The author brought up an interesting point about the mathematics curriculum, which we have probably all encountered before.
Arbitrary: All students need to be informed of the arbitrary by someone else. It belongs to the realm of memory.
Necessary: Some students can become aware of what is necessary without being informed of it by someone else, and it belongs to the realm of awareness.
I believe most people would agree that the fun of mathematics lies in the "necessary" part, where we have to potential to discover on our own. However, not "all students have the awareness to be able to work [necessary] out, only that someone is able to work this out without the need to be informed it", so the duty of math teachers are to guide students into the process of mathematical awareness.
When designing the unit plan for my long practicum, I paid lots of attention to the order in which I introduce new concepts. In the exponent chapter, I re-arranged the teachers' notes and put "negative exponent" before "quotient rule of exponent". My rationale is: if students are unlikely to discover $a^{-n} =\frac{1}{a^n}$, at least I can guide them to prove $\frac{a^n}{a^m} = a^{n-m}$, which builds on what they have just learned about the negative exponent. Being able to prove this instead of memorize it, should be the focus of our mathematics curriculum.
The author is concerned with "the amount of classroom time given over to the arbitrary compared with where the mathematics actually lies". There is no perfect solution to this problem. I do believe that students need a certain amount of "math facts" before they can really become aware of something. What we can do it to be direct about the arbitrary side of mathematics, and teaching in an educating awareness way whenever we can.
Good. I like the thoughtful example from your exponents class!
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