You may know I’m not a fan of PEMDAS.
Today I discovered another reason we should ban it.
Parentheses means…
In addition to neglecting the opposite of exponents (roots), PEMDAS is ambiguous on “parenthesis.”
The P that stands for parenthesis also means brackets and braces.
“Everybody knows this,” you might think. And you’d be right.
But what they don’t know is that P also means anything isolated – like things under radicals, inside absolute value bars and embedded in numerators and denominators.
Parentheses are on a different level, too.
Furthermore, parenthesis (and all its brothers) aren’t actual operations. So they don’t even belong with a list of operations.
They are bossy snot-boxes that tell you what to do if things aren’t so obvious.
Parenthesis are the trump card in the game of arithmetic. (tweet this)
So students are clueless.
I gave a quiz in College Algebra. Students were asked to plug x = 2 into the equation y = |x – 3| .
Many of the students calculated this as y = 5.
Here’s how:
- y = |x – 3|
- y = |2 – 3|
- y = |2 + 3|
- y = |5|
- y = 5
What happened at step #3?
Yup – that was my question too.
And the response every time was, “Absolute value bars mean to change everything to positive, so the -3 turns into +3.”
If students understood isolation, this might not happen.
Furthermore, if absolute value was included in the order of operations, this might not happen.
And if we didn’t shove a confusing and incorrect mnemonic device like PEMDAS down their throats, this might not happen!
Do you have a reason to ban PEMDAS? Share in the comments!
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This may be as much a response to your previous PEMDAS post as to this one.
Firstly, you suggest that PEMA is a fundamental property of the operations. It’s not, at all– for example, there’s no property of multiplication or addition which forces you to do multiplication first. At first glance, distributivity seems to imply we do multiplication first, but it doesn’t: we could just as well write the distributive law as (x*y)+(x*z)=x*y+z, using “PASDME”. (Yes, parenthesis do need to come first)
At the end of the day, whatever order of operations we use is a linguistic convention, and like any other linguistic convention (e.g., grammar) there’s a roughly agreed-upon set of rules which help us to transform syntactic notions (e.g., the expression 3+4*7÷12) into something with semantic meaning. Getting rid of PEMDAS won’t help people understand this. Although, to be fair, the experiment you suggested in the previous post might: it (could be) an interesting lesson in notational convention and the difference between what we write and what we mean.
Now, as for all of the other operations you mention in this post–blaming this on PEMDAS is wrong. The misunderstanding of (e.g.) absolute value comes from a failure to understand that absolute value is a unary operator which has semantic, not syntactic meaning–it acts on the thing represented by the expression below it, not on the expression itself. Gaining an intuition for this (even without being able to put it in fancy words) is a huge leap in abstraction; abstraction, not rules, is the thing that makes math hard.
Moreover, we really do need a notational convention before we get to fancy things like radicals and absolute value (at least, in the path school math is currently taught), so whatever convention you come up with, you’ll have P and then the letters ADEMS and some directional convention (e.g., “from the left”), but operators which “group” will by nature be left out of this, since they show up at a later point in the student’s education. Perhaps, teaching kids to explore notational conventions will help them separate the symbols from their meaning… But the difference between syntax and semantics is something I see graduate math students stumble over from time to time, so I don’t think you’re going to see these problems go away at lower levels, especially with such a simple change as getting rid of PEMDAS.
Thanks for your thoughtful comments, Cory. I have speculated that you can indeed switch the operations up as you’d like – provided that we all agree. Alas, I’ve not dug into this enough to agree with it (nor to disagree).
Regardless, our tradition is to keep the multiplication over addition – and exponents over multiplication.
And for sure, the neglect of “roots” is a deal breaker for me.
My annoyance at PEMDAS comes from the memorization part. They KNOW that multiplication comes before division, right? That addition comes before subtraction?
2*3-4+5 is very often solved as 6 – 9 = -3 because the multiplication is obviously first but “addition comes next.”
Indeed, Curmudgeon, if we promoted an underlying understanding instead of asking them to memorize and do as they’re told!
I often tell my students that there are a lot of invisible things in math. A variable standing alone has an invisible
‘one’ as a coefficient. Numerators and denominators have invisible parentheses. All variables have an exponent, but sometimes it’s an invisible ‘one’. It helps, I think.
Funny idea: xkcd.com had one comic that gave a different mnemonic for order of operations: Please Email My Dad A Shark. hee
Chakolate – good point! Alas, because we ask them to memorize (as if they’re computers that can etch the algorithms in their carbon based hard-drives) we can’t expect them to think about the “invisible” parts.
In fact, if we treat them like algorithm following computers, we can’t ask them to think at all.
Hopefully some of the students are seeing that PEMDAS is a guideline, not a rule. But I see too much of the opposite. #sigh
I quit using PEMDAS for order of operations a few years ago, and my class scores on this skill on standardized test went up! Instead I use models and real-world situations to show students how expressions are simplified from most grouped to least grouped. The kids didn’t have to memorize and never made the mistake of doing every multiplication before every division, and every addition before every subtraction.
I even wrote about how I quit using PEMDAS and there were some respondants who were actually ANGRY that someone dare say there was better methods to teaching order of operations than PEMDAS.
Nice post!