What is arbitrary and necessary in teaching (math)?

1. What does Hewitt mean by "arbitrary" and "necessary"? How do you decide, for a particular lesson, what is arbitrary and what is necessary?

According to Dave Hewitt, in a learning setting, "arbitrary" means that students are informed of the learning materials through external sources, such as teachers, books, the internet, and many others; "necessary" means that students do not need to be informed of the learning materials through external sources. Often times, they can work through the questions and figure out themselves. As Hewitt writes in his article, what is "arbitrary" is in the "realm of memory", and what is "necessary" is in the "realm of awareness". I have encountered this kind of thinking and reasoning during my short practicum. There were times when students asked why it must be this way and not that, in which I hear my teacher say, "Sometimes, things are just easier this way." I think many of us new and experienced teachers have asked ourselves this kind of question many times during our teaching, and many of these examples have been cleverly brought up by Hewitt.

For my teaching, personally, I would consider math symbols, notations, or any naming conventions as something that is arbitrary. I am unsure of how I can explain why 2 is bigger than 1, for instance. Number symbols (e.g. 1, 2, 3,...) are universal across the world, so explaining that to students would be difficult for me because we have been taught and memorized this way since the beginning.

I would consider proofs, properties and relationships to be necessary because in many of these cases, they can be derived and worked out by students; students within "awareness" that is. For example, understanding why a parabola is shaped a certain way can be understood by graphing and interpreting the formula because parabolic formulas can be revised. Similarly, one can probably also work out why Speed = distance / time, or why chemical formula is written this way because these formulas can be derived and solved from real-life applications, whereas symbols and notations do not.

2. How might this idea influence how you plan for your lessons, and particularly, how you decide "Who does the math" in your math class?

Hewitt's idea of what is arbitrary and what is necessary really blows my mind on thinking to myself and asking "why". There might even be questions that some university professors and teachers cannot answer because they themselves learned concepts arbitrarily. As a teacher, I will try to dig deeper on the materials being taught in class for the curious minds. I might explain, on the side, why we do math a certain way while showing and explaining that there are alternative methods to do math differently, but it would just be harder for some students to grasp (although they are not limited to solving math their own way as long as they derive a right answer in the end). I know that not all students are capable of understanding the "necessary" component to math, but as long as teachers have an answer to "why", they can use simple hints so students can really ponder on it and "think about it".