I have always had difficulty with directionality in mathematics. When I was in elementary school, I did not understand that < and > were two different signs; I was as likely to read
5 > 2
as “two is less than five” as I was to read “five is greater than two”. I remember multiple choice questions:
The symbol > means:
a) less than
b) greater than
c) equal to
And I would have no clue how to answer the question (other than the fact that choice C was obviously wrong). That symbol, > , meant both ‘greater than’ and ‘less than’ to me. I had no problem doing the actual math; I could put the right symbol in the math problems, correctly identifying 80,835 as bigger than 35,808. However, the multiple choice question shown above (or directions that required me to “write the ‘greater than’ sign”) floored me. FINALLY, someone taught me to read math from left to right. I wish I knew who had set me straight, so I could thank that person.
What does this have to do with the node title? Well, I have some problems with this particular mnemonic. As someone who has been teaching middle school math to middle school math students for fifteen years, I feel I have the authority to complain. Here’s the beef: Please Excuse My Dear Aunt Sally, or PEMDAS, as it is sometimes known, implies chronological order to the steps. That is, one would expect to perform operations in parentheses (Please) first, before operations involving exponents, (Excuse) which is actually the correct assumption, but then one might continue in that vein and expect to perform all multiplication (My) before division (Dear), and all addition (Aunt) before subtraction (Sally), and that is simply not how it should be done. Oddly enough, this didn't bother me when I was in school; I didn't notice the problem until I tried to teach Order of Operations using the mnemonic.
The thing to remember is that we do math the same way we read English; from left to right. The rules for order of operations state that multiplication and division are actually on the same tier, and whichever comes first in the problem should be done first. The same holds true for addition and subtraction.
Consider the problem:
18 – 2 x 6 - 10 ÷ 5 + 4
Done correctly, it looks like this:
18 – 2 x 6 - 10 ÷ 5 + 4
18 - 12 - 2 + 4
6 - 2 + 4
4 + 4
= 8
But if we insist on adding before subtracting, because that’s what Aunt Sally tells us to do, we get
18 – 2 x 6 - 10 ÷ 5 + 4
18 - 12 - 2 + 4
18 - 12 - 6
6 - 6
= 0
Which is not the same answer at all. It’s zero, for God’s sake. Don’t even get the math nerds on this site started on zero.
So here’s what I do, to avoid this confusion: I put the following notes on the board:
Order of Operations
1st ( )
2nd exponents
3rd *------> x , ÷
4th *------> + , -
I explain that all the steps are performed from left to right, and with that in mind kids who have memorized Please Excuse My Dear Aunt Sally or PEMDAS can use those devices to remember that parentheses are first, etc. etc., as long as they keep in mind that division can be done before multiplication and subtraction before addtion if they show up first in line. Those little arrows in my notes are reminders to start at the left side of the problem and work across, dividing or multiplying (subtracting or adding...) as you go. A few examples of how problems can go wrong (like the one illustrated above), and a little practice, and my students are good to go.
I know order of operations can be more involved than what I've shown here, what with brackets and roots and the like, but I'm teaching Pre-Algebra and Algebra I. We start small.