There’s a lovely forest near my house. It’s a wonderful place that looks exceptional in the autumn, where the fallen leaves of the trees cover the path in a flurry of orange, red, and yellow. I love running there because it’s so peaceful.
Imagine that I told you I would show you this forest. After hearing me wax poetic about it, you’re excited to see it. We get to the forest, and I show you the path that goes through. We walk along it, and after a while you ask if we can get off the path to see the forest in its more “natural” state.
Puzzled, I ask, “But this path is the forest. There’s nothing else of interest other than what’s on the path anyway.”
We might not use the same words, but this is how a lot of us view mathematics. There’s a path (the curriculum), and following it is the only way to learn about mathematics. Forget about going off-path. That’s not even a thought that crosses your mind!
Unless you are really into mathematics, chances are you haven’t seen the wonderful little niches that the subject has to offer. This is unfortunate, but it’s a consequence of the fact that we tend to look at mathematics in terms of the path forged by the curriculum. It’s also not a problem which is limited to mathematics. Almost any subject will have this standard “path” that most people end up associating with the subject itself.
If I could send one message to my younger self, it would this. Don’t make the mistake of seeing the path as the subject itself. It’s only one particular way of looking at a subject, but there are so many more available. It just takes a willingness to look past the usual offering.
Unlike what we’re taught in school, mathematics isn’t a linear subject. Sure, it’s probably a good idea to learn about arithmetic before you learn algebra, but it’s not always as clear. The web of mathematics is thick and highly-connected, which means there are many paths you can take through the subject. Just because there’s a clear trail that has been created by countless curriculums does not mean you are forced to take that same path. In fact, I would encourage you to explore more. Look for those smaller connections. They can be as interesting as the regular path.
My hope here is to encourage you that mathematics is not only the curriculum you learn in school. It has so many other aspects that are off the path, if only you start exploring.
To me, this indicates two things. First, it means that we need to spread the message through our educational institutions, because it’s important that students see mathematics as more than only a curriculum. Second, it suggests that a way to get people interested in mathematics is to find something that they are attracted to. The key point is that this may not lie on the main path, but who cares? I’m more concerned with getting people to see mathematics as it is: an ensemble of many ideas, not just a linear path.
It’s worth wandering off the path every so often to see what else is on offer.