To Mnemonic or Not To Mnemonic: The Problem with PEMDAS

I’m sure by now we are all sick of seeing all of math problem posts that circulate around all of the social media sites pretty regularly. They usually begin with “Only x% of people get this correct!” What follows is a reasonably simple orders of operation problem which people in the comments argue about. The arguments typically center around the rules they learned for the orders of operation. 

“Well, according to PEMDAS...”

Ugh! And the debates over the meaning of this horrible mnemonic tool then ensue. The interesting thing is, based on what country you are born in, you may have learned a different mnemonic for performing computation. PEMDAS isn’t universal. If you are in Australia, India, or the United Kingdom, you may have learned BODMAS. If you are in Canada or New Zealand, you have have learned BEDMAS. Different people learn a different mnemonic, and these mnemonic tools are a big part of the reason why people have misconceptions about the order of operations.

I’m going to focus on PEMDAS for this discussion. I’m sure most everyone that leaned it in grade school remembers what PEMDAS stands for: 

P = parentheses,
E = exponents,
M = multiplication,
D = division,
A = addition, and
S = subtraction.

The point of this tool being that it helps you remember the order in which the perform the operations from left to right. The problem is, if you take the tool literally, you will perform the calculations incorrectly. PEMDAS has given many the misconception that you perform multiplication first, then division, then addition, and lastly subtraction. This is not correct though.

Why is this not correct? Because, in reality, those four operations are really just two operations. Multiplication and division are inverse operations, and addition and subtraction are inverse operations (with subtraction and division kind of being imperfect inverses, but that’s another conversation). As a result, each pair of operations should be treated one in the same. 5 - 2 is equivalent to 5 + (-2) and 6 ÷ 2 is equivalent to 6 × ½. There is an additive equivalent for every subtraction expression and there is a multiplicative equivalent for every division expression. Therefore addition and subtraction should be performed together and multiplication and division should be performed together.

So, the correct orders of operation are actually:

1. Grouping symbols,
2. Exponents, 
3. Multiplication OR Division (from left to right)
4. Addition OR Subtraction (from left to right)

We can apply it to the problem below as an example. Notice in the final steps, we perform the subtraction before addition, because when moving left to right, subtraction comes first.

This is one reason why I am generally not a fan of teaching mnemonics in this manner. Although these tools are developed with well meaning, they can bring upon misconceptions or even fall short. 

Take FOIL as another example. It’s used to teach polynomial multiplication. You multiply the first terms (F), then the outer (O), then the inner (I), and then the last terms (L). This seems like a solid way to learn the sequence of multiplication steps needed before finally simplifying the polynomial, but the problem is, it only works for multiplying two binomials. If a student learns FOIL and is then presented with multiplying a trinomial by a binomial or with multiplying two trinomials, they may be at a loss.

A better approach would be to build off of the their understanding of the Distributive Property and extend it to multiplication with polynomials. When multiplying polynomials you are performing the distributive properly multiple times in sequence. We could also return to the array method that students learned as a conceptual stepping stone in elementary school which is an excellent visual to make sense of the algebraic nature of polynomial multiplication. Teaching it in either of these manners removes the restrictions that FOIL has and expands it to cover all polynomial cases.

Returning to PEMDAS, similarly, ensuring students understand the inverse and interrelated relationships of addition and subtraction and multiplication and division would help to alleviate misconceptions regarding these arithmetic processes. Exploring the conceptual aspects of the orders of operation would be far better for understanding than just memorizing a problematic mnemonic. Or, if a mnemonic is necessary, perhaps something more reliable, like PEMA (Please Eat More Avocados??). Focusing on the stronger operations of multiplication and addition (reinforcing that subtraction is a form of addition and division is a form of multiplication).

But, in the end, these mnemonics are there to aid procedural skill, and as I have mentioned before, conceptual understanding is more powerful than memorizing procedure. Focusing on truly understanding the concept will typically alleviate any need for tricks to help memorize procedure.