Now realize, following Aunt Sally is purely a matter of convention. In that sense, PEMDAS is arbitrary. Furthermore, in my experience as a mathematician, expressions like 8÷2×4 look absurdly contrived. No professional mathematician would ever write something so obviously ambiguous. We would insert parentheses to indicate our meaning and to signal whether the division should be carried out first, or the multiplication.
The last time this came up on Twitter, I reacted with indignation: It seemed ridiculous that we spend so much time in our high-school curriculum on such sophistry. But now, having been enlightened by some of my computer-oriented friends on Twitter, I’ve come to appreciate that conventions are important, and lives can depend on them. We know this whenever we take to the highway. If everyone else is driving on the right side of the road (as in the U.S.), you would be wise to follow suit. The same goes if everyone else is driving on the left, as in the United Kingdom. It doesn’t matter which convention is adopted, as long as everyone follows it.
Likewise, it’s essential that everyone writing software for computers, spreadsheets and calculators knows the rules for the order of operations and follows them. For the rest of us, the intricacies of PEMDAS are less important than the larger lesson that conventions have their place. They are the double-yellow line down the center of the road — an unending equals sign — and a joint agreement to understand one another, work together, and avoid colliding head-on. Ultimately, 8 ÷ 2(2+2) is less a statement than a brickbat; it’s like writing the phrase “Eats shoots and leaves” and concluding that language is capricious. Well, yes, in the absence of punctuation, it is; that’s why we invented the stuff.
So on behalf of all math teachers, please excuse us for drilling your younger selves on this tedium. My daughters spent weeks on it each school year for several years of their education, as if training to become automatons. No wonder so many students come to see math as an inhuman, meaningless collection of arbitrary rules and procedures. Clearly, if this latest bout of confusion on the internet is any indication, many students are failing to absorb the deeper, essential lesson. Perhaps it’s time to stop excusing dear Aunt Sally and instead embrace her.
Better still would be to teach everyone how to write unambiguous math expressions, and then all of this would go away. For those students destined to become software designers, writing code that can handle ambiguous expressions reliably whenever they arise, by all means, exhume Aunt Sally from her crypt. For everyone else, let’s spend more time teaching our students the more beautiful, interesting and uplifting parts of mathematics. Our marvelous subject deserves better.