Pam  00:01
Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam. 

Kim  00:07
And I'm Kim. 

Pam  00:07
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 know we can mentor students to think and reason like mathematicians. Not only are algorithms really not helpful in teaching mathematics, but rotely repeating steps actually keeps us from being the mathematicians we can be.

Kim  00:32
Which would be horrible. We don't want to keep people from being (unclear). 

Pam  00:34
Never! We believe in empowering and cultivating learning and reasoning. Bam! 

Kim  00:39
Okay, so we're going to dive right in.

Pam  00:42
Dive! 

Kim  00:42
Because last week, we started chatting about Order of Operations, which is a super popular topic in all of our groups. I don't know if everyone knows this. But we have a teacher Facebook group. We've got a group for Journey, which is our online implementation support. We have a leader Facebook group. All the Facebook. I spend some time in there. And in each of those groups, there are great questions where people really do want to help their students have more understanding about a ton of topics. So Order of Operations has come up several times. Actually, Carlo, in our Journey group said, "Sixth graders will be starting with Order of Operations. How would you teach this topic using thinking and reasoning as opposed to a mnemonic and the list of steps?" I love that. That's such a great question.

Pam  01:27
Nice.

Kim  01:27
And also, I found that Jennifer Badda Smith had said, "Any ideas for seventh grade Order of Operations number strings?" Which we call Problem Strings, but we'll let that go. Because these people are really thinking about making meaning. And so, let's start (unclear).

Pam  01:43
Absolutely. Yes, Carlo and Jennifer, thank you. And a ton of others. Just for you, today, we are going to do at least a problem string toward Order of Operations. So, in the last episode, we talked all about it, issues, defining it. Today, let's get at what to do to develop Order of Operations. Alright, to do this, today, we're going to give a hat tip to our friend, Nat Banting. Who might be one of the funniest secondary math teachers I've ever met. 

Kim  02:13
Oh, that's good. 

Pam  02:14
Yeah, you would like him. There's definitely some snark there. Yeah. We both appreciate a good sense of snark. Anyway, so Nat Banting has a structure, a routine structure that he calls something like, "Oops, I meant..." I hope I have that right. "Oops. Oops, instead, I meant this..." And I'm going to respectfully suggest that that is a twist on facilitating a Problem String structure. So, that we can do a problem string in a Nat Banting, "Oops, I meant..." style. So, let me give you an example, and then Nat you can shoot back at me what you thought about my idea of how "Oops, I meant..." could kind of belong as a way. Not the only way, for sure. But it's sort of a twist on facilitating a Problem String. Alright. Kimberly, you ready?

Kim  03:05
Yep. Oh, you're going to give me a problem?

Pam  03:07
I'm going to give you problems. Yep. Okay, you walk in the store with $20.00. This is important. $20.00. 

Kim  03:14
Is it my $20.00 or are you giving me $20.00. 

Pam  03:17
You have $20.00 in your hands. I don't know where you get it from.

Kim  03:19
Okay, I'll use mine.

Pam  03:21
Maybe I handed it to you. I don't know. But it's in your hand. (unclear) $20.00. Now I know that that's a little bit maybe some of our students don't have a lot of experience with money. To which, you and I loudly say, then give them experience with money. So, here's an example of that, where we can acknowledge they're swiping with cards all over the place. We're going to actually do this problem not expecting they come with experience. But that part of it could be they're gaining experience. 

Both Pam and Kim  03:44
Okay. 

Pam  03:45
You have $20.00. Alright, you walk in the with that store that $20.00, and you buy 4 pens. Not pencils, Kim, pens.

Kim  03:52
Okay. 

Pam  03:53
You buy 4 pens, And they're $0.50 each. 

Kim  03:57
Okay.

Pam  03:58
And a $3.00 notebook. 

Kim  04:01
Okay.

Pam  04:01
How much do you have left?

Kim  04:03
Four $0.50... Do you care how I'm thinking about it? Four $0.50... 

Pam  04:06
Well, tell us about it. Yeah.

Kim  04:08
$2.00. And about a notebook. That's $3.00. So, I have spent $5.00. You want to know what I have left? 

Pam  04:14
Yes. (unclear).

Kim  04:15
Oh, so if I spent $5.00, and I had a $20.00, I got $15.00 bucks left.

Pam  04:19
Okay. So, as you were talking, I might write on the board. You said 4 times $0.50. So, I write I wrote 4 "parentheses" 0.5. And then, you said you're going to add to that $3.00? Yeah. And then, you said I've got to subtract all that from $20.00. And so, I wrote 4 "parenthesis" 0.5 "closed parenthesis" plus 3. And then, I put parentheses around all of that, and I wrote in front of it $20.00 subtract. 

Kim  04:48
Okay.

Pam  04:48
Would you agree that that is what you just did? 

Kim  04:51
Yes. 

Pam  04:52
Okay. And I might even then write next to it $20.00 subtract. And you had said 4 times $0.50 cents was $2.00. So, $20.00 subtract "parenthesis" $2.00 plus $3.00. And then, I might write that's equivalent to $20.00 minus $5.00, which is equivalent to $15.00. So, I kind of have this run on sentence thing here where I have kind of this longer expression, and then I simplified the expression the way you said to, and then I simplified it again. And I just kind of kept going until you said you had $15.00 left. 

Kim  05:20
Yep. (unclear).

Pam  05:21
Next question. Oh, go ahead. 

Kim  05:22
Oh, you know, I don't want to interrupt the middle of your Problem String. 

Pam  05:25
Well, I mean. We often interrupt each other.

Kim  05:28
(unclear) I did not... As you were reading back to me what I did. I did not actually put brackets. But to be completely accurate, I could have gotten $15.00 without them. But to be completely accurate about what we're communicating, I did multiply the four $0.50 and the $3.00, and then take it from the $20.00 So, that's,

Pam  05:52
Yeah, you did stuff first. 

Kim  05:53
Yeah. 

Pam  05:54
Oh, so what you're saying is, is if you look at $20.00 subtract, grouping, 4 times 0.5 plus $3.00, that looks like you did something with the 20 first.

Kim  06:04
Yeah, I mean, I would get 15, but if we're talking about the opportunity to communicate clearly about what I actually did, perhaps I should have used a bracket.

Pam  06:15
Well, and so to be clear, if I'm doing this in class, I don't actually care what the student wrote down, I'm going to write down on the board, a way that we have agreed as a society communicates what you did.

Kim  06:26
Yep. 

Pam  06:27
Cool. Next problem.

Kim  06:28
Okay. 

Pam  06:29
You got $20.00. You walk into a store. You're going to buy a $0.50 pen and a $3.00 notebook. 

Kim  06:36
Okay. 

Pam  06:37
You got them in your hand. 

Kim  06:38
Yep. 

Pam  06:39
And then, you realize you got 4 kids, Pam. In case you were being me now because I have 4 kids. You need 4 of those. How much money do you have after you do that.

Kim  06:52
(unclear) be $3.50 for one of your kids, but you have 4 of them. So double it $7.00. Double it again is $14.00. So, you're spending $14.00, so that means you have $6.00 left. 

Pam  07:05
Okay, and so as you were saying that I wrote down. You said $3.50 per kid, and so I wrote down 0.5 plus 3 because it felt like you did that first. So, even though you didn't say it.

Kim  07:18
(unclear) $0.50 plus $3.00?

Pam  07:19
Mmhm.

Kim  07:19
Yep, that's what I did.

Pam  07:20
Yep.

Kim  07:20
Mmhm.

Pam  07:20
Yep. So, $0.50 plus $3.00. It felt like you did that first, so I wrote that down. And then, I put parentheses around that. 

Kim  07:24
Yep. 

Pam  07:25
And I wrote times 4, even though you found times 4 by doubling and doubling again. 

Kim  07:29
Sure. 

Pam  07:30
Okay, so now I have 0.5 plus $3.00, parentheses around that with a 4 out front. 

Kim  07:35
Yep. 

Pam  07:36
And then, you had to take that from the $20.00, so then to the left of that, I wrote $20.00 minus.

Kim  07:41
Mmhm. 

Pam  07:42
So, on my paper right now, or on the board, whatever, I have $20.00 minus 4 times the quantity 0.5 plus $3.00. And then, I could simplify that. I'm not going to do it maybe for the podcast, but with students, I would simplify that. And then... Well, maybe I will. So, I would have $20.00, subtract 4, times $3.50. And you said 4 times $3.50 was double $7.00, which is $14.00. And so, $20.00 minus $14.00 is $6.00. So, I now have kind of this running equation where we sort of did what you did, and we're now like, Did we did we communicate? Yes. And that was the outcome that you got. Next problem. How about you... Now, I have to think for just a second. I can't read my own writing. You did what you just talked about. So, you picked up the notebook and the pen. You realize you needed 4 of them. But as you walked up to the person to pay, they said, "Did you know there's a coupon?" And, Kim, you're kind of frugal.

Kim  08:43
I am.

Pam  08:44
So, you said, "I'll take that coupon." Well, the coupon was $0.75 off.

Kim  08:49
Tell me again, I bought 4 pens.

Pam  08:52
You bought a pen and a notebook, and then realize you needed 4 of them.

Kim  08:56
Yeah. Okay. Pen and a notebook. Sorry. Yep.

Pam  08:58
And then, you realized you needed 4 of them.

Kim  09:00
Yep.

Pam  09:00
And you have $20.00 to pay that, but there's a coupon for 0.75.

Kim  09:06
Yep. 

Pam  09:06
Okay, so how much money are you walking out with?

Kim  09:10
So, I bought the pen and the notebook. That was $3.50. I needed 4 of those sets, so that's $14.00. But I use a coupon for $0.75, so that said I spent $13.25. Actually, I probably... Yeah. So, that's $13.25 I owe, and I'm taking that from the $20.00. Which means, I have $3.75 left.

Pam  09:37
Say that again?

Kim  09:39
Sorry, $3,25. No. No, I have a pen the notebook. That's $3.50 

Pam  09:47
Yeah. 

Kim  09:48
I got to buy that 4 times, that's $14.00.

Pam  09:50
Yep. 

Kim  09:52
I took the coupon off, so that's $0.75 off. You said the coupon was $0.75.

Pam  09:57
Mmhm. 

Kim  09:58
So, I'm only spending $3.25. 

Pam  10:02
$13.25

Kim  10:04
$13.25 Sorry, I even wrote that. So, I'm spending $13.25 and I have $20.00. 

Pam  10:10
There we go. 

Kim  10:11
So, I have $6.75 left.

Pam  10:14
There. Yep. Okay, cool. 

Kim  10:15
Sorry. Did I say that? 

Pam  10:16
You're good. You're good. So, on the board, I actually have exactly what you had for the problem before. Yep. The $20.00 minus 4 times the quantity 0.5 plus $3.00.

Kim  10:25
Yep. 

Pam  10:26
But then, I put parentheses around all of that, and I... Oh, wait. Not all. No, not all of that My bad. Sorry, the 4 times the quantity 0.5 plus $3.00, that was your $14.00. And you're like, "Yeah, but I don't have to pay $14.00. I only have to pay $0.75 less than that. So, that 4 times a quantity 0.5 plus $3.00, I then subtracted 0.75. And I put parentheses around all of that and took that from $20.00. So, I've got $20.00 subtract all of that stuff. And then, somebody in the class is going to say, "Isn't it kinda like you did what you did before, but you just can add that $0.75 back?" And now, we can talk about why $20.00, subtract that, subtract 0.75. Subtract, subtract. Ends up being adding that $0.75. Because you had $6.00 before. And now, with that $0.75 coupon, it's just $6.75.

Kim  11:23
Yeah. 

Pam  11:24
And so, it's a nice kind of way of sort of talking about how are we communicating? Okay, last problem. 

Kim  11:30
Okay. That's a lot of pens and notebooks I'm buying.

Pam  11:33
You're doing a great job. It is a lot of pens and notebooks.

Kim  11:37
Beginning of the school year. 

Pam  11:38
Is it killing it a little bit that it wasn't pencils and notebooks?

Kim  11:40
No, it's fine. Pencils come in a pack, and they're not $0.50, so pens are more realistic because they're cheap and horrible.

Pam  11:48
And now, you know how Kim feels about pens. Okay. Kim, you're going to do a similar thing that you have the $0.50 pen, and the $3.00 notebook. So, I've just written down $0.50 plus $3.00. 

Kim  11:59
Yep. 

Pam  11:59
And you need 4 of them, so now I've got the 4 outside the parentheses of that. 

Kim  12:03
Okay.

Pam  12:04
You got the coupon for $0.75 off. Subtract the $0.75 cents from that. 

Kim  12:09
Yep. 

Pam  12:09
Hey, but wait. You owe me $2.00.

Kim  12:15
Okay, so I had $13.25, but I owed you $2.00, and now I only have... So, I had $6.75. I was spending $13.25, which means I had $6.75. But I owe you $2.00, and now I only have $4.75.

Pam  12:28
And now, we could have a conversation about where do you want to put that subtract $2.00. And that subtract $2.00 could go in a couple of different places, at least. We could subtract it right off the bat. You could have said, "Well, I had $20.00, and I just decided, here's your $2.00. Now, I have $18.00." Right? We could have done it right off the bat. We could have subtracted the $20.00 at the very end of the whole transaction. We could have subtracted the $20.00 inside of the $20.00 minus if we were careful about how we did that. We would have to reason about what to do with that $2.00 And we could do it in several different ways and make sense of why it's sometimes it's minus $2.00 and sometimes it's plus $2.00. 

Kim  13:06
Yeah.

Pam  13:07
And I'm not even going to say too much more about that. And that... Oh! Here I gave a hat tip to Nat, and then I so didn't even do it! Nat, I'm so embarrassed! Okay, Kim. Ya'll, do you ever do this? Get up in front of your students, and you have a plan, and then it kind of? So, what I just did. What I just did was I went kind of into Problem String mode, not like Nat's, "Oops, I meant." So, let me try to just repeat quickly what I would have done in Nat Bantings, "Oops, I meant..." facilitation. I could have said, "Hey, Kim, you walked in the store, and you bought 4 pens at $0.50 each and a $3.00 notebook. You had $20.00 how much did you spend?" Once she did that, then I could said, "Oops. Oops, I meant... I meant you walked in the store..." And then, I would get the second problem. And then, I could have gone. "Oops, oops, I meant there was a coupon." And then, I could have said, "Oops, I meant you owe me $2.00. How much did you have at the end?" So that would have been a better hat tip to Nat Bantings, "Oops, I meant..." facilitation. Sorry, Nat. And I think it would totally work well to do it either way. But I think it's kind of fun to do the "Oops, I meant..." And students tend to kind of smile a little bit, and they kind of know what's coming. And it's a fun way to kind of play a little bit with this Problem String structure. Let me back out a little bit and say, What did we just do? I gave you kind of a scenario of what was happening, and as you actually did the thing. So, this is important teachers. You actually have the kids do the thing, then you represent what they did, and say, "Hey, this is the way we would communicate those words I just said and the thing you did in your head. This is how I could communicate what you just did. We use these symbols to communicate what you just knew to do. This is how we use those symbols." And as the string goes along, "Oops, I meant something different," you do it, and then we communicate how we would represent that, how we as a society have decided that if we wrote it this way, we would all know we mean it to be that way. That is a Problem String to help. Quickly, I'll just give you kind of a, after you've done that in context. Now, I would do that in several different contexts. And I'm making up a few of those. I'll be putting them out on social media. Ya'll, follow me on social media. We'll put them out. But as you do that in context, you could also then do things not in context. You could say, "Alright, we've done a few of these in a store and wherever. Here's one where I just say, "The number 3, I want you to square it. Then, I want you to add 6. Oops, oops, I meant add 6 to 3, and then square that. Square, the total. Oops, I meant..." And then you just keep switching it up. And as they do the thing, you represent the way it would look, so we make sure we're communicating. In other words, if this is the code. So, now you want to be able to look at what's on the board, what you've written. If this is the code, what did the person mean? What are they trying to communicate? That's the purpose of Order of Operations. And then, kids can get good at, "Oh, I can recognize when my brain does this, it looks like that. So, now when I see that, if that's the code, what did the person actually mean?" Please, teachers, don't expect that the first time you do that, "Oh, bam! Kids have got Order of Operations. Give them the test!" No! Because there's a lot for them to fuss with. There's a lot for them to build in their brains about Order of Operations. There's the social part about what the notation means. There's a logical mathematical part about what they were actually doing. And then, there is the decoding part. Remember, the first is they did it, you represented it. Now they're like, "Okay, that's what the code could look like." Now, you're going to come backwards, and you're going to go, "If this is what the code looks like, what was happening?" And, in fact, I didn't do it in this particular Problem String because I don't usually do this in the first go around. The second or third Problem String could have as a last problem. In other words, I keep saying, "Do this." I represent. "Oops, I meant..." Now, I represent it. "Oops, I meant..." Now, I represent it. The very last problem could be, here's the code. What was happening in the store? Now, I just show you the code. You tell me what was going on in the store, and how much money did you walk out with? That transition. So, don't expect that the end of one short experience, kids are going to be coming out with it exactly right. And you might say, "I don't know, Pam. When I just gave them PEMDAS, and I just had a mimic me, they were all getting right answers by the end of the day. And I will believe you. And then, I will tell you, it didn't stick. And it didn't work for most kids because they didn't actually understand what was going on, and all they did was mimic and rote memorize. There were a few kids that probably were able to reason through it, and kind of figure stuff out, and they were able to move on. And maybe that was you, math teachers. But all kids, all kids, can math if we help them actually reason, build the reasoning as they're going, understand what's happening, and make sense of Order of Operations as a communication tool. Alright, Kim. 

Kim  18:14
Yes. 

Pam  18:16
Let's talk about what are some things that we can do with students?

Kim  18:22
Yeah. So, even with very, very young students, when they make a statement that's mathematical, you can say as you're writing, "Mathematicians communicate that like this." So, even in a very young age, we can have students say something and we add grouping symbols to help them see something for the first time. We're not expecting them to use a grouping symbols. We're not expecting them to go back to their desk or table and record with grouping symbols. But as we hear them say, "First, this, then this..." we can add those. 

Pam  19:03
Yeah. "First I added these two things together, and then I added that thing last." The teacher can put parentheses around those first two numbers. And again, like you said, we're not expecting students to do it, but they are seeing, "Oh, I can help communicate by using those grouping symbols." Nice.

Kim  19:16
Right, Because those symbols will be new to them at some point. And if we can let them know early on, "Hey, this is how mathematicians communicate what you just said..."

Pam  19:26
Bam!

Kim  19:27
We do that with kindergartners.  So, the other thing that we can do is have them, like you said,  when they see stuff, we are talking about what is this trying to communicate? And we can dig into the conversation about what do you think that means? And what does that look like, so that they start to understand that there is a language that we're using in mathematics.

Pam  19:54
Yeah, very nice. And let me give you, listeners, one example, one tiny example of something that could come out of the strings that we did today. You might have noticed that I said I would write down the expression that represented what Kim did, then I would write equals, and I would write down sort of the next thing she did. I would kind of simplify things. I would combine things that she had combined. Equals. And then, the next one, and then the answer. Which means, I now have a string of equivalent expressions. And I'm purposely using the equal sign to mean that this thing on the left is equivalent to the next thing, I am not using the equal sign incorrectly as a "Do it" symbol, or as a "And then, I did..," I'm using it as a, "These two expressions are equivalent." That's what the equal sign means. So, I'm using that equal sign correctly, which is sending the message to students. This is what the equal sign means. This is how it's communicating. This is what it's communicating is that these two are equivalent. So, when we really care for... We should do a whole episode, Kim, on just the meaning of the equal sign, and how it's not a "Do it" symbol. 

Kim  20:59
Well, and...

Pam  20:59
That is fantastic.

Kim  21:00
...can we add in that sometimes we shy away from some of this notation until it's time to open up and say we're going to talk about Order of Operations. And so, you know, all of a sudden, kids are seeing all this stuff for the first time, and they're like, "Wait, what do I do first? And why am I supposed to do that first?" So, we're talking about layering on symbols along the way, so that these are not new symbols that they've never seen before.

Pam  21:24
And it doesn't have to be introduced in the chapter you're going to test it.

Kim  21:27
Right, right. 

Pam  21:28
Let's enter this in far earlier in order to communicate. 

Kim  21:33
Yep. 

Pam  21:33
And for Pete's sake, don't make it unnecessarily hard!

Kim  21:38
(unclear) Yeah, there's no need. Yeah.

Pam  21:40
Yeah. Go ahead. 

Kim  21:41
There's no need to create a string of numbers. We've seen it so often that if kids can do these really hard ones, then that must mean that they understand Order of Operations. And we're trying to see if they can follow along under the guise of do they get it? And we're going to say who's that really helping? 

Pam  21:58
Yeah, Like if you have a couple kids in your room that it's like a puzzle to them, and they really enjoy these 20 operation long, with all the crazy symbols, and that's fun for them to figure out the code, by all means, let them. But we're just going to suggest, though, that it's not helpful.

Kim  22:18
Right.

Pam  22:19
Don't make it unnecessarily hard.

Kim  22:20
(unclear) Tricky. Quizy. 

Pam  22:22
Yeah. Yeah. If you want it to be puzzley, put it in a puzzley kind of thing. If we're trying to help kids really communicate well, do enough to communicate well. Make it be about communication. Alright. Ya'll, we're going to do more with PEMDAS in the future, so keep keep tuning in. And thank you for being here and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Thank you for helping spread the word that Math is Figure-Out-Able!