Arbitrary and Necessary

This is honestly one of those things that I always knew about but never thought about until I became a teacher. Of course it makes sense that in math, or any subject, there are things we need to learn through being told such as definitions and what a symbol means. To me, and I’m sure a lot of people, a circle is a circle because it is a circle, if that makes sense. We understand what a circle is because we were told, not because we miraculously figured out its name. Maybe if we could all speak Latin, we can figure out what some of the scientific names mean. We never knew that ^ stands for exponents or * stands for multiplication. We even had to learn that 3 * x is the same as 3x, which isn’t very intuitive. Meanwhile, if we learned arbitrarily that if 3x = 6, and x = 2, we might be able to figure out what 3x + 2 = 6 is. I really like the distinction between the two types of knowledge.

In terms of my own classroom, I would definitely focus on teaching the arbitrary knowledge traditionally. I would even give specific handouts or have posters for those types of knowledge, since it would be unfair for my students to come up with them on their own. For example, I would show them the meaning of lengths and widths of a shape, but it wouldn’t be necessary to show them the formula of a square. When the students learn the meaning of those basic definitions, formulas of shapes may be derived through visualization and trial and error. I think it would be very fun for the students, since it is literally just doing LEGO with math. We give them the basic building blocks (i.e. basic definitions), but the students would need to build the structure (necessary knowledge), ideally without the instruction manual (me teaching them how to do it). By framing necessary knowledge as a final goal to achieve, while giving them the tools to get to it, student inquiry is then created for a nice change of pace from the traditional I-write-you-write.