Order of Operation Issues

I mentioned this earlier this week but decided I wanted to look into the topic of order of operations due to the way it is traditionally taught.  All of the posters hanging in the hallway made by elementary students were all based off of PEMDAS - "Please Excuse My Dear Aunt Sally".

Each operation was on a separate line beginning with parenthesis, exponents, multiplication, division, addition, with subtraction at the bottom.  Each poster was identical.

One big problem with teaching it this way is that when students reach high school, they believe the multiplication is done before division and addition before subtraction. They don't understand that multiplication and division are done in order as you read the problem left to right. The same applies to addition and subtraction.

Too many elementary students never get the part about addition or subtraction depending on which one you get to first when reading the equation from left to right or multiplication or division  under the same circumstances.  When they arrive in high school, they apply the order of operations incorrectly due to this misconception.

In addition its rather difficult to undo the incorrectly learned methods when they arrive in my classes.  I know at least one 6th grade teacher who works to undo the damage but often times it is so ingrained in them that they cannot unlearn PEMDAS to do the order correctly

I have actually had students argue with me that they did the math right after doing all the multiplication followed by the division, then the addition, and finally the subtraction.  I've tried reteaching it but honestly it is so ingrained in them that it is very difficult to undo the learning.

Furthermore, when you start adding a variety of different grouping symbols and multiple exponents, the student use of this mnemonic breaks down because it does not state what to do with multiple grouping signs. I have to remind my students to work from the inner most pair of parenthesis, brackets, or other grouping symbols because PEMDAS only talks about parenthesis. 

In high school we tend to use more complex equations where we might have 2 + 4^2 all divided by 3^2 and you have to discuss implied groupings due to the way things are written.  My students do not understand implied parenthesis because they've spent their elementary years seeing only cases that follow PEMDAS.

I believe the first step is to make sure elementary teachers understand the actual way the order of operation works so they are not just teaching students the same "Please Excuse My Dear Aunt Sally" they learned.  They need to include the left to right statement to make it more complete.

Let me know what you think.  I'd love to hear.