Cementing


A common thread I see between many young students who don’t seem to “get” mathematics is that they aren’t told to look at the way they are taught mathematics as only that: a way. Unfortunately, the impression that is made on them is that mathematics is a strict set of rules that cannot be broken and must be followed every moment.

Imagine instead I was learning the English language, and that I wanted to express the emotion of anger. When I ask this, the teacher tells me, “You can say that you’re ‘angry’.”

Here’s my question: wouldn’t it be absurd if I went along my whole life using that one word to express how I was feeling? Instead of using the variety of synonyms that the English language provides in order to bring more nuance to my state of mind, I’d be repeating the same word over and over again, because I know that it will generally express what I want. In essence, it’s a “safe” option. It then gets cemented in my brain through repeated use, and I am stuck in this cycle forever.

I hope you agree that this wouldn’t be a very good thing to do to a student. Making them only see one way of looking at the world and not using any other words to describe anger robs them of expressing themselves, since it can never fully capture the particular shade of anger in that moment. However, we are doing the exact same thing to our students in mathematics!

Consider one of the earliest functions that are exposed to students: $y = f(x)$. This is taught to be a function of the independent variable $x$. After this is taught, example upon example is given using this form. However, students are then taught that functions don’t have to look like $f(x)$, but can be in the form of $f(y)$ as well, with an independent variable $y$. This isn’t necessarily a problem, but from the countless examples and repetition in the previous form, using this other method becomes confusing. Then, to make matters worse, they are taught that one can often switch between these forms by manipulating the variables.

As one can imagine, this confuses many people. How can $y$ turn from a dependent variable to an independent variable by just arranging the equation in a different manner? When do we have to use $f(x)$, and when should we use $f(y)$? These questions may be obvious to us, the tutors and teachers, but it isn’t so obvious to the students. I’ve seen it myself.

In my eyes, the solution is to make mathematics more dynamic. Instead of introducing the variables $x$ and $y$ as the de facto standard for the rest of their mathematical lives, show them that the variables we use are just their for us, the people doing the mathematics. It doesn’t really matter if something is called $x$, $y$, or $lambda$, they’re just symbols. From there, I believe students will have more confidence in using variables that aren’t your traditional $x$ and $y$. The way in which I try to help this with the students I tutor is to use the terms “horizontal” and “vertical” axis instead of the $x$ or $y$ axis. My hope is that this will get them out of their standard way of mathematics and make them realize that a lot of the things you do in mathematics is more out of convention than of need.

Obviously, the end goal is to get students using mainly just $x$ and $y$ for a lot of mathematics. However, the point is that these are just the traditional ways of doing things. There’s no requirement that the y-axis be the vertical axis. As such, I’d much rather change the names of variables and get them to think in ways that aren’t the usual, just to keep them aware that all of mathematics does not revolve around $x$ and $y$.

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