The Problem with PEMDAS


I am all for mnemonic devices as they are proven effective memory aids, especially for long-term retention of information.  Two new ones I learned while researching this blog entry are shown below:


Knuckle mnemonic for the number of days in each month of the Gregorian Calendar. Each projecting knuckle represents a 31-day month.

For example, the first 15 digits of the mathematical constant pi (3.14159265358979) can be encoded as "Now I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics"; "Now", having 3 letters, represents the first number, 3.

One of the first mnemonic devices I ever learned was "ROY Grew Blue Indigo Violets" as a way to remember the correct order for the colors of the rainbow: Red, Orange, Yellow, Blue, Indigo, & Violet.  I loved it because it was made extra easy since three of the seven colors were actually part of the sentence I memorized.  It has worked like a charm all these years as I've never forgotten it.

However, there is danger in using a simplistic mnemonic device to recall information that is more complex and intricate than just a simple list of words or string of numbers.  I'm lookin' at you PEMDAS (or GEMDAS, as is now in vogue).  PEMDAS is an oft-used mnemonic device to help remember the correct order of operations - PEMDAS (a.k.a. "Please Excuse My Dear Aunt Sally") stands for Parenthesis, Exponents, Multiplication, Division, Addition, & Subtraction.  There have always been issues with this device:  for example, in an expression like 

{2 - (9+4) * 6} / 2 

students were often confused by having both parentheses and curly brackets in the same problem (there's no C or B in PEMDAS!).  Hence the "newer" version: GEMDAS, where the G stands for "grouping symbols" and is supposed to include braces, brackets, curly brackets, parentheses, etc. Problem is, it doesn't help the student know which set to do FIRST since that's not part of the mnemonic (Answer: work from the inside out.)  Here's the real problem with PEMDAS:  it only tells part of the story; and, the part that's left out is the most important part.  

Learning the "order of operations" (also known as operator precedence) is an educational goal for mathematics -- currently being tremendously emphasized in Algebra classes all over the country -- that ensures that there is agreement on the correct answer when multiple operations are required.  Because of our nation's current obsession with standardized testing and uniformity, the pressure is on both students and teachers to "cover" as much information as possible in the shortest amount of time in order to "practice" the tests for months and months before the real test day comes, results of which all money depend.  As a result of the unfortunate trend, too many teachers have taken a tragic shortcut to teaching students the order of operations and just said, "Follow PEMDAS" by way of instruction.  

Now, you may be saying to yourself, "Well, if the student gets the right answer on the test, then what's the harm?"  Here's the thing:  OFTEN, by following PEMDAS without understanding some of the backstory and important details not included in the little sentence about my female family member, the student will NOT get the problem correct.  

Here are two examples to prove my point:

Ex. 1.  Given the numeric expression 12 - 5 + 4 students who have only been taught PEMDAS rotely will proceed thusly:

12 - 5 + 4 (decide to do Addition first since it comes before Subtraction in PEMDAS)

12 - 9 (now do Subtraction since it's the operation that comes last in PEMDAS)

3 (incorrect result!)

Ex. 2.  Take a look at the example shown below:

Order of operations example problem

Do you notice anything?  That's right:  although PEMDAS clearly states that Division is to be performed BEFORE Subtraction, this worked example (which DOES show the correct solution) violates that rule... TWICE!  Look carefully.  In the step just to the left of the professor's pointing hand, it is clear that the two operations left in play are subtraction (in the form of both -18 - 9 and 7 - 4) and division (the numerator divided by the denominator in the fraction shown).  However, in the very next step (the one shown just to the right of the pointing hand) it is clear that the two subtraction problems are completed FIRST!  Only then is the division, -27 / 3, performed and the correct result of -9 is achieved.  Division was done LAST!  If a student had actually followed PEMDAS in this problem, s/he would have gotten the wrong answer.

So what IS the rest of the story of the order of operations?  What are the missing key elements that one should have filed in the memory banks along with the priority list of operations?  Here are three keys to understanding the order of operations:

1.  All of the operations are binary -- meaning we can only operate on two numbers at a time.  Basically, the list of operations is just a way to figure out which two numbers to start with, then which two, then which two, until you finally get to one result at the end of the process.  What this means is that sometimes, like with complicated rational expressions, division can't happen until the numerator and denominator are simplified into one number each (as in Ex. 2 above).

2.  The order of importance is made up!  We (people) realized that, depending upon which set of two numbers you start with, would result in different answers to the same problem!  To fix this and make it so we'd all be able to agree, we (people) created the "rule" of which operations to do first, second, third, etc.  If we suddenly found ourselves on planet OtherEarthia, the rule might be completely different!

3.  Remember, here in the United States, we read from left to right -- math problems are worked in the same order they are read.  That means operations that share the same level of importance (e.g. addition & subtraction) are worked in the order that you come to them in the problem as you read, NOT in the addition first- subtraction second order of the mnemonic (as in Ex. 1 above).

If students are taught the order of operations with these important elements first, then, and only then, PEMDAS may be offered as a way to solidify the little details of which operations take precedence over which.

In researching elements for this blog, I came across a wonderful online Order of Operations Millionaire game that I think students would LOVE (and a way to sneak in the practice that is required in order to make the learning complete).  I'll leave you with my victory graphics shown below  Enjoy!  As always I'd love to hear from you: contact me.

Order of Operations Millionaire game
Rimwe WINS!!


Order of Operations overview

Teacher site where PEMDAS graphic came from

History of the order of operations (Math Forum)

Fabulous Order of Operations Millionaire game


The Solver Blog

Author:  Dr. Diana S. Perdue

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