Reflecting on "Arbitrary and Necessary Pt. 1" by Dave Hewitt

Dave Hewitt’s "arbitrary and necessary" framework encourages math teachers to distinguish between elements students must memorize and those they can find out through their existing or received awareness. I appreciate that Hewitt put vocabulary to something I have often encountered in my math education. I remember being so confused when I joined a Math 9 class two weeks late and not knowing what the dot between numbers meant. The important backstory is that I self-studied Math 8 after my Grade 7 teacher noticed my advanced math aptitude. With Hewitt's theorizations, I can now understand that I was missing out on the arbitrary knowledge (conventions) of math having never been taught them by an instructor. 

With this anecdote, it is clear that arbitrary is not synonymous with unimportant. Like any language, Math requires a few basic rules to communicate and understand. I think the issue in my own math education was that arbitrary and necessary were never clearly distinguished. We learn these two aspects of math differently and therefore we must teach them differently. It's okay that some things in our classrooms simply need to be memorized; but, I think students have begun to think of math as all memorization, and that's a paradigm shift I would like to adopt.

For example, when I teach fractions in the future, I will define "numerator" and "denominator" as the necessary language for this topic. It will get a one-sentence explanation. This will free up time to focus on what Hewitt calls the "real math" -- exploring relationships and patterns and finding out the reasoning behind mathematical conclusions, formulas, etc. This idea strongly connects to Skemp's relational and instrumental learning, and that reaffirms many of the changes I want to implement in my own classroom. In short, epiphanies, conclusions, and aha! moments are far more impactful when you get there independently or with a nudge in the right direction. This in turn emphasizes and incentivizes deeper understanding over rote memorization.