PEMDAS is Wrong: Why Students Misunderstand the Order of Operations

Many of my middle- and high-school students remember PEMDAS, an acronym meant to represent the order of operations for evaluating mathematical expressions, from a previous class. “Please Excuse My Dear Aunt Sally,” they say (or “Please Eat My Doritos and Salsa,” one clever student told me). “Parentheses, Exponents, Multiplication, Division, Addition, Subtraction,” they say. This is great, right? These students remember the order of operations, right? Isn’t PEMDAS amazing?

No, PEMDAS is not amazing.  PEMDAS is wrong. Or, rather, many students’ understanding of the order of operations is wrong and PEMDAS is to blame.

Let’s take a look at a simple example: 4 - 3 + 10 ÷ 5 × 2.

Since there are no parentheses or exponents, PEMDAS leads many students to think we should begin by evaluating multiplication. Thus:

4 - 3 + 10 ÷ 5 × 2 =

4 - 3 + 10 ÷ 10

Next, these students would move on to division:

4 - 3 + 10 ÷ 10 =

4 - 3 + 1

Then, addition:

4 - 3 + 1 =

4 - 4

And, lastly, subtraction:

4 - 4 = 0

An elegant solution! And all with the help of PEMDAS! What a wonderful invention!

Or, that is what one would say if this solution were correct.

For many students, PEMDAS implies that the order of operations consists of six steps, one for each letter of the acronym. In fact, the order of operations consists of four steps:

1. Evaluate operations within parentheses

2. Evaluate exponents

3. Evaluate multiplication and division from left to right

4. Evaluate addition and subtraction from left to right

Importantly, multiplication and division are evaluated as part of the same step, as are addition and subtraction in the next step. Let’s try our example again, this time using the order of operations. Since there are no parentheses or exponents, we will begin with the third step, “evaluate multiplication and division from left to right”:

4 - 3 + 10 ÷ 5 × 2 =

4 - 3 + 2 × 2 =

4 - 3 + 4

Next, we finish up by applying the fourth step, “evaluate addition and subtraction from left to right”:

4 - 3 + 4 =

1 + 4 = 5

This time, we have arrived at the correct solution.

I have encountered far too many students who misunderstand the order of operations in precisely this way. Fortunately, I have a solution to this problem: rather than beginning a lesson on the order of operations by introducing PEMDAS, it should only be introduced after working through many examples using the four-step order of operations as it is written above. Once students come to understand the order of operations through practice, only then may PEMDAS be introduced to serve its intended purpose: not as a set of instructions, but as a mnemonic device to assist with memory.

Eric Bennett is the author of Elementary Math for Computer Science with Python, a textbook for elementary-aged students. He has taught mathematics in the Austin area since 2005.