Order of Operations - Mnemonics

Mnemonics

Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally" with the first letter of each word creating the acronym PEMDAS. Canada uses BEDMAS and the UK uses BIDMAS or BODMAS. In Canada and other English speaking countries, Parentheses may be called Brackets, or symbols of inclusion and Exponentiation may be called either Indices, Powers or Orders, which have the same precedence as Roots or Radicals. Since multiplication and division are of equal precedence, M and D are often interchanged, leading to such acronyms as BOMDAS.

These mnemonics may be misleading when written this way, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order "addition first, subtraction afterward" would also give the wrong answer.

The correct answer is 9 (and not 5, which we get when we add 3 and 2 first to get 5,and then subtract it from 10 to get the final answer of 5), which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two.

An alternative way to write the mnemonic is:

P

E

MD

AS

Or, simply as PEMA, where it is taught that multiplication and division inherently share the same precedence; and that addition and subtraction inherently share the same precedence. PEMA is one of the mnemonics taught in New Zealand. This makes the equivalence of multiplication and division and of addition and subtraction clear.

Another potentially misleading aspect of this mnemonic is the inclusion of P for parentheses. First of all, parentheses are grouping symbols, not operation symbols. Also, within the grouping symbols, there may be expressions involving several operations, which would need to be evaluated according to the correct order or operations which appear AFTER the P in the mnemonic. It is probably better to teach that grouping symbols are not operations themselves, but rather are used to change the precedence of operations from the default.

Once logs are introduced they should be given the same precedence as exponents.