Yes, she votes, as do millions just like her
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The first one is in my opinion correct, juxtaposition first.
But your second one is just wrong.
6/2(1+2) is not the same as 6/2+4. And 6/2+4 is not the same as 6/6, in the case of 6/2+4 you would divide first, for 3+4=7, which is completely wrong. 6/2+4 should have been 6/(2+4) if you're doing it that way.
And, although I don't agree with it, I'll make the other argument for completeness too, which is strict adherence to PEMDAS. This is how people are getting the alleged "correct" answer:
6/2(1+2)
6/2(3)
3(3) = 9
This treats juxtaposition as straight multiplication, and moves left to right. This is how many NA teachers teach, but not how most mathematicians, engineers, and the like do things.
That's where the whole issue comes from. The statement is made to be intentionally misleading, and if you use PEMDAS you'll get 9, but PEMDAS is oversimplified, and only almost always holds up. But if you write a confusing statement made specifically to break PEMDAS, you end up with this mess.
Sure. It has been many years since I was familiar with order of operations (I think that's what the PEDMAS refers to, isn't it?). As an English major undergraduate I got a B in college algebra,. which is pretty good considering my background (math retarded). As I'm sure you know, if you don't use math skills you lose them.
What's most interesting is that you start off with what looks like a math problem, and end up with a completely bizarre human history problem. If I've got this right--and I think I do--this is how it goes:
All math-writing dudes, since basically forever, use implied multiplication (multiplication by juxtaposition). They consider it too obvious to mention. Eventually, but not that long ago (within the last 100-150 years) they debate whether they should make a formalized convention that multiplication (explicit, as opposed to implied) takes precedence over division. This debate is left unresolved, yet in that debate implied multiplication is 100%, still too obvious to mention.
In the 1980s, you get the widespread rollout of calculators to students, which I think hits American schools first. These calculators barf when you try to use implied multiplication, insisting on the consistent use of operators. The next generation of calculators, some have fixed this problem (giving priority to implied multiplication over both division and explicit multiplication, which is up to this point, universally considered correct), some still barf, and some "fixed the glitch" by adhering to strict PEMDAS (which is not correct) and hiding the process, and some use the correct PEJMDAS while adding brackets to your equation, thereby notifying you of how they were treating it.
Somewhere in this mess, a conspiracy of North American teachers, frustrated by these conflicting returns, successfully lobby many calculator-making companies to ditch PEJMDAS (the right one) and only adhere to PEMDAS (the wrong one). Going forward, the majority of calculators (or possibly just the ones sold in America) are PEMDAS. To put it plainly, American teachers bullied calculator companies to only give them calculators that abandon thousands of years of standard notational convention. So now, from this time onward, at least the teachers and the calculators are in agreement. They're wrong, but they're in agreement. Eventually Common Core comes along and exacerbates this problem: All Must Agree, Even in Wrongness.
The American students, taught by this weird Cult of PEMDAS, go on to insist on their ideology. 95% of the world rolls along as they always have, with PEJMDAS, but some notable heavies--such as Google Calculator, which obviously has oversized importance due to widespread availability and usage--continue to spread the cult of PEMDAS.
It's fascinating. A tale of innovation, groupthink, authority, conformity, memetic disease, exceptionalism, midwittery, and dogged insistence.
I don't have a high enough understanding to state any of that definitively, but I'd say your mostly over the target.
Seems like, more or less. That's my least/favorite part. Bruh, teachers should not be writing the rules of math. I get PEMDAS is a nice mnemonic, and mostly works. But if it's not completely clear, it's not the best way. But they seem so obsessed by it that they want calculators to use it because it's what they're teaching. That's moronic.
I think mathematics is a complicated enough subject that maybe high school teachers (maybe even including middle school teachers, not sure who was lobbying) shouldn't be (re)writing the rules. Look, you're technically math professionals, in that you get paid for your math. But you don't use math. You're not in the math field, you're in the teaching field. Are they dumb? No, I think math teachers tend to be cooler than the teachers for plenty of other subjects, even. No hate. But they're certainly not at the level of the field where they should be writing the rules for things they don't even understand.
Weird thing, I think it was at the end of that first video I linked, Wolfram Alpha mostly uses PEMDAS...mostly. But in certain cases ends up using something that looks more like PEJMDAS. It all depends on your input; some phrasing involving variables (correctly) solves juxtaposition first.
The whole thing is pretty crazy.
Also, imagine the sheer fucking arrogance to tell a calculator company to change their methodology because it's what they teach.