Order of Operations

A guest post by Rachel Chou

I recently saw this in my Facebook feed:

I had many questions:

1. Why is this posted in a facebook group called “Grandma’s recipes?!”
2. 462,000 people cared enough to leave a comment?
3. 6,900 people cared enough to re-share it?

Most importantly: I hope no one is posing this question to schoolchildren!

I decided to google the phrase “facebook order of operations meme”, and I was shocked to find the following: a Slate article, a Reddit thread, a Newsweek article, and even a New York Times article. It appears that annoying order of operation problems have riled up so many of our citizens, that this conflict appears to count as news.  

I taught middle school and high school math for 23 years. In this post, I share my thoughts on this meme.

Here are some ways in which I have seen operations trouble in a high school setting. Students struggle to discern the difference between:

1. 3x² and (3x)²
2. log(a + b) and log a + log b

For item (1), I do think it’s important that students be explicitly taught the convention that exponentiation should be executed prior to multiplication. What is communicated in the first expression is that the number 3 is being multiplied by x², whereas is in the second expression 3x is being squared. This is an important convention that must be shared. Likewise, it is important to emphasize the difference between –x² and (-x)².

Item (2) can be put in the category of order of operations only because it is a basic example of “Parentheses first!” But when kids mess this up, it’s not typically because their order-of-operations instruction didn’t sink in, but rather because they are overgeneralizing what they learned about multiplication distributing over addition to all operations distributing over addition. (I discussed this here.) This doesn’t turn out to be a symptom of students not properly applying the order of operations.

Item (3) is interesting because this isn’t a typical task given to students, and yet, it is something students are apt to get confused about. At the beginning of a school year, I often pose this question to 9th graders:

True/False:

Invariably, many students write “True.” My suspicion is that this is also not a symptom of not knowing the order of operations. I think most students do understand that the left side is communicating 2/3 divided by 5 while the right side is communicating 2 divided by 3/5. The mistake instead is more related to the fact that students are not often taught to think about what an operation means prior to executing it. In this case, instead of slowing down to think meaningfully, they react quickly. The two things look the same! 

Were a student to slow down and think about the meaning of the left-hand expression (for example splitting 2/3 of a pizza among 5 people), they would quickly realize that the answer is one fifth of 2/3, a small number, much less than 1! For many kids, it would be harder to contextualize what 2/(3/5) could possibly mean. Were a student to be trained to slow down, she might think about her art teacher having 2 meters of a pretty ribbon, and each student needing 3/5 m of ribbon to finish a particular art project. She would realize that the result of 2/(3/5) tells us the number of students for whom the art teacher can provide the right amount of ribbon. Well, clearly, the art teacher can provide ribbon for more than 1 student!

They would conclude that the left side of (2/3)/5 = 2/(3/5)  is much less than 1, and the right side of the equation is a bit more than 3 (which a student might reason upon realizing that 2=10/5.) They are obviously not equal! The equation is false.

This is difficult, but only because we spend a lot of time in school teaching students how to execute computations, and not enough time discussing why we might want to execute this computation. It would be much more productive to ask middle schoolers to slow down and consider what operations mean, rather than have them slog through endless and overly complicated drills on this topic.

Back to:

A debate ensued online as to whether or not the answer is 1 or 16, but I don’t see this as anyone’s failure to remember any rules that they studied in grade school. The failure that I see is in the problem writer. Three Reddit comments are apropos of my point:

 “It’s amazing how so much can be written on a math situation that never comes up after 3rd grade…” (I would correct this reader to possibly 7th or 8th grade!)

“This is why the division sign isn’t used after instruction in arithmetic. Fractions are much easier to understand.”

“This is made to confuse people, any sane person who wanted to express a computation like this wouldn’t write it in this manner.”

Exactly. No one ever communicates this way after elementary school. Many would agree that one purpose of English courses in middle school and high school is to teach students to communicate clearly and coherently, but this goal should not be limited to the English classroom! Math educators everywhere should want their students to communicate clearly and thoughtfully in a math context as well, and the creator of this meme completely failed. As I see it, they meant one of two things:

but the fact that I have no idea doesn’t mean I missed something in elementary school, it means the meme’s author didn’t communicate well. The traditional division symbol is rarely used, in favor of fractions, and there is a reason: the fraction bar makes the intended grouping of the terms explicit. 

I recently saw this on a high school homework assignment: 

                  Simplify:

What goal did the educator who authored this task have in mind? Were students in this classroom being encouraged to slow down and consider a context in which such a computation serves a purpose? Were the students being asked to think and reason? Or were the students being asked to blindly slog through a task, written in a way that confuses so many educated adults, that the New York Times decided to write on the topic?

When tasks are assigned to students, the purpose should be to challenge them to learn something new, entice them with a particularly elegant result, or provide practice with important skills and routines. In looking at this task, I see nothing useful, and instead I see a task that teaches some kids that math is a collection of arbitrary rules, that math is boring and rote practice, and worst of all, that math isn’t for them.