Jeremy CoteScientific and mathematical thinking for the curious.
https://jeremycote.me/
Familiar Forms<p>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.</p>
<p>However, once you do have an equation, the first step is <em>not</em> 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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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?</p>
<p>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.</p>
<p>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.</p>
<p>Remember, there’s nothing <em>wrong</em> 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.</p>
<p>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 <em>ignore</em> 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.</p>
Mon, 21 Jan 2019 00:00:00 +0000
https://jeremycote.me/familiar-forms
https://jeremycote.me/familiar-formsSnapping Into Focus<p>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.</p>
<p>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 <a href="https://jeremycote.me/2017/06/12/translation">before</a>.) I find it helpful while trying to understand what’s going on within an equation. All equations have a story to tell.</p>
<p>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 <em>focusing</em> on a specific argument. Only once you’ve interacted with it will you start feeling comfortable with the specific equations and expressions.</p>
<p>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 <em>gotten</em> it, there’s no going back. The concept just makes sense now.</p>
<p>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.</p>
<p>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 <em>sense</em>, 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.</p>
<hr />
<p>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.</p>
<p>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.</p>
<p>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 <em>explanation</em> 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.</p>
<p>If you want to really understand an idea, at some point it <em>will</em> 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 <em>particular</em> 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.</p>
<p>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.</p>
Fri, 18 Jan 2019 00:00:00 +0000
https://jeremycote.me/snapping-into-focus
https://jeremycote.me/snapping-into-focusEscaping the Path<p>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.</p>
<p>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.</p>
<p>Puzzled, I ask, “But this path <em>is</em> the forest. There’s nothing else of interest other than what’s on the path anyway.”</p>
<hr />
<p>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!</p>
<p>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.</p>
<p>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.</p>
<p>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 <em>not</em> 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.</p>
<p>My hope here is to encourage you that mathematics is <em>not</em> only the curriculum you learn in school. It has so many other aspects that are off the path, if only you start exploring.</p>
<p>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 <em>they</em> 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 <em>many</em> ideas, not just a linear path.</p>
<p>It’s worth wandering off the path every so often to see what else is on offer.</p>
Mon, 14 Jan 2019 00:00:00 +0000
https://jeremycote.me/escaping-the-path
https://jeremycote.me/escaping-the-pathGive Yourself A Gift<p>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.</p>
<p>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 <em>completely</em> 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.</p>
<p>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.</p>
<p>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.</p>
<p>My goal isn’t to live in <em>just</em> 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 <em>you</em> 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.</p>
<p>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.</p>
<h2 id="the-value-of-long-term">The value of long term</h2>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>But what if you don’t want to start a project? What if you just want to relax?</p>
<p>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 <em>become</em> 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.</p>
<p>So sure, don’t take on projects if you’re uninterested. But I really would recommend starting <em>something</em>. Remember, you want to give your future self a gift.</p>
<h2 id="projects-as-the-natural-long-term-item">Projects as the natural long term item</h2>
<p>So why am I constantly pushing you to do <em>projects</em>? 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.</p>
<p>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. <em>This</em> 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?</p>
<p>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.</p>
<p>We don’t often think about it, but we <em>become</em> 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.</p>
<p>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.</p>
<p>You become your future self. Wouldn’t it be nice to give that future self a gift?</p>
Fri, 11 Jan 2019 00:00:00 +0000
https://jeremycote.me/give-yourself-a-gift
https://jeremycote.me/give-yourself-a-giftNot The Usual Outreach<p>What is a science or mathematics education good for?</p>
<p>One way to answer that question would be to say that <em>teaching</em> 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?</p>
<p>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.</p>
<p>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 <em>default</em> options. In fact, I want to discuss something a bit different.</p>
<p>That’s the area of outreach.</p>
<p>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.</p>
<p>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 <a href="https://www.quantamagazine.org">Quanta Magazine</a>, or shows such as <a href="https://www.pbs.org/show/pbs-space-time/">PBS Spacetime</a>, <a href="https://www.pbs.org/show/eons/">Eons</a>, <a href="http://www.3blue1brown.com/">3Blue1Brown</a>, <a href="https://kurzgesagt.org/">Kurzgesagt</a>, 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.</p>
<p>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.</p>
<p>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 <em>like</em> scientists and mathematicians. Simply put, scientists and mathematicians are people too.</p>
<p>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, <em>you don’t have to!</em>. 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 <em>aware</em>.</p>
<p>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.</p>
<p>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.</p>
<h2 id="blogs">Blogs</h2>
<p>This should come as no surprise. Blogs are <em>fantastic</em> 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.</p>
<p>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 <em>do</em> 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.</p>
<p>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.</p>
<p>This seems like a good place to point out that outreach isn’t <em>just</em> 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 <em>just</em> 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 <em>absolutely</em> counts as outreach.</p>
<p>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.</p>
<p>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: <a href="https://andrewgelman.com/">Statistical Modeling, Causal Inference, and Social Science</a>. 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).</p>
<p>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.</p>
<h2 id="video">Video</h2>
<p>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.</p>
<p>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.</p>
<p>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).</p>
<h2 id="illustration">Illustration</h2>
<p>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.</p>
<p>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.</p>
<p>The nice thing about webcomics is that they don’t have to be pretty. My webcomic <a href="https://www.jeremycote.me/handwaving">Handwaving</a> 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.</p>
<p>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.</p>
<hr />
<p>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 <em>teaching</em>. 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 <em>any</em> 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 <em>need</em> to teach.</p>
<p>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 <em>all</em> 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.</p>
<p>For myself, I do this through writing a blog and a webcomic. My blog posts tend to <em>not</em> be about teaching particular bits of science and mathematics. It’s just not as interesting to me. Rather, I enjoy writing essays about my <em>experience</em> 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.</p>
<p>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.</p>
<p>So those are the ways <em>I</em> 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 <em>do</em> focus on this kind of work.</p>
<p>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.</p>
<p>There’s room for so many different formats. Don’t worry about doing the “usual” thing. Do what’s interesting to <em>you</em>.</p>
Mon, 07 Jan 2019 00:00:00 +0000
https://jeremycote.me/not-the-usual-outreach
https://jeremycote.me/not-the-usual-outreachChasing The Carrot<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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 <em>want</em> 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 <a href="jeremycote.me/handwaving">Handwaving</a>, 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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<hr />
<p>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?”</p>
<p>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.</p>
<p>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 <em>you</em>. 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.</p>
<p>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 <em>excited</em> 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.</p>
<hr />
<p>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?</p>
<p>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.</p>
Fri, 04 Jan 2019 00:00:00 +0000
https://jeremycote.me/chasing-the-carrot
https://jeremycote.me/chasing-the-carrotDiscovering Mathematics<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>Obviously, this isn’t ideal. In a best-case scenario, you would go through material quickly <em>and</em> 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).</p>
<p>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.</p>
<p>If working through ideas and concepts on your own is so difficult, why do I recommend it?</p>
<p>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 <em>more</em> 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.</p>
<p>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.</p>
<p>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.</p>
<p>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 <em>knew</em> it was correct, since I spent a lot of time working it out.</p>
<p>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 <em>sense</em> to me. That’s a powerful feeling in mathematics.</p>
<p>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 <em>comfortable</em> with the results.</p>
<p>There’s a lot to be said about feeling comfortable in mathematics. When you’re comfortable, results are easy to accept because you <em>know</em> 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.</p>
<p>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 <em>really</em> want to internalize the ideas, nothing beats taking out a pencil and working through them yourself.</p>
<p>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 <em>useful</em> 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.</p>
<p>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 <em>why</em> this is true is the real fun. Mathematics is about the “why” <em>behind</em> 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.</p>
Mon, 31 Dec 2018 00:00:00 +0000
https://jeremycote.me/discovering-mathematics
https://jeremycote.me/discovering-mathematicsBag of Examples<p>As a student in mathematics and physics, I’m part of two different worlds. On the one hand, proofs and abstraction come from the side of mathematics. On the other hand, physics is where concrete examples and applications are the norm. In physics (at least, within the scope of undergraduate education), we only care about the mathematical tools that we can apply to a given problem.</p>
<p>Of the two camps, I find myself identifying more with the mathematics side. That’s just because I like abstracting past one application and finding the link between a lot of concepts. I realize that we do this in physics as well, but I find myself at home within the mathematical proofs versus the handwaving that often happens in physics (which isn’t always a bad thing!). I’m definitely not a pure mathematician. I still like to link what I’ve learned with the world, so perhaps I fit best into the “applied mathematics” camp.</p>
<p>The reason I bring this up is because I’ve been thinking of the way we introduce examples in mathematics. If you’re in an advanced mathematics class, chances are the professor will first go over some definitions and perhaps a few theorems before giving examples. Even then, the examples only illustrate the idea, so they aren’t the focus of the class. Instead, the focus is on the proofs and how to go from argument to argument.</p>
<p>This makes sense. After all, advanced mathematics tends to involve abstracting past examples and capturing the general case. This means the examples are less important than the underlying characteristics they share. This of course is the strength of mathematics. When you can look past the immediate features of a specific example and see what many examples have in common, you can come up with theorems that apply more generally.</p>
<p>This came to light in my abstract algebra class. There, I learned about a mathematical object called a ring. The specifics aren’t important, but what <em>is</em> important is that a ring is something we are <em>all</em> familiar with. The integers under addition and multiplication form a ring. The real numbers form a ring. If we look at linear algebra, <em>nxn</em> matrices form a ring. The point is that even if you have no idea of what a ring is, you have likely worked within a ring in your mathematical journey.</p>
<p>As such, the underlying structure of a ring was there, beneath your feet, all this time. Instead though, you studied arithmetic and linear algebra as separate concepts. It turns out though that they are both <em>examples</em> of rings. Therefore, by studying the properties of rings, we can capture the core ideas of matrices and integers or real numbers in one go.</p>
<p>That being said, there’s a downside to abstracting to higher and higher levels. Each time you go up in abstraction, you lose certain features. If you climb a mountain, the landscape of trees you see from the summit will look smooth. Descend the mountain though, and you find that there’s a staggering amount of diversity present within the forest. What looked uniform from above can break up into unique parts upon closer inspection.</p>
<p>I think we can make a similar comparison with abstraction and examples. Sure, abstraction is great in the sense that it captures <em>everything</em> we want in one sweep. But the price we pay for it is that we don’t have specific examples in mind when working through the mathematics. This might not sound like a bad thing, but it makes it difficult to apply our knowledge to specific scenarios.</p>
<p>This is something I also learned in my abstract algebra class. As great as it is to study rings, integral domains, ideals, and fields, it’s also important to find <em>examples</em> for these specific objects. Examples clarify definitions and make the abstractions we study easier to visualize. Without them, it’s difficult to attach meaning to our objects of study.</p>
<p>It’s tempting while studying mathematics to jump straight to the abstraction. I’ve been there, and I know the feeling. You want to do it because you figure that knowing the general case will be a lot better than any specific one. However, this ends up not being true at all. In fact, I would venture to say that specific examples provide the footholds necessary to be comfortable with abstraction. When I feel at ease with a concept, it’s usually because I’ve internalized a specific example and I can see how that example stems from the general case.</p>
<p>Unfortunately, advanced mathematics courses often prioritize proofs and abstraction over concrete examples. This is emphasized through the kinds of problems that are assigned and the amount of class time dedicated to examples.</p>
<p>On the one hand, going through a bunch of examples can seen repetitive in class. Furthermore, problems which involve examples tend to be easier and don’t involve proofs. This isn’t a bad thing, but it does mean that the students don’t get to practice abstract thinking as much. I would venture to say that this is a big reason why professors don’t assign these problems as often.</p>
<p>On the other hand, I find that focusing only on the abstractions prevents you from playing with an idea in specific settings. I found the lack of examples in my classes a hindrance when it came to working on problems that dealt with specific examples. It might seem like applying the general knowledge you know to a specific example would be easy, but I can assure you it’s not. Furthermore, it’s once you succeed in applying the general knowledge to a concrete case that the idea becomes familiar. I would argue that knowing abstract knowledge without being able to <em>apply</em> it anywhere is next to useless. You need to have a balance of both the abstract and the concrete to thrive.</p>
<p>This is why I’ve begun thinking about my “bag of examples”. It’s great to know important mathematical results, but if I can’t <em>illustrate</em> them with an example, it becomes difficult to communicate them. Plus, working through a specific example tends to be the easiest way to grasp an idea.</p>
<p>Good examples become fertile ground for experimentation. When you have a concrete example, you can see how the result you’ve proved works in this case. The best examples can even inform the general result.</p>
<p>I know that my bag of examples is <em>very</em> empty at the moment. However, I want to build it up. Knowing mathematical proofs is great, but having a bag of good examples that you can pull out at any time is an under-appreciated asset.</p>
<p>Finally, I don’t want to forget about the related category of <em>counterexamples</em>. These can be just as important as examples, because they remind us that mathematics can be misleading if we look at just a few cases. Counterexamples force us to be more careful in our hasty generalizations and to remember that the final arbiter of truth is through a proof.</p>
<p>As such, my goal for now is to start amassing examples and counterexample stop illustrate various mathematical ideas. I want to find the shining jewel of examples for any idea. It’s great to keep on learning new material, but sometimes it’s worth pausing and building up a bag of examples.</p>
Mon, 24 Dec 2018 00:00:00 +0000
https://jeremycote.me/bag-of-examples
https://jeremycote.me/bag-of-examplesOnly Numbers and Algebra<p>Learning mathematics in school and doing mathematics in general are <em>not</em> the same thing.</p>
<p>This might seem obvious, but I worry a lot about students that don’t have a chance to realize this before they are turned off from mathematics forever. The reason is that the message which is sent to students throughout their years in elementary and secondary school is that mathematics is all about numbers, but this is false.</p>
<p>Sure, a lot of mathematics involves numbers, but <strong>it’s a mistake to make the leap that mathematics is all about numbers</strong><sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>. Mathematics is a way of thinking. There’s <em>so</em> much more than equations and formulas within mathematics.</p>
<p>Students get a quick glimpse of this when they are young. During their first years being exposed to mathematics, they learn about shapes and patterns. There’s no algebra involved. Their sense of numbers is only beginning to get sharpened, so the curriculum is focused on other areas of mathematics. Notice how they are <em>still</em> learning without needing to transform everything into equations.</p>
<p>Fast-forward a few years, and the focus has shifted. Now, students are getting used to doing arithmetic, and applying this knowledge to algebra. From here on out, most of the emphasis is on getting students to be proficient with equations. Students learn about probability and statistics, graphing equations, solving quadratics, doing word problems, solving trigonometric relations, exploring vectors, and thinking about geometry. However, in all of these subjects, the emphasis is almost always on using algebra to solve problems. Even in the case of geometry, the use of symmetry is often substituted for brute-force equations. And what do students learn equations are for? Plugging in numbers to get an answer out. The theme of a problem becomes finding the right equation that will spit out the desired answer.</p>
<p>I’m not saying that we should ban equations from ever being used in class. They are a great tool to get a handle on the essence of a problem. <strong>But this total emphasis on algebra shoves aside other areas of mathematics.</strong> Areas like graph theory and it’s parent field, discrete mathematics, are quite accessible to students at the secondary level, and do away with a lot of algebra<sup id="fnref:2"><a href="#fn:2" class="footnote">2</a></sup>. The point isn’t to do away with algebra, but to look at some subjects which don’t emphasize its use.</p>
<p>What if the student didn’t have to touch an equation, unless absolutely necessary?</p>
<p>I think this change would help some students see that mathematics isn’t just about “finding the number” to solve a problem. Rather, it’s a method of thinking about how we can boil down the essence of a question or a problem into something we can manipulate. This is not limited to the areas I highlighted above. Heck, I can even see a project where students learn how to use equations in order to create mathematical art. Even <em>this</em> is different, because it removes the emphasis on equations (though they are still there!) and more toward creation.</p>
<hr />
<p>Students tend to have strong opinions on mathematics, which is informed by their experience in secondary school. I’m hoping that we can do more to remove the idea of mathematics being all about numbers and equations. It’s so much more, and I think diversifying our offering to students is key to changing that mindset that some students come away with. For some, this is the mindset they will carry through for the rest of their lives. Even if they don’t study mathematics anymore, I want them to leave with a fair view of the subject, not one informed only by solving endless equations.</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>In fact, I would argue that students don’t even get to encounter the mathematics of numbers, which is the study of number theory. This isn’t part of the curriculum (at least where I’m from). <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>I also realize that we <em>do</em> use a lot of numbers in discrete mathematics. But it’s of a slightly different type. Instead of formulas, we often compute quantities like permutations and combinations which can be visualized and don’t require as many equations. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Mon, 17 Dec 2018 00:00:00 +0000
https://jeremycote.me/only-numbers-and-algebra
https://jeremycote.me/only-numbers-and-algebraNot Necessary<p>In mathematics, the terms “necessary” and “sufficient” have technical meanings. These terms come about when looking at two statements <em>P</em> and <em>Q</em>. If we say that <em>P</em> is sufficient for <em>Q</em>, then that means if <em>P</em> is true, <em>Q</em> automatically has to be true (<em>P</em> implies <em>Q</em>). On the other hand, if <em>P</em> is only necessary for <em>Q</em>, having <em>P</em> be true doesn’t mean <em>Q</em> has to be true (but the other way works, so <em>Q</em> implies <em>P</em>). If we have the <em>P</em> is both necessary and sufficient for <em>Q</em>, that means having one gives us the other for free. They are tied together and are inseparable.</p>
<p>The reason I bring this up is because I know some students have a tendency to “go overboard” in their quest for good grades. They will do <em>anything</em> to get the best grade, and anything less than perfect is unacceptable. This leads them to working more than everyone else, risking burnout so that they can get the best grades. If you’re a student reading this, you might recognize yourself in these words. I know I do.</p>
<p>The issue is that we’ve turned grades into a “necessary” condition to being considered “good” in school. This is dangerous, because grades are <em>not</em> under your total control. Yes, your work is what gets graded, but it’s your teacher who makes the final decision. Even in subjects where grading schemes are more rigid, like mathematics and science, your teacher is the one assigning your grades. As such, they are external factors that can’t be fully controlled through hard work.</p>
<p>Yet, we convince ourselves that we <em>can</em> control them. We turn the goal of perfect grades into a necessary condition to being a success. When this inevitably doesn’t work out, we then feel like failures. Worse, we might resolve to work even <em>more</em>, just to make sure it doesn’t happen again. This pushes us further into the cycle, creating an unattainable goal.</p>
<p>We need to stop looking at great grades as a necessary condition to being a good student. Sure, we need to work hard and give our best, but we should view <em>that</em> as a sufficient condition to being a good student. We can always give our best effort. That’s something we can control, since it’s an internal choice. Unlike getting perfect grades, it is realistic to say that we will do our best in every assignment and test we do. We have to learn that this is enough.</p>
<p>The beauty of this strategy is that <em>we</em> get to decide if we are a success. Did you do your best and worked hard for that last assignment? If so, then you can look at yourself as a successful student. Notice how this isn’t contingent on getting good grades. Instead, it’s about keeping your focus on aspects of your learning that you can control.</p>
<p>Remember, trying to control external factors in your life is a hopeless pursuit. It may work from time to time, but overall it is a losing strategy. Instead, apply your effort to things which you <em>can</em> control, like your effort. You will find that it makes you much more relaxed, leading to better work.</p>
Mon, 10 Dec 2018 00:00:00 +0000
https://jeremycote.me/not-necessary
https://jeremycote.me/not-necessary