In mathematics, notation is simultaneously everything and nothing. It isn’t difficult to imagine another alien species havig the same notions of calculus as we do, but without the symbols of integration or differentiation. It might seem so natural now to see the expression $\partial x$, but that’s only because we’ve spent years working with these symbols, forging a connection between concepts and notation. Due to this, it can seem entirely natural to look at notation and instantly understand what it’s about as a concept, rather than just symbols. This is quite similar to our experience with foreign languages, where the words and characters look alien to us, yet our own languages seem so obvious.
I’ve been thinking about notation after working with some students who seemed to be struggling with certain concepts. I was wondering why they couldn’t just see the same things as I could on the paper. I know that some struggle with seeing the connection between $f(x)$ and $y$ in an equation, even though of course they mean the same thing in this setting. Another variation to this issue is when a text or a problem refers to multiple functions, and names them $f(x)$, $g(x)$, and $h(x)$. It might seem natural to us to have these be the names for arbitrary functions, but this sudden spring of notation onto students can be deeply confusing when they aren’t used to it. The consequence is that it seems random and without explanation, so then students start believing that part of mathematics is just like that: innately confusing.
In my mind, unless we’re talking about probability, mathematics should not seem random.
In fact, mathematics is all about investigating structure. To do this, however, we absolutely require definitions and notation. But, the thing that is often lost on students is that we create these definitions and notation! It’s there because they’ve (mostly) stood the test of time of being good for problems. As such, I think it’s critical that we get students on board with the notation, and to be capable of moving fluidly between notations and concepts.
For students in their secondary education, this means being familiar with the idea of an equation, and not being tied to the notation itself. As an explicit example, this means having students being aware of parabolic functions apart from the “It’s the one with $x^2$ in it!” I want students to be comfortable with saying that $d(t) = t^2$ is just as much of a parabola as $y = x^2$ is. The notation is important, but the specific symbols aren’t.
Here’s another example that I find really tests whether a student understands the concept of what a function is. If they’re given the function $f(x) = x + 2$ and are asked for the value of the function when $x=2$, many will write $f(x) = 2 + 2=4$. Of course, the function value is correct, but the problem is that this isn’t $f(x)$, it’s $f(2)$. This isn’t a huge deal, but it teases out the apparent weakness with the notation $f(x)$. From what I’ve seen with many secondary students, they don’t translate finding the value of a function to putting that value into the notation of $f(x)$. I’d argue that this implies they aren’t fully grasping what $f(x)$ means, but I also think it could simply be a lack of explanation. To remedy this, we need to put more emphasis on explaining the concepts behind the notation, so that the students will be on board with using it. If we don’t do this, we create more problems for ourself down the road when more complex notation comes along and students aren’t ready to fluidly jump from one set of notation to the next.
Overall, notation isn’t important in and of itself. But to do mathematics and learn new topics, it’s crucial to be able to understand what a certain notation means and how to use it.