Jeremy Côté

Narrow Paths and Open Forests

Imagine you’re walking through a forest for the first time. This forest has trails, so you follow one of them. Since the forest is unfamiliar to you, the only thing you have in mind for your walk is the destination. You know that if you follow the path correctly, you will get to where you were hoping to go when you checked the map at the welcome station.

You’re moving along the path, and suddenly you see that the path is blocked by a bunch of trees which have fallen on the ground. Not only that, but it looks like the path hasn’t been used in a while, so wild brush has grown around the place, obscuring the trail you saw on the map.

The only option is to go off the path and navigate around it until the path becomes walkable again. You step off the path and start walking through the forest. Immediately, the whole place seems different. It feels uncertain, like when navigating a familiar space in the dark. You don’t like the feeling, and as soon as you see the path start up again, you breathe a sigh of relief and hop back on it, continuing your walk.

This scenario is probably familiar to anyone who likes to explore new places. The lack of knowledge you have at the beginning means you like to stay on the designated paths, because those have been thought out beforehand. You don’t have to worry about where this path will bring you. The destination is fixed, and you simply have to follow.

Contrast this to the open forest. Sure, you can walk through the forest, but the notion of a path disappears. Instead, you just have ground you can cover. There are few guidelines, and you have little idea of what will come up.

On the one hand, this makes the open forest scary. Without any idea of where you will go or what will happen, a layer of uncertainty is applied, making any journey a bit more stressful. On the other hand, the open forest has no rules. If you want to cross a small stream by walking through it, go right ahead. No one is stopping you. The decisions of where to go and how to do it are totally up to you.

Narrow paths bring certainty and ease, but they give up control and freedom.

It’s no surprise that we can find parallels with this idea of open forests and narrow paths all over our lives. We are constantly operating on narrow paths or blazing new trails.

One area where this clearly shows up is in mathematics education.

Students learn early on that the procedure is one of the most important things to follow. If you follow the procedure the teacher provided, you will do well on the test, allowing you to “upgrade” to a new procedure and repeat the cycle over again. Following the path set out by the teacher is a way of guaranteeing that you will succeed. That is, in the sense of passing the course.

It’s ironic then how little these procedures have to do with coming up with original mathematics or solving new problems. Most of the time, you can’t just rely on worn paths to solve a problem. New ideas have to be injected into the system, and this comes from going off the path.

What does this look like in practice? For mathematics, this often means tweaking an existing technique to see if it can be applied to a new situation, or finding a brand new way to tackle a problem. This is the very definition of going off the path.

And yet, we rarely emphasize this to students. Or we say it, but then we turn around and give tests that ask them to follow the path. Our actions don’t match our words.

Procedures are useful, just like paths are. If you don’t want to think too much about what you’re doing, following the path is perfect. This happens for even those who regularly stray off the path. For example, researchers will use the techniques we have learned before. When applying those old techniques, they won’t spend time trying to be wholly original and going off the path. The technique works fine, so following the path is good enough. The difference is that they are using this path to get themselves quickly to an area where they can then exit the usual path and find something new.

Another issue with paths is that they can give the illusion of familiarity of the whole region, when really they only provide familiarity of the path. If you walk along the same path every day in a forest, do you really know the whole forest? More likely, you know the small chunk of forest that surrounds the path. Not only that, but you know this part of the forest only from the perspective of someone on the path. You don’t even know what it’s like to view this same patch of forest from off the path.

For students, this represents the issue with learning only what the teacher presents. Students get the impression of knowledge of the whole subject, even though they only have a tiny sliver of it. Just like knowing the path in the forest doesn’t give you knowledge of the whole forest, knowing the content that a teacher presented doesn’t make you aware of everything about that subject. This is important to remember because it offers an argument for veering off the usual path. By leaving the certainty and familiarity, you give yourself a chance to see the area from another perspective, and this helps form a better picture of what you are seeing.

Do you find yourself panicking as soon as you veer off the path and into the wild forest? When traces of civilization vanish, are you comfortable, or are you scared?

When your knowledge isn’t stable, you tend to stick on the path. After all, if you stick on the path, your knowledge remains certain and useful. It’s when you get perturbed a bit that things get more complicated.

The question is: Are you resilient enough to keep going even when you are off the path?

Since we were young, school has taught us that the way to be successful is to follow the path. While decent advice, it misses the fact that new contributions and new insights require us to leave the path. Furthermore, if we only ever stay on the path, we don’t really know the forest. We just know the path in the forest. This might sound like a semantics game, but it makes a world of difference.

If your goal is to get an understanding of the whole forest, you can’t do it from only walking the paths within it. At some point, you need to swallow your fear of uncertainty and step off the path. It’s uncomfortable, it’s sometimes scary, but the reward is that you will see things that nobody who chooses to stay on the path will ever see.

I think that’s a worthwhile reward.