Jeremy CoteScientific and mathematical thinking for the curious.
https://jeremycote.me/
The Grit to Push Through<p>If you ask someone what the point of a mathematics or science degree is, chances are they will tell you a tale about becoming a great problem-solver and seeing the world through new eyes. This has become a sort of battle cry for many who want to encourage people to learn about science and mathematics. The problem-solving skills you develop during these degrees allows you to be valuable in a wide range of careers later on.</p>
<p>While this is true, I would argue that it’s not one of the main skills you learn as a student. Instead, the skill you develop is <em>persistence</em>.</p>
<p>Let me tell you a story. When I was taking a quantum mechanics class, the professor assigned homework from a textbook. A few of the problems were marked as “very difficult”. When I began working on them, I knew I was in for a long calculation. It’s not that the problem was difficult so much as it was time-consuming. I even knew what I needed to do, but it just took forever (and it wasn’t clear where to start).</p>
<p>Multiple times, I felt like giving up. I wanted to find a shortcut, some way to make this less painful to do. If I was being rational, I could have decided that my time was being wasted on such a problem. I would only lose a few points, so it wouldn’t be the end of the world.</p>
<p>Of course, I have the “lovely” problem that I can’t hand in work that isn’t completed to the best of my ability, so skipping the question because it was too tedious wasn’t an option. Even with the hours ticking by, I gritted my teeth and finished the question.</p>
<p>Was it worth the extra time to get a few more points? Not really. The tedious part was a bunch of algebra, which also meant that the problem wasn’t any more illuminating when I finished. In the moment, it felt like a thankless task. However, the benefit came later. What I learned from doing a problem like this is that I <em>can</em> get through it with perseverance. If I set my mind to it, I can get a problem done. This is what I believe to be one of the best skills I’ve acquired through my science and mathematics degrees. Being unreasonable and pushing through the tedium and difficult parts of a problem to see it to the end is important. If not, you will tend to give up when you <em>should</em> push through.</p>
<p>Having the grit to push through is a skill that’s much more applicable than to just mathematics and science. Grit is an essential part of doing work that is important to us. Whether it’s writing, drawing, dancing, practicing a sport, making music, working on a business, or doing science and mathematics, grit is what helps us make breakthroughs when everyone else has given up. Plus, while it can be argued that others have more skill or talents from genetics or the environment, <em>you</em> control your decision to continue working when it seems useless.</p>
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<p>This idea of developing grit during a science or mathematics degree is also why I don’t like having tests with time limits. Think about it. If you establish a time limit, you’re telling students to give up after this point. But isn’t it more impressive if the student keeps on working until they succeed? Sure, it might mean they have more trouble than others, but I would want to have that person on my team before the person that gives up after a few minutes.</p>
<p>One might object and say that people would all just stay until they get everything right, so the class average would be 100 (barring any mistakes). I don’t think this would be true, since my experience is that most students tend to give up quickly when they don’t know what to do. They don’t want to sit and think when they are stuck.</p>
<p>The point I want to emphasize here is that problem-solving skills are great, but I think developing grit is a skill that isn’t recognized as much as it should be. Of course, I’m not saying that we should persevere to the point of delusion, but being able to push past the initial point of discomfort is something we should all want to do. That’s why I think it’s one of the most important benefits of doing a science or mathematics degree, since you’re frequently put in the position of struggle. You learn that being stuck isn’t a bad thing, and is often temporary. You learn that giving up shouldn’t be your initial instinct, but one that is only considered after all other options are exhausted.</p>
<p>I know that this will be something I carry with me throughout my life, even if I don’t stay within the areas of science and mathematics forever. I’m thankful for learning this skill no matter where life takes me.</p>
Mon, 12 Nov 2018 00:00:00 +0000
https://jeremycote.me/the-grit-to-push-through
https://jeremycote.me/the-grit-to-push-throughBehind the Equations<p>In secondary school, students in physics learn about the kinematics equations. These equations describe the motion of objects under a constant acceleration (often gravity). There are several equations, which describe the relationships between acceleration, speed, position, and time. In particular, here is one of the equations:</p>
<p><em>x(t) = x<sub>0</sub> + v<sub>0</sub>t + at<sup>2</sup>/2.</em></p>
<p>This equation lets us find the position at any time <em>t</em>, since the other parameters are known. You might even recognize this as the equation of a parabola.</p>
<p>Students learn about this equation and the others during their first course in physics. They are then encouraged to write down all the equations, and decide on which one to choose based on the parameter that is missing. In the above case, the speed of the object at time <em>t</em> isn’t present.</p>
<p>I once was working with a student, and I could see that they did not see the link between the equations. Wanting to probe this a bit further, I asked, “Isn’t it a bit strange that this <em>exact</em> arrangement of terms describes how the position of an object changes with time? For example, why in the world do we need to have a <em>t<sup>2</sup></em> term in the equation?”</p>
<p>The student agreed that it <em>did</em> seem strange. At the very least, it wasn’t obvious as to why this combination of terms produced the right answer. After all, if you just look at the equation, there’s not much telling you if this is the “right” combination<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>. So what’s going on here?</p>
<p>Of course, the answer is that the equations <em>do</em> come from somewhere. They come from analyzing Newton’s equation <em>F=ma</em> under the presence of a constant acceleration. But if that’s it, why don’t students learn about this first? After all, they encounter Newton’s equation early on.</p>
<p>The issue is that one needs to use calculus in order to give a satisfactory derivation of the kinematics equations. In secondary school, students don’t have this background of calculus yet, so they cannot follow the steps (even though they are quite simple). As a result, students are presented only with the final answer. The derivation is left out. The implicit message is that the derivation isn’t that important.</p>
<p>The main consequence of this is that students end up seeing these equations as “magic”. In other words, they feel that there’s <em>no way</em> they would have been able to come up with these equations on their own. This is untrue, but the lack of derivation forces them to accept the equations on authority.</p>
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<p>I hope you can agree with me that this is not a good situation to be in. Accepting equations on authority is in the <em>opposite</em> spirit of both subjects. This encourages students to memorize equations without understanding where they come from and why they are true. After all, if you <em>know</em> where an equation comes from, you don’t have to worry as much about memorizing it. If you forget it, you can always work it out again.</p>
<p>We are doing a disservice to students by forgetting about working out the details. We are teaching them that it’s more important to know how to use an equation than where it comes from. We are sending the message that equations and formulas are something to be recalled, but the proofs themselves don’t matter.</p>
<p>Yes, it will take more time to present the material. That means there won’t be as much time to do practice problems, and the pace might be slower. But students <em>want</em> to learn about the explanations behind the equations! I’ve found that students are interested in finding the connection between the formulas they have and the concepts they learn. It’s much more satisfying to be able to connect these in one’s mind, and students agree with me.</p>
<p>Therefore, we <em>need</em> to move beyond only presenting formulas. This will require a lot of work to create proofs of equations that students learn. It will also require creativity in the presentation, which is where the expertise of the teacher will come in.</p>
<p>I’m not suggesting that <em>everything</em> needs to be proved. Even in university, some proofs are skipped due to time constraints. <strong>But don’t let results stand on their own.</strong> If you don’t have time to prove them, give the students the appropriate resources so that they can look at the connections on their own. Do all you can to make sure that students aren’t forced to accept statements and equations out of the blue. If the students don’t have the requisite background, <em>explain that to them</em>. Don’t just say that “it works”. Give some intuition so that their explanation of a concept doesn’t only involve stating the equation.</p>
<p>Our purpose in teaching these concepts should not be about the results. It’s about the links between concepts and the way of thinking that is important. Proofs exemplify these principles, while the formula at the end is just a nice endpoint. The real learning comes through understanding <em>where</em> this equation originates, not the fact that it works.</p>
<p>Anyone can learn to substitute numbers into an equation and get an answer. But <em>why</em> does this equation do the thing you want it to do? Is this form particular, or are there different ones that can be used? Does the equation seem surprising? If so, can a student work from first principles to get back to that equation? These are the questions that should be asked more often in the classroom.</p>
<p>Behind every equation lurks an explanation. Don’t be fooled into thinking the equation itself is the point. Always shine a spotlight <em>behind</em> an equation to illuminate its origin and the reason it works.</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>The one thing that you could argue is that the dimensions of both sides of the equation are the same. If you look at each term, they all have the dimensions of length, so that matches up. It doesn’t explain why there is a <em>1/2</em> in front of a term, though. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Mon, 05 Nov 2018 00:00:00 +0000
https://jeremycote.me/behind-the-equations
https://jeremycote.me/behind-the-equationsQuantities in Context<p>One of the differences between physics and mathematics is that mathematicians don’t tend to care about the units they are working with. In fact, they will usually consider all quantities as unitless<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>. This makes it easy to compare quantities, because one only has to look at the number itself. If you have two numbers, 5 and 9, you know that 9 is the larger quantity.</p>
<p>In physics, however, the situation isn’t quite the same. That’s because our quantities have dimensions attached to them. As such, it doesn’t make sense to say that 5 L is larger than 3 m, since they don’t describe the same property of a system. Therefore, in physics we require more than just a comparison between the value of numbers themselves. We want the <em>dimensions</em> of the quantities to match up as well.</p>
<p>Things can get tricky though, because different people use different <em>units</em> to describe the same dimension. For our notion of length, we have plenty of units, from the metre to the yard to the light year. They all represent length, but there’s a <em>huge</em> difference between one metre and one light year. <strong>From this, we conclude that in addition to requiring quantities to be in the same dimension in order to compare them, they <em>also</em> need to have the same units.</strong></p>
<p>A related notion is that of a quantity being “large”. If anyone tells you that a quantity is large, your first question should be, “Compared to what?”</p>
<p>There is no such thing as an “absolute” size. In other words, a quantity can only be large compared to something else. You might think that 300,000,000 m is an enormous length, but light travels that distance in about one second, which means that a light <em>year</em> is about 31.5 million times this length. As such, it doesn’t make sense to only say that 300,000,000 m is large. It needs to be compared to something else. Only then can the notion of “large” have meaning. (Think of it like an inequality. You can’t have an inequality with only <em>one</em> quantity. You need two.)</p>
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<p>If this seems like it can get messy with all of the different units people use, you are correct. This is why many physicists like to use dimensionless quantities such as ratios. If the ratio involves two quantities with the same dimension, the ratio will “cancel out” the dimensions, leaving a dimensionless quantity. This is useful because it means one doesn’t have to worry about the units involved in the problem. No matter what units you use to measure my weight and your weight, the <em>ratio</em> of our weights will be the same no matter what instrument we use.</p>
<p>The next time you hear someone saying that a quantity is large, make sure to remind yourself what they are comparing their quantity too. Without doing this, there’s a chance for misunderstanding or manipulation. Therefore, don’t jump to conclusions when numbers are thrown around with the implication that they are large or small. Demand another number to compare it to!</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Actually, theoretical physicists like to do this too, since everyone agrees that dealing with units can be annoying. This is why you might see physicists saying that the speed of light is <em>c=1</em>. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Mon, 29 Oct 2018 00:00:00 +0000
https://jeremycote.me/quantities-in-context
https://jeremycote.me/quantities-in-contextBlack Boxes<p>Why is the area of a circle given by πr<sup>2</sup>?</p>
<p>I’m not asking why it’s in this specific form. Rather, I want to know why this is true. Can you tell me? Can you <em>convince</em> me?</p>
<p>Let’s take something a bit more concrete. I bet you use a lamp every day to light up something in your home. Can you explain how the lamp works? What makes the bulb shine? How does the electricity work to create this light?</p>
<p>These are all questions that have answers.</p>
<p>You <em>know</em> they have answers. I’m not asking technical questions here. Just a simple explanation for how the lamp works would please me. You don’t have to start talking about the various particles that make up the lamp, or how light behaves as a wave and interacts with the environment such that we are able to see.</p>
<p>Still nothing?</p>
<p>I’m not surprised. To be honest, I can’t even give answers to some of these simple questions.</p>
<p>You might think that we should be able to answer these types of questions. After all, we <em>do</em> use these things every single day. We should know how they work, right? And yet, most of us don’t know the inner workings of these machines and processes. We just know that they work, and that’s enough for us.</p>
<p>There’s a technical term for this in science (computer science in particular): a black box. This expression refers to a process or a device which we can give an input and get an output, but the inside of the black box remains unknown. The only feedback we get is the output.</p>
<p>This is not something we want in science. We would <em>much</em> rather have a process in which we knew each step along the way and how it went from step to step. However, in the absence of anything else, a black box that gives results is still useful. We aren’t going to throw away something that works just because we don’t know much about it! Black boxes signal that we have more to learn about (starting with the inside of the black box).</p>
<p>We all carry around our own black boxes. These are processes that we <em>know</em> happen around us, yet we don’t have a clue what the inner workings are like. From our cars to our refrigerators to the internet, most of us don’t <em>actually</em> have any idea how these things work. We might be able to jumble along together an ad-hoc explanation, but these tend to be wrong and not thought out at all.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1535730025/Incorrectly_Correct.png" alt="Incorrectly Correct" /></p>
<p>When reflecting on this though, we often don’t care that we carry around these black boxes. It’s unreasonable to expect us to be knowledgeable about <em>everything</em> we use, so who cares if we don’t know how our car works? We know how to drive it, and that’s all that matters.</p>
<p>To a certain degree, that’s true. Often, we are able to get by with only the knowledge of how to <em>use</em> our black box. We don’t need to know how it works. As long as we steer the car and follow the signs on the road, we trust that the car will do its thing and not malfunction.</p>
<p>The issue is more about our <em>perception</em> of these black boxes. How many black boxes do you think you have? Chances are, the number you gave is too small. <em>Way</em> too small. The simple truth is that we use black boxes throughout all of our lives, often without realizing it. This is of particular interest when we consider education.</p>
<h2 id="black-boxes-everywhere">Black boxes everywhere</h2>
<p>So we are in agreement that people <em>do</em> use black boxes in their lives. As such, it shouldn’t be a surprise that students use black boxes in their education. These are of particular use in subjects such as science and mathematics, where one can get many answers without knowing the underlying concepts. What <em>seems</em> like learning is just a focus on the outputs.</p>
<p>The danger with black boxes in education is that they are seductive. They represent a way to gather a lot of “surface-level” knowledge without digging deep to think about the concepts themselves. This means that if a student is having difficulty, it’s much easier to learn how to use a black box than to go through the longer process of absorbing the content.</p>
<p>I see this quite often in my field of physics. Physics uses a lot of mathematics, but it’s not as concerned about the mathematical concepts. This means that a lot of the <em>tools</em> of mathematics are transported to physics, and can be used without knowing the theory beneath.</p>
<p>Physicists encounter a lot of differential equations. However, professors teaching physics don’t often care about <em>how</em> one solves the equations. Instead, there are the staple differential equations, such as the simple harmonic oscillator, which every physics student knows and memorizes before they are done their education. Can they all explain the process of <em>finding</em> this solution (apart from telling someone to “plug it in and see”)? Probably not.</p>
<p>Here’s another example, this one not only about physics. If you take the function <em>x<sup>n</sup></em>, what is the derivative? Any student who has taken a first course in calculus will tell me that it’s <em>nx<sup>n-1</sup></em>. This is correct. But can the student then go on to explain <em>why</em> this is true? Perhaps the student who is in their first calculus class can (because they are in the midst of working with the definition), but I bet that many others who have taken <em>many</em> calculus courses and have long-memorized the power rule cannot. Instead, they might say something like, “That’s just how it is.”</p>
<p>Why does this happen? Why do we go from deep, underlying knowledge to trading it in for a black box that produces the right answer each time? The reason, I suspect, is because it’s much easier to remember the power rule than working it out from first principles each time. In fact, <em>no one</em> does that, because it’s a waste of time. Once we know the rules of the game, there’s no need to go back and rederive everything.</p>
<p>The issue occurs when we go for so long without looking at the first principles argument and only remember the rules themselves. <em>This</em> is what we want to avoid. It’s at this point that our knowledge goes from deep understanding to being a black box. We then cease to be knowledgeable about the subject. Instead, we become proficient at <em>using</em> the tools from the subject.</p>
<p>There’s a difference here, and it’s one that isn’t highlighted enough in school. Knowing how to use the tools of a subject to solve problems is a skill, but it’s <em>not</em> the same as understanding how those tools were developed. This is critical, because it informs how we make decisions about what to teach students. Do we want to focus on giving them skills to solve problems, or do we want to emphasize the concepts beneath? I don’t think we should focus only on one, but we are deluding ourselves if we think that schools (particularly early on) are emphasizing the importance of deep understanding. From my perspective, the priority is skill first, deeper understanding second. This aligns <em>precisely</em> with the use of a black box.</p>
<p>Students aren’t incentivized to dig deeper and develop more of an understanding of their subject. They’re incentivized to solve problems quickly and know how to do a lot of things. The byproduct of this is that black boxes are used to keep up.</p>
<p>Again, I’m not saying that the black boxes aren’t useful. They <em>are</em>, but if we want to do more than pay lip-service to the idea that students should have a deep understanding of their subject, we <em>need</em> to highlight this tendency to default to black boxes. On the other hand, if our priority is to only develop the skills of students, then fine, we can keep on using black boxes. We just can’t have it both ways.</p>
<h2 id="what-are-your-black-boxes">What are your black boxes?</h2>
<p>I hope I’ve convinced you that black boxes are everywhere. Now, I want you to think of your own life. What are your black boxes? We all have them, and my objective here is to get you to <em>think</em> about what they are.</p>
<p>At some point, we all hit a black box where we just don’t know how something works. This isn’t a bad thing. In fact, it’s a good exercise to see how deep you can go. Chances are, you won’t go far with most things. You will only be able to go deeper with the subjects you are passionate about. That’s okay and normal.</p>
<p>Now, think about your black boxes. Can you push past them and get a better understanding of the underlying mechanics? Pick a few that you want to get past, and start learning about them. Read a book on the subject, or ask a friend who is knowledgeable. I warn you that this is difficult, painful work. Understanding something isn’t a trivial task, so make sure you <em>really</em> want to learn.</p>
<p>You will know that you aren’t using a black box anymore when you can explain the idea or concept to someone who has no idea how it works. <em>This</em> is what you should strive for. If you can explain the concept (and not just recite it from a book), there’s a good chance you aren’t using a black box anymore.</p>
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<p>This is my goal for education, both for myself and for the students I work with. My motto is “mathematics and science without black boxes”. More than anything, I want to help students <em>understand</em> what they are learning, not just how to use the tools to solve problems. The pendulum in education has swung too far in the direction of building skills without knowing the underlying concepts. My aim is to help nudge it back in the other direction.</p>
<p>There’s nothing wrong with building problem-solving skills. But we miss out on a large portion of the value of education when we <em>only</em> look to develop our skills. If you ask mathematicians, they will tell you some variant of “mathematics is art”. Many <em>won’t</em> tell you that they do mathematics to only solve problems in the world. Instead, it will be about gaining a deeper understanding. In essence, they are trying to push through their own black boxes. Why? Because they value knowledge in addition to solving problems.</p>
<p>Mathematics and science doesn’t have to <em>only</em> be about being skilled with tools. It’s an opportunity to inspect one’s black boxes, and work at opening each one up to peer inside.</p>
Mon, 22 Oct 2018 00:00:00 +0000
https://jeremycote.me/black-boxes
https://jeremycote.me/black-boxesThe Priority of Education<p>Is not learning.</p>
<p>Here’s a question. What’s the best outcome that can happen when you take a course?</p>
<p>The most common answer (and one I would give myself) is to get 100%. To do everything perfectly, never making a mistake. Most students agree that this would be the ultimate goal. How could anything be better than getting a perfect grade?</p>
<p>When you look back on a course, what do you think about? Chances are, the final grade you got is a good indicator of how you thought about the course. It’s the only tangible metric left, so we base our reflections on that one number. The number is <em>supposed</em> to encapsulate our knowledge anyway, so where’s the harm in using the grade as a proxy for how we felt about the course?</p>
<p>The problem is that you never even mentioned that you took a course in order to <em>learn</em>.</p>
<p>It’s worth thinking about this, because we go through tons and tons of classes in school worrying only about our grades. At the end of the day, who cares if <em>you</em> think you learned a lot in a course? If your final grade doesn’t reflect this, no one else will try and dig deeper to say, “Yes, you might not have finished with a good grade, but you’ve learned so much and grown throughout the course.” As such, the incentive is to turn away from learning, and focus on getting good grades.</p>
<p>Good grades unlock new opportunities. <em>Saying</em> you learned a lot in a course but only getting an average grade doesn’t convince anyone. Therefore, it’s natural that we learn to focus on getting good grades instead of learning in general. If learning happens during a course, that’s great, but it isn’t something to focus on.</p>
<p>Think about how you feel once you’ve written the final exam for your course. Do you think, “How can I fit these new concepts and principles into the way I think?” I’d predict your thoughts go along the lines of, “I’m finally done with this course! I just hope I did enough to get a good grade.” These two attitudes illustrate the difference between going to school to learn and going to school to get good grades.</p>
<p>It may be surprising, but you <em>can</em> get good grades in school without learning. It’s possible. You don’t even have to turn into a memorization machine. You just have to know what to focus on at the right time, and be a good performer on tests.</p>
<p>In a sense, we <em>do</em> learn something in school. We learn that society rewards those with good grades, which means we have a choice. We can either focus <em>only</em> on getting good grades to impress others, or we can dig deeper to do the difficult work of learning. That means being able to explain topics to others. It means being able to think through a problem without just wondering what special fact or result is needed to resolve it. Learning is a different kind of skill, and it’s one that is too often in short supply at school.</p>
<p>I’m not sitting on my high-horse here. I feel the pressure to get good grades just as much as you, the reader. When I’m overwhelmed by the amount of work to be done, I retreat to my defaults: get good grades, don’t worry about learning itself. I <em>know</em> this is the wrong mentality, but I also know that a lot of my academic future relies on getting good grades.</p>
<p>My advice is simply this. As much as we spend time focusing on getting good grades, you <em>need</em> to take time to think about if you’re also learning. This is more difficult than it seems when you’re getting good grades. That’s because you will be able to answer many different problems that involve the concepts you looked at in class. However, like I wrote above, a key indicator of learning material is being able to explain it on your own. If you don’t have your notes, can you give an impromptu explanation to a friend? How about someone who knows nothing about the subject? If you can create an explanation to that sort of person, then you’re doing alright in terms of learning.</p>
<p>I think many people will be surprised to find that they <em>can’t</em> do this. If that’s you, don’t worry! It’s an indication that you have more to learn. Take it as an opportunity to go back and try to soak in the information from your class, but also remember to <em>explain</em> it to others.</p>
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<p>The priority of our education system isn’t aligned with <em>learning</em>. It’s aligned with getting good grades so that you can stand out. Therefore, if you want to be more than just the student who can get 100% in your class, then it’s up to <em>you</em> to focus on the more intangible aspects of learning. No one will look out for you, or care that you’re doing “more” work. However, the difference <em>will</em> show up in the long run, because you will understand the topics deeply versus those who only prepared enough to ace the final exam.</p>
Mon, 15 Oct 2018 00:00:00 +0000
https://jeremycote.me/priority-of-education
https://jeremycote.me/priority-of-educationA Splash of Colour<p>When giving a presentation, it’s difficult to present ideas in science or mathematics <em>without</em> the use of equations. It’s possible, but unless you’re exploring a geometry problem, you’re probably out of luck. If you want to get a message across to your audience that is more substantive than a bunch of emphatic adjectives about science, you need to use equations.</p>
<p>Unfortunately, equations in presentations don’t have a habit of looking (How shall we say it?) <em>nice</em>. Instead, they are either crammed in beside plots of data or formatted in a way that makes them difficult for even scientists to decode. It sometimes makes me wonder if scientists spend all of their time on research and forget that it’s important to make things presentable for others.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1535058616/FinalVersion.png" alt="Not quite what you would call a "final" version." class="centre-image" /></p>
<p>It doesn’t have to be this way. In particular, I’ve learned that we can make equations much more readable when giving a presentation. It’s a small change that makes a big difference, allowing your audience to follow the equations without getting lost. (And let’s be honest. Getting lost happens a lot when listening to presentations!)</p>
<p>Let’s pretend you have a slide that looks like this.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1533407621/TypicalSlide.png" alt="The typical slide." class="centre-image" /></p>
<p>There’s nothing particularly <em>wrong</em> with the slide. However, everything is neutral, which means that no elements jump out at the audience. The equation is there with all of the variables defined, but everything has the same emphasis. This could be made a lot better.</p>
<p>First, if you’re anything like me, you enjoy a classic colour pallet of black and white. You feel at home with the pre-colour era of media, and find there’s nothing wrong with black and white newspapers. If this resonates with you, don’t worry. I feel the same way. However, this isn’t the same century, and we’ve been able to add actual <em>colour</em> to our presentations. As such, we’re going to do something that might seem absurd, childish, or even (gasp!) <em>unprofessional</em>.</p>
<p>We’re going to colour the equations.</p>
<p>You heard me right. Even worse, I’m not talking about changing the colour of the equation from black to blue. I’m talking about changing the colour of each individual <em>variable</em>. If we take the slide from above and put it through a colour transformation, we might get something like this.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1533407620/ColouredEquation.png" alt="Eye-popping colour." class="centre-image" /></p>
<p>The advantage of this slide is clear. Instead of a neutral equation with the variables explained, the additional bit of colour gives the audience a connection between the variable and it’s definition. The connection is clearer, with the colours guiding the eye without needing to read.</p>
<p>“But wait,” you say. “I don’t want to turn my slides into a kaleidoscope of colour!”</p>
<p>I share your concern. Thankfully, we have this wonderful area of art called colour theory which deals with exactly this problem. By using an appropriate colour palette (such as the one from <img src="https://color.adobe.com/create/color-wheel/" alt="Adobe" />), you can make your slides look just as good as they did in black and white. Plus, you get the additional benefit that the audience can locate objects of interest on your slides with less difficulty. Which, in the end, is what we want.</p>
<p>Presentations are an important part of getting your groundbreaking research into the public eye. Unfortunately, a lot of presentations include slides with equations that can seem impenetrable. By adding colour, not only do you ease the separation of an equation into its components, but you <em>also</em> get the nice bonus of making your slides more friendly.</p>
<p>What more could you want?</p>
Mon, 08 Oct 2018 00:00:00 +0000
https://jeremycote.me/splash-of-colour
https://jeremycote.me/splash-of-colourOutside the Curriculum<p>Do you feel like you’re not getting enough out of what you do in class? Does mathematics feel boring, just a bunch of rules that you follow without more or less knowing <em>why</em>?</p>
<p>I don’t blame you. This isn’t necessarily your teacher’s fault either. Instead, it’s a mentality that we’ve adopted with respect to your education. Take a bunch of mathematical concepts that are easy to test, and make students like you do lots of problems. If you can answer the problems correctly, you get good grades and move on. If not, we’ll likely <em>still</em> let you move on, even though you have no business studying more things when you haven’t mastered these concepts first.</p>
<p>It’s a bad situation on both sides. If you’re good in class, then you can get bored from the repetition of ideas. On the other hand, if you struggle, you’re spending most of the time in class scratching your head and wondering how in the world these symbols all work.</p>
<p>There needs to be a better way to do this.</p>
<p>I don’t have all the answers. I realize that I can’t start implementing radical ideas in the classroom (mainly because I’m only a tutor). <em>But</em>, I know that the world of mathematics is vast and wonderful, and in school you only get to see a <em>tiny</em> sliver of it. Perhaps I’m being too cynical, but I would argue that what you see isn’t even part of the “greatest hits”. There are some good parts in the curriculum, but there are parts that leave me wondering why someone would ever want to teach students this as a requirement.</p>
<p>My point is that you deserve better. I think it’s fine to say that you dislike (or even hate) a certain subject. However, you have to at least dive into the subject a little bit. In school, I don’t think you learn enough about mathematics in order to give it an honest rating. That’s not your fault. It’s the fault of an old educational system that needs to be updated.</p>
<p>Let’s be realistic here. Changing the way an institution functions within the next few years you’re in school isn’t going to happen. The change will occur over many years, as schools and governments get more pressure to change how we do things.</p>
<p>So what can you do now?</p>
<p><strong>The number one thing you can do is to explore the wide space of mathematics.</strong> Instead of only thinking about mathematics inside the context of your classes, learn about more topics online. Dive into some articles on a subject that seems interesting. Watch a video online (which there are many great introductory ones). Ask your teacher for topics. Do <em>something</em> is my point. The only way to learn more is to go out and find more resources.</p>
<p>The most surprising thing you will find is that mathematics <em>does not</em> look like the kind of stuff you do at school! This will be a shock, I know. Right now, mathematics seems like its a mixture of formulas and finding the right numbers to plug into these formulas. But within the wider scope of mathematics, this is not the focus at all. The world of mathematics is filled with a lot more creativity than you would expect from the work you do in school.</p>
<p>I want to be clear here. The skills you develop in your mathematics classes <em>are</em> important. However, they are just that: skills. The wider world of mathematics is concerned with how you can <em>use</em> these tools to find out new truths about various objects. Sadly, this isn’t something you often get to see in your mathematics classes. This is why I’m telling you to explore the world of mathematics outside of your classroom. Only then will you get a flavour of what mathematics is all about.</p>
<p>If you do that, then you can form your opinions on mathematics. But I think that once you see how much <em>more</em> there is to mathematics than what you see in school, you will find that mathematics isn’t boring at all. Only <em>certain topics</em> aren’t as interesting. And that’s alright. Just don’t give up on mathematics because of a few topics that aren’t as engaging as the others.</p>
<p>The lesson I want you to draw from this is that mathematics has something for <em>everyone</em>. Schools don’t cater to this fact, but it’s true. Therefore, don’t give up on mathematics. Go out and explore! I guarantee you that there will be <em>something</em> you find interesting.</p>
Mon, 01 Oct 2018 00:00:00 +0000
https://jeremycote.me/outside-the-curriculum
https://jeremycote.me/outside-the-curriculumThrough the Minefield<p>When mathematics makes sense, each piece seems to fit together. There’s no question about how to do things, because it’s all natural. This is what happens when you become good at algebraic manipulations. No matter how hairy the expression, you’re able to deal with it. Double-decker fractions aren’t frightening. Sure, it might be tedious to work through, but it’s doable. It’s sort of like strolling through a path in a meadow. Nothing is blocking you, and the way forward is clear.</p>
<p>Contrast this to when mathematics <em>doesn’t</em> make sense. When this happens, it’s like trying to navigate a minefield. Each step is uncertain. You worry about how one wrong move will ruin everything.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1535297327/Handwaving/Published/Mathematical_Minefield.png" alt="Navigating the minefield is a scary thing." class="centre-image" /></p>
<p>Both experiences happen to everyone. There are times when learning goes smoothly, and times where it is anything but smooth. Of course, some people experience one side of this coin more than others. For those that find mathematics to be a minefield on average, it’s not surprising that they will grow to dislike mathematics. If <em>I</em> had to do something which made no sense to me every single day, it wouldn’t take long before I became frustrating with it, too.</p>
<p>The question then becomes: how do we move the needle so that more people find mathematics to be something they understand?</p>
<p>I want to be clear here that I’m not saying those who enjoy mathematics don’t experience any difficulties. There are times when the concepts are challenging. However, the difference is that they have enough forward momentum that they keep on pushing through the difficulties.</p>
<p>To solve this problem, I think there are two related points that need to be addressed.</p>
<h2 id="does-it-make-sense">Does it make sense?</h2>
<p>The first is that mathematics needs to make <em>sense</em>. I would say this is one of the most important aspects of teaching that we don’t emphasize. Mathematics isn’t a bunch of random rules that we need to memorize and apply in just the right way so that the answer is magically spit out. Instead, mathematics is built on a few key rules (axioms) and uses logic to build more complex structure. This means you don’t <em>need</em> to memorize everything. By knowing a few key rules, you can branch out and do a lot more. Furthermore, you should be able to connect what you’re doing back to those axioms.</p>
<p>The way we mess this up is by chopping ideas up into these arbitrary categories that make the concepts seem distinct, even though they aren’t. The most egregious example that comes to mind is the insistence in secondary school to show students how to solve a system of two linear equations by using elimination, substitution, or comparison. These methods are presented to students as different ways to solve an equation, but it’s often not pointed out that these are essentially doing the <em>same</em> thing. Sure, the methods might be slightly different, but there’s not much separating them. After all, comparison is just a special case of substitution, and elimination isn’t that much different either.</p>
<p>What’s the key insight here? Instead of focusing on three arbitrary “methods” to solve a system, the main emphasis should be on the fact that you can do basically anything to an equation, as long as you do it to both sides. <em>That’s</em> the key insight, not these three arbitrary methods.</p>
<p>When we emphasize different methods and forget to mention how they are all similar, students can get confused and think that you’re only allowed to use certain ones depending on the situation. I know this because I’ve worked with students who had this impression. It’s no fault of their own, because these methods are listed as distinct. Why should they expect them to be linked?</p>
<p>At it’s core, mathematics is a discipline that makes sense. Not to me in particular, but to anyone who is willing to sit down and chew through the arguments. It’s not always easy, but the results are accessible. Therefore, instead of emphasizing rules and procedures without talking about the underlying parts, we should focus <em>first</em> on the underlying mechanisms and show how they give rise to our basic rules. My prediction is that this would shift the mindset of students from “mathematics is a bunch of rules that I need to apply in <em>just</em> the right way” to “mathematics is a subject that makes sense if I carefully follow the arguments.”</p>
<h2 id="taking-a-small-step">Taking a small step</h2>
<p>The second is that we need students to take small steps. If you’re like me, you’ve tried to work through a problem or a piece of mathematics and became confused when an author suddenly took a large step. The result is that you become disoriented, since the step was too big for your tastes. This can happen to everyone. We all have our own preferred pace in which to tackle a problem. As such, it’s no surprise to me that the “default” step size which is present in textbooks and in teaching practices can be too large for some.</p>
<p>The result is that some students become bewildered and are unable to follow. And since a classroom isn’t often made to suit the needs of one student but of thirty, it means they can be left behind on the mathematical journey. For the student, the only way to catch up is to take what is said as a given and just commit it to memory. Instead of going over each argument in detail and <em>thinking</em> about it, they have to accept it without further investigation. This reinforces the notion that mathematics is a bunch of facts that need to be accepted, instead of a series of reasoned arguments. Who can blame a student for holding this view when they have been left behind?</p>
<p>This is both an easy and a difficult fix. It’s easy in the sense that we know how to help the student. They need to sit down with the material and go through it at their own pace, taking steps that seem reasonable to them. I often catch myself going too fast when working with a student, and when I do I try to slow down, because the explanation isn’t for <em>me</em>. It’s for them. On the other hand, the fix is difficult because teachers can’t give <em>each</em> student this opportunity at all times. The reality is that a classroom is made to serve many students, which means the time each one gets with the teacher is limited.</p>
<p>As a tutor, I get the opportunity to work with the students on a one-on-one manner. This helps, but not everyone has access to a tutor. The best advice I could then give to a student is to see after class if they can go through the arguments that were presented in class. It’s during this time where they can see if the steps taken were acceptable. If not, the student needs to work through the confusion, or else they will be forced to accept the results without understanding them.</p>
<p>Ideally, a student would go through any claim with small enough steps that each one seemed obvious. Sure, that means it might take longer to understand a result, but I would argue it’s preferable to taking the knowledge at face value without understanding the arguments. (Of course, this doesn’t necessarily translate into better grades.)</p>
<hr />
<p>The unfortunate reality is that I see students who look at mathematics as a minefield, with every step an uncertain one. The reason this happens is because we’ve taught them to value facts over the arguments that <em>link</em> those facts. It’s the links which are so much more important, but since they aren’t emphasized on exams, students don’t internalize them. The result is that a student might get good marks, but this doesn’t mean they understand the mathematics.</p>
<p>My goal as a tutor is to help bridge this link. Instead of getting students to take big, uncertain steps through what looks like a minefield, I want them to take smaller steps through a meadow. At its core, mathematics is understandable. We just need to stop focusing on the results and more on the underlying mechanisms.</p>
Mon, 24 Sep 2018 00:00:00 +0000
https://jeremycote.me/through-the-minefield
https://jeremycote.me/through-the-minefieldVisuals in Mathematics<p>There’s no doubt that writing is a useful tool. If anything, I’m biased <em>towards</em> writing. I write every day, so I know what it means to use words to craft an explanation. If you can use the right words in the right arrangement, almost everything becomes clear.</p>
<p>That being said, there’s still a difference between writing and <em>communicating</em>. As much as I love writing about physics and mathematics, I realize that using this medium to craft explanations can be problematic. This is why I rarely write pieces with long calculations without using something in addition to words. It’s not that writing is <em>bad</em>. Rather, it’s that writing on its own isn’t great at communicating mathematics and physics concepts.</p>
<p>Thankfully, we have it a lot better today than in the past. If you look at older texts on mathematics or physics, you will see that <em>everything</em> was communicated using words. The end result is that learning required both a desire to understand a new concept <em>and</em> the patience to decode the text. This isn’t great for someone who is having a difficult time with the concept itself. As we know, learning can be difficult, so it should be our priority to craft explanations that lead students to understanding without needless barriers.</p>
<p>Remember, I <em>like</em> writing. I’m not saying we should quit writing explanations about mathematics and physics just because writing isn’t the best medium. I’m suggesting that we should complement our writing with other media.</p>
<p>In particular, consider the under-appreciated diagram. A diagram conveys both words <em>and</em> the relationships between them. If you’re working through a mechanics problem, it’s often helpful to draw a diagram. This lets you see the different constraints on the system and lets you set up the coordinate system. Sure, this could be described in words, but writing it out would be tedious and wouldn’t convey the idea in as simple of a format as a diagram. The best thing about a diagram is that it’s <em>visual</em>, which means you can consume it quickly. You don’t have to parse through a paragraph of text while simultaneously building up the diagram in your mind. Instead, you get the diagram as part of the explanation. This prevents you from building the <em>wrong</em> diagram in your mind, and it’s better at conveying the message than a paragraph.</p>
<p>Here’s another example. Suppose I wanted to convey the fact that the total revenue from an event was the sum of the sales from the three ticket types. I might say that the first type of ticket cost ten dollars, the second type cost fifteen dollars, and the third ticket cost twenty dollars. To find the total revenue, one simply has to multiply the number of the first type of tickets sold by the cost of that ticket, and do the same for the other two ticket types. Taking the sum would give the total revenue.</p>
<p>There’s nothing wrong with writing it out like this. If you’re like me though, the above paragraph is a bit of a mess to follow, with everything being spelled out in words. Instead, we could just label the revenue as <em>R</em>, and the number of tickets sold by each type as <em>a</em>, <em>b</em>, and <em>c</em>. Then, we could skip the confusing paragraph and write:</p>
<p><em>R = 10a + 15b + 20c</em>.</p>
<p>We were able to compress our long paragraph of explanation into one line that explains what each variable represents, and then an equation giving the relationship. Even better we didn’t <em>lose</em> anything by compressing our explanation. In fact, I think the equation makes it even more clear, since we can imagine how the revenue will change based on the tickets sold. Plus, we have developed methods of manipulating equations using the rules of algebra. As far as I know, we don’t have an “algebra of paragraphs”.</p>
<p>I bring this example up to show that we already avoid using words when we can. I’m not suggesting something novel here. We don’t use equations in mathematics to make the lives of students difficult when they start out. We do it because it makes our lives easier in the long-term. In this same way, I think we should be placing our focus on making mathematics more visual.</p>
<p>It pays to be careful here. When I’m talking about mathematics being “visual”, I don’t think we should only do mathematics by drawing diagrams and sketches. That’s making the pendulum swing too far in one direction. What I would like to see is an emphasis on drawing pictures to accompany an explanation. Keep the algebra and the definitions there if you have to, but don’t stop there. Use visuals to convey an idea whenever you can. Students will thank you for it.</p>
<p>When I read a mathematics textbook, one of the first things given in each chapter is a list of definitions. Because of the nature of mathematics, these definitions are technical. However, what is often forgotten is that definitions tend to stem from some sort of observation. That initial observation should be given to the students. In particular, if the observation is visual, that should be shown. I can’t tell you how many times I’ve read through several definitions only to find myself scratching my head. The times where I was able to understand quickly was when the author included a diagram showing the idea in a visual manner.</p>
<p>If you’re reading about functions and you come across the terms “one-to-one” and “onto”, the definitions can seem cryptic at first. Sure, they are clear, but what do they <em>mean</em>? The best way to give students a visceral feel of what’s happening is by drawing two sets and showing how the elements in one set are mapped to the other. By including this diagram, the idea of a function being “one-to-one” or “onto” becomes clear. It’s not that the definition was inadequate. It’s that definitions can be difficult to parse, whereas a (good) visual leads to near-instant understanding. (Of course, some visuals can be confusing, but I would argue that’s the fault of the author.)</p>
<hr />
<p>I’ve only given a few examples here, but I hope you have been able to think of others by reading this essay. I think we can sometimes get in the habit of using words instead of visuals because words are easy to type and visuals take longer to prepare. However, my goal here is to convince you that including more visuals in your explanations will make it a lot easier for people to follow you.</p>
<p>It’s not that visuals will automatically transform your explanations into world-class pieces. It’s that visuals will let students absorb your ideas without needing to decode a bunch of text first. I’ve written about this before, but when a student is learning, the best thing they can have is a foothold from where they are to where they need to go. Without that foothold, it’s difficult to get to the next level. The gap becomes too great, and students just get frustrated. Visuals can be those footholds.</p>
<p>How does this change the way I’m going to teach and craft explanations? The biggest change is that I’m not going to let myself slide into writing because it’s comfortable. I love writing, but writing on its own isn’t enough to make a student follow. When I’m trying to learn, the best combination I’ve found is to mix written text with a lot of visuals. This means you get the advantage of being to explain details in depth with words, but you also get the benefit of sprinkling the explanation with visuals. This both breaks up the text (giving the reader natural “break points”) and let’s them ponder over what they’ve read in terms of a visual. I’ve found that this works well for me when I am learning, which is why I will do my best to include more visuals in any exposition piece I write.</p>
<p>At the end of the day, mathematics is about <em>ideas</em>. It also just so happens that we are wired to understand pictures and drawings very well. The saying about a picture being worth a thousand words exists for a reason. Writing is comfortable for me, but it’s not the best tool to use for explaining mathematics and physics concepts. It works well if you use it in conjunction with lots of visuals. But without them, you risk losing readers in a sea of words.</p>
<p>It’s difficult holding a bunch of new information in your head, so make the job easier on the reader by giving them the visuals they need.</p>
<h2 id="endnotes">Endnotes</h2>
<p>One thing I didn’t mention in the main piece was that I am aware that there are some people who have visual impairments. This means using visuals wouldn’t be helpful to them. I’m not sure how to address this, and it’s something I still have to think about. Just because <em>I</em> haven’t had experience with students who are visually impaired doesn’t mean they should be shut out from this discussion.</p>
<p>The second point I wanted to mention is that there’s a whole other wave of mathematics explanations which use animations and movement. I have a lot more thoughts on this, and it will be the subject of a future piece.</p>
<p>Finally, I realize that there’s a certain irony in not including any visuals in this essay. However, my argument is that I’m not explaining a concept in mathematics or physics. Don’t get too mad at me!</p>
Mon, 17 Sep 2018 00:00:00 +0000
https://jeremycote.me/visuals-in-mathematics
https://jeremycote.me/visuals-in-mathematicsThe Necessary Details<p>As a student in science, you’re taught how to understand the details, the gory bits of an argument or a concept. When you learn about Kepler’s law of equal areas being swept out in equal times, you’re not <em>just</em> told that fact, but you <em>prove</em> it. Each part of the argument is explained, and you get a full explanation.</p>
<p>This is great, but the problem is that we don’t get to learn how to <em>explain</em> a concept. We’re given all of the details, but the truth is that they aren’t all useful when explaining the idea to someone else. The other person probably doesn’t care about the technical details. They want the big picture, so focusing on the minutiae doesn’t help them. The result is that they become disinterested.</p>
<p>The details <em>are</em> important, but it depends on the situation. If you’re trying to learn a subject, then sure, look at all of the technical details. However, if you want an overview of an idea, getting into the details isn’t as important.</p>
<p>Achieving this balance is crucial when trying to communicate an idea. What are your goals? Do you want the person to have enough knowledge to explain the concept themselves, or do you want them to understand the <em>idea</em>? You also need to consider what <em>they</em> are looking for. Without aligning these two objectives, attempts to explain science won’t go well.</p>
<p>I’ve noticed this difficulty when trying to describe my work to my family. They have no background in physics, so my explanations can’t be technical. I therefore have to find other ways to explain what I do. Do they understand the details when I’m done explaining? Of course not, but that’s not my goal. My goal is to get them to the point where they could summarize what I do in a few sentences instead of blankly saying, “He studies <em>something</em> in science.”</p>
<p>As I’ve gone through my undergraduate degree, I’ve realized that this is something which isn’t emphasized nearly enough. Perhaps it’s the result of my specific program or the university I attend, but there’s not a lot of emphasis on outreach and explaining what it is we do to a broader audience. Maybe that’s because it doesn’t seem like an “important” part of the job of a scientist, but I wholeheartedly disagree with that sentiment. On a practical level, scientists are mostly funded by government agencies or academic institutions, which means this is public money. As such, I would argue we have an obligation as scientists to explain what we are doing to the public.</p>
<p>On a more philosophical level, I think it’s important to do this because science affects all of our lives. We learn about how the world around us works, and we get to be curious about our place in the universe. Explaining science is a worthy endeavour, and yet science students aren’t prepared to do this. Instead, they focus on working through problem sets.</p>
<p>It all begins with choosing which details are necessary. Of course, we could just say, “If you’re interested in science, look at a textbook!” This might help in terms of disseminating information, but it doesn’t take into account the amount of knowledge the average person has. A standard textbook will likely be too advanced for them, and it’s not designed to inform. It’s designed to give all the details, which is more than a person often wants.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_600/v1536588530/Handwaving/Published/NecessaryDetails.png" alt="This is probably a *bit* overboard." /></p>
<p>If you’re a science student, my wish is that you take the time to explain what you’re learning to others. In particular, try to explain your research or what you’re learning to those who have little experience in science. By doing this, you will get to practice the art of giving just the right amount of detail. You get the benefit of getting more people to learn about science, and I think you will find that your own knowledge of the subject will become more stable. After all, one of the best ways to internalize a concept is to explain it to others.</p>
<p>Just because we aren’t taught this in university, it doesn’t mean it’s worthless. It might be one of the more important things we do.</p>
Mon, 10 Sep 2018 00:00:00 +0000
https://jeremycote.me/the-necessary-details
https://jeremycote.me/the-necessary-details