Discovery and Delight

I’ve been thinking a lot about where one finds joy in a subject, and how the goal of educators should be to create situations in which this is most likely to happen. In particular, I’ve been reflecting on the way this is achieved in specific subjects during the likes of one’s elementary and secondary education, before choosing a career path to embark on. My question is simple. Are we doing all we can to deliver delight to students? Like always, I want to focus on my particular interests, which encompasses science and mathematics.

I think it’s fruitful to compare the two, because they both have a quantitative aspect to them, and a similar type of mindset is required for each.

From what I’ve observed, the science classes seem to produce much more delight in students than mathematics. I’m not saying that students chatter excitedly about their science classes, but what I’m saying is that there is a certain purpose to learning, where it makes sense to learn a particular example. Whether it’s learning about how machines are composed of simpler ones, or how the force of gravity creates parabolic trajectories for objects in free fall, the purpose is clear. It’s to explain the way the world works, and it’s not that difficult to get students on board with this goal and have them delight in the fact. After all, it’s quite fun being able to explain to one’s family why objects accelerate at the same rate, or why certain reactions occur faster than others. As such, I’d argue that there can be a delight in learning about science just from this one aspect, regardless of if the subject is physics, chemistry, biology, or the many other wonderful topics in science.

If I’m honest, I think this is why I pursued science in the first place, and in particular, physics. It was a subject in which I thought it was so cool to be able to understand the world with only a few principles such as Newton’s laws, and so I was motivated to continue learning. What I want to point out here is that, yes, I did have some good science teachers, but the school I went to didn’t have a bunch of resources, so it’s not like I was in a top-tier science program. Instead, it was more that the subject itself (and the way it was taught), allowed one to become interested without too much extra effort.

Mathematics, on the other hand, wasn’t something that many regarded as enjoyable. Even I, who did well in my mathematics classes, didn’t particularly love it. I thought it was an interesting subject, but I never thought I wanted to study it later on after secondary school. Instead, mathematics was something I did to get better at analyzing situations, but I wasn’t there to learn mathematics because I delighted in it.

So why was that? Why was mathematics not filled with as much delight as there was in my science classes?

I think the answer is simply, purpose.

I’ve worked with many secondary school students, and I’ve heard this over and over from them. “Why are we learning this?” They don’t understand why the things they are learning are required of them. The emphasis is important here, because it’s not that none of this is inherently uninteresting. Instead, it’s because there’s no reason why it should be required of the students to learn a lot of what is taught.

My perspective on this is as follows. I find that within mathematics, a little bit of everything gets taught, but there’s no theme or sense weaved in between the subjects. One goes from geometry to solving equations to proofs without there ever being a why. Instead, there’s always this underlying sense that there is more to learn, but without the motivation. As such, it becomes difficult for the students to latch on and take delight in the learning process, because they’re being told that all of this skill in mathematics that they are building up has a purpose and that it is applicable in the “real world”, but the truth is that mathematics isn’t about applications. It’s about the connections between structures, the creativity of the mind, and the consistency of starting with axioms and building those structures. In the end, that’s where the delight in mathematics comes from. It’s not from the “context” paragraphs that are slapped onto word problems to give them what is supposed to be real world application. The delight comes from the mathematics themselves, and this is the fact that needs to be realized by all of us who teach. There is a joy in the pure abstract world of mathematics, and that’s what we need to cultivate.

Every time I get on this train of thought, I think about Paul Lockhart’s, “A Mathematician’s Lament”. It’s a great read, and something I recommend to anyone who thinks about mathematics education. In particular, I find that his comparison between teaching random mathematical facts and forcing students to learn how to read sheet music to be particularly apt. As I said above, the real joy and delight of mathematics comes from the subject itself, not the applications to the outside world. It’s very similar to other forms of art. If you look at someone who plays music, they usually do it because they delight in the act of playing music itself, not because they want to make it big. They don’t become musicians only to impress others. It’s because they simply love music. Likewise for mathematics, the goal shouldn’t primarily be to get students “prepared” for real life after secondary school. Instead, it should be to show students how wonderful the connections in mathematics are.

I’m not saying that mathematics has nothing to say about a lot of problems. Of course it does, but if we emphasize those (or worse, only emphasize the methods without context at all), we send the message to students that mathematics is only a collection of facts. But that is doing such a disservice to mathematics that I can’t ignore it.

If we want students to find more joy and delight in mathematics, it’s up to us to foster the sense of wonder within its various subjects. This means taking the time to point out the many connections between ideas, and to shy away from merely showing how to calculate things. After all, the goal of mathematics is to solve the problem, not to repeatedly do it over and over again.

Integrating the Gaussian Function

As I take my first quantum mechanics class, I’ve come to find that integrating probability density functions are a pain. From only a few problems I’ve worked on, the integrals are long and tedious to do. If there’s a single theme present in these integrals, it’s the strategy of integrating by parts. However, I wanted to show a specific integration today, because it’s quite ingenious and allows one to integrate a function that would otherwise be very difficult.

But first, a little context. The integrals that one sees in quantum mechanics are essentially known as probability density functions. The idea is that, at each value of the function, one has an associated probability density. Multiplying this by a small interval $dx$ gives one a probability. This also means that any one point doesn’t have a probability. Instead, one requires an interval in order to get a probability. Formally, we get the following:

Additionally, we have some properties that we want for our probability. First, we want any probability to be greater or equal to zero, and second, we want the probability over the whole space that we are interested in to add up to one. These are reasonable requirements if we want to capture what we mean by probability when talk about it in everyday life.

What this second condition usually means though is that we have to normalize a function. We have to integrate over the whole space, and introduce a constant such that the integral then gives one.

This is where my quantum mechanics assignment comes in. I had to normalize what is known as the Gaussian distribution, and the integral looks like this:

My job was to solve for the constant A. At first glance, this integral doesn’t look too bad. However, after a few moments, you’ll realize that this isn’t as easy as it looks. Indeed, the only way I know of to solve this is to use neat trick. To start, we’ll square both sides of the equation. Since the one won’t change, let’s just look at how the left-hand side changes.

This might not seem any better, but what if we change one of the variables of integration on the right-hand side? If we go from $x$ to $y$, we get the following:

Now everything is coming together. If you’ve taken a calculus course and seen other coordinate systems, you should immediately recognize this to be a good candidate for switching into polar coordinates. Doing so gives us the following:

This is something that is much easier to integrate, since we have the factor of $r$ in the denominator. Performing the integration gives us:

This is quite a nice expression, because we know that this is equal to one (from the fact that we need the equation to be normalized to one). Therefore, we get $A = \sqrt{\frac{\lambda}{\pi}}$.

Without squaring our original integral, it would have been very difficult to evaluate. However, by seeing this clever workaround, we were able to turn a difficult integral into one we could evaluate without too much trouble.

The Sweet Spot

Despite what many schools actually do, I think most of us can agree that learning is highly personal. What works for you might not work for me, and there’s nothing wrong with that. Thankfully, there’s more than one way to learn a subject.

However, what I fear happens early on in school is that, if a student struggles with the explanations and teaching style of a particular teacher, the student writes off the subject they are learning as unattainable. This can only be made worse when one sees the other students around them succeeding, while the ideas in the subject can’t seem to sink in to the student. It wouldn’t be surprising to me if this is a frustrating experience that also makes one have bad memories of a particular subject.

I’m pretty sure I’ve mentioned this before, but I’m eternally thankful to my parents for helping me learn arithmetic early on (as well as to my own, perhaps slightly better than average ability), which meant that the early subjects in mathematics during elementary school was easy for me. Without this boost, I feel like I would have been like many of my friends, where mathematics and science were subjects that one “survived”, but never really liked. From this small advantage, I was able to turn it into a whole education trajectory, where I’m now studying mathematics and physics in university.

The tragedy here is twofold. First, there’s the simple fact that mathematics is not linear. Despite what one may think after looking at a school curriculum, there’s not a required progression. Of course, it’s always nice to know arithmetic to speed up calculations within other fields of mathematics, and learning algebra is definitely a prerequisite to calculus, but should you take calculus first, or a class on proofs? Should you study graph theory before probability, or linear algebra before number theory? The answers to these questions depend heavily on the subjects, but the point I want to illustrate is that you don’t necessarily have a set progression in which you have to learn one subject before the next. Instead, you have a range of fields, and many can be studied independently of the others (though of course, one can always find interesting connections between fields).

Coming back to arithmetic, the problem with this is that arithmetic is now seen as the first “hurdle” to get over while learning mathematics at school. If you can’t get a grasp of arithmetic, then it’s as if you’ll be incapable of being good at other branches of mathematics (or, at least, you’ll have a lot of difficulty). But as I mentioned in the last paragraph, there are other fields of mathematics where the reliance on arithmetic may not be as great, so the student would be fine. However, the way mathematics progression is set up now, these small discrepancies between students now grow very large after only a few years, which begs the question, “Why arithmetic first?” If we started with a different subject, would that create a different set of students who seem to “excel” at mathematics? My feeling is that it would to some extent, which means we need to be careful in treating the beginning of mathematics education as a hurdle.

The second part of the tragedy is that, as I mentioned in the beginning, learning is personal. As such, there are different ways to teach subjects to a student. Different ways will work for different students, so the most critical thing to do when you don’t understand a subject is to try and learn it in a different way. If your class has a large emphasis on equations and solving abstract systems of equations, perhaps you need to try and transform this into a visual. It’s incredible how helpful it can be to have a visual when solving abstract equations. It might not make the manipulations easier, but it gives clarification as to why a particular strategy is used. And often, that can be the big difference.

I know that I’ve rambled on about both “higher” mathematics and more elementary mathematics, but my message is the same. If you’re having difficulty with a subject (which happens at every level), look for a different way to understand the material. You might look for a visual, or perhaps you rather work with only the abstract equations. Maybe diagrams help. The possibilities are numerous, so don’t feel like you have to learn things a certain way.

What I’m referring to here is your sweet spot, which is the amount of abstraction that you feel comfortable with while learning. I feel like I have a good grasp on calculus, but if you make it too mathematical and abstract, I get lost. It’s the same story with physics. I like to have some mathematical sophistication, but I’m not able to go off the deep end with it. Therefore, when learning a new subject, I try to find explanations and material that’s at my level. I know there’s no point trying to learn physics in the most mathematical and abstract way for now, because I’ll get lost. Instead, I start at my level, and I let myself become familiar with the subject. Then, when I start feeling more comfortable, I can extend my studies to the more sophisticated concepts. There’s no use to metaphorically bang your head against a brick wall when you simply don’t understand a concept. Use different or easier material as a stepping stone for what you want to learn.

This is exactly what I had to do when I worked in the physics department of my university this summer. I had to learn general relativity, so did I take the most mathematical and sophisticated book on general relativity? Of course not! I used the much more introductory book by Sean Carroll (which is great, mind you) to help me get through the initial hurdle of learning the subject. Without this book, it would have been extremely difficult to understand what was going on in my work. However, since this book was made for people like myself (who were just starting to learn), it made my journey into general relativity much easier.

Therefore, my advice is simple. Figure out what your sweet spot is when learning, and don’t feel afraid to seek out alternative explanations to what you get in class. Almost always, there’s another way to understand a concept, so use that alternative perspective to help you learn. Remember, you’re not necessarily like every other student in your class, so chances are high that you’ll get a class where the explanations the professor uses doesn’t mesh well with you. When this happens, seek the alternatives, and this will help you a lot as you advance your studies.

Linear versus Cyclic Permutations

One aspect of probability I’ve always found to be a little tricky is the part where you need to count things. In theory, this sounds easy enough. After all, it’s just looking at the complete list of things you’re studying, and enumerating them, right?

Well, we know that things become much more subtle when you have a big number to count and you have to be careful when using the “tricks” of multiplication in order to avoid duplication. This is the part that has sometimes seemed straightforward, while at other times being totally mystifying.

If you’ve taken an introductory probability class, you’ve undoubtedly come across the notion of permutations. This concept simply deals with answering the question, “How many ways can I order these items I have?” Just to briefly review, let’s look at an example. Suppose I want to know how many different ways I can order four items: a, b, c, and d. The answer is given by $4!$, which can be seen below.

This makes sense, and it’s usually the image we have in mind when thinking about permutations. This readily generalizes to $n$ items, where the number or permutations is $n!$.

But here’s a question. What are the possible orderings of the four items from above if I arrange them in a circle like this?

Suddenly, the idea has shifted. Now, the absolute placement of the object doesn’t matter (eg. Is it first, second, third, or fourth?), but it’s placement relative to the other objects matters. If we look at object a, the only thing that matters is that it is in between objects $d$ and $b$. How the circle is oriented doesn’t matter. As such, these scenarios are now equivalent.

The question now becomes, “How do we address this change in possibilities within our expression?”

If we consider the circles I drew above, you may notice that there are four of them. This is not a coincidence. The reason is that if we have a certain configuration of the circle with $n$ objects comprising the circle, we can rotate the circle $n$ times without changing the relative positions of the objects. This is because the circle possesses rotational symmetry. As such, the number of ways you can permute the four objects is the the number of ways we can permute them when they are in a line, divided by $n$. Mathematically, this looks like: Here, I’ve used the symbol $P_{cyclic}$ to represent the permutations that can be made in a cycle.

This wasn’t meant to be a long post, but it’s something that I thought was interesting, since (without thinking about) we usually only consider the linear case when looking at permutations. I wanted to show here that circular permutations are just as easy as regular permutations.