The order of operations includes two types of rules: those that are based on the way the operations work, and those that are arbitrary. My friend @harrisonalg from the Twitter chat #mathchat and I have been discussing this.

You can explain the truly arbitrary elements of PEMDAS (the left to right of AS and MD) through an experiment. Allow students, independently, to do these two problems any way they want, ignoring any stupid arbitrary rule they might have previously memorized:

- 3 – 2 + 8 – 3 + 4
- 2 x 7 ÷ 2 x 6 ÷ 3

The idea is that they will come up with many different answers:

- 3 – 2 + 8 – 3 + 4 could be any of -14, 0, 10 or others
- 2 x 7 ÷ 2 x 6 ÷ 3 might be 7/18, 7/2, 14, etc.

If they were on a team building a bridge with these calculations, things wouldn’t work so well.

### Enter the Stupid Arbitrary Rule (SAR).

Because we need to all come up with the same answer, we need a rule to follow. Really, it can be any stupid arbitrary rule (SAR). But we agreed, at some point in history, to all follow the “left to right” thing once we were down to addition & subtraction or multiplication & division.

It’s important to note that kids didn’t get to be part of that agreement ** we** made. Just like they don’t get to vote in elections.

Is it fair? Probably not. They would probably do a better job of choosing leaders as well as determining the order of operations. But that’s the way things likes SARs work.

### PEMDAS doesn’t cover it all!

To throw a few more kinks into it, check out how wikipedia explains some special cases of the order of operations. Factorials aren’t covered in PEMDAS, neither are the order of exponents.

Here’s another fun one: even calculators do things differently (and arbitrarily). Check out the way the the TI-92 and the TI-30XII handles exponents. Clearly you get to make up your own order of exponents!

### Let them play king: they create the rules!

Let your students determine the order of operations for addition & subtraction and multiplication & division – just for the class. Make sure to write the rule down and tack it to the wall so everyone remembers what they decided. In your class, ** and your class only**, they are to follow that rule.

Remember the back of the book, as well as your answer keys or teacher edition textbook, will now be totally wrong. But it’s worth it.

Doing it this way they might start to understand which pieces of PEMDAS are dependent upon the way the operations are constructed (PEMA) and which pieces of PEMDAS are arbitrary (left to right).

Once the class determines their special arbitrary rule, practice creating expressions whose result is some important number. Some options could be:

- The age of one of the kids
- Someone’s favorite number
- Your age (be careful)
- The age of some other grown-up they know
- The last four digits of their phone number
- The number part of their street address

Create the expression using the Class Rule as well as the stupid arbitrary rule. Then translate and have a good laugh. I just did my age with my “Bon Rule.”

Bon Rule: addition and subtraction go from left to right, and multiplication comes before division.

My age is 120 ÷ 2 x 3.

If I did this according to the SAR, I’d be dead. Or really famous.

### Your turn!

Can you teach the order of operations like this with your kids? If you classroom school, how much trouble are you going to get in when you become the math teaching rebel?

Share your experiences in the comments!

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This is ridiculous. Sure, let’s teach children to do it in away that they get a completely different answer from what they will need to get when out in the world. That way, they will be completely confused on what the “agreed upon” way is. I’m not buying your argument.

Fair enough, JG.

And that’s a normal response from people who live in a world where we start crushing novel thinking and curiosity out of students in the first grade.

I propose that we bring back novel thinking – maybe not in the 7th grade. But we must stop teaching 1st graders that there is a RIGHT answer rather than various ways to get AN answer.

I’m with JR on this one. I think the risk that students will get confused and not remember which is the

agreedorder of operations is too high.On the other hand, having a discussion about the fact that the rule is really arbitrary is quite healthy, I think. That is the sort of thing I wish

myteachers had taught, rather than making feel like I must be stupid if I didn’t understand why the rules are the way they are.That “making me feel like I must be stupid” tactic is exactly how grownups work to eliminate the novel thinking in young kids.

So damaging.

Thanks for stopping by, Peter.

As a freshmen math teacher, I have students who have been exposed to PEMDAS before but always seem to forget the rules. So to begin the lesson I have pairs of students work on different problems, stating each step. Once completed they share with the class their steps and thought process. Other groups cannot comment until all groups have presented. Then we discuss why would anyone put parenthesis first (because it is grouped and a good starting place). We discuss the rules as created thousands of years ago so that we all can compute the same answers no matter where we live. Students then practice the correct way and once understood create their own rules, create a problem and have a friend solve it. No one is confused and they realize we all need to agree on one set of rules to function world wide:)

Brilliant!

As a high school teacher you’re having to manage years of “math baggage” of incoming students. If it was negative, or some topics were not firmly learned in a positive way, you’re faced with a huge challenge.

It looks like you’ve figured how to make it work. That’s so awesome to read – thank you for sharing this method!

I struggled for years in math because not once did any of my teachers understand my frustration at how “made up” these kinds of maths were. They’d repeat the lines from the book, give us the tests, and wait for summer. Having really shitty teachers does a disservice to mathematics.

Now, as a fully grown man with a doctorate in mathematics, I can say with pride that my confusion wasn’t because I was too stupid to understand it (in fact what was responsible for my problem learning it was a keen and rare philosophical insight which I would later go on to write about in my doctoral dissertation!) but rather it was due to having crappy teachers not give a crap about their children’s /understanding/ of the material. In a school system where grades matter more than understanding, the child will fail every time; he’ll fail even if he graduates.

Good math teachers connect with and create understanding in their students. Kudos to you, Bon, for breaking the poor mold imposed on teachers. By doing this you’re giving your kids a fair chance at exploring the beauty of the world of mathematics. No regrets in that.

Good on you,

Adam S.

Thanks so much, Adam, for sharing your story and for the kind words. I’m certainly looking forward to perusing your dissertation – you have me curious!

I realize you wrote this blog entry 2 years ago, so my comment is hardly timely. However, the PEMDAS algorithm is not at all arbitrary.

I’ll grant you PEMDAS is difficult for students to remember and that it causes quite a bit of frustration. However, allowing them to do it any other way isn’t about allowing student decision making or ‘not crushing novel thinking’. Doing so will only damage their ability to understand math.

Like it or not, math is a logical structure with firm rules. Progress in understand comes from mastering the rules and understanding their interactions.

The priority of the order of operations is not arbitrary. It exists because some operations are shortcuts or condensations of others. Multiplication has a higher priority than addition because multiplication is condensed addition.

3 + 2 x 5 = 13, not 25, because 2 x 5 is actually 2 + 2 + 2 + 2 + 2. So 3 + 2 x 5 = 3 + 2 + 2 + 2 + 2 + 2. By expanding the problem, the students can see that multiplication is merely a shortcut for addition.

If you let the students do 3 + 2 x 5 as 3 + 2 = 5 and 5 x 5 = 25, there is no way you can show them how to expand the multiplication as addition.

Worse yet, allowing ‘choice’ in order of operations will kill the students ability to see patterns because terms will not have consistent values. If 3 + 2 x 5 = 13, then 4 + 2 x 5 = 14 because we are merely adding 1 more, which can be shown as 1 + 3 + 2 x 5. But the other way gives 4 + 2 = 6 and 6 x 5 = 30, which is not 1 more than the wrong answer of 25 we got from doing doing the first problem the wrong way.

Exponents come before multiplication for the same reason. Exponents are condensed multiplication so in an expression like 3 x 2^5, doing the multiplication before the exponents means you cannot expand the the expression. 3 x 2^5 = 3 x 2 x 2 x 2 x 2 x 2 = 96. If 3 x 2^5 = 3 x 2 = 6 and then 6^5, how can you show the expansion? 6 x 6 x 6 x 6 x 6? Where did all the 6′s come from?

I understand your desire to create student engagement and reduce frustration, but in this case you are just wrong. Teaching the students that PEMDAS is a social justice matter in which they have been deprived of their choice does nothing but mathematically handicap them.

Well, John, I’m pretty sure PEMDAS (regardless of how you feel about it) is not going out of fashion any time soon – so your timing is still good!

There are many good reasons for the order of operations. But there are some serious problems with forcing the PEMDAS mnemonic. Even if we decide that it’s not arbitrary, it still leaves out roots, logs and absolute values.

And I think Keith Devlin might take up the argument of multiplication being repetitive addition: http://www.maa.org/external_archive/devlin/devlin_01_11.html