Jeremy Côté

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!