Math is hard when you don't remember it

"5+5+5-5+5-5+5x0=? (I bet more people will get this wrong:P) Give a try! 40, 0, 20, 15" -poll on facebook

I saw this poll pop up in my Facebook feed the other day. As predicted by its author, the vast majority of people did, in fact, incorrectly answer "0", including several of my friends. Enough so that I thought it might be useful to write a little bit about why, exactly, this is. But first, a little anecdote of my own.

You see, math actually IS hard. Maybe it's no harder than any other discipline, but then, most things ... are hard! Many of you who know me won't be surprised to know that math comes fairly naturally to me. While my primary school classmates struggled and studied to learn the material, I simply answered the questions and looked around for something else to do (this had plenty of negative repercussions for me later in life, but that's a topic for another time). The clean, simply beauty of math was a perfect match for my brain. Almost everyone I know feels this way about something, be it music, art, communicating with others, or captioning cute pictures of cats to put on the internet. Nature, nurture, both, neither... again, topics that can be discussed more later -- point is, I was good at math, and I knew it.

Fast forward 20-ish years, to me, starting work on my Master's degree, after a 10-year lapse of being out of a scholastic environment. Within weeks of beginning classes, I was forced to beg for help from an old friend... for math. Years of disuse had left me so rusty that even I needed help with some fairly basic differentiation and integration that I would've had no problem with a decade earlier. The problem wasn't one of logic. The problem was the rules. Along with all that stuff that just came naturally were lists and lists of rules: differentiation rules, integration rules...some more intuitive than others. With a little help from my friend, and a little help from the world wide web, I relearned the rules I needed, and solved the math problems.

Which brings me back to the Facebook problem. Most of us learned PEMDAS (or a mnemonic like "Please excuse my dear Aunt Sally") in school. PEMDAS means:
Parentheses (or brackets, or braces)
Exponents
Multiplication, Division
Addition, Subtraction

This list tells you the order of operations-what order to interpret the terms of the equation in. It is read from the top down, and items on the same line have the same level of priority (in other words, it will never matter if you add first or subtract first - the result will be the same either way). The equation in the Facebook problem has no parentheses or exponents, so we can ignore those. There is multiplication, however. PEMDAS tells us that we must do the multiplication before any addition or subtraction, which takes us from 5+5+5-5+5-5+5+5x0 to 5+5+5-5+5+5-5+0. We've computed the result of 5x0, which we know is 0, and replaced it in the equation. Now we have only addition and subtraction left, which we can perform in any order we like.
5+5+5-5+5+5-5+0
5+5+5-5+5+5-5
10+5-5+5+5-5
15-5+5+5-5
10+5+5-5
15+5-5
20-5
15

PEMDAS is actually a parsing rule or guide: it tells the interpreter (you) how to turn a set of numbers and symbols ("5+5+5-5+5+5-5+5x0") into concepts ("what is the result of adding five of a thing to five more of a thing?"). PEMDAS is merely a human convention, it has nothing to do with universal laws of numbers, because those universal laws are based purely on concepts, not the textual symbols that the equation is notated in. But as it is a convention, agreed upon by the international math community, it must be obeyed -- any other interpretation is, by definition, incorrect. Ignoring this convention is no different, conceptually, from ignoring the convention of denoting male and female restrooms with their well-known pictographs. You can claim ignorance of the convention when you're arrested in the opposite gender's bathroom, but it won't help your court case (at least, not as much as you might hope).

Without PEMDAS, math would actually be much more challenging to read and write. Every equation would require level-upon-level of nested brackets to isolate terms and ensure that it was interpreted properly. Trust me, you don't want that.