Math time with Roman statue profile pictures

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I recently saw the following photo on the internet.

It was posted by Twitter user @lporiginalg, an account with a Roman statue of Socrates profile picture that mostly posts conservative takes and makes fun of liberals. The idea is that the teacher is showing young kids two different ways to parse the pictured expression, and lporiginalg and his followers want to point and laugh because there is only one right answer.

Every once in a while a picture like this is posted and righteous internet users cry that America’s education system is falling apart. They often insinuate that “woke liberals” are making our kids dumber because they declare all answers equally right, even the wrong ones. Normally I roll my eyes at this dance, but this one was particularly annoying.

The way we write math is completely arbitrary. Notation and order of operations is not something that we can “get wrong” the way people imply here. Conventions exist to make our lives easier. For example, I type a lot of my math and end up writing things like $1 / 2(x + 1)$ to mean $\frac{1}{2(x + 1)}$. This is faster to type and saves some keystrokes by avoiding parentheses. Some of my colleagues might think I mean $\frac{x + 1}{2}$, but we would pretty quickly understand what the other meant and adjust accordingly. Neither way is “wrong,” it’s just an ambiguous expression without a convention telling you what it means.

We teach students PEMDAS because because it’s a useful, common convention. It isn’t “The Law of Math, as Ordained by Our Father in Heaven, Hallowed be His Name.” As we see in the picture, PEMDAS doesn’t even resolve every ambiguity unless you also mention working left-to-right.

The important lesson about order of operations is that a student understands they need an order of operations when parsing an expression. Then they should understand how PEMDAS+left-to-right parsing is normally used and how to spot ambiguities, but the conceptual understanding is more important than any algorithm we give them. To the extent that this picture illustrates that, I think it reflects well on the lesson. To the extent that people use this picture to heckle a teacher, I think the whole affair reflects poorly on the hecklers.