Jeremy Côté

The Art of Explanation

I’ve enjoyed science and mathematics for a long time. I love the feeling of understanding a concept through its fundamental constituents. It feels like you’re getting at something deep, unavailable to those who just look over the topic with a cursory glance. In a sense, learning about science and mathematics has helped me learn how to think.

As much as I might romanticize it though, there is no getting around the fact that learning is difficult work. If you want to really understand a topic, it’s not enough to flip through a book. For science and mathematics, you need to sit down and dedicate a decent amount of time to understanding the subtleties at play. There’s nothing special about science or mathematics either. Take any subject you want, and you will find that there is a ton to be learned. It might be fashionable to have a cursory understanding of a bunch of topics, but the real treat comes when you dig deep to understand what is going on.

This work is tiring. I sometimes find myself frustrated, unable to understand something that I think I should be able to understand. I have to take breaks, come back to the ideas with fresh eyes and a clear head, and go at it some more. I love it, but learning isn’t without its difficulties.

No matter what you are studying, explanations will be everywhere. We are able to find more explanations for ideas than ever before. Whether they come from essays, books, videos, or any other medium, the resources available to us are vast. As such, there isn’t a lot of difficulty in finding explanations for topics we are interested in. Instead, the difficulty is in finding good explanations.

At first, this might seem like a minor problem. Isn’t there one good explanation? If someone finds a really good way to explain an idea, shouldn’t everyone else just point to that resource?

The issue is that, at least in science and mathematics, explanations come in many different flavours. Furthermore, they don’t all address the same kind of audience. This is an absolutely crucial distinction to make. A huge part of whether an explanation flops is if it is crafted with a specific audience in mind. If you try to teach advanced mathematics to someone who isn’t comfortable with arithmetic, you probably aren’t sharing your explanation with the right audience.

One brilliant example of this is this video on quantum computation. Physicist Talia Gershon of IBM spends some time talking about the research she is doing in the field of quantum computing at five different levels. She starts with a young child, moves to a teenager, then an undergraduate, then a graduate student, and finally ends with a physicist. In each one, she explains a bit about what she is doing and what the core ideas of her research are. I highly recommend watching the video, because it shows just how important crafting an explanation for your audience is.

Simplifying, not dumbing down

If you watch the video, you can clearly see that she isn’t dumbing down the content for those without the background. Instead, she uses analogies and comparisons to engage the person and gets them to understand a small part of what she knows.

She never “talks down” to someone who doesn’t know a lot. Instead, she figures out how to connect with them. Rather than presenting her own explanation, she figures out what clicks with them, and then crafts her explanation accordingly. I think we often miss this when teaching or giving explanations. We spend a bunch of time coming up with our own brilliant ideas without looking to see how they land with others. Teaching requires an acknowledgement of what the other person knows. If we ignore that, there’s a good chance we will give explanations that completely miss the other person.

One thing that I find useful is the idea of narrowing your focus when giving an explanation. There’s no doubt in my mind that many physics explanations are too advanced for someone without a background to understand in all of their subtleties. I think that’s inevitable. Rather than trying to cover it all but at a “lower” level, what if we lessened the scope of our explanation? Then, we could still keep the explanation detailed and specific.

The reason I think this is better is because we get to transfer valuable knowledge. Instead of conveying metaphors or analogies that are only approximately true, we can give a full explanation for a small idea. The more I think about this, the more I think it’s the way to go. I’m not saying that analogies and metaphors aren’t useful, but I do think we should strive for a full explanation when we can. As such, we should think about how much we can narrow our focus when giving an explanation.

The click

This brings us back to the question of what makes an explanation “click”. I’ve seen a lot of explanations given over the years in my education. Some I didn’t fully get, some I got but didn’t really remember after a day or two, and some have stuck with me for a long time. First, there’s something to be said about repetition. If you encounter the explanation over and over again, even if it isn’t great you will probably remember it. What I’m interested in is how come I can remember proofs and explanations that I’ve only seen once or twice.

I think it has to do with a mix of simplicity and novelty. The explanations that I would classify as “great” are those that capture an idea in a concise way that leaves no doubt in my mind how it works. I’ve worked through proofs in mathematics which I could see at the end gave the answer I wanted, but I never felt like I had a full grasp of the result. Conversely, there are some explanations that I understood immediately (or close to), and never forgot. These are the kinds of explanations that I seek to recreate.

I realize that this idea of an explanation “clicking” is inherently subjective. In mathematics, some proofs will make perfect sense to me, while others will leave me baffled and confused. It’s not that any of them are wrong. Rather, there are just some that are easier for me to understand. What this means is that I want to find the explanations that I can relate to. Conversely, if I want to communicate a proof to someone else, I better take into account their background if I want them to get the idea.

The medium matters

This brings us to the next part of the equation. An explanation has to be given in some sort of medium. This is obvious, but it really matters when trying to think about the best way to communicate an idea. For the longest time, I’ve been attached to the romantic notion of writing words on the page. I like writing, so I wanted everything to happen through words.

However, I’m starting to understand that mathematical and scientific understanding don’t always lend themselves to writing (at least, in my experience). Can you give a good explanation on the page? Absolutely. In fact, that’s how scientific and mathematical research is done. And yet, I’ve never been great at learning an idea just on paper or through the words of an author.

I think it has to do with the fact that physics and mathematics aren’t always “made” for words on a screen. This isn’t the medium in which understanding happens. If I think to my experience learning physics and mathematics, I can pinpoint two different instances in which learning occurs. The first comes from a good explanation that a teacher gives. Whether that is on the board in front of the class or one-on-one. I think the great thing about these kinds of explanations is that they are more dynamic than reading a textbook. I’ve said it before, but the main drawback of a textbook is that it won’t respond to you when you pose it a question. If you get stuck, you need to either bang your head against the wall until you realize what’s going on, find someone else to explain the idea to you, or find another textbook which treats the topic in a slightly different way.

I realize that other media such as videos and animation don’t have this back-and-forth available to them either, but videos will often go over points in multiple ways and reiterate ideas in a way that textbooks won’t. Likewise, you can find sites that write about mathematics in a more casual way and don’t mind circling around the same point again and again, but this isn’t the norm in a concise textbook. As such, I think we can easily run the risk of being too concise with our words when we write about mathematics. Videos and animation can remedy this a bit, but the real factor is how you view your work in the first place. If you’re trying to make someone learn an idea, it’s probably worth taking the extra time to circle around the idea and make sure that both the content and the “big idea” are articulated.

The second place where most of my learning happens is when I’m working on a problem on my own. When I’m struggling with homework and spend hours going through a bunch of questions, I learn more than just the ideas. I learn how they are put into practice. This is truly invaluable when it comes to learning.

As someone trying to craft explanations, I realize that I can’t suddenly transform people into understanding my ideas fully. To do this, they need to do the work themselves. However, what it does mean is that I can provide the framework to encourage them to think about an idea in greater depth. If I can achieve that, I’ve done my job right.

Honestly, the reason I spend so much time trying to figure out what makes an explanation great is because I want to learn more. When I spend the time crafting an explanation, that is time that I spend learning about the ideas. The result is that I become more familiar with them, and that’s satisfying in and of itself. Being able to share it with others is icing on the cake.

What I hope is clear though is that a good explanation has several factors. It needs to be targeted at a specific audience in order to be the most effective. Then, instead of simplifying a bunch of ideas, focus on communicating one idea in a more in depth way. Narrow your focus and keep the detail. People are smart. They can get an idea if you guide them along.

Be aware that an explanation won’t work for everyone. That doesn’t mean you did a terrible job. Rather, it could just mean it’s an explanation that will work well for a specific person. That’s not a failure. What you need to do is find that person.

Finally, it’s worth considering what medium you use for your explanations. You might have a sentimental attachment to a specific medium, but remember what your ultimate goal is: explanation. If you want to do your best to achieve that goal, using various media could be worthwhile.

We have an enormous opportunity now with the amount of resources we have to craft explanations. From video to animation to drawings to text, we have a lot of tools at our disposal. It’s time to think about how we can best use all of them together to communicate scientific and mathematical ideas.