Brackets and Orders, not that anybody calls “(” a bracket or “^” an “order”.

It's "to the order of" actually. 2² is 2 to the order of 2

not that anybody calls “(” a bracket

Yes we do. They're ALL Brackets - square, round, squiggly...

Why do people put bot pairs of multiplication and division, and addition and subtraction on the acronym?

Because it's intended to be used as a checklist

Do you really follow that order with the associative operations?

Yep

They're all just different mnemonics for the same rules.

BODMAS (or BEDMAS), not BOMDAS. (unless they did that in some random state). I'm an Australian Maths teacher

multiplication and division are the same operation

No they're not, but they are both binary operators.

in order to stop confusing people, it should just be: BOMA.

Leaving out D and S confuses people about where to do them in the order. It's intended to be used as a checklist

Was anyone else ever taught it as BOMDAS as opposed to BODMAS?

I don't think so, and I've seen Maths textbooks from all over the world. Only the U.S. wants to reverse the order of DM from everyone else's acronym. 🙄

I’ll never understand these approaches to learning. They require remembering the phrase, and then require remembering how the phrase translates to the rules you need to remember

Yeah, exactly, but the U.S. seems to have a chip on it's shoulder about always doing everything differently to the whole rest of the world. "Maths? We're not going to use BEDMAS, and we're not going to call them Brackets, and...".

OMG! That's the first time I've actually seen AI get order of operations right! 😂

You don’t necessarily have to do parentheses first

Yes, you do necessarily have to do it first

What matters is that the things inside the parentheses are a group that you can’t break apart

And outside, and you must do them first. You haven't finished Brackets until you have 5(8-5)=15.

10÷2+3-2*(2+1) you can do the division first

only because you've separated that part with a plus sign

then multiplications

There aren't any, only Addition and Brackets (and Subtraction inside Brackets)

I mean, there are very few ambiguous cases

There are precisely none which are ambiguous

so it would be clearer

That's because it's already clear as is, as per the rules of Maths.

I think someone told me that order of operations is like a natural law

It's a natural consequence of the definitions of the operators. e.g. Multiplication is shorthand for repeated Addition - 2x3=2+2+2 - so if you don't do it before addition you end up with wrong answers. The order of operations rules is in fact just breaking everything down into Addition and Subtraction and then solving from there.

not a convention

There are some conventions, like left to right, but in that case that's only because students tend to make mistakes with signs when they don't go from left to right, so it's there to preserve teachers sanity.