Hey, everyone. Welcome to this next episode of Your Daily Equation. Today, I’m going to give a brief episode and mix things up a little bit. Rather than talking about some deep insight into physics or mathematics that emerges from equations, instead I’m going to focus upon a simple issue in math pedagogy, math education, that irritates me, bothers me. I’d be interested to hear from you all your perspective. If I’m looking at this the wrong way, I’d be happy to have my perspective changed. And this is an issue that I also see cropping up online every now and then as a simple problem in arithmetic, gets sent around with people arguing as to what is the right answer.
That kind of equation, arithmetic problem I’m talking about, it’s pretty straightforward. Let me just pick one at random, things like eight, say, minus two, and then we have a variety of other operations, say, divided by two times three plus four. A little equation like that gets kicked around and people argue online as to what the right answer to this problem is. And there was a real interesting conceptual issue here, which is that the answer depends on the order of operations. So if I was going to title this little episode, it would be Order of Operations, and that’s an interesting subject for sure.
So let’s just look at it in this particular example and let me just show how depending on how you interpret this equation, you will get different answers. Let me just copy it down a couple of times so I can perhaps do it a few times. Maybe I can paste it in another time here.
For instance, if you interpreted this to mean that, say, you did the 2 divided by 2 first in this expression and you got a 1 there. And then if you did that times the 3 together, you’d get a 3. Then if you had the 8 minus the 5 gives you a 5 plus the 4 would give you 9. That’s one answer you’d get from interpreting the sentence, the mathematical sentence in that manner.
However, there are many other things that you could do. For instance, if we come over, and let’s choose a different color over here, if did the 8 minus 2 first you’d get your 6. If you did your 6 divided by 2, that would get you a 3. If you then multiplied the 3 by the 3, that would give you a 9. And if you then added the 4 to that, that would give you 13 in this particular case.
And what other things can you do? Well, I don’t know. If, say, you did the 3 plus the 4 first, you got a 7, and then you did the 7 times a 2, you’d get a 14 over here. And if you did, say, the 8 minus 2 to get a 6 and they you do a 6 divided by the 14, you’d have 6 over 14, which would be 3 divided by 7. So a completely different answer.
And again, so there is an interesting conceptual point that this little example illustrates, that the order of operations in a mathematical sentence matters to it. Now here’s the part that I become irritated by. My daughter is in school and in school they teach her, and I don’t know, maybe they taught this to me, too and I blocked it out or something from when I was a kid, in school, they teach you this thing called PEMDAS.
PEMDAS is an arbitrary rule that tells you the order of operations to employ in any given mathematical sentence. So the P, parentheses, E, exponents, M and D, multiplication division, and the A and S, addition and subtraction. So the idea is if you’re given some sort of mathematical sentence like this, you employ PEMDAS to figure out which order of operations to employ. And indeed, this top one over here, I believe I did in effect follow the PEMDAS prescription. There weren’t any parentheses. There weren’t any exponents, but I did my multiplication and my division first, and when you have both multiplication and division, you follow the order of those operations from left to right as they’re written in the problem. So indeed I did my division first, then I did my multiplication over here, and then I did my addition and subtraction, whereas in these other examples that did something else. Fine. Okay.
Now, here’s the thing. Point number one. I’ve been in this business being a professional physicist. Say you become a professional when you get your doctorate. So I’ve been a professional for more than 30 years, and I’m pretty certain that not a single time in my entire career have I ever made use of PEMDAS, and in fact, I have to say, if a graduate student, I’d be more forgiving on an undergraduate student, but if a graduate student came into my office and actually wrote down on the blackboard this original mathematical sentence that we had here from the get go. If I had a graduate student come into my office and write down the left hand side of this, I would say to that student, “Don’t talk to me that way. Get out of my office”. I probably wouldn’t be that mean about it, but I’d say we physicists don’t speak to each other this way. We don’t speak in ambiguous mathematical sentences. We talk in precise mathematical sentences, and we don’t rely upon some arbitrary acronym PEMDAS in order to make sense of the mathematical sentences we write down.
Instead, I would say to that graduate student, “Are you kidding me? Where are your parentheses?” We use parentheses to indicate precisely the order of operations that we have in mind. So in fact, if the interpretation of this mathematical sentence was meant to be the one that comes out of PEMDAS, I would want my student and what we professionals always do to write down some parentheses. So I would do, say, a parenthesis like this and then I would do another one say over here, and that would tell me to do the 2 divided by 2 first to get my 1, then do my 1 times my 3 to get 3, and then do my 8 minus 3 is 5 plus 4 equals 9. That’s it.
We simply use parentheses to make it very precise and clear what it is that we mean with our mathematical sentences. And indeed, if the second interpretation is the one, say, that the graduate student had in mind, the graduate student would simply write 8 minus 2 with some nice parentheses around it, and then if the dividing by 2 was meant to happen next, they could put another parentheses, and to make it more clear, they could put, say, a square bracket around it. And then if we want it to be even more precise, we could say the times 3 is next for those parentheses, and then we would add our 4. So we get our 8 minus 2 is 6 divided by 2 is 3 times 3 is 9 plus . is equal to 13, and that’s it. No PEMDAS necessary. We just make certain that our mathematical sentences are clear by using parentheses to bracket those operations that we want to happen first or second or third and so forth.
So what’s my gripe here? My gripe is the following. When my daughter learned this PEMDAS order of operations thing in school, they spent, and maybe I’m exaggerating, but they spent weeks on it and they kept giving my daughter problem after problem after problem, all sorts of ambiguous-looking arithmetical problems, and it was up to her to apply this PEMDAS rule to figure out what the answer to a given problem was meant to be. And all these problems, they’re got you problems, they’re meant to try to trip her up. And at the end of this long exercise that went on at least for days and maybe weeks, and I think it was even in two different years in school, they repeated it, at the end of this process, what did my daughter take away? She took away the notion that math is all about applying arbitrary rules to make sense of a collection of arbitrary symbols. It was kind of for me so, so kind of heartbreaking. I felt so dejected by it because that’s not what math is. Math is not about these arbitrary rules. Math is about the wondrous patterns that we can encapsulate with the precise mathematical sentences that we write down.
So if it was up to me, what would I do? I would advocate look, give the students a couple of examples like the example I just gave you, show them this absolutely important concept that the order of operation does matter, but with a couple of examples, kids will get that, and then tell them, show them that by using parentheses, you can make clear the order of operations that you have in mind when you write down a mathematical sentence. Period. End of story. And from that point forward, you can move on to the beautiful qualities of math that can articulate the patterns of the world.
So the point is at any opportunity, we should take away the quality of math that you apply rules. You do apply rules, but we should take away that quality. We should not emphasize that quality when there’s an opportunity to emphasize the beauty of math, the power of math to encapsulate wondrous and universal patterns. We should focus in that direction. And here’s a case where you’d never need to introduce PEMDAS. Just introduce the concept of the ambiguity of the order of operations and then this idea of using brackets or parentheses to make sense, precise, mathematical sense of every single mathematical sentence you write down. And that’s what we professionals do. We don’t use PEMDAS.
Anyway, so that’s my point. I’d be interested to hear from you guys if you have a different view on this. Do you make use of PEMDAS in the work that you do or in the analysis that you undertake? I think computer programmers may have a slightly different perspective as you can make your code perhaps a little tighter if you make use of these sort of arbitrary rules, but I’d be interested to hear that perspective directly as well.
But that’s the episode for today. Math is about patterns, beautiful, universal patterns that we can encapsulate with precise mathematical sentences, and that’s where the beauty and wonder comes from, not from applying arbitrary rules over and over and over again. That’s not math. That’s drudgery. Anyway, that is the point for today. That is Your Daily Equation. Really I would like to hear from me if you have a different perspective on this, but until next time, until the next episode, take care.