Problems With PEMDAS Or Please Excuse My Dear Aunt Sally!

By the time my students get to high school, they've memorized the order of operations but still do not understand that it can be used fluidly.  Students are taught the pneumonic PEMDAS or, Please Excuse My Dear Aunt Sally or something similar as a way to help them remember the order of operations but too many of the students have a disconnect between the pneumonic and performing the order of operations.

I've found there are situations when I can ignore the rules and others where I have to adhere to them in order to get the right answer. For instance, if the equation uses the distributive property, it is sometimes better to divide rather than distribute because it makes it easier and more likely to get the correct answer because students often forget the second term inside the parenthesis.

On the other hand, students love multiplying the coefficient and the value of x before squaring it in a problem like 2(3)^2.  I admit that I remind them that the 3^2 has to be carried out first before they can multiply the answer by two to get 18 but they could easily write the problem as 2*3*3 and get the correct answer.  

In addition, the P in PEMDAS can limit students because it stands for parentheses so when they run into any other type of grouping symbol, they don't know what to do because they aren't the proper shape.  Since multiplication and division are right next to each other, students often perform all the multiplication first and then all the division rather than doing them as you come across them in the equation.  The same applies to addition and subtraction.

One solution I've seen suggested is the GEMA one where G stands for grouping symbols because it includes square brackets, parentheses, and even implied grouping symbols such as division in fractions, or even absolute value.  E stands for exponential operations, M for multiplicative operations, and A for additive operations.  

The A for additive operations is really more appropriate since subtraction can be written as adding a negative number but unfortunately, many students do not understand that when they enter high school.  I've talked about subtracting the number or adding a negative and my students ask me what I'm talking about.  They don't see the two as being interchangeable.  The same applies to the idea of M being used for multiplicative operations.  They don't see that dividing by 12 is the same as multiplying by 1/12.

I think this is one reason students have difficulty applying the order of operations to any problem.  They don't fully understand the relationships between multiplication and division and addition and subtraction.  This might be because in elementary school, they are taught that multiplication is repeated addition and division is repeated subtraction.  Then once they get to signed numbers they see subtraction as subtraction and not as adding a negative number.

Perhaps when we teach anything in high school math, we should take time to talk about how we use order of operations to solve problems as a way of reinforcing it's use.  Let me know what you think, I'd love to hear.  Have a great day.