I think this speaks to why I have a total of 5 years of college and no degree.

Starting at about 7th grade, math class is taught to every single American school child as if they're going to grow up to become mathematicians. Formal definitions, proofs, long sets of rules for how you manipulate squiggles to become other squiggles that you're supposed to obey because that's what the book says.

Early my 7th grade year, my teacher wrote a long string of numbers and operators on the board, something like 6 + 4 - 7 * 8 + 3 / 9. Then told us to work this problem and then say what we came up with. This divided us into two groups: Those who hadn't learned Order of Operations on our own time who did (six plus four is ten, minus seven is three, times eight is 24, plus three is 27, divided by nine is three) Three, and who were then told we were wrong and stupid, and those who somehow had, who did (seven times eight is 56, three divided by nine is some tiny fraction...) got a very different number, and were told they were right. Terrible method of teaching, because it alienates the students who need to do the learning right off the bat. And this basically set the tone until I dropped out of college for the second time.

Meanwhile, I'm over in the corner like

Having read your article, I contend it should be:
P(arentheses)
E(xponents)
M(ultiplication)D(ivision)
A(ddition)S(ubtraction)
and strong juxtaposition should be thrown out the window.

Why? Well, to be clear, I would prefer one of them die so we can get past this argument that pops up every few years so weak or strong doesn't matter much to me, and I think weak juxtaposition is more easily taught and more easily supported by PEMDAS. I'm not saying it receives direct support, but rather the lack of instruction has us fall back on what we know as an overarching rule (multiplication and division are equal). Strong juxtaposition has an additional ruling to PEMDAS that specifies this specific case, whereas weak juxtaposition doesn't need an additional ruling (and I would argue anyone who says otherwise isn't logically extrapolating from the PEMDAS ruleset). I don't think the sides are as equal as people pose.

To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).

But again, I really don't care. Just let one die. Kill it, if you have to.

Forgot the algebra using fruit emoji or whatever the fuck.

Bonus points for the stuff where suddenly one of the symbols has changed and it's "supposedly" 1/2 or 2/3 etc. of a banana now, without that symbol having been defined.

I don't remember everything, but I remember the first two operations are exponents then parentheses. Edit: wait is it the other way around?

Hi, I’m stupid, is it 1+2 first, then multiple it by 2, then divide 6 by 6?

Or is it 1+2, then divide 6 by 2, then multiple?

I think it’s the first one but I’ve got no idea.

You guys are doing it all wrong: ask chatgpt for the correct answer and paste it here. Done.

Who needs to learn or know anything really?

When I used to play WoW years ago I'd always put -6 x 6 - 6 = -12 in trade chat and they would all lose their minds. Adding that incorrect solution usually got them more riled up than having no solution.

While I agree the problem as written is ambiguous and should be written with explicit operators, I have 1 argument to make. In pretty much every other field if we have a question the answer pretty much always ends up being something along the lines of "well the experts do this" or "this professor at this prestigious university says this", or "the scientific community says". The fact that this article even states that academic circles and "scientific" calculators use strong juxtaposition, while basic education and basic calculators use weak juxtaposition is interesting. Why do we treat math differently than pretty much every other field? Shouldn't strong juxtaposition be the precedent and the norm then just how the scientific community sets precedents for literally every other field? We should start saying weak juxtaposition is wrong and just settle on one.

This has been my devil's advocate argument.

My years out of school has made me forget about how division notation is actually supposed to work and how genuinely useless the ÷ and / symbols are outside the most basic two-number problems. And it's entirely me being dumb because I've already written problems as 6÷(2(1+2)) to account for it before. Me brain dun work right ;~;

You lost me on the section when you started going into different calculators, but I read the rest of the post. Well written even if I ultimately disagree!

The reason imo there is ambiguity with these math problems is bad/outdated teaching. The way I was taught pemdas, you always do the left-most operations first, while otherwise still following the ordering.

Doing this for 6÷2(1+2), there is no ambiguity that the answer is 9. You do your parentheses first as always, 6÷2(3), and then since division and multiplication are equal in ordering weight, you do the division first because it's the left most operation, leaving us 3(3), which is of course 9.

If someone wrote this equation with the intention that the answer is 1, they wrote the equation wrong, simple as that.

I found a few typos. In the 2nd paragraph under the section "strong feelings", you use "than" when it should be "then". More importantly, when talking about distributive properties, you say x(x+z)=xy+xz. I believe you meant x(y+z)=xy+xz.

Otherwise, I enjoyed that read. I'm embarrassed to say that I did think pemdas meant multiplication came before division, however I'm proud to say that I've unconsciously known that it's important to avoid the ambiguity by putting parentheses everywhere for example when I make formulas in spreadsheets. Which by the way, spreadsheets generally allow multiplication by juxtaposition.

I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher

I just finished your article and wow! I'm definitely going to save it and share it the next time I come across another one of those viral problems. It was incredibly thorough and well researched, you clearly put a lot of energy and effort into it and it blew me away. It was really refreshing to see someone articulate themselves so passionately with supporting research. I look forward to reading more of your work!! 👏

I guess if you wrote it out with a different annotation it would be

‎ ‎ 6

-‐--------‐--------------

2(1+2)

=

6

-‐--------‐--------------

2×3

=

6

--‐--------‐--------------

6

=1

I hate the stupid things though

The order of operations is not part of a holy text that must be blindly followed. If these numbers had units and we knew what quantity we were trying to solve for, there would be no argument whatsoever about what to do. This is a question that never comes up in physics because you can use dimensional analysis to check to see if you did the algebra correctly. Context matters.