e.g. you made a comment about teaching the "scientific" method to kids, and I can tell you as a teacher that we already do (but the OP never looked at any Maths textbooks, nor asked any Maths teachers) :-)
We do teach children how to solve this. It's not children who get it wrong - it's adults that get it wrong! Cos they've forgotten the rules of Maths (in this case The Distributive Law and Terms).
Probably a cop-out from the teacher
No, that's an actual convention of Maths, to make sure people (who don't know better) obey the actual rule of left associativity.
Look, this is not the only case where semantics and syntax don’t always map
Syntax varies, semantics doesn't. e.g. in some places colon is used for division, in others an obelus, but regardless of which notation you use, the interpretation of division is immutable.
they use different semantics for a notation that for you seems to have clear meaning
They might use different notation, but the semantics is universal.
You need to accept that human communication isn’t as perfectly unambiguous as mathematics (writing math down using notation is a way of communicating)
Writing Maths notation is a way of using Maths, and has to be interpreted according to the rules of Maths - that's what they exist for!
implicit multiplication
There's no such thing as "implicit multiplication"
Some place it above explicit multiplication and division,
Which is correct, seeing as how we're solving brackets, and brackets always come first.
But if you place it as equal to it’s explicit counterparts, then you’d sweep left to right giving you an answer of 9
Which is wrong.
Since those are both valid interpretations of the order of operations
No, they're not. Treating brackets as, you know, brackets, is the only valid interpretation. "Multiplication" refers literally to multiplication signs, of which there are none in this problem.
But in reality nobody would write an equation like this
Yes they would. a(b+c) is the standard way to write a factorised term.
6⁄2(1+2) ⇒ 6⁄2*3 ⇒ 6⁄6 ⇒ 1
You're more patient than me to go to that trouble! 😂 But yeah, looks good. Just one technicality (and relates to how many people arrive at the wrong answer), the 2x3 should be in brackets. Yes if you had a proper fraction bar it wouldn't matter, but that's what's missing with inline writing, and is compensated for with brackets (and brackets can't be removed unless there's only 1 term inside). In your original comment, it does indeed look like 6/(2x3), but, to illustrate the issue with what you wrote, as soon as I quoted it, it now looks like (6/2)x3 in my comment.
why @wischi went through the effort of writing a rather detailed blog on this is because the order of operations most people are taught doesn’t cover juxtaposition.
The order of operations rules do cover it. Did you not notice that the OP never referenced a single Maths textbook? Because, had that been done, the whole house of cards would've fallen down. See my Fact Check posts doing exactly that.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
They didn't - neither of them is a rule of Maths.
When the @onion said there were two different sets of rules, you know as well as I do that they meant strong vs. weak juxtaposition
Neither of which is a rule in Maths, as had already been pointed out. The 2 relevant rules, both of which never got mentioned in the whole discussion, are The Distributive Law and Terms.
You’re right that in reality nobody would write an equation like this
No, that was wrong - they really would. 2(1+2) is the standard way to write a factorised term. i.e. a(b+c).
hole that exists in the standard order of operations
There's not a hole in the order of operations - there's a hole in people's memories where The Distributive Law and Terms used to be. You'll notice no students ever get this wrong, because they remember all the rules.
given no other context to do so, you would either have to pick a side
That "side" being follow the rules of Maths - works every time. :-)
In dark mode you press it to go light mode, and in light mode you press it to go dark mode, but yeah, strictly speaking should've used a button rather than a switch for that usage.
Just an update on this. There's been an update to the issue on the 10th of January with "Under Investigation. This issue is currently being investigated...", which is a nice change from "more info needed", or even worse "this isn't a bug it's a feature", so they've managed to reproduce the bug and are looking into it.
This means if you see the 17.8.4 update available message, the fix isn't in there, so don't update yet. I'm using 17.8.0 - I'm not sure if the bug was introduced in 17.8.1, 17.8.2, or 17.8.3, but 17.8.0 is definitely safe to use.
P is 2(2+2)=(2x2+2x2)=(4+4)=8
If you do 8/2 when you still have brackets, then you just did division before brackets and disobeyed order of operations rules. You also broke the rule of Terms, since 2(4) is a single term.
I see you didn't read my thread then. I have a section on calculators, which points out that WolframAlpha does it wrong. i.e. they also break the rule of Terms.