If you know anyone over 40 on the internet, you have probably come across some image with literally hundreds of thousands of likes/comments/tweets/whatever:
What is 2 x 2 + 3 x 4 ? 99% of people won’t get this!
There are only 5 actual comments, made tends of thousands of times over:
40
28
16
Some completely erroneous number.
It’s 16! PEMDAS/BODMAS order of operations! People are soooo stupid!
[Rant irrelevant to the crux of this article begins here. Scroll down if you’re well-adjusted]
It’s one of those reminders about how the inevitability of death, which I personally find completely unfair, is probably of some benefit to society. “Nobody cares about what you had for dinner!” shouts the incredulous 55 year old woman as she types “16” into the box along with a hundred thousand other people. Nobody bothers actually explaining one answer verses the others beyond “because my math teacher said so”. It’s some sort of oblivious democracy where everyone feels the need to give their 2 cents with zero regard for the fact that people have been saying quite explicitly the absolute identical thing thousands and thousands of times over. My infuration over human nature aside, the obvious question is, who’s right?
40, 28, and 16 are all right. There is an instinct in any number of you to shout at me, “No, only 16 is right! Order of operations!” I know for a fact you got better grades than I did in school. But your reaction is a testament to the failure of our education system. It’s not you’re fault; you were raised to think like that. Raised by people who think that getting a right answer is more important than understanding a concept. Raised by people who think getting A’s means you’re smart, not that understanding things means you’re smart. I won’t even concede that smart people have a better chance of getting A’s. The system’s way too far gone for that. They do a little, but at this point the association is mostly meaningless.
Order of operations is a practice in mathematics that is crammed down the throats of children sometime around middle school. In my experience teaching math, it takes a pretty substantial amount of time for most kids that age to really internalize why order of operations is taught, if they ever actually do. For most kids, the lesson cannot be taught in a couple weeks. It can be shoved down throats in a couple weeks, sure, but not taught effectively. Most kids, when conditioned effectively, can memorize a ton of rules and regurgitate them for you on a test. The value we place on testing is inversely proportional to our insecurity with our ability to educate children. At the present time, that means that schools are primarily focused on tests, and not learning–and why fixes like Common Core are completely incapable of fixing this problem–so we tend to demand memorization more than understanding. But I digress again.
[Actual content begins here]
Order of operations is based on the fact that mathematical expressions can be interpreted ambiguously. 2 x 2 + 3 x 4 is an ambiguous statement. If we add 2 and 3 before multiplying, we get 40; if we multiple 2 and 2 and then add 3 before multiplying by 4, we get 28; if we multiply both numbers first and then add, we get 16. Mathematically, all of these options are correct. Go ahead and double-check–I do make mistakes from time to time. Because you can get three different answers for this same equation, it helps to clear up this ambiguity so that we aren’t left to speculate as to which answer we need to choose from. In this case, parentheses are used as the notation to clarify. (2 x 2) + (3 x 4) is unequivocally 16. 2 x (2 + 3) x 4 is unequivocally 40. In the absence of explicit un-ambiguity, a there arose a convention referred to by the mnemonic in the States as PEMDAS, and BODMAS in the British world. These stand for Parentheses (or Brackets), Exponents (or Orders), Multiplication & Division, and Addition and Subtraction, the order in which one solves an equation in the absence of specific, indicative notations.
To reiterate, PEMDAS is a convention. It’s something you can appeal to in the absence of direct orders. It also does not exist in the real world beyond middle school math homework, making it completely pointless. That is not to say teaching ambiguity in mathematical statements is useless. Far from it! That is a very necessary lesson in mathematics and logic. Understanding that typing “2 x 2 + 3 x 4” will get you 28 when you do it one at a time on a simple calculator but “16” on a more advanced machine is eminently practical, to say nothing of the logic-building and mathematical-understanding values in the subject. But PEMDAS doesn’t do this. It’s a rule that is expected to be followed on a test that has no actual existence in math itself. Let’s make a word problem, where we take an example of mathematics and apply it to the constraints of reality:
A grocer is selling oranges out of boxes. One box fits 2 oranges in each row, and it has 2 rows; the other box fits 4 oranges in a row and has 3 rows. How many oranges does the grocer have for sale? Show how you solved this problem.
The grocer has 16 oranges for sale. 2 oranges x 2 oranges + 3 oranges x 4 oranges -> 2 x 2 + 3 x 4 = 16.
Another problem: A grocer is selling oranges out of boxes. The grocer also has some extra oranges, and decided to bundle them in with boxed oranges. One box fits 2 oranges in each row, and it has 2 rows. The grocer adds 3 extra oranges to each box. He has 4 of these bundled boxes all together. How many oranges does the grocer have for sale? Show how you solved this problem.
The grocer has 28 oranges for sale. 2 oranges x 2 oranges + 3 oranges x 4 -> 2 x 2 + 3 x 4 = 28
Another problem: A grocer is selling oranges out of boxes. He has boxes that are holding 2 bags of oranges, each of which contains 2 oranges. He opts for putting another 3 oranges in each bag. He puts out 4 boxes. How many oranges does the grocer have for sale? Show how you solved this problem.
The grocer has 40 oranges for sale. 2 x 2 oranges + 3 oranges x 4 -> 2 x 2 + 3 x 4 = 40
Awkwardness of phrasing aside, the same equation can be used to explain three distinct scenarios. You can yell “order of operations” all you want, but the actual number of oranges is rooted in reality that can be at odds with the convention of order of operations.
What ultimately matters is the reality. If your math contradicts reality, then you’re doing the math wrong.
The “answer” is that the expression is ambiguous, and can equal a number of answers depending upon how it is interpreted. To narrow this down to a single solution, ambiguity must be eliminated in the statement.” But that’s not what’s going to be in the back of the book or on the test, so you’ll only see it show up on those posts once in a blue moon.