PEMDAS is wrong. Here is why (and how to avoid this common mistake)

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If you follow PEMDAS to the letter, you are going to get the problem wrong in some cases. Algebra students need to know the correct order of operations to build a good foundation for higher level math classes.

This article tells you what’s wrong with the PEMDAS mnemonic (Please Excuse My Dear Aunt Sally, used to remember Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).

I’ll also go over how Algebra students can avoid making this mistake.

Why is Order of Operations Important?

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Order of Operations is necessary in every math class up to the Calculus level and beyond.

It’s really a Pre-Algebra concept that bleeds into Algebra. Teachers expect students to know Order of Operations rules.

Many students think they know it, but they get it wrong. And if you get Order of Operations wrong, the whole problem is wrong. Most students don’t even know where to find the error.

The PEMDAS mnemonic is often to blame.

The reason for mistakes? It is taught incorrectly.

PEMDAS vs Order of Operations

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Let’s get these terms straight first.

The vocabulary isn’t the same.

Because we follow Common Core Standards, there’s a big push to use common, universal language. Not everyone says Please Excuse My Dear Aunt Sally or PEMDAS, believe it or not.

The correct terminology is Order of Operations.

And that’s how I teach.

If we continue to use these cutesy nicknames or Kindergarten-level language, the concept loses its directness.

When I tutor, I ask students about PEMDAS, Please Excuse My Dear Aunt Sally, and Order of Operations. That’s just to help with their recollections to see how they were taught in the past. But I teach the correct way from there out.

What is wrong with PEMDAS?

Thing is, Order of Operations is not a difficult concept. Most students go into it with confidence, often brushing it off and saying they already know it.

What’s wrong is that Order of Operations isn’t always in order.

Sometimes it’s Please Excuse Dear My Sally Aunt, going back to the PEMDAS mnemonic for Order of Operations.

The order for multiplying/dividing and adding/subtracting is reversible.

Students may be called to divide before multiplying or subtract before adding.

It all depends on the expressions.

Since expressions are read from left to right, students MUST do the operation that appears first.

PEMDAS wouldn’t be so bad if there was more of an emphasis on the areas of interchangeableness.

The reversible rule in the Order of Operations has always been that way. Unfortunately, it just hasn’t been communicated well. Myself included.

I first learned in college and in professional learning groups with other teachers (where teachers meet together)

PEMDAS was created with the expectation that people know about the interchangeableness.

The Real Explanation Behind PEMDAS

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And why is Adding/Subtracting and Multiplying/Dividing Interchangeable?

Simply put, it’s because dividing and subtracting almost don’t exist.

Hang with me.

When you’re dividing, you’re really multiplying by a fraction.

And when you’re subtracting, you’re really adding a negative number.

So, Order of Operations is really four steps:

  • Parentheses
  • Exponents
  • Multiply/Divide
  • Add/Subtract

I’ve had students to the point of arguing with me that a math problem was right when it was really wrong because of an incorrect understanding PEMDAS.

Understanding the inner workings and the why behind it all will help.

Algebra and Order of Operations

Algebra students need to know how to do each step in the Order of Operations, so teachers start with reviewing exponent rules, working with parentheses, and the basics of multiplying, dividing, subtracting, and adding.

Then, the course usually goes a little more in-depth in the steps to simplifying expressions and solve equations.

In pre-Algebra, students see one-step or two-step equations. If a student can handle that, then they are most likely following correct Order of Operations. Algebra builds on that by adding multiple steps to find a solution.

The Algebra component of Order of Operations is in justifying. Students have to prove why they are doing certain actions.

Why Are Some Order of Operations Problems So Hard?

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Algebra expressions and equations have the potential to get really complex.

But, trust me, hard Order of Operations problems are not to intimidate you.

Teachers present this information with an expectation that students go beyond the basic Order of Operations.

Order of Operations should be so second nature that Algebra students can see these complex ideas and they can take it little by little.

Students can’t just look at the expression or equation and automatically get overwhelmed.

Take it one step at a time.

I find some struggling students don’t know when to stop or end their math problem.

You should stop when the expression or equation is fully simplified. The way that you know is there will be nothing else you can do.

Be careful not oversimplify. I grade some papers where the student initially have the right answer, but the student randomly adds their own steps and ends up getting the problem wrong.

These students don’t need more practice in this case, they need to work on better understanding the concepts.

How is Order of Operations Presented in Standardized Testing?

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Order of Operations shows up on more complex levels in standardized tests, particularly the ACT or SAT college entrance exam.

In theory, class material should be on a level higher than the standardized test.

A standardized test is just seeing if you learn, or mastered, the standard.

It’s the teacher’s job to teach above and beyond that standard.

There aren’t many overly complex problems in class.

Order of Operations isn’t really a standalone standard. It’s a concept coupled with the expressions standards. It can be time-consuming to separate and focus on.

On the test, students are usually given a list of steps to solve or simplify. There’s usually a missing step or a mistake for the students to identify.

I see a lot of this in open-ended, non-multiple choice problems.

Final Thoughts

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In 2019, PEMDAS took center-stage after a viral math problem circulated on social media.

It showed that a lot of people make the common mistakes with the PEMDAS mnemonic.

Order of Operations allows the student to break down their problem and look at the little things.

I suggest students momentarily forget everything else at the time while they are working each step in the Order of Operations.

Don’t overthink it.

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