Student misconceptions about PEMDAS

Simplify 6/2*(1+2).

A Common Incorrect Answer. According to PEMDAS, we should handle the parentheses first. So 6/2*(1+2) = 6/2*3. Next, there are no exponents, so we should proceed to multiplication. So 6/2*3 = 6/(2*3) = 6/6. Finally, we move to division, and we obtain the answer 6/6 =1.

The above answer is incorrect and (even worse) arises from a natural but unfortunate misconception of the way that children are commonly taught order of operations. If you don’t see the misconception, please give it some thought before continuing.

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The mnemonic PEMDAS, commonly taught in the United States, stands for

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

I personally never learned this memorization trick when I was in school. What I do remember, from learning BASIC computer programming around 1980, was the mnemonic My Dear Aunt Sally. I’m told that in the United Kingdom (and perhaps elsewhere in the English-speaking world) schoolchildren are taught BIMDAS, where B stands for Brackets and I stands for Indices.

Unfortunately, all of these memorization devices suffer from a common flaw: they do not indicate that multiplication and divison have equal precedence, and that addition and subtraction have equal precedence. In other words, the order of operations really are

Parentheses

Exponents

Multiplication and Divison (left to right)

Addition and Subtraction (left to right)

Therefore, the correct answer to the above problem is

6/2*(1+2) = 6/2*3 = (6/2)*3 = 3*3 = 9.

In brief, though not intended by teachers, PEMDAS and BIMDAS perhaps promote the misconception that multiplication takes precedence over division and addition takes precedence over subtraction. To avoid this misconception, one of my colleagues suggests that PEMDAS be taught more visually as

P
E
MD
AS

so that students will have a better chance of remembering that MD and AS should have equal precedence.

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