Response to "Arbitrary and Necessary Part 1"

In this article, Dave Hewitt describes his thoughts on the "arbitrary" and "necessary" sides of mathematics education. From what I could gather, "arbitrary" is concerned with the things being taught that are spawned out of thin air, such as terminology, conventions, notation, and other similar abstractions. As someone who is a big fan of standardization as a means to get everyone on the same page, and abstraction to condense ideas and hide messy implementation details "under-the-hood" (similar to abstraction in a computer science sense), I feel that there is an equal amount of value with the idea of abstraction. I appreciate the article for eventually acknowledging that students will likely benefit more from learning and memorizing arbitrary mathematics content for many of the same reasons that I believe: it is a top-down issue of how people have communicated in mathematics for the longest time. While Hewitt goes on about how memorizing is not doing mathematics, I would argue that it is a key set-up step that allows for a richer, more successive form of mathematics. In terms of "necessary" math content, this is likely akin to the whole "progressive-relational-Lockhart-y" form of mathematics that makes up the other side of the grand spectrum of mathematics curricula. I didn't think that there was another way to expand on that further, but here we are. In any case, I see this as being a little more important, mainly because the process of awareness is one so integral to mathematical thinking. To have students thinking is a recurring theme lately with all this talk around Thinking Classrooms, and there are certainly benefits of being more independent learners and deeper thinkers in not just a mathematical context, but also as lifelong learners.

The article's second half goes into ideas on how to think about the arbitrary and necessary when planning one's teaching. For me, I can at least say for sure that there needs to be an interplay -- a dance -- between both of these things. Don't get me wrong; I still believe in promoting the necessary aspect of things, and that students should be taught how to fish instead of being given the fish. If they are given the fish, they should spend a bit of time reflecting on how the fish got there before feasting away, though many people will likely just take the fish and think later or never. That being said, sometimes a fish is needed in the immediate moment. Along that line, I think that it is worth exploring this dance of arbitrary and necessary a bit more as I plan my lessons and units. With every unit in the new BC curriculum, there still exists a wide plethora of terminology and conventions that students have to get comfortable with. I feel that it is okay to impose these on students, AS LONG AS they are given the time and space to truly wrestle with what they represent, and why the conventions are the way they are. This would be one type of activity or exercise that goes into my own lessons -- the idea of getting students' feet wet so they can get a real grasp of the arbitrary, so that they may come up with their own. The reading mentions that arbitrary knowledge can be made in multiple ways with "students making up their own" being one, which I feel is worth exploring in a lesson. In a previous EDUC 450 blog post of mine, I went into a possible activity where students could make up their own *arbitrary* terminology, and I still think that it has merit as an activity where students got to contribute something to the mathematics community while applying necessary thinking. To sum up what I am saying in this paragraph, the article has mostly reinforced my existing beliefs that there needs to be a bit of both in order to have a rich and meaningful educational experience for students. The main *new* element to my planning is really to just ensure that this is the case by providing outlets for necessary thinking while establishing the matter-of-fact of the arbitrary.

One last thing that I thought about was the order of arbitrary and necessary teaching. I have arranged the possible orders into three types:

1. Pre-Necessary: Explain the arbitrary knowledge and wisdoms before getting students to think and be aware, in order to set a foundation first.
2. During-Necessary: When we allow students to work through ideas on their own or when we explore ideas together, arbitrary knowledge can be illuminated as the situations arise.
3. Post-Necessary: We give students and ourselves the freedom to explore and work out the necessaries, and then illuminate what we have discovered using the arbitrary terms.

Personally I am a fan of the first strategy just because it is how things were always done traditionally and I'm admittedly most comfortable with it. There is also a benefit of letting the students be aware of the conventions/terminology/notations first, and then allowing them to immerse themselves with expanding further on them to generate necessary mathematics in a beautiful dance of the established and the unexplored. In fact, this is something I did for a lesson on solving trigonometric equations with identities: I provided advice to keep in mind (arbitrary), but gave students the chance to discover why exactly the advice would be useful in their exploration of the questions of the chapter. However, there is merit in the other strategies; they opt to not bog down students with the memorization details and choose to prioritize a more experiential form of learning new ideas. There are lesson and unit implementations that can follow a Thinking Classroom, lab-like way of discovery-first by encouraging students to think and be aware together to notice patterns, and then consolidating what they have discovered afterwards. I would say that none of these things should be done in a full 0% or 100% manner, but it would be wise to consider which order applies best for the lesson/unit at hand.

Small edit: I also see many teachers leaving certain things for formula sheets. This is another implementation of arbitrary teaching that curbs the mental load of memorization, which I've found that no one really objects to. Especially in an age of instantaneous information, it does feel a bit silly if we have to dock points from students for having to search up matter-of-fact information that could've been a Google search away (as opposed to being critical with the process where they use this matter-of-fact information). Thus, the article has influences on how I separate the "need to know on the test" with the "it will be on the formula sheet"; with the sheet, we can ask richer questions and students can spend more time on being aware instead of being bogged down by trivia.