Arbitrary and Necessary

Hewitt's idea of arbitrary and necessary made me rethink how I usually plan math lessons and even whole units. I realized I often try to "explain everything well," but I do not always separate what students actually need to remember from what they should be able to figure out on their own. This distinction changes the way I look at my planning. If something is arbitrary, like notation or vocabulary, then I can simply introduce it clearly and move on. But if something is necessary, then I need to create moments where students can actually see why it must be true. That means I cannot jump straight to formulas or polished steps just because it feels efficient. I need to slow down at the right places and design experiences that let the idea appear naturally.

This also reshapes how I plan a unit. Instead of listing pages to cover or examples to finish, I start thinking about the "awareness moments" I want students to have. For example, in geometry, I want students to notice angle relationships by cutting and rearranging shapes. In algebra, I want them to understand why certain operations preserve equality instead of memorizing slogans like "two negatives make a positive." During my short practicum, I often heard my SA say that he really did not like students using the phrase cross multiplication. What exactly is cross multiplication? We are not doing it because we need to memorize a rule. We do it because we are multiplying or dividing both sides of the equation by the same number. These moments require time and space, so my pacing needs to be flexible rather than packed with every exercise. I also find myself paying more attention to which parts of a lesson might overload students with things they can't reason out yet. If I ask them to memorize too many arbitrary details before giving them something meaningful to reason through, they lose motivation.

Overall, Hewitt's framework pushes me to design lessons where understanding is not handed to students but discovered step by step. It reminds me that math is not just a collection of rules but something that makes sense when students have the chance to look for structure. As I keep planning future lessons, I want to build a classroom where students remember the arbitrary because they have to, but they understand the necessary because they saw it happen.