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Depending on where you went to school, you may have memorized one of several acronyms for the order of operations.  In the United States it is usually PEMDAS ("Please Excuse My Dear Aunt Sally"). In much of the UK and Canada, it is BIDMAS or BEDMAS. Some classrooms use GEMA.  At an international school like ours, our students arrive having learned several of these, which reveals something important: the mnemonic is not the math.

Every one of these memory devices, in every language, stands for the same underlying order — and following that order faithfully will get you the correct answer to an expression.  Yet many students still arrive at the wrong answer.  The trouble is that they follow the letters of the mnemonic literally, rather than the mathematical structure those letters are only standing in for.  The memory aid is a model of that structure, and like any model there are places where it breaks down, which can lead to misconceptions.

Every so often a problem like 6 ÷ 2(1 + 2) sweeps across social media, and thousands of commenters split into two furious groups, half insisting the answer is 1, the other half insisting that it's 9.  While these disagreements can become heated, mathematicians don't worry too much about them.  When I ask mathematician friends about them, they tell me that the expression is written ambiguously, and simply writing it more clearly will get rid of the ambiguity.  The lesson of the viral problem is not "learn your order of operations better."  It is that the grouping — deciding what belongs together — is what's important.

Let me bust the most common myths directly.

Take the M and D in PEMDAS, which stand for "multiply" and "divide."  Because the M is listed first, students often assume they must always multiply before they divide — and, in the same way, that the A ("add") must always come before the S ("subtract") . Neither is true. Multiplication and division are inverse operations that sit at the very same level, and so are addition and subtraction; within each pair, you simply work from left to right. Even when students are specifically taught this, they often forget and fall back on the order of the letters in the acronym.

These problems are the predictable result of teaching math as a strict, top-to-bottom checklist.  The mnemonic accidentally invents two rules that do not exist.

The GEMA mnemonic addresses this problem directly. GEMA stands for Grouping symbols (including parentheses and brackets, but also the vinculum, aka fraction bar). Next is E, for exponents.  Third is M for multiplication, which includes division (remember: 8 ÷ 2 can be written 8 x ½ ). Last comes A, addition, which includes subtraction (15 - 5 = 15 + (-5)).

So the order of operations is not just a list to memorize, but rather it reflects the structure of arithmetic.  Some operations are simply more powerful than others, and need to be done first.  Exponents are repeated multiplication; multiplication is repeated addition.  Because exponentiation grows a number faster than multiplication does, and multiplication faster than addition, we handle the most powerful operations first.  That is not a rule someone invented to trip students up; it is a description of how the operations are built out of one another.

For example, go ahead and match the diagram below with one of these two expressions:

• 10 ∙ 4 + 5
• 10 ∙ (4+5)

It's the underlying mathematics that makes these two different, not a human-chosen order.  It is the mathematical structure that we are trying to teach, not just a set of rules that we are asking students to obey.