Why Everyone Gets This Wrong Anyway
Most kids learn the acronym and then immediately forget it the moment a problem gets more than three steps. PEMDAS exists, sure, but the real issue is that students treat every operation as equally important. They don't actually internalize that some operations override others. I've sat through enough practice sessions to know this pattern by heart. A student will see 3 + 4 × 5 and instantly write 35 because they read left to right without pausing to check whether multiplication is hiding somewhere in the expression. The standard sequence goes this way: parentheses first, then exponents, then multiplication and division working left to right as a single tier, then addition and subtraction also working left to right as a single tier. That last part is where most mistakes happen. Multiplication and division share the same priority level, which means you don't do all multiplication before all division. You do them in the order they appear. Same thing with addition and subtraction. Here is a concrete example that trips people up regularly: 12 ÷ 3 × 2. The correct answer is 8, not 2. You divide first because it appears on the left, giving you 4, then multiply by 2. If you blindly multiply before dividing because someone told you M comes before D, you end up with 2 and you are wrong. The acronym makes this worse because it literally spells out M before D, even though they are equal in precedence.
The Exponent Trap That Nobody Warns About
Exponents apply only to the immediate term they are attached to unless parentheses say otherwise. This is not obvious to sixth graders. Take the expression 3². The correct evaluation is 9, not 9. The exponent applies to 3 first, giving you 9, and then the negative sign sits outside that result. But if you write (3)², the exponent applies to the entire grouped value, and the answer becomes 9. Students routinely conflate these two forms because they look similar on paper but produce opposite results. I ran into this specific problem last year when a student kept getting textbook answers wrong on a worksheet. The problem was 5 2³ × 4. She was computing 2³ as if it were (2)³ or (5 2)³ depending on her mood that day. Once I had her rewrite every exponent expression with explicit grouping before touching any other operation, her accuracy jumped from about 40 percent to roughly 85 percent within two weeks. The workaround was mechanical and boring, which is exactly why it worked.
What Actually Works In Practice
The most reliable method I have seen students use consistently is circle-each-operation with priority levels. Write out the full expression, then draw a circle around every set of parentheses. Next circle all exponents. Then go through multiplication and division left to right, circling each one. Finally circle addition and subtraction left to right. Evaluate each circled operation before moving to the next ring. It takes longer at first, maybe 45 seconds per problem instead of 10, but it builds the correct habit. After about a month of this drill, the circling becomes internal and they no longer need the paper marks. Another practical tip involves fractions inside expressions. When a fraction bar appears in a problem like (6 + 4) ÷ (2 × 3), the fraction bar itself acts as a grouping symbol. The entire numerator and the entire denominator are each treated as if they sit inside invisible parentheses. Students miss this constantly. They will evaluate 6 + 4 ÷ 2 × 3 without recognizing that the horizontal line changes the priority structure entirely.
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When This Method Falls Apart
Order of operations as taught in sixth grade works fine for standard arithmetic expressions, but it breaks down the moment you introduce variables, nested functions, or programming notation. In algebra, expressions like 2x + 3 become ambiguous without agreed-upon conventions about implicit multiplication. Some curricula treat 2x as having higher precedence than explicit multiplication, which is inconsistent with the strict PEMDAS model. A calculator programmed with strict left-to-right rules will give different answers than one that assumes implicit multiplication binds tighter. This inconsistency shows up on standardized tests more often than teachers admit. The real bottleneck with teaching order of operations at this level is that it rewards memorization over understanding. Students can parrot PEMDAS without grasping why the hierarchy exists. The hierarchy exists because of how mathematics structures repeated operations. Multiplication is repeated addition. Exponents are repeated multiplication. Parentheses exist to force a reordering of the natural sequence. When students understand that chain, they do not need to memorize an acronym. Most sixth graders are not ready for that level of abstraction, which is why the acronym persists as a crutch. If a student is struggling, skip the acronym entirely and just teach them to identify and solve grouped expressions first, then handle exponents, then everything else left to right. It is less catchy but it produces fewer errors. The circle method I described above does exactly this without requiring any memory work.
