When something doesn’t make sense, it’s easy to look at it and think that it is just a collection of random rules. Unfamiliar topics look incomprehensible, so naturally we miss the structure behind them. This doesn’t mean that the structure isn’t there. It’s simply that you can’t see it because you don’t understand the topic well enough.
This happens all the time in mathematics. On the one hand, this is very odd. Mathematics is the study of structure through logic, so you would think that there’s no risk of having “random” rules. On the other hand, mathematics is part of the fabric of our educational system, which means it’s studied by students who need to take tests. As a result, understanding can sometimes be thrown out of the window in favour of efficiency.
The pattern I see goes like this. A student comes across a topic that doesn’t quite make sense. They could spend a lot of time figuring out how it works, but that would be long and they have a bunch of other work that needs to get done. Instead, they find out how to perform the needed steps without understanding the theory behind it. (Teachers may even present material in this way.)
This leads to an “operational” understanding without a conceptual foundation. This is a fancy way of saying that the student knows what to do, but now why. This might not seem like a big deal, but it has a lot of ramifications in the world of mathematics.
Take the concept of fractions. We teach students that there are different procedures for combining fractions. If they are adding or subtracting fractions, students first need a common denominator, and then they need to add the numerators while keeping the denominators the same.
One could ask the question of why this is the way you need to do it. I can think of other ways to define addition, so why do we use this particular one?
Once you get this question as a teacher, there are several approaches you could take. I’ll go through two of them here. You could tell the student that it’s simply how mathematics is. This uses authority as a reason (not a very good reason in my opinion), and it will likely produce students who follow the rules. Alternatively, you could show students that the way we define addition is the only sensible way to do it. This could be done by showing them how other definitions could make for some weird equalities that we don’t want.
Personally, I like the latter approach. The idea is to show students two things. First, mathematics is a malleable field. We choose its rules. You can play by your own rules if you want (though that might not be fruitful). Second, mathematics should make sense. We shouldn’t just present rules to students and expect them to make sense. We need to do some work to get students on board. I presented the example of fraction addition, but what about fraction division? If you recall, diving by a fraction requires us to “flip” (take the reciprocal of) the fraction in the denominator and then multiply through. If you just think of this in the abstract, it doesn’t make much sense. However, once you notice that this “rule” comes from the fact that the multiplicative inverse of a fraction is its reciprocal, things start to come together.
Coming up with a bunch of rules for how to do mathematics is fine, but it shouldn’t be a replacement for understanding. When that happens, students then become reliant on finding the right “trick” for a given situation, instead of exercising some thought about what to do to solve the problem.