder to create learning experiences that deliver these same results and equations through surprise.

Remember, you know everything beforehand. You’re not the audience for this surprise. It’s the students who don’t know the punchline, who are blank slates. You wouldn’t tell a joke by giving away the ending, would you? But this is exactly what we do in a lot of our mathematics and science classes. We forget to build up to the moment of surprise! We waste countless opportunities to surprise and delight students with results that they would never have guessed. Class gets reduced to taking notes of a bunch of results, and there’s no context to them.

This is more than a complaint about our educational system. It’s an observation about learning in general. Surprise cements the memory of a lesson into the students’ minds. It isn’t surprising to go to class, take notes, and listen as the teacher goes through a little bit of theory and works out problems. But it is surprising if you work through a bunch of unrelated concepts and then find that they all share this beautiful link in between them. Of course, you knew this as the teacher, but the students don’t know. This makes the reveal so much more powerful than if you told students right at the beginning.

As the teacher, the students have a natural inclination to take anything you say as the word of law. If you write an equation on the board, the students will all bend over and write it down, even if they have no clue what it means. This is a terrible way to go about teaching. Yes, it transmits the information in an efficient manner, but it doesn’t mean the students will understand it. On the other hand, if students are working for a long time on smaller cases of a larger problem, there’s a good chance that the introduction of the magical relation that captures exactly what they are trying to do will be more surprising.

I’m not saying that you should just withhold the information from students. Forcing them to slog through problems isn’t always a good thing. But, you have all the ingredients concerning the topic, so use them well! You get to choose how they are presented, and this is what makes all the difference.

Think the theorem you are teaching today is too boring or bland? You have to be more creative with the presentation! Every lemma, conjecture, result, and theorem has some surprising connection or insight associated to it. To the students, topics aren’t so obvious when first learning them. Use this to craft your lessons, to weave surprise into the classroom atmosphere. It could be as simple as asking students what they think will happen during a science experiment, or perhaps taking a few minutes to set up some counter-intuitive scenario. At minimum, you need to avoid teaching your class in the same manner every single day.

By changing up the way things are done in your class, students will have to be nimble and ready for surprises. There is no way I can undersell the important of this, so I will say it again: education in mathematics and science needs more surprise. Efficiency is great, but we have to remember that students are served better if we give them learning experiences that stick with them, versus paying lip service to the fact that the material was covered.

I know, this isn’t an easy wish. You already have plenty of topics you need to cover, and so little time during the year. What I ask of you is to try it, at least for a few classes. It’s not practical to transform every class into one that’s filled with surprise, but I recommend that students should get to have a surprise every time a new concept is introduced. I know that there’s at least one thing that you can say about the topic that will be surprising to students. If we want our students to be more engaged and enjoy our classes, it’s our responsibility to deliver these surprises.

Being Second Atop the Mountain

Doing research isn’t an easy thing to do. There’s a reason that not everyone is an academic. Trying to bang one’s head up against the wall of science isn’t most people’s idea of a fun time.

That being said, when you do get an idea of a direction it can go, it’s exciting. You start gaining momentum, and before you know it, you’ve gotten a paper drafted up. Soon, you will be able to publish it, inserting yourself into the scientific literature. You will be able to stake your claim on the metaphorical mountain and declare, “I was able to come up with this!”

Of course, that’s until you see this.

Second atop the mountain.

Suddenly, all of your progress seems like a waste of time. While you were busy carving your own path up the mountain, someone had already beaten you to it.

This is frustrating, to say the least. Depending on what kind of work you did, it might all be useless now. Hopefully, you can still salvage the work, make it more general. However, that doesn’t change the fact that some of your results (perhaps even your key results) aren’t new.

One might honestly wonder how this can happen, particularly now, when we have such great search engines that can index more information than we could possibly ever consume. Are you just lazy for not checking the literature beforehand?

The truth is a bit more complicated. As historians know, preserving documents into the future isn’t easy. This is definitely true when considering the fact that we’ve gone from paper to digital, which means that a bunch of papers in the literature needed to be digitized. This isn’t too difficult, but getting these older papers catalogued is. Even with the power of search engines, it can be a hassle to find older papers, since they aren’t catalogued well. You need to dive deep down the rabbit hole of references from other papers to locate it. At that point, you better hope that you can find a PDF of the paper, because a lot of the time there’s a paywall that you can’t leapfrog.

Suffice to say, searching for older papers isn’t easy. That’s a big part of the reason why I ended up “discovering” an older paper that had part of my project. It’s not that the literature wasn’t searched, it’s that papers can be buried several layers deep and hidden from view. This is compounded by the fact that physics is a dense field with a lot of people.

Still, it makes me appreciate the scientists who did work before we had most papers digitized. The problems now are niche, but before digital finding any paper could be a challenge. I imagine you needed to have someone who was skilled at scouring the literature to make sure that you were doing new work.

The moral of the story is simple. The literature is vast, and so you need to be careful when starting a new project. If you do find that you’re repeating older work, try to see if you can build upon it in someway. Mistakes happen, so don’t get into a rage if you find yourself treading up an already-climbed mountain.

Where Are Your Weaknesses?

Not too happy about factoring.

During one of my calculus classes in university, we were running behind in terms of class content. Within the last few days of class, the professor announced that the topic of Taylor and McLaurin series and expansions weren’t going to be part of the final exam, but he would hold an optional “extra” class for those who were interested.

I was interested, but I also live far away from the university, which meant I didn’t want to drive a total of eighty minutes for fifty-five minutes of class. So I didn’t go. I figured it wouldn’t be a big deal, since it was just a small topic in my calculus class.

Unfortunately, I was dead-wrong. As any physics student knows, being able to write a function as a power series is a very useful technique to know, and is used all over physics. The reason is simple. It can be difficult to solve the differential equations that are encountered in physics, and using a power series expansion can allow us to solve problems to a great approximation. Knowing how to use this technique to express a function as a polynomial is powerful.

Because I skipped that class, I didn’t get to see this topic in detail. As such, I continued through my education with only a rough idea of how it worked. This meant that when professors would tell us to expand a function as a power series in order to solve a problem, I would always be slightly behind, not quite sure how to do it. I knew that it involved derivatives and factorials, but it was clear even to myself that I wouldn’t be able to do it on my own.

The problem here is one that many students face. They end up not really understanding a concept, or miss it for whatever reason during the semester, and then go on with their education with this missing gap in their knowledge. That’s fine, but these gaps do show up later on. At that point, it’s usually more annoying to go back and learn the concept, so students either try to fake their way through understanding, or fail.

It doesn’t have to be this way. I’ve decided to put in some time to look at expanding a function as a power series, because I know that it’s an important skill that I need in my toolbox. It won’t be something I’ll figure out in five seconds, but it’s a good investment of time.

I am sure these same kinds of weaknesses exist for you. Perhaps there was a concept that kind of “slipped through the cracks” for you when you first came across it, and you just haven’t thought about it in a while. These are your weak points, and it takes honesty to admit that they are there. Furthermore, it takes a certain amount of willingness to say, “I’m not satisfied with acting like I know this. I want to really understand it.” It’s not easy, but it’s important. I will keep on beating the same drum: in mathematics and science, concepts build on top of each other. If you don’t have a strong foundation, it is difficult to learn about new concepts. It’s possible, but your understanding will be riddled with holes. If you don’t believe me, find a topic that you know next to nothing about, and then find some lecture notes or a textbook aimed for an advanced audience. I’m willing to bet that almost none of it will make sense to you. That’s because you don’t have the foundational experience necessary to jump into these resources.

If you want to get better at anything, it’s crucial that you identify your weaknesses, and then work to improve them. This latter part is just as important as the former. Saying you have weaknesses is one thing, but working hard to address them is a different challenge. However, if you are willing to put the work in, it is doable.

Don’t do like I did. Don’t miss a concept and let it go ignored for years. It will come back to haunt you, so you might as well put in the work to understand it.