You can explain the truly arbitrary elements of PEMDAS (the left to right of AS and MD) through an experiment. Allow students, independently, to do these two problems any way they want, ignoring any stupid arbitrary rule they might have previously memorized:Include here a few order of operations-type problems.
Enter the Stupid Arbitrary Rule (SAR).While I agree that it is an arbitrary rule, it's far from stupid and, for me, it highlights one of the reasons why schools exist; that is, telling kids how the world they are about to enter works and what its rules are. But then I get this:
Because we need to all come up with the same answer, we need a rule to follow. Really, it can be any stupid arbitrary rule (SAR). But we agreed, at some point in history, to all follow the “left to right” thing once we were down to addition & subtraction or multiplication & division.
It’s important to note that kids didn’t get to be part of that agreement we made. Just like they don’t get to vote in elections. Is it fair? Probably not. They would probably do a better job of choosing leaders as well as determining the order of operations. But that’s the way things likes SARs work.You have to stop that crap right at the source. How can anyone say "that agreement we made" and conclude that it probably wasn't fair that kids can't be part of that decision?
First, of course, is the "we" thing. There is no "we" and "they" here and nobody waved their scepter around declaring that henceforth All Students Will Do It This Way. The order of operations didn't exist at some point in time, but then neither did algebraic notation. There weren't exponents until fairly recently (they were written words), someone had to have been the first to use a zero and place value ... you can go on. The point is that someone started using a notation, explained what it meant and how it worked and others decided it was easier and fell in with the crowd.
Enter the modern student, spoiled silly and clutching his cellphone and fantasies of being a "Digital Native" who can multitask and has no use for That Boring Crap.
What has "fair" got to do with it? Why is this pubescent psycho-babble coming from the only adult in the room?
And when he says students would probably do a better job of electing leaders, you have just heard the sound of a deluded mind. It's typical in education, echoing the "noble savage" mentality. So many teachers harbor this idea that kids know so much more than we stupid adults, that if we only took off the restraints, they'd be teaching themselves calculus in no time. They're better than we were, smarter than we were, and by golly just look at how responsible they'd be.
This is a huge disservice and only feeds the disillusionment with school and learning - "Why are you screwing me over? This is so UNFAIR."
And to then make up new rules for mathematics, post them in the classroom and keep using them? You've just gotten through telling them that all the rules are stupid and arbitrary and you want to have them invent, and then use, more stupid and arbitrary ones?
I'll stick with the valuable, useful and arbitrary ones and I'm always looking for a new way to demonstrate them ... like this image I found (might be Dy/Dan's):