Ending the PEMDAS Debate: Clearing Up the Confusion Once and For All

If you've been involved in teaching math, you've likely encountered the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). While it's a helpful tool for remembering the order of operations, many teachers, parents, and even students have debated its finer points. Specifically, confusion often arises around the sequence of multiplication and division, as well as addition and subtraction. In this blog, we’ll clarify these rules, offer helpful tips for teaching them, and provide resources for further exploration.

The Heart of the Controversy: Multiplication vs. Division

A common misconception with PEMDAS is that multiplication must always be performed before division. This is not true. Multiplication and division, along with addition and subtraction, are performed from left to right, as they appear in the expression.

Consider this example:

12 ÷ 3 × 4

If you follow the left-to-right rule:

12 ÷ 3 = 4
4 × 4 = 16

However, if you incorrectly perform multiplication before division:

3 × 4 = 12 

12 ÷ 12 = 1

Clearly, these two approaches yield very different results (16 and 1), and only the first one is correct based on the left-to-right rule. Teaching students to read equations in the correct order is crucial to avoid mistakes.

This left-to-right rule is also true for addition and subtraction:

10 − 4 + 3

Following the left-to-right rule:

10 − 4 = 6 

6 + 3 = 9

If addition is done before subtraction:

4 + 3 = 7 

10 − 7 = 3

Again, different answers demonstrate why order matters.

PEMDAS Alternatives: GEMS and GEMA

Many educators have shifted from PEMDAS to alternative acronyms like GEMS (Grouping, Exponents, Multiply/Divide, Subtract/Add) or GEMA (Grouping, Exponents, Multiplicative, Additive). These alternatives avoid the misconception that multiplication comes before division by grouping multiplication and division, and addition and subtraction, as equals. The rule is simple: go left to right, treating these operations with equal importance.

For example, GEMS can be broken down as follows:

• G for Grouping (which includes parentheses, brackets, and fraction bars)
• E for Exponents
• M for Multiply/Divide (whichever comes first, from left to right)
• S for Subtract/Add (whichever comes first, from left to right)

These systems simplify teaching by making it clear that the order of multiplication and division, as well as addition and subtraction, is determined by their position in the equation—not by some inherent precedence of multiplication over division or addition over subtraction.

Why Is This Important?

The order of operations ensures consistency in math. Without a standardized approach, different people could arrive at different results for the same problem. As teacher Aaron Nunley pointed out, "The reason we do order of operations like this is that we are moving from simplest to complex operations. Multiplication is fast adding. Exponents are fast multiplication." This logical progression guarantees that everyone solves equations in the same way, avoiding confusion and inconsistency.

How Singapore Math and U.S. Curriculum Handle PEMDAS

In Singapore Math’s "Math in Focus" curriculum, the left-to-right rule for multiplication and division is taught, but some teachers might misinterpret this as a strict hierarchy of multiplication over division. It’s essential that both educators and students understand that while multiplication and division are closely related, they must be performed in the order they appear in the expression.

In the U.S., the left-to-right rule is widely taught and reinforced in standardized tests like Texas's STAAR assessments. This further demonstrates that the approach is not unique to one curriculum but a standard in modern math education.

Helpful Resources for Educators and Students

Here are some helpful resources to explore this topic further and access tools that clarify the order of operations:

1. Khan Academy - Introduction to Order of Operations
   Khan Academy provides excellent explanations and practice exercises for students to master the order of operations, including interactive examples and video tutorials.
2. Mathnasium - The Order of Operations in Math: A Kid-Friendly Guide
   This resource is designed for parents and students, explaining PEMDAS in a straightforward and accessible way. It's perfect for younger learners and includes tips for avoiding common mistakes.
3. Nix the Tricks - Order of Operations
   "Nix the Tricks" offers a guide to teaching math concepts without relying too much on memorized rules like PEMDAS. It’s a great resource for fostering conceptual understanding rather than rote memorization.
4. Math is Fun - Order of Operations
   This website breaks down the order of operations with simple examples, showing how multiplication and division are performed left to right, along with practice problems for reinforcement.
5. Khan Academy - PEMDAS Video Example
   A more in-depth video tutorial on PEMDAS and how to apply it correctly in various math problems.

Conclusion: Clarity is Key

While PEMDAS remains a useful tool for remembering the order of operations, it’s essential to teach students that multiplication and division, as well as addition and subtraction, are performed based on their appearance in the expression—left to right. Whether you use PEMDAS, GEMS, or GEMA, the goal is to ensure that students understand the logic behind the rules.

The debate over PEMDAS might persist in classrooms, but the reality is simple: following the left-to-right rule will always lead to the correct result. By teaching this approach and utilizing the resources above, we can help students avoid confusion and develop a deeper understanding of math.