It was at the first meeting of the Challenger Middle School Mathematics Department that I first heard of GEMDAS. What an obviously good idea!

Remember PEMDAS

PEMDAS is a mnemonic device used to remember the order of operations. The order of operations is a set of rules that specify in which order to perform the mathematical operations in an expression. Such a convention is necessary so that everyone in the world gets the same results when working a math problem (assuming they don’t make a mistake!).

Youre history Aunt Sally (but fess up before you go)

You're history Aunt Sally (but fess up before you go) (Nono via Picasa)

PEMDAS is also remembered as “Please Excuse My Dear Aunt Sally” (what for, no one is quite sure). It means that when evaluating an expression that includes more than one operator, anything in parentheses is evaluated first, followed by exponents, then multiplication and/or division (whichever comes first, left to right), and finally addition and/or subtraction (whichever comes first, left to right). That is the order of operations.

GEMDAS is Better

At the aforementioned department meeting, Ms. Hertzog, a math teacher at Challenger, said something about GEMDAS being superior to PEMDAS because with PEMDAS some learners get it stuck in their heads that parentheses are the only grouping symbols that need to be taken into account, or else they get confused when some other grouping symbol is used instead of parentheses. She made no claim to inventing GEMDAS, but apparently heard about it at a workshop somewhere.

In GEMDAS, the G stands for Grouping symbol, and all the other letters keep the same meanings that they have in PEMDAS.  So with GEMDAS, learners are better able to keep in mind that ALL expressions in, on, or under grouping symbols need to be evaluated first. Grouping symbols include parentheses (), brackets [], braces {}, the vinculum ____ (which is the technical name for what most people call a fraction bar, which is also a repetition symbol used in decimal notation) , and the radical (also known as the square root symbol).

Google hits for PEMDAS outnumber hits for GEMDAS by a bit more than 10 to 1 (40,400 to 3,030, as of the date of this posting) so clearly PEMDAS is still a more popular mnemonic that GEMDAS.

I for one am switching over to using and teaching GEMDAS, and I have the impression that all the math teachers at my school are going to do the same so we are using a common vocabulary.  It is traditional to teach and use PEMDAS, but tradition is no reason to keep an old idea around when a better alternative exists. It’s time to get rid of PEMDAS. Long live GEMDAS!

  1. Sure does generalize the word “parentheses”. Now we just need a fun little phrase for this acronym “GEMDAS” (Unless there already is one) …Giving everyone my delicious apple sauce…I don’t know. I bet there are some other ideas for said word.

  2. Thanks for checking it out April. I like “Give Everyone My Delicious Apple Sauce!” or “Give Everyone More Delicious Apple Sauce”. Any other ideas out there?

  3. Just read your post today. It’s funny, I just taught order of operations to my students last week and suggested we use GEMDAS rather than PEMDAS so we were sure to address all grouping symbols and not just parentheses. I find no matter how much I emphasize it I still find students performing M before D and A before S rather than working out M and D from left to right before A and S from left to right. Perhaps we need GE(MD)(AS) or GE (DM)(SA).

  4. Yeah it’s still “okay” but just not as good as GEMDAS for the reasons I outlined in the post. If you are able to wrap your head around the important idea that the P does not only apply to parentheses then stick with it.

  5. Thanks for reading Zach. You point out another problem with the mnemonic method. It goes to show that there is no substitute for true understanding in place of blind use of a mnemonic device.

  6. I believe it needs to change completely. By using either one we are teaching 6 steps instead of 4 steps.

    We use Gary Enjoys MaD AntS. This is only 4 words.
    Grouping symbols, Exponents, and so forth. We put this on a poster with a boy burning ants with a magnifying glass. If we keep using PEMDAS or GEMDAS we are still teaching six steps instead of 4 steps.

  7. Thank you for sharing your ideas Leslie.

    Well, that poster about the ants is inappropriate, but I can see how it would definitely work!

    I am not convinced there is any advantage to your method. There is no way to shorten the number of steps in evaluating an expression using the order of operations. The number of steps is the number of steps, no matter what method we use to teach, learn and remember it.

  8. I was looking for a mneomic for GEMDAS when my 5th grader came home with it. Another trick he was using was drawing vertical columns down on either side off addition and subtraction symbols so that everything else is pretty much in their own group. Then, after doing the proper grouping symbols it is all just left-to-right and the vertical bars assist in doing addition and subtraction last. Nice.

    Now if only he’d get the implied multipication when a number is outside a group. Example: 5(66-64)


  9. GEMDAS is nice, but like Leslie I think it still adds to confusion. I think Leslie could possibly do better by saying MuDdy AntS instead of MaD AntS, since there may be confusion about the A in MaD standing for Addition. Plus, this loses the ant cruelty aspect of her mnemonic device…unless by muddy you mean

  10. Is / a grouping symbol? What are the rules behind that? If it is, then is 4/2^2 supposed to evaluate the division first or the items on either side first? If its the items on either side, then this is 4/4 = 1. Otherwise, if we look at this as 4/2*2 is the 2*2 part of the group or just 2? In other words can we rewrite as 2*4/2 or as 2/2*4? Please provide a link to something that explains how to handle the various division symbols as grouping symbols and whether 4/2^2 is not the same as 4/2*2. I guess a/bc could = c/ba or it could = a/(bc)depending on perspective, but what about a/b where there is a leading 1? IN this case we actually have a/b = a/1b, so based on a/bc = ac/b, we can say a/1b=ab/1? I don’t think so. You tell me.

  11. I have heard it called that and I agree. It means “simplify both the top and bottom before proceeding.” 4/2^2 means you cannot do anything with the 4 or the 2^2 until the expression is simplified because they are in a “group.” You have to square 2 then divide that result into 2 before anything else can be done with that expression.

  12. GEMDAS is still a faulty way of teaching order of op’s. Multiplication and division are on the same level as well as addition and subtraction. Each should be calculated from left to right. Of course there is associativity and commutative properties and distributive. All of which are inherent in operations if one understand’s their group theory. PEMDAS and GEMDAS still fall short of what you/we should be teaching. Teach the math, not faulty and incomplete rules that frustrate everyone.

  13. Sure it’s faulty. Just like pretty much everything else we do. But, you have to do “something” and I think it is better to at least have some sort of mnemonic like this to work from rather than nothing at all. If you have an idea of how the order of operations can be taught better without ever mentioning GEMDAS or the like, I would love to hear about it. Of course we “teach the math.”

    How about GE[MD][AS], with the brackets indicating the L-R orders. Or perhaps overlining the MD and the AS.

  14. Quick question, does absolute value also fall in that category of grouping along with parenthesis and brackets?

  15. Great question. Yes, because nothing can go through the absolute value symbol unless you can somehow simplify and resolve what is enclosed in the absolute value bars. Sometimes you have to accept that there is more than one result.

  16. You know, my math teacher taught it to me as GEMDAS too. To make it easier for the less quick kids to remember though, and to make sure it was clear that the Multiply/Divide and Add/Subtract were one thing each, she had me use GEMS. Which is Grouping; Exponents; Multiply; and Subtract. So just GEMS. It’s easier to remember for me as GEMS, and also, it groups the Add/Subtract and Multiply/Divide bit together.

  17. I’ve come up with a phrase for GEMDAS that I like because of the message and the play on words.

    Getting Educated Makes Dollars and Sense

    I really would like to find 2 words for M/D and A/S which would work even if the order were to be swapped around. In the meantime, I use a set of arrows much like the syncing symbol, around the M D and the A S.

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