Learning mathematics is an additive process. What I mean by this is that new mathematics often builds on what came before. Learning mathematics isn’t exactly a linear journey, but it’s a good enough rough approximation.
In order to go from one concept to the next, it’s useful to offer analogies to explain what is happening. For example, when you first learn to count, you learn about the natural numbers. These are positive integers that grow in size by one each time. It might be difficult to understand at first, but with a bit of practice, most people become good at doing this. However, teachers then throw you a curve ball with negative numbers. What in the world does this mean? It seems ridiculous at first. For the counting numbers, you were able to point to these quantities. They made sense and were clear. Negative numbers, on the other hand, don’t seem to be as tangible.
But then your teacher explains how negative numbers can be thought of as moving across the familiar number line in the opposite direction (usually to the left). Just like the number “after” zero is one, the number “before” (to the left of) zero is -1. All you get with negative numbers is a way to go past zero to the left when you’re doing things like addition and subtraction.
The analogy here is very straightforward: negative numbers act just like positive numbers, but they lie on the other side of the familiar number line. If you have a good understanding of the number line, extending to to include negative numbers becomes easier.
What I want to highlight here is that I didn’t mention anything about additive inverses, which is the technical way to introduce negative numbers in structures known as rings. I didn’t mention that, not because I’m trying to deceive a student, but because I know that it’s not helpful. What is helpful is an analogy that takes a student from a place they are familiar with to one that they aren’t.
Here’s another, more advanced example. If you’ve taken a course in linear algebra before, you know that matrices are a big part of the subject. You learn that matrices can be added, multiplied and are useful for solving many simultaneous equations. However, have you ever thought of what it means to take the exponential of a matrix? In other words, if you have a square matrix M, what is eM?
It turns out that the way to go about this is to use the fact that the exponential function can be written as a power series, which is just a polynomial that is infinite in length. What’s nice about this is that we know how to deal with squaring, cubing, or raising a matrix to any power. Just do a bunch of matrix multiplication! Barring some technical details, this is how you can define the exponential of a matrix. Here, the analogy is that the exponential of a matrix behaves similar to taking the exponential of a real number, but the way to compute it requires a power series.
Mathematics is littered with examples like this. They pop up all over the place, since we often look for generalizations of ideas we know. The familiar factorial operation n! = n(n-1)…(2)(1) can also be written in a more complicated looking form using integrals and exponential functions. This new form behaves just like the usual factorial when n is an integer, but it turns out that the integral isn’t limited to integer inputs. Instead, one can use any real number. That means you can calculate interesting expressions such as π!, even though we aren’t using the usual form for the factorial.
There are advantages and disadvantages to using analogies while generalizing ideas in mathematics. The upside is that it gives you a foothold to getting to a new concept. Without an analogy, learning something new can be quite jarring. You end up wondering what in the world this new idea has to do with everything else you learned, and the experience tends to be one of confusion. I’ve written about this multiple times before, because I find that it’s critical to give yourself footholds while learning. When I don’t understand what’s happening at all (I have nothing to latch onto), it’s difficult for me to follow and the experience is often frustrating. Therefore, I try to always find some sort of analogy to help me through the initial stage of learning.
That being said, analogies don’t automatically make you a better learner. In fact, analogies can be harmful if you don’t approach them with the right mindset. I’ve found that when we are very comfortable with a concept, we end up creating our own analogies. These are simplified versions of what is going on, and they make a lot of sense to us. The crucial thing to realize though is that part of the reason they make sense to you is because you have the background. Sure, your analogy might be short and sweet, but if it only makes sense to those who know the subject inside and out, it won’t exactly help others learn. As such, when I hear an expert on a subject give an analogy, I don’t outright accept it. I approach it with caution, since I don’t want to be led astray.
This is an important point to reiterate: analogies can and will lead you astray if you follow them too literally. Analogies are there to be helpful and to give you a foothold to a subject. They don’t replace the subject. I think of it as similar to trying to understand the message of a book by looking at chapter summaries. Sure, you can get the essence of the book in a much shorter text, but there are bound to be nuances that you miss. That’s just the nature of compression. In the same way, analogies in mathematics often compress the technical details. This is helpful for initial understanding, but it doesn’t replace the fact that you need to do the difficult work of learning the subject.
If there’s one idea I want you to take away from this essay, it’s that analogies in mathematics are very helpful, but they don’t replace the rest of learning. Almost any concept in mathematics can be explained with an analogy, but there’s always more to that explanation. Therefore, use those analogies to your advantage as a starting point, but don’t rely on them too much. Doing so will likely lead you down the path of misconceptions, which gets worse and worse as time goes by and you hold on to these incorrect ideas.
At the end of the day, your goal is to learn some new mathematics. Analogies can bring you part of the way, but you have to be willing to do the rest of the work.