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?
Not The Usual Outreach
What is a science or mathematics education good for?
One way to answer that question would be to say that teaching is a good use of such a degree. The idea makes sense, since a degree should give you a lot of knowledge in the subject. And, once you’ve gone through the challenges of completing the courses necessary for your degree, wouldn’t teaching the material be the next natural step?
Another route is to do research. That’s the objective that most students in graduate school are shooting for. They’ve not only enjoyed learning about a specific area of knowledge, but they also want to push the boundaries of it. This isn’t a path for everyone, but it does present some interesting opportunities in terms of working for either a university, a private company, or a government institution.
For the most part, I would say that these are the two ideas that pop into mind when thinking about what a science or mathematics degree can be used for. Teaching and doing research. However, while these are both rewarding endeavours in their own way, I want to push back against these as being the default options. In fact, I want to discuss something a bit different.
That’s the area of outreach.
Traditionally, outreach has been thought of as an extension of teaching. It might not be the usual classroom variety, but it’s still teaching in some way. That means giving explanations and helping more people learn about (in this case) science or mathematics. The end goal is to get more people learning about science through actually teaching them.
In this sense, we usually see outreach in the form of writing and video. Journalists interview scientists and investigate various issues around science, and video producers craft documentaries and other shows that communicate the wonders of science or mathematics. Here, I’m thinking of either publications such as Quanta Magazine, or shows such as PBS Spacetime, Eons, 3Blue1Brown, Kurzgesagt, and many others. All try to explain science or mathematics to (often) a broader audience, and this comes in the form of teaching the reader or viewer.
However, the point I want to get across in this piece is that science and mathematics outreach doesn’t have to go through the usual routes of journalism or documentary production. There are so many other ways to do outreach, and I think we do a disservice to students in these fields by not presenting them these other options.
Before I go into the options specifically, I want to lay out my philosophy regarding what outreach should be. In my eyes, outreach isn’t merely teaching. Sure, teaching is great, but it’s not the only important thing. Rather, I believe that the goal of outreach should be about getting science and mathematics into the public conversation. It’s not just about getting people to understand the issues and latest research (though that too is great). It’s also about communicating what it means to do science and mathematics. It’s to get people thinking like scientists and mathematicians. Simply put, scientists and mathematicians are people too.
When viewed in this way, there are many more options that open in terms of what outreach could mean. If you don’t fancy yourself as a teacher who wants to go through a specific idea like you would find in a class, you don’t have to!. That’s the great thing about taking this broader view of outreach. The goal isn’t to make people learn. It’s to make people aware.
By doing this, the side effect is that people will become more invested in what science is, which will hopefully motivate them to learn more. As such, outreach can be seen as a motivating force for getting people to learn more about science and mathematics, even if the learning isn’t happening through the outreach itself.
With that out of the way, here are some different forms of outreach that I would argue do just as well of a job as traditional teaching to spread the love for science and mathematics.
This should come as no surprise. Blogs are fantastic at communicating science to the public. There are many reasons for this. First, a blog is a “living” entity. What I mean by this is that it’s updated regularly, which means things don’t go out of date. Sure, a post could be a few years old, but the author can then write a new post, or even update the old one. The result is that, unlike a book, blogs are never “finished”. This means someone who reads a blog can keep on being apprised of the latest information within a certain field.
Second, a blog gives readers a unique perspective. Unlike a book which presents a topic, a blog can also give the reader information on the writer. For example, a blog might discuss issues about mathematics, but also on what it means to do research in mathematics. This wouldn’t often be considered topical in a book, but in a blog it feels more natural. The result is that a reader gets to also have insights into the work itself, not just the products of research.
Third, blogging comes in a lot of varieties, which lends itself well to discussing a bunch of different issues around mathematics and science. For example, there are many blogs I follow which focus on academia and navigating that world as either a student or a young researcher. This gives readers an inside look into what it means to go on this journey. Since I’m planning on following the academia route, these kinds of blogs are very interesting to me, and allow me to understand the inner workings of science and mathematics research that I otherwise would have trouble finding.
This seems like a good place to point out that outreach isn’t just to the general public. I know we all have this mythical idea of a “general public”, but the truth is that everyone is somewhere along their own path with regards to science and mathematics. Some might be further along than others, but you don’t have to do outreach just to the public. In fact, it can be as useful to write and share knowledge to those that are only a bit behind you in their journey. This absolutely counts as outreach.
Fourth, blogs can vary in size while still maintaining a cohesive whole. Whereas a book has a minimum length (you don’t see to many 500-word books unless you’re looking at a children’s book), blog posts can have whatever length the author wants. The advantage here is that they can go in whatever depth they wish. If the author only has enough ideas to write a 1000-word post, they don’t have to agonize about how to turn it into a book. Instead, they can write up their thoughts and publish it as is. This means it’s easier for authors to share what they know, and removes the need for artificial constraints.
Fifth, blogs can be a place for various scientists and mathematicians to comment on the work of others. This might not seem important, but it allows for those not in that specific field to see what various people think of certain work. One blog in particular comes to mind: Statistical Modeling, Causal Inference, and Social Science. This blog is written by several authors, and most posts are short and involve studies that have problems in them or suggestions for how to make them better. The purpose of these posts isn’t for me to “learn” like I would in a traditional class. However, I still get a lot of value out of these small posts, and they have made me think more about statistics than I would have just from a regular class (which I’ve also taken).
Taken together, blogs are a very important part of scientific and mathematical outreach. I follow blogs that are about specific themes (like quantum mechanics, cosmology, etc.), but I also follow blogs which discuss the lives of specific researchers. This means I can get different types of posts, and what I read depends on what I’m looking for on a specific day.
The next category is video. Of course, just like in the case of writing a blog, you can absolutely use video to present a concept in detail like in a traditional lecture. In fact, this is probably preferable to a text, since the viewer can follow along as the person goes through a derivation or problem.
However, there are many other opportunities to show parts of science and mathematics which aren’t just teaching. In particular, video is good for showing “behind the scenes” of what’s it’s like to do research. You can use video to show an experiment, to show what it’s like to work at a specific place, or many other aspects of being a researcher. The idea here is to show the viewer what it means to be a scientist or mathematician. Instead of having the public view researchers as people who produce papers, they get to see what goes on in the background. I think it’s silly to let the perception of researchers shine through only a stack of papers. The job itself is plenty interesting, and video allows one to capture this in a nice way.
Video is also good for interviewing. Just because you in particular aren’t a researcher doesn’t mean you can’t speak to those who are. In this way, you’re still doing outreach, and the interviews can shed light both on certain concepts and the process behind the research. Interviewing is also good for the audio format (such as podcasts).
Finally, we have illustrations. In particular, I’m thinking about what we would classify as science and mathematics webcomics, but “traditional” science illustrators are invaluable as well. These tend to be drawings and cartoons that discuss mathematical and scientific topics to a broader audience using pictures and humour. Often, the illustrations will employ analogies and metaphors to get a subject across.
With webcomics, the idea (at least to me when I draw my comics) is to get people interested in the ideas of science and mathematics. It’s not that they should necessarily get every single joke and reference. Instead, it’s about making them curious about these fields through good (or bad) illustrations. When I make my comics, I try to link it to some concept within the two fields, and my hope is that this will get people to dig deeper or to at least reflect on the mathematical ideas.
The nice thing about webcomics is that they don’t have to be pretty. My webcomic Handwaving is made using a simple app on iOS and employs stick figures as the main characters. There’s nothing artistic about my drawings, but that’s not the point. The point is to give the reader of taste of science and mathematics, and I think my webcomic accomplishes that.
Of course, there’s also traditional illustration that you see being done for magazines and online publications. That kind of illustration is definitely nicer-looking than my webcomic, and it’s an important area as well. It’s fine to say that science and mathematics allows us to see the beauty of the universe, but if we go around and never make nice visualizations or illustrations, how can we expect others who don’t have the necessary background to get that feeling? We can’t, and so I would argue that illustration also serves the purpose of drawing people into these two fields.
I just mentioned three specific media above, but my point applies more generally. My goal here isn’t to say that you should start a blog, make videos, or create a webcomic. Instead, my point is to argue that outreach in science and mathematics doesn’t have to be limited to teaching. It should be about transmitting the joy of these subjects to your audience. Do you really like science? Are you passionate about analysis or topology? If so, finding any way to communicate that joy is what matters. Maybe that means meeting with younger students to tell them about your experience in the field. Maybe it is starting blog. It’s up to you, just don’t feel like you need to teach.
If we could move away from focusing on teaching, we would see that there are so many options for outreach. And the most important point is that they are all valuable. There isn’t necessarily a hierarchy that has teaching at the top and everything else further down. Instead, each different aspect of outreach plays a role in getting people interested about science and mathematics.
For myself, I do this through writing a blog and a webcomic. My blog posts tend to not be about teaching particular bits of science and mathematics. It’s just not as interesting to me. Rather, I enjoy writing essays about my experience around the two subjects and the connections I’ve seen between them. My hope with each piece is that the reader will see that mathematics and science are interwoven and are bursting with connections once you dig a little deeper.
For my webcomic, the goal is to bring a lighter side to science and mathematics. I want to put a smile on the reader’s face, or perhaps get a laugh out of them. I want to show how there are humourous aspects to these two subjects and that they aren’t just serious all the time. More than anything, I think the webcomic format is a very good way at transmitting information in a compressed form.
So those are the ways I do outreach. As you can see, neither of them really counts as “teaching”. Sure, I might sneak in a bit of learning here or there, but that’s not the focus. And that’s alright. I don’t feel the pressure to teach in my work because I know there are many other resources who do focus on this kind of work.
Outreach is a multifaceted beast. Thinking about it as synonymous with teaching is limiting yourself in what you can do. Don’t think of outreach as teaching. Instead, think of it as spreading awareness of science and mathematics. If you do that, I’m confident you will find a variety of ways to share your passion for these subjects.
There’s room for so many different formats. Don’t worry about doing the “usual” thing. Do what’s interesting to you.
Chasing The Carrot
When you want to form a new habit, how do you go about it? Do you purchase equipment in the hope that spending money will “force” you to stay consistent? Perhaps you try to stay accountable by enlisting the help of a friend. Maybe you announce a project publicly, to show that you’re serious, or sign up for a class on a subject you’re interested in. This can apply to many situations, from getting better at writing, running every day, drawing, learning mathematics or science, playing a sport, or any other activity that’s important to you. Each one requires consistency, and a habit is the best way to build that consistency.
What you might notice from the above is that the ways we go about incentivizing ourselves usually have to do with some kind of reward. If we want to get better at running, we sign up for a race. If we want to learn some advanced mathematics, we sign up for a class that has assignments and tests to complete. If we want to write more, we set a word count and join a writing group. Each of these is what I would I call a “carrot”. It’s a way to prod us forward in a way that we want. The methods reward us for doing a good job, so we don’t have to rely on our own willpower every day.
I have no doubt that this is a good way to get started. In fact, like I mentioned above, this is how school works. Classes might not be the most exciting thing in the world, but you’re supposed to work hard so that at the end you get a good grade (the reward). Many of the systems in our lives have these rewards baked in. Put in some effort now, and get rewarded later.
I have nothing against these systems, and I’ve used them many times myself in order to move the needle of my habits in the right direction. However, now that I’m past the point of the beginner and into the realm of someone who does a specific activity consistently, I realize that chasing the carrot may not be the best way to go about this.
What is the ultimate goal with any activity? This is a question you need to consider and ponder, because it informs what you will do in the long term. I’ll illustrate my answer with the activities of writing, drawing, and doing science and mathematics. Every day, I write and I draw. I’ve gotten to the point where it’s not even really a question of whether I want to draw. I now always want to make more work. Each day, I do more to explore my thoughts about science and mathematics through these media. The result is that my writing and drawing are things I love to do even without an external reward. Sure, I do post my work on my blog and my webcomic Handwaving, but I don’t have analytics and frankly I don’t know how many people see my work. It’s just not on my mind. I’m just focused on doing my best work every day, and producing something of value to publish here.
In essence, I’ve removed the need for an external carrot. I don’t crave a huge audience for my work, because I’ve never had it in the first place! In that way, it’s freeing. I don’t have to think about how my work will be received by everyone. I can just focus on what I’m doing, and making it the best it can be.
In my educational journey, I’ve made a similar transformation. The external carrot for school is grades. They are the currency that determine what kind of opportunities you can have later on. They can finance your education, and give you a chance to go to certain prestigious schools. As such, there’s a huge incentive to watch your grades with religious zeal.
When I started my undergraduate degree, I was just like this. I obsessed over my grades, making sure I got the best marks I could possible eek out of any class. However, I realized that all this obsessing was stressing me out. No matter how well I did in class, I was only “satisfied” if I got 100 on a given assignment or test, and otherwise I became steadily more disappointed. This is a bit ridiculous. As such, I decided to make a dramatic shift: I would no longer look at my grades. I decided this several years ago, and now I could not tell you what my average is, or how well I did in a given class. I simply don’t know.
I decided to stop chasing the carrot. Of course, I still work as hard as always in the work I do at school, but I don’t worry about the grades. They will take care of themselves. It’s my effort that matters. I know that this isn’t a luxury everyone can take (as far as I know, my grades are still very good), but it has helped me stress out about tests and grades much less. They don’t inhabit any mental space of mine, leaving me to focus on other things.
The question I’ve been pondering for a long time is, “What makes me show up every day to do the work that I do?”
For me, I write, draw, read, journal, run, and do school work every day. These are the steady rhythms in my life. Each day will include these six activities, and I don’t do them for external carrots. (Yes, I do have external rewards for doing my school work, but I also do it because I’m interested in the subject.) I do them because they are important to me, whether or not the rest of the world knows what I’m doing or if they care. Each day, I get to work on my creativity and ability to discuss science and mathematics. This is a reward in and of itself.
My challenge to you is to think about what you do in your life, and what the split is between chasing external carrots and building habits that are important to you. There’s nothing wrong with chasing carrots, but my argument is that it isn’t what you want to do in the long term. It’s so much better to do things every day because you want to do them, rather than because you feel like you have to do them. Accountability and consistency are important, but I’ve found that these come from dedicating oneself to some specific activities and learning to love the process of doing them without any external carrot.
This is particularly relevant for those that are in school. Sure, sometimes we have to simply get an assignment done, no matter how boring it is. But think about how great it would be if you could get up every day and be excited about the work you need to do for school? That’s what you should be shooting for on average, and that means you need to align yourself with a subject that interests you. If you can find that, you will be well set up for studying because of internal motivation.
At the end of the day, this isn’t any different than the usual discussion of intrinsic versus extrinsic motivation. The slight difference is that I’m tell you to start with the latter (because it’s good at keeping you accountable), but work to transition to the former. The former is what will keep you working on whatever you love for years. Like I mentioned above, I run every day. What’s more surprising is that I haven’t run a race since 2015. That’s nearly four years ago, and I don’t really feel a desire to do one. I just love to run, and isn’t that enough?
That’s my advice to you. Find something you enjoy doing, and make it be enough. Don’t worry about being the best, or having the most views or being the most prestigious. Just focus on doing the work you enjoy, and have that be enough. No carrot needed.
There are plenty of ways to enjoy mathematics. You can attend a classroom lecture, you can read a textbook, you can look at a news article, you can watch a video, or you could just play with some concepts yourself. There’s not one way in particular that is better than any other. Rather, each one has its own advantages and disadvantages. It depends what you’re looking to get out of your session.
Despite this, I want to highlight one method in particular that I think is crucial when learning mathematics. That is the method of discovering more mathematics on your own.
In terms of pure productivity, it makes little sense. After all, you’re likely going to hit a bunch of roadblocks when learning new material on your own. When you’re going through a textbook and trying to decipher the ideas without the help of a teacher, it’s easy to get stuck. As soon as you have a question, you don’t have an expert on hand to consult. Your choices are to skip ahead or struggle with the concept until it makes sense. Either way, you won’t be blazing through a textbook when learning on your own.
Obviously, this isn’t ideal. In a best-case scenario, you would go through material quickly and understand it. However, that’s not going to happen if you’re trying to push your boundaries. By definition, you’re trying to understand something that you didn’t know before. As a result, progress will be slow (at least at first).
I’m speaking about this from my own experience. I recently took an independent course on advanced topics in quantum mechanics, which means I learned on my own from a textbook. That’s not a bad thing, but it meant I had to learn the topics without a lot of outside help. I’ll admit that it was a challenge at times, but it went well overall.
If working through ideas and concepts on your own is so difficult, why do I recommend it?
The first reason is that you start learning how to learn on your own. This is an important skill, because once your education is over, you shouldn’t stop learning. If anything, it will be even more important to keep learning as new developments occur. Once you’re out of school, you won’t have a group of teachers waiting to help you out and lead you by the hand while you learn. It will be up to you to figure it out, which means the sooner you start learning how to learn on your own, the easier the transition will be.
That’s the practical reason why you should try to learn on your own. It will help you in the future and free yourself from needing a teacher to go through topics with you. By finding your own path through material, you will figure out what’s important and what isn’t.
However, the reason I think learning on your own is so important is that discoveries in mathematics stick with you longer when you come up with them on your own.
I’ll give you an example. This summer, I was doing research in gravitational theory. Like in any research project, there were papers I needed to consult. In particular, there was one paper that was very important. When I read it, I could have just accepted the results and gone on with my research. Instead, I spent multiple hours going through all the calculations, doing each of them one by one. I did this because we used a different sign convention than the author, but it also had the side effect of helping me understand what was happening. By the end of all the calculations, I understood where each term came from. It wasn’t just something I needed to accept on faith. Rather, I knew it was correct, since I spent a lot of time working it out.
This sounds like a minor thing, but it actually changed the way I saw the subject. It wasn’t something that I knew was probably correct but had no clue how to do. By doing those calculations, it made sense to me. That’s a powerful feeling in mathematics.
If we want to apply this to learning in a classroom, my suggestion is that you spend more time trying to make sense of everything your teacher says. When they say that a particular calculation yields this specific result, can you see that? Do you know how it comes about, or do are you trusting that your professor did their algebra right? The point isn’t to find mistakes. The point is to be comfortable with the results.
There’s a lot to be said about feeling comfortable in mathematics. When you’re comfortable, results are easy to accept because you know why they are true. You can look past the final answer and into the steps that led to the answer. The answer itself ceases to be important. It’s the knowledge that you know how to get from the beginning to the end of the problem that’s crucial.
This is where learning on your own comes in. Even if you’re in a class with a teacher who leads you through the material, it’s important to discover mathematical results on your own. When your teacher gives you a result, it’s easy to forget it. After all, what’s so special about that result? It’s just another thing to remember. However, when you spend hours working through a calculation to get to an answer, you remember it longer. At least, this has been my experience while working within physics. It’s well and good to read papers and textbooks and have people present results to you, but if you really want to internalize the ideas, nothing beats taking out a pencil and working through them yourself.
It’s a lesson I’m learning over and over as I go through my education. The lazy way to learn mathematics is to listen to someone tell you the answer, or read it in a text. The long and painful but ultimately useful way is to go at it on your own. It’s not easy, and I can say from experience that it will result in you getting mad many times, but it’s the only way I’ve found that ends with you remembering the results and not just knowing them.
And in the end, isn’t that part of what mathematics is about? Knowing that the square root of two is irrational is good, but understanding why this is true is the real fun. Mathematics is about the “why” behind the results, not the results themselves. As such, when you take the time to discover the mathematics on your own, it will have a larger impact than if you passively consumed it from someone else.