Pairing Simple Examples With Complex Machinery
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.
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. ↩
One of the most important things you can do when learning a new subject is attempting it with your best effort. While this sounds simple, so many people think they are too good for this step and skip it.
I was reminded of this at the beginning of last semester. One of my professors said, “When you read the chapters in the book, do the examples that are there. It’s extremely important that you do. It might be tempting to go through and read the solution of a problem and then say, ‘Oh yes, that makes sense.’ However, doing this won’t help you out in the long run.”
I couldn’t have put it better myself. This is exactly the kind of issue I run into all the time, and I know that it can affect others just as much. We have this idea that reading implies learning, but there are many ways to read a textbook and learn next to nothing about solving problems. This includes skipping through the exercises and examples, preferring to just read the solutions. This is good up until the point that you are faced with a question on your own, such as on a test. At that point, you may find that the problems are not so easy.
It’s simple to say, “Oh yes, this solution makes sense. I get it.” It’s much more difficult to think of what that solution is on your own. But this is what doing the examples in a textbook is for! They are there to allow you to think of the solution without a helping hand. The solutions are for after you’ve done this step.
This is so important, because I see people all the time take ten seconds to read a problem before saying, “I don’t know what to do.” Well, of course you don’t know what to do! You’ve barely had time to think about it at all.
The purpose of doing all these exercises is to give you a sense of what works and what doesn’t. As a consequence, you start to gain an eye for what a solution might look like. When I tell the students I tutor how I solved the problem they are struggling with, they look up at me in disbelief, as if they would never be able to do what I did. But they’re wrong. I didn’t do anything special. In fact, I did something I’ve done a bunch of times. That doesn’t mean I was as good when I was in the same year as them. It just means I’ve been at it longer and have put the work in to make those problems trivial.
It seems like a waste of time, but the real key to learning and succeeding in your studies is to attempt to do problems on your own, without referring to your text. I’m not suggesting you memorize every single formula in physics. Rather, I’m talking about learning to come up with your own strategies to solve problems. From there, it’s just a lot of practice and refinement. If you prefer to skip the work and read the solutions, this skill won’t be developed. You’ll always be looking for the “trick” to solve problems, instead of applying techniques. Therefore, it’s in your best interest to work through as many problems as you can. It will pay off in your future.
Permission to Learn
When you were in elementary school, did you get a say in what you learned? How about in secondary school? Odds are, you were given a set of classes, and your time at school looked something like this:
We all had these instances when we wondered, “Why am I learning this?” We were then either told that we had no choice, or that that there were benefits to learning that material that we couldn’t see right now. One of these must have seemed reasonable, because you continued on.
We then get to university, where the classes are more varied, yet we still have to take certain ones. Hopefully by this point you like the program that you’re in, so this doesn’t seem as bad.
My question to you is this. Have you ever decided to learn something despite it not being a part of a class? Did you ever decide to learn something just because you were curious?
If the answer is “no”, I encourage you to ask yourself why you haven’t. I’m sure that there must have been something that piqued your interest at one point. Why didn’t you follow up on it?
The point I want to get at here is that, now more than ever, we don’t need permission to learn something new. Sure, what you’re curious about might not be on the test in class. It might be unrelated. But you’re curious, aren’t you? That’s enough, and it’s why you need to learn on your own.
This doesn’t tend to be an instinct, because our education system doesn’t prepare us for this. From the get-go, we’re told what to study. Forget about everything else, it’s not as important to you. Focus on what you need to know for the test.
That’s fine advice if you want to get good marks, but it doesn’t help you grow. We have so many resources available now that you can learn pretty much anything you want. Why not use that opportunity to move away from the curriculum you got from school and learn something that interests you?
We can all use the reminder. We don’t have to be in a class to learn something new. Giving yourself permission to learn something new is a powerful first step to growing much more than you thought possible.
Motivating New Tools
One of the great things about mathematics is that there tend to be multiple ways to solve a problem. This implies two things. First, there is no “correct” way to finding a solution. As long as the approach is logically sound and produces the answer, it is a good method to solve the problem. Second, multiple methods imply a fastest method. This method might take less work, be more obvious, or even be simpler. Of course, keep in mind that this is a subjective measure of “simple”. One method that might be simple or obvious to one student could seem bizarre to another. It’s a matter of perspective, yet the fact remains that a given student will find one approach to a problem easier than another.
If one approach is “easier” than another, it lends itself well to developing new mathematics. In particular, a lot of mathematics that is taught in school (in particular, secondary school) is there because it solves a historical problem. (Whether this historical problem had any practical use is a different question!) In other words, a lot of the tools of mathematics are taught to students because they solved a problem.
In post-secondary education, the standard example would be of integral calculus. How much area is under a given curve, and how does this relate to the derivative of that function? These kinds of questions are answered with the tools of calculus, and have applications all over science.
The problem, from the student’s point of view, is that this isn’t how the new tools tend to be presented. Instead, they are given the new tools without the prior motivation, which can make learning these new tools seem pointless. This is even more poignant when students are learning techniques to solve problems they have already learned how to solve. I can hear the question: Why do I need to learn about a new way to solve a problem I know how to solve?
This question isn’t answered as often as it should be. Remember, before a student gets into pure mathematics (where the emphasis is more on proofs and results), mathematics is computational in nature. This means the student is learning how to use new tools to solve problems. While “real-world” examples are given, they don’t tend to capture the heart of why these tools are useful.
Here’s an example where the motivation for the tools is good. Students in physics start by learning the kinematic equations of motion, which hold when the acceleration is constant (such as the case of gravity on the surface of the Earth). Once students learn calculus, they are able to tackle problems in which the acceleration continuously changes. Armed with these tools, they can now address complex problems that their old tools were not equipped to solve. The use of calculus is clear to a physics student. It’s not that it let’s you solve equations of motion. It’s that calculus is able to model the continuous change that is present in a lot of our world. That’s the point of calculus, in a nutshell. How can we quantify change, even when it isn’t constant? Physics students get to experience this sort of motivation.
On the other hand, a lot of students don’t get to see the motivation underlying their tools. Instead, they are introduced to a new rule or concept, and then are assigned practice in order to get better at applying the rule. There’s little mention as to why these rules are as they are, or why these tools are important.
I want to note here that I’m not advocating for motivating mathematics with “real-world” applications. I don’t care if the mathematics that one learns has any inherent use, because that’s not the point (at least, not the whole point) of mathematics. What I am interested in is the question, “What made a mathematician so inclined to invent this mathematical tool?” That’s an interesting question that isn’t dependent on an external application.
Teachers need to find these reasons and explain it to the students. It will achieve two important functions. First, it will be easier for students to answer conceptual questions, because they will know why this tool was needed. By knowing the “why”, students associate mathematical tools with a specific purpose. The second function is that students will be more on board with a new method. Have you ever tried teaching a new method of solving a problem and got a lot of resistance from students who prefer the older method? That’s not an accident. Either both methods are equally good (in which case the students have a point about not wanting to learn a new method), or else the new method is better. But, you aren’t teaching it in a way that convinces them to come on board with you. The latter part here is crucial. If the new method is truly better, you need to show that to the students. I always think back to addition and multiplication. If you have a bunch of objects you need to count, we all now know that it’s so much easier to arrange them into a rectangular array so that we can multiply them. We would much rather arrange 800 objects into 40 rows of 20 and perform (40)(20)=800 instead of counting each object individually. The “tool” of using arrays with multiplication just blows addition out of the water! You need to replicate this experience with your students for all topics.
If you feel like your students aren’t understanding why a new mathematical tool is so much better than an older one, I would argue that it wasn’t presented in a way that made this fact clear. Sure, in hindsight it might be clear, but by the time students have this benefit of hindsight, they will have long-completed the class. In order to get them to understand how a certain tool solves a problem the “best”, presenting the historical need for this tool is a great way to go. It’s not that teaching things in a historical manner is the best way to teach, but history does give us some guidance as to what problems can motivate a new tool. Use this as guidance when you’re planning your new lesson, and chances are the students will be both more on board with you, and will have a better conceptual idea of why they need to learn this new tool.