If you want to teach people methods to solve equations, limits, integrals etc. speed with basic algebraic operations is necessary.
Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.
I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.
TL;DR: algebra != arithmetic AKA real math doesn't use numbers
I'm only good at _arithmetic_ because of making Warhammer 40k armies (true story bro).
I'm good at _algebra_ because I was taught well, on top of a knack.
Speed with basic algebraic operations was very helpful in many places but speed with arithmetic operations has only been helpful in board games.
I don't think anyone here would disagree with your point about algebra, but I think a lot of people such as myself would disagree that pre-K memorization of arithmetic helps with algebra later.
But strong arithmetic fundamentals are absolutely necessary for strong algebra. I've watched kids struggle with basic algebra because when they don't instantly recognize that 7x8=56, they also don't recognize that 7zx8z=56z(edit: squared).
Edit: thanks to the reply; HN ate my superscript 2. Apparently it doesn't like the unicode multiplication x, either: × ² ?
Hmm. Alright, I can see solid arithmetic being good for introducing the concept of a variable...
...but the only time I ever saw significant numbers when actually doing math was in toy problems that deliberately chose weird, big coefficients, where the arithmetic part was by far the least significant.
Performing a bunch of calculations for tabletop wargaming is basically the same as learning multiplication tables and solving related problem sets. It should help every time that for example you have to simplify a polynomial involving fractions and similar operations.
As I stated in another post, I don't know when it is neurologically ideal to learn arithmetics, it seems something that would be important to study carefully (personally I learnt before grade school, when I was 3-4 years old, but I didn't learn to read until I was 6, something that is often taught earlier).
Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.
I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.