Let's stop making excuses for dear Aunt Sally. You know who I mean: Please Excuse My Dear Aunt Sally. Many students can still remember the mnemonic PEMDAS for the rules of order of operations, but PEMDAS omits or confuses some situations. Most prominently, even my current college algebra textbook does not make it clear that in the M and D of PEMDAS multiplication and division are of equal priority, so a division is performed before a multiplication if the division appears to the left of the multiplication.
I like to invite my students to use all their calculators on expressions like 300 - 200/50*10 to see which calculators obey the rules of operations and which do not. I am always surprised when students first turn to the calculator on their cell phone to answer this.
PEMDAS doesn't address fraction bars, nor that with multiple grouping symbols you start with the innermost grouping symbols first. And I suppose it's too much to expect that PEMDAS address that in -32 the negation precedes the exponentiation; by the way, Excel disagrees with this.
Speaking of exponentiation, what do you do with x y z ? Exponentiation is not associative. Johnny Lott in A Problem Solving Approach to Mathematics says exponentiation is done in order from right to left (page 276), although Excel disagrees with this one too.
So there's more to the rules of order of operation than PEMDAS. But if you are getting a little tired of Aunt Sally, why not switch to Please Email My Dad A Shark, by the authors of xkcd.com
Anyone have any good ideas on teaching order of operations?