Arbitrary vs. Necessary in the Math Curriculum

This article allows its readers to think about things that we generally tend to ignore under the umbrella of ‘that’s the way it is”, or ‘that’s the convention’, or even ‘true-because-teacher-says-so’. My question if what if I let the students break the convention. For example, what if I let them write the y-coordinate before they write the x-coordinate. Would it be looked upon as breaking the rules, or would it give the students some power or control over Math, I wonder. Moreover, I wouldn’t want to answer ‘the-why’ questions from the students via ‘that’s the convention’. I would like to support their curiosity and get them explore ‘why that’s the way it is’. However, there is a fine line between satisfying curiosity, and exploring for purpose and getting confused, i.e. as an educator, I’d need to be careful for not getting the students confused by sharing too much information and getting them exposed to it without support.

One of the ways for defining arbitrary and necessary is by considering what can/cannot be solved. In other words, ‘arbitrary’ is information (or “received wisdom”) that may not be be worked out and is in the "realm of memory", whereas ‘necessary’ includes the properties and relationships that are in the "realm of awareness", and can be worked out, as described in the article. The decision of what’s arbitrary and what's necessary can vary from topic to topic. The question of what arbitrary and necessary are under a given topic can be responded to by considering the answers. For instance, the aspects of the lesson that can be summarized as “symbols, notation, and convention” are arbitrary, and the properties and relationships that carry good explanation to satifsy the ‘why’ are necessary. However, if one excels at explaining the ‘why’, what’s arbitrary can become necessary, as Hewitt outlines why New York is New York.

Furthermore, the differentiation between arbitrary and necessary of course influences the lesson plans, as this decision would greatly impact how much time should be spent on what and why. As teachers, we don’t want to spent too much time to what's arbitrary. Instead, our focus should be on problem based and inquiry based learning within the given topic. It's related to what's we talked about previously regarding regarding instrumental and relational thinking. We want to teach the students to think mathematically under the given topic.