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.

Pairing Simple Examples With Complex Machinery

More complicated than necessary.

Teaching a new concept within mathematics or science isn’t easy. It requires taking students outside of their comfort zones to try and understand how we model phenomena that is more complex than previously seen. This tends to require new tools and techniques, which means students have to shed their old tools in favour of these new ones. This can leave students disgruntled, particularly those who were attached to the old method of solving problems.

While teaching a new concept, one often revisits an older problem that didn’t require these new tools, and shows that the new tools can also solve the problem. This is done quite a bit, because there is a limited amount of time during a class, and older problems can be worked through more quickly. As such, they are good candidates for the first example of a new tool.

However, a problem can occur if a teacher only uses these older examples and doesn’t move on to newer, more difficult examples. The reason is that the students will see new machinery being introduced for no good reason. They will ask, “Why do I need this new machinery to solve a problem I can already do without these tools?” If the new tools aren’t vastly more efficient, this can be a valid point from a student, even if the teacher knows there are other reasons to learn this tool.

For example, consider classical mechanics. The first classes of classical mechanics involves vectors and calculus. One has to keep track of where the forces point at all times, and deal with each component separately. It works, but as the complexity of a problem increases, using this vector calculus method of solving classical mechanics problems becomes both tedious and somewhat intractable.

Then, one learns about a new technique to solve mechanics problems. This is the topic of Lagrangian mechanics, where vectors are nowhere in sight. Instead, generalized coordinates play a big role. The important point is that this method is much better than vector calculus in general. As such, it’s easy to showcase the power of Lagrangian mechanics when analyzing the motion of a bead on a spinning hoop (for example), which would be a messy affair using vectors. As a student, I thought, “Obviously, this is way better than what we were doing before. Lagrangian mechanics is a great tool to know how to use!”

But Lagrangian mechanics isn’t the endpoint. After learning about the Lagrangian and the Euler-Lagrange equations of motion, the Hamiltonian is used. For myself, this is where things got murkier. The Hamiltonian approach to mechanics was described to me as a more “sophisticated” approach to classical mechanics. I believe that, but in my eyes, the practicality of the Hamiltonian was lost on me. It was an alternative way to use the Lagrangian, but it didn’t feel any better. I wasn’t sold on why this new tool was useful.

I think the reason came down to not having any examples where I could say, “This is where I need to use the Hamiltonian1.” The Lagrangian method was adequate for me, so I stuck with it and paid the Hamiltonian method little attention.

This is an issue that can arise when teaching a lot of mathematics. We develop a tool with the students, they get better with that tool, and then we tell them, “Hey, there’s actually another tool you need to learn, and it will be useful.” Then we fail to deliver on this promise when we rely on simple examples and past problems to showcase the new tool. That’s not enough! If we want students to be on board with the new tools we teach them, then we have to push the students’ boundaries a bit. Put them into a situation where the old method fails, which will force them to try the new tool. Show them that this is what the tool really excels at, and that there isn’t a better way to solve the problem. Note here that I’m not suggesting that we tell students this is the way or the only way to solve a problem. Rather, we should be able to come up with examples where the “reasonable” strategy to solve a problem is with this new tool. If you don’t do this, you will get students who stick with the old method of solving a problem because it’s the method they are used to, and they won’t get to see how good another method can be. Your job is to at least give students a glimpse of that alternative way.

There’s one last important thing that I want to say: I despise test questions that “force” a method of solving on the student. There is only one good side to this, and it has to do with the fact that forcing students to use a certain method will give them a hint as to how they can solve the problem. Apart from that, I don’t think it’s productive to force or insist that students use a particular method to solve problems. Of course, there’s nothing wrong with having students solve problems in a particular way during an assignment (after all, that’s for practice). Tests, on the other hand, should be left wide-open in terms of solving strategies. A test should seek to answer the question, “Can a student use appropriate steps to get to the answers?” I realize that there’s the issue of covering the whole curriculum, but then I would suggest writing problems that steer students into a particular way of solving, but without forcing them to. This way, if the tool was introduced in a way that made it clear why it existed and what kind of problems it simplified, the students will be able to make that connection and solve the problem during tests.

  1. I do realize that the Hamiltonian is used a lot in quantum mechanics, but the context of what I was learning in class wasn’t quite the same. The Hamiltonian in quantum mechanics is used to solve the Schrödinger equation, which isn’t quite like the procedures used in Lagrangian mechanics.