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As I was tutoring one of my students about algebra, I could see that he was struggling with making sense of the rules of how to manipulate equations in order to solve for a given variable. The equation involved fractions and like terms, and looked something like this:

He was having trouble with doing his steps in order. For example, he was dividing to solve for “x” before he placed all the other terms on the other side.

To be honest, it was a bit of a challenge for me to address, because I had so much experience working with basic equations such as these that the process was basically automatic. I knew the steps I took to get there, though, so that’s what I told him.

“You know how put the like terms together on one side, so take all the “x” terms and put them on one side of the equation, and then divide by that factor. You can’t divide before putting everything on the other side. And when terms switch sides, you have to switch signs, unless you’re trying to solve for “x” and you have a factor in front of it, in which case you have to divide by that factor.”

Okay, I didn’t really say all of that. But I did try and get the gist of the message to him.

Once I corrected the order of the steps he was taking, he muttered to me, “These rules are stupid.”

I’ve thought about this phrase for a while now, because it indicates a certain view of mathematics. Mainly, that it’s a bunch of rules that need to be followed. However, I strongly oppose this view. Instead, I think it would do us well to remember what a lot of mathematics demonstrates: relationships. Furthermore, it shows how these relationships are necessary if the original assumptions are true.

Let’s take something absurdly simple. Imagine you have two points in a flat space of $R^2$ (a sheet of paper, basically). Now, if I asked you to find a straight line that connected these two points, you would quickly find that there is only one way for you to do this. It’s inescapable. This is what is called an axiom (in this case specifically, an axiom for Euclidean geometry).

From this foundation (with other axioms), we then can figure out a whole bunch of things about lines, shapes, and the angles between them. We aren’t exactly inventing rules. A better way to think of it is that we are following the logical conclusions of things we’ve already accepted as true.

If we agree that this and this are true, then by extension, that must be true.

This kind of thinking can be applied to equations that need solving. For this situation, the idea is that an equation represents a sort of balance. Therefore, if we introduce something on one side of the equation, the only way we could conceivably keep this state of balance is to introduce the exact same something on the other side of the equation.

From this realization, we can then figure out how to solve the original equation shown up above. To do so, we’ll bring all of the “x” terms on the left side, and the rest of the elements on the right.

From here, we can put the terms together using common denominators, which yields:

Lastly, to solve for “x” we need to bring the factor on the “x” term to the denominator of the right side. In terms of “balancing” the equation, we need to divide by $-\frac{7}{5}$ on both sides of the equation in order to solve for “x”. Doing so yields:

Now, finding this wasn’t done by applying rules, per se. Instead, it was about following the logical end of our equation. Given our initial equation, we could only modify it in identical ways on both sides, eventually finding our answer.

In secondary school mathematics, the emphasis is on learning theorems and formulas, which are often regarded as “rules”. This is unfortunate, because it then makes the student worry about remembering how to use every single rule, leading to these proclamations of the rules in mathematics being stupid.

I believe a way to combat this is to get students to see that mathematics isn’t governed by rules, but that these are necessary conclusions that we have to make if we accept more basic axioms of mathematics. They aren’t negotiable, just as it makes no sense to say that breathing is a “rule” for humans. It’s not that breathing is a rule, it’s that there’s simply no other way to power our cells for the long term without some kind of aerobic capacity.

My wish is that this view of mathematics is pushed forward. Personally, I don’t remember ever hearing about axioms in secondary school, and these are supposed to be the bedrock of how we formulate most mathematical ideas (perhaps not inductive proofs, but those aren’t usually covered in secondary school). Instead of thinking about how mathematics just has these rules which work, it’s much more gratifying to think about them as a collection of necessary conclusions given certain information.

If we did this, perhaps I wouldn’t hear my students I tutor bemoaning how the rules of mathematics are stupid. They would understand that the theorem has to be true, just as surely as any triangle will have inside angles adding up to 180 degrees.

Let’s avoid dumping rules and formulas onto students, and start showing them how and why these exist.