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.
Visuals in Mathematics
There’s no doubt that writing is a useful tool. If anything, I’m biased towards 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.
That being said, there’s still a difference between writing and communicating. 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 bad. Rather, it’s that writing on its own isn’t great at communicating mathematics and physics concepts.
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 everything was communicated using words. The end result is that learning required both a desire to understand a new concept and 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.
Remember, I like 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.
In particular, consider the under-appreciated diagram. A diagram conveys both words and 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 visual, 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 wrong diagram in your mind, and it’s better at conveying the message than a paragraph.
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.
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 R, and the number of tickets sold by each type as a, b, and c. Then, we could skip the confusing paragraph and write:
R = 10a + 15b + 20c.
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 lose 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”.
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.
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.
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.
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 mean? 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.)
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.
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.
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.
At the end of the day, mathematics is about ideas. 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.
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.
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 I haven’t had experience with students who are visually impaired doesn’t mean they should be shut out from this discussion.
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.
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!