TL;DR:
A mnemonic stores an order or a list. It does not store the meaning behind it.
PEMDAS is useful, but read literally it gets
8 ÷ 2 × 4wrong. Multiplication and division share a level, and so do addition and subtraction.Pair every mnemonic with two or three worked examples, including one that catches the usual trap.
Almost everyone who has taken a maths class remembers "Please Excuse My Dear Aunt Sally". It spells PEMDAS, the order for evaluating an expression: parentheses, exponents, multiplication, division, addition, subtraction. It is a very good mnemonic. It is also a good example of how a mnemonic can mislead you when you trust it more than the idea behind it.
What a mnemonic does well
A mnemonic is a handle. It compresses a list that has no natural order in your head into something you can say in one breath: an acronym (PEMDAS), a rhyme ("i before e, except after c"), or a sentence made of the first letters. It works because a short, rhythmic, or silly phrase is easier to recall than a bare list.
That makes mnemonics good for arbitrary sequences and labels: the order of the planets, the cranial nerves, the parts of a lab report. There is nothing to understand about why the planets are in that order from the Sun. You just need the order.
What it cannot do
A mnemonic holds the sequence, not the reason. If the rule has a reason and exceptions, the handle will drop both. You remember the letters and apply them mechanically, and the places where the rule bends are exactly where you go wrong.
Worked example: PEMDAS and where it trips you
Take the expression 8 ÷ 2 × 4. Read PEMDAS literally and you do M before D, so you multiply 2 × 4 = 8 first and then compute 8 ÷ 8 = 1. That is wrong.
The real rule is that multiplication and division sit on the same level, and you go left to right. So 8 ÷ 2 = 4, then 4 × 4 = 16. The answer is 16.
The same goes for addition and subtraction. Take 6 − 2 + 3. Left to right: 6 − 2 = 4, then 4 + 3 = 7. Doing the addition first would give 2 + 3 = 5, then 6 − 5 = 1, which is wrong.
Now a case where the letters work fine: 3 + 4 × 2². Exponent first: 2² = 4. Then multiplication: 4 × 4 = 16. Then addition: 3 + 16 = 19. Check by plugging in: if you added first, you would get 7 × 4 = 28, which is not the same, so the order really matters.
Expression | Literal PEMDAS (wrong) | Correct order | Answer |
|---|---|---|---|
8 ÷ 2 × 4 | 2 × 4 = 8, then 8 ÷ 8 = 1 | left to right: 8 ÷ 2 = 4, then 4 × 4 | 16 |
6 − 2 + 3 | 2 + 3 = 5, then 6 − 5 = 1 | left to right: 6 − 2 = 4, then 4 + 3 | 7 |
3 + 4 × 2² | same as correct | 2² = 4, 4 × 4 = 16, 3 + 16 | 19 |
The fix is a small change to the handle: draw PEMDAS as four levels, with M and D sharing one and A and S sharing one. And attach the reason. The order exists because multiplication is repeated addition and exponents are repeated multiplication, so higher operations bind tighter.
How to use a mnemonic well
Understand the thing first: Know what the items are and why they go in that order, if there is a reason. The mnemonic then serves as a retrieval cue.
Build or choose a handle: Acronyms, rhymes, and silly sentences all work. A handle you made yourself is often easier to keep than a standard one.
Test the handle on examples: Try it on at least one normal case and one tricky case, as with
8 ÷ 2 × 4above.Test yourself without the handle: Later, solve problems without looking at it. If you only get them right with the phrase, you have the order but not the idea.
A quick test for any mnemonic
Ask two questions. Does the thing I am memorizing have a reason? If so, learn the reason first. Could the handle give a wrong answer in an unusual case? If so, write that case next to it. A mnemonic that passes both questions is a safe shortcut. One that fails either is a trap waiting for the test.
Mistakes to avoid
Memorizing a mnemonic and skipping the explanation it stands for.
Applying it mechanically when the problem has a twist.
Inventing a mnemonic for something you could easily understand instead.
Collecting so many handles that you cannot recall which goes with which.
Forgetting what the letters stand for. An acronym with lost letters is worthless.
Your plan for this week
List the mnemonics you currently use in your classes.
For each, write one sentence on why the order is what it is, or note that it is arbitrary.
Write two worked examples for any that you have ever got wrong.
Quiz yourself on the examples a few days later with the mnemonic hidden.
Frequently asked questions
Are mnemonics cheating?
No. They are a legitimate memory tool. They only become a problem when they replace understanding on topics where understanding is possible.
Is it better to make my own?
Often yes, because the effort of making one helps you remember it, and you can use words you find funny. Check that it still gives the right order.
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