The Priority of Education
Is not learning.
Here’s a question. What’s the best outcome that can happen when you take a course?
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?
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 supposed to encapsulate our knowledge anyway, so where’s the harm in using the grade as a proxy for how we felt about the course?
The problem is that you never even mentioned that you took a course in order to learn.
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 you 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.
Good grades unlock new opportunities. Saying 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.
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.
It may be surprising, but you can 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.
In a sense, we do learn something in school. We learn that society rewards those with good grades, which means we have a choice. We can either focus only 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.
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 know this is the wrong mentality, but I also know that a lot of my academic future relies on getting good grades.
My advice is simply this. As much as we spend time focusing on getting good grades, you need 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.
I think many people will be surprised to find that they can’t 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 explain it to others.
The priority of our education system isn’t aligned with learning. 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 you 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 will show up in the long run, because you will understand the topics deeply versus those who only prepared enough to ace the final exam.
A Splash of Colour
When giving a presentation, it’s difficult to present ideas in science or mathematics without 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.
Unfortunately, equations in presentations don’t have a habit of looking (How shall we say it?) nice. 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.
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!)
Let’s pretend you have a slide that looks like this.
There’s nothing particularly wrong 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.
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 colour to our presentations. As such, we’re going to do something that might seem absurd, childish, or even (gasp!) unprofessional.
We’re going to colour the equations.
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 variable. If we take the slide from above and put it through a colour transformation, we might get something like this.
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.
“But wait,” you say. “I don’t want to turn my slides into a kaleidoscope of colour!”
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 ), 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.
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 also get the nice bonus of making your slides more friendly.
What more could you want?
Outside the Curriculum
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 why?
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 still let you move on, even though you have no business studying more things when you haven’t mastered these concepts first.
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.
There needs to be a better way to do this.
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). But, I know that the world of mathematics is vast and wonderful, and in school you only get to see a tiny 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.
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.
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.
So what can you do now?
The number one thing you can do is to explore the wide space of mathematics. 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 something is my point. The only way to learn more is to go out and find more resources.
The most surprising thing you will find is that mathematics does not 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.
I want to be clear here. The skills you develop in your mathematics classes are important. However, they are just that: skills. The wider world of mathematics is concerned with how you can use 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.
If you do that, then you can form your opinions on mathematics. But I think that once you see how much more there is to mathematics than what you see in school, you will find that mathematics isn’t boring at all. Only certain topics 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.
The lesson I want you to draw from this is that mathematics has something for everyone. 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 something you find interesting.
Through the Minefield
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.
Contrast this to when mathematics doesn’t 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.
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 I 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.
The question then becomes: how do we move the needle so that more people find mathematics to be something they understand?
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.
To solve this problem, I think there are two related points that need to be addressed.
Does it make sense?
The first is that mathematics needs to make sense. 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 need 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.
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 same 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.
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. That’s the key insight, not these three arbitrary methods.
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?
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 first 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 just the right way” to “mathematics is a subject that makes sense if I carefully follow the arguments.”
Taking a small step
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.
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 thinking 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?
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 me. It’s for them. On the other hand, the fix is difficult because teachers can’t give each 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.
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.
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.)
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 link 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.
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.