In reading Hewitt's article, I was immediately taken with the distinction that he makes. I was especially persuaded by his description of students who have arbitrary of memory-based approaches to concepts which are in fact necessary, such as "two negatives make a positive." This is something that I have frequently witnessed, in both classroom and cultural settings.
One example which comes to mind is the "do you remember basic algebra" posts which regularly crop up on social media:
This problem circulated across various social media platforms with wide engagement, largely consisting of people arguing in the comments about PEMDAS or BEDMAS (or various other versions of the mnemonic). The answer to the question, of course, is that the solution is unclear. The question is intentionally misleading, obscuring the grammar of the arithmetic to cause controversy. The issue would be moot, though, if more people understood Order of Operations as an arbitrary, grammatical tool.
In terms of teaching, I am a firm believer in making the distinction between arbitrary and necessary clear to students. I have absolutely been a student asked to recall some arbitrary piece of information and frustratingly "coached" to the right answer when I couldn't remember. No one wants that.
Additionally, I believe that revealing the arbitrary nature of much of the mathematics curriculum helps students consider the discipline as a body of research and communication, as opposed to some universal collection of truths handed down to the teacher from god. It can demystify mathematics for students to realize why variables are conventionally called x and y, for instance.
Something I am cognizant of as a teacher is the fact that just because something is necessary does not mean that all or even any students have the awareness to realize it. I worry in my classroom about judging student awareness, especially when planning units/lessons well in advance of meeting my students. Hopefully, this is something that I will be able to adjust to quickly. Additionally, I am trying to build flexibility into my lessons so that I can help build student awareness if need be, or make the question more complex if students quickly grasp the intended necessity.
Excellent. Very thoughtful and perceptive commentary on this article. I agree that it is difficult to judge the mathematical awareness of your class before actually meeting them, though with experience, you will begin to have a baseline expectation that you can work from for each age group. I expect that you will very soon get a feel for your practicum classes! Lots of listening in on groups as they work on open-ended problems helps greatly, and there will be adjustments to make as you see where they are and what challenges and support they need.
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