Mathematical Reflections on “Please Excuse My Dear Aunt Sally”

From the Marshall Memo #428

In this article in Teaching Children Mathematics, third-grade teacher and Eastern Illinois University professor Kyungsoon Jeon reports that when most teachers are asked about order of operations, they immediately think PEMDAS – “Please Excuse My Dear Aunt Sally” (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) – but can’t explain why order of operations matters. They also tend to make mistakes because they don’t know some of the finer points of PEMDAS.

To help with this common problem, Jeon suggests several steps. First, check out basic knowledge of PEMDAS by solving these problems:

7 – 3 + 11

2 – 5 • 4 + 1

2 – 3 • 4 + 5 • 2 – 1 + 5

2 + 16 ÷ 4 • 2 + 8 

She finds that many teachers struggle with these but can master them once they’ve reviewed PEMDAS.

Second, do this operation with a pencil and then with a calculator:    3 x 4 – 8 ÷ 2.   Doing each operation in order produces the answer 2, whereas most calculators (which are programmed to use PEMDAS) produce the answer 8. Also, explore the difference parentheses can make:  (3 x 4) – (8 ÷ 2) and 3 x (4 – 8) ÷ 2 and think about why getting two different answers to the same problem clarifies for students the need to have rules for order of operations.

Finally, Jeon suggests trying to describe a real-life situation that could be represented by 5 + 8 x 6. Many teachers can solve the expression correctly, but can’t come up with a correct real-life scenario. An example of a misconceived answer: “Ms. Harper’s classroom has five boys and eight girls, and they each get a total of six cookies for participating in the fundraiser for Hurricane Katrina victims. How many cookies would be given out?” This is a good word problem for (5 + 8) x 6. 

Here’s a correct word problem for 7 – 3 + 11: “Hannah went to pick apples at an orchard. She picked 7 apples and gave three apples to her younger sister, Erin. Then their mom gave Hannah 11 apples. How many apples does Hannah have now?” 

These problems help clarify one of the misconceptions with PEMDAS: that addition must be done before subtraction. Jeon (citing Golembo, 2000) says a more helpful way of thinking of the mnemonic is:

P       Parenthesis

E       Exponents

MD   Multiplication or Division (whichever is first from left to right)

AS    Addition or Subtraction (whichever is first from left to right)

“The universally agreed-upon system helps students see that the order of multiplication and division is interchangeable, as is that for addition and subtraction, as long as they perform the operations from left to right,” says Jeon. 

“Reflecting on PEMDAS” by Kyungsoon Jeon in Teaching Children Mathematics, February 2012 (Vol. 18, #6, p. 370-377), http://www.nctm.org; the author can be reached at 

kjeon@eiu.edu.