If you look at a typical mathematics class, you will find that they follow a similar rhythm. First, students are presented with a definition. A few examples of how that definition applies might be given, and then the rest of the class is spent proving results based on this definition.
It’s a familiar recipe, and it works well for transmitting information. If you asked students to give a presentation on the topic, I would predict that most students would follow this layout.
There’s nothing bad with this layout. However, I think there is a key missing ingredient, and it has consequences for how students perceive mathematics. It’s the answer to the question, “Where does this concept/definition/axiom/theorem fit in the grand scheme of mathematics?”
We don’t put enough emphasis on answering this, to the detriment of students. There are two main consequences of failing to answer this question. One, it makes students unaware of their place within the “space” of mathematics. And two, it makes mathematics feel somewhat arbitrary and disconnected.
Locating yourself within the web of mathematics
In physics, we have a notion called “phase space”. In classical mechanics, phase space refers to your coordinates in both position and momentum space. Since we live in three spatial dimensions, there are three position coordinates and three corresponding momentum coordinates, for a total of six coordinates. These six numbers describe all that matters when analyzing a system. Phase space is six-dimensional (when analyzing a single particle), so it’s difficult to visualize. However, if you know the tools of mathematics, you can still analyze phase space and figure out what the particle is doing.
I think a similar structure exists in all disciplines, and this structure constitutes the “totality” of the subject. You can think of it as the space where a discipline (like mathematics) “lives”. I like to think of this space as a web, since there are many connections between various topics. The key observation is that very few mathematical facts exist in isolation. In general, whenever you learn something new about a specific topic, you’re growing the space of mathematics that you know about.
This is crucial to understand, because it means that there’s utility in thinking about what you’re learning at a higher level. When you learn about logic and implications, it’s easy to dismiss this as a niche corner of mathematics that only has applications to riddles about islanders who either tell the truth or lie. However, a broader understanding of mathematics makes you realize that logic permeates all the we do in mathematics. It lets us fashion theorems, helps us prove them with various techniques, and gives us different ways to think about the same problem (through equivalent truth tables).
The unfortunate part is that a student learning the subject isn’t always privy to this “extra” information. Instead, they’re taught the minimum so that everything is self-contained. This might be good for focusing in class and not worrying about more material than is necessary, but it means that students often aren’t thinking about the “big picture”. While this won’t affect their performance in class, it will affect how they see mathematics.
To address this, we need to become more serious about adding a “motivation” section to classes. This is sometimes done by professors, but frankly I think the motivation is missed in most classes (at least, it was to me). In an ideal scenario, I’d like to exit a class after I learn a new idea and be capable of saying, “Here is what I learned, and this is why it’s important.”
The latter doesn’t “give” you anything in the sense of mathematical skill, but it does let you situate yourself in the web of mathematics, and I think this is a valuable thing in and of itself.
Just a bunch of rules
Hearing the above heading as a description of mathematics is always disheartening for me. I know that mathematics isn’t just a bunch of arbitrary rules. In fact, only a small part of mathematics is arbitrary1, and that’s often to establish a convention when ambiguity arises. Apart from that, mathematics is a connected collection of islands that are linked through logic. Unless you go very deep, you can find explanations for most mathematical facts. It’s definitely not just a bunch of rules.
At first though, this isn’t clear at all. However, the cool thing about mathematics is that as you get deeper into the subject and learn more, you find that it gets easier. What once looked like a bunch of rules actually stems from a single explanation.
This is why it’s so important to teach the context around mathematics. When we know where a definition comes from, we start to see how it’s not fully arbitrary. This turns mathematics from something that looks disconnected to a coherent framework.
A good example of this is when you study results that look like they were “reverse-engineered”. When I was studying abstract algebra, I learned about a result called Eisenstein’s criterion. It was a way of determining if a particular polynomial was irreducible. The details don’t matter, but the important bit was that the requirements for this result seemed quite ridiculous. Another example is that a number is divisible by three if the sum of its digits is divisible by three. At first glance, there’s no reason why this should work. It seems kind of random. However, once you understand how this result is a consequence of modular arithmetic, everything becomes clear.
We can’t let definitions and results float around as isolated items that only belong in one specific class. We need to make sure that the connections are brought to light. There is a lot of room for improvement within the classroom, but I also think there’s a huge opportunity for those who communicate online through blogs and videos.
When I learn new mathematical ideas now, I try to always ask myself where this idea fits within the mathematical web. It’s not that I don’t see any value in learning ideas in isolation. Rather, I think the best part of mathematics is seeing how everything is interconnected. If you’re willing to put in the effort, the web becomes clear.
It just takes a choice to move past the bare minimum of what is asked in school.
Though, we have to be careful by exactly what we mean here. Do humans make up the rules of mathematics? Sure. Therefore, in some sense, mathematics is arbitrary. ↩