Order of operations seems more rote memorizable than figure-out-able, or is it? In this episode Pam and Kim discuss the pitfalls of teaching order of operations as an acronym to memorize, and offer a better alternative.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.
And I'm Kim.
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. Hey, Kim.
So, I've been hanging on to this particular comment for quite a while because we knew that it would be kind of a fun topic to talk about.
But it would take a minute. So we're going to start today with me sharing what the comment was. So, we got this message in one of our groups. I can't remember. Probably the teacher Facebook group, where Kim said, "Okay, I have officially listened to almost all of the episodes..." I wonder if she's kept up since then. And she said, "The amount of times that I've been like, 'Oh, crud. I should have taught that.' Yikes." She said, "I've been enjoying them so much. And being that my name is Kim also, sometimes I'm listening, and when you say Kim's name, I have stop myself from answering." Isn't that cute? She said, "Now, that I've been listening, I'm fairly certain that these questions are not answered in your episodes." So, she's keeping track. So, here's a couple of questions. She said, "One, what is the 'Figure-Out-Able-ness..." Is that a word? Figure-Out-Able-ness?
Works for me.
"...of PEMDAS." And then, she also said, "What's your thought process when you want to write a Problem String?" So, two different questions. And then, she says, "Thanks for all that you and Kim do. It's made a huge difference in how I'm approaching planning for my third grade class." So that's super fun. So, hey, Kim!" (unclear)
(unclear). So, the Kim's of the world are super thoughtful.
They take good notes. Good attention to detail. Mmhm. Nice, nice.
We try. So, Kim, we sure wish that you were in Journey because we do work there with writing Problem Strings. So, it's a big topic. But maybe that will be fun to us to do in an episode. And we'd have to dial that back. Because it's a lot of thinking that goes into it. But in the meantime, we are going to tackle your question about PEMDAS today. So many people have asked the same thing.
Yeah, so PEMDAS. PEMDAS. But love the fact, Kim. Thank you for your comments, and your suggestions, and wonders. Love the fact your name is Kim. And thanks for listening, Kim. Kim. Pick a Kim. Any Kim. I don't think I've ever met Kim I don't like. Several Kim's in my life that I like. And I do every once in a while remember the way Robin, our friend, says your name. Kim.
So, PEMDAS. And some people are like, "What does that even mean?" And it's kind of funny because my husband was looking over my shoulder, and he read Please Excuse My Dear Aunt Sally. And he was like, "What is that?" That's a way of remembering PEMDAS. And in fact, I didn't as a student ever hear PEMDAS. I only heard "Please Excuse My Dear Aunt Sally". So, if you say the letters of those names. Please Excuse My Dear Aunt Sally, that is P-E-M-D-A-S. PEMDAS
Well, that's what you and I grew up with maybe.
But there are some... People would call them improvements, maybe. But other acronyms. Our Canadian friends probably use GEMDAS. That's another popular one. And in the UK, I just recently learned there are BODMAS and BIDMAS. Like, all these acronyms that, you know, we're trying to make sense of.
And these are all good faith attempts to help students memorize and spit stuff out for Order of Operations.
And let's just upfront admit Order of Operations, for many students, appears to be completely random, and nonsensical, and difficult. And teachers feel that, and so there's this good faith effort to go, "Okay. I know this is hard. Let me make it easy for you. I'm going to come up with these initials that stand for this stuff, and then you're going to use that to help you solve these problems. Here it is. I'm going to clearly, clearly tell you, show you. Now, you mimic."
Both Pam and Kim 04:26
Is a fine thing to do if math is fake math.
If math is a thing to rote memorize and mimic, then by all means, let's help students rote memorize and mimic it well, and correctly, and easily, and not make them guess, not make them, you know... Yeah. Guess what's in my head. Let's just tell them what it is clearly and give them things to practice. And bam. They can be successful at that mimicking.
What if it's not? What were you going to say?
Yeah, I was going to say before we move into the episode too far, let's just ask people, you know, just to think about for a second like what did Order of Operations make sense to you when you were growing up? Or does it make sense to you now? And how did you think about it? Why did it make sense, if it made sense? Or maybe why didn't it if it didn't make sense? Well, and also curious about if that's something that you say? Do you hold PEMDAS in your head now? Or do you think and reason about operations now? That's interesting.
we're super curious. We'd love to hear from you. Pick a social media platform and just let us know. Like, when you were student, did you understand Order of Operations? Did PEMDAS make sense? Was math fake to you, and so you were super glad that a teacher gave you BIDMAS or GEMDAS? Or, you know like, "Just do this," and you were like, "I'm going to do that. Good, I'm done with my homework. Moving on to real stuff. Yeah, Yeah, yeah. What do you do? Yeah. Yeah, and so I think for this particular one was written...
So, we have all probably seen social media posts, right? Pick any social media. Where these crazy memes about which answer do you think? And so, typically, there's going to be some equation or expression, actually. And it'll say, "Which answer do you think?" Is it 1? Or is it 9?" And then, there's this slew of comments where people are arguing back and forth about which one it is. So, I actually just looked not too long ago, and there was one there. And one of the comments was that somebody said, "The Order of Operations was invented in 1912." And people were screaming at each other. And he was like, "Are you serious? We're still talking about this?" But another comment was really fantastic. And they said, "This was made to confuse people. Any sane person who wanted to express this, wouldn't write it this way." Which I thought was really interesting that somebody was calling out the confusing nature of some of these posts that people are going back and forth arguing about. Yeah.
...6 divided by 2, "parenthesis", 1 plus 2.
And if you just like 6 divided by 2, "parenthesis" 1 plus 2, this person's point was, "You've written it that way to confuse people. Any sane person who wanted to express that thing wouldn't write it that way. They would be more clear in the way they write it."
And that is the point of the work that we do with Order of Operations, or PEMDAS, or pick your thing.
Now, wait a minute. What does that mean? The point is communication.
The point is that we want to get something done. We want to have a particular outcome, given some inputs, and we want to have there not be confusion in what we want done, how. And so, we had to figure out ways of communicating. Order of Operations. Would you consider that Order of Operations is all about communication. Now, that's.... Maybe I'll just say that. This, Order of Operations, is tricky because like many parts of mathematics, it's twofold. There are bits of Order of Operations that are social convention, and there are bits of Order of Operations that are logical mathematical. Because there's this mix of two different types of knowledge that are imbued in what's happening, that can get mixed up and messed up, and it can appear to be all social, all convention. And we've talked long and hard in other episodes on the podcast about the difference between social knowledge and logical mathematical knowledge. If something is social knowledge, you have to just tell people. If we've decided by convention something is true, then you have to tell people. There's no way they can figure it out on their own by reasoning because somebody chose it to be that way. Like, we just defined it by convention to be that way. But if something is logical mathematical, telling someone is, most of the time, not enough. They actually have to deal with the relationships and get their brains to think in that way, and then they own it enough to actually do something with it. And so, Order of Operations is not cut and dry.
Because it has that mix. So, a big question that I think we can all consider as we are trying to teach. Or if you're a teacher who teaches Order of Operations. If you could have in your mind that the emphasis, the big overarching question is, how will you know what I mean? Well, we're going to have to understand each other's language in order to know what we mean by that. I'll tell you just a funny example. Some of you know I taught... I taught. I played semi-professional basketball in Switzerland right out of high school. So, right out of high school, I went to Switzerland. I played on this professional basketball team. I had a blast. I learned some Italian. I learned some German. We had some people from Yugoslavia when that was still a country on the team. And like all this crazy stuff going on. And one day, I was walking down the street. I think I was leaving practice or something. And I noticed across the street, one of the gentleman who was on the... There was a board that owned the team. And I saw him across the street, and I waved at this gentleman like, "Hey! Hi!" You know like, "I recognize you." And I waved. Well, I waved in such a way...I didn't know. I found this out later...means "Come here" in Italian. And so, I'm like, acknowledging, "Hey, I see you! I know who you are!" And he looks at me. He goes, "Oh!" And I could see. Like, this look on his face was like, "Oh!" And it was almost a little bit like, "Are you really telling me to come here?" Now, I didn't know that till later, right? But it was kind of this look like, "Oh, I'm being commanded to do something. Here I come." And he marched across this busy street, and then he says like, "What do you need?" And I was like, "Um, hi?" So, we weren't communicating. In mathematics, we need to have a language that we can communicate what we mean. So, Kim, let's do an example. Ready?
If I said to you pick a number. And you can pick any number you want. Maybe pick a number that's kind of smallish.
Don't be over 100. Okay, so pick a number.
Okay. Now, divide that by 2.
Okay. Now subtract 4.
And then, double all of that.
Double my final amount?
Okay, what did you get?
Now, do I really care? But what if I wanted to give anybody a number and do those things but, "Hey, I actually got to go do some things. Will you do that while I'm gone?" And I can't just tell you those things in order. I actually need you to do those things that way. But I got to leave the room, and so I need you to do it. How can I communicate to you that I want you to take a number, and I want you add 8 to it, then divide that by 2, then take all of that, subtract 4 from that, and then take all of that and multiply it by 2. And you might be interested to know that I'm actually looking at what I wrote on my paper, not what I had planned to have you. I'm not looking at the words I wrote initially. I'm looking at a way that I represented that mathematically.
And now, that I've looked at 2, "parentheses", n plus 8, all divided by 2, subtract 4. So, I've got the quantity of n plus 8. N's your number. N plus 8 divided by 2 minus 4, all times 2. That's what I have written down as the instruction that I might leave somebody to do to solve that problem.
Yeah. I like that you're calling it communication, and that it almost feels like an instruction.
We would all want to follow the same instructions in order to end up with the same thing at the end. You know, if we're building something, we need to follow the same instruction.
And we need to be able to communicate. And if I say something that is unclear. If I'm, like, "Do this..." Well, in fact, at some point, I think I said, pick a number, add 8, divided by 2, you could have said, "Wait, divide the 8 by 2 or divide my number plus 8 divided by 2?"
And so, I would want to be specific about that. I would want to clearly communicate my intent for you to do those things. And once I've clearly communicated what I want you to do, well, then we can actually look at what I asked you to do. And there's a possibility we could simplify what you need to do. So, if anybody wrote down what I just said, 2, times the quantity, n plus 8 divided by 2. That subtract 4. Well, in fact, maybe I'll say it even a little bit differently. N plus eight, that divided by 2. Then take that quantity and subtract 4. Then, take all of that, and multiply that by 2. Well, if you do that, you notice that there's some twos that I'm dividing by and multiplying by that I could divide out. And I could end up with a simplified your number plus 8, subtract 8. Well, your number plus 8 subtract 8 means it's just your number. So, Kim, you said you got 4. Did you start with 4?
Bam! So, you could try this with any number. Like, you could start with 3 and do all that stuff. Or you could just know that you're going to end up with your number at the end because of the way I kind of set it up.
So, you can actually simplify it if you understand mathematics and the properties that are involved, the relationships that are involved. That can be a useful thing. Let me actually just make a kind of historical note, if I may. So, most of us have heard of Rene Descartes. Descartes was the guy who said, "I think, therefore I am." He was into philosophy. He was kind of a renaissance man. He did a lot of different things. One of the things that he did was a lot of work in mathematics. And he was very prolific. He wrote a lot of things in mathematics. And because of that, people from around the world at that time were trying to read his proofs, his writings, his thoughts about mathematics, and so they needed to understand what he meant. And so, they would ask him, "Hey, when you write it that way, what do you mean?" Because at that point, we didn't have a lot of social convention. We had different ways of representing, doing different things. In fact, I'll just pop in that when calculus was invented in two different places, one Great Britain and one in Germany, Leibniz and Newton, were both... I just switched them, by the way, where they were. They were both inventing calculus. The way they notated calculus was different. That was interesting. Like, they were using different notation because as they were inventing it, they weren't using each other's notation. We had to sort of get to a place where we as a society began to use the same notation. It's about communicating your ideas. And so, if we would consider that that is the main upshot, that the biggest reason to help students have a motivation to learn Order of Operations is not so that they can do...Kim, we've seen them...these worksheets of 15 operation long problems, where it's almost like a test of stamina. Can you make it through this like really complicated thing? Ya'll, is that really the point? The point is, when I give you a mathematical expression that I intend for you to do in one way, that it means these things, can you read that? Are we communicating? Do you understand what this notation means?
So, again, there's there's sort of two parts to it. One is there's the part that is the social, how we have decided to write multiplication. And man alive is anybody else out there with me. Raise your hand right now if the fact that we have so many different ways to write multiplication is a little nuts to you. For example, when you learn multiplication, don't we use that little x symbol? Like the times symbol is kind of a little x? It's kind of a... It's not really an x, it's a little tiny x. That we use that. But boy, then when we get to kind of algebra, we don't typically use that anymore because now we have x's as variables, and so that gets confusing. We have the tiny x. It's multiplication. And the x is a variable, so then we tend to use a dot that means multiplication. But don't forget that when we have all that stuff going on, we often put parentheses or even brackets to mean multiplication. Ah! Like, we have all these different symbols to mean. And then, we're on a computer, so now we use an asterisk, right? And so, all these different things. Why is that? Well, ya'll, because we developed different symbols at different times to suit the needs of the time. And as a society, we've kind of adopted all of them. And we haven't yet said as a society, "No, let's not use that one. It's too confusing." And frankly... I don't know this for sure, but this is a bet on my on my part. I think the asterisk came into being because we needed to have an easy symbol using computers.
Computers. Yeah, yeah, yeah.
Yeah, I don't think we ever used asterisk to mean multiplication before keyboards, before we had to like type stuff. Alright, so notation has this part of it that is social, that we just decided we're going to make these symbols to mean these things. But there's also this logical mathematical part of it. And, Kim, when you and I were talking about doing this podcast, I believe it was you that said, "Well, wait a minute. Isn't it interesting..." I don't know if you want to talk about or if you want me to talk about it. This idea of multiplicative and additive and how they're kind of related.
Oh, yeah, yeah.
Yeah. Yeah, that's sort of the
So, when kids are early learners, they move from learning addition, or addition and subtraction, into learning multiplication and division. Addition and subtraction comes first. But with the Order of Operations, we're asking them to think about multiplicative ideas, and then think about additive ideas. So, in other words, when we're talking about doing exponents, that is repeated multiplication. And then, we want them to do the kind of one-step multiplication. But then, after that, we say, "Okay, now handle the addition and subtraction." (unclear)
PEMDAS of it.
Yeah, the PEMDAS, right? Parenthesis, and then Exponents. And then, that's P-E. And then, M is multiplication, division. That's sort of...
I think maybe where we get it wrong sometimes is we want kids to think about grouping symbols or parentheses and exponents. But what we don't necessarily help them understand is that we're moving from the more sophisticated thinking to the less sophisticated thinking within an expression. So, if kids understood multiplicative type things, and then additive type things, we can help them understand why exponents first. And then, why multiplication, division next. And why addition and subtraction at the end.
Yeah, and I'll back up even a little bit more. If the only thing we've ever done in addition and subtraction is have kids mimic algorithms for addition and subtraction, we actually haven't built their additive reasoning. And then, we go to multiplication and division, and if the only thing we've done is have kids mimic multiplication... Well, first of all, rote memorize multiplication facts, and then mimic the long multiplication and division algorithms, and so we haven't built their multiplicative reasoning, then all of a sudden, we dumped them in these crazy things, where these all those symbols on the page, and we say, "Make sense of that. Oh, by the way, and here is PEMDAS. That will help you." Like, if they're not already reasoning multiplicatively, and they see these numbers all crazy on the page? They're not reasoning additively or multiplicatively, and they see all these crazy numbers of the page? Ah! Like, it just becomes, "Really, this is math? Fun. Can I go to recess now?" And I don't think we can blame kids at that moment because it literally then just looks like a bunch of gibberish that they're supposed to rote memorize. There are parts of it that we need to just tell kids, "Hey, this is what this multiplication symbol means." But that is not sufficient. That is the bare minimum. We have to build their reasoning, and we have to help them logically mathematically get a feel for why we're even doing Order of Operation problems at all. And if I can just suggest, it's about communication.
Sure. Yeah. Well, and then you throw in this tricky part about if multiplication and division are both within multiplicative thinking, which one do I do first? The idea that multiplication problems can be done with the commutative property in mind, but division can't be. You get a completely different result. And then, same with addition and subtraction. I can solve an addition problem, you know, in either order because of the commutative property, but subtraction has to go in a certain way. So, all of these big ideas of mathematics are at play at the same time.
Yeah. Yeah, nicely said. So, today, we thought we'd give you just a thing to think about. And we're going to talk more about PEMDAS. We're going to give you some suggestions of things to do. But here is a way that you can help students get this big idea of communication, and maybe less of a, "Go plug and chug through 29 of these crazy problems" and more of a, "Ooh, let's actually build some ideas and reasoning," with a little bit of play involved as well. So, big shout out to Robert Kaplinski, a colleague, a genuine good guy. We really enjoy Robert. He's done an amazing job moving math education forward. One of the things, and only one because Robert said several good things, is his openmiddle.com site.
Where he's created a bunch of Open Middle problems, and he has...
What's the word I want? Curate. Yeah
He's curated a bunch of contributions from a lot of people around the world, who have thrown in some Open Middle problems. And I'll throw out an example of an Open Middle problem that I think you could use to help build a lot of really nice things for students with Order of Operations. So.
Do you want to talk about what Open Middle problems are first, in general?
Golly, will I do a good job of that? Would you do a better job?
I'll start, and you can add on.
So, in Open Middle problems, each problem will say, "Use these particular digits." 1 through 9. 1 through 5. 0 through 7. Whatever the prescribed digits are.
For that particular problem. Mmhm.
And you will have an expression with some blank spots. And the idea is that you are to fill those blank spots with the digits that you're told to use, in order to do something. Get the highest number. Get the lowest number. Get the closest to whatever. And so, you basically play with the digits that you're allowed to use to solve lots of problems in the goal of doing a particular outcome. So, if the goal is get closest to 100. It's kind of like our... You know one of our favorite games is close to 100. You have certain digits that you're rearranging, so that you're often solving problems to get towards the closest 100. So, he has a ton on his site of different expressions that you can use to reach...
Mmhm. For different goals.
...different goals. Yeah.
Different standards. Different (unclear).
To build different big ideas. Yeah, it's a fine way to get kids kind of playing with an idea. And one of those is Order of Operations. So, here's a particular one for Order of Operations. There's others, but here's one, where it says blank. So, you're going to put a digit in this blank. Blank plus. Now we have "parenthesis", blank times blank. And I'll repeat this in a minute. So, blank plus "parenthesis" blank times blank, end the parentheses.
So, that quantity in parentheses is going to be raised to a power, but that power is a blank. And then, you're going to subtract something from all of that.
Yep. So, you have five blanks total.
Five blanks total. You've got blank plus. That's the first blank. Then, in parentheses, you have blank times blank. Those are two blanks. Then you're raising it to a power. That's a blank. And then, you're going to subtract blank.
Okay. So, if I may, I'm just going to. On my paper, I literally have blank plus parenthesis, open parenthesis, blank times blank, close parentheses, raised to the blank, minus blank.
Okay. Now, the instructions are use the numbers 1 through 5 to get the greatest amount for the entire expression. So, just to be clear, if you've never done this before, I could literally just put 1 in the first blank, 2 in the second blank, 3 in the third blank, 4 in the fourth blank, 5 in the fifth blank, and then I could figure out what I have, and I could go, "How high did I get?" And then, I could reverse those and put 5 in the first blank and 4 in the second blank. And I could then just figure it out. And so, a student who really isn't thinking much at all or strategically at all, could just throw numbers in there, figure out what it is, come out with an expression. Or come out with an
(unclear) In other words, they could tinker and participate.
answer. They could just not tinker, and just throw things in, and just get answers. Or what we really wanted to do is to tinker.
Right. But what I'm saying is everybody could participate in that.
Even if they aren't thinking strategically.
So, somebody who's messing and just kind of, "Let me try this." Solve. "Let me try this." Solve. Even if it's not strategic, it's something that your kids can access at all different ways that they're going to access it.
I guess to me tinkering and messing means that you're being strategic. And I hear you saying, "No, they're just like throwing them in there." (unclear)
You could just... Yeah. So maybe you're just plugging in numbers. Yeah.
Yeah. So, let's talk about that kid who just throws numbers in there, and then figures out the expression. Well, they're getting practice with Order of Operations. So, as they do it, then you could be giving them feedback to go, "Yes, you've done that correctly. How high?" So, our goal here is to get the greatest amount, you could say, "Do you think that's the greatest amount you could get?" And then, you could walk away and come back. Or you could say, "Do you think you could get something bigger than that?" and walk away. And so, then, kids can start to think strategically. Or maybe they started from the beginning, but they could start to think, "Hmm, I've got..."
If our listeners...
I'm not going to give it away. I'm not going to give it away.
Okay. But if they hadn't had a chance to pause the podcast and tinker themselves.
This is a good time to do that.
My bad. I was getting excited. So, pause the podcast. Pause the podcast. Use the digits 1 through 5.
Put them in those blanks, and see if you can find the greatest amount. What are you thinking about? Then, come back. Alright, so come back. And now, we're actually not going to give what we think is the answer for where the digits 1 through five could go in here. But I do want to raise some things that you could think about as you're deciding, as you're strategically deciding where to put numbers. Do you want to put the largest number somewhere? We're trying to get the largest amount. Where could you put the largest number that has the biggest impact? And if you're not sure, could you move that largest number around and see where it had the biggest impact? Where can you put the smallest number, so that it will have the least impact? If we're trying to get the biggest outcome, maybe the smallest number we want to put somewhere, so it doesn't really grow the outcome very much. Vice versa, what if the instruction was put the numbers 1 through 5 in there, so you wanted the smallest amount? Well, then, you can kind of asking that question. Does it land in one of those addition places? Does it land in one of those multiplication places? Does it land in one of those subtraction places? Or raised to the exponent place? Where does it make the most difference? And we'll let you play with that. Because it's kind of fun. And then, you can tell us on social media where you think those numbers belong in there. We'll throw this out on my social media, and you can I respond, and tell what you're thinking about. Maybe even what you tried first before you found the answer that you thought was the best.
Maybe like a week from now, we can put the highest amount that we got. Not put the answer. But can you...
Just the result
...(unclear) higher than this number.
We'll think about that.
Nice. I like it. I like it.
Alright, so some major takeaways from today is that it's about communication, right? That's the point is way to communicate mathematically what we're trying to accomplish. And there are both logical and social parts to what we're talking about with Order of Operations. So, there's really some stuff to tell which is the notation, but most of this is about development. It's really about multiplicative relationships and additive relationships. So, there's some work to be done there. And multiplication is nested within exponents, and addition is nested within multiplication.
Big ideas that are happening. Ya'll, have some fun with Order of Operations. Thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!