Distinguishing variables from parameters

Imagine the following dialog. Professor: f is a function of a real variable x that takes a real parameter k. Student: What’s a parameter? Professor: It’s a constant that can vary. Student: Then if it can vary, isn’t it a variable? Professor: Sorta, but no not really. This conversation plays out over and over, and unfortunately it often ends as it does above, with the student confused. Here’s how I believe the conversation should continue. Professor: You’re absolutely right that f is a function of two variables, x and k. But usually k is fixed in the context of a specific application and x is not. A different application might have a different, but also fixed, value of k. So it is helpful to think of f(x; k), a function of x with a parameter k, rather than f(x, k), a function of two variables. The former carries more information, giving a hint as to how the numbers are used. Is there really a difference between a parameter and a variable? In a reductionistic sense, no. But in a…