When you first start solving a problem in mathematics, the goal is often to find a way to express the problem as some sort of differential equation. During this initial search, you don’t care how the equation looks. It’s more important to get it written down so that you can proceed.
However, once you do have an equation, the first step is not to try and solve it. That’s a rookie mistake. Instead, the question you should be asking yourself is, “Can I put this equation into a form I recognize?” Asking yourself this question can save a ton of time in solving a problem. After all, if you can recognize the form of the equation, then you know the answer without doing any more work.
This might seem like an edge case that never happens in practice, but that’s not true. In particular, mathematicians have studied the solutions of many ordinary and partial differential equations, and know the answers. Therefore, if you’re working with a differential equation (which is almost always the case in physics), you might be able to save yourself a lot of time if you recognize the form of the equation.
For example, any student in physics who has taken more than a few courses will recognize the differential equation representing simple or damped harmonic motion. Physics students come across it all the time. This equation comes up when considering swinging pendulums, motion of a spring, electrical circuits, stability of circular orbits, and even in the Schrödinger equation. It’s what you might call a pervasive equation.
I can guarantee that professors don’t go over the solution to this equation after perhaps the first semester of physics. The reason is that students learn how to solve this differential equation, so there’s no need to go through all the work again and again. Instead, they identify the equation, and then give the solution.
However, it might not always be obvious that an equation satisfies the differential equation for harmonic motion. If there are a bunch of constants littered everywhere in the equation (due to the physical situation), it can be difficult to see the underlying equation. How do we deal with this so that we can try and identify an equation?
The trick is to change variables and bundle up constants together as much as you can. If your equation has constants littered everywhere, see if you can divide the constants out so that you have less in total. In the same vein, if you can see a simple change of variables that will allow you to “absorb” some of your constants in the differential equation, that can also help in simplifying the equation.
The goal here is to try to make your equation as generic as possible. That’s often the best way to compare it to the known equations in mathematics which have solutions. When you look at solutions to differential equations, they won’t be given in terms of parameters like the mass of a particle. The constants will be generic. Therefore, it’s often in one’s best interest to “clean up” a differential equation as soon as possible in order to make it recognizable.
Remember, there’s nothing wrong with ploughing ahead and solving the equation right off. It can still work. It’s just that the constants present in an equation that are specific to the problem can muddy the waters of the solution. By dividing constants out and changing variables, the equation will shed its “particular” qualities, showing only the essence underneath. Then, one can save time by identifying it with a known differential equation.
The point when solving a physical problem isn’t to go through all of the mathematical detail for no reason. If a solution is already known, there’s no point to ignore that. Use the fact that you can recognize solutions to speed up your problem solving. In the end, it’s the physical solution itself that matters.
Snapping Into Focus
Learning new ideas in mathematics or science isn’t always easy. Heck, I would venture to say that most of the time it’s difficult. I imagine the experience is the same whether or not you consider yourself to be “good” in a given subject. That’s because, on some level, we are all in the same situation when it comes to learning. We need to figure out how to integrate new knowledge into our existing worldview.
In particular, I find that mathematical ideas and equations can be the toughest aspects of learning new material. The challenge for me always revolves around the question, “How can I restate these equations and expressions into words that I can understand?” (I’ve written about a similar idea of translating from words to equations before.) I find it helpful while trying to understand what’s going on within an equation. All equations have a story to tell.
I’ll be honest: even as someone who has seen a lot of mathematics, if you drop me inside a derivation without any background, the probability of having me understand what’s going on converges to zero. Mathematics requires context, and it requires focusing on a specific argument. Only once you’ve interacted with it will you start feeling comfortable with the specific equations and expressions.
It’s during the end of this period of struggle where something interesting happens. Just as you’re starting to to figure out what’s going on, things seem to “snap” into focus. The best way I can describe it is through an analogy with running in the fog. When you’re in the fog, you can’t see anything. The light attenuates quickly, and you end up seeing only twenty or so metres in front of you. However, if you climb a hill, there’s this moment where you break through the fog, get above it, and can see everything. While studying mathematics, this is where an idea clicks into place and everything makes sense. The great thing is that once you’ve gotten it, there’s no going back. The concept just makes sense now.
This moment is something I search for all the time, both in myself and in others. As a tutor, there’s nothing that makes me happier than seeing the student I’m working with suddenly exclaim that a concept is now clear to them. It’s the reason I tutor students in the first place. Sure, it’s a job, but it’s also rewarding to witness these moments where concepts snap into focus.
I love this feeling because it illustrates the difference between receiving information and internalizing it. As a student, I have many different classes, each with their own set of assignments, tests, and lectures. In an ideal world, I would be focused during each one. However, if you are (or were) a student, you know that this isn’t the case. Most of the time, we are distracted, not focused, or aren’t engaging with the material more than what is needed to pass the test. You might “understand” the material fine for the course, but I would argue that having this deep understanding where ideas snap into focus is a different situation. When this happened, it became so clear to me that I didn’t have to worry about forgetting it. The idea just made sense, and I felt like I could hold the idea in my head without effort. Contrast this to the feeling one gets when studying the day before an exam, and I think you will see what I mean.
Having this experience is great, but it’s also a lot of work. You need to engage with it, making sure each point makes sense.
Because of this, I can only reasonably commit to fully understanding a few ideas at a time. It depends on the number of ideas you can juggle in your head. Furthermore, I’ve found that engaging with the ideas from a class isn’t enough. In order to get the perfect alignment which is characteristic of something snapping into focus, I need to perform a deep dive. This can be done through writing or teaching.
This isn’t practical to everyone. We don’t all enjoy writing, and producing these pieces takes a lot of work. As such, there are other strategies you might want to employ. First, you can work through related problems that highlight this specific idea. An idea can seem fuzzy in the abstract but be clear when applied to a problem. As such, practice problems can be useful. Second, see if you can explain the concept without any extra help from a textbook to lecture notes. If you can do that, then there’s a good chance that the idea will snap into focus for you soon (or already has). Beware though: you need to make sure the explanation is clear to you. Often, we can be tempted to take the shortcut of merely parroting what is said by the teacher, but that won’t help here.
If you want to really understand an idea, at some point it will have to snap into focus. That’s non-negotiable. The act of snapping into focus is just a milestone in learning. As such, we should be thinking about how to get there, and the strategies we should use to do it. Like I’ve written above, going through problems and trying to explain the topic yourself are good strategies. Another one though that is important is asking someone else. Sometimes, it’s just a particular explanation that is holding you back from understanding. If you limit yourself to just what your teacher says, than you will be in trouble when they say something you can’t figure out. Finding an alternative explanation is the best way to go when this happens. This could be from a friend, from a textbook, or even from the teacher. The point is that sometimes we just don’t understand a particular route, and a different explanation is all it takes to snap into focus.
Most importantl of all, remember that learning is more than just showing up to class and getting a passing grade (or even a good grade!). It’s about struggling with a concept until finally the fog clears and everything falls into place.
Escaping the Path
There’s a lovely forest near my house. It’s a wonderful place that looks exceptional in the autumn, where the fallen leaves of the trees cover the path in a flurry of orange, red, and yellow. I love running there because it’s so peaceful.
Imagine that I told you I would show you this forest. After hearing me wax poetic about it, you’re excited to see it. We get to the forest, and I show you the path that goes through. We walk along it, and after a while you ask if we can get off the path to see the forest in its more “natural” state.
Puzzled, I ask, “But this path is the forest. There’s nothing else of interest other than what’s on the path anyway.”
We might not use the same words, but this is how a lot of us view mathematics. There’s a path (the curriculum), and following it is the only way to learn about mathematics. Forget about going off-path. That’s not even a thought that crosses your mind!
Unless you are really into mathematics, chances are you haven’t seen the wonderful little niches that the subject has to offer. This is unfortunate, but it’s a consequence of the fact that we tend to look at mathematics in terms of the path forged by the curriculum. It’s also not a problem which is limited to mathematics. Almost any subject will have this standard “path” that most people end up associating with the subject itself.
If I could send one message to my younger self, it would this. Don’t make the mistake of seeing the path as the subject itself. It’s only one particular way of looking at a subject, but there are so many more available. It just takes a willingness to look past the usual offering.
Unlike what we’re taught in school, mathematics isn’t a linear subject. Sure, it’s probably a good idea to learn about arithmetic before you learn algebra, but it’s not always as clear. The web of mathematics is thick and highly-connected, which means there are many paths you can take through the subject. Just because there’s a clear trail that has been created by countless curriculums does not mean you are forced to take that same path. In fact, I would encourage you to explore more. Look for those smaller connections. They can be as interesting as the regular path.
My hope here is to encourage you that mathematics is not only the curriculum you learn in school. It has so many other aspects that are off the path, if only you start exploring.
To me, this indicates two things. First, it means that we need to spread the message through our educational institutions, because it’s important that students see mathematics as more than only a curriculum. Second, it suggests that a way to get people interested in mathematics is to find something that they are attracted to. The key point is that this may not lie on the main path, but who cares? I’m more concerned with getting people to see mathematics as it is: an ensemble of many ideas, not just a linear path.
It’s worth wandering off the path every so often to see what else is on offer.
Give Yourself A Gift
A characteristic trait of students is that we tend to think in the short term. Our lives have natural milestones: semesters, midterms, due dates for assignments, final exams, and summer and winter breaks. These lead to students having a certain mindset with respect to time. For the most part, we think about our lives in terms of weeks and (maybe) months.
For example, I’m writing this (not at the time of publication) in a week that I have a test, assignments due next week, a presentation I have to prepare, and a grant application I need to write. These are all within the span of a few weeks, so that’s how I’m thinking of my future. I’m not completely blind to time after that, but the majority of my attention is focused on these items. The result is that I’m always thinking about the short term. I plan my time in terms of these things that are due. I think you can agree that it would be too easy to let our whole life during a semester be led by these requirements. I can envision an alternative version of myself just responding to homework assignments and tests from week to week, never thinking about the longer term.
I think this is a mistake. It might seem like the way to go in the moment, but neglecting your longer term future is a mistake. Unfortunately, it’s only one that you will notice much later, and correcting it will only manifest even further down the line.
I don’t want this to happen with me. I know that as long as I’m in school I’ll be held to these short term requirements, but I make sure that there’s more to my life than that. In particular, I try to always keep some longer term projects in mind. That way, I don’t get stuck trying to satisfy the various urgent and short term responsibilities and ignoring the long term ones.
My goal isn’t to live in just the future or just the present. It’s to keep an eye on both, and make sure they each get their appropriate amount of focus. The reality is that the short term also tends to be the source for more urgent items, which is why we forget about the long term. I’ve found that school emphasizes this short term, to the detriment of everything else. This freedom means it’s up to you to find something of value to give your future self. A gift won’t appear out of the ether. What you do now will inevitably affect your future self, which is why it’s worth thinking about the kind of steps you’re taking now.
In this essay, I want to explore this idea in more detail. Basically, I want to argue for cultivating this long term mindset. The more you can think of your present investment as a gift to your future self, the better.
The value of long term
First off, why is there so much value in the long term? Sure, I keep on telling you that it’s important, but there’s no reason to take my word for it. The reason I argue for the long term boils down to the amount of influence you can make. This influence occurs whether you are a student studying physics and mathematics (like myself), or a designer, marketer, businessperson, or anything else.
In the short term, you can tackle small tasks and deal with the daily trials of life. Think about work tasks, homework assignments, tests, and the like. These are all important and do affect your trajectory.
However, there are some projects which just take a lot more time. If you’re like me and want to do science and mathematics communication, this isn’t something you can just jump into. As such, these kinds of projects require perseverance over long time scales. This means you need to plan ahead, and do more than react to incoming requirements of your time. A long term project takes a lot of effort, but it can exert a lot more influence than a shorter term project.
For myself, I have two long term projects that are separate from my education. I have my blog and my webcomic, which I work on every day. I make sure to carve out time to work on them because I know that the net affect of consistently showing up over years and years will be greater than the investment I make every day to do a bit more work on them. This is my gift to my future self. I’m keeping the blog and webcomic going in a consistent manner, and these become assets that gains momentum. If I didn’t work on them each day, this momentum would take a lot longer to grow.
One implication of thinking in the long term is that you need to be already comfortable with your short term requirements. When I work on my blog and webcomic each day, this takes time. I would estimate I spend roughly an hour to ninety minutes each day working on them. This is time I can’t use for other things, such as homework or studying. As I write these words, I have a test tomorrow that I need to prepare for. And yet, I’m still here, writing and drawing for my future self.
To get to this point, you need to have some sort of order in your life already. I wouldn’t be able to do this if I had ten assignments due the next day and a test to study for. There would just be too much short term work required of me. Therefore, it’s through budgeting my time properly and getting my short term requirements done that I can work on these longer term projects.
If you’re in school like myself, it might feel like each day is a battle to get everything done for the next day. I would submit that you’re in an unstable situation. If you feel like you’re barely holding on, ask yourself if there’s anything you can do to change this feeling. Obviously, there are some people who have particular circumstances, but there are often find pockets of time that we can reclaim back. Heck, just managing our time better can be enough to get started on longer term projects.
But what if you don’t want to start a project? What if you just want to relax?
These are fair questions, but I would answer them the same way I would of those who wonder when they will find time to exercise. It might not be fun at first, but if you do it long enough, it will become fun. Even if it isn’t, you need to think of it as a gift to your future self. Don’t think about it as something you want to do right now. Think of it as something your future self will thank you for.
So sure, don’t take on projects if you’re uninterested. But I really would recommend starting something. Remember, you want to give your future self a gift.
Projects as the natural long term item
So why am I constantly pushing you to do projects? Well, there’s nothing special about projects. However, as a student who is immersed in the short term world, I’ve found projects to be the natural long term item. Projects are great because they require planning, consistently showing up, and executing over a longer period of time. This gives you skills that you won’t get from the steady cycle of assignments, tests, and final exams.
For myself, it’s why I like writing about my experience in science and mathematics. I get a space in which I can reflect, explain, and work through the various ideas I’ve come across. Through writing, I become better at explaining what I know and laying it out in an interesting way. Writing my blog gives me a chance to build an asset for my future, a proof that I know how to think and explain ideas. In particular, my hope is to become a professor one day and teach, which is why I find writing to be a great practice for this. Each day, I get to think about various ideas and see how I can give them their best exposition. As such, while I’m not a professor right now, I’m building myself up to the point where, when the time comes, I will be more ready than if I did nothing. This is the kind of project you should look for. What can you build that will move the needle in the right direction for what you want to do in the future?
Obviously, I can’t tell you what that means, since we all have our particular situations. This requires introspection and reflection. Once you’ve come up with an idea though, you want to find a way to do a little bit each day (or as consistently as possible). The idea is to always take a small step in the right direction. From day to day, it won’t be noticeable. But over the long term, you will find yourself with something you never could have accomplished in the short term.
We don’t often think about it, but we become our future selves. This means that what you do today will affect what your future is like. As such, you have a choice. You can react to the short term events and never think about the future, or you can be proactive in giving your future self a gift. This is so important in school, where we often find ourselves riding along and doing what we’re told. That’s fine, but what happens when you get your degree? You’re dumped into the world, and forced to figure it out on your own.
That sounds like a bad situation to me. Instead, what if you started building something now? What if you spent a little bit of time every day working on building an asset, a gift, for your future self? I might not enjoy every writing session I have, but I sure am happy when I look at the hundreds of thousands of words I’ve written. Looking back at my past, I’m glad I made the choice each day to write. If you want to finish school and have more than just a degree, I would suggest thinking about this. The day-to-day investment is small, but it really does pay off in the future.
You become your future self. Wouldn’t it be nice to give that future self a gift?