**This is Day 9 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.**

*“Anytime somebody is absolutely certain about something, they are almost always absolutely wrong.”*

~ Jeff Lindsay

Absolute values are weird.

The popular way of handling them is, “Get rid of all the signs.”

And if you’re taking the absolute value of a real number, that works for the most part.

But if you’re working with variables, it might not.

### Working with Numbers

Suppose I ask you:

- Pick your favorite number.
- Subtract 100 from it.
- Take off the sign (so if it’s negative, make it positive).

You now have a positive number. But at step 2, you probably didn’t. (The top 10 favorite numbers are all pretty small.)

Using the world’s most popular favorite number (7), we would do this:

- Pick your favorite number: 7
- Subtract 100 from it: 7-100 = -93
- Take off the sign: 93

### Working with Letters

Now let’s look at an expression: |x-100|

The absolute value bars indicate to “take off the sign” – our popular definition.

Many people make |x-100| into x+100.

Compare this to our example above:

- Pick your favorite number. (We’ll call it x…)
- Subtract 100 from it. (…so that would be x-100…)
- Take off the sign. (…or do |x-100|.)

For our number 7, we got 93.

But if we claim that |x-100| is the same as x+100, then 93 is the same a 7+100.

### Everything You Know is Wrong

Okay, no it’s not. (But that’s a great Weird Al song!)

It all comes back to the squirrelly order of operations (you know – PEMDAS/crazy Aunt Sally). Absolute value bars are a method of isolation. Which means the stuff inside gets ranked as parenthesis.

So you have to “do the inside first,” as they say.

Which means you can’t just get rid of them that easily.

### The Bright Side

People believe that absolute value bars turn all negatives into positives.

This is a pretty good way to think, at least in life.

So the next time you’re feeling yucky, grab a couple of sticks or pool noodles and hold them to either side of you.

Now smile – you’re positive!

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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