So I’m going through this “algebra for incredibly stupid people” book so I can help my nephew and niece out with their math problems, and I’m reviewing all that stuff I haven’t touched in literally decades and I’m having a bit of trouble with order of operations.

I thought I was taught “Please Excuse My Dear Aunt Sally”, which would mean doing all your addition before your subtraction. I’m getting a few of these sample questions wrong by doing that, though - it seems I’d get their answer if I just went left to right adding or subtracting as I come to it. The book is a little fuzzy on this. I assume it’s right and I’ve been doing it wrong my whole life?

For example, for 16 - 2 * 4 + 3 I get 5 and they get 11.

Addition and subtraction are performed in the order that they appear. Always have been as far back as I can remember. Of course multiplication and division come first.

You have no parentheses. You have no exponents. You DO have multiplication, so you do that next, and get the 16 - 8 + 3 referenced above. You don’t have division. You have addition and subtraction, but their order doesn’t matter, just go left to right. The order of multiplication or division doesn’t matter either, you just have to do it before the addition or subtraction (unless the addition or subtraction is in brackets, in which case they jump to the top of the order.)

Yeah, I get that. No problem with the rest of it. And now I get that you do the addition and the subtraction together. What I don’t get is the people saying it shouldn’t matter.

See, this is what got her confused in the first place. The proper order is:
Parentheses
Exponents
Multiplication/Division
Addition/Subtraction
BTW EristicKallistic, the mistake that you made was that (-1) * 8 +3 != -1 * (8 + 3)

I think studying up on it is making me forget how I know how to do it. If I’d gone with my gut and just skimmed through those problems I would have done it the right way.

Yes. I never was taught the mnemonic and always remembered that multiplication is the same level of precedence as division, and addition and subtraction are on the same level of precedence, too. And, in that case, you go from left to right.

ETA: Argh…except that it doesn’t really matter with multiplication and division, as 10x4/2 is the same whether you do (10x4)/2 or 10x(4/2). Whereas, 10-4+2 is equal to 10 + (-4) + 2 and not 10-(4+2), which is 10-4-2.

Just remember that addition IS subtraction. 5 - 8 is the same thing as 5 + -8. So do them in the order they appear from left to right. (Same goes for multiplication and division.)

It matters with multiplication and division the same way as it matters with addition and subtraction. Compare (10/4)x2 = 5 to 10/(4x2) = 1.25.

The subtlety is that it doesn’t matter if you rewrite everything into just multiplication and reciprocation, same as it doesn’t matter if you rewrite everything into just addition and negation. But what gets negated/reciprocated when you rewrite subtraction/division is highly dependent on how you group things; to properly interpret our subtraction/division notation, you have to avoid clumping things together when negating/reciprocating, which [because we stick the - and / signs on the left of the things they negate/reciprocate] amounts to the same thing as “Go from left to right”.

Yes.
Any string of numbers (or terms) added or subtracted gives the same result.
Ditto, any strng of terms multiplied or divided gives the same result.
Commutative means you can shuffle the order.

As pointed out above, 16-8+3 does not mean 16 - (8+3), it means (+16) + (-8) + (+3)

For Example:
Similarly, you have to be clear on order of operation in multiplication/division.
23/56, unless brackets of some other rule intervens, means:
(2)(3)(1/5)(6).
Unless someone states the question differently, it does not mean divide by (56), it means divide by 5 and mutiply your result by 6. The trouble is that simple in-line formula writing does not convey the same subtlety as a fancy mathematical equation. This is the source of your confusion. OTOH, the people posing the problem deliberately wrote it this way to ensure the lesson gets across. I bet you’ll never forget it this time…

Once you undestand this, it is obvious that any string of addition/subtraction(s) can be done in any order, and any string of multiplication/division(s) can be done in any order.

You just have to settle all the multiplication/divisions before tackling addition/subtraction.