Xavier Questions: PEMDAS (or is it PE(MD)(AS)?)
Updated: Dec 17, 2021
A middle school math teacher sees an equation his fifth grade daughter scribbled on a piece of paper: 82-5x2+337x3=1,473. Knowing an order of operations issue when he sees one, he quickly reworks the problem using Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS): 82-10+1,011=
When his daughter sees it; she tells his father he's wrong and the answer is 1,473. After some back and forth she bets him a PlayStation 5 she can prove she is right. Completely confident in himself, the dad makes the bet. She pulls up a Facebook post and five minutes later her father is buying a scalped PS5 off eBay for $800.
How can this be? PEMDAS is how math is done on Planet Earth! Right? It is objectively true. Right?
I recently fell down this rabbit hole on Facebook which started a quirky quest worthy, I believe, of inclusion in The New Yorker or The Atlantic. I wanted to find the mathematical proof of Please Excuse My Dear Aunt Sally; a mnemonic device used to teach elementary students a convention for the order of operations. I'll be honest; I can't remember being taught an order of operations growing up so I found myself simply solving these arithmetic problems from left to right. It's natural since that's the way I read: but so many people enjoy passionately telling me how wrong and uneducated people like me are.
Requests for mathematical proofs of this went unanswered so I went to the Googles. Here I discovered there is no mathematical proof that PEMDAS is THE way to do math. It's a convention rather than a universally accepted law of mathematics; a general agreement on how to solve equations in the face of ambiguity.
Where did this ambiguity come from?
Yes if we have two of something and add two more we now have four of whatever that something is. However, as math becomes more complex than simple arithmetic and gets into the realm of algebra the author's intent becomes important. And this is where printing and publishing comes into play and is at the root of the origins of PEMDAS.
Without being rooted to something real, math is just a game. Beyond simple 1+1=2 or 2x2=4, the rules of the game are not as universal as we've been taught they are. Instead, the rules are established by the author of the equation. As publishing allowed mathematicians to publish books for other math nerds to solve problems, each author would spend the first few pages explaining his conventions for solving the problems contained within.
As universal education became the norm, this led to the rise in need for textbooks. This opened the door for non-math related influences to effect the order of operations. Having a standard convention reduced printing costs by eliminating the need for the author to spell out how to do the equations in his text. It also assisted teachers who had to instruct a variety of subjects by having a standard to go by. And so, in the US, sometime between the late 1800s and 1920s PEMDAS became a thing.
Houston, we have a problem...
It turns out that PEMDAS does not completely reduce ambiguity. If given a rule to follow, young students will not deviate from it. Which produces a problem since mathematically speaking multiplication and division have the same associative properties. Same goes for addition and subtraction. Which means after solving for parenthesis and exponents, the next step should be solving multiplication and division problems as they appear from left to right. For example:
M/D: 9-2+1 (4/4 then 1x9)
A/S: 8 (9-2 then 7+1)
The answer is 8 as PEMDAS is understood by those educators writing standardized tests.
However, students will often work the problem:
Which will result in a wrong answer as scored by a machine. This produces a faulty impression that our schools are failing to properly teach math. Which is why there are some teachers who argue that PEMDAS shouldn't be taught given how it leads to an incomplete understanding of the order of operations. Making their case stronger is the fact that high level, theoretical mathematicians do not bother themselves with order of operations conventions preferring to solve equations.
So how did the girl win her bet?
She won because of two things:
1) When math is disconnected from real life problems it's only a game with the correct outcome subjectively determined by the equation's author's intentions and/or choice of conventions.
2) Real life problems that can be solved by math do not always lend themselves to orders of operations.
In the above example, the daughter was trying to figure out a real life problem: how much a fundraiser for her scout troop raised. She had raised $82, but lost $5. Her father had promised to double what she raised. The rest of her scout troop raised $337. Another dad had promised to triple the amount raised. She was trying to figure out how much money was raised and jotted it down as she worked through the problem in her head: 82-5x2+337x3=*. Which she then solved from left to right to get at $1,473. (Which is how she would input it into a single step calculator. EDIT: You can try this at home, Microsoft's Calculator (v10.2103.8.0) in standard mode returns the result 1,473. So does this online calculator at iCalculator.)
Since the equation was tied to a problem with an objective outcome, the "unconventional" math was correct. The answer cannot be 1,083 because that's not the answer to the question the equation is being used to solve. It may fly in the face of convention; but it worked.
WRITING PROMPT: A group of people are trapped in a Saw-esque escape room and the pin to the door is the solution to some problem. However, the author of the equation is one of the "uneducated" who solves problems left to right.
In the end, this problem leaves people unsettled because it reveals that math can be ambiguous and at times even subjective. In a Slate article titled "What Is the Answer to That Stupid Math Problem on Facebook?" Tara Haelle observes: Of course, the fervor with which some people debate basic arithmetic may be a proxy: There’s less at stake in a math debate than a potentially friendship-ending political debate. Arguing over multiplication may even be a way to make a subtle political point, using others’ “wrong” answers to reinforce a broader worldview, such as that the United States has poor math education.
And that is what, I believe, is at core of these "gotcha" math problems. It serves as a proxy to allow us to feel superior over others. It's a socially acceptable way to "punch down" at people we think are dumb or even evil (if we're assuming they belong to that other political death cult). I know as I queried social media for the mathematical proof or underlying logic to PEMDAS I received some ridicule. Despite posting links to actual high-level mathematical websites (including one from Harvard), I was met with comments (from strangers) regarding my foolishness or even "belief" in science.
Which is why when I think about my writing prompt I see three people each with their own solution to the puzzle. Two of them college graduates, perhaps even engineers, coming up with two different answers to an ambiguous equation. The only thing that can bring them together as they fight over why their solutions aren't releasing the lock is their derisive mockery of the high school drop-out who insists the pin is solving the equation left to right.
Then as the timer is about to expire and the room pumped full of toxic gas he decides: "Screw it, I'm gonna try it." And with a second to spare, the timer stops. No gas is released. And the door opens.
* I completely understand that the best way for her to have represented the equation would be: 3(2(82-5)+337).