This year I have had quite a few students who are pushing themselves mathematically and doing a lot of great thinking. This also equates to making me think as well, and I really need that. Now a student has me thinking about PEMDAS, how it is taught, and how it is a coveted norm of mathematics.

One thing I have always struggled with when talking with students about PEMDAS is how to make it meaningful for students without pulling out the “This is the way it is” card. I don’t want it to become a memorization practice only, I really want students to understand why we do operations in a particular order.

One thing I have used and students understand (not surprising because I adopted this because of a student) is that multiplication implies “groups of.” I’m sure many pure mathematicians might frown upon this, or the fact that it also leads to repeated addition, but students really grasp the concept and start visualizing order of operations.

3×4+2

3 groups of 4 plus 2

4+4+4+2

14

When students think of it this way, they don’t want to add 4 and 2 and then multiply that by 3. This is also true when they see 2+3×4, the “groups of” thinking prevents them from adding 2 and 3. Is this just replacing a trick with a trick? It took me a few years to even accept this type of thinking for my students- I don’t want to provide a new crutch for students.

This all leads me to the other day, when a student was solving the equation 3X + 2 = 8. He was unsure of what to do, so I asked him what step he would try. He repled: “I would divide by 3 since I want to solve for X.” I immediately balked, that answer went against all of my mathematical fiber, but I tried to not show it visibly. I told him to go ahead and try his idea. This is what was produced:

Wait now! Hold on, that is the answer I expected, but not attained in the way I assumed it would be. I can’t tell you how many times I have used the last in- first out type of thinking with students to solve equations, and I expected seeing something like this:

But all of these provide the same solution! Is that a fluke? Some weird oddity that can’t be reproduced? It had me thinking of possible counter-examples and what else it could imply. Does order of operation last in-first out thinking absolute in solving equations? Or is it something like the standard addition algorithm that we accept as the best method and ignore others? My student gave me another example to consider later that hour.

This problem involved the area of a trapezoid. They wanted him to algebraically solve it for b2. I was very curious what he was going to do, and instead of starting with the parenthesis, so subtracting b1 (which is what I assumed he would do because of his previous reply), he told me he needed to distribute before solving. His work was this:

Which again produced the correct solution after he cleaned up the compound fraction. Once again his work produced the right answer without doing the expected procedure:

So currently I am rethinking approaches to solving equations. How will not using the standard approach effect him later on? I can see how trying some sort of variation on this with powers will be messy. I’m totally open to suggestions and comments on this, as it has my brain working on overload ATM.

My students also want to solve 3x+2=8 beginning with division by three. However, they assume that since you would only add/subtract one time, you also only need to divide one time, and we wind up with x+2=8/3. Obviously that isn’t cool.

As long as they’re dividing by three correctly, I don’t care at all if that’s their first step. I do point out that they then have more fractions to deal with, but that’s ok too.

My only point with order of operations is that math is about communication. We have to have a way to agree on what operations get done first, or we would get different answers to the same problem. Actually, the Math Antics video does a good job of presenting that idea.

Interesting. Your student seems to understand distribution of division and multiplication over subtraction and addition — this is very sophisticated thinking that many of my university students doing higher calculus don’t get! I would definitely not try to squash it!

The next stage is for him to be flexible enough to try a different approach if needed, and to understand the approaches of others. I’d be encouraging him to think of multiple approaches to solving the equation. Indeed, I’d encourage the other students to do so as well. While the standard in-out version is often the most *efficient* method, it does cloud the fact that really, we’re just utilising the laws that numbers live by to help us figure out the solution to our problem. There’s nothing stopping us using different laws.

As to your explanation of why * is before + — I wouldn’t worry so much! That is precisely the explanation I give for why. I just make sure I use words that say it’s not the only interpretation of *. “One way to interpret *, at least for whole numbers, is as a whole lot of + all together. So of course you have to do it first because it’s already a whole lot of stuff all at once.” Incidentally, it’s the same reason why powers are before * — one way to interpret them is as a whole lot of * all at once, so of course it’s before *.

That’s the distributive property right there! (3x+3)/3 is the same as (3x+3)•(1/3). Distribute and you’ve got what the kid got. So nice.

I have actually ditched order of operations this year. In fact, I started the school year by claiming I don’t believe in it. We do number talks warm ups and when students share different strategies if I see one that does not follow the order of operations we discuss it. The idea being that order of operations leads us to think there is only one way. As long as we use mathematical sense making we can get the correct answer other ways. The trick then becomes teaching students how to make mathematical sense of problems. Your idea of multiplication as groups is exactly that. I have students use integer tiles to show what the problem says. It is interesting with some problems that if you read them they sound exactly the same, but if you represent them the meaning is different. 3(4 + 2) and 3(4) +2. With so many math properties like distributive, commutative, and associative there are so many more ways to simplify an equation than the order of operations. I focus on terms a lot. If you separate an expression by terms you can be more flexible in your thinking, in the way you move numbers around and simplify. A great way to reinforce this is to do error analysis. I take actual work of students and they analyze it. In sensemaking, it is important to understand what works, what doesn’t, and most importantly why does it (or doesn’t it) make mathematical sense.

This “not the order of operations” reminds me of what I wrote a while ago https://blogs.adelaide.edu.au/maths-learning/2015/09/11/the-reorder-of-operations/