Hewitt Article on arbitrary and necessary

One of the things that I have always loved about math is that you don’t have to remember much. Most people would probably find this sentence peculiar, odd or strange for a couple of different reasons. First, most people don’t love math. They hate it with a passion. Second, most people think math is all about remembering formulas or theorems or procedures, but it is not. It’s about figuring them out. I’ve always loved this about math because as a student, if I couldn’t remember something, say what to do when multiplying two logs together, I would try different things, different approaches, different rules, and see what worked. I would just put some numbers in for my variables and verify. These math rules are the “necessary.” Knowing them qualifies as or constitutes mathematical understanding.

The “arbitrary” have a key role in mathematics as well and this is where most of the small amount of remembering required in mathematics lives, in my opinion. In my own teaching, when it comes to the arbitrary, I usually frame it in a linguistic context. For example, talking about degrees and radians, I describe these not as mathematical ideas to be understood but as mere English vocabulary – definitions to be learned. And if you go do math in another language in another part of the world, they likely will have different arbitraries than we do. Arbitraries need not be worried about because they offer little in the way of mathematical challenge. They are the definitions that precede the theorems and can be written on the board and left up there during the test. In my own teaching, to keep the arbitrary from getting too boring or cumbersome to remember, I like to give it a bit of historical context. For example, when talking about degrees, I tell students about the Babylonians and their love for the number 60. I find this just makes it easier to remember that there are 360 degrees in a circle. Before reading this article I had never heard that they multiplied it by 6 because of the ratio of the perimeter of a hexagon to the radius of a circumscribed circle.