One short math problem set the internet ablaze this weekend—and incited charged arguments on math teacher Twitter, too.
It was this equation:
8 ÷ 2(2+2) = ?
oomfies solve this pic.twitter.com/0RO5zTJjKk
— ##ia (@pjmdolI) July 28, 2019
Everyone agreed on the first step: Solve inside the parentheses, for 2+2=4. But after that, people split down two paths. Some multiplied first, while others divided, leading to different answers—1 and 16.
The seemingly simple problem is designed to trip up the solver over order of operations: the sequence in which you complete the different components of an equation.
In the U.S., most students learn order of operations by memorizing the mnemonic “Please Excuse My Dear Aunt Sally"—or PEMDAS, for parentheses, exponents, multiplication, division, addition, subtraction. As many educators have pointed out, the problem with that device is that it seems to spell out that multiplication (“M”) comes before division (“D”). But the order of multiplication and division is arbitrary; the two operations hold the same place in this sequence. So teachers usually offer an addendum: When multiplication and division appear next to each other, they should be solved from left to right.
Following this logic, there is only one solution: 8÷2=4, 4 x 4=16. And some math teachers came down hard on this side, arguing that 16 was the only correct answer.
Getting tired of this problem 😴. Ans = 16. Acronyms like PEMDAS, BOMDAS, FOIL, foster lots of confusion & HAVE NO PLACE in the #math classroom. And this simple problem (now gone viral) is clear evidence of this. #MTBoS #ITeachMath #math #ElemMathChat #MSMath #HSMath #algebra pic.twitter.com/HRCLllhGGu
— Tim Brzezinski (@dynamic_math) August 3, 2019
So, case closed?
Well, not quite. Math is a language, just like English. Multiplication signs, division signs, and other symbols are conventions. They provide a grammar and syntax—a way of parsing a math sentence so that it makes sense. And as with English sentences, math sentences are sometimes written in ways that aren’t clear.
“This is like a maths version of the sentence, ‘He fed her cat food,’” Hannah Fry, an associate professor in the mathematics of cities at University College London, told the Daily Mail in an article about the viral equation. “Does it mean the man gave some food to a cat? Or—slightly darker—fed some cat food to a woman. It’s impossible to tell from the information we’ve been given.”
“No professional mathematician would ever write something so obviously ambiguous,” Steven Strogatz, a professor of mathematics at Cornell, wrote in the New York Times. “We would insert parentheses to indicate our meaning and to signal whether the division should be carried out first, or the multiplication.”
And although most Americans have been taught that PEMDAS is the final word on order of operations, there is ongoing debate in the mathematical community about how certain conventions should be interpreted. This 2013 article in Slate (written after another, similarly unsolvable equation took the internet by storm) outlines some of the open questions.
Are there lessons to be learned for the classroom? On Twitter, many educators pointed to the importance of context. When writing problems, math teachers should be clear in what they’re asking of their students, teachers said. And there’s value in teaching students how to be clear communicators of math, too.
Why keep flogging this dead horse. The truth is proper syntax matters. It is too bad that the calculators whose images started this whole mess assume they understand ambiguous expressions.
— “Dr. Chris” Brownell (@cbrownLmath) August 5, 2019
Surely the point is that we shouldn’t write calculations in ways that can be seen as ambiguous?
— Steve Collins (@SteveC_Maths) August 5, 2019
Lack of context aside, can’t this stem a conversation about clarity in communicating math? I stress so often to my students that anyone should be able to read their writing about math (in words or symbols) and understand what they mean. This clearly doesn’t achieve that goal.
— Shannon Ramos (@SRamos82) August 6, 2019
But others argued that the focus on parsing symbols and syntax is misplaced. Dan Meyer, the chief academic officer at Desmos, a math education company that created a free, online graphic calculator, wrote on Twitter that math teachers “should be horrified to see this kind of ‘math’ trending.”
“There is an existing sense that success in math class means that a person should calculate quickly and accurately with formulas,” Meyer said, in an interview with Education Week. “The particular problem that went viral, and the discussion around it, reinforced that particular definition of math. It would be as though a teacher of literature saw an issue about comma splices go viral.”
Equations like this one, that test knowledge of memorized scripts, imply “that the point of mathematics [is] to trip up other people with stupid rules,” Amie Wilkinson, a mathematician at the University of Chicago, told the New York Times.
Meyer would want to see more discussion around problems that require flexible thinking. He offered the “four fours” problem as an example: Students have four number fours, and can use whatever operations necessary to get to another number, such as 10.
“That prioritizes creativity, whereas [the viral problem] prioritizes a pedantic obsession with mathematical syntax,” said Meyer.
Still, some were glad that the problem got so many people talking about math at all.
“I think it’s a door opener to interesting conversations with people who might not otherwise be interested,” tweeted Laurie Rubel, a mathematics education professor at the City University of New York.
A version of this news article first appeared in the Teaching Now blog.